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Biologically Plausible Local Learning Rules for the Adaptation of the Vestibulo-Ocular Reflex Olivier Coenen* Terrence J. Sejnowski Computational Neurobiology Laboratory Howard Hughes Medical Institute The Salk Institute P.O.Box 85800 San Diego, CA 92186-5800 Stephen G. Lisberger Department of Physiology W.M. Keck Foundation Center for Integrative Neuroscience University of California, San Fransisco, CA, 94143 Abstract The vestibulo-ocular reflex (VOR) is a compensatory eye movement that stabilizes images on the retina during head turns. Its magnitude, or gain, can be modified by visual experience during head movements. Possible learning mechanisms for this adaptation have been explored in a model of the oculomotor system based on anatomical and physiological constraints. The local correlational learning rules in our model reproduce the adaptation and behavior of the VOR under certain parameter conditions. From these conditions, predictions for the time course of adaptation at the learning sites are made. 1 INTRODUCTION The primate oculomotor system is capable of maintaining the image of an object on the fovea even when the head and object are moving simultaneously. The vestibular organs provide information about the head velocity with a short delay of 14 ms but visual Signals from the moving object are relatively slow and can take 100 ms to affect eye movemen.ts. The gain, a, of the VOR, defined as minus the eye velocity over the head velocity (-if h), can be modified by wearing magnifying or diminishing glasses (figure 1). VOR adaptation, absent in the dark, is driven by the combination of image slip on the retina and head turns. ·University of California, San Diego. Dept. of Physics. La Jolla, CA, 92037. Email address: oli vier@helmholtz.sdsc.edu 961 962 Coenen, Sejnowski, and Lisberger During head turns on the first day of wearing magnifying glasses, the magnified image of an object slips on the retina. After a few days of adaptation, the eye velocity and hence the gain of the VOR increases to compensate for the image magnification. We have constructed a model of the VOR and smooth pursuit systems that uses biologically plausible local learning rules that are consistent with anatomical path ways and physiological recordings. The learning rules in the model are local in the sense that the adaptation of a synapse depends solely on signals that are locally available. A similar model with different local learning rules has been recently proposed (Quinn et at., Neuroscience 1992). xl.O Spectacles 1 B 1 I; 1.4 1.2 z < 1.0 <.:) on 9 , , , l· o . Gain = 1.01 + 0.68(1 . e-<1020 t) • ~ :: r ~--:---:---:----:-----m. __ ___ 0.6 [ + Gain" = O.~ + 027 Ie" -0 1J t) " II xO.5 SpectaCles on xl.O Spectacles off ~ Gain = 1.01 + 0.68 Ie -0._ tl " " ~---------------, f xO.5 Spectacles off • • , , I ! ! I I • ,L-....L.---.J'_....l,_....L, _...L' _..L.' _.I..-' o 23456780 234567 TIME IDaysl Figure 1: Tune course of the adapting VOR and its recovery of gain in monkeys exposed to the longterm influence of magnifying (upper curves) and diminishing (lower curves) spectacles. Different symbols obtained from different animals, demonstrating the consistency of the adaptive change. From Melvill Jones (1991), selected from Miles and Eighmy (1980). 2 THEMODEL Feedforward and recurrent models of the VOR have been proposed (Fujita, 1982; Galiana, 1986; Kawato and Gomi, 1992; Quinn et al., 1992; Arnold and Robinson, 1992; Lisberger and Sejnowski, 1992). In this paper we study a static and linear version of a previously studied recurrent network model of the VOR and smooth pursuit system (Lisberger, 1992; Lisberger and Sejnowski, 1992; Viola, Lisberger and Sejnowski, 1992). The time delays and time constants associated with nodes in the network were eliminated so that the time course of the VOR plasticity could be more easily analyzed (figure 2). The model describes the system ipsilateral to one eye. The visual error, which carries the image retinal slip velocity signal, is a measure of the performance of both the VOR and smooth pursuit system as well as the main error signal for learning. The value at each node represents changes in its firing rate from its resting firing rate. The transformation from the rate of firing of premotor signal (N) to eye velocity is represented in the model by a gain Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex 963 o : Gains P : Purkinje Cell N : Vestibular Nucleus g : Desired gain • h Inhibitory Visual error: mossy fibers -(g h + e) • e eye velocity Visual error: climbing fibers Figure 2: Diagram of the VOR and smooth pursuit model. The input and output of the model are, respectively, head velocity and eye velocity. The model has three main parts: the node P represents an ensemble of Purkinje cells from the ventral paraflocculus of the cerebellum, the node N represents an ensemble of flocculus-target neurons in the vestibular nucleus, and the visual inputs which provide the visual error signals in the mossy and climbing fibers. The capital letter gains A and D, multiplying the input signals to the nodes, are modified according to their learning rules. The lower case letters b, v, and 9 are also multiplicative gains, but remain constant during adaptation. The traces represent head and eye velocity modulation in time. The visual error signal in the climbing fibers drives learning in node N but does not constitute one of its inputs in the present model. of -1. The gain of the VOR in this model is given by ~=:. We have not modeled the neural integrator that converts eye velocity commands to eye position signals that drive the motoneurons. 3 LEARNING RULES We have adopted the learning rules proposed by Marr (1969), Albus (1971) and Ito (1970) for adaptation in the cerebellum and by Lisberger (1988), Miles and Lisberger (1981) for plasticity in the brain stem (figure 3). These are variations of the delta rule and depend on an explicit representation of the error signal at the synapses. Long term depression at mossy fiber synapses on Purkinje cells has been observed in vitro under simultaneous stimulation of climbing fibers and mossy fibers (Ito, Sakurai and Tongroach, 1982). In addition, we have included a learning mechanism for potentiation of mossy fiber head velocity inputs under concurrent mossy fiber visual and head velocity inputs. Although the climbing fiber inputs to the cerebellum were not directly represented in this model (figure 2), the image velocity signal carried by the mossy fibers to P was used in the model to achieve the same result. There is good indirect evidence that learning also occurs in the vestibular nucleus. We have adopted the suggestion of Lis berger (1988) that the effectiveness of the head velocity input to some neurons in the vestibular nucleus may be modified by head velocity input in 964 Coenen, Sejnowski, and Lisberger ~ Learning ( InPut) (Error) Rate x Signal x Signal Cerebellum (P): A qA X ( Head ) ( Mossy fiber ) Velocity x Visual signal qA x h x -v(gh + e) qA X h x -v[(g - D)h + P] ex: h2 Vestibular nucleus (N): b qD X ( Head ) (Climbing fiber Purkinje ) Velocity x Visual signal Signal qD x h x [(1 - q)(gh + e) - qP] qD X h x [(1 - q)(g - D)h + (1 - 2q)P] oc h2 where P A - bD - (g - D)v . h I-b+v Figure 3: Learning rules for the cerebellum and vestibular nucleus. The gains A and D change according to the correlation of their input signal and the error signal to the node, as shown for ~ at the top. The parameter q determines the proportion of learning from Purkinje cell inputs compared to learning from climbing fiber inputs. When q = I, only Purkinje cell inputs drive the adaptation at node N; if q = 0, learning occurs solely from climbing fiber inputs. association with Purkinje cells firing. We have also added adaptation from pairing the head velocity input with climbing fiber firing. The relative effectiveness of these two learning mechanisms is controlled by the parameter q (figure 3). Learning for gain D depends on the interplay between several signals. If the VOR gain is too small. a rightward head turn P (positive value for head velocity) results in too small a leftward eye turn (a negative value for eye velocity). Consequently, the visual scene appears to move to the left (negative image slip). P then fires below its resting level (negative) and its inhibitory influence on N decreases so that N increases its firing rate (figure 4 bottom left). This corrects the VOR gain and increases gain D according to figure 3. Concurrently, the climbing fiber visual signal is above resting firing rate (positive) which also leads to an increase in gain D. Since the signal passing through gain A has an inhibitory influence via Ponto N, decreasing gain A has the opposite effect on the eye velocity as decreasing gain D. Hence, if the VOR is too small we expect gain A to decrease. This is what happens during the early phase of learning (figure 4 top left). 4 RESULTS Finite difference equations of the learning rules were used to calculate changes in gains A and D at the end of each cycle during our simulations. A cycle was defined as one biphasic Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex 965 Desired gain 9 = 1.6 Magnitude Magnitude 2 2 1.75 G t.75 G 1.5 / 1.5 D t.25 D 1.25 1 1 0.7!1 0.75 A 0.5 A 0.5 0.25 0.25 0 20 40 60 80 10lime 0 2000 4000 6000 8000 10(J;ime A, D & VOR gain G vs time Amplitude Amplitude 2 2 1.5 / 1.5 1 N 1 N 0.5 0.5 ~ 40 60 80 100 Time 2000 4000 6000 ~.5 ~.5 P P ·1 ·1 P & N responses to a head turn during learning vs time Figure 4: Simulation of change in gain from 1.0 to 1.6. Top: Short-term (left) and long-term (right) adaptation of the gains A, D and G. Bottom: Changes on two time scales of P and N responses to a head turn of amplitude 1 during learning. The parameters were v = 1.0, b = .88, T = .!lA = 10. , f1D and q = .01. head velocity input as shown in figure 2. We assumed that the learning rates were so small that the changes in gains, and hence in the node responses, were negligibly small during each iteration. This allowed the replacement ~f A(t) .and D(t) by their values obtained on the previous iteration for the calculations of A and D. The period of the iteration as well as the amplitude of the head velocity input were chosen so that the integral of the head velocity squared over one iteration equaled l. For the simulations shown in figure 4 the gain G of the VOR increased monotonically from 1 to reach the desired value 1.6 within 60 time steps. This rapid adaptation was mainly due to a rapid decrease in A, as expected from the local learning rule (figure 3), since the learning rate 'f/A was greater than the learning rate 'f/D. Over a longer time period, learning was transferred from A to D: D increased from 1 to reach its final value 1.6 while the VOR gain stayed constant. Transfer of learning occurs when P fires in conjunction with a head turn. P can have an elevated firing rate even though the visual error signal is zero (that is, even if the VOR gain G has reached the desired gain g) because of the difference between its two other inputs: the head velocity input through A and the eye velocity feedback input through b. It is only when these two inputs become equal in.amplit~lde that P firing goes to zero. It can be shown that when learning settles (when D and A equal zero) D = g, A = bg, and P = O. With these values for A and D, the two other inputs to P are indeed equal in amplitude: one equals Ah, while the other equals b( -1 )Dh. During the later part of learning, gain A is driven in the opposite direction (increase) than during the earlier 966 Coenen, Sejnowski, and Lisberger part ( decrease). This comes from a sign reversal of the visual error input to P. After the first 60 time steps, the gain has reached the desired gain due to a rapid decrease in A, this means that any subsequent increase in D, due to transfer of learning as explained above, will cause the gain of the VOR G to become larger than the desired gain g, hence the visual error changes sign. In order to compensate for this small error, gain A increases promptly, keeping G very close to the desired gain. This process goes on until A and D reach their equilibrium values stated above. The short and long-term changes in P and N responses to a velocity step are also shown. As the firing of P decreased with the adaptation of A, the firing rate of N increased to the right level. 5 OVERSHOOT OF THE VOR GAIN G In this section we show that for some ranges of the learning parameters, the gain G in the model overshoots the desired value g. Since an overshoot is not observed in animals (figure I), this provides constraints on the parameters. The parameter q in the learning rule for the vestibular nucleus (node N, gain D), determines the proportion of learning from Purkinje cell inputs compared to learning from climbing fiber inputs. When q = 1, only Purkinje cell inputs drive the adaptation at node N; if q = 0, learning at N occurs solely from climbing fiber inputs. These two inputs have quite different effects on learning as shown in figure 5. Asymptotically, P goes to 0, and D goes to 9 if q = 1; and P can only differ from ° if q = 0. The gain has an overshoot for any value of q different than 0, as shown in figure 6. Nevertheless, its amplitude is only significant for a limited extent in the parameter space of q and r (graph of figure 6). The overshoot is reduced with a smaller q and a larger r. One possibility is that q is chosen close to ° and r > I, that is TJA > 7JD. These conditions were used to choose parameter values in the simulations (figure 4). 6 DISCUSSION AND CONCLUSION The VOR model analyzed here is a static model without time delays and multiple time scales. We are currently studying how these factors affect the time course of learning in a dynamical model of the VOR and smooth pursuit. In our model, learning occurs in the dark if P #- 0, which has not been observed in animals. One way to avoid learning in the dark when P is firing would be to gate the learning by a visual input, such as that provided by climbing fibers. The responses of vestibular afferents to head motion can be classified into two categories: phase-tonic and tonic. In this model, only the tonic afferents were represented. Both afferent types encode head velocity, while the phasic-tonic responds to head acceleration as well. The steady state VOR gain can also be changed by altering the relative proportions of phasic and tonic afferents to the Purkinje cells (Lisberger and Sejnowski, 1992). We are currently investigating learning rules for which this occurs. The model predicts that adaptation in the cerebellum is faster than in the vestibular nucleus, and that learning in the vestibular nucleus is mostly driven by the climbing fiber error signals. The model shows how the dynamics of the whole system can lead to long-term adaptation Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex 967 Desired gain g = 1. 6 q = 1 q=O Magnibtde Magnibtde 2.5 1 1.75 G G 0 1.5 1.15 1 0 A 0.75 0.5 A 0.5 0.15 0 100 200 300 400 sooTime 0 50lime 100 200 300 400 A, D & VOR gain G vs time Amplitude Amplitude 1.5 2.5 1 N N 300 400 500 Time 100 P 200 300 400 500 Time ·1 ·1 P & N responses to a head turn during learning vs time Figure 5: Effect of q on learning curves for gain increase. Left: q = 1 leads to an (wershoot in the VOR gain G above the desired gain. D increases up to the desired gain, P starts from 0 and asymptotically goes back to 0; both indicate that learning is totally transferred from P to N. Right: With q = 0, there is no overshoot in the VOR gain, but since A decreases to a constant value and D only increases very slightly, learning is not transfered. Consequently, P firing rate stays constant after an initial drop. E (I-b+v) (D ) q (I-b) - 9 (2q-I)-rv 10 Figure 6: Overshoot f. of the VOR gain G as a function of q and r. The parameter q is the proportion of learning to node N (vestibular nucleus), coming from the P node (cerebellum) compared to learning from climbing fibers. The parameter T is the ratio of the learning rates TJA and TJD. No overshoot is seen in animals, which restricts the parameters space of q and r for the model to be valid. Note that the overshoot diverges for some parameter values.' which differs from what may be expected from the local learning rules at the synapses because of differences in time scales and shifts of activity in the system during learning. This may reconcile apparently contradictory evidence between local learning rules observed in vitro (Ito, 1970) and the long-term adaptation seen in vivo in animals (Miles and Lisberger, 1981). 968 Coenen, Sejnowski, and Lisberger Acknowledgments O.c. was supported by NSERC during this research. References Albus, J. S. (1971). A theory of cerebellar function. Math. Biosci., 10:25-61. Arnold, D. B. and Robinson, D. A. (1992). A neural network model of the vestibulo-ocular reflex using a local synaptic learning rule. Phil. Trans. R. Soc. Lond. B, 337:327-330. Fujita, M. (1982). Simulations of adaptive modification of the vestibulo-ocular reflex with an adaptive filter model of the cerebellum. Biological Cybernetics, 45:207-214. Galiana, H. L. (1986). A new approach to understanding adaptive visual-vestibular interactions in the central nervous system. Journal of Neurophysiology, 55:349-374. Ito, M. (1970). Neurophysiological aspects of the cerebellar motor control system. Int.J.Neurol., 7:162-176. Ito, M., Sakurai, M., and Tongroach, P. (1982). Climbing fibre induced depression of both mossy fibre responsiveness and glutamate sensitivity of cerebellar purkinje cells. J. Physiol. Lond., 324:113-134. Kawato, M. and Gomi, H. (1992). The cerebellum and VORlOKR learning models. Trends in Neuroscience, 15 :445-453. Lisberger, S. G. (1988). The neural basis for learning of simple motor skills. Science, 242:728-735. Lisberger, S. G. (1992). Neural basis for motor learning in the vestibulo-ocularreflex ofprimates:IV. The sites of learning. In preparation. Lisberger, S. G. and Sejnowski, T. J. (1992). Computational analysis suggests a new hypothesis for motor learning in the vestibulo-ocular reflex. Technical Report 9201, INC, Univ. of California, San Diego. Marr, D. (1969). A theory of cerebellar cortex. J. Physiol., 202:437-470. MelviIl Jones, G. M. (1991). The Vestibular Contribution, volume 8 of Vision and Visual Dysfunction, chapter 2, pages 293-303. CRC Press, Inc., Boston. General Editor: J. R. Cronly-Dillon. Miles, E A. and Eighmy, B. B. (1980). Long-term adaptive changes in primate vestibulo-ocular reflex.l. Behavioural observations. Journal of Neurophysiology, 43:140&-1425. Miles, F. A. and Lisberger, S. G. (1981). Plasticity in the vestibulo-ocular reflex: A new hypothesis. Ann. Rev. Neurosci., 4:273-299. Quinn, K. J., Baker, J., and Peterson, B. (1992). Simulation of cerebellar-vestibular interactions during VOR adaptation. In Program 22nd Annual Meeting. Society for Neuroscience. Quinn, K. J., Schmajuk, N., Jain, A., Baker, J. E, and Peterson, B. W. (1992). Vestibuloocular reflex arc analysis using an experimentally constrained network. Biologtcal Cybernetics, 67: 113-122. Viola, P. A., Lisberger, S. G., and Sejnowski, T. J. (1992). Recurrent eye tracking network using a distributed representation of image motion. In Moody, 1. E., Hansen, S. J., and Lippman, R. P., editors, Advances in Neural Information Processing Systems 4, San Mateo. IEEE, Morgan Kaufmann Publishers.
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Combining Neural and Symbolic Learning to Revise Probabilistic Rule Bases J. Jeffrey Mahoney and Raymond J. Mooney Dept. of Computer Sciences University of Texas Austin, TX 78712 mahoney@cs.utexas.edu, mooney@cs.utexas.edu Abstract This paper describes RAPTURE a system for revising probabilistic knowledge bases that combines neural and symbolic learning methods. RAPTURE uses a modified version of backpropagation to refine the certainty factors of a MYCIN-style rule base and uses ID3's information gain heuristic to add new rules. Results on refining two actual expert knowledge bases demonstrate that this combined approach performs better than previous methods. 1 Introduction In complex domains, learning needs to be biased with prior knowledge in order to produce satisfactory results from limited training data. Recently, both connectionist and symbolic methods have been developed for biasing learning with prior knowledge lFu, 1989; Towell et a/., 1990; Ourston and Mooney, 1990]. Most ofthese methods revise an imperfect knowledge base (usually obtained from a domain expert) to fit a set of empirical data. Some of these methods have been successfully applied to real-world tasks, such as recognizing promoter sequences in DNA [Towell et ai., 1990; Ourston and Mooney, 1990]. The results demonstrate that revising an expert-given knowledge base produces more accurate results than learning from training data alone. In this paper, we describe the RAPTURE system (Revising Approximate 107 108 Mahoney and Mooney Probabilistic Theories Using Repositories of Examples), which combines connectionist and symbolic methods to revise both the parameters and structure of a certainty-factor rule base. 2 The Rapture Algorithm The RAPTURE algorithm breaks down into three main phases. First, an initial rule-base (created by a human expert) is converted into a RAPTURE network. The result is then trained using :ertainty-factor backpropagation (CFBP). The theory is further revised through network architecture modification. Once the network is fully trained, the solution is at hand-there is no need for retranslation. Each of these steps is outlined in full below. 2.1 The Initial Rule-Base RAPTURE uses propositional certainty factor rules to represent its theories. These rules have the form A ~ D, which expresses the idea that belief in proposition A gives a 0.8 measure of belief in proposition D [Shafer and Pearl, 1990]. Certainty factors can range in value from -1 to + 1, and indicate a degree of confidence in a particular proposition. Certainty factor rules allow updating of these beliefs based upon new observed evidence. Rules combine evidence via probabilistic sum, which is defined as a EB b - a + b - abo In general, all positive evidence is combined to determine the measure of belief(MB) for a given proposition, and all negative evidence is combined to obtain a measure of disbelief (MD). The certainty factor is then calculated using C F = M B + MD. RAPTURE uses this formalism to represent its rule base for a variety of reasons. First, it is perhaps the simplest method that retains the desired evidence-summing aspect of uncertain reasoning. As each rule fires, additional evidence is contributed towards belief in the rule's consequent. The use of probabilistic sum enables many small pieces of evidence to add up to significant evidence. This is lacking in formalisms that use only MIN or MAX for combining evidence [Valtorta, 1988]. Second, probabilistic sum is a simple, differentiable, non-linear function. This is crucial for implementing gradient descent using backpropagation. Finally, and perhaps most significantly, is the widespread use of certainty factors. Numerous knowledgebases have been implemented using this formalism, which immediately gives our approach a large base of applicability. 2.2 Converting the Rule Base into a Network Once the initial theory is obtained, it is converted into a RAPTURE -network. Building the network begins by mapping all identical propositions in the rule-base to the same node in the network. Input features (those only appearing as rule-antecedents) become input nodes, and output symbols (those only appearing as rule-consequents) become output nodes. The certainty factors of the rules become the weights on the links that connect nodes. Networks for classification problems contain one output for each category. When an example is presented, the certainty factor for each of the categories is computed and the example is assigned to the category with the Combining Neural and Symbolic Learning to Revise Probabilistic Rule Bases 109 1.3 I I .2t' , , I , Figure 1: A RAPTURE NETWORK highest value. Figure 1 illustrates the following set of rules. ABC~D E~D C~G EF~G HI~C As shown in the network, conjuncts must first pass through a MIN node before any activation reaches the consequent node. Note that each of the conjuncts is connected to the corresponding MIN mode with a solid line. This represents the fact that the link is non-adjustable, and simply passes its full activation value onto the MIN node. The standard (certainty-factor) links are drawn as dotted lines indicating that their values are adjustable. This construction shows how easily a RAPTURE-network can model a MYCIN rulebase. Each representation can be converted into the other, without loss or corruption of information. They are two equivalent representations of the same set of rules. 2.3 Certainty Factor Backpropagation Using the constructed RAPTURE-network, we desire to maximize its predictive accuracy over a set of training examples. Cycling through the examples one at a time, and slightly adjusting all relevant network weights in a direction that will minimize the output error, results in hill-climbing to a local minimum. This is the idea behind gradient descent [Rumelhart et al., 1986), which RAPTURE accomplishes with Certainty Factor Backpropagation (CFBP), using the following equations. If Uj is an output unit ApWji = 7Jopj(1 ± LWjkOpk) k#-i (1) 110 Mahoney and Mooney (2) If Uj is not an output unit hpj = L: hpk wkj(1 ± EWjkOpk) (3) kmin i;tk The "Sigma with circle" notation is meant to represent probabilistic sum over the index, and the ± notation is shorthand for two separate cases. If WjiOpi ~ 0, then is used, otherwise + is used. The kmin subscript refers to the fact that we do not perform this summation for every unit k (as in standard backpropagation), but only those units that received some contribution from unit j. Since a unit j may be required to pass through a min or max-node before reaching the next layer (k), it is possible that its value may not reach k. RAPTURE deems a classification correct when the output value for the correct category is greater than that of any other category. No error propagation takes place in this case (hpj = 0). CFBP terminates when overall error reaches a minimal value. 2.4 Changing the Network Architecture Whenever training accuracy fails to reach 100% through CFBP, it may be an indication that the network architecture is inappropriate for the current classification task. To date, RAPTURE has been given two ways of changing network architecture. First, whenever the weight of a link in the network approaches zero, it is removed from the network along with all of the nodes and links that become detached due to this removal. Further, whenever an intermediate node loses all of its input links due to link deletion, it too is removed from the network, along with its output link. This link/node deletion is performed immediately after CFBP, and before anything new is introduced into the network. RAPTURE also has a method for adding new nodes into the network. Specific nodes are added in an attempt to maximize the number of training examples that are classified correctly. The simple solution employed by RAPTURE is to create new input nodes that connect directly, either positively or negatively, to one or more output nodes. These new nodes are created in a way that will best help the network distinguish among training examples that are being misclassified. Specifically, RAPTURE attempts to distinguish for each output category, those examples of that category that are being misclassified (Le. being classified into a different output category), from those examples that do belong in these different output categories. Quinlan's ID3 information gain metric [Quinlan, 1986] has been adopted by RAPTURE to select this new node, which becomes positive evidence for the correct category, and negative evidence for mistaken categories. With these new nodes in place, we can now return to CFBP, where hopefully more training examples will be successfully classified. This entire process (CFBP followed by deleting links and adding new nodes) repeats until all training examples are correctly classified. Once this has occurred, the network is considered trained, and testing may begin. Combining Neural and Symbolic Learning to Revise Probabilistic Rule Bases 111 Soybean Test Ac:eurac:y ..i ,-' , :-:...n.oo-+-+---Lf--,.:---:-f--':........-::-t~=-"'--+-1 ,.' I,' ,,' I"~ 70.00--+--+-_,-"-• ...,..::. f---~ ''--+-----+----1---I, . ', ,,' . ~ I,' 015.00-++-+---+.--+----+---+---+-/ . ,' I .' I 6O.00'-++-+_..;...<f-__ --+ __ -+ __ -+ __ (1 ,. ':' n,oo'---H-J(.<-",'-;,-.' -+---+----+----+-" ## I~ ~ . SO.OO i,' 41.OO--:O'*'00:-----:20.00:I-::----:00'*00=-----,6O.IIO±:-----:1O.::I-:00,-:----.""..... O'CII--,O~.oo.------:!~'=----7.lco.:i-;CII;-------;I:-.\50.=OO 1hinil:p Figure 2: RAPTURE Testing Accuracy 3 Experimental Results To date, RAPTURE has been tested on two real-world domains. The first of these is a domain for recognizing promoter sequences in strings of DNA-nucleotides. The second uses a theory for diagnosing soybean diseases. These datasets are discussed in detail in the following sections. 3.1 Promoter Recognition Results A prokaryotic promoter is a short DNA sequence that precedes the beginnings of genes, and are locations where the protein RNA polymerase binds to the DNA structure [Towell et al., 1990]. A set of propositional Horn-clause rules for recognizing promoters, along with 106 labelled examples (53 promoters, 53 non-promoters) was provided as the initial theory. In order for this theory to used by RAPTURE it had to be modified into a certainty factor format. This was done by breaking up rules with multiple antecedents, into several rules. In this fashion, each antecedent is able to contribute some evidence towards belief in the consequent. Initial certainty factors were assigned in such a way that if every antecedent (from the original rule) were true, a certainty factor of 0.9 would result for the consequent. To test RAPTURE using this dataset, standard training and test runs were performed, which resulted in the learning curve of Figure 2a. This graph is a plot of average performance in accuracy at classifying DNA strings over 25 independent trials. A single trial consists of providing each system with increasing numbers of examples to use for training, and then seeing how well it can classify unseen test examples. This graph clearly demonstrates the advantages of an evidence summing 112 Mahoney and Mooney system like RAPTURE over a pure Horn-clause system such as EITHER, a pure inductive system such as ID3, or a pure connectionist system, like backprop. Also plotted in the graph, is KBANN [Towell et a/., 1990], a symbolic-connectionist system that uses standard backpropagation, and RAPTURE-O, which is simply RAPTURE given no initial theory, emphasizing the importance of the expert knowledge. For this dataset, CFBP alone was all that was required in order to train the network. The node addition module was never called. 3.2 Soybean Disease Diagnosis Results The Soybean Data comes from [Michalski and Chilausky, 1980] and is a dataset of 562 examples of diseased soybean plants. Examples were described by a string of 35 features including the condition of the stem, the roots, the seeds, as well as information such as the time of year, temperature, and features of the soil. An expert classified each example into one of 15 soybean diseases. This dataset has been used as a benchmark for a number of learning systems. Figure 2b is a learning curve on this data comparing RAPTURE, RAPTURE-O, backpropagation, ID3, and EITHER. The headstart given to RAPTURE does not last throughout testing in this domain. RAPTURE maintains a statistically significant lead over the other systems (except RAPTURE-O) through 80 examples, but by 150 examples, all systems are performing at statistically equivalent levels. A likely explanation for this is that the expert provided theory is more helpful on the easier to diagnose diseases than on those that are more difficult. But these easy ones are also easy to learn via pure induction, and good rules can be created after seeing only a few examples. Trials have actually been run out to 300 examples, though all systems are performing at equivalent levels of accuracy. 4 Related Work The SEEK system [Ginsberg et a/., 1988) revises rule bases containing M-of-N rules, though can not modify real-valued weights and contains no means for adding new rules. Valtorta [Valtorta, 1988) has examined the computational complexity of various refinement tasks for probabilistic knowledge bases, and shows that refining the weights to fit a set of training data is an NP-Hard problem. Ma and Wilkins [Ma and Wilkins, 1991) have developed methods for improving the accuracy of a certainty-factor knowledge base by deleting rules, and they report modest improvements in the accuracy of a MYCIN rule base. Gallant [Gallant, 1988) designed and implemented a system that combines expert domain knowledge with connectionist learning, though is not suitable for multi-layer networks or for combination functions like probabilistic sum. KBANN [Towell et a/., 1990] uses standard backpropagation to refine a symbolic rule base, though the mapping between the symbolic rules and the network is only an approximation. Fu [Fu, 1989) and Lacher [Lacher, 1992) have also used backpropagation techniques to revise certainty factors on rules. However, the current publications on these two projects do not address the problem of altering the network architecture (i.e. adding new rules) and do not present results on revising actual expert knowledge bases. Combining Neural and Symbolic Learning to Revise Probabilistic Rule Bases 113 5 Future Work The current method for changing network architecture in RAPTURE is restricted to adding new input units that directly feed the outputs. We hope to incorporate newer techniques for creating and linking to hidden nodes, in order to improve the range of architectural changes that it can make. Another area requiring further research concerns the differences between certaintyfactor networks and traditional connectionist networks. Further comparison of the RAPTURE and KBANN approaches to knowledge-base refinement are also indicated. Finally, in recent years, certainty-factors have been the subject of considerable criticism from researchers in uncertain reasoning [Shafer and Pearl, 1990]. However, the basic revision framework in RAPTURE should be applicable to other uncertain reasoning formalisms such as Bayesian networks, Dempster-Shafer theory, or fuzzy logic [Shafer and Pearl, 1990]. As long as the activation functions in the corresponding network implementations of these methods are differentiable, backpropagation techniques should be employable. 6 Conclusions Automatic refinement of probabilistic rule bases is an under-studied problem with important applications to the development of intelligent systems. This paper has described and evaluated an approach to refining certainty-factor rule bases that integrates connectionist and symbolic learning. The approach is implemented in a system called RAPTURE, which uses a revised backpropagation algorithm to modify certainty factors and ID3's information gain criteria to determine new rules to add to the network. In other words, connectionist methods are used to adjust parameters and symbolic methods are used to make structural changes to the knowledge base. In domains with limited training data or domains requiring meaningful explanations for conclusions, refining existing expert knowledge has clear advantages. Results on revising three real-world knowledge bases indicates that RAPTURE generally performs better than purely inductive systems (ID3 and backpropagation), a purely symbolic revision system (EITHER), and and purely connectionist revision system (KBANN). The certainty-factor networks used in RAPTURE blur the distinction between connectionist and symbolic representations. They can be viewed either as connectionist networks or symbolic rule bases. RAPTURE demonstrates the utility of applying connectionist learning methods to "symbolic" knowledge bases and employing symbolic methods to modify "connectionist" networks. Hopefully these results will encourage others to explore similar opportunities for cross-fertilization of ideas between connectionist and symbolic learning. Acknowledgements This research was supported by the National Science Foundation under grant IRI9102926, the NASA Ames Research Center under grant NCC 2-629, and the Texas Advanced Research Program under grant 003658114. We wish to thank R.S. Michal114 Mahoney and Mooney ski for furnishing the soybean data, M. Noordewier, G.G. Towell, and J.W. Shavlik for supplying the DNA data, and the KBANN results. References [Fu, 1989] Li-Min Fu. Integration of neural heuristics into knowledge-based inference. Connection Science, 1(3):325-339, 1989. [Gallant, 1988] S.1. Gallant. Connectionist ex'")ert systems. Communications of the Association for Computing Machinery, 31:152-169, 1988. [Ginsberg et al., 1988] A. Ginsberg, S. M. Weiss, and P. Politakis. Automatic knowledge based refinement for classification systems. Artificial Intelligence, 35:197-226, 1988. [Lacher, 1992] R.C. Lacher. Expert networks: Paradigmatic conflict, technological rapprochement. Neuroprose FTP Archive, 1992. [Ma and Wilkins, 1991] Y. Ma and D. C. Wilkins. Improving the performance of inconsistent knowledge bases via combined optimization method. In Proceedings of the Eighth International Workshop on Machine Learning, pages 23-27, Evanston, IL, June 1991. [Michalski and Chilausky, 1980] R. S. Michalski and S. Chilausky. Learning by being told and learning from examples: An experimental comparison of the two methods of knowledge acquisition in the context of developing an expert system for soybean disease diagnosis. Journal of Policy Analysis and Information Systems, 4(2):126-161, 1980. [Ourston and Mooney, 1990] D. Ourston and R. Mooney. Changing the rules: a comprehensive approach to theory refinement. In Proceedings of the Eighth National Conference on Artificial Intelligence, pages 815-820, Detroit, MI, July 1990. [Quinlan, 1986] J. R. Quinlan. Induction of decision trees. Machine Learning, 1(1):81-106,1986. [Rumelhart et al., 1986] D. E. Rumelhart, G. E. Hinton, and J. R. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland, editors, Parallel Distributed Processing, Vol. I, pages 318-362. MIT Press, Cambridge, MA, 1986. [Shafer and Pearl, 1990] G. Shafer and J. Pearl, editors. Readings in Uncertain Reasoning. Morgan Kaufmann, Inc., San Mateo,CA, 1990. [Towell et al., 1990] G. G. Towell, J. W. Shavlik, and Michiel O. Noordewier. Refinement of approximate domain theories by knowledge-based artificial neural networks. In Proceedings of the Eighth National Conference on Artificial Intelligence, pages 861-866, Boston, MA, July 1990. [Valtorta, 1988] M. Valtorta. Some results on the complexity of knowledge-base refinement. In Proceedings of the Sixth International Workshop on Machine Learning, pages 326-331, Ithaca, NY, June 1988.
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Transient Signal Detection with Neural Networks: The Search for the Desired Signal Jose C. Principe and Abir Zahalka Computational NeuroEngineering Laboratory Department of Electrical Engineering University of Florida, CSE 447 Gainesville, FL 32611 principe@synapse.ee.ufl.edu Abstract Matched filtering has been one of the most powerful techniques employed for transient detection. Here we will show that a dynamic neural network outperforms the conventional approach. When the artificial neural network (ANN) is trained with supervised learning schemes there is a need to supply the desired signal for all time, although we are only interested in detecting the transient. In this paper we also show the effects on the detection agreement of different strategies to construct the desired signal. The extension of the Bayes decision rule (011 desired signal), optimal in static classification, performs worse than desired signals constructed by random noise or prediction during the background. 1 INTRODUCTION Detection of poorly defined waveshapes in a nonstationary high noise background is an important and difficult problem in signal processing. The matched filter is the optimal linear filter assuming stationary noise [Thomas, 1969]. The application area that we are going to discuss is epileptic spike detection in the electrocorticogram (ECoG), where the matched filtering produce poor results [Barlow and Dubinsky, 1976], [Pola and Romagnoli, 1979] due to the variability of the spike shape and the ever changing background (the brain electric activity). Recently artificial neural networks have been applied to spike detection [Eberhart et ai, 1989], [Gabor and 688 Transient Signal Detection with Neural Networks: The Search for the Desired Signal 689 Seydal, 1992], but static neural network architectures were chosen. Here a static multilayer perceptron (MLP) will be augmented with a short term memory mechanism to detect spikes. In the way we utilized the dynamic neural net, the ANN can be thought of as an extension of the matched filter to nonlinear models, which we will refer to as a neural template matcher. In our implementation, the ANN looks directly at the input signal with a time window larger than the longest spike for the sampling frequency utilized. The input layer of the dynamic network is a delay line, and the data is clocked one sample at a time through the network. The desired signal is "I" following the occurrence of a spike. With this strategy we teach the ANN to produce an output of" 1" when a waveform similar to the spike is present within the time window. A spike will be recognized when the ANN output is above a given threshold. Unlike the matched filter, the ANN does not require a single, explicit waveform for the template (due to the spike shape variability some form of averaging is needed to create the "average" spike shape which is normally a poor compromise). Rather, the ANN will learn the important features of the transient class by training on many sample spikes, refining its approximation with each presentation. Moreover, the ANN using the sigmoid nonlinearity will have to necessarily represent the background activity, since the discriminant function in pattern space is established from information pertaining to all input classes. Therefore, the nonstationary nature of the background can be accommodated during network training and we can expect that the performance of the neural template matcher will be improved with respect to the matched filter. We will not address here the normalization of the ECoG, nor the issues associated with the learning criterion [Zahalka, 1992]. The purpose of this paper is to delve on the design of the desired signal, and quantify the effect on performance. What should be the shape of the desired sinal for transient detection, when on-line supervised learning is employed? In our approach we decided to construct a desired signal that exists for all time. We shall point out that the existence of a desired signal for every sample will simplify supervised learning, since in principle the conventional backpropagation algorithm [Rumelhart et aI, 1986] can be utilized instead of the more time consuming backpropagation through time [Werbos, 1990] or real-time recurrent learning [Williams and Zipzer, 1989]. The simplified learning algorithm may very well be one of the factors which will make ANNs learn on-line as adaptive linear filters do. The main decision regarding the desired signal for spike detection is to decide the value of the desired signal during the background (which for spikes represent 99% of the time), since we decided already that it should be "1" following the spike. Similar problems have been found in speech recognition, when patterns that exist in time need to be learned [Unnikrishnan et aI, 1991] [Watrous et aI, 1990]. Here we will experimentally compare three desired signals (Figure 1): Desired signal 1- Extrapolating the Bayes rule for static patterns [Makhoul], we will create a target of zero during the background, and a value of 1 following the spike, with a duration equal to the amount of time the spike is in the input window. Desired signal 11- During the background, a random, uniformly distributed (between -0.5 and 0.5), zero mean target signal. Same target as above following the spike. 690 Principe and Zahalka Desired signal 111- During the background the network will be trained as a one step predictor. Same targe,t as above following the spike. +~--+-------o --~I-foI----'desired output -lr---------------(I). Hardlimited 01+ 1 signal desired output . \, tnput +~---------+~.-~~---+ +0. o -1~--------~-------l~------------------(II). Random noise signal (III). Prediction signal Figure 1. Desired signals considered. 2 NEURAL NETWORK ARCmTECTURE AND TRAINING One ECoG channel was sampled at 500 Hz (12 bit AID converter) and was preprocessed for normalization between -1 and 0.4, and DC removal before being presented to the ANN [Zahalka, 1992]. An epileptic spike has a duration between 20 and 70 msec. The dynamic network used for this application is a time delay neural network (TDNN) consisting of an input layer with 45 taps, 12 hidden processing elements (PEs) and one linear output PE. The input window corresponds to 90 msec, so even the longest spike is fully present in the window during at least 10 samples. The training set consists of 60 hand picked 2 sec. segments containing one spike each, embedded in 6,860 points of background activity. In principle the data could be streamed on-line to the ANN, provided that we could create also on-line the desired signal. But at this point of the research we preferred to control the segment length and choose well defined spikes to study training issues. The test data set consists of another set (belonging to the same individual) of 49 spikes embedded in 6,970 samples. A spike was defined when the ANN output was above 0.9. The ANN was trained with the backpropagation algorithm [Rumelhart et al 1986]. The weights were updated after every sample (real-time mode). A momentum term (a.=0.9) was used, and 0.1 was added to the sigmoid derivative to speedup learning [Fahlman, 1988]. The training was stopped when the test set performance decreased. The network typically learned in less than 50 presentations of the training set. All the Transient Signal Detection with Neural Networks: The Search for the Desired Signal 691 results presented next use the same trainingltest sets, the same learning and stop criterion and the same network topology. 3 RESULTS Desired Signal I. We begin with the most commonly used desired signal for static classification, the hardlimited 01+ 1 signal of Figure 1 (I), but now extended in time (0 during the background, 1 after the occurrence of the spike). This desired signal has been shown to be optimal in the sense that it is a least square approximation of the Bayes decision rule [Makhoul, 1991 J. However it performs poorly for transient detection (73% of correct detections and 8% of false positives). We suspect that the problem lies in the different waveshapes present in the background. When an explicit 0 value is given as the desired signal for the background, the network has difficulties extracting common features to all the waves and biases the decision rule. Samples of the network input and the corresponding output are shown in Figure 2(1). Notice that the ANN output gets close to "1" during high amplitude background, and fails to be above 0.9 during small amplitude spikes. In Table 1, the test set performance is better than the training set due to the fact that the spikes chosen for training happen to include more difficult cases. Table 1. Performance Results. Desired Training Set Training Set Test Set Test Set Signal detections false positives detections false positives 0<--> +1 38/60 = 63% 2/38 = 5% 36/49 = 73% 3/36 = 8% noise 54/60 = 90% 0154 = 0% 46/49 = 94% 1/47 = 2% prediction 50/60 = 83% 1150 = 2% 45/49 = 92% 2/45 = 4% Detections mean number of events in agreement between the human expert and the ANN (normalized by the number of spikes in the set). False positives mean the number of events picked by the ANN, but not considered spikes by the human expert (nonnalized by the number of ANN detections) The "random noise" desired signal is shown in Figure 1 (II). This signal consists simply of uniformly distributed random values bounded between -0.5 and +0.5 for the nonspike regions and again a value of "+ 1" after the spikes. This approach is based on the fact that the random number generator will have a zero mean distribution. Therefore, during training over the nonspike regions, the errors backpropagated through the network will normally average out to zero, yielding in practice a "don't care" target for the background. This effectively should give the least bias to the decision rule. The net result is that consistent training is only performed over the spike patterns, allowing the network to better learn these patterns and provide improved performance. The results of this configuration are very 692 Principe and Zahalka promising, as can be seen in Table 1 and in Figure 2(11). The "noise" signal performs better than all of the other desired signal configurations examined (94% correct detections and 2% false detections). D." II. 1\ 1\ .' Ok ~~~~V-\f"1t . Input to the network ~jPL~ if;. ZL:~, ( --- 100-02 a.t~ ~ ~o -0.2 Output of network (I) 0/+1 signal 0 .7 5 0 ." D·"b A II ~}\fJrrwrifO o • ".. "l!1 A 1\ ""./\ _ r 11 ~~~;~y ~VdWV'V 7~00 . mput to the l~~twork ~~. ~ ~. ~~k~jUl~ • 100 200-0 300 .00 500 Uutput ot' network (II) noise desired signal Input for the network. :~:~f' h 1M II " " ~ ~~~~ vV~v \[0V\J wrtJ' \.ft o Output for the network (III) prediction signal Figure 2. Input and output of neural network The last signal configuration is the prediction paradigm. The network is performing one-step prediction during the nonspike portions of the signal and a saturation value of "+ 1" is added to the desired signal after the spike, as in Figure 1(111). The rationale behind such a configuration is that the desired signal is readily available and may require fewer network resources than a target of "0", decreasing the bias in the decision rule for the spike class. The results are given in Table 1, where we see a marked improvement over the hardlimited desired signal (92% correct detection and 4% false positives). 4 COMPARISON WITH MATCHED FILTER The template for the matched filter was formed by averaging five spikes from the training set. While averaging reduces morphological "crispness," it enhances the robustness of spike shape representation, since the spike shape is variable. The 45 point template is correlated continuously with the time signal from the test set, hence Transient Signal Detection with Neural Networks: The Search for the Desired Signal 693 no correction to the nonwhite nature of the ECoG is being done. Performance in both the matched filter and the ANN approach will be dependent upon the threshold levels chosen for detection, which determines the receiver operating characteristic (ROC) for the detector, but requires large data sets to be meaningful. Table 2 shows a partial result, wht;~;,! bvth detectors are compared in terms of false positives for 90% detection rate and detection rate for 2 false positives. The results presented for the ANN are for the random noise desired signal. In both cases, we see the superiority of the ANN approach. Table 2. Comparison ANNlMatched filter 90% detection rate 2 false positives NN MF NN MF o false +'s 21 false +'s 92% 13% 5 CONCLUSIONS The ultimate conclusion of this experimental work is that neural networks can be used to implement transient detectors, outperforming the conventional matched filter for the detection of spike transients embedded in the ECoG. However, the difficulty in the ANN approach comes in how to setup the training, and the desired signal. Although the training was not discussed here, we would like to point several issues that are important. When a cost function that is sensitive to the a priori probability of the classes is used (as the error signal in backpropagation) the proper balance of background versus spike data sizes is relevant for adequate training. Our results show that at a low spike concentration of 4% (ratio of spike samples over background samples in the training set), the network never learns the waveform and always outputs the value chosen to represent the background. For our application, these problems disappear at an 18% concentration level. Another important issue is the selection of the class exemplars for training. Transient detection is a one class classification problem (A versus the universe), rather than a two class classification problem. It is not possible to cover appropriately the unconstrained background with examples, and even when a large number of background waves is utilized (in our case 80% of the training samples belonged to the background) the training produced bad results. We found out that only the waves similar in shape to the spikes are important to bound the spike class in feature space. As a solution, we included in the training set the false positives, i.e. waves that the system detects as spikes but the human expert classifies as background, to improve the detection agreement with the human [Zahalka, 1992]. The issues regarding the choice of the desired signal are also important. We found experimentally that the best desired signal for our case is the one that uses random noise during the background. We can not, at this point, explain this result 694 Principe and Zahalka theoretically. This desired signal is also the one that uses fewer network resources (i. e. number of hidden units) without degrading the performance [Zahalka, 1992]. This result came as a surprise since the 0/1 signal has been shown to be optimal for static patterns in the sense that makes the classifier approximate the Bayes decision rule. The explanation may be simply a matter of local minima in the performance surface of the network trained with the 0/1 desired signal, but it may also reflect deeper causes. With the OIl desired signal, the network weights are updated equally with the information of the background waveforms and with the information regarding the spikes. The training paradigm should emphasize the spikes, since the spikes are the waves we are interested in. Moreover, since the background class is unconstrained, many more degrees of freedom are necessary to represent well the background. When not enough hidden units are available, the network biases the discriminant function, and the performance is poor. In the prediction paradigm the background is selected as the desired signal and the required number of hidden nodes to predict the next sample is much smaller (actually only three hidden nodes are sufficient to keep the reported performance [Zahalka, 1992]). The network resources naturally self organize as two template matching nodes and one prediction node. However, we have found out that the signal has to be properly normalized in the sense that the target for the spike ("I") must be outside the range of the input signal voltages (that is the reason we normalize the input signal between -1 and 0.4). From a classification point of view we "do not care" what the net output is provided it is far from the target of "1". The random noise target with zero mean achieves this goal very easily, because the error gradient during the background averages out to zero. Therefore the weights reflect primarily the information containing in the shape of the spike. We found out that only two hidden nodes are sufficient to keep the reported performance. [Zahalka, 1992]. It seems that the theory of time varying signal classification with neural networks is not a straight forward extension of the static classification case, and requires further theoretical analysis. An implication of this work regards the role of supervised learning in biological neural networks. This work shows that during non-interesting events, there is no need to provide a target to the neural assembly (noise, which is so readily available in biological systems, suffices). Re-enforcement stimulus are only needed during or after relevant events, which is compatible with the information processing models of the olfactory bulb [Freeman and DiPrisco, 1989]. Therefore, looking at learning and adaptation in biological systems as a signal detection instead of a classification problem seems promising. Acknowledgments This work has been partially supported by NSF grants ECS-9208789 and DDM8914084. References Barlow J. S. and Dubinsky J., (1976) "Some Computer Approaches to Continuous Transient Signal Detection with Neural Networks: The Search for the Desired Signal 695 Automatic Clinical EEG Monitoring," in Quantitative Analytic Studies in Epilepsy, Raven Press, New York, 309-327. Eberhart R., Dobbins R., Weber W., (1989) "Casenet: a neural network tool for EEG waveform classifiaction", Proc IEEE Symp. Compo Based Medical Systems, Minneapolis, 60-68, 1989. Fahlman S.,(1988) "Faster learning variations on backpropagation: an empirical study", Proc. 1988 Connectionist Summer School, Morgan Kaufmann, 38-51. Freeman W., DiPrisco V., (1986) "EEG spatial pattern differences with discriminated odors manifest chaotic and limit cycle attractors in olfactory bulb of rabbits", in Brain Theory, Ed. Palm and Aertsen, Springer, 97-120. Gabor A., Seydal M., (1992) "Automated interictal EEG spike detection using artifical neural networks", Electroenc. Clin. Neurophysiol., (83),271-280. Makhoul J., (1991) "Pattern recognition properties of neural networks" ,Proc. 1991 IEEE Workshop Neural Net. in Sig. Proc., 173-187, Princeton. Pola P. and Romagnoly 0., (1979) "Automatic analysis of interictal epileptic activity related to its morphological aspects", Electroenceph. Clin. Neurophysiol., #46, 227-231. Rumelhart,D.E., Hinton,G.E. and Williams,R.J. (1986) "Learning internal representations by error propagation. in Parallel Distributed Processing (Rumelhart, McClelland, eds.), ch. 8, Cambridge, MA. Thomas J., (1969) "An Introduction to statistical Communication Theory", Wiley. Watrous R., Ladendorf B., Kuhn G., (1990) "Complete gradient optimization of a recurrent network applied to b,d,g discrimination", 1. Acoust. Soc. Am. 87 (3), 1301-1309. Werbos, P.J. (1990) "Backpropagation through time: what it does and how to do it", Proc. IEEE, vol 78, nolO, 1550-1560. Williams,R.J. and Zipser, D. (1989) "A learning algorithm for continually running fully recurrent neural networks. in Neural Computation, vol. 1 (2). Unikrishnan K., Hopfield J., Tank D., (1991) "Connected-Digit Speaker-dependent speech recognition using a neural network with time delayed connections", IEEE Trans. Sig Proc., vol 39, #3, 698-713. Zalahka A., (1992) "Signal detection with neural networks: an application to the recognition of epileptic spikes", Master Thesis, University of FLorida.
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Information Theoretic Analysis of Connection Structure from Spike Trains Satoru Shiono· Cen tral Research Laboratory Mi tsu bishi Electric Corporation Amagasaki, Hyogo 661, Japan Michio Nakashima Cen tral Research Laboratory Mi tsu bishi Electric Corporation Amagasaki, Hyogo 661, Japan Satoshi Yamada Central Research Laboratory Mitsu bishi Electric Corporation Amagasaki, Hyogo 661, Japan Kenji Matsumoto Facul ty of Pharmaceu tical Science Hokkaidou University Sapporo, Hokkaidou 060, Japan Abstract We have attempted to use information theoretic quantities for analyzing neuronal connection structure from spike trains. Two point mu tual information and its maximum value, channel capacity, between a pair of neurons were found to be useful for sensitive detection of crosscorrelation and for estimation of synaptic strength, respectively. Three point mutual information among three neurons could give their interconnection structure. Therefore, our information theoretic analysis was shown to be a very powerful technique for deducing neuronal connection structure. Some concrete examples of its application to simulated spike trains are presented. 1 INTRODUCTION The deduction of neuronal connection structure from spike trains, including synaptic strength estimation, has long been one of the central issues for understanding the structure and function of the neuronal circuit and thus the information processing ·corresponding author 515 516 Shiono, Yamada, Nakashima, and Matsumoto mechanism at the neuronal circuitry level. A variety of crosscorrelational techniques for two or more neurons have been proposed and utilized (e.g., Melssen and Epping, 1987; Aertsen et. ai., 1989). There are, however, some difficulties with those techniques, as discussed by, e.g., Yang and Shamma (1990). It is sometimes difficult for the method to distinguish a significant crosscorrelation from noise, especially when the amount of experimental data is limited. The quantitative estimation of synaptic connectivity is another difficulty. And it is impossible to determine whether two neurons are directly connected or not, only by finding a significant crosscorrelation between them. The information theory has been shown to afford a powerful tool for the description of neuronal input-output relations, such as in the investigation on the neuronal coding of the visual cortex (Eckhorn et. ai., 1976; Optican and Richmond, 1987). But there has been no extensive study to apply it to the correlational analysis of action potential trains. Because a correlational method using information theoretic quantities is considered to give a better correlational measure, the information theory is expected to offer a unique correlational method to overcome the above difficulties. In this paper, we describe information theory-based correlational analysis for action potential trains, using two and three point mutual information (MI) and channel capacity. Because the information theoretic analysis by two point MI and channel capacity will be published in near future (Yamada et. ai., 1993a), more detailed description is given here on the analysis by three point MI for infering the relationship among three neurons. 2 CORRELATIONAL ANALYSIS BASED ON INFORMATION THEORY 2.1 INFORMATION THEORETIC QUANTITIES According to the information theory, the n point mutual information expresses the amount of information shared among n processes (McGill, 1955). Let X, Y and Z be processes, and t and s be the time delays of X and Y from Z, respectively. Using Shannon entropies H, two point MI between X and Y and three point MI, are defined (Shannon, 1948; Ikeda et. ai., 1989): I(Xt : Ys ) I(Xt : Y, : Z) H(Xt) + H(Y,) - H(Xt, Y,), H(Xt) + H(Y,) + H(Z) - H(Xt, y,) -H(Y" Z) - H(Z, Xt) + H(Xt, Y" Z). I(Xt : Ys : Z) is related to I(Xt : Y,) as follows: (1 ) (2) (3) where I(Xt : YsIZ) means the two point conditional MI between X and Y if the state of Z is given. On the other hand, channel capacity is given by (r = s - t), CC(X: Yr) = maxI(X: Yr). p(x,) ( 4) We consider now X, Y and Z to be neurons whose spike activity has been measured. Information Theoretic Analysis of Connection Structure from Spike Trains 517 Two point MI and two point conditional MI are obtained by (i, j, k = 0, 1), (X Y) ~ ( I) ( )1 p(Yj,Tlxi) I : T = L....J P Yj,T Xi P Xi og (.)' . . P YJ,T I,J (5) I(X YIZ) ~( I)()l p(xi,t,Yj"lzk) t:, = L....J P Xi,t, Yj" Zk P Zk og (x. Iz ) ( . Iz ). . . •. P I,t k P YJ,s k I,J,'" (6) where x, Y and z mean the states of neurons, e.g., Xl for the firing state and Xo for the non-firing state of X, and p( ) denotes probability. And three point MI is obtained by using Equation (3). Those information theoretic quantities are calculated by using the probabilities estimated from the spike trains of X, Y and Z after the spike trains are converted into time sequences consisting of 0 and 1 with discrete time steps, as described elswhere (Yamada et. al., 1993a). 2.2 PROCEDURE FOR THREE POINT MUTUAL INFORMATION ANALYSIS Suppose that a three point MI peak is found at (to, so) in the t, s-plane (see Figure 1). The three time delays, to, So and r = So - to, are obtained. They are supposed to be time delays in three possible interconnections between any pair of neurons. Because the peak is not significant if only one pair of the three neurons is interconnected, two or three of the possible interconnections with corresponding time delays should truly work to produce the peak. We will utilize I(n : m) and I(n : mil) (n, m, I = X, Y or Z) at the peak to find working interconnections out of them. These quantities are obtained by recalculating each probability in Equations (5) and (6) over the whole peak region. If two neurons, e.g., X and Y, are not interconnected either I(X : Y) or I(X : YIZ) is equal to zero. The reverse proposition, however, is not true. The necessary and sufficient condition for having no interconnection is obtained by calculating I( n : m) and I( n : mil) for all possible interconnection structures. The neurons are rearranged and renamed A, Band C in the order of the time delays. There are only four interconnection structures, as shown in Table 1. I: No interconnection between A and B. A and B are statistically independent, i. e., p(aj,bj ) = p(aj)p(bj ), I(A: B) = O. The three point MI peak is negative. II: No interconnection between A and C. The states of A and C are statistically independent when the state of B is given, i.e., p(ai' cklbj) = p(adbj)p(Cklbj), I(A : CIB) = O. The peak is positive. III: No interconnection between Band C. Similar to case II, because p(bj , cklai) = p(bjlai)p(cklai), I(B: CIA) = O. The peak is positive. IV: Three in terconnections. The above three cases are considered to occur concomitantly in this case. The peak is positive or negative, depending on their relative contributions. Because A and B should have an apparent effect on the firing-probability of the postsynaptic neurons, I(A : B), I(A : CIB) and I(B : CIA) are all non-zero except for the case where the activity of B completely coincides with that of A with the specified time delay (in this case, both I(A : CIB) and I(B : CIA) are zero (see Yamada et. al., 1993b)). 518 Shiono, Yamada, Nakashima, and Matsumoto Table 1. Interconnection Structure and Information Theoretic Quantities Interconnection I: ~ II: ~ III: @ IV: ~ ~@ Structure ~@cl@ ctJ@ 2 point MI I(A:B) =0 >0 >0 >0 I(A:C) ~O >0 >0 ~O I(B:C) ~O >0 >0 ~O 2 point condition MI I(A:B I C) >0 ~O ~O ~O I(A:CIB) >0 =0 ~O >0 I(B:C I A) >0 ~O =0 >0 3 point MI I(A:B:C) + + + or From what we have described above, the interconnection structure for a three point MI peak is deduced utilizing the following procedure; (a) A negative 3pMI peak: it corresponds to case I or IV. The problem is to determine whether A and B are interconnected or not. (1) If I(A : B) = 0, case I. (2) If I(A : B) > 0, case IV. (b) A positive 3pMI peak: it corresponds to case II, III or IV. The existence of the A-C and B-C interconnections has to be checked. (1) If I(A : CIB) > ° and I(B : CIA) > 0, case IV. (2) If I(A : CIB) = ° and I(B : CIA) > 0, case II. (3) If I(A : CIB) > ° and I(B : CIA) = 0, case III. (4) If I(A : CIB) = ° and I(B : CIA) = 0, the interconnection structure cannot be ded nced except for the A - B interconnection. This procedure is applicable, if all the time delays are non-zero. If otherwise, some of the interconnections cannot be determined (Yamada et. ai., 1993b). 3 SIMULATED SPIKE TRAINS In order to characterize our information theoretic analysis, simulations of neuronal network models were carried out. We used a model neuron described by Information Theoretic Analysis of Connection Structure from Spike Trains 519 the Hodgkin-Huxley equations (Yamada et. ai., 1989). The used equations and parameters were described (Yamada et. al., 1993a). The Hodgkin-Huxley equations were mathematically integrated by the Runge-Kutta-Gill technique. 4 RESULTS AND DISCUSSION 4.1 ANALYSIS BY TWO POINT MUTUAL INFORMATION AND CHANNEL CAPACITY The performance was previously reported of the information theoretic analysis by two point MI and channel capacity (Yamada et. ai., 1993a). Briefly, this anlytical method was compared with some conventional ones for both excitatory and inhibitory connections using action potential trains obtained by the simulation of a model neuronal network. It was shown to have the following advantages. First, it reduced correlational measures within the bounds of noise and simultaneously amplified beyond the bounds by its nonlinear function. It should be easier in its crosscorrelation graph to find a neuron pair having a weak but significant interaction, especially when the synaptic strength is small or the amount of experimental data is limited. Second, channel capacity was shown to allow fairly effective estimation of synaptic strength, being independent of the firing probability of a presynaptic neuron, as long as this firing probability was not large enough to have the overlap of two successive postsynaptic potentials. 4.2 ANALYSIS BY THREE POINT MUTUAL INFORMATION The practical application of the analysis by three point MI is shown below in detail, using spike trains obtained by simulation of the three-neuron network models shown in Figures 1 and 2 (Yamada et. ai., 1993b). The network model in Figure 1(1) has three interconnections. In Figure 1(2), three point MI has two positive peaks at (17ms, 12ms) (unit "ms" is omitted hereafter) and (17,30), and one negative peak at (0,12). For the peak at (17,12), the neurons are renamed A, B and C from the time delays (Z as A, Y as B and X as C), as in Table 1. Because only I(B : CIA) ..:. 0 (see Figure 1 legend), the peak indicates case III with A-+B (Z-+Y) (s = 12) and A-+C (Z-+X) (t = 17) interconnections. Similarly, the peak at (17,30) indicates Z -+X and X -+Y (s - t = 13) interconnections, and the peak at (0,12) indicates Z-+Y and X -+Y interconnections. The interconnection structure deduced from each three point MI peak is consistent with each other, and in agreement with the network model. Alternatively, the three point MI graphical presentation such as shown in Figure 1(2) itself gives indication of some truly existing interconnections. If more than two three point MI peaks are found on one of the three lines, t = to, s = So and s-t = TO, the interconnection with the time delay represented by this line is considered to be real. For example, because the peaks at (17, 12) and (17, 30) are on the line of t = 17 (Figure 1(2)), the interconnection represented by t = 17 (Z-+X) are considered to be real. In a similar manner, the interconnections of s = 12 (Z-+Y) and s - t = 12 (X -+Y) are obtained. But this graphical indication is not complete, and thus the calculation of two point MI's and two point conditional MI's should be always 520 5hiono• Yamada. Nakashima. and Matsumoto (1) (2) 0.0010 -so Neuron X ~ Neuron Y DNeuronZ o so t (ms) Figure 1. Three point Ml analYsis of simulated spike trains. (1) A. three-neuron network model with Z .... X Z .... Y and X .... Y interconnections. The total number of spikes; X:4000• y:5400• Z:3150. (2) Three poinl Ml analysis of spike trains. Three point Ml has two positive peaks al (17.12) and (17.30). and one negative peak at (0.12). For the peak al (17. 12) the neurons are renamed (Z as A. Y as B and X as C). Two point Ml and two point conditional M1 for the peak at (17. 12) are: I(A: B) == 0.03596. I(A: C) == 0.06855• I(B : C) == 0.01375• I(A : BIC) == 0.02126. I(A : CIB) == 0.05376. I(B : CIA) == 0.00011. So. I( B : CIA) .:. o. indicating case ~.oo10 111 (see Table 1) with A .... B (Z .... Y) and A .... C (Z .... X) interconnections. Similarly, for the peaks at (17,30) and at (0,12). Z .... X and X .... Y interConnections. and Z .... Y and X .... Y interconnections are obtalned, respectively. performed for connrma.tion. The nelwork model in Figure 2(1) has four interCOnnections. Three point Ml has ftVe major peaks: four positive peaks at (17. -12). (17. 30). (_24.-12) and (1 7 • 12 ) and one negative peak at (0.10). The peaks at (17. -12). (17. 12) and (17. 30) Me on the line 01 t == 17 (Z .... X). the peaks at (17, -12) and (-24. -12) are on Il\e line 01 s == -12 (Z ... Y). the peaks at (17.12) and (0. 10) are on the line of s == 12 (Z .... Y). and the peaks at (-24. -12), (0.10) and (17. 30) are on the line of Information Theoretic Analysis of Connection Structure from Spike Trains 521 (1) (2) 0.0008 -0.0008 Neuron X ~ Neuron Y ~euronz o t (ms) 50 Figure 2. Three point MI analysis of simulated spike trains. (1) A three-neuron network model with Z-+X Z-+Y, Z~Y and X-+Y interconnections. The total number of spikes; X:4300, Y:5150, Z:4850. (2) Three point MI analysis of spike trains. Three point MI has five major peaks, four positive peaks at (17, -12), (17,12), (17,30) and (-24, -12), and one negative peak at (0,10). s - t = 12 (X -+Y). The calculation of two point MI and two point conditional MI for each peak gives the confirmation that each three point MI peak was produced by two interconnections. Namely, their calculation indicates Z-+X (t = 17), Z~Y (s = -12), Z-+Y (s = 12) and X-+Y (s - t = 12) interconnections. There are also some small peaks. They are considered to be ghost peaks due to two or three interconnections, at least one of wllich is a combination of two interconnections found by analyzing the major peaks. For example, the positive peak at (-7, -12) indicates Z~Y and X-+Y interconnections, but the latter (s - t = -5) is the combination of the Z -+ X interconnection (t = 17) and the Z -+ Y interconnection (s = 12). The interconnection structure of a network containing an inhibitory intercolLllectioll or consisting of more than four neurons can also be deduced, although it becomes more difficult to perform the three point MI analysis. 522 Shiono, Yamada, Nakashima, and Matsumoto References A. M. H. J. Aertsen, G. L. Gerstein, M. K. Habib & G. Palm. (1989) Dynamics of neuronal firing correlation: modulation of" effective connectivity". J. N europhY6iol. 61: 900-917. R. Eckhorn, O. J. Griisser, J. Kremer, K. Pellnitz & B. Popel. (1976) Efficiency of different neuronal codes: information transfer calculations for three different neuronal systems. Bioi. Cyhern. 22: 49-60. K. Ikeda, K. Otsuka & K. Matsumoto. (1989) Maxwell-Bloch turbulence. Prog. Theor. Phys., Suppl. 99: 295-324. W. J. McGill. (1955) Multivariate information transmission. IRE Tran6. Inf. Theory 1: 93-111. W. J. Melssen & W. J. M. Epping. (1987) Detection and estimation of neural connectivity based on crosscorrelation analysis. Bioi. Cyhern. 57: 403-414. L. M. Optican & B. J. Richmond. (1987) Temporal encoding of two-dimensional patterns by single units primate inferior temporal cortex. III. Information theoretic analysis. J. NeurophY6iol. 57: 162-178. C. E. Shannon. (1948) A mathematical theory of communication. Bell. Syst. Techn. J. 27: 379-423. S. Yamada, M. Nakashima, K. Matsumoto & S. Shiono. (1993a) Information theoretic analysis of action potential trains: 1. Analysis of correlation between two neurons. Bioi. Cyhern., in press. S. Yamada, M. Nakashima, K. Matsumoto & S. Shiono. (1993b) Information theoretic analysis of action potential trains: II. Analysis of correlation among three neurons. submitted to BioI. Cyhern. W. M. Yamada, C. Koch & P. R. Adams. (1989) Multiple channels and calcium dynamics. In C. Koch & I. Segev (ed), Methods in Neuronal Modeling: From Synapses to Neurons, 97-133, Cambridge, MA, USA: MIT Press. X. Yang & S. A. Shamma. (1990) Identification of connectivity in neural networks. Biophys. J. 57: 987-999.
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On the Use of Projection Pursuit Constraints for Training Neural Networks Nathan Illtl'ator'" Comput.er Science Department Tel-Aviv Universit.y Ramat.-A viv, 69978 ISRAEL and Inst.itute for Brain and Neural Systems, Brown University nin~math,tau.ac.il Abstract \Ve present a novel classifica t.ioll and regression met.hod that combines exploratory projection pursuit. (unsupervised traiuing) with projection pursuit. regression (supervised t.raining), t.o yield a. nev,,' family of cost./complexity penalLy terms. Some improved generalization properties are demonstrat.ed on real \vorld problems. 1 Introduction Parameter estimat.ion becomes difficult. in high-dimensional spaces due t.o the increasing sparseness of t.he dat.a. Therefore. when a low dimensional representation is embedded in t.he da.t.a. dimensionality l'eJuction methods become useful. One such met.hod - projection pursuit. regression (Friedman and St.uet.zle, 1981) (PPR) is capable of performing dimensionality reduct.ion by composit.ion, namely, it constructs an approximat.ion to the desired response function using a composition of lower dimensional smooth functions, These functions depend on low dimensional projections t.hrough t.he data . • Research was support.ed by the N at.ional Science Foundat.ion. the Army Research Office, and the Office of Naval Researclr . 3 4 Intrator When the dimensionality of the problem is in the thousands, even projection pursuit methods are almost alwa.ys over-parametrized, t.herefore, additional smoothing is needed for low variance estimation. Explol'atory Projection Pursuit (Friedman and Thkey, 1974; Friedman, 1987) (EPP) may be useful for t.hat. It searches in a high dimensional space for structure in the form of (semi) linear projections with constraints characterized by a projection index. The projection index may be considered as a universal prior for a large class of problems, or may be tailored t.o a specific problem based on prior knowledge. In this paper, the general for111 of exploratory projection pursuit is formulated to be an additional constraint for projection pUl'suit regression. In particular, a hybrid combination of supervised and unsupervised artificial neural network (ANN) is described as a special case. In addition, a specific project.ion index that is particularly useful for classification (Int.rator, 1990; Intrator and Cooper, 1992) is introduced in this context. A more detailed discussion appears in Intrator (1993). 2 Brief Description of Projection Pursuit Regression Let (X, Y) be a pair of random variables, X E Rd , and Y E R. The problem is to approximate the d dimensiona.l surfa('e I(x) = E[Y'IX = x} from n observations (Xl, YI), ... , (Xu, Yn). PPR tries t.o approximate a funct.ion 1 by a sum of ridge functions (functions that are constant. along lines) 1(:1') ~ L gj(af x). j=l The fit.t.ing procedure alt.ernat.es between a.n estimation of a direction a and an estimat.ioll of a smoot.h funct.ion g. such that at. iterat.ion j, t.he square average of t.he resid uals l'ij(xd = 1'ij-l - 9j((IJ xd is minimized. This process is init.ialized by setting 1'jO = !Ii. Usually, the initial values of aj a.re t.aken to be the first few principal component.s of the data. Estimation of the ridge functions call be achieved by various nonparamet.ric smoothing techniques such as locally linear functions (Friedman and Stuetzle, 1981), k-nearest neighbors (Hall. 1989b), splines or variable degree polynomials. The smoot.hness const.raint. imposed on !1, implies t.hat. t.he actual projection pursuit is achieved by minimizing at. it.erat.ioJl j. t.lte sum II i= 1 for some smoothness measure C. Although PPR cOllverg('s t.o the desired response function (Jones, 1987), the use of non-paramet.ric function estimat.ion is likely to lead to ovel'fitt.ing. Recent results (Hornik, 1991) suggest. that a feed forward net.work archit.ecture with a single On the Use of Projection Pursuit Constraints for Training Neural Networks 5 hidden layer and a rat.her general fixed activat.ion function is a universal approximator. Therefore, the use of a non-parametric single ridge function estimation can be avoided. It is thus appropriate to concentrate on the est.imation of good projections. In the next section we present a general framework of PPR architecture, and in sect.ion 4 we restrict it. t.o a feed-forward architecture with sigmoidal hidden units. 3 EstiInating The Projections Using Exploratory Projection Pursuit Explorat.ory projection pursuit ·is based on seeking interesting projections of high dimensional data points (Krllskal, 1969; Switzer, 1970; Kruskal, 1972; Friedman and Tukey, 1974; Friedman, 1987; Jones and Sibson, 1987; Hall, 1988; Huber, 1985, for review). The notion of interesting projections is motivated by an observation t.hat for most. high-dimensional data clouds, most low-dimensional projections are approximat.ely normal (Oiaconis alld F!'('edlllan, 1984). This finding suggests that the important information in the data is conveyed in t.hose direct.ions whose single dimensional project.ed dist.ribution is far from Gaussian. Variolls projection indices (measures for t.he goodrwss of a. projl-'ction) differ on the assumptions about the nature of deviation from norl1lality, (Iud ill their comput.ational efficiency. They can be considered as different priOl's mot.ivat.ed by specific assumptions on t.he underlying model. To partially decouple the search for a projection vectol' from the search for a nonparametric ridge function, we propose to add a penalty term, which is based on a pl'Oject.ion index, t.o t.he energy minimizat.ion associated wit.h the estimation of the ridge functions and t.he projections. Specifically, let p( a) be a projection index which is minimized for project.ions wit.h a certain deviation fl'0111 normality; At the j'th iterat.ion, we minimize the sum L 1}( .r;) + (,'(gj) + p(aj). i When a concurrent minimizat.ion ovet' several project.ions/functions is practical, we get a penalty t.erm of t.he form B(j) = L[C(gj) + p(aj )]. j Since C and p may not be linear, t.he more general measure t.hat does not assume a step",Tise approach, but. instead seeks I projections and ridge functions concurrently, is given by B(f) = C(9J," ·,gd + p(a.J, .. . ,ad, In practice, p depends implicit.ly 011 t.he t.raining dat.a, (t.he empirical density) and is therefore replaced by its empirical measure ii. 3.1 Some Possible Measures Some applicable projection indices are disc.ussed in (Huber, 1985; Jones and Sibson, 1987; Friedman, 1987; Hall, 1989a; Intrator, 1990). Probably, a.ll the possible 6 Intrator measures should emphasize some form of deviation from normality but the specific type may depend on the problem at hand. For example, a measure based on the Karhunen Loeve expansion O"Iougeot et al., 1991) may be useful for image compression with autoassociative net.works, since in this case one is int.erested in minimizing the L2 norm of tlH' dist.ance between t.he reconst.ructed image and the original one, and under mild condit.ions, t.he Karhunen Loeve expansion gives the optimal solution. A different type of prior knowledge is required for classificat.ion problems. The underlying a'5sumption then is that the data is clustered (when projecting in the right direct.ions) and that t.he classification may be achieved by some (nonlinear) ma.pping of these clustel·s. In such a case, the projection index should emphasize multi-modality as a specific deviation from normality. A projection index that emphasizes multimodalities in the projected distribution (without relying on the class la.bels) has recently been int.roduced (Intrator, 1990) and implemented efficiently using a variant of a biologically motivated unsupervised network (Intrat.or and Cooper, 1992). Its int.egration into a back-propagat.ion classifier will be discussed below. 3.2 Adding EPP constraints to baek-propagatioll network One way of adding SOllie prior knowledge int 0 the archi t.ecLme is by 111lllll1llZmg the effective number of parameters llsing weight. sharing, ill which a single weight is shared among many connections in the network (\\'aibel et. al., 1989; Le Cun et aI., }989). An ext.f'nsion of t.his idea is the "soft. \',·eight. sharing" which favors irregularities in the weight distribution in the form of mult.imodality (Nowlan and Hinton, 1992). This penalty improved generalization results obtained by weight elimination penalt.y. Bot.h t.hese wet.hods make an explicit. assumption about the structure of t.he weight. space, but. wit.h 110 regarJ to the structure of the illput space. As described in the context of project.ion pursuit. regression. a penalt.y term may be added t.o the energy funct.ional minimized by error back propagation, for the purpose of mea<;uring direct.ly t.he goodness of t.he projections sOllght by the network. Since our main int.erest. is in reducing ovedHt.ing fOI' high dimensional pl'Oblems, our underlying assumpt.ion is t.hat. t.ile slll-faCf.' fUllct.ion to be estirnat.ed can be faithfully represented using a low dimensiollal composition of sigmoidal functions, namely, using a back-propagation net.work in which t.he number of hidden units is much smaller t.han the number of input unit.s. Therefore, t.he penalty term may be added only to the hidden layer. The synapt.ic modification equat.ions of the hidden units' weights become OWij fJt -c [ot(w, .1') aWij 0P(Wl, .... wn) +-----OU'ij +(Contrihul,ion of cost/complexity t.erms)]. An appl'Oach of t.his type has lWl'1I used in ima.ge compl'cssion, wit.h a penalty aimed at minimizing tIl<' ent.ropy of the projected distribution (Bichsel and Seitz, 1989). This penalt.y eel'tainly measures deviat.ion from normality, since entropy is maximized for a Gaussian distribution. On the Use of Projection Pursuit Constraints for Training Neural Networks 7 4 Projection Index for Classification: The Unsupervised BCM Neuron Intrator (1990) has recently shown that a variant of the Bienenstock, Cooper and Munro neuron (Bienenstock et al., 1982) performs exploratory projection pursuit using a projection index that measures multi-modality. This neuron version allows theoretical analysis of some visual deprivation experiments (lntrat.or and Cooper, 1992), and is in agreement. with the vast experimental result.s on visual cortical plasticity (Clothiaux et al., 1991). A network implementation which can find several projections in parallel while ret.aining its computational efficiency, was found to be applicable for extracting features from very high dimensional vector spaces (Intrator and Gold, 1993; Int.rator et al., 1991; Intrator, 1992) The activity of neuron k in the network is Ck = Li XiWik + WOk. The inhibited activity and threshold of the k'th neuron is given by C/.: = (1(Ck -II LCj), j'f;/.: 8 ~ k E[''l] - m = . cj,: . The threshold e~~l is the point. at. which the modificat.ion function </J changes sign (see Intrator and Cooper, 1992 for further det.ails). The function </J is given by </J(c, 8/11} = c(c - 8 m }. The risk (projection index) for a single neuron is given by R( Wk) = -{ ~ E[c2] ~ E2(c~]}. The total risk is the sum of each local risk. The negative gradient. of the risk that leads to the synaptic modification equations is given by OWjj E[A..( 8 j} '( ~ ) ~ A.( ~ 8-k) '( -) ] at = IJ) Cj, - m (1 Cj l!j 11 L <p Cl', - III (1 Ck Xi . k'f;j This last equa.tion is an a.dditional pellalt.y to t.he energy minimizat.ion of the supervised net.work. Not.e that there is an int.eract.ion between adjacent neurons in the hidden layer. In practice, t.he st.ochast.ic version of t.he different.ial equat.ion can be used as the learning ntle. 5 Applications Vve have applied t.his hybrid classification met.hod to various speech and image recognition problems in high dimensional space. In one speech application we used voiceless stop consonant.s extracted from the TIMIT database as training tokens (Intrator and Tajchman, 1991). A det.ailed biologically motivated speech representation was produced by Lyoll's cochlear model (Lyon, 1982; Slaney, 1988). This representation produced 5040 dimensions (84 channels x 60 t.ime slices). In addition t.o an init.ial voiceless st.op, each t.oken cont.ained a final vowel from the set [aa, ao, er, iy]. Classificat.ion of t.he voiceless stop consonant.s using a test set that included 7 vowels [uh, ih, eh, ae, ah, uw, ow] produced an average error of 18.8% 8 Intrator while on the same task classification using back-propagation network produced an average error of 20.9% (a significant difference, P < .0013). Addit.ional experiments on vowel tokens appear in Tajchman and Intrator (1992). Another application is in the area of face l·ecognit.ion from gray level pixels (Intrator et al., 1992). After aligning and normalizing the images, the input was set to 37 x 62 pixels (total of 2294 dimensions). The recognition performance was tested on a subset of t.he MIT Media Lab database of face images made available by Turk and Pent.land (1991) which cont.ained 27 face images of each of 16 different persons. The images were taken under val'ying illumiuation and camera location. Of the 27 images available, 17 randomly chosen ones served for tl'aining and the remaining 10 were used for test.iug, U siug all ensemble average of hybrid networks (Lincoln and Skrzypek, 1990; Pearlmut.t.er and Rosenfeld, 1991; Perrone and Cooper, 1992) we obtained an errOl' rat.e of 0.62% as opposed to 1.2% using a similar ensemble of back-prop networks. A single back-prop network achieves an error between 2.5% to 6% on this data. The experiments were done using 8 hidden units, 6 SUl11l11ary A penalty that allows the incol'porat.ioll of additional prior information on the underlying model was presC'llt.ed. This prior was introduced in t.he context of projection pursuit regression, classificat.ioll, aud in the context of back-propagation network. It achieves pa.rt.ial decoupling of est.illIat.ion of t.he ridge fuuctions (in PPR) or the regression function in back-propagat.ion net. from t.he est.imatioll of t.he projections, Thus it is potentially useful in reducing problems associat.ed wit.h overfitting which are more pronounced in high dimensional dat.a. Some possible projection indices were discllssed and a specific projection index that is particula.rly useful for classificat.ion was pt'esented in this ('on text. This measure that emphasizes multi-modality in the projected distribut.ioll, was found useful in several very high dimensional problems. 6.1 Ackllowledglueuts I wish to t.hank Leon Cooper, Stu Gel1lan anJ Michael Pel'l'one for many fruitful conversations and to t.he referee for helpful comments. The speech experiments were performed using the comput.at.ional facilit.ies of the Cognitive Science Department at Browll University. Research was supported by the National Science Foundation, t.he Army Research Office, a.nd t.he Office of Naval Research. References Bichsel, M. and Seit.z, P. (1989). Minimuln class ent.ropy: A maximum informat.ion approa.ch t.o layered netowrks. ,\"cllmi Ndworks, 2:133-141. Bienenstock, E. L .. Cooper, L. N., and ~'ltlHro, P. W. (198'2). Theory for t.he development of neuron select.ivit.~,: orientat.ioll s~wcificit.y allel binocular int.eract.ion in visual cortex. Journal Nctll'Oscicllct'. 1::32- 48. On the Use of Projection Pursuit Constraints for Training Neural Networks 9 Clothiaux, E. E., Cooper, L. N., and Bear, M. F. (1991). Synaptic plasticity in visual cortex: Comparison of theory with experiment. Joumal of Neurophysiology, 66:17851804. Diacollis, P. and Freedman, D. (1984). Asymptotics of graphical projection pursuit. Annals of Statistics, 12:793-815. . Friedman, J. H. {1987}. Exploratory projection pursuit. Journal of the American Statistical Association, 82:249-266. Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. Journal of the Ame"ican Statistical Association, 76:817-823. Friedman, J. H. and Tukey, J. W. (1974). A projection pursuit algorithm for exploratory data analysis. IEEE Tmnsactions on Compute,-s, C(23):881-889. Hall, P. (1988). Estimating t.he direction in which data set is most interesting. PJ'Obab. Theo,'Y Rei. Fields, 80:51-78. Hall, P. (1989a). On polynomial-based projection indices for exploratory projection pursuit.. The Annals of Statistics, 17:589-605. Hall, P. (1989b). On projection pursuit. regression. The Annals of Statistics, 17:573-588. Hornik, K. (1991). Approximat.ioll capabilities of lUult.ilayer feedforward networks. Neural Netwo,'ks,4:251-257. Huber, P. J. (1985). Project.ion pursuit.. (wit.h discussion). The Annals of Statistics, 13:435-475. Int.l'ator, N. (1990). Featllre extract.ion llsing an ullsupervised neural network. In Touretzky, D. S .. EHman, J. L., Sejnowski. T. J., and Hint.on, G. 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Y. and Bruckstein, A., edit.ors, Pmceedings of the 8th Ismeli Conference on AICll, pages 113-123. Elsevier. Intrator, N., Reisfeld, D., and YeshUl'u 11 , Y. (1992). Face recognition using a hybrid supervised/unsupervised neural network. Preprint. Intrator, N. and Tajchman, G. (1991). Supervised and unsupervised feature extraction from a cochlear model for speech recognition. In Juang, B. H., Kung, S. Y., and Kamm, C. A., editors, Neuml NetwOJ·J.;s for Signal Pmcessing - Proceedings of the 1991 IEEE WOJ'kshop, pages 460-469. IEEE Press, New York, NY. Jones, L. (1987). On a conjecture of huber concerning t.he cOllvergE'nce of projection pursuit regression. Annals of Statistics. 15:880-882. Jones, M. C. and Sibson, R. (198i). What. is projection pursuit? (with discussion). J. Roy. Statist. Soc .. Ser. A(150):1 ·-36. 10 Intrator Kruskal, J. B. (1969). 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When lletworks disagree: Generalized ensemble method for neural net.works. In Mammone, R. J. and Zeevi, Y., editors, Neural Networks: Theory (mel .4ppiicn/.iol1s, volume 2. Aca.demic Press. Slaney, M. (1988). Lyoll's cochlear model. Technical repOl·t, Apple Corporat.e Library, Cupertino, CA 95014. Switzer, P. (1970). Numerical c1assificat.ion. In BarIlelt.. V., edit.or, Geostatistics. Plenum Press, New York. Tajchmall, G. N. and Intrator, N. (1992). Phonet.ic classification of T1MIT segments preprocessed with lyoll's cochlear model using a sllperviscd/un:mpenrised hybrid neural network. [n P"oct'Cdings Itllenwliolwl COII/CI'CtlCC on Spoh'/l Language Processing, Banff, Albert.a, Canada. Turk, M. and Pent.land, A. (1991). Eigc'lIfaces for recognit.ion. 1. 0/ Cognitive Netl1'Oscience, 3:71-86. Waibel, A., Hanazawa, T., Hinton, G., Shikano, K., and Lang, I<. {1989}. Phoneme recognition using time-delay neura.! net.works. IEEE Transoctions on ASSP, 37:328339.
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Assessing and Improving Neural Network Predictions by the Bootstrap Algorithm Gerhard Paass German National Research Center for Computer Science (GMD) D-5205 Sankt Augustin, Germany e-mail: paass<Dgmd.de Abstract The bootstrap algorithm is a computational intensive procedure to derive nonparametric confidence intervals of statistical estimators in situations where an analytic solution is intractable. It is applied to neural networks to estimate the predictive distribution for unseen inputs. The consistency of different bootstrap procedures and their convergence speed is discussed. A small scale simulation experiment shows the applicability of the bootstrap to practical problems and its potential use. 1 INTRODUCTION Bootstrapping is a strategy for estimating standard errors and confidence intervals for parameters when the form of the underlying distribution is unknown. It is particularly valuable when the parameter of interest is a complicated functional of the true distribution. The key idea first promoted by Efron (1979) is that the relationship between the true cumulative distribution function (cdf) F and the sample of size n is similar to the relationship between the empirical cdf Fn and a secondary sample drawn from it. So one uses the primary sample to form an estimate Fn and calculates the sampling distribution of the parameter estimate under Fn. This calculation is done by drawing many secondary samples and finding the estimate, or function of the estimate, for each. If:Fn is a good approximation of F, then H n , the sampling distribution of the estimate under Fn , is a generally good approximation to the sampling distribution for the estimate under F. H n is 196 Assessing and Improving Neural Network Predictions by the Bootstrap Algorithm 197 called the bootstrap distribution of the parameter. Introductory articles are Efron and Gong (1983) and Efron and Tibshirani (1986). For a survey of bootstrap results see Hinkley (1988) and DiCiccio and Romano (1988). A neural networks often may be considered as a nonlinear or nonparametric regression model Z = g/3(Y) + € (1) which d~fines the relation between the vectors Y and z of input and output variables. The ter:n € can be interpreted as a random 'error' and the function g/3 depends on some Ul_kown parameter f3 which may have infinite dimension. Usually the network is used to determine a prediction Zo = g/3(Yo) for some new input vector Yo. If the data is a random sample, an estimate P differs from the true value of f3 because of the sampling error and consequently the prediction g~(yo) is different from the true prediction. In this paper the bootstrap approach is used to approximate a sampling distribution of the prediction (or a function thereof) and to estimate parameters of that distribution like its mean value, variance, percentiles, etc. Bootstrapping procedures are closely related to other resampling methods like cross validation and jackknife (Efron 1982). The jackknife can be considered as a linear approximation to the bootstrap (Efron, Tibshirani 1986). In the next section different versions of the bootstrap procedure for feedforward neural networks are defined and their theoretical properties are reviewed. Main points are the convergence of the bootstrap distribution to true theoretical distribution and the speed of that convergence. In the following section the results of a simulation experiment for a simple backprop model are reported and the application of the bootstrap to model selection is discussed. The final section gives a short summary. 2 CONSISTENCY OF THE BOOTSTRAP FOR FEEDFORWARD NEURAL NETWORKS Assume X (n) := (Xl, ... , xn) is the available independent, identically distributed (iid) sample from an underlying cdf F where Xi = (Zi' Yi) and Fn is the corresponding empirical cdf. For a given Yo let T} = T}(g/3(Yo)) be a parameter of interest of the prediction, e.g. the mean value of the prediction of a component of Z for Yo. The pairwise bootstrap algorithm is an intuitive way to apply the bootstrap notion to regression. It was proposed by Efron (1982) and involves the independent repetition of following steps for b = 1, ... , B. 1. A sample XbCn) of size n is generated from Fn. Notice that this amounts to the random selection of n elements from X(n) with replacement. 2. An estimate TJb is determined from X;(n). The resulting empirical cdf of the TJb, b = 1, ... ,n is denoted by H B and approximates the sampling distribution for the estimate TJ under Fn. The standard deviation of HE is an estimate of the standard error of T}(Fn) , and [Hnl(a), Hn 1(1- a)] is an approximate (1 - 2a) central confidence interval. 198 Paass In general two conditions are necessary for the bootstrap to be consistent: • The estimator, e.g. fib has to be consistent . • The functional which maps F to HB has to be smooth. This requirement can be formalized by a uniform weak convergence condition (DiCiccio, Romano 1988). Using these concepts Freedman (1981) proved that for the parameters of a linear regression model the pairwise bootstrap procedure is consistent, i.e. yields the desired limit distribution for n, B --+ 00. Mammen (1991) showed that this also holds for the preuictive distribution of a linear model (i.e. linear contrasts). These results hold even if the errors are heteroscedastic, i.e. if the distribution of fi depends on the value of Yi. The performance of the bootstrap for linear regression is extensively discussed by Wu (1986). It turns out that the small sample properties can be different from the asymptotic relations and the bootstrap may exhibit a sizeable bias. Various procedures of bias correction have been proposed (DiCiccio, Romano 1988). Beran (1990) discusses a calibrated bootstrap prediction region containing the prediction g,6(YO)+f with prescribed probability a. It requires a sequence of nested bootstraps. Its coverage probability tends to a at a rate up to n- 2 • Note that this procedure can be applied to nonlinear regression models (1) with homoscedastic errors (Example 3 in Beran (1990, p.718) can be extended to this case). Biases especially arise if the errors are heteroscedastic. Hinkley (1988) discusses the parametric modelling of dependency of the error distribution (or its variance) from Y and the application of the bootstrap algorithm using this model. The problem is here to determine this parametric dependency from the data. As an alternative Wu (1986) and Liu (1988) take into account heteroscedasticity in a nonparametric way. They propose the following wild bootstrap algorithm which starts with a consistent estimate /3 based on the sample X(n). Then the set of residuals (il , ... ,in) with fi := Zi g~(yd is determined. The approach attempts to mimic the conditional distribution of Z given Yi in a very crude way by defining a distribution Oi whose first three moments coincide with the observed residual fi: J udOj(u) = 0 J u2dOi (u) = f; J u3dOi (u) = f~ (2) Two point distributions are used which are uniquely defined by this requirement (Mammen 1991, p.121). Then the following steps are repeated for b = 1, ... , B: 1. Independently generate residuals fi according to Oi and generate observations z; := 9p(Yi) + fi for i = (1, ... , n). This yields a new sample X;(n) of size n. 2. An estimate fit is determined from Xb(n). The resulting empirical cdf of the fit is then taken as the bootstrap distribution H B which approximates the sampling distribution for the estimate fI under Fn. Mammen (1991, p.123) shows that this algorithm is consistent for the prediction of linear regression models if the least square estimator or M-estimators are used and discusses the convergence speed of the procedure. Assessing and Improving Neural Network Predictions by the Bootstrap Algorithm 199 The bootstrap may also be applied to nonparameric regression models like kerneltype estimators of the form g(y) = [g?~:~~(~lJ (3) with kernel [{ and bandwidth h. These models are related to radial basis functions discussed in the neural network literature. For those models the pairwise bootstrap does not work (HardIe, Mammen 1990) as the algorithm is not forced to perform locall:J., ~raging. To account for heteroscedasticity in the errors of (1) HardIe (1990, p.103) advocates the use of the wild bootstrap algorithm described above. Under some regularity conditions he shows the convergence of the bootstrap distribution of the kernel estimator to the correct limit distribution. To summarize the bootstrap often is used simply because an analytic derivation of the desired sampling distribution is too complicated. The asymptotic investigations offer two additional reasons: • There exist versions of the bootstrap algorithm that have a better rate of convergence than the usual asymptotic normal approximation. This effect has been extensively discussed in literature e.g. by Hall (1988), Beran (1988), DiCiccio and Romano (1988, p.349), Mammen (1991, p.74). • There are cases where the bootstrap works, even if the normal approximation breaks down. Bickel and Freedman (1983) for instance show, that the bootstrap is valid for linear regression models in the presence of outliers and if the number of parameters changes with n. Their results are discussed and extended by Mammen (1991, p.88ff). 3 SIMULATION EXPERIMENTS To demonstrate the performance of the bootstrap for real real problems we investigated a small neural network. To get a nonlinear situation we chose a "noisy" version of the xor model with eight input units YI, ... ,Ys and a single output unit z. The input variables may take the values 0 and 1. The output unit of the true model is stochastic. It takes the values 0.1 and 0.9 with the following probabilities: p(y = 0.9) = 0.9 if Xl + X2 + X3 + X4 < 3 and X5 + X6 + X7 + Xs < 3 p(y = 0.9) = 0.1 if Xl + X2 + X3 + X4 < 3 and X5 + X6 + X7 + Xs > 3 p(y = 0.9) = 0.1 if Xl + X2 + X3 + X4 > 3 and X5 + X6 + X7 + XS < 3 p(y = 0.9) = 0.9 if Xl + X2 + X3 + X4 > 3 and X5 + X6 + X7 + Xs > 3 In contrast to the simple xor model generalization is possible in this setup. We generated a training set X( n) of n = 100 inputs using the true model. We used the pairwise bootstrap procedure described above and generated B = 30 different bootstrap samples X;(n) by random selection from X(n) with replacement. This number of bootstrap samples is rather low and only will yield reliable information on the central tendency of the prediction. More sensitive parameters of 200 Paass Input vectors Y1 Yg 10110110 01110110 1 1 1 101 1 0 00001110 10001110 01001110 11001110 00101110 10101110 01101110 1 1 101 1 1 0 00011110 10011110 01011110 11011110 00111110 ... true expected value Bootstrap Distribution of Prediction 0.0 0.5 t---------tl ~ H r------mJ-; ...---------.,14 I~ .... .A l!:rt A.. I· ~ zill t------::zs:~~ .A I!H l{l}-i ~ .A value predicted by the original backprop model ~------f--llnH percentiles of the 10 25 50 75 90 bootstrap distribution 1.0 Figure 1: Box-Plots of the Bootstrap Predictive Distribution for a Series of Different Input Vectors the distribution like low percentiles and the standard deviation can be expected to exhibit larger fluctuations. We estimated 30 weight vectors ~b from those samples by the backpropagation method with random initial weights. Subsequently for each of the 256 possible input vectors Yi we determined the prediction g~1> (Yi) yielding a predictive distribution. For comparison purposes we also estimated the weights of the original backprop model with the full data set X (n) and the corresponding Assessing and Improving Neural Network Predictions by the Bootstrap Algorithm 201 Table 1: Mean Square Deviation from the True Prediction INPUT HIDDEN MEAN SQUARE DIFFERENCE TYPE UNITS BOOTSTRAP DB FULL DATA DF training 2 0.18 0.19 inputs 3 0.17 0.19 4 0.17 0.19 ne I-training 2 0.30 0.34 inputs 3 0.35 0.38 4 0.37 0.42 Table 2: Coverage Probabilities of the Bootstrap Confidence Interval for Prediction HIDDEN UNITS 2 3 4 predictions. FRACTION OF CASES WITH TRUE PREDICTION IN [q2S, q7S] [q10, qgo] 0.47 0.44 0.43 0.77 0.70 0.70 For some of those input vectors the results are shown in figure 1. The distributions differ greatly in size and form for the different input vectors. Usually the spread of the predictive distribution is large if the median prediction differs substantially from the true value. This reflects the situation that the observed data does not have much information on the specific input vector. Simply by inspecting the predictive distribution the reliability of a predictions may be assessed in a heuristic way. This may be a great help in practical applications. In table 1 the mean square difference DB := (~L:~=1 (Zi - qSo)2) 1/2 between the true prediction Zi and the median qso of the bootstrap predictive distribution is compared to the mean square difference Ds := (1. L:?=l(Zi - Zi,F)2)1/2 between the true prediction and the value Zi,F estimated with full data backprop model. For the non-training inputs the bootstrap median has a lower mean deviation from the true value. This effect is a real practical advantage and occurs even for this simple bootstrap procedure. It may be caused in part by the variation of the initial weight values (cf. Pearlmutter, Rosenfeld 1991). The utilization of bootstrap procedures with higher order convergence has the potential to improve this effect. Table 2 list the fraction of cases in the full set of all 256 possible inputs where the true value is contained in the central 50% and 80% prediction interval. Note that the intervals are based on only 30 cases. For the correct model with 2 hidden units the difference is 0.03 which corresponds to just one case. Models with more hidden units exhibit larger fluctuations. To arrive at more reliable intervals the number of 202 Paass HIDDEN UNITS 2 3 4 Table 3: Spread of the Predictive Distribution MEAN INTERQUARTILE RANGE FOR TRAINING INPUTS NON-TRAINING INPUTS 0.13 0.11 0.11 0.29 0.35 0.37 bootstrap samples has to be increased by an order of magnitude. If we use a model with more than two hidden units the fit to the training sample cannot be improved but remains constant. For nontraining inputs, however, the predictions of the model deteriorate. In table 1 we see that the mean square deviation from the true prediction increases. This is just a manifestation of 'Occam's razor' which states that unnecessary complex models should not be prefered to simpler ones (MacKay 1992). Table 3 shows that the spread of the predictive distribution is increased for non-training inputs in the case of models with more than two hidden units. Therefore Occam's razor is supported by the bootstrap predictive distribution without knowing the correct prediction. This effect shows that bootstrap procedures may be utilized for model selection. Analoguous to Liu (1993) we may use a crossvalidation strategy to determine the prediction error for the bootstrap estimate ~b for sample elements of X (n) which are not contained in the bootstrap sample X; (n). In a similar way Efron (1982, p.52f) determines the error for the predictions g~b(Y) within the full sample X(n) and uses this as an indicator of the model performance. 4 SUMMARY The bootstrap method offers an computation intensive alternative to estimate the predictive distribution for a neural network even if the analytic derivation is intractable. The available asymptotic results show that it is valid for a large number of linear, nonlinear and even nonparametric regression problems. It has the potential to model the distribution of estimators to a higher precision than the usual normal asymptotics. It even may be valid if the normal asymptotics fail. However, the theoretical properties of bootstrap procedures for neural networks - especially nonlinear models - have to be investigated more comprehensively. In contrast to the Bayesian approach no distributional assumptions (e.g. normal errors) are have to be specified. The simulation experiments show that bootstrap methods offer practical advantages as the performance of the model with respect to a new input may be readily assessed. Acknowledgements This research was supported in part by the German Federal Department of Reserach and Technology, grant ITW8900A 7. Assessing and Improving Neural Network Predictions by the Bootstrap Algorithm 203 References Beran, R. (1988): Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements. Journal of the American Statistical Association. vol. 83, pp.687-697. Beran, R. (1990): Calibrating Prediction Regions. Journal of the American Statistical Association., vol. 85, pp.715-723. Bickel, P.J., Freedman, D.H. (1981): Some Asymptotic Theory for the Bootstrap. The Annals of Statistics, vol. 9, pp.1l96-1217. Bickel, P.J., Freedman, D.H. (1983): Bootstrapping Regression Models with many Parame~ers. In P. Bickel, K. Doksum, J .C. Hodges (eds.) A Festschrift for Erich Lehmann. Wadsworth, Belmont, CA, pp.28-48. DiCiccio, T.J., Romano, J.P. (1988): A Review of Bootstrap Confidence Intervals. J. Royal Statistical Soc., Ser. B, vol. 50, pp.338-354. Efron, B. (1979): Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics, vol 7, pp.1-26. Efron, B. (1982): The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia. Efron, B., Gong, G. (1983): A leisure look at the bootstrap, the jackknife and crossvalidation. A merican Statistician, vol. 37, pp.36-48. Efron, B., Tibshirani (1986): Bootstrap methods for Standard Errors, Confidence Intervals, and other Measures of Statistical Accuracy . Statistical Science, vol 1, pp.54-77. Freedman, D.H. (1981): Bootstrapping Regression Models. The Annals of Statistics, vol 9, p.1218-1228. HardIe, W.(1990): Applied Nonparametric Regression. Cambridge University Press, Cambridge. HardIe, W., Mammen, E. (1990): Bootstrap Methods in Nonparametric Regression. Preprint Nr. 593. Sonderforschungsbereich 123, University of Heidelberg. Hall, P. (1988): Theoretical Comparison of Bootstrap Confidence Intervals. The Annals of Statistics, vol 16, pp.927-985. Hinkley, D .. (1988): Bootstrap Methods. Journal of the Royal Statistical Society, Ser. B, vol.50, pp.321-337. Liu, R. (1988): Bootstrap Procedures under some non i.i.d. Models. The Annals of Statistics, vo1.16, pp. 1696-1708. Liu, Y. (1993): Neural Network Model Selection Using Asymptotic Jackknife Estimator and Cross-Validation Method. This volume. MacKay, D. J. C. (1992): Bayesian Model Comparison and Backprop Nets. In Moody, J .E., Hanson, S.J., Lippman, R.P. (eds.) Advances in Neural Information Processing Systems 4. Morgan Kaufmann, San Mateo, pp.839-846. Mammen, E. (1991): When does Bootstrap Work: Asymptotic Results and Simulations. Preprint Nr. 623. Sonderforschungsbereich 123, University of Heidelberg. Pearlmutter, B.A., Rosenfeld, R. (1991): Chaitin-Kolmogorov Complexity and Generalization in Neural Networks. in Lippmann et al. (eds.): Advances in Neural Information Processing Systems 3, Morgan Kaufmann, pp.925-931. C.F.J. Wu (1986): Jackknife, Bootstrap and other Resampling Methods in Regression Analysis. The Annals of Statistics, vol. 14, p.1261-1295.
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Self-Organizing Rules for Robust Principal Component Analysis Lei Xu l ,2"'and Alan Yuillel 1. Division of Applied Sciences, Harvard University, Cambridge, MA 02138 2. Dept. of Mathematics, Peking University, Beijing, P.R.China Abstract In the presence of outliers, the existing self-organizing rules for Principal Component Analysis (PCA) perform poorly. Using statistical physics techniques including the Gibbs distribution, binary decision fields and effective energies, we propose self-organizing PCA rules which are capable of resisting outliers while fulfilling various PCA-related tasks such as obtaining the first principal component vector, the first k principal component vectors, and directly finding the subspace spanned by the first k vector principal component vectors without solving for each vector individually. Comparative experiments have shown that the proposed robust rules improve the performances of the existing PCA algorithms significantly when outliers are present. 1 INTRODUCTION Principal Component Analysis (PCA) is an essential technique for data compression and feature extraction, and has been widely used in statistical data analysis, communication theory, pattern recognition and image processing. In the neural network literature, a lot of studies have been made on learning rules for implementing PCA or on networks closely related to PCA (see Xu & Yuille, 1993 for a detailed reference list which contains more than 30 papers related to these issues). The existing rules can fulfil various PCA-type tasks for a number of application purposes. "'Present address: Dept. of Brain and Cognitive Sciences, E10-243, Massachusetts Institute of Technology, Cambridge, MA 02139. 467 468 Xu and Yuille However, almost all the previously mentioned peA algorithms are based on the assumption that the data has not been spoiled by outliers (except Xu, Oja&Suen 1992, where outliers can be resisted to some extent.). In practice, real data often contains some outliers and usually they are not easy to separate from the data set. As shown by the experiments described in this paper, these outliers will significantly worsen the performances of the existing peA learning algorithms. Currently, little attention has been paid to this problem in the neural network literature, although the problem is very important for real applications. Recently, there have been some success in applying t:te statistical physics approach to a variety of computer vision problems (Yuille, 1990; Yuille, Yang&Geiger 1990; Yuille, Geiger&Bulthoff, 1991). In particular, it has also been shown that some techniques developed in robust statistics (e.g., redescending M-estimators, leasttrimmed squares estimators) appear naturally within the Bayesian formulation by the use of the statistical physics approach. In this paper we adapt this approach to tackle the problem of robust PCA. Robust rules are proposed for various PCArelated tasks such as obtaining the first principal component vector, the first k principal component vectors, and principal subspaces. Comparative experiments have been made and the results show that our robust rules improve the performances of the existing peA algorithms significantly when outliers are present. 2 peA LEARNING AND ENERGY MINIMIZATION There exist a number of self-organizing rules for finding the first principal component. Three of them are listed as follows (Oja 1982, 85; Xu, 1991,93): m(t + 1) = m(t) + aa(t)(xy - m(t)y2), (1) m(t + 1) = m(t) + aa(t)(xy - m(~~~(t)y2), (2) m(t + 1) = m(t) + aa(t)[y(x - iI) + (y - y')X]. (3) where y = m(t)T x, iI = ym(t), y' = m(tf iI and aa(t) 2:: 0 is the learning rate which decreases to zero as t -- 00 while satisfying certain conditions, e.g., Lt aa(t) = 00, Lt aa(t)q < 00 for some q> 1. Each of the three rules will converge to the principal component vector i almost surely under some mild conditions which are studied in detail in by Oja (1982&85) and Xu (1991&93). Regarding m as the weight vector of a linear neuron with output y = mT x, all the three rules can be considered as modifications of the well known Hebbian rule m(t + 1) = m(t) + aa(t)xy through introducing additional terms for preventing IIm(t)1I from going to 00 as t -- 00. The performances of these rules deteriorate considerably when data contains outliers. Although some outlier-resisting versions of eq.(l) and eq.(2) have also been recently proposed (Xu, Oja & Suen, 1992), they work well only for data which is not severely spoiled by outliers. In this paper, we adopt a totally different approach-we generalize eq.(1),eq.(2) and eq.(3) into more robust versions by using the statistical physics approach. To do so, first we need to connect these rules to energy functions. It follows from Xu (1991&93) and Xu & Yuille(1993) that the rules eq.(2) and eq.(3) are respectively Self-Organizing Rules for Robust Principal Component Analysis 469 on-line gradient descent rules for minimizing J1 (m), J2(m) respectivelyl: I N -T ::::T J ( -) = _ "'(-'!' -. _ m Xixi m) 1 m L..J x, X, -T _ N i=l m m (4) N hem) = ~ L !Iii - uill 2 . i=1 (5) It has also been proved that the rule given by eq.(l) satisfies (Xu, 1991, 93): (a) hTh2 2: 0,E(hJ)T JJ(h1) 2: 0, with hI = iy-my2, h2 = iy- mo/.m y2 ; (b) E(hl)TE(h3) > 0, with h3 = y(i-iI)+(y-y')i; (c) Both J1 and h have only one local (also global) minimum tr(~) - iI'r-i, and all the other critical points (i.e., the points satisfy 8Jakm) = 0, i = 1,2) are saddle points. Here ~ = E{ii t}, and i is the eigenvector of r- corresponding to the largest eigenvalue. That is, the rule eq.(l) is a downhill algorithm for minimizing J1 in both the on line sense and the average sense, and for minimizing J2 in the average sense. 3 GENERALIZED ENERGY AND ROBUST peA We further regard J1(m), J2(m) as special cases of the following general energy: N J(m) = ~L Z(ii, m), Z(ii' m) 2: 0. i=1 where Z(ii' m) is the portion of energy contributed by the sample ii, and Following (Yuille, 1990 a& b), we now generalize energy eq.(6) into E(V, m) = L:f:1 Vi Z(ii' m) + Eprior(V) (6) (7) (8) where V = {Vi, i = 1, .. " N} is a binary field {\Ii} with each \Ii being a random variable taking value either 0 or 1. \Ii acts as a decision indicator for deciding whether ii is an outlier or a sample. When \Ii = 1, the portion of energy contributed by the sample ii is taken into consideration; otherwise, it is equivalent to discarding ii as an outlier. Eprior(V) is the a priori portion of energy contributed by the a priori distribution of {Vi}. A natural choice is N EpriorCV) = 11 1:(1- Vi) (9) i=1 This choice of priori has a natural interpretation: for fixed m it is energetically favourable to set \Ii = 1 (i.e., not regarding ii as an outlier) if Z(ii' m) < yfii (i.e., lWe have J1(ffi) 2: 0, since iTi m"fm = lIiW sin2 (Jxm 2: o. 470 Xu and Yuille the portion of energy contributed by Xi is smaller than a prespecified threshold) and to set it to 0 otherwise. Based on E(V, m), we define a Gibbs distribution (Parisi 1988): 1 [- -] P[V m] = _e-{3E V,m 'z ' (10) where Z is the partition function which ensures Lv Lm pry, m] = 1. Then we compute Pmargin(m) 1 L -{3 ~ {V,z(x"m)+T/(l-V,)} e L..J, Z _ v ! II L e-{3{V,z(x"m)+T/(l-V,)} = _1_e-{3EeJJ (m). Z . Zm , V,={O,l} EeJj(m) = -1 Llog{1 + e-{3{z(x"m)-T/}}. (3 i (11) (12) Eel! is called the effective energy. Each term in the sum for Eel I is approximately z(xi,m) for small values of Z but becomes constant as z(xi,m) -+ 00. In this way outliers, which are more likely to yield large values of z( Xi, m), are treated differently from samples, and thus the estimation m obtained by minimizing EeJj(m) will be robust and able to resist outliers. Ee! f (m) is usually not a convex function and may have many local minima. The statistical physics framework suggests using deterministic annealing to minimize EeJj(m). That is, by the following gradient descent rule eq.(13), to minimize EeJj(m) for small (3 and then track the minimum as (3 increases to infinity (the zero temperature limit): _( ) _() (~ 1 oz(xi,m(t)) m t + 1 = m t lYb t) ~ 1 + e{3(z(x"m(f))-T/) om(t) . , (13) More specifically, with z's chosen to correspond to the energies hand J2 respectively, we have the following batch-way learning rules for robust peA: _ ( ) _ ( ) ( ) ~ 1 ( _ m( t) 2) () m t + 1 = m t + lYb t ~ 1 + e{3(z(x"m(t))-T/) XiYi - m(t)Tm(t)Yi' 14 z met + 1) = met) + abet) ~ 1 + e{3(Z(;"m(f))-T/) [Yi(Xi - ild + (Yi - yDXi]. (15) , For data that comes incrementally or in the on-line way, we correspondingly have the following adaptive or stochastic approximation versions -( 1) -C) () 1 (met) 2) m t + = m t + aa t 1 + e{3(z(x"m(t))-17) XiYi m(t)T met) Yi , (16) met + 1) = met) + aa(t) 1 + e{3(Z(;"m(t))-17) [Yi(Xi - iii) + (Yi - YDXi]. (17) Self-Organizing Rules for Robust Principal Component Analysis 471 It can be observed that the difference between eq.(2) and eq.(16) or eq.(3) and eq.(17) is that the learning rate G'a(t) has been modified by a multiplicative factor 1 G'm(t) = 1 + e{j(Z(tri,m(t))-")' (18) which adaptively modifies the learning rate to suit the current input Xi. This modifying factor has a similar function as that used in Xu, Oja&Suen(1992) for robust line fitting. But the modifying factor eq.(18) is more sophisticated and performs better. Based on the connecticn between the rule eq.(I) and J1 or J2 , given in sec.2, we can also formally use tile modifying factor G'm(t) to turn the rule eq.(I) into the following robust version: met + 1) = met) + G'a(t) 1 + e{j(Z(;.,m(t))-,,) (iiYi - m(t)yi), (19) 4 ROBUST RULES FOR k PRINCIPAL COMPONENTS In a similar way to SGA (Oja, 1992) and GHA (Sanger, 1989) we can generalize the robust rules eq.(19), eq.(16) and eq.(17) into the following general form of robust rules for finding the first k principal components: mj(t + 1) = mj(t) + G'a(t) 1 + e{j(Z(tr)n,m;(t))-,,) ~mj(xi(j), mj(t», (20) j-l Xi(O) = ii, ii(j + 1) = Xi(j) - L Yi(r)mr(t), Yi(j) = mJ (t)ii(j), (21) r=l where ~mj(ii(j), mj(t», Z(Xi(j), mj(t» have four possibilities (Xu & Yuille, 1993). As an example, one of them is given here dmj(xi(j), mj(t» = (Xi(j)Yi(j) - mj(t)Yi(j)2), ( .. (.) .. (t» .. (')T - (.) Yi(j)2 Z Xi J ,mj = Xi J Xi J - mj(t)Tmj(t)' In this case, eq.(20) can be regarded as the generalization of GHA (Sanger, 1989). We can also develop an alternative set of rules for a type of nets with asymmetric lateral weights as used in (Rubner&Schulten, 1990). The rules can also get the first k principal components robustly in the presence of outliers (Xu & Yuille, 1993). 5 ROBUST RULES FOR PRINCIPAL SUBSPACE Let M = [ml, .. " mk], ~ = [¢1, .. " ¢k], Y = [Yl, .. " Ykf and y = MT X, it follows from Oja(1989) and Xu(1991) the rules eq.(l), eq.(3) can be generalized into eq.(22) and eq.(23) respectively: (22) 472 Xu and Yuille Mu = y, y = MTa (23) In the case without outliers, by both the rules, the weight matrix M(t) will converge to a matrix MOO whose column vectors mj, j = 1,"" k span the k-dimensional principal subspace (Oja, 1989; Xu, 1991&93), although the vectors are, in general, not equal to the k principal component vectors ¢j, j = 1, ... , k. Similar to the previously used procedure, we have the following results: (1). We can SllOW that eq.(23) is an on-line or stochastic approximation rule which minimizes the energy 13 in the gradient descent way (Xu, 1991& 93): N J3 (ffi) = ~ L: IIXi - ai ll 2 , a = My, Y' = MT iI. (24) i=l and that in the average sense the subspace rule eq.(22) is also an on-line "down-hill" rule for minimizing the energy function Ja. (2). We can also generalize the non-robust rules eq.(22) and eq.(23) into robust versions by using the statistical physics approach again: M(t + 1) = M(t) + GA(t) 1 + e!3(I//-U.1I2_'1) [Yi(Xi - ildT - (fii - Y1)iT]' (25) 1 -,..fJ'-~ M(t + 1) = M(t) + GA(t) 1 + e!3(l/x.-u;1/2_'1) [y,Xi - YiY, M(t)] (26) 6 EXAMPLES OF EXPERIMENTAL RESULTS Let x from a population of 400 samples with zero mean. These samples are located on an elliptic ring centered at the origin of R3 , with its largest elliptic axis being along the direction (-1,1,0), the plane of its other two axes intersecting the x - Y plane with an acute angle (30°). Among the 400 samples, 10 points (only 2.5%) are randomly chosen and replaced by outliers. The obtained data set is shown in Fig.1. Before the outliers were introduced, either the conventional simple-variance-matrix based approach (i.e., solving S¢ = A¢, S = k L~l iiX[) or the unrobust rules eqs.(I)(2)(3) can find the correct 1st principal component vector of this data set. On the data set contaminated by outliers, shown in Fig.l, the result of the simplevariance-matrix based approach has an angular error of ¢p by 71.04°-a result definitely unacceptable. The results of using the proposed robust rules eq.(19), eq.(16) and eq.(17) are shown in Fig.2(a) in comparison with those of their unrobust counterparts- the rules eq.(I), eq.(2) and eq.(3). We observe that all the unrobust rules get the solutions with errors of more than 21° from the correct direction of ¢p. By contrast, the robust rules can still maintain a very good accuracy-the error is about 0.36°. Fig.2(b) gives the results of solving for the first two principal component vectors. Again, the unrobust rule produce large errors of around 23°, while the robust rules have an error of about 1. 7° . Fig.3 shows the results of soIling for the 2-dimensional principal subspace, it is easy to see the significant improvements obtained by using the robus.t rules. Self-Organizing Rules for Robust Principal Component Analysis 473 " , , ' , , ,~ \\' "~"114" ., ~ • • J t • , f ., ~ • I 2 • • J • Figure 1: The projections of the data on the x - y, y z and z x planes, with 10 outliers. Acknowledgements We would like to thank DARPA and the Air Force for support with contracts AFOSR-89-0506 and F4969092-J-0466. We like to menta ion that some further issues about the proposed robust rules are studied in Xu & Yuille (1993), including the selection of parameters 0', j3 and 1], the extension of the rules for robust Minor Component Analysis (MCA) , the relations between the rules to the two main types of existing robust peA algorithms in the literature of statistics, as well as to Maximal Likelihood (ML) estimation of finite mixture distributions. References E. Oja, J. Math. Bio. 16, 1982,267-273. E. Oja & J. Karhunen, J. Math. Anal. Appl. 106,1985,69-84. E. Oja, Int. J. Neural Systems 1, 1989,61-68. E. Oja, Neural Networks 5, 1992, 927-935. G. Parisi, Statistical Field Theory, Addison-Wesley, Reading, Mass., 1988. J. Rubner & K. Schulten, Biological Cybernetics, 62, 1990, 193-199. T.D. Sanger, Neural Networks, 2, 1989,459-473. L. Xu, Proc. of IJCNN'91-Singapore, Nov., 1991,2368-2373. L. Xu, Least mean square error reconstruction for self-organizing neural-nets, Neural Networks 6, 1993, in press. L. Xu, E. Oja & C.Y. Suen, Neural Networks 5, 1992,441-457. L. Xu & A.L. Yuille, Robust principal component analysis by self-organizing rules based on statistical physics approach, IEEE Trans. Neural Networks, 1993, in press. A.L. Yuille, Neural computation 2, 1990, 1-24. A.L. Yuille, D. Geiger and H.H. Bulthoff,Networks 2, 1991. 423-442. 474 Xu and Yuille --. .. (a) (b) Figure 2: The learning curves obtained in the comparative experiments for principal component vectors. (a) for the first principal component vector, RAl, RA2, RA3 denote the robust rules eq.(19), eq.(16) and eq.(17) respectively, and U AI, U A2, U A3 denote the rules eq.(l), eq.(2) and eq.(3) respectively. The horizontal axis denotes the learning steps, and the vertical axis is (Jm(t)¢Pl with (Jx,y denoting the acute angle between x and y. (b) for the first two principal component vectors, by the robust rule eq.(20) and its unrobust counterpart GHA. U Akl, U Ak2 denote the learning curves of angles (Jml(t)¢Pl and (Jm2(t)¢P2 respectively, obtained by GHA . RAk 1, RAk2 denote the learning curves of the angles obtained by using the robust rule eq.(20). In both (a) & (b), i pj , j = 1,2 is the correct 1st and 2nd principal component vector respectively. t t 1 _______ _ ........ Figure 3: The learning curves obtained in the comparative experiments for for solving the 2-dimensional principal subspace. Each learning curve expresses the change of the residual er(t) = L:J=ll!mj(t) - L:;=l(mj(tf ipr)¢prI12 with learning steps. The smaller the residual, the closer the estimated principal subspace to the correct one. SU Bl, SU B2 denote the unrobust rules eq.(22) and eq.(23) respectively, and RSU Bl, RSU B2 denote the robust rules eq.(26) and eq.(25) respectively.
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Intersecting regions: The key to combinatorial structure in hidden unit space Janet Wiles Depts of Psychology and Computer Science, University of Queensland QLD 4072 Australia. janetw@cs.uq.oz.au Mark Ollila, Vision Lab, CITRI Dept of Computer Science, University of Melbourne, Vic 3052 Australia molly@vis.citri.edu.au Abstract Hidden units in multi-layer networks form a representation space in which each region can be identified with a class of equivalent outputs (Elman, 1989) or a logical state in a finite state machine (Cleeremans, Servan-Schreiber & McClelland, 1989; Giles, Sun, Chen, Lee, & Chen, 1990). We extend the analysis of the spatial structure of hidden unit space to a combinatorial task, based on binding features together in a visual scene. The logical structure requires a combinatorial number of states to represent all valid scenes. On analysing our networks, we find that the high dimensionality of hidden unit space is exploited by using the intersection of neighboring regions to represent conjunctions of features. These results show how combinatorial structure can be based on the spatial nature of networks, and not just on their emulation of logical structure. 1 TECHNIQUES FOR ANALYSING THE SPATIAL AND LOGICAL STRUCTURE OF HIDDEN UNIT SPACE In multi-layer networks, regions of hidden unit space can be identified with classes of equivalent outputs. For example, Elman (1989) showed that the hidden unit patterns for words in simple grammatical sentences cluster into regions, with similar patterns representing similar grammatical entities. For example, different tokens of the same word are clustered tightly, indicating that they are represented within a small region. These regions can be grouped into larger regions, reflecting a hierarchical structure. The largest 27 28 Wiles and Ollila groups represent the abstract categories, nouns and verbs. Elman used cluster analysis to demonstrate this hierarchical grouping, and principal component analysis (PCA) to show dimensions of variation in the representation in hidden unit space. An alternative approach to Elman's hierarchical clustering is to identify each region with a functional state. By tracing the trajectories of sequences through the different regions, an equivalent finite state machine (FSM) can be constructed This approach has been described using Reber grammars with simple recurrent networks (Cleeremans, ServanSchreiber & McClelland, 1989) and higher-order networks (Giles, Sun, Chen, Lee, & Chen, 1990). Giles et a1. showed that the logical structure of the grammars is embedded in hidden unit space by identifying each regions with a state, extracting the equivalent finite state machine from the set of states, and then reducing it to the minimal FSM. Oustering and FSM extraction demonstrate different aspects of representations in hidden unit space. Elman showed that regions can be grouped hierarchically and that dimensions of variation can be identified using PCA, emphasizing how the functionality is reflected in the spatial structure. Giles et al. extracted the logical structure of the finite state machine in a way that represented the logical states independently of their spatial embedding. There is an inherent trade off between the spatial and logical analyses: In one sense, the FSM is the idealized version of a grammar, and indeed for the Reber grammars, Giles et a1. found improved performance on the extracted FSMs over the trained networks. However, the states of the FSM increase combinatorially with the size of the input. If there is information encoded in the hierarchical grouping of regiOns or relative spatial arrangement of clusters, the extracted FSM cannot exploit it. The basis of the logical equivalence of a FSM and the hidden unit representations is that disjoint regions of hidden unit space represent separate logical states. In previous work, we reversed the process of identifying clusters with states of a FSM, by using prior knowledge of the minimal FSM to label hidden unit patterns from a network trained on sequences from three temporal functions (Wiles & Bloesch, 1992). Canonical discriminant analysis (CDA, Cliff, 1987) was then used to view the hidden unit patterns clustered into regions that corresponded to the six states of the minimal FSM. In this paper we explore an alternative interpretation of regions. Instead of considering disjoint regions, we view each region as a sub-component lying at the intersection of two or more larger regions. For example, in the three-function simulations, the six clusters can be interpreted in terms of three large regions that identify the three possible temporal functions, overlapping with two large regions that identify the output of the network (see Figure 1). The six states can then be seen as combinations of the three function and two output classes (Le, 5 large overlapping regions instead of 6 smaller disjoint ones). While the three-function simulation does provide a clear demonstration of the intersecting structure of regions, nonetheless, only six states are required to represent the minimal FSM and harder tasks are needed to demonstrate combinatorial representations. 2 SIMULATIONS OF THE CONJUNCTION OF COLOR, SHAPE AND LOCATION The representation of combinatorial structure is an important aspect of any computational tasi( because of the drastic implications of combinatorial explOSion for scaling. The intersection of regions is a concise way to represent all possible combinations of different items. We demonstrate this idea applied to the analysis of a hidden unit space c " c &. E o u ft .!::! c o c " U ":2 .. ; Intersecting regions: The key to combinatorial structure in hidden unit space 29 r - - - -~ r- - ---, r - - - -., , " , ..l----r' 1 If -l----[tt' ~ .1.. - - r-r --; , , Xl 'J I: ::ftl RI I 1 I • .J. J J 1(1) AI 1 'lli-~--=--t-f-~ I • ~ ...:r -h= e -, r f ~-+- -r r -.. I , f I I r 'I ,-:1 r I I f I f , I ~·.i ,I : I J r J , r r xo, I 1:_ '1 _ RO: 1 II I I I : .e AO r i.' r ~ , r I ,} r I J' I' ~ .-1 J J ,i: I JI :- ~ r I L ~. -r- , 1 1 I '- _~.J L __ ~J L ___ ~ Firse CIDOnicaJ componCDI Figure 1. Intersecting regions in hidden unit space. Hidden unit patterns from the threefunction task of Wiles and Bloesch (1992) are shown projected onto the first and third canonical components. Each temporal function, XOR, AND and OR is represented by a vertical region, separated along the first canonical component. The possible outputs, 0 and 1 are represented by horizontal regions, separated down the third canonical component. The states of the finite state machine are represented by the regions in the intersections of the vertical and horizontal regions. (Adapted from Wiles & Bloesch, 1992, Figure lb.) 30 Wiles and Ollila representation of conjunctions of colors, shapes and locations. In our task, a scene consists of zero or more objects, each object identified by its color, shape and location. The number of scenes, C, is given by C = (s/+l)l where s,/, and 1 are the numbers of shapes, features and locations respectively. This problem illustrates several important components: There is no unique representation of an object in the input or output - each object is represented only by the presence of a shape and color at a given location. The task of the network is to create hidden unit representations for all possible scenes, each containing the features themselves, and the binding of features to position. The simulations involved two locations, three possible shapes and three colors (100 legitimate scenes). A 12-20-12 encoder network was trained on the entire set of scenes and the hidden unit patterns for each scene were recorded. Analysis using CDA with 10 groups designating all possible combinations of zero, one or two colors showed that the hidden unit space was partitioned into intersecting regions corresponding to the three colors or no color (see Figure 2a). CDA was repeated using groups designating all combinations of shapes, which showed an alternative partitioning into four intersecting regions related to the component shapes (see Figure 2b). Figures 2a and 2b show alternate two-dimensional projections of the 20-dimensional space. The analyses showed that each hidden unit pattern was contained in many different groupings, such as all objects that are red, all triangles, or all red triangles. In linguistic terms, each hidden unit pattern corresponds to a token of a feature, and the region containing all tokens of a given group corresponds to its abstract type. The interesting aspect of this representation is that the network had learnt not only how to separate the groups, but also to use overlapping regions. Thus given a region that represents a circle and one representing a triangle, the intersection of the two regions implies a scene that has both a circle and a triangle. Given suitable groups, the perspectives provided by CDA show many different abstract types within the hidden unit space. For example, scenes can be grouped according to the number of objects in a scene, or the number of squares in a scene. We were initially surprised that contiguous regions exist for representing scenes with zero, one and two Objects, since the output units only require representations of individual features, such as square or circle, and not the abstraction to "any shape", or even more abstract, "any object". It seems plausible that the separation of these regions is due to the high dimensionality provided by 12-20-12 mappings. The excess degrees of freedom in hidden unit space can encode variation in the inputs that is not necessarily required to complete the task. With fewer hidden units, we would expect that variation in the input patterns that is not required for completing the task would be compressed or lost under the competing requirement of maximally separating functionally useful groups in the hidden unit space. This explanation found support in a second simulation, using a 12-8-12 encoder network. Whereas analysis of the 12-20-12 network showed separation of patterns into disjoint regions by number of objects, the smaller 12-8-12 network did not. Over all, our analyses showed that as the number of dimensions increases, additional aspects of scenes may be represented, even if those aspects are not required for the task that the network is learning. Intersecting regions: The key to combinatorial structure in hidden unit space 31 , ..... -"'" First canonical component ..,.-----, .~ .. "-.. -.. -.. "'''., t~· r JJ i I ' A' ~.. 1 \.' \. ' \ • • \ • \ 4 2 ~, , I I , .. ! \ ...... . \. : ••• ~ •••••••• \.~ j I \i --; , fI ,II ., : --; .... ., . "' .. --",: • • . ' • • ~ : -.. : -.. ,t -. 3 ,. -.. 'I ...... !~ ... First canonical component 2A LEGEND Region 1 Any scene with red 2 Any scene with blue 3 Any scene with green 4 Any scene with 0 or 1 color A Scenes with red & green objects B Scenes with red & blue objects C Scenes with 1 red object D Scenes with 2 red objects E Scenes with green & blue objects F Scenes with 1 green object G Scenes with 2 green objects H Scenes with 1 blue object I Scenes with 2 blue objects J Scenes with no objects 2s LEGEND Region 1 Any scene with a triangle 2 Any scene with a circle 3 Any scene with a square 4 Scenes with 0 or 1 object A Scenes with a triangle & a circle B Scenes with a triangle & a square C Scenes with a single triangle D Scenes with 2 triangles E Scenes with a circle & a square F Scenes with a single circle G Scenes with 2 circles H Scenes with a single square I Scenes with 2 squares J Scenes with no objects Figure 2. CDA plots showing the representations of features in a scene. A scene consists of zero, one or two objects, represented in terms of color, shape and location. 2a. Patterns labelled by color: Hidden unit patterns form ten distinct clusters, which have been grouped into four intersecting regions, 1-4. For example, the hidden unit patterns within region 1 all contain at least one red object, those in regions 2 contain at least one blue one, and those in the intersection of regions 1 and 2 contain one red and one blue object. 2b. Patterns labelled by shape: Again the hidden unit patterns form ten distinct clusters, which have been grouped into four intersecting regions, however, these regions represent scenes with the same shape. 2a and 2b show alternate groupings of the same hidden unit space, projected onto different canonical components. The two projections can be combined in the mind's eye (albeit with some difficulty) to form a four dimensional representation of the spatial structure of intersecting regions of both color and shape. 32 Wiles and Ollila 3 THE SPATIAL STRUCTURE OF HIDDEN UNIT SPACE IS ISOMORPHIC TO THE COMBINATORIAL STRUCTURE OF THE VISUAL MAPPING TASK In conclusion, the simulations demonstrate how combinatorial structure can be embedded in the spatial nature of networks in a way that is isomorphic to the combinatorial structure of the task, rather than by emulation of logical structure. In our approach, the representation of intersecting regions is the key to providing combinatorial representations. If the visual mapping task were extended by including a feature specifying the color of the background scene (e.g., blue or green) the number of possible scenes would double, as would the number of states in a FSM. By contrast, in the hidden unit representation, the additional feature would involve adding two more overlapping regions to those currently supported by the spatial structure. This could be implemented by dividing hidden unit space along an unused dimension, orthogonal to the current groups. The task presented in this case study is extremely simplified, in order to expose the intrinsic combinatorial structure required in binding. Despite the simplifications, it does contain elements of tasks that face real cognitive systems. In the simulations above, individual Objects can be clustered by their shape or color, or whole scenes by other properties, such as the number of squares in the scene. These representations provide a concise and easily accessible structure that solves the combinatorial problem of binding several features to one object, in such a way as to represent the individual object, and yet also allow efficient access to its component features. The flexibility of such access processes is one of the main motivations for tensor models of human memory (Humphreys, Bain & Pike, 1989) and analogical reasoning (Halford et aI., in press). Our analysis of spatial structure in terms of intersecting regions has a straightforward interpretation in terms of tensors, and provides a basis for future work on network implementations of the tensor memory and analogical reasoning models. Acknowledgements We thank Simon Dennis and Steven Phillips for their canonical discriminant program. This work was supported by grants from the Australian Research Council. References Cleeremans, A., Servan-Schreiber, D., and McClelland, J.L. (1989). Finite state automata and simple recurrent networks, Neural Computation, 1, 372-381. Cliff, N. (1987). Analyzing Multivariate Data. Harcourt Brace Jovanovich, Orlando, Florida. Elman, J. (1989). Representation and structure in connectionist models. CRL Technical Report 8903, Center for Research in Language, University of California, San Diego, 26pp. Giles, C. L., Sun, G. Z., Chen, H. H., Lee, Y. C., and Chen, D. (1990). Higher Order Recurrent Networks. In D.S. Touretzky (ed.) Advances in Neural Information Processing Systems 2, Morgan-Kaufmann, San Mateo, Ca., 380-387. Intersecting regions: The key to combinatorial structure in hidden unit space 33 Halford, G.S., Wilson, W.H., Guo, J., Wiles, J. and Stewart, J.E.M. Connectionist implications for processing capacity limitations in analogies. To appear in KJ. Holyoak & J. Barnden (Eds.), Advances in Connectionist and Neural Computation Theory, Vol 2: Analogical Connections. Norwood, NJ: Ablex, in press. Humphreys, M.S., Bain, J.D., and Pike, R. (1989). Different ways to cue a coherent memory system: A theory of episodic, semantic and procedural tasks, Psychological Review, 96 (2), 208-233. Wiles, J. and Bloesch, A. (1992). Operators and curried functions: Training and analysis of simple recurrent networks. In J. E. Moody, S. J. Hanson, and R. P. Lippmann (Eds.) Advances in Neural Information Processing Systems 4, Morgan-Kaufmann, San Mateo, Ca.
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A Parallel Gradient Descent Method for Learning in Analog VLSI Neural Networks J. Alspector R. Meir'" B. Yuhas A. Jayakumar D. Lippet Bellcore Morristown, NJ 07962-1910 Abstract Typical methods for gradient descent in neural network learning involve calculation of derivatives based on a detailed knowledge of the network model. This requires extensive, time consuming calculations for each pattern presentation and high precision that makes it difficult to implement in VLSI. We present here a perturbation technique that measures, not calculates, the gradient. Since the technique uses the actual network as a measuring device, errors in modeling neuron activation and synaptic weights do not cause errors in gradient descent. The method is parallel in nature and easy to implement in VLSI. We describe the theory of such an algorithm, an analysis of its domain of applicability, some simulations using it and an outline of a hardware implementation. 1 Introduction The most popular method for neural network learning is back-propagation (Rumelhart, 1986) and related algorithms that calculate gradients based on detailed knowledge of the neural network model. These methods involve calculating exact values of the derivative of the activation function. For analog VLSI implementations, such techniques require impossibly high precision in the synaptic weights and precise modeling of the activation functions. It is much more appealing to measure rather than calculate the gradient for analog VLSI implementation by perturbing either a ·Present address: Dept. of EE; Technion; Haifa, Israel tpresent address: Dept. of EE; MIT; Cambridge, MA 836 A Parallel Gradient Descent Method for Learning in Analog VLSI Neural Networks 837 single weight (Jabri, 1991) or a single neuron (Widrow, 1990) and measuring the resulting change in the output error. However, perturbing only a single weight or neuron at a time loses one of the main advantages of implementing neural networks in analog VLSI, namely, that of computing weight changes in parallel. The oneweight-at-a-time perturbation method has the same order of time complexity as a serial computer simulation of learning. A mathematical analysis of the possibility of model free learning using parallel weight perturbations followed by local correlations suggests that random perturbations by additive, zero-mean, independent noi~e sources may provide a means of parallel learning (Dembo, 1990). We have pre :Tiously used such a noise source (Alspector, 1991) in a different implement able learning model. 2 Gradient Estimation by Parallel Weight Perturbation 2.1 A Brownian Motion Algorithm One can estimate the gradient of the error E(w) with respect to any weight WI by perturbing WI by OWl and measuring the change in the output error oE as the entire weight vector w except for component Wl is held constant. E(w + OWl) - E(w) OWl This leads to an approximation to the true gradient g:l: oE oE = + O([owd) OWl OWl (1) (2) For small perturbations, the second (and higher order) term can be ignored. This method of perturbing weights one-at-a-time has the advantage of using the correct physical neurons and synapses in a VLSI implementation but has time complexity of O(W) where W is the number of weights. Following (Dembo, 1990), let us now consider perturbing all weights simultaneously. However, we wish to have the perturbation vector ow chosen uniformly on a hypercube. Note that this requires only a random sign multiplying a fixed perturbation and is natural for VLSI. Dividing the resulting change in error by any single weight change, say OWl, gives oE OWl which by a Taylor expansion is E(w + ow) - E(w) OWl leading to the approximation (ignoring higher order terms) (3) (4) 838 Alspector, Meir, Yuhas, Jayakumar, and Lippe (5) An important point of this paper, emphasized by (Dembo, 1990) and embodied in Eq. (5), is that the last term has expectation value zero for random and independently distributed OWi since the last expression in parentheses is equally likely to be +1 as -1. Thus, one can approximately follow the gradient by perturbing all weights at the same time. If each synapse has access to information about the resulting change in error, it can adjust its weight by assuming it was the only weight perturbed. The weight change rule (6) where TJ is a learning rate, will follow the gradient on the average but with the considerable noise implied by the second term in Eq. (5). This type of stochastic gradient descent is similar to the random-direction Kiefer-Wolfowitz method (Kushner, 1978), which can be shown to converge under suitable conditions on TJ and OWi. This is also reminiscent of Brownian motion where, although particles may be subject to considerable random motion, there is a general drift of the ensemble of particles in the direction of even a weak external force. In this respect, there is some similarity to the directed drift algorithm of (Venkatesh, 1991), although that work applies to binary weights and single layer perceptrons whereas this algorithm should work for any level of weight quantization or precision - an important advantage for VLSI implementations - as well as any number of layers and even for recurrent networks. 2.2 Improving the Estimate by Multiple Perturbations As was pointed out by (Dembo, 1990), for each pattern, one can reduce the variance of the noise term in Eq. (5) by repeating the random parallel perturbation many times to improve the statistical estimate. If we average over P perturbations, we have oE 1 p oE 8E 1 P W (EJE) (owf) OWl = P L oif.l = 8Wl + P L ?= 8Wi OwPl p=l p=l&>l (7) where p indexes the perturbation number. The variance of the second term, which . . . IS a nOise, v, IS I where the expectation value, <>, leads to the Kronecker delta function, off, . This reduces Eq. (8) to A Parallel Gradient Descent Method for Learning in Analog VLSI Neural Networks 839 2 1 P W (OE)2 < II > = p2 LL OW. p=li>l z (9) The double sum over perturbations and weights (assuming the gradient is bounded and all gradient directions have the same order of magnitude) has magnitude O(PW) so that the variance is O(~) and the standard deviation is (10) Therefore, for a fixed variance in the noise term, it may be necessary to have a number of perturbations of the same order as the number of weights. So, if a high precision estimate of the gradient is needed throughout learning, it seems as though the time complexity will still be O(W) giving no advantage over single perturbations. However, one or a few of the gradient derivatives may dominate the noise and reduce the effective number of parameters. One can also make a qualitative argument that early in learning, one does not need a precise estimate of the gradient since a general direction in weight space will suffice. Later, it will be necessary to make a more precise estimate for learning to converge. 2.3 The Gibbs Distribution and the Learning Problem Note that the noise of Eq. (7) is gaussian since it is composed of a sum of random sign terms which leads to a binomial distribution and is gaussian distributed for large P. Thus, in the continuous time limit, the learning problem has Langevin dynamics such that the time rate of change of a weight Wk is, (11) and the learning problem converges in probability (Zinn-Justin, 1989), so that as~mpto~ically Pr(w) <X exp[-,BE(w)] where ,B is inversely proportional to the nOIse vanance. Therefore, even though the gradient is noisy, one can still get a useful learning algorithm. Note that we can "anneal" Ilk by a variable perturbation method. Depending on the annealing schedule, this can result in a substantial speedup in learning over the one-weight-at-a-time perturbation technique. 2.4 Similar Work in these Proceedings Coincidentally, there were three other papers with similar work at NIPS*92. This algorithm was presented with different approaches by both (Flower, 1993) and (Cauwenberghs, 1993). 1 A continuous time version was implemented in VLSI but not on a neural network by (Kirk, 1993). 1 We note that (Cauwenberghs, 1993) shows that multiple perturbations are not needed for learning if D.w is small enough and he does not study them . This does not agree with our simulations (following) 840 Alspector, Meir, Yuhas, Jayakumar, and Lippe 3 Simulations 3.1 Learning with Various Perturbation Iterations We tried some simple problems using this technique in software. We used a standard sigmoid activation function with unit gain, a fixed size perturbation of .005 and random sign. The learning rate, T/, was .1 and momentum, Q, was o. We varied the number of perturbation iterations per pattern presentation from 1 to 128 (21 where 1 varies from 0 to 7). We performed 10 runs for each condition and averaged the results. Fig. 1a shows the average learning curves for a 6 input, 12 hidden, 1 output unit parity problem as the number of perturbations per pattern presentation is varied. The symbol plotted is l. ~ f I I I pa"ty 6 avg10 ----::-1 ~------~ -------~I-j 50 100 150 replication 6 avg , 0 7 7 1 7 7 7 7 7 ,. 7 1 7 :lllil • I III ;;:-;-, ~ 1 .. ........ . 7 .. ..... .. •• 7 • 3 3 3 :I 3 3 3 3 3 3 33 3 3 3 3 3 10 15 20 :I 3 33 3 3 3 3 3 3 I I ~- J 2S Figure 1. Learning curves for 6-12-1 parity and 6-6-6 replication . There seems to be a critical number of perturbations, Pc, about 16 (1 = 4) in this case, below which learning slows dramatically. We repeated the measurements of Fig. 1a for different sizes of the parity problem using a N-2N-1 network. We also did these measurements on a different problem, replication or identity, where the task is to replicate the bit pattern of the input on the output. We used a N-N-N network for this task so that we have a comparison with the parity problem as N varies for roughly the same number of weights (2N 2 + 2N) in each network. The learning curves for the 6-6-6 problem are plotted in Fig. lb. The critical value also seems to be 16 (l = 4). perhaps b ecause we do not d ecrease 6w and 11 as learning proceeds. He did not check this for large problems as we did. In an implementation, one will not be able to reduce 6w too much so that the effect on the output error can be measured. It is also likely that multiple perturbations can be done more quickly than multiple pattern presentations, if learning speed is an issue. He also notes the importance of correlating with the change in error rather than the error alone as in (Dembo, 1990). A Parallel Gradient Descent Method for Learning in Analog VLSI Neural Networks 841 3.2 Scaling of the Critical Value with Problem Size To determine how the critical value of perturbation iterations scales, we tried a variety of problems besides the N-N-N replication and N-2N-1 parity. We added N2N-N replication and N-N-1 parity to see how more weights affect the same problem. We also did N-N-N /2 edge counting, where the output is the number of sign changes in an ordered row of N inputs. Finally we did N-2N-N and N-N-N hamming where the output is the closest hamming code for N inputs. We varied the number of perturbation iterations so that p = 1,2,5,10,20,50,100,200,400. Edge N-N-N/2 Parity N-2N-1 i i 0 . I I 0 I I lOll - ... ... , .... "'" ... .., ... , .... --~ ~ i i Hamming N-2N-N Replication N-2N-N ~ ~ 10) .tOO 100 100 '000 200 .00 100 eoo '000 --Figure 2. Critical value scaling for different problems. Fig. 2 gives a feel for the effective scale of the problem by plotting the critical value of the number of perturbation iterations as a function of the number of weights for some of the problems we looked at. Note that the required number of iterations is not a steep function of the network size except for the parity problem. We speculate that the scaling properties are dependent on the shape of the error surface. If the derivatives in Eq. 9 are large in all dimensions (learning on a bowl-shaped surface), then the effective number of parameters is large and the variance of the noise term will be on the order of the number of weights, leading to a steep dependence in Fig. 2. If, however, there are only a few weight directions with significantly large error derivatives (learning on a taco shell), then the noise will scale at a slower rate than the number of weights leading to a weak dependence of the critical value with problem size. This is actually a nice feature of parallel perturbative learning because it means learning will be noisy and slow in a bowl where it's easy, but precise and fast in a taco shell where it's hard. The critical value is required for convergence at the end of learning but not at the start. This means it should be possible to anneal the number of perturbation iterations to achieve an additional speedup over the one-weight-at-a-time perturba842 Alspector, Meir, Yuhas, Jayakumar, and Lippe tion technique. We would also like to understand how to vary bw and 11 as learning proceeds. The stochastic approximation literature is likely to serve as a useful guide. 3.3 Computational Geometry of Stochastic Gradient Descent error weight I '" o o o fW;Y:t~~f~;s~~:ii~ '" :" :, H' • • • ••• _ ., , ' ., Figure 3. Computational Geometry of Stochastic Gradient Descent. Fig. 3a shows some relevant gradient vectors and angles in the learning problem. For a particular pattern presentation, the true gradient, gb, from a back-propagation calculation is compared with the one-weight-at-a-time gradient, go, from a perturbation, bWi , in one weight direction. The gradient from perturbing all weights, gm, adds a noise vector to go. By taking the normalized dot product between gm and gb, one obtains the direction cosine between the estimated and the true gradient direction. This is plotted in Fig. 3b for the 10 input N-N-l parity problem for all nine perturbation values. The shaded bands increase in cos (decrease in angle) as the number of perturbations goes from 1 to 400. Note that the angles are large but that learning still takes place. Note also that the dot product is almost always positive except for a few points at low perturbation numbers. Incidentally, by looking at plots of the true to one-weight-at-a-time angles (not shown), we see that the large angles are due almost entirely to the parallel perturbative noise term and not to the stepsize, bw. 4 Outline of an analog implementation Fig. 4 shows a diagram of a learning synapse using this perturbation technique. Note that its only inputs are a single bit representing the sign of the perturbation and a broadcast signal representing the change in the output error. Multiple perturbations can be averaged by the summing buffer and weight is stored as charge on a capacitor or floating gate device. A Parallel Gradient Descent Method for Learning in Analog VLSI Neural Networks 843 An estimate of the power and area of an analog chip implementation gives the following: Using a standard 1.2J,.tm, double poly technology, the synapse with about 7 to 8 bits ofresolution and which includes a 0.5 pf storage capacitor, weight refresh (Hochet, 1989) and update circuitry can be fabricated with an area of about 1600 J,.tm2 and with a power dissipation of about 100 J,.t W with continuous self-refresh. This translates into a chip of about 22000 synapses at 2.2 watts on a 36 mm2 die core. It is likely that the power requirements can be greatly reduced with a more relaxed refresh technique or with a suitable non-volatile analog storage technology. S WI,j P (Perturbation anneal) , I .. , I,J Teach . I umming &, Integrating , buffer ~cw ~ --L I,j ~~ "-+.-~ I· k I., I, I, aW. 1,1 ~ pertur6~r----. Synapse' ----_. Figure 4. Diagram of perturbative learning synapse. We intend to use our noise generation technique (Alspector, 1991) to provide uncorrelated perturbations potentially to thousands of synapses. Note also that the error signal can be generated by a simple resistor or a comparator followed by a summer. The difference signal can be generated by a simple differentiator. 5 Conclusion We have analyzed a parallel perturbative learning technique and shown that it should converge under the proper conditions. We have performed simulations on a variety of test problems to demonstrate the scaling behavior of this learning algorithm. We are continuing work to understand speedups possible in an analog VLSI implementation. Finally, we describe such an implementation. Future work will involve applying this technique to learning in recurrent networks. Acknowledgment We thank Barak Pearhuutter for valuable and insightful discussions and Gert Cauwenberghs for making an advance copy of his paper available. This work has 844 Alspector, Meir, Yuhas, Jayakumar, and Lippe been partially supported by AFOSR contract F49620-90-C-0042, DEF. References J. Alspector, J. W. Gannett, S. Haber, M.B. Parker, and R. Chu, "A VLSI-Efficient Technique for Generating Multiple Uncorrelated Noise Sources and Its Application to Stochastic Neural Networks", IEEE Trans. Circuits and Systems, 38, 109, (Jan., 1991). J. Alspector, A. Jayakumar, and S. Luna, "Experimental Evaluation of Learning in a Neural Microsystem" in Advances in Neural Information Processing Systems 4, J. E. Moody, S. J. Hanson, and R. P. Lippmann (eds.) San Mateo,CA: MorganKaufmann Publishers (1992), pp. 871-878. G. Cauwenberghs, "A Fast Stochastic Error-Descent Algorithm for Supervised Learning and Optimization," in Advances in Neural Information Processing Systems, San Mateo, CA: Morgan Kaufman Publishers, vol. 5, 1993. A. Dembo and T. Kailath, "Model-Free Distributed Learning", IEEE Trans. Neural Networks Bt, (1990) pp. 58-70. B. Flower and M. Jabri, "Summed Weight Neuron Perturbation: An O(n) Improvement over Weight Perturbation," in Advances in Neural Information Processing Systems, San Mateo, CA: Morgan Kaufman Publishers, vol. 5, 1993. B. Hochet, "Multivalued MOS memory for Variable Synapse Neural Network", Electronics Letters, vol 25, no 10, (May 11, 1989) pp. 669-670. M. Jabri and B. Flower, "Weight Perturbation: An Optimal Architecture and Learning Technique for Analog VLSI Feedforward and Recurrent Multilayer N etworks", Neural Computation 3 (1991) pp. 546-565. D. Kirk, D. Kerns, K. Fleischer, and A. Barr, "Analog VLSI Implementation of Gradient Descent," in Advances in Neural Information Processing Systems, San Mateo, CA: Morgan Kaufman Publishers, vol. 5, 1993. H.J. Kushner and D.S. Clark, "Stochastic Approximation Methods for Constrained and Unconstrained Systems", p. 58 ff., Springer-Verlag, New York, (1978). D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning Internal Representations by Error Propagation", in Parallel Distributed Processing: Ezplorations in the Microstructure of Cognition. Vol. 1: Foundations, D. E. Rumelhart and J. L. McClelland (eds.), MIT Press, Cambridge, MA (1986), p. 318. S. Venkatesh, "Directed Drift: A New Linear Threshold Algorithm for Learning Binary Weights On-Line", Journal of Computer Science and Systems, (1993), in press. B. Widrow and M. A. Lehr, "30 years of Adaptive Neural Networks. Perceptron, Madaline, and Backpropagation", Proc. IEEE 78 (1990) pp. 1415-1442. J. Zinn-Justin, "Quantum Field Theory and Critical Phenomena", p. 57 ff., Oxford University Press, New York, (1989). PART XI COGNITIVE SCIENCE
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A Model of Feedback to the Lateral Geniculate Nucleus Carlos D. Brody Computation and Neural Systems Program California Institute of Technology Pasadena, CA 91125 Abstract Simplified models of the lateral geniculate nucles (LGN) and striate cortex illustrate the possibility that feedback to the LG N may be used for robust, low-level pattern analysis. The information fed back to the LG N is rebroadcast to cortex using the LG N 's full fan-out, so the cortex-LGN-cortex pathway mediates extensive cortico-cortical communication while keeping the number of necessary connections small. 1 INTRODUCTION The lateral geniculate nucleus (LGN) in the thalamus is often considered as just a relay station on the way from the retina to visual cortex, since receptive field properties of neurons in the LGN are very similar to retinal ganglion cell receptive field properties. However, there is a massive projection from cortex back to the LGN: it is estimated that 3-4 times more synapses in the LG N are due to corticogeniculate connections than those due to retinogeniculate connections [12]. This suggests some important processing role for the LGN, but the nature of the computation performed has remained far from clear. I will first briefly summarize some anatomical facts and physiological results concerning the corticogeniculate loop, and then present a simplified model in which its function is to (usefully) mediate communication between cortical cells. 409 410 Brody 1.1 SOME ANATOMY AND PHYSIOLOGY The LG N contains both principal cells, which project to cortex, and inhibitory interneurons. The projection to cortex sends collaterals into a sheet of inhibitory cells called the perigeniculate nucleus (PGN). PGN cells, in turn, project back to the LGN. The geniculocortical projection then proceeds into cortex, terminating principally in layers 4 and 6 in the cat [11, 12]. Areas 17, 18, and to a lesser extent, 19 are all innervated. Layer 6 cells in area 17 of the cat have particularly long, non-end-stopped receptive fields [2]. It is from layer 6 that the corticogeniculate projection back originates.1 It, too, passes through the PGN, sending collaterals into it, and then cont.acts both principal cells and interneurons in the LGN, mostly in the more distal parts of their dendrites [10, 13]. Both the forward and the backward projection are retinotopically ordered. Thus there is the possibility of both excitatory and inhibitory effects in the corticogeniculate projection, which is principally what shall be used in the model. The first attempts to study the physiology of the corticogeniculate projection involved inactivating cortex in some way (often cooling cortex) while observing geniculate responses to simple visual stimuli. The results were somewhat inconclusive: some investigators reported that the projection was excitatory, some that it was inhibitory, and still others t.hat it had no observable effect at all. [1, 5,9] Later studies have emphasized the need for using stimuli which optimally excite the cortical cells which project to the LGN; inactivating cortex should then make a significant difference in the inputs to geniculate cells. This has helped to reveal some effects: for example, LGN cells with corticogeniculate feedback are end-stopped (that is, respond much less to long bars than to short bars), while the end-stopping is quite dearly reduced when the cortical input is removed [8]. One study [13] has used cross-correlat.ion analysis between cortical and geniculate cells to suggest that there is spatial structure in the corticogeniculate projection: an excitatory corticogeniculate interaction was found if cells had receptive field centers that were close to each other, while an inhibitory interaction was found if the centers were farther apart. However, the precise spatial structure of the projection remains unknown. 2 A FEEDBACK MODEL I will now describe a simplified model of the LGN and the corticogeniculate loop. The very simple connection scheme shown in fig 1 originated in a suggestion by Christof Koch [3] that the long receptive fields in layer 6 might be used to facilitate contour completion at t.he LGN level. In the model, then, striate cortex simple cells feed back positively to the LGN, enhancing the conditions which gave rise to their firing. This reinforces, or completes, the oriented bar or edge patterns to which they are tuned. Assuming that the visual features of interest are for the most part oriented, while much of the noise in images may be isotropic and unoriented, enhancing the oriented features improves the signal-to-noise ratio. 1 In all areas innervated by the LGN. A Model of Feedback to the Lateral Geniculate Nucleus 411 OO[i] ///~ ~~ D--~~ RETINA LGN ~ 00 CELLS [f] ~~ VI CELLS Figure 1: Basic model connectivity: A schematic diagram showing the connections between different. pools of units in the single spatial frequency channel model. LGN cells first filter the image linearly through a center-surround filter (V 2G), the result of which is then passed through a sigmoid nonlinearity (tanh). (In the simulations presented here G was a Gaussian with standard deviation 1.4 pixels.) VI cells then provide oriented filtering, which is also passed through a nonlinearity (logistic; but see details in text) and fed back positively to the LGN to reinforce detected oriented edges. VI cells excite LGN cells which have excitat.ory connections to them, and inhibit those t.hat have inhibitory connections to them. Inhibition is implicitly assumed to be mediated by interneurons. (Note that there are no intracortical or intrageniculate connections: communication takes place entirely through the feedback loop.) See text for further details. For simplicity, only striate cortex simple "edge-detecting" cells were modeled. Two models are presented. In the first one, all cortical cells have the same spatial frequency characteristics. In the second one, two channels, a high frequency channel and a low frequency channel, interact simultaneously. 2.1 SINGLE SPATIAL FREQUENCY CHANNEL MODEL A srhematic diagram of the model is shown in figure 1. The retina is used simply as an input layer. To each input position (pixel) in the retina there corresponds one LGN unit. Linear weights from the retina to the LGN implement a '\l2C filter, where G(x,y) is a two-dimensional Gaussian. The LGN units then project to eight. different pools of "orientation-t.uned" cells in VI. Each of these pools has as many units as t.here are pixels in the input "retina". The weights in the projection forward to VI represent eight rotations of the template shown in figure 2a, covering 360 degrees. This simulates basic orientation tuning in VI. Cortical cells then feed 412 Brody back positively to the geniculus, using rotations of the template shown in fig 2(b). The precise dynamics of the model are as follows: Ri are real-valued retinal inputs, Li are geniculate unit outputs, and V; are cortical cell outputs. Gji represent weights from retina LGN, Fji forward weights from LGN VI, and Bji backward weights from VI LGN. o:,/3,,,,(,TCl and TC2 are all constants. For geniculate units: dlL L _J =-"111.+ G .. Q· + B ' LVIe dt I J JI~~ J~ Lj = tanh(/j ) i Ie While for cortical cell units: dVj = -o:v' + ~ Y·L· - /3(~ IY·IL·)2 dt J L.....J JI I L.....J JI I i i Here gO is the logistic function. "receptIYe field le.atII" ••••••• ••••••• ••••••• 0000000 0000000 0000000 V;={ g(Vj - Tcd o • • • • • • • • • • • • • • • •••••• 0000000 o 0 0 0 000 o 0 0 0 000 (b) if vi > TC2 otherwise Figure 2: Weights between the LGN and VI. Figure 2(a): Forward weights, from the LGN to VI. Each circle represents the weight from a cell in the LGN; dark circles represent positive weights, light circles negative weights (assumed mediated by interneurons). The radius of each circle represent.s the strength of the corresponding weight. These weights create "edge-detecting" neurons in VI. Figure 2(b): Backwards weights, from VI back to the LGN. Only cells close to the contrast edge receive strong feedback. In the scheme described above many cortical cells have overlapping receptive fields, both in the forward projection from the geniculus and in the backwards projection from cortex. A cell which is reinforcing an edge within its receptive field will also partially reinforce the edge for retinotopically nearby cortical cells. For nearby cells with similar orientation tuning, the reinforcement will enhance their own firing; they will then enhance the firing of ot.her, similar, cells farther along; and so on. That is, the overlapping feedback fields allow the edge detection process to follow contours (note that the process is tempered at the geniculate level by actual input from the retina). This is illustrated in figure 3. A Model of Feedback to the Lateral Geniculate Nucleus 413 Figure 3: Following contours: This figure shows the effect on the LGN of the feedback enhancement. The image on the left is the retinal input: a very weak, noisy horizontal edge. The center image is the LGN after two iterations of the simulation. Note that initially only certain sectors of the edge are detected (and hence enhanced). The rightmost image is the LGN after 8 iterations: the enhanced region has spread to cover the entire edge through the effect of horizontally oriented, overlapping receptive fields. This is the final stable point of the dynamics. 2.2 MULTIPLE SPATIAL FREQUENCY CHANNELS MODEL In the model described above the LGN is integrating and summarizing the information provided by each of the orientation-tuned pools of cortical cells.2 It does so in a way which would easily ext.end to cover other types of cortical cells (bar or grating "detectors" , or varying spatial frequency channels). To experiment simply with this possibility, an extra set of eight pools of orientation-tuned "edge-detecting" cortical cells was added. The new set's weights were similar to the original weights described above, except t.hey had a "receptive field length" (see figure 2) of 3 pixels: the original set had a "receptive field length" of 9 pixels. Thus one set was tuned for detecting short edges, while the other was tuned for detecting long edges. The effect of using both of these sets is illustrated in figure 4. Both sets interact nonlinearly to produce edge detection rather more robust than either set used alone: the image produced using both simultaneously is not a linear addit.ion of those produced using each set separately. Note how little noise is accepted as an edge. The same model, running with the same parameters but more pixels, was also tested on a real image. This is shown in figure 5. 3 DISCUSSION ON CONNECTIVITY A major function fulfilled by the LG N in this model is that of providing a communicat.ions pathway between cortical cells, both between cells of similar orientation but different location or spatial frequency tuning, and between cells of different orienta2 A function not unlike that suggested by Mumford [7], except that here the "experts" are extremely low-level orient.ation-tuned channels. 414 Brody ~::. ' ''1 :r" Figure 4: Combined spatial frequency channels: The leftmost image is the retinal input, a weak noisy edge. (The other three images are "summary outputs", obtained as follows: the model produces activations in many pools of cortical cell units; the activations from all VI units corresponding to a particular retinotopic position are added together to form a real-valued number corresponding to that position; and this is then displayed as a grey-scale pixel. Since only "edge-detecting" units were used, this provides a rough estimate of the certainty of there being an edge at that point.) Second from left we see the summary output of the model after 20 iterations (by which time it has stabilized), using only the low spatial frequency channel. Only a single segment of the edge is detected. Third from left is the output after 20 iterations using only the high frequency channel. Only isolat.ed, short, segment.s of the edge are detected. The rightmost image is the output using both channels simultaneously. Now the segments detected by the high frequency channel can combine with the original image to provide edges long enough for the low frequency channel t.o detect and complete into a single, long continuous edge. tion tuning: for example, these last compete to reinforce their particular orientation preference on the geniculus. The model qualitatively shows that such a pathway, while mediated by a low-level representation like that of the LGN, can nevertheless be used effectively, producing contour-following and robust edge-detection. We must now ask whether such a function could not be performed without feedback. Clearly, it could be done without feedback to the LGN, purely through intracortical connections, since any feedback net.work can in principle be "unfolded in time" into a feedforward network which performs the same computation- provided we have enough units and connections available. In other words, any sugg{'st.ed functional role for corticogeniculate feedback must not only include an account of the proposed computation performed, but also an account of why it is preferable to perform that computation through a feedback process, in terms of some efficiency measure (like the number of cells or synapses necessary, for example). There can be no other rationale, apart from fortuitous coincidence, for constructing an elaborate feedback mechanism to perform a computation that could just as well be done without it. With this view in mind, it. is worth re-stating that in this model any two cortical cells whose receptive fields overlap are connected (disynaptically) through the LG N. How many connections would we require in order to achieve similar communication if we only used direct connections between cortical orientation-tuned cells instead? In monkey, each cell's receptive field overlaps with approximately 106 others [4]- thus, ~·1~ . . ' . ~ \ . , , , .... ~ :o~, ;...=:: A Model of Feedback to the Lateral Geniculate Nucleus 415 .. ::~:.:... , ·;:~Hk.;~.~ i e '. 5*1 ' romw gg- SF' " ,6, 'CrT5S ff',.P.-... ~ .. II., .... ·;i.;\'"W.,.,-W, ... , ........... __ ...... ,..... .-' ... ' ,'., ......... '. ,'., ,. il t ~\ }: / .:.-r. :: fi w~~\ r·~:,,~~,· •• . , • • • ...;y;; .. ~ _ . .. ....... -.••• .r •••••• ~ 1 :' ' ........ ~., .. " .... ; .. ~ ~ ::' """'" . '::~?~:t;\ ' :I:~i~J:~~:·::;.:;~:;:;:;~~;~:~:~I:~:; :;:';:.~: _:', :l~~s't]~k.il~:~~~~~~: Figure 5: A real image: The top image is the retinal input. Stippling is due to printing only. The center image is that obtained through detecting the zero-crossings of v 2c. To reduce spurious edges. a minimum slope threshold was placed on the point of the zero-crossing below which edges were not accepted. The image shown here was the best that could be obtained through varying both the width of the Gaussian G and the slope threshold value. The last image shows the summary output from the model, using two simultaneous spatial frequency cha.nnels. Note how noise is reduced compared to the center image, straight lines are smoother, and resolution is not impaired, but is better in places (group of people at lower left. or "smoke stacks" atop launcher). 416 Brody any cortical cell would need to synapse onto at least 106 cells. If the information can be sent via the LGN, geniculate cell fan-out can reduce the number of necessary synapses by a significant factor. It is estimated that geniculate cells (in the cat) synapse onto at least 200 cortical cells (probably more) [6], reducing the number of necessary connections considerably. 4 BIOLOGY AND CONCLUSIONS In section 1.1 I noted one important study [8J which established that corticogeniculate input reduces firing of geniculate cells for long bars; this is in direct contradiction to the prediction which would be made by this model, where the feedback enhances firing for long features (here, edges). Thus, the model does not agree with known physiology. However, the model's value lies simply in clearly illustrating the possibility that feedback in a hierarchical processing scheme like the corticogeniculate loop can be utilized for robust, low-level pattern analysis, through the use of the cortex-+LGN-+cortex communications pathway. The possibility that a great deal of different types of information could be flowing through this pathway for this purpose should not be left unconsidered. Acknowledgements The author is supported by fellowships from the Parsons Foundation and from CONACYT (Mexico). Thanks are due to Michael Lyons for careful reading of the manuscript. References [IJ Baker, F. H. and Malpeli, J. G. 1977 Exp. Brain Res. 29 pp. 433-444 [2J Gilbert, C.D. 1977, J. Physiol., 268, pp. 391-421 [3J Koch, C. 1992, personal communication. [4J Hubel, D.H. and Wiesel, T. N. 1977, Proc. R. Soc. Lond. (B) 198 pp. 1-59 [5] Kalil, R. E. and Chase, R. 1970, J. Neurophysiol. 33 pp. 459-474 [6] Martin, K.A.C. 1988, Q. J. Exp. Phy. 73 pp. 637-702 [7] Mumford, D. 1991 Bioi. Cybern. 65 pp. 135-145 [8] Murphy, P.C. and Sillito, A.M. 1987, Nature 329 pp. 727-729 [9] Richard. D. et. al. 1975, Exp. Brain Res. 22 pp. 235-242 [10] Robson, J. A. 1983. J. Compo Neurol. 216 pp. 89-103 [11] Sherman, S. M. 1985. Prog. in Psychobiol. and Phys. Psych. 11 pp. 233-314 [12J Sherman, S.M. and Koch, C. 1986, Exp. Brain Res. 63 pp. 1-20 [13J Tsumoto, T. et. al. 1978, Exp. Brain Res. 32 pp. 345-364
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History-dependent Attractor Neural Networks Isaac Meilijson Eytan Ruppin School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel-A viv University, 69978 Tel-Aviv, Israel. Abstract We present a methodological framework enabling a detailed description of the performance of Hopfield-like attractor neural networks (ANN) in the first two iterations. Using the Bayesian approach, we find that performance is improved when a history-based term is included in the neuron's dynamics. A further enhancement of the network's performance is achieved by judiciously choosing the censored neurons (those which become active in a given iteration) on the basis of the magnitude of their post-synaptic potentials. The contribution of biologically plausible, censored, historydependent dynamics is especially marked in conditions of low firing activity and sparse connectivity, two important characteristics of the mammalian cortex. In such networks, the performance attained is higher than the performance of two 'independent' iterations, which represents an upper bound on the performance of history-independent networks. 1 Introduction Associative Attractor Neural Network (ANN) models provide a theoretical background for the understanding of human memory processes. Considerable effort has been devoted recently to narrow the gap between the original ANN Hopfield model (Hopfield 1982) and the realm of the structure and dynamics of the brain (e.g., Amit & Tsodyks 1991). In this paper, we contribute to the examination of the performance of ANNs under cortical-like architectures, where neurons are typically 572 History-dependent Attractor Neural Networks 573 connected to only a fraction of their neighboring neurons, and have a low firing activity (Abeles et. al. 1990). We develop a general framework for examining various signalling mechanisms (firing functions) and activation rules (the mechanism for deciding which neurons are active in some interval of time). The Hopfield model is based on memoryless dynamics, which identify the notion of 'post-synaptic potential' with the input field received by a neuron from the neurons active in the current iteration. We follow a Bayesian approach under which the neuron's signalling and activation decisions are based on the current a-posteriori probabilities assigned to its two possible true memory states, ±1. As we shall see, the a-posteriori belief in +1 is the sigmoidal function evaluated at a neuron's generalized field, a linear combination of present and past input fields. From a biological perspective, this history-dependent approach is strongly motivated by the observation that the time span of the different channel conductances in a given neuron is very broad (see Lytton 1991 for a review). While some channels are active for only microseconds, some slow-acting channels may remain open for seconds. Hence, a synaptic input currently impending on the neuron may influence both its current post-synaptic membrane potential, and its post-synaptic potential at some future time. 2 The Model The neural network model presented is characterized as follows. There are m 'random memories' elJ , 1 < iJ ::; m, and one 'true' memoryem +1 = e. The (m + 1)N entries of these memories are independent and identically distributed, with equally likely values of +1 or --1. The initial state X has similarityP(Xi = ei) = (1+€)/2, P(Xi = -ed = (1 - €)/2, independently of everything else. The weight of the synaptic connection between neurons i and j (i -j:. j) is given by the simple Hebbian law m+l Wij* = L elJielJj (1) 1J=1 Each neuron receives incoming synaptic connections from a random choice of K of the N neurons in the network in such a way that if a synapse exists, the synapse in the opposite direction exists with probability r, the reflexivity parameter. In the first iteration, a random sample of Ll neurons become active (i.e., 'fire'), thus on the average nl = LlK/N neurons update the state of each neuron. The field //1) of neuron i in the first iteration is N j .(l) - ~ ~ w. .. * [..I·(l)X· I ~ IJ IJ 1 J' nl j=1 (2) where Iij denotes the indicator function of the event 'neuron i receives a synaptic connection from neuron j', and I/t ) denotes the indicator function of the event 'neuron j is active in the t'th iteration'. Under the Bayesian approach we adopt, neuron i assigns an a-priori probability ,\/0) = p(ei = +1\Xi) = (1 + {Xi)/2 to having +1 as the correct memory state and evaluates the corresponding a-posteriori probability ,\/1) = p(ei = +1\Xi , fi(I), which turns out to be expressible as the 574 Meilijson and Ruppin sigmoidal function 1/( 1 + exp( -2x)) evaluated at some linear combination of Xi and fi(I). In the second iteration the belief A/I) of a neuron determines the probability that the neuron is active. We illustrate two extreme modes for determining the active updating neurons, or activation: the random case where L2 active neurons a.re randomly chosen, independently of the strength of their fields, and the censored case, which consists of selecting the L2 neurons whose belief belongs to some set. The most appealing censoring rule from the biological point of view is tail-censoring, where the active neurons are those with the strongest beliefs. Performance, however, is improved under interval-censoring, where the active neurons are those with midrange beliefs, and even further by combining tail and interval censoring into a hybrid rule. Let n2 = L2 f{ / N . The activation rule is given by a function C : [~, 1] -+ [0, 1]. Neuron j, with belief A/I) in +1, becomes active with probability C(maxp/I), 1Aj (1»)), independently of everything else. For example, the random case corresponds to C = In and the tail-censored case corresponds to C(A) = 1 or 0 depending on whether max(A, 1 - A) exceeds some threshold. The output of an active neuron j is a signal function S(A/ I ») of its current belief. The field f/ 2) of neuron i in the second iteration is N f ·(2) - ~ "" w. .. * /- ·I ·(2)S(A ·(1») (3) , L...J I} I}} } • n2 j=1 Neuron i now evaluates its a-posteriori belief Ai(2) = p(ei = +1IXi,Ii(l),f/I\I/2»). As we shall see, Ai(2) is, again, the sigmoidal function evaluated at some linear combination of the neuron's history Xi, XJi(I), f/ I ) and 1/2). In contrast to the common history-independent Hopfield dynamics where the signal emitted by neuron j in the t'th iteration is a function of /jet-I) only, Bayesian history-dependent dynamics involve signals and activation rules which depend on the neuron's generalized field, obtained by adaptively incorporating /j(t-I) to its previous generalized field. The final state X/ 2 ) of neuron i is taken as -lor +1, depending on which of 1 - A/2) and A/2) exceeds 1/2. For nI/N, ndN, m/N, K/N constant, and N large, we develop explicit expressions for the performance of the network, for any signal function (e.g., SI(A) = Sgn(A1/2) or S2(A) = 2A - 1) and activation rule. Performance is measured by the final overlap e" = ~ L eiX/2) (or equivalently by the final similarity (1+e")/2). Various possible combina.tions of activation modes and signal functions described above are then examined under varying degrees of connectivity and neuronal activity. 3 Single-iteration optimization: the Bayesian approach Consider the following well known basic fact in Bayesian Hypothesis Testing, Lemma 1 Express the prior probability as 1 pee = 1) = ---:-1 + e- 2x (4) History-dependent Attractor Neural Networks 575 and assume an observable Y which, given e, is distributed according to Yle", N(lle, (12) (5) ~or some constants Il E (-00,00) and (12 E (0,00). Then the posterior probability IS 1 P(C = llY = y) = «) ). I, 1 + e-2 x+ 1-'/0'2 y (6) Applying this Lemma to Y = fi(I), with Il = e and (12 = .ill. = al, we see that nl ~i(1) = p(ei = 11Xi , fi(I)) = __ ---:-_1_--:-:---:1 + e- 2f("Y(f)X,+f,(1)/0:'1) , (7) where i( f) = if log ~:!:~. Hence, pee = 11Xi , f/ 1)) > 1/2 if and only if 1/1) + an( f)Xi > O. The single-iteration performance is then given by the similarity (8) 1 ;f~ (;." +1(f)fo1) + 1; '~ (;." -1(f)fo1) = Q(e, at) where <I> is the standard normal distribution function. The Hopfield dynamics, modified by redefining Wii as mi(e) (in the Neural Network terminology) is equivalent (in the Bayesia.n jargon) to the obvious optimal policy, under which a neuron sets for itself the sign with posterior probability above 1/2 of being correct. 4 Two-iterations optiInization For mathematical convenience, we will relate signals and activation rules to normalized generalized fields rather than to beliefs. We let h x - S x - C max 1 1 (( 1 1)) ( ) (1 + e- 2CX ) , p( ) 1 + e-2cx ' 1 + e- 2cx (9) for C = f/ -Jal. The signal function h is assumed to be odd, and the activation function p, even. In order to evaluate the belief ~/2), we need the conditional distribution of li(2) given Xi, 1/1) and 1/1), for ei = -1 or ei = + 1. We adopt the working a.ssumption that the pair of random variables (Ii (1), h(2)) has a bivariate normal distribution given ei, 1/1) and Xi, with ei, 1/1) and Xi affecting means but not va.riances or correlations. Under this working assumption, fi(2) is conditionally normal given (ei,1/ 1),Xi,I/1)), with constant variance and a mean which we will identify. This working assumption allows us to model performance via the following well known regression model. 576 Meilijson and Ruppin Lemma 2 If two random variables U and V with finite variances are such that E(VIU) is a linear function of U and Var(VIU) is constant, then and E(VIU) = E(V) + Cov(U, V) (U - E(U» Var(U) Letting U = fi(l) and V = f/ 2), we obtain (10) A/2) = pee; = llXi, h(l>, li(l), 1/2» = (12) 1 1 + exp{ -2 [E (// 1) la1 + {(E)Xi ) + f*;;af (fi(2) - bXa/1) - afi (1»)]} 1 1 + exp{ -2 [ (q( E) b(f*T;af) I/ 1») Xi + (afl - a(f:;af») li(l) + f* ;;af 1/2 )] } which is the sigmoidal function evaluated at some generalized field. Expression (12) shows that the correct definition of a final state Xi (2), as the most likely value among +1 or -1, is X .(2) _ S [( () _ b(E" - af)I.Cl») . (~_ a(E" - aE») J.".(1) E" - aE f.(2)j , gn E{ f 2 I X, + 2 h + 2 I T al T T (13) and the performance is given by p(X/2) = eilei) = 1; E4> (R + {(E)N) + 1 ~ E4> (R -{(E)~) = (14) Q( E, a") where the one-iteration performance function Q is defined by (8), and " m a =n" m (15) We see that the performance is conveniently expressed as the single-iteration optimal performance, had this iteration involved n" rather than nl sampled neurons. This formula yields a numerical and analytical tool to assess the network's performance with different signal functions, activation rules and architectures. Due to space restrictions, the identification of the various parameters used in the above formulas is not presented. However, it can be shown that in the sparse limit arrived at by fixing al and a2 and letting both J{ 1m and N I J{ go to infinity, it is always better to replace an iteration by two smaller ones. This suggests that Bayesian History-dependent Attractor Neural Networks 577 updating dynamics should be essentially asynchronous. We also show that the two-iterations performance Q (c, -L+-L(~)2) is superior to the performance CIt 1 cw2 ~ Q (2Q( c, al) - 1, (2) of two independent optimal single iterations. 5 Heuristics on activation and signalling U 1 -1 -4 Figure 1: A typical plot of R(x) = ¢l(X)/¢O(x). Network parameters are N = 500, K = 500, n1 = n2 = 50 and m = 10. By (14) and (15), performance is mostly determined by the magnitude of (col< - ac)2. It can be shown that (16) and ~a = 1 00 p(x)¢o(x)dx (17) where ¢l and ¢o are some specific linear combinations of Gaussian densities and their derivatives, and ~ a = n2/ K is the activity level. High performance is achieved by maximizing over p and possibly over h the absolute value of expression (16) ~eeping (17) fixed. In complete analogy to Hypothesis ~esting in Statistics, where Wa takes the role of level of significance and (c· - ac)wa the role of power, p(x) should be 1 or a (activate the neuron or don't) depending on whether the field value x is such that the likelihood ratio h(X)¢l(X)/¢O(x) is above or below a given threshold, determined by (17). Omitting details, the ratio R(x) = ¢l(X)/¢O(x) looks as in figure 1, and converges to -00 as x --+ 00 . We see that there are three reasonable ways to make the ratio h(X)¢l(X)/¢O(x) large: we can take a negative threshold such as t1 in figure 1, activate all neurons with generalized field exceeding /33 (tail-censoring) and signal hex) = -Sgn(x), 578 Meilijson and Ruppin or take a positive threshold such as t2 and activate all neurons with field value between /31 and /32 (interval-censoring) and signal h(x) = Sgn(x). Better still, we can consider the hybrid signalling-censoring rule: Activate all neurons with absolute field value between /31 and /32, or beyond /33' The first group should signal their preferred sign, while those in the second group should signal the sign opposite to the one they so strongly believe in ! 6 Numerical results Performance predicted experimental Random activation 0.955 0.951 Tail censoring 0.972 0.973 Intervalj'Hybrid censoring 0.975 0.972 Hopfield - zero diagonal 0.902,0.973 Independent ,( () diagonal 0.96 Independent zero diagonal 0.913 Table 1: Sparsely connected, low activity network: N = 1500, J{ = 50, nl = n2 = 20,m = 5. I • I i ' i '.-1 :)..~ .. / . ··f .~. . ... : ...... i , ' i ..... ~ ...... . 0.95 ~ .... . ........................... . 0.85 ',' ~. . "' .. . ...... ?( ....... . ..... . ' . -- First Iteration .... Random Activation > • • •• - Tail-censoring .... . - .. Interval-censoring ....... : Hybrid censoring-signalling -'-... 0.80 L-~_--'-_~_'--~_-'-_-'----.J'--~_--' 0.0 20000.0 40000.0 60000.0 80000.0 100000.0 K Figure 2: Performance of a large-scale cortical-like 'columnar' ANN, at different values of connectivity J{, for initial similarity 0.75. N = 105 , nl = n2 = 200, m = 50. The horizontal line denotes the performance of a single iteration. History-dependent Attractor Neural Networks 579 Our theoretical performance predictions show good correspondence with simulation results, already at fairly small-scale networks. The superiority of historydependent dynamics is apparent. Table 1 shows the performance achieved in a sparsely-connected network. The predicted similarity after two iterations is reported, starting from initial similarity 0.75, and compared with experimental results averaged over 100 trials. Figure 2 illustrates the theoretical two-iterations performance of large, low-activity 'cortical-like' networks, as a function of connectivity. We see that interval-censoring can maintain high performance throughout the connectivity range. The performance of tail-censoring is very sensitive to connectivity, almost achieving the performance of interval censoring at a narrow low-connectivity range, and becoming optimal only at very high connectivity. The superior hybrid rule improves on the others only under high connectivity. As a cortical neuron should receive the concomitant firing of about 200 - 300 neurons in order to be activated (Treves & Rolls 1991), we have set n = 200. We find that the optimal connectivity per neuron, for biologically plausible tail-censoring activation, is of the same order of magnitude as actual cortical connectivity. The actual number nN / K of neurons firing in every iteration is about 5000, which is in close correspondence with the evidence suggesting that about 4% of the neurons in a module fire at any given moment (Abeles et. a!. 1990). References [1] J.J. Hopfield. Neural networks and physical systems with emergent collective abilities. Proc. Nat. Acad. Sci. USA, 79:2554,1982. [2] D. J. Amit and M. V. Tsodyks. Quantitative study of attractor neural network retrieving at low spike rates: I. substrate-spikes, rates and neuronal gain. Network, 2:259-273, 1991. [3] M. Abeles, E. Vaadia, and H. Bergman. Firing patterns of single units in the prefrontal cortex and neural network models. Network, 1:13-25, 1990. [4] W. Lytton. Simulations of cortical pyramidal neurons synchronized by inhibitory interneurons. J. Neurophysiol., 66(3):1059-1079, 1991. [5] A. Treves and E. T. Rolls. What determines the capacity of autoassociative memories in the brain? Network, 2:371-397, 1991.
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Learning Spatio-Temporal Planning from a Dynamic Programming Teacher: Feed-Forward N eurocontrol for Moving Obstacle A voidance Gerald Fahner * Department of Neuroinformatics University of Bonn Romerstr. 164 W -5300 Bonn 1, Germany Rolf Eckmiller Department of Neuroinformatics University of Bonn Romerstr. 164 W-5300 Bonn 1, Germany Abstract Within a simple test-bed, application of feed-forward neurocontrol for short-term planning of robot trajectories in a dynamic environment is studied. The action network is embedded in a sensorymotoric system architecture that contains a separate world model. It is continuously fed with short-term predicted spatio-temporal obstacle trajectories, and receives robot state feedback. The action net allows for external switching between alternative planning tasks. It generates goal-directed motor actions - subject to the robot's kinematic and dynamic constraints - such that collisions with moving obstacles are avoided. Using supervised learning, we distribute examples of the optimal planner mapping over a structure-level adapted parsimonious higher order network. The training database is generated by a Dynamic Programming algorithm. Extensive simulations reveal, that the local planner mapping is highly nonlinear, but can be effectively and sparsely represented by the chosen powerful net model. Excellent generalization occurs for unseen obstacle configurations. We also discuss the limitations of feed-forward neurocontrol for growing planning horizons. *Tel.: (228)-550-364 342 FAX: (228)-550-425 e-mail: gerald@nero.uni-bonn.de Learning Spatio-Temporal Planning from a Dynamic Programming Teacher 343 1 INTRODUCTION Global planning of goal directed trajectories subject to cluttered spatio-temporal, state-dependent constraints - as in the kinodynamic path planning problem (Donald, 1989) considered here - is a difficult task, probably best suited for systems with embedded sequential behavior; theoretical insights indicate that the related problem of connectedness is of unbounded order (Minsky, 1969). However, considering practical situations, there is a lack of globally disposable constraints at planning time, due to partially unmodelled environments. The question then arises, to what extent feed-f )rward neurocontrol may be effective for local planning horizons. In this paper, we put aside problems of credit assignment, and world model identification. We focus on the complexity of representing a local version of the generic kinodynamic path planning problem by a feed-forward net. We investigate the capacity of sparse distributed planner representations to generalize from example plans. 2 ENVIRONMENT AND ROBOT MODELS 2.1 ENVIRONMENT The world around the robot is a two-dimensional scene, occupied by obstacles moving all in parallel to the y-axis, with randomly choosen discretized x-positions, and with a continuous velocity spectrum. The environment's state is given by a list reporting position (Xi,Yi) E (X,Y), X E {0, ... ,8}, Y = [y-,y+], and velocity (0, Vi) ; Vi E [v- ,v+] of each obstacle i. The environment dynamics is given by (1) Obstacles are inserted at random positions, and with random velocities, into some region distant from the robot's workspace. At each time step, the obstacle's positions are updated according to eqn.(l), so that they will cross the robot's workspace some time. 2.2 ROBOT We consider a point-like robot of unit mass, which is confined to move within some interval along the x-axis. Its state is denote~. by (xr,xr) E (X,X);X = {-1,0, I}. At each time step, a motor command u E X = {-I, 0, I} is applied to the robot. The robot dynamics is given by Xr(t + 1) zr(t + 1) = xr(t) + u(t) = zr(t) + xr(t + 1) . (2) Notice that the set of admissible motor commands depends on the present robot state. With these settings, the robot faces a fluctuating number of obstacles crossing its baseline, similar to the situation of a pedestrian who wants to cross a busy street (Figure 1). 344 Fahner and Eckmiller dyno.MiC obsto.cles o o robot gOo.l Figure 1: Obstacles Crossing the Robot's Workspace 3 SYSTEM ARCHITECTURE AND FUNCTIONALITY Adequate modeling of the perception-action cycle is of decisive importance for the design of intelligent reactive systems. We partition the overall system into two modules: an active Perception Module (PM) with built-in capabilities for short-term environment forecasts, and a subsequent Action Module (AM) for motor command generation (Figure 2). Either module may be represented by a 'classical' algorithm, or by a neural net. PM is fed with a sensory data stream reporting the observed sens~ infor~ Perception Moclule lon9terM goal roloot state JJJJ interno.l representa tion Action Moclule JJ Motor COMMancl Figure 2: Sensory-Motoric System Architecture dynamic scene of time-varying obstacle positions. From this, it assembles a spatioLearning Spatio-Temporal Planning from a Dynamic Programming Teacher 345 temporal internal representation of near-future obstacle trajectories. At each time step t, it actnalizes the incidence function occupancy(x, k) = { _11 (x = Xi and - s < Yi(t + k) < s) for any obstacle i otherwise, where s is some safety margin accounting for the y-extension of obstacles. The incidence furlction is defined on a spatio-temporal cone-shaped cell array, based at the actual rc bot position: Ix - xr(t)1 ~ k ; k = I, .'" HORIZON (3) The opening angle of this cone-shaped region is given by the robot's speed limit (here: one cell per time step). Only those cells that can potentially be reached by the robot within the local prediction-/planning horizon are thus represented by PM (see Figure 3). The functionality of AM is to map the current PM representation to x I, ,. ..... 4Ir / T / r2J i--';" (~o 0 [3]-[3]~ 0 ~@] 0 x ... ~ ./ [5J ,~I---I£J T T .... , o 1 2 3 Figure 3: Space-Time Representation with Solution Path Indicated an appropriate robot motor command, taking into account the present robot state, and paying regard to the currently specified long-term goal. Firstly, we realize the optimal AM by the Dynamic Programming (DP) algorithm (Bellman, 1957). Secondly, we use supervised learning to distribute optimal planning examples over a neural network. 4 DYNAMIC PROGRAMMING SOLUTION Given PM's internal representation at time t, the present robot state, and some specification of the desired long-term goal, DP determines a sequence of motor commands minimizing some cost functional. Here we use HORIZON cost{u(t), ... ,u(t+HORIZON)} = L:: (xr(t + k) - xo)2 + c u(t + k)2 , (4) k=O 346 Fahner and Eckmiller with xr(t + k) given by the dynamics eqns.(2) (see solution path in Figure 3). By xo, we denote the desired robot position or long-term goal. Deviations from this position are punished by higher costs, just as are costly accelerations. Obstacle collisions are excluded by restricting search to admissible cells (x, X, t + k )admiuible in phase-space-time (obeying occupancy(x,t+k) = -1). Training targets for timet are constituted by the optimal present motor actions uopt(t), for which the minimum is attained in eqn.( 4). For cases with degenerated optimal solutions, we consistently break symmetry, in order to obtain a deterministic target mapping. 5 NEURAL ACTION MODEL For neural motor command generation, we use a single layer of structure-adapted parsimonious Higher Order Neurons (parsiHONs) (Fahner, I992a, b), computing outputs Yi E [0,1] ; i = 1,2,3. Target values for each single neuron are given by yfe& = 1, if motor-action i is the optimal one, otherwise, yfe& = 0. As input, each neuron receives a bit-vector x = Xl, ... ,XN E {-I, I}N, whose components specify the values of PM's incidence function, the binary encoded robot state, and some task bits encoding the long-term goal. Using batch training, we maximize the loglikelihood criterion for each neuron independently. For recall, the motor command is obtained by a winner-takes-all decision: the index of the most active neuron yields the motor action applied. Generally, atoms for nonlinear interactions within a bipolar-input HON are modelled by input monomials of the form N 1]Ot = II xji ; Cl' = Cl'l ... Cl'N E n = {O, I}N . i=1 (5) Here, the ph bit of Cl' is understood as exponent of Xi. It is well known that the complete set of monomials forms a basis for Boolean functions expansions (Karpovski, 1976). Combinatorial growth of the number of terms with increasing input dimension renders allocation of the complete basis impractical in our case. Moreover, an action model employing excessive numbers of basis functions would overfit trainig data, thus preventing generalization. We therefore use a structural adaptation algorithm, as discussed in detail in (Fahner, I992a, b), for automatic identification and inclusion of a sparse set of relevant nonlinearities present in the problem. In effect, this algorithm performs a guided stochastic search exploring the space of nonlinear interactions by means of an intertwined process of weight adaptation, and competition between nonlinear terms. The parsiHON model restricts the number of terms used, not their orders: instead of the exponential size set {1]Ot : Cl' En}, just a small subset {1]{3 : /3 ESC n} of terms is used within a parsimonious higher order function expansion ye,t(x) = f [2: w{31]{3(X)] ; w{3 E 1R . {3ES (6) He~'e, f denotes the usual sigmoid transfer function. parsiHONs with high degrees of sparsity were effectively trained and emerged robust generalization for difficult nonlinear classification benchmarks (Fahner, I992a, b). Learning Spatia-Temporal Planning from a Dynamic Programming Teacher 347 6 SIMULATION RESULTS We performed extensive simulations to evaluate the neural action network's capabilities to generalize from learned optimal planning examples. The planner was trained with respect to two alternative long-term goals: XO = 0, or XO = 8. Firstly, optimal DP planner actions were assembled over about 6,000 time steps of the simulated environment (fa.irly crowded with moving obstacles), for both long-term goals. At each time step, optimd motor commands were computed for all 9 x 3 = 27 available robot states. From this bunch of situations we excluded those, where no collision-free path existed within the planning horizon considered: (HORIZON = 3). A total of 115,000 admissible training situations were left, out of the 6,000 x 27 = 162,000 one's generated. Thus, out of the full spectrum of robot states which were checked every time step, just about 19 states were not doomed to collide, at an average. These findings corrobate the difficulty of the choosen task. Many repetitions are present in these accumulated patterns, reflecting the statistics of the simulated environment. We collapsed the original training set by removing repeated patterns, providing the learner with more information per pattern: a working data base containing about 20.000 different patterns was left. Input to the neural action net consisted of a bit-vector of length N = 21, where 3 + 5 + 7 bits encode PM's internal representation (cone size in Figure 3), 6 bits encode the robot's state, and a single task bit reports the desired goal. For training, we delimited single neuron learning to a maximum of 1000 epochs. In most cases, this was sufficient for successful training set classification for any of the three neurons (Yi < .8 for yfe& = 0, and Yi > .8 for yfe& = 1 ; i = 1,2,3). But even if some training patterns were misclassified by individual motor neurons, additional robustness stemming from the winner-takes-all decision rescued fault-free recall of the voting community. To test generalization of the neural action model, we par" ... II> .-.-.. a. '0 II> ... ... .. " ., .. .... " ., e '" o II> 0-.. ... C II> " ... II> a. 6 a)-HON 0 c 9)-HON + 9)-HON C 93-HON )( 5 llO-HON '" llO-HON • UO-HON 0 + 3 o 2 o + o )( + t .. O~----~----~----~----~----~----~----~~ o 2 4 6 a 5ize of tra~nin9 set 10 12 "1000 Figure 4: Generalization Behavior 14 titioned the data base into two parts, one containing training patterns, the other 348 Fahner and Eckmiller containing new test patterns, not present in the training set. Several runs were performed with parsiHONs of sizes between 83 and 110 terms. Results for varying training set sizes are depicted in Figure 4. Test error decreases with increasing training set size, and falls as low as about one percent for about 12,000 training patterns. It continues to decrease for larger training sets. These findings corrobate that the trained architectures emerge sensible robust generalization. To get some insight into the complexity of the mapping, we counted the number of terms which carry a given order. The resulting distribution has its maximum at order 3, exhibits many terms of orders 4 and higher, and finally decreases to zero for oruers exceeding 10 (Figure 5). This indicates that the planner mapping considered is highly nonlinear. '" u c (1/ ::I IT (II ... .... (II > ... .., ., .... (II .. o .25 r------~----_,_----__,_----_..,.-___, averaged over several ne~~orks ~ 0.2 0.15 0.1 0.05 o~------~----~~~ ______ -+ ________ ~ __ ~ o 5 10 order 15 Figure 5: Distribution of Orders 20 7 DISCUSSION AND CONCLUSIONS Sparse representation of planner mappings is desirable when representation of complete policy look-up tables becomes impracticable (Bellman's "curse of dimensionality"), or when computation of plans becomes expensive or conflicting with real-time requirements. For these reasons, it is urgent to investigate the capacity of neurocontrol for effective distributed representation and for robust generalization of planner mappmgs. Here, we focused on a new type of shallow feed-forward action network for the local kinodynamic trajectory planning problem. An advantage with feed- forward nets is their low-latency recall, which is an important requirement for systems acting in rapidly changing environments. However, from theoretical considerations concerning the related problem of connectedness with its inherent serial character (Minsky, 1969), the planning problem under focus is expected to be hard for feed-forward nets. Even for rather local planning horizons, complex and nonlinear planner mapLearning Spatio-Temporal Planning from a Dynamic Programming Teacher 349 pings must be expected. Using a powerful new neuron model that identifies the relevant nonlinearities inherent in the problem, we determined extremely parsimonious architectures for representation of the planner mapping. This indicates that some compact set of important features determines the optimal plan. The adapted networks emerged excellent generalization. We encourage use of feed-forward nets for difficult local planning tasks, if care is taken that the models support effective representation of high-order nonlinearities. For growing planning horizons, it is expected that feed-forward neurocontrol will run into limitatioml (Werbos, 1992). The simple test-bed presented here would allow for inser tion a.Dd testing also of other net models and system designs, including recurrent networks. Acknowledgements This work was supported by Federal Ministry of Research and Technology (BMFTproject SENROB), grant 01 IN 105 AID) References E. B. Baum, F. Wilczek (1987). Supervised Learning of Probability Distributions by Neural Networks. In D. Anderson (Ed.), Neural Information Processing Systems, 52-61. Denver, CO: American Institute of Physics. R. E. Bellman (1957). Dynamic Programming. Princeton University Press. B. Donald (1989). Near-Optimal Kinodynamic Planning for Robots With Coupled Dynamic Bounds, Proc. IEEE Int. Conf. on Robotics and Automation. G. Fahner, N. Goerke, R. Eckmiller (1992). Structural Adaptation of Boolean Higher Order Neurons: Superior Classification with Parsimonious Topologies, Proc. ICANN, Brighton, UK. G. Fahner, R. Eckmiller. Structural Adaptation of Parsimonious Higher Order Classifiers, subm. to Neural Networks. M. G. Karpovski (1976). Finite Orthogonal Series in the Design of Digital Devices. New York: John Wiley & Sons. M. Minsky, S. A. Papert (1969). Perceptrons. Cambridge: The MIT Press. P. Werbos (1992). Approximate Dynamic Programming for Real-Time Control and Neural Modeling. In D. White, D. Sofge (eds.) Handbook of Intelligent Control, 493-525. New York: Van Nostrand.
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Single-iteration Threshold Hamming Networks Isaac Meilijson Eytan Ruppin Moshe Sipper School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, 69978 Tel Aviv, Israel Abstract We analyze in detail the performance of a Hamming network classifying inputs that are distorted versions of one of its m stored memory patterns. The activation function of the memory neurons in the original Hamming network is replaced by a simple threshold function. The resulting Threshold Hamming Network (THN) correctly classifies the input pattern, with probability approaching 1, using only O(mln m) connections, in a single iteration. The THN drastically reduces the time and space complexity of Hamming Network classifiers. 1 Introduction Originally presented in (Steinbuch 1961, Taylor 1964) the Hamming network (HN) has received renewed attention in recent years (Lippmann et. al. 1987, Baum et. al. 1988). The HN calculates the Hamming distance between the input pattern and each memory pattern, and selects the memory with the smallest distance. It is composed of two subnets: The similarity subnet, consisting of an n-neuron input layer connected with an m-neuron memory layer, calculates the number of equal bits between the input and each memory pattern. The winner-take-all (WTA) subnet, consisting of a fully connected m-neuron topology, selects the memory neuron that best matches the input pattern. 564 Single-iteration Threshold Hamming Networks 565 The similarity subnet uses mn connections and performs a single iteration. The WTA sub net has m 2 connections. With randomly generated input and memory patterns, it converges in 8(m In(mn)) iterations (Floreen 1991). Since m is exponential in n, the space and time complexity of the network is primarily due to the WTA subnet (Domany & Orland 1987). We analyze the performance of the HN in the practical scenario where the input pattern is a distorted version of some stored memory vector. We show that it is possible to replace the original activation function of thf' neurons in the memory layer by a simple threshold function, and completely discard the WTA subnet. If the threshold is properly tuned, only the neuron standing for the 'correct' memory is likely to be activated. The resulting Threshold Hamming Network (THN) will perform correctly (with probability approaching 1) in a single iteration, using only O(m In m) connections instead of the O( m 2 ) connections in the original HN. We identify the optimal threshold, and measure its performance relative to the original HN. 2 The Threshold Hamming Network We examine a HN storing m + 1 memory patterns e", 1 ~ jJ ~ m + 1, each being an n-dimensional vector of ±1. The input pattern x is generated by selecting some memory pattern ~I-' (w.l.g., ~m+l), and letting each bit Xi be either ~f or -~f with probabilities a and (I - a) respectively, where a > 0.5. To analyze this HN, we use some tight. approximations to the binomial distribution. Due to space considerations, their proofs are omitted. Lemlna 1. Let X,...., Bin(n,p). If Xn are integers such that limn-+oo~ = /3 E (p, 1), then P(X > xn) ~ 1 - p exp{ -n[/3ln /3 + (1 _ /3) In 1 - /3]} (1 ~ )v211'n/3(1 - /3) p 1 - p in the sense that the ratio between LHS and RHS converges to 1 as n ~ 00. the special case p = ~, let G{/3) = In 2 + /31n/3 + (1- /3) In{1 - /3), then P(X> x ),...., -:--_e7'xp,-,{=--;:-=n=G::::(/3:::::)==} =. n ,...., (2 ~ )V211'n/3(1 - 13) Lenllna 2. (1) For (2) Let Xi ,...., Bin(n,~) be independent, 'Y E (0,1), and let Xn be as in Lemma 1. If m = (2 - ~)\h7rn/3(I- 13) (ln~) enG ({3), (3) then (4) Lenllna 3. Let y,...., Bin(n,Q') with a >~, let (Xi) and 'Y be as in Lemma 2, and let T} E (0,1). Let Xn be the integer closest to nf3, where /3=a_/a(l-a)z _~ V n 1) 2n (5) 566 Meilijson, Ruppin, and Sipper and zTj is the T] - quantile of the standard normal distribution, i.e., T] = -e- x /2dx 1 jZ'I 2 vf2; -00 Then, if Y and (Xd are independent P (max(X1 , X 2 ,"', Xm) < Y) 2:: P(max(X1 , X 2,"', Xm) < Xn ~ Y) => ,T] as n -+- 00, for m as in (3). (6) (7) Based on the above binomial probability approximations, we can now propose and analyze a n-neuron Threshold Hamming Network (THN) that classifies the input patterns with probability of error not exceeding f, when the input vector is generated with an initial bit-similarity a: Let Xi be the similarity between the input vector and the j'th memory pattern (1 ~ j < m), and let Y be the similarity with the 'correct' memory pattern ~m+l. Choose, and T] so that ,T] 2:: 1 f, e.g., , = T] = VI="f; determine f3 by (5) and m by (3). Discard the WTA subnet, and simply replace the neurons of the memory layer by m neurons having a threshold Xn , the integer closest to nf3. If any memory neuron with similarity at least Xn is declared 'the winner', then, by Lemma 3, the probability of error is at most f, where 'error' may be due to the existence of no winner, wrong winner, or multiple wmners. 3 The Hamming Network and an Optimal Threshold Hamming Network We now calculate the choice of the threshold Xn that maximizes the storage capacity m = men, f, a). Let </J (eI» denote the standard normal density (cumulative distribution function), and let r = </J/(l- eI» denote the corresponding failure rate function. Then, Lenuna 4. The opt.imal proportion between the two error probabilities is which we will denote by 8. Proof: 1 - , r(zTj) ---- ~ -r==~==~--~ 1 T] vna(1 - a) In 6 ' (8) Let AI = max(X1,X2,""Xm), and let Y denote the similarity with the 'correct' memory pattern, as before. We have seen that P(M < x) ~ exp{ -m J exp{ -nG({3)} 1 } . Since G'(f3) = In (1~ ~)' then by Taylor expansion 21!"n{3(1-{3)(2-~) I-' exp{-n[G(f3 + X-XO)]} P(AI<x)=P(M<xo+x-xo)~exp{-m n 1 }~ V211'nf3(1 - (3)(2 - fj) exp{ -nG(f3) - (x - xo) In (1~{3)} (-..L )",0-" exp{ -m V211'nf3(1 _ (3)(2 _ ~) } =, l-fj (9) Single-iteration Threshold Hamming Networks 567 (in accordance with Gnedenko extreme-value distribution of type 1 (Leadbetter et. al. 1983)). Similarly, P(Y < x) = exp{ln P(Y < Xo + x - xo)} ~ ~(z) x - Xo x- Xo P(Y < xo)exP{«1>$(z) Jna(l- a)} = (1-7])exp{r(z) Jna(1- a)} (10) where ~ is the standard normal density function, «1> is the standard normal cumulative distribution function, «1>$ = 1 «1> and r = -J. is the corresponding failure rate function. The probability of correct recognition using a threshold x can now be expressed as P(M < x)P(Y ~ x) = ,(6)"'0-"'(1- (1-7])exp{r(z) x - Xo }) (11) Jna(l- a) We differentiate expression (11) with respect to Xo - x, and equate the derivative at Xo = x to zero, to obtain the relation between , and 7] that yields the optimal threshold, i.e., that which maximizes the probability of correct recognition. This yields 1'(Z) 1 7] ,= exp{--} Jna(l-a)ln~ 7] We now approximate r(z) ( ) 1 - , ~ - In , ~ 1 7] Jna(l- a)ln 4 and thus the optimal proportion between the two error probabilities is 1 -; r(z) -- ~ = {yo 1 7] jna(1 - a) In ~ o Based on Lemma 4, if the desired probability of error is (, we choose {Jf. ,=I-I+{Y' _ 1t 7] (1 + {y) (12) (13) (14) (15) We start with, = 7] = ..;r=f, obtain {3 from (5) and {y from (8), and recompute 7] and, from (15). The limiting values of j3 and, in this iterative process give the maximal capacity m and threshold x n . We now compute the error probability t( m, n, a) of the original HN (with the WTA subnet) for arbitrary tn, n and a, and compare it with (. Lemma 5. For arbitrary n, a and t, let m, {3", 7] and {y be as calculated above. Then, the probability of error ((m, n, a) of the HN satisfies 1- e- 61n 6 {y6 ((m,n,a)~r(I-{Y) -.L ({y)lH(1+6 (16) {yIn 1-{3 1 + 568 Meiiijson, Ruppin, and Sipper where (17) is the Gamma function. Proof: P(Y ~ M) = LP(Y ~ x)P(M = x) = x LP(Y ~ x)[P(M < x+ 1) - P(M < x)] ~ x L P(Y ~ xo)e- 6(xo-x)ln 6 x (18) We now approximate this sum by the integral of the summand: let b = ~ and c = 6ln ~. We have seen that the probability of incorrect performance of the WTA subnet is equal to P(Y :S M) ~ L P(Y ~ xo)e-c(xo-x)[(P(M < xo))b(ro-r-l) - (P(M < xo))b(ro-r)] ~ x Now we transform variables t = bY In ~ to get the integral in the form This is the convergent difference between two divergent Gamma function integrals. ~e perform inte~rat~on by parts to obtain a representation as an integr~l wi~h rK2 mstead of t-(1+ 2) m the mtegrand. For 0 ~ K2 < 1, the correspondmg mtegral converges. The final result is then 1 - e- C c 1 c (1 - 7]) r(l - -)(1n -)1iib C In b 'Y (21 ) Hence, we have 1 -61n -1L 1 - e l-{J P(Y ~ M) ~ (1-7]) -L r(l- 6)(ln _)6 ~ 6ln l-f3 'Y 1 - e- 6ln 6 (f6)6 r(l - 6) -L (1 6)1+6 f 6In 1_/3 + (22) Single-iteration Threshold Hamming Networks 569 % error -+ predicted predicted experimental experimental" threshold , m THN HN THN HN ! 133 , 145 2.46 0.144 2.552 0.103 (1 'Y = 1.03 (1 'Y = 1.0 1 T/ = 1.46) 1 T/ = 1.552) 134 • 346 3.4 0.272 3.468 0.253 (1 'Y = 1.37 (1 'Y = 1.373 1-T/=2.1l) 1 T/ = 2.168) 135 , 825 4.714 0.494 4.152 0.485 (1 'Y = 1.776 (1 'Y = 1.606 1 11 = 2.991) 1 11 = 2.576) 136 , 1970 6.346 0.857 6.447 0.863 (1 'Y = 2.274 (1 'Y = 2.335 1 T/ = 4.167) 1 T/ = 4.162) Table 1: The performance of a HN and optimal THN: A comparison between calculated and experimental results (a = 0.7,n = 210). as claimed. Expression (22) is presented as K(f, 8, (3)f, where K(f, 8, (3) is the factor (:::; 1) by which the probability of error f of the THN should be multiplied in order to get the probability of error of the original HN with the WTA subnet. For small 8, K is close to 1, however, as will be seen in the next section, K is typically larger. 4 Numerical results The experimental results presented in table 1 testify to the accuracy of the HN and THN calculations. Figure 1 presents the calculated error probabilities for various values of input similarity a and memory capacity m, as a function of the input size n. As is evident, the performance of the THN is worse than that of the HN, but due to the exponential growth of m, it requires only a minor increment in n to obtain a THN that performs as well as the original HN. To examine the sensitivity of the THN network to threshold variation, we have fixed a = 0.7, n = 210, m = 825, and let the threshold vary between 132 and 138. As we can see in figure 2, the threshold 135 is indeed optimal, but the performance with threshold values of 134 and 136 is practically identical. The magnitude of the two error types varies considerably with the threshold value, but this variation has no effect on the overall performance near the optimum. These two error probabilities might as well be taken equal to each other. Conclusion In this paper we analyzed in detail the performance of a Hamming Network and a Threshold Hamming Network. Given a desired storage capacity and performance, we described how to compute the corresponding minimal network size required. The THN drastically reduces the time and connectivity requirements of Hamming Network classifiers. 570 Meilijson, Ruppin, and Sipper epsilon (error probability) epsilon ( error probability) epsilon (error probability) alpha=0.6,m=103 0.0001 0.0003 0.000 0.002 THN~ 0.007 HN -+0.14 0.37 800 1000 1200 1400 1600 1800 2000 2200 n (network size) alpha=0.7,m=106 0.0001 0.0003 0.000 THN~ HN -+0.14 0.37 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 n (network size) a1pha=0.8,m=109 0.0001....,.-----------r--------..,...---, 0.0003 0.000 0.37 160 180 200 220 240 260 n (network size) THN~ HN -+280 300 320 Figure 1: Probability of error as a function of network size: three networks are depicted, displaying the performance at various values of (}' and m. For graphical convenience, we have plotted log ~ versus n. Single-iteration Threshold Hamming Networks 571 10 9 8 7 6 % error 5 4 3 2 1 0 132 133 THN performance epsilon ~ 1 - gamma +1 - eta -e134 135 threshold 136 137 Figure 2: Threshold sensitivity of the THN (a = 0.7, n = 210, m = 825). References [1] K. Steinbuch. Dei lernmatrix. /(ybernetic, 1:36-45, 1961. 138 [2] \iV.K. Taylor. Cortico-thalamic organization and memory. Proc. of the Royal Society of London B, 159:466-478, 1964. [3] R.P. Lippmann, B. Gold, and M.L. Malpass. A comparison of Hamming and Hopfield neural nets for pattern classification. Technical Report TR-769, MIT Lincoln Laboratory, 1987. [4] E.E. Baum, J. Moody, and F. Wilczek. Internal representations for associative memory. Biological Cybernetics, 59:217-228, 1987. [5] P. Floreen. The convergence of hamming memory networks. IEEE Trans. on Neural Networks, 2(4):449-457, 1991. [6] E. Domany and H. Orland. A maximum overlap neural network for pattern recognition. Physics Letters A, 125:32-34,1987. [7] M.R. Leadbetter, G. Lindgren, and H. Rootzen. Extremes and related properties of random sequences and processes. Springer-Verlag, Berlin-HeidelbergNew York , 1983.
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Reinforcement Learning Applied to Linear Quadratic Regulation Steven J. Bradtke Computer Science Department University of Massachusetts Amherst, MA 01003 bradtke@cs.umass.edu Abstract Recent research on reinforcement learning has focused on algorithms based on the principles of Dynamic Programming (DP). One of the most promising areas of application for these algorithms is the control of dynamical systems, and some impressive results have been achieved. However, there are significant gaps between practice and theory. In particular, there are no con vergence proofs for problems with continuous state and action spaces, or for systems involving non-linear function approximators (such as multilayer perceptrons). This paper presents research applying DP-based reinforcement learning theory to Linear Quadratic Regulation (LQR), an important class of control problems involving continuous state and action spaces and requiring a simple type of non-linear function approximator. We describe an algorithm based on Q-Iearning that is proven to converge to the optimal controller for a large class of LQR problems. We also describe a slightly different algorithm that is only locally convergent to the optimal Q-function, demonstrating one of the possible pitfalls of using a non-linear function approximator with DP-based learning. 1 INTRODUCTION Recent research on reinforcement learning has focused on algorithms based on the principles of Dynamic Programming. Some of the DP-based reinforcement learning 295 296 Bradtke algorithms that have been described are Sutton's Temporal Differences methods (Sutton, 1988), Watkins' Q-Iearning (Watkins, 1989), and Werbos' Heuristic Dynamic Programming (Werbos, 1987). However, there are few convergence results for DP-based reinforcement learning algorithms, and these are limited to discrete time, finite-state systems, with either lookup-tables or linear function approximators. Watkins and Dayan (1992) show that the Q-Iearning algorithm converges, under appropriate conditions, to the optimal Q-function for finite-state Markovian decision tasks, where the Q-function is represented by a lookup-table. Sutton (1988) and Dayan (1992) show that the linear TD(A) learning rule, when applied to Markovian decision tasks where the states are representated by a linearly independent set of feature vectors, converges in the mean to Vu , the value function for a given control policy U. Dayan (1992) also shows that linear TD(A) with linearly dependent state representations converges, but not to Vu , the function that the algorithm is supposed to learn. Despite the paucity of theoretical results, applications have shown promise. For example, Tesauro (1992) describes a system using TD(A) that learns to play championship level backgammon entirely through self-playl. It uses a multilayer perceptron (MLP) trained using back propagation as a function approximator. Sofge and White (1990) describe a system that learns to improve process control with continuous state and action spaces. Neither of these applications, nor many similar applications that have been described, meet the convergence requirements of the existing theory. Yet they produce good results experimentally. We need to extend the theory of DP-based reinforcement learning to domains with continuous state and action spaces, and to algorithms that use non-linear function approximators. Linear Quadratic Regulation (e.g., Bertsekas, 1987) is a good candidate as a first attempt in extending the theory of DP-based reinforcement learning in this manner. LQR is an important class of control problems and has a well-developed theory. LQR problems involve continuous state and action spaces, and value functions can be exactly represented by quadratic functions. The following sections review the basics of LQR theory that will be needed in this paper, describe Q-functions for LQR, describe the Q-Iearning algorithm used in this paper, and describe an algorithm based on Q-Iearning that is proven to converge to the optimal controller for a large class of LQR problems. We also describe a slightly different algorithm that is only locally convergent to the optimal Q-function, demonstrating one of the possible pitfalls of using a non-linear function approximator with DP-based learning. 2 LINEAR QUADRATIC REGULATION Consider the deterministic, linear, time-invariant, discrete time dynamical system given by :Z:t+l f(:Z:t,Ut) A:Z:t + BUt Ut U :Z:t, where A, B, and U are matrices of dimensions n x n, n x m, and m x n respectively. :Z:t is the state of the system at time t, and Ut is the control input to the system at 1 Backgammon can be viewed as a Markovian decision task. Reinforcement Learning Applied to Linear Quadratic Regulation 297 time t. U is a linear feedback controller. The cost at every time step is a quadratic function of the state and the control signal: rt r(zt, ud x~Ext + u~Fut, where E and F are symmetric, positive definite matrices of dimensions n x nand m x m respectively, and Z' denotes z transpose. The value Vu (xe) of a state Zt under a given control policy U is defined as the discounted sum of all costs that will be incurred by using U for all times from t onward, i.e., Vi,(ze) = 2::o'Y'rt+i, where 0 :s: 'Y :s: 1 is the discount factor. Linear-quadratic control theory (e.g., Bertsekas, 1987) tells us that Vi, is a quadratic function of the states and can be expressed as Vu(zd = z~Kuzt, where Ku is the n x n cost matrix for policy U. The optimal control policy, U~, is that policy for which the value of every state is minimized. We denote the cost matrix for the optimal policy by K-. 3 Q-FUNCTIONS FOR LQR Watkins (1989) defined the Q-function for a given control policy U as Qu(z, u) = r(z, u) + 'YVu(f(x, u)). This can be expressed for an LQR problem as Qu(z, u) r(z, u) + 'YVu(f(z, u)) Zl Ez + u' Fu + 'Y(Az + BU)' Ku(Az + Bu) [ ]' [ E + 'YA' Ku A 'YA' Ku B 1 [ ] Z,U 'YB' Ku A F + 'YB' Ku B z, u , where [z,u] is the column vector concatenation of the column vectors z and u. Define the parameter matrix H u as H [E+'YAIKU A u 'YB' Ku A Hu is a symmetric positive definite matrix of dimensions (n + m) x (n + m). 4 Q-LEARNING FOR LQR (1) (2) The convergence results for Q-learning (Watkins & Dayan, 1992) assume a discrete time, finite-state system, and require the use of lookup-tables to represent the Q-function. This is not suitable for the LQR domain, where the states and actions are vectors of real numbers. Following the work of others, we will use a parameterized representation of the Q-function and adjust the parameters through a learning process. For example, Jordan and Jacobs (1990) and Lin (1992) use MLPs trained using backpropagation to approximate the Q-function. Notice that the function Qu is a quadratic function of its arguments, the state and control action, but it is a linear function of the quadratic combinations from the vector [z,u]. For example, if z = [Zb Z2], and 1.1. = [1.1.1], then Qu(z,u) is a linear function of 298 Bradtke the vector [x~, x~, ut, XIX2, XIUl, X2Ul]' This fact allows us to use linear Recursive Least Squares (RLS) to implement Q-Iearning in the LQR domain. There are two forms of Q-Iearning. The first is the rule \Vatkins described in his thesis (Watkins, 1989) . Watkins called this rule Q-Iearning, but we will refer to it as optimizing Q-Iearning because it attempts to learn the Q-function of the optimal policy directly. The optimizing Q-Iearning rule may be written as Qt+I(Xt, Ut) = Qt(:et, Ut) + a [r(:et, ut) + 'Y mJn Qt(:et+l, a) - Qt(:et, Ut)] , (3) where Qt is the tth approximation to Q". The second form of Q-Iearning attempts to learn Qu, the Q-function for some designated policy, U. U mayor may not be the policy that is actually followed during training. This policy-based Q-learning rule may be written as Qt+I (:et, Ut) = Qt(:et, Ut) + a [r( :et, Ut) + 'YQd :et+l, U :et+l) - Qt( :et, ue)] , (4) where Qt is the t lh approximation to Qu. Bradtke, Ydstie, and Barto (paper in preparation) show that a linear RLS implementation of the policy-based Q-Iearning rule will converge to Qu for LQR problems. 5 POLICY IMPROVEMENT FOR LQR Given a policy Uk, how can we find an improved policy, Uk+l? Following Howard (1960) , define Uk+l as Uk+lX = argmin [r(x, '1.£) + 'Y11ul< U(:e, '1.£))]. u But equation (1) tells us that this can be rewritten as Uk+I:e = argmin QUI< (:e, u). u We can find the minimizing '1.£ by taking the partial derivative of QUI«:e, u) with respect to '1.£, setting that to zero, and solving for u. This yields '1.£ = -'Y (F + 'YB' KUI<B)-l B' KUI<A:e. , ., V' UI<+l Using (2), Uk+l can be written as Uk+l = -H:;/ H21 . Therefore we can use the definition of the Q-function to compute an improved policy. 6 POLICY ITERATION FOR LQR The RLS implementation of policy-based Q-Iearning (Section 4) and the policy improvement process based on Q-functions (Section 5) are the key elements of the policy iteration algorithm described in Figure 1. Theorem 1, proven in (Bradtke, Reinforcement Learning Applied to Linear Quadratic Regulation 299 Y dstie, & Barto, in preparation), shows that the sequence of policies generated by this algorithm converges to the optimal policy. Standard policy iteration algorithms, such as those described by Howard (1960) for discrete time, finite state Markovian decision tasks, or by Bertsekas (1987) and Kleinman (1968) for LQR problems, require exact knowledge of the system model. Our algorithm requires no system model. It only requires a suitably accurate estimate of HUk • Theorem 1: If (1) {A, B} is controllable, (2) Un is stabilizing, and (3) the control signal, which at time step t and policy iteration step k is UJ,-Xt plus some "exploration factor", is strongly persistently exciting, then there exists a number N such that the sequence of policies generated by the policy iteration algorithm described in Figure 1 will converge to UX when policy updates are performed at most every N time steps. Initialize the Q-function parameters, HII • t = 0, k = o. do forever { } Initialize the Recursive Least Squares estimator. for i = 1 to N { } • Ut = UkXt + et, where et is the "exploration" component of the control signal. • Apply Ut to the system, resulting in state Xt+l. • Define at+l = UkXt+l. • Update the Q-function parameters, H k using the Recursive Least Squares implementation of the policy-based Q-learning rule, equation (4). • t=t+1. Policy improvement based on Hk : Initialize parameters Hk+l = Hk . k=k+1 Figure 1: The Q-function based policy iteration algorithm. It starts with the system in some initial state Xo and with some stabilizing controller Uo. k keeps track of the number of policy iteration steps. t keeps track of the total number of time steps. i counts the number of time steps since the last change of policy. vVhen i = N, one policy improvement step is executed. Figure 2 demonstrates the performance of the Q-function based policy iteration algorithm. We do not know how to characterize a persistently exciting exploratory signal for this algorithm. Experimentally, however, a random exploration signal generated from a normal distribution has worked very well, even though it does not meet condition (3) of the theorem. The system is a 20-dimensional discrete time approximation of a flexible beam supported at both ends. There is one control point. The control signal is a scalar representing acceleration to be applied at that point. Uo is an arbitrarily selected stabilizing controller for the system. Xo is a random 300 Bradtke point in a neighborhood around 0 E n20. \Ve used a normal random variable with mean 0 and variance 1 as the exploratory signal. There are 231 parameters to be estimated for this system, so we set N = 500, approximately twice that. Panel A of Figure 2 shows the norm of the difference between the current controller and the optimal controller. Panel B of Figure 2 shows the norm of the difference between the estimate of the Q-function for the current controller and the Q-function for the optimal controller. After only eight policy iteration steps the Q-function based policy iteration algorithm has converged close enough to U~ and Q~ that further improvements are limited by the machine precision. A 1 ... 03,...... ...... ..,......_ ........ -.......,. ....... _ ....... ...., '00 10 I '\ 0.1 \ 0.01 . = ,..0) i.\ .ill 1...04 ~ 1..05 1 ~::: \. 1..06 1...09 \ :::~ \, 1 •• 2 ~' ''~~ ........... ----' ... u y ~ ---- -. 1~1" O!--.....-.~IO----=2'::-0 ............ --f::10 ......... ------!40::--~30 k. number of poIi"" ileration ste,. B 10+00,...... ............... _-.-_ ....... _-...._........, 100 \ 10 I \ = I~! i\ .. , '..0< ~ 1..03 :::: \\ & 1..01 1..09 L ~ _ ~ ~ ~/'.A~ . 1 .. 10 -- ~ -- "v-'.... ..""'-"1 , .. u 1,.-12 1 .. 13 1 .. 14 O!--.....-."7::IO---:20::---30:!::--~""~'--!30 k. number of poll"" Iteration .t~ Figure 2: Performance of the Q-function based policy iteration algorithm on a discretized beam system. 7 THE OPTIMIZING Q-LEARNING RULE FOR LQR Policy iteration would seem to be a slow method. It has to evaluate each policy before it can specify a new one. Why not do as VVatkins' optimizing Q-Iearning rule does (equation 3), and try to learn Q- directly? Figure 3 defines this algorithm precisely. This algorithm does not update the policy actually used during training. It only updates the estimate of Q-. The system is started in some initial state :Z:o and some stabilizing controller Uo is specified as the controller to be used during training. To what will this algorithm converge, if it does converge? A fixed point of this algorithm must satisfy [ ]' [ H 11 H 12] [ ] :z:, u H21 H22 :Z:, u = :z:'E:z:+u'Eu+'Y[A:z:+Bu,a]' [~~~ ~~~] [A:z:+Bu,a), (5) where a = -H:;/ H21(A:z:+Bu). Equation (5) actually specifies (n+m)(n+m+ 1)/2 polynomial equations in (n + m)(n + m + 1)/2 unknowns (remember that Hu is symmetric). We know that there is at least one solution, that corresponding to the optimal policy, but there may be other solutions as well. As an example of the possibility of multiple solutions, consider the I-dimensional system with A = B = E = F = [1) and l' = 0.9. Substituting these values into Reinforcement Learning Applied to Linear Quadratic Regulation 301 Initialize the Q-function parameters, ilu. Initialize Recursive Least Squares estimator. t = o. do forever { } • Ut = UOXt + et, where et is the "exploration" component of the control signal. • Apply Ut to the system, resulting in state Xt+ 1. A -1 A • Define at+1 = -H22 H 21 Xt+1. • Update the Q-function parameters, fIt, using the Recursive Least Squares implementation of the optimizing Q-Iearning rule, equation (3). • t=t+1. Figure 3: The optimizing Q-learning rule in the LQR domain. Uo is the policy followed during training. t keeps track of the total number of time steps. equation (5) and solving for the unknown parameters yields two solutions. They are [ 2.4296 1.4296 1.4296] d [ 0.3704 2.4296 an -0.6296 -0.6296] 0.3704 . The first solution is Q-. The second solution, if used to define an "improved" policy as describe in Section 5, results in a destablizing controller. This is certainly not a desirable result. Experiments show that the algorithm in Figure 3 will converge to either of these solutions if the initial parameter estimates are close enough to that solution. Therefore, this method of using Watkins' Q-learning rule directly on an LQR problem will not necessarily converge to the optimal Q-function. 8 CONCLUSIONS In this paper we take a first step toward extending the theory of DP-based reinforcement learning to domains with continuous state and action spaces, and to algorithms that use non-linear function approximators. We concentrate on the problem of Linear Quadratic Regulation. We describe a policy iteration algorithm for LQR problems that is proven to converge to the optimal policy. In contrast to standard methods of policy iteration, it does not require a system model. It only requires a suitably accurate estimate of Hu/c. This is the first result of which we are aware showing convergence of a DP-based reinforcement learning algorithm in a domain with continuous states and actions. We also describe a straightforward implementation of the optimizing Q-Iearning rule in the LQR domain. This algorithm is only locally convergent to Q-. This result demonstrates that we cannot expect the theory developed for finite-state systems using lookup-tables to extend to continuous state systems using parameterized function representations. 302 Bradtke The convergence proof for the policy iteration algorithm described in this paper requires exact matching between the form of the Q-function for LQR problems and the form of the function approximator used to learn that function. Future work will explore convergence of DP-based reinforcement learning algorithms when applied to non-linear systems for which the form of the Q-functions is unknown. Acknowledgements The author thanks Andrew Barto, B. Erik Ydstie, and the ANW group for their contributions to these ideas. This work was supported by the Air Force Office of Scientific Research, Bolling AFB, under Grant AFOSR-89-0526 and by the National Science Foundation under Grant ECS-8912623. References [1] D. P. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, Englewood Cliffs, N J, 1987. [2] S. J. Bradtke, B. E. Ydstie, and A. G. Barto. Convergence to optimal cost of adaptive policy iteration. In preparation. [3] P. Dayan. The convergence ofTD(A) for general A. Machine Learning, 1992. [4] R. A. Howard. Dynamic Programming and Markov Processes. John Wiley & Sons, Inc., New York, 1960. [5] M. 1. Jordan and R. A. Jacobs. Learning to control an unstable system with forward modeling. In Advances in Neural Information Processing Systems 2. Morgan Kaufmann Publishers, San Mateo, CA, 1990. [6] D. L. Kleinman. On an iterative technique for Riccati equation computations. IEEE Transactions on Automatic Control, pages 114-115, February 1968. [7] L.-J. Lin. Self-improving reactive agents based on reinforcement learning, planning and teaching. Machine Learning, 1992. [8] D. A. Sofge and D. A. White. Neural network based process optimization and control. In Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, Hawaii, December 1990. [9] R. S. Sutton. Learning to predict by the method of temporal differences. Alachine Learning, 3:9-44, 1988. [10] G. J. Tesauro. Practical issues in temporal difference learning. Machine Learning, 8{3/4):257-277, May 1992. [11] C. J. C. H. Watkins. Learning from Delayed Rewards. PhD thesis, Cambridge University, Cambridge, England, 1989. [12] C. J. C. H. Watkins and P. Dayan. Q-Iearning. Machine Learning, 1992. [13] P. J. Werbos. Building and understanding adaptive systems: A statistical/numerical approach to factory automation and brain research. IEEE Transactions on Systems, Man, and Cybernetics, 17(1):7-20, 1987.
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The Computation of Stereo Disparity for Transparent and for Opaque Surfaces Suthep Madarasmi Computer Science Department University of Minnesota Minneapolis, MN 55455 Daniel Kersten Department of Psychology University of Minnesota Ting-Chuen Pong Computer Science Department University of Minnesota Abstract The classical computational model for stereo vision incorporates a uniqueness inhibition constraint to enforce a one-to-one feature match, thereby sacrificing the ability to handle transparency. Critics of the model disregard the uniqueness constraint and argue that the smoothness constraint can provide the excitation support required for transparency computation. However, this modification fails in neighborhoods with sparse features. We propose a Bayesian approach to stereo vision with priors favoring cohesive over transparent surfaces. The disparity and its segmentation into a multi-layer "depth planes" representation are simultaneously computed. The smoothness constraint propagates support within each layer, providing mutual excitation for non-neighboring transparent or partially occluded regions. Test results for various random-dot and other stereograms are presented. 1 INTRODUCTION The horizontal disparity in the projection of a 3-D point in a parallel stereo imaging system can be used to compute depth through triangulation. As the number of 385 386 Madarasmi, Kersten, and Pong points in the scene increases, the correspondence problem increases in complexity due to the matching ambiguity. Prior constraints on surfaces are needed to arrive at a correct solution. Marr and Poggio [1976] use the smoothness constraint to resolve matching ambiguity and the uniqueness constraint to enforce a 1-to-1 match. Their smoothness constraint tends to oversmooth at occluding boundaries and their uniqueness assumption discourages the computation of stereo transparency for two overlaid surfaces. Prazdny [1985] disregards the uniqueness inhibition term to enable transparency perception. However, their smoothness constraint is locally enforced and fails at providing excitation for spatially disjoint regions and for sparse transparency. More recently, Bayesian approaches have been used to incorporate prior constraints (see [Clark and Yuille, 1990] for a review) for stereopsis while overcoming the problem of oversmoothing. Line processes are activated for disparity discontinuities to mark the smoothness boundaries while the disparity is simultaneously computed. A drawback of such methods is the lack of an explicit grouping of image sites into piece-wise smooth regions. In addition, when presented with a stereogram of overlaid (transparent) surfaces such as in the random-dot stereogram in figure 5, multiple edges in the image are obtained while we clearly perceive two distinct, overlaid surfaces. With edges as output, further grouping of overlapping surfaces is impossible using the edges as boundaries. This suggests that surface grouping should be performed simultaneously with disparity computation. 2 THE MULTI-LAYER REPRESENTATION We propose a Bayesian approach to computing disparity and its segmentation that uses a different output representation from the previous, edge-based methods. Our representation was inspired by the observations of Nakayama et al. [1989] that midlevel processing such as the grouping of objects behind occluders is performed for objects within the same "depth plane" . As an example consider the stereogram of a floating square shown in figure 1a. The edge-based segmentation method computes the disparity and marks the disparity edges as shown in figure lb. Our approach produces two types of output at each pixel: a layer (depth plane) number and a disparity value for that layer. The goal of the system is to place points that could have arisen from a single smooth surface in the scene into one distinct layer. The output for our multi-surface representation is shown in figure 1c. Note that the floating square has a unique layer label, namely layer 4, and the background has another label of 2. Layers 1 and 3 have no data support and are, therefore, inactive. The rest of the pixels in each layer that have no data support obtain values by a membrane fitting process using the computed disparity as anchors. The occluded parts of surfaces are, thus, represented in each layer. In addition, disjoint regions of a single surface due to occlusion are represented in a single layer. This representation of occluded parts is an important difference between our representation and a similar representation for segmentation by Darrell and Pentland [1991]. 3 (a) (b) The Computation of Stereo Disparity for Transparent and for Opaque Surfaces 387 Il1IIII -Layer 4 ~ -Layer 2 Layer 4 Figure I: a) A gray scale display of a noisy stereogram depicting a floating square. b. Edge based disp. = 0 method: disparity computed and disparity discontinuity computed. c. MultiSurface method: disparity computed, surface grouping . performed by layer assigndlsp. = 4 d d' . f h ment. an lspanty or eac layer filled in. ALGORITHM AND SIMULATION METHOD We use Bayes' [1783] rule to compute the scene attribute, namely disparity u and its layer assignment 1 for each layer: ( IldL dR) = p(dL,dRlu, I)p(u, I) p u,' p( dL , dR ) where dL and dR are the left and right intensity image data. Each constraint is expressed as a local cost function using the Markov Random Field (MRF) assumption lGeman and Geman, 1984], that pixels values are conditional only on their nearest neighbors. Using the Gibbs-MRF equivalence, the energy function can be written as a probability function: 1 E(.,) p(x) = -e-"Z where Z is the normalizing constant, T is the temperature, E is the energy cost function, and x is a random variable Our energy constraints can be expressed as E = >'D VD + >'s Vs + >'G VG + >'E VE + AR VR where the>. 's are the weighting factors and the VD, Vs, VG, VE, VR functions are the data matching cost, the smoothness term, the gap term, the edge shape term, and the disparity versus intensity edge coupling term, respectively. The data matching constraint prefers matches with similar intensity and contrast: VD = t [Idr - dfl +.., .2: I(df - dr) - (d~ - df)l] , JENi with the image indices k and m given by the ordered pairs k = (row(i), col(i)+uC,i), m = (row(j) , col(j) + UCii), M is the number of pixels in the image, Ci is the layer classification for site i, and Uli is the disparity at layer I. The.., weighs absolute intensity versus contrast matching. The >'D is higher for points that belong to unambiguous features such as straight vertical contours, so that ambiguous pixels rely more on their prior constraints. 388 Madarasmi, Kersten, and Pong cost (b) depth difference depth difference Figure 2: Cost function V s. a) The smoothness cost is quadratic until the disparity difference is high and an edge process is activated. b) In our simulations we use a threshold below which the smoothness cost is scaled down and above which a different layer assignment is accepted at a constant high cost. Also, if neighboring pixels have a higher disparity than the current pixel and are in a different layer, its )..D is lowered since its corresponding point in the left image is likely to be occluded. The equation for the smoothness term is given by: M L Vs = LL L V,(uu, U'j)a, i 1 jEN. where, Ni are the neighbors of i, V, is the local smoothness potential, a, is the activity level for layer I defined by the percent of pixels belonging to layer I, and L is the number layers in the system. The local smoothness potential is given by: if (a - b)2 < Tn otherwise where JJ is the weighting term between depth smoothness and directional derivative smoothness. The ~k is the difference operation in various directions k, and T is the threshold. Instead of the commonly used quadratic smoothness function graphed in figure 2a, we use the (7 function graphed in figure 2b which resembles the Ising potential. This allows for some flexibility since )..5 is set rather high in our simulations. The VG term ensures a gap in the values of corresponding pixels between layers: This ensures that if a site i belongs to layer C., then all points j neighboring i for each layer 1 must have different disparity values ulj than uCia' The edge or boundary shape constraint VE incorporates two types of constraints: a cohesive measure and a saliency measure. The costs for various neighborhood configurations are given in figure 3. The constraint VR ensures that if there is no edge in intensity then there should be no edge in the disparity. This is particularly important to avoid local minima for gray scale images since there is so much ambiguity in the matching. The Computation of Stereo Disparity for Transparent and for Opaque Surfaces 389 • - same layer label D -different layer label cost = 0 cost = 0.2 cost = 0.25 cost == 0.5 cost == 0.7 cost == I Figure 3: Cost function VE. The costs associated nearest neighborhood layer label con~gurations. a) Fully cohesive region (lowest cost) b) Two opaque regions with straight hne boundary. c) Two opaque regions with diagonal line boundary. d) Opaque regions with no figural continuity. e) Transparent region with dense samplings. f) Transparent region with no other neighbors (highest cost). Layer 3 layer labels Wire-frame plot of Layer 3 Figure 4: Stereogram of floating cylinder shown in crossed and uncrossed disparity. Only disparity values in the active layers are shown. A wireframe rendering for layer 3 which captures the cylinder is shown. The Gibbs Sampler [Geman and Geman, 1984] with simulated annealing is used to compute the disparity and layer assignments. After each iteration of the Gibbs Sampler, the missing values within each layer are filled-in using the disparity at the available sites. A quadratic energy functional enforces smoothness of disparity and of disparity difference in various directions. A gradient descent approach minimizes this energy and the missing values are filled-in. 4 SIMULATION RESULTS After normalizing each of the local costs to lie between 0 and 1, the values for the weighting parameters used in decreasing order are: .As, .AR, .AD, .AE,.AG with the .AD value moved to follow .AG if a pixel is partially occluded. The results for a randomdot stereogram with a floating half-cylinder are shown in figure 4. Note that for clarity only the visible pixels within each layer are displayed, though the remaining pixels are filled-in. A wire-frame rendering for layer 3 is also provided. Figure 5 is a random-dot stereogram with features from two transparent frontoparallel surfaces. The output consists primarily of two labels corresponding to the foreground and the background. Note that when the stereogram is fused, the percept is of two overlaid surfaces with various small, noisy regions of incorrect matches. Figure 6 is a random-dot stereogram depicting many planar-parallel surfaces. Note 390 Madarasmi, Kersten, and Pong Figure 5: Random-dot stereogram of two overlaid surfaces. Layers 1 and 4 are the mostly activated layers. Only 5 of the layers are shown here. ")51 " . , ·.ii layer labeb Figure 6: Random-dot stereogram of multiple flat surfaces. Layers 4 captures two regions since they belong to the same surface (equal disparity). ... ~ _ _ 1 - ~ -~ Layer I - _ - -iJ2~JI!5iw ~4j. -, 7 An Layer 2 -Layer 3 - _ ..... ........Laye~o;II~_:-~_IIIII;;C;=;:;F;_~.~2i:4::3="":;;"--~~ Layer 5 .. ). layer labels that there are two disjoint regions which are classified into the same layer since they form a single surface. A gray-scale stereogram depicting a floating square occluding the letter 'C' also floating above the background is shown in figure 7. A feature-based matching scheme is bound to fail here since locally one cannot correctly attribute the computed disparity at a matched corner of the rectangle, for example, to either the rectangle, the background, or to both regions. Our VR constraint forces the system to attempt various matches until points with no intensity discontinuity have no disparity discontinuity. Another important feature is that the two ends of the letter 'C' are in the same "depth plane" [Nakayama et al., 1989] and may later be merged to complete the letter. Figure 8 is a gray scale stereogram depicting 4 distant surfaces with planar disparity. At occluding boundaries, the region corresponding to the further surface in the right image has no corresponding region in the left image. A high .AD would only force these points to find an incorrect match and add to the systems errors. The.AD reduction factor for partially occluded points reduces the data matching requirement for such points. This is crucial for obtaining correct matches especially since the images are sparsely textured and the dependence on accurate information from the textured regions is high. A transparency example of a fence in front a bill-board is given in figure 9. Note
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Forecasting Demand for Electric Power Jen-Lun Yuan and Terrence L. Fine School of Electrical Engineering Cornell University Ithaca, NY 14853 Abstract We are developing a forecaster for daily extremes of demand for electric power encountered in the service area of a large midwestern utility and using this application as a testbed for approaches to input dimension reduction and decomposition of network training. Projection pursuit regression representations and the ability of algorithms like SIR to quickly find reasonable weighting vectors enable us to confront the vexing architecture selection problem by reducing high-dimensional gradient searchs to fitting single-input single-output (SISO) subnets. We introduce dimension reduction algorithms, to select features or relevant subsets of a set of many variables, based on minimizing an index of level-set dispersions (closely related to a projection index and to SIR), and combine them with backfitting to implement a neural network version of projection pursuit. The performance achieved by our approach, when trained on 1989, 1990 data and tested on 1991 data, is comparable to that achieved in our earlier study of backpropagation trained networks. 1 Introduction Our work has the intertwined goals of: (i) contributing to the improvement of the short-term electrical load (demand) forecasts used by electric utilities to buy and sell power and ensure that they can meet demand; 739 740 Yuan and Fine (ii) reducing the computational burden entailed in gradient-based training of neural networks and thereby enabling the exploration of architectures; (iii) improving prospects for good statistical generalization by use of rational methods for reducing complexity through the identification of good small subsets of variables drawn from a large set of candidate predictor variables (feature selection); (iv) benchmarking backpropagation and neural networks as an approach to the applied problem of load forecasting. Our efforts proceed in the context of a problem suggested by the operational needs of a particular electric utility to make daily forecasts of short-term load or demand. Forecasts are made at midday (1 p.m.) on a weekday t ( Monday - Thursday), for the next evening peak e(t) (occuring usually about 8 p.m. in the winter), the daily minimum d(t + 1) (occuring about 4 a.m. the next morning) and the morning peak m( t + 1) (about noon ). In addition, on Friday we are to forecast these three variables for the weekend through the Monday morning peak. These daily extremes of demand are illustrated in an excerpt from our hourly load data plotted in Figure 1. 4600 4400 f 4200 ~ i J 4000 3600 3400 0 5am 5 lam I~am 10 15 ' lam Rpm 6am Ilam 9jlm 20 2S 30 35 40 45 50 number of houri Figure 1: Hourly demand for two consecutive days showing the intended forecasting variables. In this paper, we focus on forecasting these extremal demands up to three days ahead (e.g. forecasting on Fridays). Neural network-based forecasters are developed which parallel the recently proposed method of slicing inverse regression (SIR) (Li [1991]) and then use backfitting (Hastie and Tibshirani [1990]) to implement a training algorithm for a projection pursuit model (Friedman [1987]' Huber [1985]) that can be implemented with a single hidden layer network. Our data consists of hourly integrated system demand (MWH) and hourly temperatures measured at three cities in the service area of a large midwestern utility during 1989-91. We use 1989 and 1990 for a training set and test over the whole of 1991, with the exception of holidays that occur so infrequently that we have no training base. Forecasting Demand for Electric Power 741 2 Baseline Performance 2.1 Previous Work on Load Forecasting Since demand is a process which does not have a known physical or mathematical model, we do not know the best achievable forecasting performance, and we are led to making comparisons with methods and results reported elsewhere. There is a substantial literature on short-term load forecasting, with Gross et al. [1987] and Willis et al. [1984] providing good reviews of approaches based upon such statistical methods as linear least squares regression -1Ild Box-Jenkins and ARMAX time series models. Many utilities rely upon the seemingly seat-of-the-pants estimates produced by individuals who have been long employed at this task and who extrapolate from a large historical data base. In the past few years there have been several efforts to employ neural networks trained through backpropagation. In two such recent studies conducted at the Univ. of Washington an average peak error of 2.04% was reported by Damborg et al. [1990] and an hourly load error of about 2.2% was given by Connor et al. [1991]. However, the accuracies reported in the literature are difficult to compare with since utilities are exposed to different operating conditions (e.g., weather, residential/industrial balance). To provide a benchmark for the error performance achieved by our method, we evaluated three basic forecasting models on our data. These methods are based on a pair of features made plausible by the scatter plots shown in Figure 2. 5000~~--~---r--~ 4800 . , 0; : 0 .. ~ ,It . ~ . . . . . : ..... : : • ~ 0° 4600 o •• ' .... ~ • •••••• '. 4b-:~. " ":" ... ", ... , ...... . :c 4400 ~ ~ ~ 4200 4000 3800 .. .. " D. : aD 10 : ..... , ... ~ .... .,..,-:JO" " " ....... ........ . . . 3~~'-OO--4000 .......... ~4-:'-5oo-:---:-5000"":---S-:'5oo m(t)MWH 50oo~--......----'r----....., 4800 '0 . . • • '," , , 4600 _ ..... .......... .. :"" .. : ....... ; . . . .Ib. ~ . \ . '.f .. ; :?' ~ 4400 ........... .. , ... ~ .... 1. .. ~ ...... ~ .. ~ ~ . ~, ~ 4200 .~.t"". 3800 . , .. • • " 3~'----~--~~--~ -SO o so 100 temperalUre(t) oP Figure 2: Evening peaks ('IUe.-Fri.,1989-90) vs. morning peaks and temperatures. 2.2 Feature Selection and Homogeneous Data Types Demand depends on predictable calendar factors such as the season, day-of-theweek and time-of-day considerations. We grouped separately Mondays, 'IUesdays through Fridays, Saturdays, and Sundays, as well as holidays. In contrast to all of the earlier work on this problem, we ignored seasonal considerations and let the network and training algorithm adjust as needed. The advantage of this was the ability to form larger training data sets. We thus constructed twelve networks, one 742 Yuan and Fine type m(t+1) e(t) d(t+1) Monday m(t-3) m(t) d(t-3) Tue.-Fri. m(t-1) m(t) d(t-1) Saturday m(t-1) m(t-1) d(t-1) Sunday m(t-2} m(t-2) d(t- 2) Table 1: Most recent peaks of a two-feature set type m(t+1} e(t) d(t+1) LLS LOESS BP LLS LOESS BP LLS LOESS BP Monday 3.78 2.45 2.42 1.73 2.43 1.59 4.40 3.30 2.69 The.-Fri. 3.01 2.44 1.98 1.89 3.04 1.65 3.29 3.81 2.49 Saturday 3.37 2.60 2.36 4.54 3.76 3.10 3.48 3.25 2.06 Sunday 4.83 3.28 3.79 4.89 2.74 3.81 4.26 2.44 3.03 Table 2: Forecasting accuracies (percentage absolute error) for three basic methods for each pair consisting of one of these four types of days and one of the three daily extremes to be forecast. Demand also depends heavily upon weather which is the primary random factor affecting forecasts. This dependency can be seen in the scatter plots of current demand vs. previous demand and temperature in Figure 2, particularly in the projection onto the 'current demand-temperature' plane which shows a pronounced "U"-shaped nonlinearity. A two-feature set consisting of the most recent peaks and average temperatures over the three cities and the preceding six hours is employed for testing all three models (Table 1). 2.3 Benchmark Results The three basic forecasting models using the two-featured set are: 1) linear regression model fitted to the data in Figure 2; 2) demand vs. temperature models which roughly model the "U-shaped" nonlinear relationship, (LOESS with .5 span was employed for scatter plot smoothing); 3) backpropagation trained neural networks using 5 logistic nodes in a single hidden layer. The test set errors are given in Table 2. Note that among these three models, BP-trained neural networks gives superior test set performance on all but Sundays. These models all give results comparable to those obtained in our earlier work on forecasting demands for Thesday-Friday using autoregressive neural networks (Yuan and Fine [1992]). Forecasting Demand for Electric Power 743 3 Projection Pursuit Training Satisfactory forecasting performance of the neural networks described above relies on the appropriate choice of feature sets and network architectures. Unfortunately, BP can only address the problem of appropriate architecture and relevant feature sets through repeated time-consuming experiments. Modeling of high-dimensional input features using gradient search requires extensive computation. We were thus prompted to look at other network structures and at training algorithms that could make it easier to explore architecture and training problems. Our initial attempt combined the dimension reduction algorithm cf SIR (Li [1991]), currently replaced by an algorithm of our devising sketched in Section 4, and backfitting (Hastie et.al [1990]) to implement a neural network version of projection pursuit regression (PPR). 3.1 The Algorithm A general nonlinear regression model for a forecast variable y in terms of a vector x of input variables and model noise €, independent of x, is gi ven by y = f({3ix,{3~x, .. ,{3~X,€) (*). A least mean square predictor is the conditional expectation E(ylx). The projection pursuit model/approximation of this conditional expectation is given in terms of a family of SISO functions 2 1,22, ", 2k by k E(ylx) = I: 2i ({3:x) + {3o. i=l A single hidden layer neural network can approximate this representation by introducing subnets whose summed outputs approximate the individual 2 j . We train such a 'projection pursuit network' with nodes partitioned into subnets, representing the 2i , by training the subnets individually in rotation. In this we follow the statistical regression notion of backfitting. The subnet 2i is trained to predict the residuals resulting from the difference between the weighted outputs of the other k - 1 subnets and the true value of the demand variable. After a number of training cycles one then proceeds to the next subnet and repeats the process. The inputs to each subnet 2i are the low-dimensional projections {3:x of the regression model. One termination criteria for determining the number of subnets k is to stop adding subnets when the projection appears to be normally distributed; results of Diaconis and Freedman point out that 'most' projections will be so distributed and thus are 'uninteresting'. The directions {3i can be found by minimizing some projection index which defines interesting projections that deviates from Gaussian distributions (e.g., Friedman [1987]). Each {3i determines the weights connecting the input to sub net 2 i .The whole projection pursuit regression process is simplified by decoupling the direction {3 search from training the SISO subnets. Albeit, its success depends upon an ability to rapidly discern the significant directions {3i. 3.2 Implementations There are several variants in the implementation of projection pursuit training algorithms. General PPR procedure can be implemented in one stage by computa744 Yuan and Fine type m(t+l) e(t) d(t+1) Monday 2.35/3.45 1.25/1.60 2.76/3.49 Tue.-Fri. 2.37/2.83 1.65/1.66 2.15/2.66 Saturday 2.67/3.16 2.78/3.96 2.57/3.04 Sunday 3.15/5.38 2.63/3.67 2.29/3.61 Table 3: Forecasting performance (training/testing percentage error) of projection pursuit trained networks tionally intensive numerical methods, or in a two-stage heuristic (finding f3i, then Bi) as proposed here. It can be implemented with or without back fitting after the PPR phase is done. Intrator [1992] has recently suggested incorporating the projection index into the objective function and then running an overall BPA. Other variants in training each Bi net include using nonparametric smoothing techniques such as LOESS or kernel methods. BP training can then be applied only in the last stage to fit the smoothed curves so obtained. The complexity of each subnet is then largely determined by the smoothing parameters, like window sizes, inherent in most nonparametric smoothing techniques. Another practical advantage of this process is that one can incorporate easily fixed functions of a single variable (e.g. linear nodes or quadratic nodes) when one's prior knowledge of the data source suggests that such components may be present. Our current implementation employs the two-stage algorithm with simple (either one or two nodes) logistic Bi subnets. Each SISO Bi net runs a BP algorithm to fit the data. The directions f3i are calculated based on minimizing a projection index (dispersion of level-sets, described in Section 4) which can be executed in a direct fashion. One can encourage the convergence of backfitting by using a relaxation parameter (like a momentum parameter in BPA ) to control the amount updated in the current direction. Training (fitting) of each (SISO) 3 i net can be carried out more efficiently than running BP based on high-dimensional inputs, for example, it is less expensive to evaluate the Hessian matrices in a Bi net than in a full BPA networks. 3.3 Forecasting Results Experimental results were obtained using the two-component feature data sets which gave the earlier baseline performance. To calibrate the performance we employed in all twelve projection pursuit trained networks an uniform architecture of three subnets ( a (1,2, 2)-logistic network), matching the 5 nodes of the BP network of Section 2. The number of backfitting cycles was set to 20 with a relaxation parameter w = 0.1. BPA was employed for fitting each Binet. The training/testing percentage absolute errors are given in Table 3. The limited data sets in the cases of individual days (Monday, Saturday, Sunday) led to failure in generalization that could have been prevented by using one or two, rather than three, subnets. Forecasting Demand for Electric Power 745 4 Dimension Reduction 4.1 Index of Level-Set Dispersion A key step in the projection pursuit training algorithm is to find for each 3 i net the projection direction f3i' an instance of the important problem of economically choosing input features/variables in constructing a forecasting model. In general, the fewer the number of input features, the more likely are the results to generalize from training set performance to test set performance- reduction in variance at the possible expense of increase in bias. Our controlled size subnet projection pursuit training algorithm deals with part of the complexity problem, provided that the input features are fixed. We turn now to our approach to finding input features or search directions based on minimizing an index of dispersion of level-sets. Li [1991] proposed taking an inverse ('slicing the y's') point of view to estimate the directions f3i. The justification provided for this so-called slicing inverse regression (SIR) method, however, requires that the input or feature vector x be elliptically symmetrically distributed, and this is not likely to be the case in our electric load forecasting problem. The basic idea behind minimizing dispersion of level-sets is that from Eq. (*) we see that a fixed value of y, and small noise £, implies a highly constrained set of values for f3ix, ... ,f3~x, while leaving unconstrained the components of x that lie in the subspace B~ orthogonal to that space B spanned by the f3i.. Hence, if one has a good number of i.i.d. observations sharing a similar value of the response y, then there should be more dispersion of input vectors projected into Bl.. than along the projections into B. We implement this by quantizing the observed y values into, say, H slices, with Lh denoting the h_th level-set containing those inputs with y-value in the h_th slice, and X-h is their sample mean. The f3 are then picked as the the eigenvector associated with the smallest eigenvalue of the centered covariance matrix: H L L (Xi - x"h)(Xi. - Xh)'. h=l xiELh 4.2 Implementations In practical implementations, one may discard both extremes of the family of H level sets (trimming) to avoid large response values when it is believed that they may correspond to large magnitudes of input components. One should also standardize initially the input data to a unit sample covariance matrix. Otherwise, our results will reflect the distribution of x rather than the functional relationship of Eq. (*). We have applied this projection index both in finding the f3i. during projection pursuit training and in reducing a high-dimensional feature set to a lowdimensional feature set. We have implemented such a feature selection scheme for forecasting the Monday - Friday evening peaks. The initial feature set consists of thirteen hourly loads from lam to 1pm, thirteen hourly temperatures from lam to 1pm and the temperature around the peak times. Three eigenvectors of the centered covariance matrix were chosen, thereby reducing a 27-dimensional feature set to a 3-dimensional one. We then ran a standard BPA on this reduced featured set and tested on the 1991 data. We obtained a percentage absolute error of 1.6% (rms error about 100 MWH), which is as good as all of our previous efforts. 746 Yuan and Fine Acknow ledgements Partial support for this research was provided by NSF Grant No. ECS-9017493. We wish to thank Prof. J. Hwang, Mathematics Department, Cornell, for initial discussions of SIR and are grateful to Dr. P.D. Yeshakul, American Electric Service Corp., for providing the data set and instructing us patiently in the lore of shortterm load forecasting. References Connor, J., L. Atlas, R.D. Martin [1991], Recurrent networks and NARMA modeling, NIPS 91. Damborg, M., M.EI-Sharkawi, R. Marks II [1990], Potential of artificial neural networks in power system operation, Proc. 1990 IEEE Inter.Symp. on Circuits and Systems, 4, 2933-2937. Friedman, J. [1987], Exploratory projection pursuit, J. Amer. Stat. Assn., 82, 249-266. Gross,G., F. Galiana [1987], Short-term load forecasting, Proc. IEEE, 75, 15581573. Hastie, T., R. Tibshirani [1990], Generalized Additive Models,Chapman and Hall. Huber, P. [1985]' Projection pursuit, The Annals of Statistics, 13, 435-475. Intrator, N. [1992] Combinining exploratory projection pursuit and projection pursuit regression with applicatons to neural networks, To appear in Neural Computation. Li, K.-C. [1991] Slicing inverse regression for dimension reduction, Journal of American Statistical Assoc., 86. Willis, H.1., J.F.D. Northcote-Green [1984], Comparison tests of fourteen load forecasting methods, IEEE Trans. on Power Apparatus and Systems, PAS-103, 1190-1197. Yuan, J-1., T.L.Fine [1992]' Forecasting demand for electric power using autoregressive neural networks, Proc. Con! on Info. Sci. and Systems, Princeton, NJ.
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Visual Motion Computation in Analog VLSI using Pulses Rahul Sarpeshkar, Wyeth Bair and Christof Koch Computation and Neural Systems Program California Institute of Technology Pasadena, CA 91125. Abstract The real time computation of motion from real images using a single chip with integrated sensors is a hard problem. We present two analog VLSI schemes that use pulse domain neuromorphic circuits to compute motion. Pulses of variable width, rather than graded potentials, represent a natural medium for evaluating temporal relationships. Both algorithms measure speed by timing a moving edge in the image. Our first model is inspired by Reichardt's algorithm in the fiy and yields a non-monotonic response vs. velocity curve. We present data from a chip that implements this model. Our second algorithm yields a monotonic response vs. velocity curve and is currently being translated into silicon. 1 Introd uction Analog VLSI chips for the real time computation of visual motion have been the focus of much active research because of their importance as sensors for robotic applications. Correlation schemes such as those described in (Delbriick, 1993) have been found to be more robust than gradient schemes described in (Tanner and Mead, 1986), because they do not involve noise-sensitive operations like spatialdifferentiation and division. A comparison of four experimental schemes may be found in (Horiuchi et al., 1992). In spite of years of work, however, there is still no motion chip that robustly computes motion under all environmental conditions. 781 782 Sarpeshkar, Bair, and Koch Motion algorithms operating on higher level percepts in an image such as zerocrossings (edges) are more robust than those that operating on lower level percepts in an image such as raw image intensity values (Marr and Ullman, 1981). Our work demonstrates how, if the edges ill an image are identified, it is possible to compute motion, quickly and easily, by using pulses. We compute the velocity at each point in the image. The estimation of the flow-field is of tremendous importance in computations such as time-to-contact, figure-ground-segregation and depth-frommotion. Our motion scheme is well-suited to typical indoor environments that tend to have a lot of high-contrast edges. The much harder problem of computing motion in low-contrast, high-noise outdoor environments still remains unsolved. We present two motion algorithms. Our first algorithm is a "delay-and-correlate" scheme operating on spatial edge features and is inspired by work on fly vision (Hassenstein and Reichardt, 1956). It yields a non-monotonic response vs. velocity curve. We present data from a chip that implements it. Our second algorithm is a "facilitate-and-trigger" scheme operating on temporal edge features and yields a monotonic response vs. velocity curve. Work is under way to implement our second algorithm in analog VLSI. 2 The Delay-and-Correlate Scheme Conceptually, there are two stages of computation. First, the zero-crossings in the image are computed and then the motion of these zero-crossings is detected. The zero-crossing circuitry has been described in (Bair and Koch, 1991). We concentrate on describing the motion circuitry. A schematized version of the chip is shown in Figure 1a. Only four photo receptors in the array are shown. The 1-D image from the array of photoreceptors is filtered with a spatial bandpass filter whose kernel is composed of a difference of two exponentials (implemented with resistive grids). The outputs of the bandpass filter feed into edge detection circuitry that output a bit indicating the presence or absence of an edge between two adjacent pixels. The edges on the chip are separated into two polarities, namely, right-side-bright (R) and left-side-brigbt (L), which are kept separate throughout the chip, including the motion circuitry. For comparison, in biology, edges are often separated into light-on edges and light-off edges. The motion circuits are sensitive only to the motion of those edges from which they receive inputs. They detect the motion of a zero-crossing from one location to an adjacent location using a Reichardt scheme as shown in Figure lb. Each motion detecting unit receives two zero-crossing inputs ZCn and ZCn+21. The ON-cells detect the onset of zero-crossings (a rising voltage edge) by firing a pulse. The units marked with D's delay these pulses by an amount D, controlled externally. The correlation units marked with X's logically AND a delayed version of a pulse from one location with an undelayed version of a pulse from the adjacent location. The output from the left correlator is sensitive to motion from location n + 2 to location n since the motion delay is compensated by the built-in circuit delay. The outputs of the two correlators are subtracted to yield the final motion signal. Figure 2a shows the circuit details. The boxes labelled with pulse symbols represent axon circuits. The 1 ZCn+1 could have been used as well. ZCn+2 was chosen due to wiring constraints and because it increases the baseline distance for computing motion. Visual Motion Computation in Analog VLSI using Pulses 783 A B PR3 BANDPASS SPATIAL FILTER M M + Mt~ Figure l-(A) The bandpass filtered photoreceptor signal is fed to the edge detectors marked with E's. The motion of these edges is detected by the motion detecting units marked with M's. (B) A single motion detecting unit, corresponding to a "M" unit in fig. A, has a Reichardt-like architecture. axon circuits generate a single pulse of externally controlled width, P, in response to a sharp positive transition in their input, but remain inactive in response to a negative transition. In order to generate pulses that are delayed from the onset of a zero-crossing, the output of one axon circuit, with pulse width parameter D, is coupled via an inverter to the input of another axon circuit, with pulse width parameter P. The multiplication operation is implemented by a simple logical AND. The subtraction operation is implemented by a subtraction of two currents. An offchip sense amplifier converts the bidirectional current pulse outputs of the local motion detectors into active-low or active-high voltage pulses. The axon circuit is shown in Figure 2b. Further details of its operation may be found in (Sarpeshkar et al., 1992). Figure 3a shows how the velocity tuning curve is obtained. If the image velocity, v, is positive, and ~x is the distance between adjacent zero-crossing locations, then it can be shown that the output pulse width for the positive-velocity half of the motion detector, tp, is tp = u8(u), (1) 784 Sarpeshkar, Bair, and Koch where Ax u=P-I--DI, v (2) and e(~) is the unit step function. If v is negative, the same eqns. apply for the negative-velocity half of the motion detector except that the signs of Ax and tp are reversed. A I-----+C V REF Figure 2-(A) The circuitry implementing the Reichardt scheme of Figure lb, is shown. The boxes labelled P and D represent axon circuits of pulse width parameter P and D, respectively. (B) The circuit details of an axon circuit that implements an ON-cell of Figure lb are shown. The input and output are 'Vin and Vout respectively. The circuit was designed to mimic the behavior of sodium and leak conductances at the node of Ranvier in an axon fiber. The pulse width of the output pulse, the refractory period following its generation, and the threshold height of the input edge needed to trigger the pulse are determined by the values of bias voltages VD, VR and VL respectively. Experimental Data Figure 4a shows the outputs of motion detectors between zero-crossings 3 and 5, 7 and 9, and 11 and 13, denoted as Mt3, Mh, and Mtll, respectively. For an edge passing from left to right, the outputs Mtll, Mt7 and Mt3 are excited in this order, and they each report a positive velocity (active high output that is above VREF)' For an edge passing from right to left, the outputs Mt3, Mt7 and Mtll are excited in this order and they each report a negative velocity (active low output that is below VREF)' Note that the amplitudes of these pulses contain no speed information and only signal the direction of motion. Figure 4b shows that the output M t3 is tuned to a particular velocity. As the rotational frequency of a cylinder with a painted edge is decreased from a velocity corresponding to a motor voltage of - 6.1 V to a velocity corresponding to a motor voltage of -1.3V, the output pulse width increases, then decreases again, as the optimal velocity is traversed through. A similar tuning curve is observed for positive motor voltages. If the distance from the surface of the spinning cylinder to the center of the lens is 0, the distance from the center of the lens to the chip is i, the radius of the spinning cylinder is R, and its frequency Visual Motion Computation in Analog VLSI using Pulses 785 ( D (I) P Delayed Pulse Time ~ ~x U ndelayed .=:: v Pulse ~~------~--~------~ Time Motion Delay 8 6 4 2 ~ a ~ ~ -2 ~ 5. -4 ~ -6 o -8 "0 C~X) ~-10~-+--~--~-+--+-~ b.O -100 a 100 .r.n Veloci ty (degj sec) Figure 3-(a) The figure shows how the overlap between the delayed and undelayed pulses gives rise to velocity tuning for motion in the preferred direction. (b) The figure shows experimental data (circles) and a theoretical fit (line) for the motion unit Mt3 's output response vs. angular velocity. of rotation is f, then the velocity, v, of the moving edge as seen by the chip is given by z v=27rfR-. o The angular velocity, w, of the moving edge, is 27rf R v w = "7. o Z (3) (4) Figure 3b shows the output pulse width of M t3 plotted against the angular velocity of the edge (w). The data are fit by a curve that com pu tes tp vs. w from (1)-( 4), using measured values of ~x = 180 pm, 0 = 310 mm, i = 17 mm, and R = 58 mm. 786 Sarpeshkar, Bair, and Koch A • , t ,. ".n .. • ..... ;' ;1 t • ' Mtll .. ,n ~.. • ", .,. I M t7 p • .. • II : :.n:;, :, ;,! Mta "u It: ~ :, Mta , "~ Mt7 t , II' e ..... lJ~'" , Itt Mtll 0.0 0.05 0.1 0.15 0.20 Time (Sec) ,.--.... > .-0 ""0 "-> lr.l N ............, (l) b.O C1:'l .,;> ..... ~ .,;> ;::j 0..,;> ::1 0 B n +5.0 . 111 .. 11. :iJ.,Q I . 1 ~ n +1.5 • n ±1.a jolll • .. , -1.3 • U' -1.5 t: : • ,I I , i D -6.I 0.0 1.0 2.0 3.0 4.0 Time (10-2 Sec) Figure 4-(A) The chip output waveforms are active-high when the motion is in a direction such that motion detectors 11, 7 and 3 are stimulated in that order. In the opposite direction (3 7 11), the output waveforms are activelow. (B) The figure demonstrates the tuned velocity behavior of the motion unit Mt3 for various motor voltages (in V). Large positive voltages correspond to fast motion in one direction and large negative voltages correspond to fast motion in the opposite direction. 3 The Facilitate-and-Trigger Scheme Although the delay-and-correlate scheme works well, the output yields ambiguous motion information due to its non-monotonic velocity dependence, i.e., we don't know if the output is small because the velocity is too slow or because it is too fast. This problem may be solved by aggregating the outputs of a series of motion detectors with overlapping tuning curves and progressively larger optimal velocities. This solution is plausible in physiology but expensive in VLSI. We were thus motivated to create a new motion detector that possessed a monotonic velocity tuning curve from the outset. Figure Sa shows the architecture of such a motion detecting unit: The need for computing spatial edges is obviated by having a photoreceptor sensitive to temporal features caused by moving edges, i.e., sharp light onsets/offsets (Delbriick, 1993). Each ON-cell fires a pulse in response to the onset of a temporal edge. The pulses are fed to the facilitatory (F) and trigger (T) inputs of motion detectors tuned for motion in the left or right directions. The facilitatory pulse defines a time window of externally controlled width F, within which the output motion pulse may be activated by the trigger pulse. Thus, the rising edge of the trigger pulse must occur within the time window set by the facilitatory pulse in order to create a motion Visual Motion Computation in Analog VLSI using Pulses 787 output. If this condition is satisfied, the output motion pulse is triggered (begins) at the start of the trigger pulse and ends at the end of the facilitatory pulse; its width, thus, encodes the arrival time difference between the onset pulses at adjacent locations due to the motion delay. Each half of the motion detector only responds to motion in the direction that corresponds to F before T. Figure 5c shows that the velocity tuning in this scheme is monotonic. It can be shown that the output pulse width for the positive half of the motion detector, tp , is given by tp=u8(u), (5) where u is given by, Ax u = F- -. (6) v The dead zone near the origin may be made as small as needed by increasing the width of the facilitatory pulse. Figure 5b shows a compact circuit that PRn A PRn+1 C Q) F ~ Facilitation ~ ....... ~ Pulse Time Q) Trigger ~ ~ Pulse ....... ~ Time ..c::: ~ ~ 0.."'0 ro·~~ ~Q) O~ ~ ;:1 Motion Delay (~X) ~ F ..c::: ~ ~ 0.."'0 ro·~~ ~ Q) ~F 0,.$ ~ Velocity (v) Figure 5-(A) The motion detecting unit uses a facilitate-and-trigger paradigm instead of a delay-and-correlate paradigm as a basis for its operation. (B) A compact circuit implements the motion detecting unit. VT is the trigger input, VF is the facilitatory input, Vout is the output and VTH is a bias voltage. (C) The output response has a monotonic dependence on velocity. implements the facilitate-and-trigger scheme. The ON-cell, subtraction and senseamplifier circuitry are implemented as in the Reichardt scheme. 788 Sarpeshkar, Bair, and Koch The facilitate-and-trigger approach requires a total of 32 transistors per motion unit compared with a total of 64 in the delay-and-correlate approach, needs one parameter to be controlled (F) rather than two (D and P), and yields monotonic tuning from the outset. We, therefore, believe that it will prove to be the superior of the two approaches. 4 Conclusions The evaluations of onsets, delays and coincidences, required for computing motion are implemented very naturally with pulses rather than by graded potentials as in all other motion chips, built so far. Both of our motion algorithms time the motion of image features in an efficient fashion by using pulse computation. Acknowledgements Many thanks to Carver Mead for his encouragement, support and use of lab facilities. We acknowledge useful discussions with William Bialek, Nicola Franceschini and Tobias Delbriick. This work was supported by grants from the Office of Naval Research and the California Competitive Technologies Program. Chip fabrication was provided by MOSIS. References W. Bair, and C. Koch, "An Analog VLSI Chip for Finding Edges from Zerocrossings.", In Advances in Neural Information Processing Systems Vol. 3, R. Lippman, J. Moody, D. Touretzky, eds., pp. 399-405, Morgan Kaufmann, San Mateo, CA,199l. T. Delbriick, "Investigations of Analog VLSI Visual Transduction and Motion Processing", PhD. thesis, Computation and Neural Systems Program, Caltech, Pasadena, CA, 1993. B. Hassenstein and W. Reichardt, "Systemtheoretische Analyse der Zeit, Reihenfolgen, und Vorzeichenauswertung bei der Bewegungsperzepion des Riisselkafers Chlorophanus", Z. Naturforsch.11b: pp. 513-524, 1956. T. Horiuchi, W. Bair, A. Moore, B. Bishofberger, J. Lazzaro, C. Koch, "Computing Motion Using Analog VLSI Vision Chips: an Experimental Comparison among Different Approaches", IntI. Journal of Computer Vision, 8, pp. 203-216, 1992. D. Marr and S. Ullman, "Directional Selectivity and its Use in Early Visual Processing", Proc. R. Soc. Lond B 211, pp. 151-180, 1981. R. Sarpeshkar, L. Watts, C. Mead, "Refractory Neuron Circuits", Internal Lab Memorandum, Physics of Computation Laboratory, Pasadena, CA, 1992. J. Tanner and C. Mead, "An Integrated Optical Motion Sensor", VLSI Signal Processing II, S-Y Kung, R.E. Owen, and J.G. Nash, eds., 59-76, IEEE Press, NY, 1986.
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Learning Control Under Extreme Uncertainty Vijaykumar Gullapalli Computer Science Department University of Massachusetts Amherst, MA 01003 Abstract A peg-in-hole insertion task is used as an example to illustrate the utility of direct associative reinforcement learning methods for learning control under real-world conditions of uncertainty and noise. Task complexity due to the use of an unchamfered hole and a clearance of less than 0.2mm is compounded by the presence of positional uncertainty of magnitude exceeding 10 to 50 times the clearance. Despite this extreme degree of uncertainty, our results indicate that direct reinforcement learning can be used to learn a robust reactive control strategy that results in skillful peg-in-hole insertions. 1 INTRODUCTION Many control tasks of interest today involve controlling complex nonlinear systems under uncertainty and noise. 1 Because traditional control design techniques are not very effective under such circumstances, methods for learning control are becoming increasingly popular. Unfortunately, in many of these control tasks, it is difficult to obtain training information in the form of prespecified instructions on how to perform the task. Therefore supervised learning methods are not directly applicable. At the same time, evaluating the performance of a controller on the task is usually fairly straightforward, and hence these tasks are ideally suited for the application of associative reinforcement learning (Barto & Anandan, 1985). IFor our purposes, noise can be regarded simply as one of the sources of uncertainty. 327 328 Gullapalli In associative reinforcement learning, the learning system's interactions with its environment are evaluated by a critic, and the goal of the learning system is to learn to respond to each input with the action that has the best expected evaluation. In learning control tasks, the learning system is the controller, its actions are control signals, and the critic's evaluations are based on the performance criterion associated with the control task. Two kinds of associative reinforcement learning methods, direct and indirect, can be distinguished (e.g., Gullapalli, 1992). Indirect reinforcement learning methods construct and use a model of the environment and the critic (modeled either separately or together), while direct reinforcement learning methods do not. We have previously argued (Gullapalli, 1992; Barto & Gullapalli, 1992) that in the presence of uncertainty, hand-crafting or learning an adequate model-imperative if one is to use indirect methods for training the controller-can be very difficult. Therefore, it can be expeditious to use direct reinforcement learning methods in such situations. In this paper, a peg-in-hole insertion task is used as an example to illustrate the utility of direct associative reinforcement learning methods for learning control under real-world conditions of uncertainty. 2 PEG-IN-HOLE INSERTION Peg-in-hole insertion has been widely used by roboticists for testing various approaches to robot control and has also been studied as a canonical robot assembly operation (Whitney, 1982; Gustavson, 1984; Gordon, 1986). Although the abstract peg-in-hole task can be solved quite easily, real-world conditions of uncertainty due to (1) errors and noise in sensory feedback, (2) errors in execution of motion commands, and (3) uncertainty due to movement of the part grasped by the robot can substantially degrade the performance of traditional control methods. Approaches proposed for peg-in-hole insertion under uncertainty can be grouped into two major classes: methods based on off-line planning, and methods based on reactive control. Off-line planning methods combine geometric analysis of the peg-hole configuration with analysis of the task statics to determine motion strategies that will result in successful insertion (Whitney, 1982; Gustavson, 1984; Gordon, 1986). In the presence of uncertainty in sensing and control, researchers have suggested incorporating the uncertainty into the geometric model of the task in configuration space (e.g., Lozano-Perez et al., 1984; Erdmann, 1986; Caine et al., 1989; Donald, 1986). Offline planning is based on the assumption that a realistic characterization of the margins of uncertainty is available, which is a strong assumption when dealing with real-world systems. Methods based on reactive control, in comparison, try to counter the effects of uncertainty with on-line modification of the motion control based on sensory feedback. Often, compliant motion control is used, in which the trajectory is modified by contact forces or tactile stimuli occurring during the motion. The compliant behavior either is actively generated or occurs passively due to the physical characteristics of the robot (Whitney, 1982; Asada, 1990). However, as Asada (1990) points out, many tasks including the peg insertion task require complex nonlinear compliance or admittance behavior that is beyond the capability of a passive mechanism. Unfortunately, humans find it quite difficult to prespecify appropriLearning Control Under Extreme Uncertainty 329 ate compliant behavior (Lozano-Perez et al., 1984), especially in the presence of uncertainty. Hence techniques for learning compliant behavior can be very useful. We demonstrate our approach to learning a reactive control strategy for peg-in-hole insertion by training a controller to perform peg-in-hole insertions using a Zebra Zero robot. The Zebra Zero is equipped with joint position encoders and a sixaxis force sensor at its wrist, whose outputs are all subject to uncertainty. Before describing the controller and presenting its performance in peg insertion, we present some experimental data quantifying the uncertainty in position and force sensors. 3 QUANTIFYING THE SENSOR UNCERTAINTY In order to quantify the position uncertainty under varying load conditions similar to those that occur when the peg is interacting with the hole, we compared the sensed peg position with its actual position in cartesian space under different load conditions. In one such experiment, the robot was commanded to maintain a fixed position under five different loads conditions applied sequentially: no load, and a fixed load of O.12Kgf applied in the ±:z: and ±y directions. Under each condition, the position and force feedback from the robot sensors, as well as the actual :z:-y position of the peg were recorded. The sensed and actual :z:-y positions of the peg are shown in Table 1. The sensed :z:-y positions were computed from the joint positions sensed by the Zero's joint position encoders. As can be seen from the table, there is a large discrepancy between the sensed and actual positions of the peg: while the actual change in the peg's position under the external load was of the order of 2 to 3mm, the largest sensed change in position was less than 0.025mm. In comparison, the clearance between the peg and the hole (in the 3D task) was 0.175mm. From observations of the robot, we could determine that the uncertainty in position was primarily due to gear backlash. Other factors affecting the uncertainty include the posture of the robot arm, which affects the way the backlash is loaded, and interactions between the peg and the en vironment. Table 1: Sensed And Actual Positions Under 5 Different Load Conditions Load Condition Sensed :c-y Position (mm) Actual :c-y Position (mm) No load position (0.0, 0.000000) (0.0, 0.0) With -y load (0.0, -0.014673) (0.0, -2.5) With +:c load (0.0,0.000000) (1.9, -0.3) With +y load (0.0,0.024646) (-2.9, -0.2) With -:c load (0.0,0.010026) (0.3,2.2) Final (no load) position (0.0,0.000000) (0.0, -0.6) Figure 1 shows 30 time-step samples of the force sensor output for each of the load conditions described above. As can be seen from the figure, there is considerable sensor noise, especially in recording moments. Although designing a controller that can robustly perform peg insertions despite the large uncertainty in sensory input 330 Gullapalli is difficult, our results indicate that a controller can learn a robust peg insertion strategy. : F, I-A--"---"-',---,---.,..-.J ~ i .!! Fy I""rv-W--..I I I F, f-'"-.-__ ...,.......,._--.Jv-,,.,.,........--..J~..,..,...,...---~- .. t ~ :II: 0. to ... + .!! My Nr-... r---U u I 1 ~ I-----"""'-...""....,,....../''"'''''"'-...,.......L-...~ .............. _,....-~o 10 10 10 1111 No Ioed LDIId In .., Loed In +. Loed In +, LDIId In -. No Ioed Figure I: 30 Time-step Samples Of The Sensed Forces and Moments Under 5 Different Load Conditions. With An Ideal Sensor, The Readings Would Be Constant In Each 30 Time-step Interval. 4 LEARNING PEG-IN-HOLE INSERTION Our approach to learning a reactive control strategy for peg insertion under uncertainty is based on active generation of compliant behavior using a nonlinear mapping from sensed positions and forces to position commands.2 The controller learns this mapping through repeated attempts at peg insertion. The Peg Insertion Tasks As depicted in Figure 2, both 2D and 3D versions of the peg insertion task were attempted. In the 2D version of the task, the peg used was 50mm long and 22.225mm (7/8in) wide, while the hole was 23.8125mm (15/16in) wide. Thus the clearance between the peg and the hole was 0.79375mm (1/32in). In the 3D version, the peg used was 30mm long and 6mm in diameter, while the hole was 6.35mm in diameter. Thus the clearance in the 3D case was 0.175mm. The Controller The controller was implemented as a connectionist network that operated in closed loop with the robot so that it could learn a reactive control strategy for performing peg insertions. The network used in the 2D task had 6 inputs, viz., the sensed positions and forces, (X, Y, e) and (Fx, Fy , Mz), three 2See also (Gullapalli et al., 1992). The 20 peg InsertIon task (X. Y,EI) Position (X. Y.EI) and Foltles (F •• Fr' M.), Cor'lrOls .... : Position commands (x. y. 8) Learning Control Under Extreme Uncertainty 331 The 3D peg Insertion task Posmon (X.Y . Z. El1 ~2)and =': .. t .. Fr' ~ . M • . My .M,) , Posmon commands (x. y. Z. 81 , 82) Figure 2: The 2D And 3D Peg-in-hole Insertion Tasks. outputs forming the position command (x, y, 8), and two hidden layers of 15 units each. For the 3D task, the network had 11 inputs, the sensed positions and forces, (X,Y,Z,E>1,E>2) and (Fx,Fy,F: , l.{r;, My,lV/z ), five outputs forming the position command (x, y, Z, 811 82), and two hidden layers of 30 units each. In both networks, the hidden units used were back-propagation units, while the output units used were stochastic real-valued (SRV) reinforcement learning units (Gullapalli, 1990). SRV units use a direct reinforcement learning algorithm to find the best real-valued output for each input (see Gullapalli (1990) for details). The position inputs to the network were computed from the sensed joint positions using the forward kinematics equations for the Zero. The force and moment inputs were those sensed by the six-axis force sensor. A PD servo loop was used to servo the robot to the position output by the network at each time step. Training Methodology The controller network was trained in a s~quence of trials, each of which started with the peg at a random position and orientation with respect to the hole and ended either when the peg was successfully inserted in the hole, or when 100 time steps had elapsed. An insertion was termed successful when the peg was inserted to a depth of 25mm into the hole. At each time step during training, the sensed peg position and forces were input to the network, and the computed control output was executed by the robot, resulting in some motion of the peg. An evaluation of the controller's performance, r, ranging from 0 to 1 with 1 denoting the best possible evaluation, was computed based on the new peg position and the forces acting on the peg as _ { max(O.O, 1.0 - O.Olllposition errorll) if all forces :S 0.5Kgf, r max(O.O, 1.0 - O.Olllposition errorll - O.lFmax) otherwise, where Fmax denotes the largest magnitude force component. Thus, the closer the sensed peg position was to the desired position with the peg inserted in the hole, the higher the evaluation. Large sensed forces, however, reduced the evaluation. Using this evaluation, the network adjusted its weights appropriately and the cycle was repeated. 332 Gullapalli 5 PERFORMANCE RESULTS A learning curve showing the final evaluation over 500 consecutive trials on the 2D task is shown in Figure 3 (a). The final evaluation levels off close to 1 after about c: 1.00 1100.00 8 J • c: • 110.00 > • .. 0.10 c: a ~ i i 8 10.10 J 10.00 III 0.40 40.00 0.20 0.0025 76 125 176 226 %75 3:26 375 <126 476 0.0025 76 125 176 226 276 3:26 376 <126 476 T,.I, T,.I, (a) (b) Figure 3: Smoothed Final Evaluation Received And Smoothed Insertion Time (In Simulation Time Steps) Taken On Each Of 500 Consecutive Trials On The 2D Peg Insertion Task. The Smoothed Curve Was Obtained By Filtering The Raw Data Using A Moving-Average Window Of 25 Consecutive Values. 150 trials because after that amount of training, the controller is consistently able to perform successful insertions within 100 time steps. However, performance as measured by insertion time continues to improve, as is indicated by the learning curve in Figure 3 (b), which shows the time to insertion decreasing continuously over the 500 trials. These curves indicate that the controller becomes progressively more skillful at peg insertion with training. Similar results were obtained for the 3D task, although learning was slower in this case. The performance curves for the 3D task are shown in Figure 4. 6 DISCUSSION AND CONCLUSIONS The high degree of uncertainty in the sensory feedback from the Zebra Zero, coupled with the fine motion control requirements of peg-in-hole insertion make the task under consideration an example of learning control under extreme uncertainty. The positional uncertainty, in particular, is of the order of 10 to 50 times the clearance between the peg and the hole and is primarily due to gear backlash. There is also significant uncertainty in the sensed forces and moments due to sensor noise. Our results indicate that direct reinforcement learning can be used to learn a reactive control strategy that works robustly even in the presence of a high degree of uncertainty. Learning Control Under Extreme Uncertainty 333 c: 1.00 , .. 0.10 i I 10.10 • 1'00.00 • 110.00 I i 10.00 0.40 40.00 0.20 20.00 0.0021 , 6 226 326 426 _ _ 726 TrW, (a) (b) Figure 4: Smoothed Final Evaluation Received And Smoothed Insertion Time (In Simulation Time Steps) Taken On Each Of 800 Consecutive Trials On The 3D Peg Insertion Task. The Smoothed Curve Was Obtained By Filtering The Raw Data Using A Moving-Average Window Of 25 Consecutive Values. Although others have studied similar tasks, in most other work on learning peg-inhole insertion (e.g., Lee & Kim, 1988) it is assumed that the positional uncertainty is about an order of magnitude less than the clearance. Moreover, results are often presented using simulated peg-hole systems. Our results indicate that our approach works well with a physical system, despite the much higher magnitudes of noise and consequently greater degree of uncertainty inherent in dealing with physical systems. Furthermore, the success of the direct reinforcement learning approach to training the controller indicates that this approach can be useful for automatically synthesizing robot control strategies that satisfy constraints encoded in the performance evaluations. Acknowledgements This paper has benefited from many useful discussions with Andrew Barto and Roderic Grupen. I would also like to thank Kamal Souccar for assisting with running the Zebra Zero. This material is based upon work supported by the Air Force Office of Scientific Research, Bolling AFB, under Grant AFOSR-89-0526 and by the National Science Foundation under Grant ECS-8912623. References [1] H. Asada. Teaching and learning of compliance using neural nets: Representation and generation of nonlinear compliance. In Proceedings of the 1990 IEEE International Conference on Robotics and Automation, pages 1237-1244, 1990. 334 Gullapalli [2] A. G. Barto and P. Anandan. Pattern recognIzmg stochastic learning automata. IEEE Transactions on Systems, Man, and Cybernetics, 15:360-375, 1985. [3] A. G. Barto and V. Gullapalli. Neural Networks and Adaptive Control. In P. Rudomin, M. A. Arbib, and F. Cervantes-Perez, editors, Natural and Artificial Intelligence. Research Notes in Neural Computation, Springer-Verlag: Washington. (in press). [4] M. E. Caine, T. Lozano-Perez, and W. P. Seering. Assembly strategies for chamferless parts. In Proceedings of the IEEE International Conference on Robotics and Automation, pages 472-477, May 1989. [5] B. R. Donald. Robot motion planning with uncertainty in the geometric models of the robot and environment: A formal framework for error detection and recovery. In Proceedings of the IEEE International Conference on Robotics and Automation, pages 1588-1593, 1986. [6] M. Erdmann. Using backprojections for fine motion planning with uncertainty. International Journal of Robotics Research, 5(1):19-45, 1986. [7] S. J. Gordon. A utomated assembly using feature localization. PhD thesis, Massachusetts Institute of Technology, MIT AI Laboratory, Cambridge, MA, 1986. Technical Report 932. [8] V. Gullapalli. A stochastic reinforcement learning algorithm for learning realvalued functions. Neural Networks, 3:671-692, 1990. [9] V. Gullapalli. Reinforcement Learning and its application to control. PhD thesis, University of Massachusetts, Amherst, MA 01003, 1992. [10] V. Gullapalli, R. A. Grupen, and A. G. Barto. Learning reactive admittance control. In Proceedings of the 1992 IEEE International Conference on Robotics and Automation, pages 1475-1480, Nice, France, 1992. [11] R. E. Gustavson. A theory for the three-dimensional mating of chamfered cylindrical parts. Journal of Mechanisms, Transmissions, and Automated Design, December 1984. [12] S. Lee and M. H. Kim. Learning expert systems for robot fine motion control. In H. E. Stephanou, A. Meystal, and J. Y. S. Luh, editors, Proceedings of the 1988 IEEE International Symposium on Intelligent Control, pages 534-544, Arlington, Virginia, USA, 1989. IEEE Computer Society Press: Washington. [13J T. Lozano-Perez, M. T. Mason, and R. H. Taylor. Automatic synthesis of finemotion strategies for robots. The International Journal of Robotics Research, 3(1):3-24, Spring 1984. [14] D. E. Whitney. Quasi-static assembly of compliantly supported rigid parts. Journal of Dynamic Systems, Measurement, and Contro~ 104, March 1982. Also in Robot Motion: Planning and Contro~ (Brady, M., et al. eds.), MIT Press, Cambridge, MA, 1982.
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Efficient Pattern Recognition Using a New Transformation Distance Patrice Simard Yann Le Cun John Denker AT&T Bell Laboratories, 101 Crawford Corner Road, Holmdel, NJ 07724 Abstract Memory-based classification algorithms such as radial basis functions or K-nearest neighbors typically rely on simple distances (Euclidean, dot product ... ), which are not particularly meaningful on pattern vectors. More complex, better suited distance measures are often expensive and rather ad-hoc (elastic matching, deformable templates). We propose a new distance measure which (a) can be made locally invariant to any set of transformations of the input and (b) can be computed efficiently. We tested the method on large handwritten character databases provided by the Post Office and the NIST. Using invariances with respect to translation, rotation, scaling, shearing and line thickness, the method consistently outperformed all other systems tested on the same databases. 1 INTRODUCTION Distance-based classification algorithms such as radial basis functions or K-nearest neighbors often rely on simple distances (such as Euclidean distance, Hamming distance, etc.). As a result, they suffer from a very high sensitivity to simple transformations of the input patterns that should leave the classification unchanged (e.g. translation or scaling for 2D images). This is illustrated in Fig. 1 where an unlabeled image of a "9" must be classified by finding the closest prototype image out of two images representing respectively a "9" and a "4". According to the Euclidean distance (sum of the squares of the pixel to pixel differences), the "4" is closer even though the "9" is much more similar once it has been rotated and thickened. The result is an incorrect classification. The key idea is to construct a distance measure which is invariant with respect to some chosen transformations such as translation, rotation and others. The special case of linear transformations has been well studied in statistics and is sometimes referred to as Procrustes analysis 50 Pattern to be classified Efficient Pattern Recognition Using a New Transformation Distance 51 prototype A Prototype B Figure 1: What is a good similarity measure? According to the Euclidean distance the pattern to be classified is more similar to prototype B. A better distance measure would find that prototype A is closer because it differs mainly by a rotation and a thickness transformation, two transformations which should leave the classification invariant. (Sibson, 1978). It has been applied to on-line character recognition (Sinden and Wilfong, 1992). This paper considers the more general case of non-linear transformations such as geometric transformations of gray-level images. Remember that even a simple image translation corresponds to a highly non-linear transformation in the highdimensional pixel spacel . In previous work (Simard et al., 1992b), we showed how a neural network could be trained to be invariant with respect to selected transformations of the input. VVe now apply similar ideas to distance-based classifiers. ''''hen a pattern P is transformed (e.g. rotated) with a transformation s that depends on one parameter a (e.g. the angle of the rotation), the set of all the transformed patterns Sp = {x I 35 such that x = s(5, P)} is a one-dimensional curve in the vector space of the inputs (see Fig. 2). In certain cases, such as rotations of digitized images, this curve must be made continuous using smoothing techniques (see (Simard et al., 1992b)). When the set of transformations is parameterized by n parameters ai (rotation, translation, scaling, etc.), Sp is a manifold of at most n dimensions. The patterns in Sp that are obtained through small transformations of P, i.e. the part of Sp that is close to P, can be approximated by a plane tangent to the manifold Sp at the point P. Small transformations of P can be obtained by adding to P a linear combination of vectors that span the tangent plane (tangent vectors). The images at the bottom of Fig. 2 were obtained by that procedure. Tangent vectors for a transformation s can easily be computed by finite difference (evaluating os(a, P)/oa); more details can be found in (Simard et al., 1992b; Simard et al., 1992a). As we mentioned earlier, the Euclidean distance between two patterns P and E is in general not appropriate because it is sensitive to irrelevant transformations of P and of E. In contrast, the distance V(E, P) defined to be the minimal distance between the two manifolds Sp and SE is truly invariant with respect to the transformation used to generate Sp and SE. Unfortunately, these manifolds have no analytic expression in general, and finding the distance between them is a hard optimization problem with multiple local minima. Besides, t.rue invariance is not 1 If the ima.ge of a "3" is translated vertica.lly upward, the middle top pixel will oscillate from black to white three times. 52 Simard, Cun, and Denker [3] True rotations of P • -15 • -7.5a=-O.2 a=-Q.l p .7.5 Transformations at p 1/ II _ •• _ ........... _ •• .Y p a=O.l Pixel space a=O.2 p T. V. Figure 2: Top: Small rotations of an original digitized image of the digit "3". Middle: Representation of the effect of the rotation in pixel space (if there were only 3 pixels). Bottom: Images obtained by moving along the tangent to the transformation curve for the same original digitized image P by adding various amounts (a) of the tangent vector (T.V.). necessarily desirable since a rotation of a "6" into a "9" does not preserve the correct classification. Our approach consists of approximating the non-linear manifold Sp and SE by linear surfaces and computing the distance D( E, P) defined to be the minimum distance between them. This solves three problems at once: 1) linear manifolds have simple analytical expressions which can be easily computed and stored, 2) finding the minimum distance between linear manifolds is a simple least squares problem which can be solved efficiently and, 3) this distance is locally invaria.nt but not globally invariant. Thus the distance between a "6" and a slightly rota.ted "6" is small but the distance between a "6" and a "9" is la.rge. The different. distan ces between P and E are represented schematically in Fig. 3. The figure represents two patterns P and E in 3-dimensional space. The ma.nifolds generated by s are represented by one-dimensional curves going through E and P respectively. The linear approximations to the manifolds are represented by lines tangent to the curves at E and P. These lines do not intersect in 3 dimensions and the shortest distance between them (uniquely defined) is D(E, P). The distance between the two non-linear transformation curves VeE, P) is also shown on the figure. An efficient implementation of the tangent distance D(E, P) will be given in the Efficient Pattern Recognition Using a New Transformation Distance 53 Figure 3: Illustration of the Euclidean distance and the tangent distance between P and E next section. Although the tangent distance can be applied to any kind of patterns represented as vectors, we have concentrated our efforts on applications to image recognition. Comparison of tangent distance with the best known competing method will be described. Finally we will discuss possible variations on the tangent distance and how it can be generalized to problems other than pattern recognition. 2 IMPLEMENTATION In this section we describe formally the computation of the tangent distance. Let the function s which map u, a to s(a, u) be a differentiable transformation of the input space, depending on a vector a of parameter, verifying s(O, u) = 'It. If u is a 2 dimensional image for instance, s(a, u) could be a rotation of u by the angle &. If we are interested in all transformations of images which conserve distances (isometry), 8(a, u) would be a rotation by a r followed by a translation by ax, a y of the image u. In this case & = (ar , ax, a y) is a vector of parameters of dimension 3. In general, & = (ao, .. " am-d is of dimension m. Since 8 is differentiable, the set Stl. = {x I 3a for which x = 8( a, 'It)} is a differentiable manifold which can be approximated to the first order by a hyperplane Ttl.. This hyperplane is tangent to Stl. at u and is generated by the columns of matrix Ltl. = 08(&~ 'It) I = [08(&, u), ... , 08(a, U)] (1) aa d=cf oao aam-l d=O which are vectors tangent to the manifold. If E and P are two patterns to be compared, the respective tangent planes TE and Tp can be used to define a new distance D between these two patterns. The tangent distance D(E, P) between E and P is defined by D(E, P) = min IIx - yW (2) xETE,yETp The equation of the tangent planes TE and Tp is given by: (3) (4) 54 Simard, Cun, and Denker where LE and Lp are the matrices containing the tangent vectors (see Eq. 1) and the vectors a E and ap are the coordinates of E' and P' in the corresponding tangent planes. The quantities LE and Lp are attributes of the patterns so in many cases they can be precomputed and stored. Computing the tangent distance (5) amounts to solving a linear least squares problem. The optimality condition is that the partial derivatives of D(E, P) with respect to a p and aE should be zero: oD(~, P) = 2(E'(aE) _ p'(ap» T LE = 0 oaE oD(~,P) = 2(p'(ap) _ E'(aE»T Lp = 0 oap (6) (7) Substituting E' and P' by their expressions yields to the following linear system of equations, which we must solve for ap and ilE: L;(E - P - Lpilp + LEaE) = 0 Lf(E - P - Lpap + LEilE) = 0 The solution of this system is (LPEL"E1L~ - L;)(E - P) = (LPEL"E1LEP - Lpp )ap (LEPLp~L; L~)(E - P) = (LEE LEPLp~LpE)aE (8) (9) (10) (11) where LEE = LfT LE , LpE = L~LE' LEP = L~Lp and Lpp = L~Lp. LU decompositions 0 LEE and Lpp can be precomputed. The most expenSIve part in solving this system is evaluating LEP (LPE can be obtained by transposing LEP). It requires mE x mp dot products, where mE is the number of tangent vectors for E and mp is the number of tangent vectors for P. Once LEP has been computed, ilp and ilE can be computed by solving two (small) linear system of respectively mE and mp equations. The tangent distance is obtained by computing IIE'(aE) - p'(ap )11 using the value of a p and ilE in equations 3 and 4. If n is the length of vector E (or P), the algorithm described above requires roughly n(mE+l)(mp+l)+3(m~+m~) multiply-adds. Approximations to the tangent distance can be computed more efficiently. 3 RESULTS Before giving the results of handwritten digit recognition experiments, we would like to demonstrate the property of "local invariance" of tangent distance. A 16 by 16 pixel image similar to the "3" in Fig 2 was translated by various amounts. The tangent distance (using the tangent vector corresponding to horizonta.l translations) and the Euclidean Distance between the original image and its translated version were measured as a function of the size k (in pixels) of the translation. The result is plotted in Fig. 4. It is clear that the Euclidean Distance starts increasing linearly with k while the tangent distance remains very small for translations as large as two pixels. This indicates that, while Euclidean Distance is not invariant to translation, tangent distance is locally invariant. The extent of the invariance can be Efficient Pattern Recognition Using a New Transformation Distance 55 Distance 10 8 6 4 2 Tangent Distance o~--.---~~~~--~~ ~ ~ ~ ~ 0 2 4 6 8 # of pixels by which image is translated Figure 4: Euclidean and tangent distances between a 16x16 handwritten digit image and its translated version as a function of the amount of translation measured in pixels. increased by smoothing the original image, but significant features may be blurred away, leading to confusion errors. The figure is not symmetric for large translations because the translated image is truncated to the 16 by 16 pixel field of the original image. In the following experiments, smoothing was done by convolution with a Gaussian of standard deviation u = 0.75. This value, which was estimated visually, turned out to be nearly optimal (but not critical). 3.1 Handwritten Digit Recognition Experiments were conducted to evaluate the performance of tangent distance for handwritten digit recognition. An interesting characteristic of digit images is that we can readily identify a set of local transformations which do not affect the identity of the character, while covering a large portion of the set of possible instances of the character. Seven such image transformations were identified: X and Y translations, rotation, scaling, two hyperbolic transformations (which can generate shearing and squeezing), and line thickening or thinning. The first six transformations were chosen to span the set of all possible linear coordinate transforms in the imn~e plane (nevertheless, they correspond to highly non-linear transforms in pixel space). Additional transformations have been tried with less success. The simplest possible use of tangent distance is in a Nearest Neighbor classifier. A set of prototypes is selected from a training set, and stored in memory. W·hen a test pattern is to be classified, the J( nearest prototypes (in terms of tangent distance) are found, and the pattern is given the class that has the majority among the neighbors. In our applications, the size of the prototype set is in the neighborhood of 10,000. In principle, classifying a pattern would require computing 10,000 tangent distances, leading to excessive classification times, despite the efficiency of the tangent distance computation. Fortunately, two patterns that are very far apart in terms of Euclidean Distance are likely to be far apart in terms of tangent distance. Therefore we can use Euclidean distance as a "prefilter" , and eliminate prototypes that are unlikely to be among the nearest neighbors. V'le used the following 4-step classification procedure: 1) the Euclidean distance is computed between the test pattern and all the prototypes, 2) The closest 100 prototypes are selected, 3) the tangent distance between these 100 prototypes and the test pattern is computed 56 Simard, Cun, and Denker 6 USPS error (%) 5 4 3 2 1 0 Human T-Dlst NNet K-NN 5 4 3 2 1 o NIST Human T -Dlst NNet Figure 5: Comparison of the error rate of tangent nearest neighbors and other methods on two handwritten digit databases and 4) the most represented label among the J( closest prototype is outputed. This procedure is two orders of magnitude faster than computing all 10,000 tangent distances, and yields the same performance. US Postal Service database: In the first experiment, the database consisted of 16 by 16 pixel size-normalized images of handwritten digits, coming from US mail envelopes. The entire training set of 9709 examples of was used as the prototype set. The test set contained 2007 patterns. The best performance was obtained with the "one nearest neig~bor" rule. The results are plotted in Fig. 5. The error rate of the method is 2.6%. Two members of our group labeled the test set by hand with an error rate of 2.5% (using one of their labelings as the truth to test the other also yielded 2.5% error rate). This is a good indicator of the level of difficulty of this task2 . The performance of our best neural network (Le Cun et al., 1990) was 3.3%. The performance of one nearest neighbor with the Euclidean distance was 5.9%. These results show that tangent distance performs substantially better than both standard K-nearest neighbor and neural networks. NIST database: The second experiment was a competition organized by the N 8,tional Institute of Standards and Technology. The object of the competition was to classify a test set of 59,000 handwritten digits, given a training set of 223,000 patterns. A total of 45 algorithms were submitted from 26 companies from 7 different countries. Since the training set was so big, a very simple procedure was used to select about 12,000 patterns as prototypes. The procedure consists of creating a new database (empty at the beginning), and classifying each pattern of the large database using the new database as a prototype set. Each time an error is made, the pattern is added to the new database. More than one pass may have to be made before the new database is stable. Since this filtering process would take too long with 223,000 prototypes, we split the large database into 22 smaller databases of 10,000 patterns each, filtered those (to about 550 patterns) and concatenated the result, yielding a database of roughly 12,000 patterns. This procedure has many drawbacks, and in particular, it is very good at picking up mislabeled characters in the training set. To counteract this unfortunate effect, a 3 nearest neighbors procedure was used with tangent distance. The organizers decided to collect the 2This is an extremely difficult test set. Procedures that achieve less than 0.5% error on other handwritten digit tasks barely achieve less than 4% on this one Efficient Pattern Recognition Using a New Transformation Distance 57 training set and the test set among two very different populations (census bureau workers for the training set, high-school students for the test set), we therefore report results on the official NIST test set (named "hard test set"), and on a subset of the official training set, which we kept aside for test purposes (the "easy test set"). The results are shown in Fig. 5. The performance is much worse on the hard test set since the distribution was very different from that of the training set. Out of the 25 participants who used the NIST training database, tangent distance finished first. The overall winner did not use the training set provided by NIST (he used a much larger proprietary training set), and therefore was not affected by the different distributions in the training set and test set. 4 DISCUSSION The tangent distance algorithm described in the implementation section can be improved/adjusted in at least four different ways: 1) approximating the tangent distance for better speed 2) modifying the tangent distance itself, 3) changing the set of transformations/tangent vectors and 4) using the tangent distance with classification algorithms other than K-nearest neighbors, perhaps in combination, to minimize the number of prototypes. We will discuss each of these aspects in turn. Approximation: The distance between two hyperplanes TE and Tp going through P and E can be approximated by computing the projection PEep) of Ponto TE and Pp(E) of E onto Tp. The distance IIPE(P) - Pp(E)1I can be computed in O(n(mE + mp» multiply-adds and is a fairly good approximation of D(E, P). This approximation can be improved at very low cost by computing the closest points between the lines defined by (E, PEep»~ and (P, Pp(E». This approximation was used with no loss of performance to reduce the number of computed tangent distance from 100 to 20 (this involves an additional "prefilter"). In the case of images, another time-saving idea is to compute tangent distance on progressively smaller sets of progressively higher resolution images. Changing the distance: One may worry that the tangent planes of E and P may be parallel and be very close at a very distant region (a bad side effect of the linear a.pproximation). This effect can be limited by imposing a constraint of the form IlaEIi < f{E and lIapli < f{p. This constraint was implemented but did not yield better results. The reason is that tangent planes are mostly orthogonal in high dimensional space and the norms of [[aEIi and !lapll are already small. The tangent distance can be normalized by dividing it by the norm of the vectors. This improves the results slightly because it offsets side effects introduced in some transformations such as scaling. Indeed, if scaling is a transformation of interest, there is a potential danger of finding the minimum distance between two images after they have been scaled down to a single point. The linear approximation of the scaling transformation does not reach this extreme, but still yields a slight degradation of the performance. The error rate reported on the USPS database can be improved to 2.4% using this normalization (which was not tried on NIST). Tangent distance can be viewed as one iteration of a Newton-type algorithm which finds the points of minimum distance on the true transformation manifolds. The vectors aE and ap are the coordinates of the two closest points in the respective tangent spaces, but they can also be interpreted for real (non-linear) transformations. If ae; is the amount of the translation tangent vector that must be added to E to make it as close as possible to P, we can compute the true translation of image E by ae,; pixels. In other words, E'(aE) and pl(ap) are projected onto 58 Simard, Cun, and Denker close points of SE and Sp. This involves a resampling but can be done efficiently. Once this new image has been computed, the corresponding tangent vectors can be computed for this new image and the process can be repeated. Eventually this will converge to a local minimum in the distance between the two transformation manifold of P and E. The tangent distance needs to be normalized for this iteration process to work. A priori knowledge: The a priori knowledge used for tangent vectors depends greatly on the application. For character recognition, thickness was one of the most important transformations, reducing the error rate from 3.3% to 2.6%. Such a transformation would be meaningless in, say, speech or face recognition. Other transformations such as local rubber sheet deformations may be interesting for character recognition. Transformations can be known a priori or learned from the data. Other algorithms, reducing the number of prototypes: Tangent distance is a general method that can be applied to problems other than image recognition, with classification methods other than K-nearest neighbors. Many distance-ba.sed classification schemes could be used in conjunction with tangent distance, among them LVQ (Kohonen, 1984), and radial basis functions. Since all the operators involved in the tangent distance are differentiable, it is possible to compute the partial derivative of the tangent distance (between an object and a prototype) with respect to the tangent vectors, or with respect to the prototype. Therefore the tangent distance operators can be inserted in gradient-descent based adaptive machines (of which LVQ and REF are particular cases). The main advantage of learning the prototypes or the tangent vectors is that fewer prototypes may be needed to reach the same (or superior) level of performance as, say, regular K-nearest neighbors. In conclusion, tangent distance can greatly improve many of the distance-based algorithms. We have used tangent distance in the simple K-nearest neighbor algorithm and outperformed all existing techniques on standard classification tasks. This surprising success is probably due the fact that a priori knowledge can be very effectively expressed in the form of tangent vectors. Fortuna.tely, many algorithms are based on computing distances and can be adapted to express a priori knowledge in a similar fashion. Promising candidates include Parzen windows, learning vector quantization and radial basis functions. References Kohonen, T. (1984). Self-organization and Associative Memory. In Springer Sedes in Information Sciences, volume 8. Springer-Verlag. Le Cun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, Vv., and Jackel, 1. D. (1990). Handwritten digit recognition with a back-propagation network. In Touretzky, D., editor, Advances in Neural Information Processing Systems 2 (NIPS *89), Denver, CO. Morgan Kaufman. Sibson, R. (1978) . Studies in the Robustness of Multidimensiona.l Scaling: Procrust.es Statistices. 1. R. Statist. Soc., 40:234-238. Simard, P. Y., LeCun, Y., Denker, J., and Victorri, B. (1992a). An Efficient Met.hod for Learning Invariances in Adaptive classifiers. In International Conference on Pattern Recognition, volume 2, pages 651-655, The Hague, Netherlands. Simard, P. Y., Victorri, B., LeCun, Y., and Denker, J. (1992b). Tangent Prop - A formalism for specifying selected invariances in an adaptive network. In Neural Information Processing Systems, volume 4, pages 895-903, San Mateo, CA. Sinden, F. and Wilfong, G. (1992). On-line Recognition of Handwritten Symbols. Technical Report 11228-910930-02IM, AT&T Bell Laboratories.
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Planar Hidden Markov Modeling: from Speech to Optical Character Recognition Esther Levin and Roberto Pieraccini A IT Bell Laboratories 600 Mountain Ave. Murray Hill, NJ 07974 Abstract We propose in this paper a statistical model (planar hidden Markov model PHMM) describing statistical properties of images. The model generalizes the single-dimensional HMM, used for speech processing, to the planar case. For this model to be useful an efficient segmentation algorithm, similar to the Viterbi algorithm for HMM, must exist We present conditions in terms of the PHMM parameters that are sufficient to guarantee that the planar segmentation problem can be solved in polynomial time, and describe an algorithm for that. This algorithm aligns optimally the image with the model, and therefore is insensitive to elastic distortions of images. Using this algorithm a joint optima1 segmentation and recognition of the image can be performed, thus overcoming the weakness of traditional OCR systems where segmentation is performed independently before the recognition leading to unrecoverable recognition errors. Tbe PHMM approach was evaluated using a set of isolated band-written digits. An overall digit recognition accuracy of 95% was acbieved. An analysis of the results showed that even in the simple case of recognition of isolated characters, the elimination of elastic distortions enhances the performance Significantly. We expect that the advantage of this approach will be even more significant for tasks such as connected writing recognition/spotting, for whicb there is no known high accuracy method of recognition. 1 Introduction The performance of traditional OCR systems deteriorate very quickly when documents are degraded by noise, blur, and other forms of distortion. Tbe main reason for sucb deterioration is that in addition to the intra-class cbaracter variability caused by distortion, the segmentation of the text into words and characters becomes a nontrivial task. In most of the traditional systems, such segmentation is done before recognition, leading to many recognition errors, since recognition algorithms cannot usually recover from errors introduced in the segmentation pbase. Moreover, in many cases the segmentation is illdefined, since many plausible segmentations migbt exist, and only grammatical and linguistic analysis can find the "rigbt " one. To address these problems, an algorithm is needed that can : • be tolerant to distortions leading to intra-class variability 731 732 Levin and Pieraccini • perform segmentation together with recogruuon, thus jointly optimizing both processes, while incorporating grammatica1llinguistic constraints. In this paper we describe a planar segmentation algorithm that has the above properties. It results from a direct extension of the Viterbi (Forney, 1973) algorithm, widely used in automatic speech recognition, to two-dimensional signals. In the next section we desaibe the basic hidden Markov model and define the segmentation problem. In section 3 we introduce the planar HMM that extends the HMM concept to model images. The planar segmentation problem for PHMM is defined in section 4. It was recently shown (Kearns and Levin, 1992) that the planar segmentation problem is NP-hard, and therefore, in order to obtain an effective planar segmentation algorithm, we propose to constrain the parameters of the PHMM. We show sufficient conditions in terms of PHMM parameters for such algorithm to exist and describe the algorithm. This approach differs from the one taken in references (Chellappa and Chatterjee, 1985) and (Derin and Elliot, 1987), where instead of restricting the problem, a suboptimal solution to the general problem was fmUld. Since in (Kearns and Levin, 1992) it was also shown that planar segmentation problem is hard to approximate, such suboptimal solution doesn't have any guaranteed bounds. The segmentation algorithm can now be used effectively not only for aligning isolated images, but also for joint recognition/segmentation, eliminating the need of independent segmentation that usually leads to unrecoverable errors in recognition. The same algorithm is used for estimation of the parameters of the model given a set of example images. In section 5, results of isolated hand-written digit recognition experiments are presented. The results indicate that even in the simple case of isolated characters, the elimination of planar distottions enhances the performance significantly. Section 6 contains the summary of this work. 2 Hidden Markov Model The HMM is a statistical model that is used to describe temporal signals G= (g(t): 1 ~t~ T, g E G c Rill in speech processing applications (Rabiner, 1989; Lee et ai., 1990; Wilpon et ai., 1990; Pieraccini and Levin, 1991). The HMM is a composite statistical source comprising a set s::: { 1, ... ,TR} of TR sources called states. The i-th state, i E S, is characterized by its probability distribution Pj(g) over G. At each time t only one of the states is active, emitting the observable g(t). We denote by s(t), s(t) E s the random variable corresponding to the active state at time t. The joint probability distribution (for real-valued g) or discrete probability mass (for g being a discrete variable) P (s(t),g(t» for t > 1 is characterized by the following property: P(s(t),g(t) I s(1:t-l),g(1:t-l»=P(s(t) I s(t-l» P(g(t) I s(t»== (1) =P(s(t) I s(t-l» ps(r)(g(t» , where s(l:t-l) stands for the sequence (s(1), ... s(t-l) l, and g(l:t-l)= (g(1), ... ,g(t-l)}. We denote by ajj the transition probability P(s(t)=j I s(t-l)=i), and by ~, the probability of state i being active at t=l, 1tj =P(s(1)=i). The probability of the entire sequence of states S=s(1:n and observations G=g(1:T) can be expressed as T P(G,S)=1ts(1)Ps(1)(g(1» n as(r-l)s(r) Ps(r)(g(t». r=2 (2) The interpretation of equations (1) and (2) is that the observable sequence G is generated in two stages: first, a sequence S of T states is chosen according to the Markovian disfribution parametrized by {a jj } and {1t;}; then each one of the states s (t), 1~~T, in S generates an observable g(t) according to its own memoryless distribution PS(I)' forming the observable sequence G. This model is called a hidLlen Markov model, because the state sequence S is not given, and only the observation sequence G is known. A particular case of this model, called a left-ta-right HMM, where ajj =0 for j<i, and Planar Hidden Markov Modeling: from Speech to Optical Character Recognition 733 1t, = I, is especially useful for speech recognition. In this case each state of the model represents an unspecified acoustic unit, and due to the "left-to-rigbt" structure, the whole word is modeled as a concatenation of such acoustic \D1its. The time spent in each of the states is not fixed, and therefore the model can take into account the duration variability between different utterances of the same word. The segmAentation problem of HMM is that of estimating the most probable state sequence S, given the observation G, S=ar~P(S I G)=ar~P(G,S). (3) s s A The problem of finding S through exhaustive search is of exponential complexity, since there exist TTl possible state sequences, but it can be solved in polynomial time using a dynamic programming approach (i.e. Viterbi algorithm). The segmentation plays a central role in all HMM-based speech recognizers, since for connected speech it gives the segmentation into words or sub-word units, and performs a recognition simultaneously, in an optimal way. This is in contrast to sequential systems, in which the connected speech is first segmented into wordslsubwords according to some rules, and than the individual segments are recognized by computing the appropriate likelihoods, and where many recognition errors are caused by tmrecoverable segmentation errors. Higher-level syntactical knowledge can be integrated into decoding process through transition probabilities between the models. The segmentation is also used for estimating the HMMs parameters using a corpus of a training data. 3 The Two-Dimensional Case: Planar HMM In this section we describe a statistical model for planar image G={g(x,y):(x,y)e Lx.Y , g e G}. We call this model "Planar HMM" (pHMM) and design it to extend the advantages of conventional HMM to the two-dimensional case. The PHMM is a composite source, comprising a set s = {(i,y), I~~R' l~y~Y R} of N=XRYR states. Each state in s is a stochastic source characterized by its probability density Pi.y(g) over the space of observations g e G. It is convenient to think of the states of the model as being located on a rectangular lattice where each state corresponds to a pixel of the corresponding reference image. Similarly to the conventional HMM, only one state is active in the generation of the (x,y)-th image pixel g (x,y). We denote by s(x,y) e s the active state of the model that generates g (x,y). The joint distribution governing the choice of active states and image values has the following Markovian property: P(g(x,y), s(x,y) I g(1:X, I:y-l), g(1:x-I,y), s(l:X,I:y-l),s(l:x-l,y»= (4) =P(g(x,y) I s(x,y» P(s(x,y) I s(x-l,y),s(x,y-l)= =PS(z.y)(g(x,y» P(s(x,y) I s(x-l,y),s(x,y-l»= where g(l:X,y-l)= {g (x,y): (x,y) e Rx.y-d, g (1:x-l,y)= {g (l,y), ... ,g (x-l,y)}, and s(l:X,l:y-l), s(1:x-l,y) are the active states involved in generating g(1:X,y-l), g(1:x-l,y), respectively, and RX,y-l is an axis parallel rectangle between the origin and the point (X,y-l). Similarly to the one-dimensional case, it is useful to define a left-toright bottom-up PHMM where P(s(x,y)=(m,n) I s(x-l,y)::=(i,j),s(x,y-l)=(k,l)):;t:{) only when i9n and l~, that does not allow for "fold overs" in the state image. The Markovian property (4) allows the lefl-t<rright bottom-up PHMM to model elastic distortions among different realizations of the same image, similarly to the way the Markovian property in left-to-right HMM handles temporal alignment We have chosen this definition (4) of Markovian property rather than others (see for example Oerin and Kelly, 1989) since it leads to the formulation of a segmentation problem which is similar to the planar alignment defined in (Levin and Pieraccini, 1992). 734 Levin and Pieraccini Using property (4), the joint likelihood of the image G = g(l:X, l:Y) and the state image S=s(l:X, l:Y) can be written as where: and x y P(G,S)= nnps(r,y)(g(x.y» r=1 y .. 1 X H Y V Y X 1I:s (I,I) n as (.x-I, I),s(r,1) n as (I,y-I),s(l,y) n n As (r-I ,y),s(r,y-I),s (r,y)' r~ y~ y~z~ A (i,j),(t,i),(m,II) =P (s (x.y) = (m,n) I s (x-l,y) = (i,j), s (x.y-l) = (k,l) ), H a(i,j),(m,II) =P(s(x.l)=(m,n) I s(x-l,l)=(i,j», v a(t,I),(m,II) = P (s (l,y) = (m, n) I s (l,y) = (k,1) ), 1I:ij =P(s(l, l)=(i,j» (5) denote the generalized transition probabilities of PHMM, Similarly to HMM, (5) suggests that an image G is generated by the PHMM in two successive stages: in the first stage the state matrix S iven~d according to the Markovian probability distribution parametrized by {A}, {a }, {a }, and {1t}, In the second stage, the image value in the (x,y)-th pixel is produced independently from other pixels according to the distribution of the s(x,y)-th state ps(z,y)(g). As in HMM, the state matrix S in most of the applications is not known, only G is observed, 4 Planar Segmentation Problem A The segmentation problem of PHMM is that of finding the state matrix S that best explains the observable image G and defines an optimal alignment of the image to the model. Solving this problem eliminates the sensitivity to inlra-class elastic distortions and allows for simulqrneous segmentation/recognition of images similarly to the onedimensional case. S can be estimated as in (3) by S = Qrgmax P (G,S). If we approach this s maximization by exhaustive search, the computational complexity is exponential, since there are (XR yR)XY different state matrices, Since the segmentation problem is NP-hard (Kearns and Levin, 1992), we suggest to simplify the problem by constraining the parameters of the PHMM, so that efficient segmentation algorithm can be found. In this section we present conditions in terms of the generalized transitiop probabilities of PHMM that are sufficient to guarantee that the most likely state image S can be computed in polynomial time, and describe an algorithm for doing that. A For the problem of finding S to be solved in polynomial time. there should exist a grouPin~G of the set s of states of the model into NG mutually exclusivel subsets of states "(P' s = U "(po The generalized transition probabilities should satisfy the two following p=1 constraints with respect to such grouping: H A (i,j),(t,I),(m,II) ;to ; a(i,j),(m,II) ;to (6) onlyifthereexistsp, l$p$NG, sucbthat(i,j),(m,n) E "(p. v v A (i,j),(t,I),(m,II) =A(i,i),(kl,I\),(m,II) ; a(t,I),(m,II) =a(tl,ld,(m.II) (7) I It is lXlssible to drop the mutually exclusiveness constraints by duplicating states, but then we have to ensure that the number of subsets, NG, should be lXl1ynomial in the dimensions of the model XR• YR , Planar Hidden Markov Modeling: from Speech to Optical Character Recognition 735 if there exists p , 1 Sp SNG. such that (k,l) , (khl l ) E Yp. Condition (6) means that the the left neighbor (i,j) of the state (m,n) in the state matrix S must be a member of the same subset Yp as (m, n). Condition (7) means that the value of transition probability A (i.j).(k.I).(Ift.,,) does not depend explicitly on the identity (k, l) of the bottom neighboring state, but only on the subset Yp to which (k,l) belongs. " Under (6) and (7) the most likely state matrix S can be found using an algorithm described in (Levin and Pieraccini, 1992). This algorithm makes use of the Viterbi procedure at two different levels. In the first stage optimal segmentation is computed for each subset yp with each image raw using Viterbi. Then global segmentation is fmmd, through Viterbi, by combining optimally the segmentations obtained in the previous stage. Although conditions (6),(7) are hard to check in practice since any possible grouping of the states has to be considered, they can be effectively used in constructive mode, i.e., chosing one particular grouping, and then imposing the constraints (6) and (7) on the generalized transition probabilities with respect to this grouping. For example, if we choose Yp= {(i,y) I IS.iSXR, y =p }, 1 Sp S YR, then the constraints (6),(7) become: H A (i.j).(l.I).(Ift.,,);;':O, a(i.i).(m.,,) ;;,:0 only for j = n , (8) and, v v A (i.i).(l.I).(Ift.,,)=A(i.i).(kl.l).(Ift.,,), a(l.l).(m.,,) =a(ll.I).(Ift.,,) for ISk lt k SXR • (9) Note that constraints (6), (7) break the symmetry between the roles of the two coordinates. Other sets of conditions can be obtained from (6) and (7) by coordinate transformation. For example, the roles of the vertical and the horizontal axes can be exchanged. A grouping and constraints set chosen for a particular application should reflect the geometric properties of the images. 5 Experimental Results The PHMM approach was tested on a writer-independent isolated handwritten digit recognition application. The data we used in our experiments was collected from 12 subjects (6 for training and 6 for test). Each subject was asked to write 10 samples of each digiL Samples were written in fixed-size boxes, therefore naturally size-normalized and centered. Each sample in the database was represented by a 16x16 binary image. Each character class (digit) was represented by a single PHMM, satisfying (6) and (7). Each PHMM had a strictly left-to-right bottom-up structure, where the state matrix S was restricted to contain every state of the model, i.e., states could not be skipped. All models had the same number of states. Each state was represented by its own binary probability distribution, i.e., the probability of a pixel being 1 (black) or 0 (white). We estimated these probabilities from the training data with the following generalization of the Viterbi training algorithm (Jelinek, 1976). For the initialization we uniformly divided each training image into regions corresponding to the states of its model. The initial value of Pj(g=I) for the i-th state was obtained as a frequency count of the black pixels in the corresponding region over all the samples of the same digiL Each iteration of the algorithm consisted of two stages: first,,,the samples were aligned with the corresponding model, by finding the best state matrix S. Then, a new frequency count for each state was used to update Pj(1), according to the obtained alignment. We noticed that the training procedure converged usually after 2-4 iterations, and in all the experiments the algorithm was stopped at the 10th iteration. The recognition was performed by assigning the test sample to the class k for which the alignment likelihood was maximal. 736 Levin and Pieraccini Table 1 shows the number of errors in the recognition of the training set and the test set for different sizes of the models. Number of states Recognition Errors XR=YR Training Test 6 78 82 8 36 50 9 35 48 10 26 32 11 21 38 12 18 42 16 36 64 Table 1: Number of errors in the recognition of the training set and the test set for different size of the models (out of 600 trials in both cases) It is worth noting the following two points. First, the test error shows a minimum for XR = YR = 10 of 5%. By increasing or decreasing the number of states this error increases. This phenomenon is due to the following: 1. The typical under/over parametrization behavior. 2. Increasing the number of states closer to the size of the modeled images reduces the flexibility of the alignment procedure, making this a trivial uniform alignment when XR = YR = 16. Also, the training error decreases monotonically with increasing number of states up to XR = Y R = 16. This is again typical behavior for such systems, since by increasing the number of states, the number of model parameters grows, improving the fit to the training data. But when the number of states equals the dimensions of the sample images, XR = YR = 16, there is a sudden Significant increase in the training error. This behavior is consistent with point (2) above. Fig. 1 shows three sets of models with different numbers of states. The states of the models in this figure are represented by squares, where the grey level of the square encodes the probability P(g=I). The (6x6) state models have a very coarse representation of the digits, because the number of states is so small. The (lOxl0) state models appear much sharper than the (16x16) state models, due to their ability to align the training samples. This preliminary experiment shows that eliminating elastic distortions by the alignment procedure discussed above plays an important role in the task of isolated character recognition, improving the recognition accuracy significantly. Note that the simplicity of this task does not stress the full power of the PHMM representation, since the data was isolated, size-normalized, and centered. On this task, the achieved performance is comparable to that of many other OCR systems. We expect that in harder tasks, involving connected text, the advantage of the proposed method will enhance the performance. Recently, this approach is being successfully applied to the task of recognition of noisy degraded printed messages (Agazzi et aL, 1993). 6 Summary and Discussion In this paper we describe a planar hidden Markov model and develop a planar segmentation algorithm that generalizes the Viterbi procedure widely used in speech recognition. This algorithm can be used to perform joint optimal recognition/segmentation of images incorporating some grammatical constraints and tolerating intra-class elastic distortions. The PHMM approach was tested on an isolated, hand-written digit recognition application. An analysis of the results indicate that even in a simple case of isolated characters, the elimination of elastic distortions enhances Planar Hidden Markov Modeling: from Speech to Optical Character Recognition 737 recognition performance significantly. We expect that the advantage of this approach will be even more valuable in harder tasks, such as cursive writing recognition/spotting, for which an effective solution using the current available techniques has not yet been found . • a ·t ,~;. .'.;; $: & ~:J .:~.::.'. , :til:::.: .:.:. ::::: ·fi .::: Figure 1: Three sets of models with 6x6, lOxlO, and 16x16 states. References O. E. Agazzi, S. S. Kuo, E. Levin, R. Pieraccini, " Connected and Degraded Text Recognition Using Planar Hidden Markov Models," Proc. Of Int. COnference on Acoustics Speech and Signal Processing, April 1993. R. Chellappa, S. Chatterjee, "Classification of textures Using Gaussian Markov Random Fields," IEEE Transactions on ASSP , Vol. 33, No.4, pp. 959-963, August 1985. 738 Levin and Pieraccini H. Derin, H. Elliot, "Modeling and Segmentation of Noisy and Textured Images Using Gibbs Random Fields," IEEE Transactions on PAMI, Vol. 9, No.1 pp. 39-55, January 1987. H. Derin, P. A. Kelly, 'Discrete-Index Markov-Type Random Processes,' in IEEE Proceedings, vol 77, #10, pp.1485-1510, 1989 G.D. Forney, "The Viterbi algorithm," Proc. IEEE. Mar. 1973. F. Jelinek, "Continuous Speech Recognition by Statistical Methods," Proceedings of IEEE, vol. 64, pp. 532-556, April 1976. M. Keams, E. Levin, Unpublished, 1992. C.-H. Lee, L. R. Rabiner, R. Pieraccini, J. G. Wilpon, "Acoustic Modeling for Large Vocabulary Speech Recognition," Computer Speech and Language, 1990, No.4, pp. 127-165. E. Levin, R. Pieraccini, "Dynamic Planar Warping and Planar Hidden Markov Modeling: from Speech to Optical Character Recognition," submitted to IEEE Trans. on PAMl. 1992. R. Pieraccini, E. Levin, "Stochastic Representation of Semantic Structure for Speech Understanding," Proceedings of EUROSPEECH 91, Vo1.2, pp. 383-386, Genova, September 1991. L.R. Rabiner, "A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition," Proc. IEEE, Feb. 1989. 1. G. Wilpon, L. R. Rabiner, C.-H. Lee, E. R. Goldman, "Automatic Recognition of Keywords in Unconstrained Speech Using Hidden Markov Models," IEEE Trans. on ASSP, Vol. 38, No. 11, pp 1870-1878, November 1990.
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Global Regularization of Inverse Kinematics for Redundant Manipulators David DeMers Dept. of Computer Science & Engr. Institute for Neural Computation University of California, San Diego La Jolla. CA 92093-0114 Kenneth Kreutz-Delgado Dept. of Electrical & Computer Engr. Institute for Neural Computation University of California, San Diego La Jolla, CA 92093-0407 Abstract The inverse kinematics problem for redundant manipulators is ill-posed and nonlinear. There are two fundamentally different issues which result in the need for some form of regularization; the existence of multiple solution branches (global ill-posedness) and the existence of excess degrees of freedom (local illposedness). For certain classes of manipulators, learning methods applied to input-output data generated from the forward function can be used to globally regularize the problem by partitioning the domain of the forward mapping into a finite set of regions over which the inverse problem is well-posed. Local regularization can be accomplished by an appropriate parameterization of the redundancy consistently over each region. As a result, the ill-posed problem can be transformed into a finite set of well-posed problems. Each can then be solved separately to construct approximate direct inverse functions. 1 INTRODUCTION The robot forward kinematics function maps a vector of joint variables to the end-effector configuration space, or workspace, here assumed to be Euclidean. We denote this mapping by f(·) : en -+ wm ~ X m , f(O) I-t x, for 0 E en (the input space or joint space) and x E wm (the workspace). When m < n, we say that the manipulator has redundant degrees--of -freedom (dot). The inverse kinematics problem is the following: given a desired workspace location x, find joint variables 0 such that f(O) = x. Even when the forward kinematics is known, 255 256 DeMers and Kreutz-Delgado the inverse kinematics for a manipulator is not generically solvable in closed form (Craig. 1986). This problem is ill-posedl due to two separate phenomena. First. multiple solution branches can exist (for both non-redundant as well as redundant manipulators). The second source of ill-posedness arises because of the redundant dofs. Each of the inverse solution branches consists of a submanifold of dimensionality equal to the number of redundant dofs. Thus the inverse solution requires two regularizations; global regularization to select a solution branch. and local regularization. to resolve the redundancy. In this paper the existence of at least one solution is assumed; that is. inverses will be sought only for points in the reachable workspace. i.e. desired x in the image of 1(·). Given input-output data generated from the kinematics mapping (pairs consisting of joint variable values & corresponding end-effector location). can the inverse mapping be learned without making any a priori regularizing assumptions or restrictions? We show that the answer can be "yes". The approach taken towards the solution is based on the use of learning methods to partition the data into groups such that the inverse kinematics problem. when restricted to each grouP. is well-posed. after which a direct inverse function can be approximated on each group. A direct inverse function is desireable. For instance. a direct inverse is computable quickly; if implemented by a feedforward network were used. one function evaluation is equivalent to a single forward propagation. More importantly. theoretical results show that an algorithm for tracking a cyclic path in the workspace will produce a cyclic trajectory of joint angles if and only if it is equivalent to a direct inverse function (Baker. 1990). That is. inverse functions are necessary to ensure that when following a closed loop the ann configurations which result in the same end-effector location will be the same. Unfortunately. topological results show that a single global inverse function does not exist for generic robot manipulators. However. a global topological analysis of the kinematics function and the nature of the manifolds induced in the input space and workspace show that for certain robot geometries the mapping may be expressed as the union of a finite set of well-behaved local regions (Burdick. 1991). In this case. the redundancy takes the form of a submanifold which can be parameterized (locally) consistently by. for example. the use of topology preserving neural networks. 2 TOPOLOGY AND ROBOT KINEMATICS It is known that for certain robot geometries the input space can be partitioned into disjoint regions which have the property that no more than one inverse solution branch lies within anyone of the regions (Burdick. 1988). We assume in the following that the manipulator in question has such a geometry. and has all revolute joints. Thus en = 1'" • the n-torus. The redundancy manifolds in this case have the topology of T n - m • n - m-dimensional torii. For en a compact manifold of dimensionality n. wm a compact manifold of dimensionality m. and I a smooth map from en to wm . let the differential del be the map from the tangent space of en at (J E en to the tangent space of wm at I( (J). The set of points in en which 1 lli-posedness can arise from having either too many or too few constraints to result in a unique and valid solution. That is, an overconstrained system may be ill-posed and have no solutions; such systems are typically solved by finding a least-squares or some such minimum cost solution. An underconstrained system may have multiple (possibly infinite) solutions. The inverse kinematics problem for redundant manipulators is underconstrained. Global Regularization of Inverse Kinematics for Redundant Manipulators 257 map to x E wm is the pre-image of x, denoted by 1-1 (x). The differential del has a natural representation given by an m by n Jacobian matrix whose elements consist of the first partial derivatives of I w.r.t. a basis of en. Define S as the set of critical points of I, which are the set of all 0 E en such that de/(O) has rank less than the dimensionality of wm. Elements of the image of S, I(S) are called the critical values. The set'R!.Wm\S are the regular values of f. For 0 E en, if 30· E 1-1(/(0)), O· E S, we call 0 a co-regular point of I. The kinematic ~napping of certain classes of manipulators (with the geometry herein assumed) can be decomposed based on the c<rregular surfaces which divide en into a finite number of disjoint, connected regions, Ci. The image of each Ci under I is a connected region in the workspace, Wi. We denote the kinematics mapping restricted to Cito be Ii; Ii : Ci --+ Wi. Locally, one inverse solution branch for a region in the workspace has the structure of a product space. We conjecture that (Wi, Tn-m ,Ci, flci) forms a locally trivial fiber bundle2 and that the Ci, therefore, form regions where the inverse is unique modulo the redundancy. Given a point in the workspace, x, for which a configuration is sought, global regularization requires choosing from among the multiple pre-image torii. Local regularization (redundancy resolution) requires finding a location on the chosen torus. We would like to effectively "mod out" the redundancy manifolds by constructing an indexed one-t<rone, invertible mapping from each pre-image manifold to a point in X m , and to obtain a consistent representation of the manifold by constructing an invertible mapping from itself to a set of n - m "location" parameters. 3 GLOBAL REGULARIZATION The existence of multiple solutions for even non-redundant manipulators poses difficult problems. U sua1ly (and often for plausible reasons) the manipulator's allowable configurations or task space is effectively constrained so that there exists only a single inverse solution (Martinetz et al., 1990; Kuperstein, 1991). This approach regularizes the problem by allowing the existence of only one-t<rone data. We seek to generalize to the multi-solution case and to learn all of the possible solutions. For an all revolute manipulator there typically will be multiple pre-image torii for a particular point in the workspace. The topology of the pre-image solution branches will generally be known from the type of geometry, although their number may not be obvious by inspection, and will usually be different for different regions of the workspace. An upper bound on the number of inverse solutions is known (Burdick, 1988); consequently, the determination of the number could be made by search for the best fit among the possibilities. The sampling and clustering approach described in (DeMers & Kreutz-Delgado, 1992) can be used to partition input-output data into disjoint pre-image sets. This approach uses samples of the forward behavior to identify the sets in the input space which map to '}. A fiber bundle is a four-tuple consisting of a base space, a fiber (here. T m - n ). a total space and a projection p mapping the total space to the base space with certain properties (here. the projection is equivalent to fi restricted to the total space). A locally trivial fiber bundle is one for which a consistent parameterization of the fibers is possible. 258 DeMers and Kreutz-Delgado "-. J.int. 2 0 -2 V ....,: , t " t ..t '" r",\ o "'.1at. 1 i ~I" r-1 ".J • .b~ 3 U • ...... • ::u .,. ... t: a U • • -" ~ : "" . .. • \ . '. \ ~ \. • ..,.1t 1 Figure 1: The two pre-image manifoldsfor positioning the end-effector of the 3-R planar manipulator at a specific (x, y) location. within a small distance of a specific location in the workspace. The pre-image points will lie on the disjoint pre-image manifolds. These manifolds will typically be separable by clustering3• Figures 1 shows two views of the two redundancy, or "self-motion" manifolds for a particular end-effector location of the 3-R planar arm. All of the input space points shown position the arm's end-effector near the same (x,y) location. Note that the input space is the 3-torus, T3. In order to visualize the space, the torus is "sliced" along each dimension. Thus opposite faces of the "cube" shown are identified with each other. 4 LOCAL REGULARIZATION The inverse kinematics problem for manipulators with redundant dofs is usually solved either by using differential methods, which attempt to exploit the redundancy by optimizing a task-<lependent objective function, or by using learning methods which regularize at training time by adding constraints equal to the number of redundant dof. The former may be computationally inefficient in that iterative solution of derivatives or matrix inversions are required, and may be unsatisfactory for real-time control. The latter is unsatisfying as it eliminates the run-time dexterity available from the redundancy; that is, it imposes prior constraints on the use of any extra dofs. Although in practice numerical, differential methods are used for redundancy resolution, it has recently been shown that simple recurrent neural networks can resolve the redundancy by optimization of certain side-constraints at run-time, (Jordan & Rumelhart, 1992), (Kindermann & Linden, 1990). The differential methods have a number of desireable properties. In general it is possible to iterate in order to achieve a solution of arbitrary accuracy. They also tend to be capable of handling very flexible constraints. Global regularization as discussed above can be used to augment such methods. For example, once a choice of a solution branch has been made, an initial starting location away from singularities can be selected, and differential methods used to achieve an accurate solution on that branch. Our work shows that construction of redundancy-parameterized approximations to di3Por end-effector locations near the co-regular values, the pre-image manifolds tend to merge. This phenomenon can be identified by our methods. Global Regularization of Inverse Kinematics for Redundant Manipulators 259 rect inverses are achievable. That is, the mapping from the workspace (augmented by a parameterization of the redundancy) to the input space can be approximated. This local regularization is accomplished by parameterizing the p~image solution branch torii. Given (enough) samples of (J points paired with their x image, a parameterization can be discovered for each branch. The method used exploits the fact that neighborhoods within each Ci map to neighborhoods, and that neighborhoods within each Wi have as pre-images a finite number of solution branches. First, all points in our sample which have their image near some initial point Xo are found. Pullin!; back to the input space by accessing the (J component of each of these «(J, x) data pairs finds the points in the pre-image set of the neighborhood of Xo. Now, because the topology of the p~image set is known (here, the torus), a self-organizing map of appropriate topology can be fit to the p~image points in order to parameterize this manifold. Neighboring torii have similar parameterizations; thus by repeating this process for a point Xl near Xo and using the parameterization of the p~image of Xo as initial conditions. a parameterization of the pre-image of Xl qualitatively similar to that for Xo can be constructed efficiently. By stepping between such "query points", Xi a set of parameterizations can be obtained. 5 THE 3-R REDUNDANT PLANAR ARM This approach can be used to provide a global and local regularization for a three-link manipulator performing the task of positioning in the plane. For this manipulator, I : T3 -... R2. The map I restricted to each connected region in the input space bounded by the co-regular separating surfaces defines Ii : T x R X 51 -... R X 51. The pre-image of a point in the workspace thus consists of either one or two 1-torii (the actual number can be no more than the number of inverse solutions for a non-redundant manipulator of the same type, (Burdick, 1988». Each torus is the pre-image of one of the restricted mappings Ii. The goal is to identify these torii and parameterize them. Figure 2 shows the input space and workspace of this arm, and their separating surfaces. These partition the workspace into disjoint annular regions,and the input space into disjoint tubular regions. The circles indicate workspace locations which can be reached in a kinematically singular configuration. The inverse image of these circles form the co-regular separating surfaces in the input space. For the link lengths used here (It = 5, /2 = 4, h = 3), there is a single self-motion manifold for workspace locations in regions A and C, and two self-motion manifolds for workspace locations in B and D. Figure 3 shows two views of the data points near the p~image manifolds for an end effector location in region A. and its parameterization by a self-organizing map (using the elastic net algorithm). Such a parameterization can be made for various locations in the workspace. Inverse kinematics can thus be computed by first locating the nearest parameterization network for a given workspace position and then choosing a configuration on the manifold. which can be done at run-time. Because the kinematics map is locally smooth, interpolation between networks and nodes on the networks can be done for greater accuracy. For convenience, a node on each torus was chosen as a canonical zero-point, and the remaining nodes assigned values based on their normalized distance from this point. Therefore all parameter values are scaled to be in the interval [0,1]. For some end-effector locations, there are two pre-image manifolds. These first need to be identified by a global partitioning, then the individual manifolds parameterized. The 260 DeMers and Kreutz-Delgado Input Space l __ ~~~~::~~ .. ~~------~~ -7t -----7t o Joint 3 Workspace .... Figure 2: The forward kinematics for a generic three-link planar manipulator. The separating sUrfaces in the input space do not depend on the value of joint angle 1, therefore the input space is shown projected to the joint 2 -joint 3 space. All end-effector locations inside regions A and C of the workspace have a single pre-image manifold in A and C, respectively, of the input space. End-effector locations inside regions B and D of the workspace have two pre-image manifolds, one in each of B and B' (resp. D and D' ) of the input space. manifolds may belong to one of a finite set of homotopy classes; that is. because they "live" in an ambient space which is a torus, they mayor may not wrap around one or more of the dimensions of the torus. Unlike in Euclidean space. where there are only two possible onedimensional manifolds. there are multiple topologically distinct types (homotopy classes) of closed loops which can serve as self-motion manifolds. Fortunately. because physical robots rarely have joints with unlimited range of motion. in practice the manifolds will usually not have wraparound. However. we should like to be able to parameterize any possibility. Appropriate choice of topology for a topology-preserving net results in an effective parameterization. Figure 4 shows two views of a parameterization for one of the self-motion manifolds shown above in Figure 1. which is the pre-image for an end-effector location in region B of Figure 2. 6 DISCUSSION The global regularization accomplished by the method described above partitions the original input/output data into sets for each of the distinct Ci regions. The redundancy parameters, t. obtained by local regularization can be used to augment this data. resulting in a transformation of the (0, x) data into (0 i, (Xi, t». Let 7 : 0 1-+ t be a function that computes a parameter value for each 0 in the input space. Let fi(O) = (fi(O), 7(0». By construction. the regularized mapping fi : 0 1-+ (x, t) is one-to-one and onto. Now, given examples from a one-to-one mapping, the inverse map h-1 (x, t) 1-+ 0 can be directly approximated by, e.g .• a feedforward neural network. Global Regularization of Inverse Kinematics for Redundant Manipulators 261 Joint 2 Anq Ie -1 1 1 Joint 3 Angle 0 .5 o i nt 1 Angle -L-_______ ~2.5 Figure 3: The data points in the llself-motion" pre-image manifold of a point in the workspace of the 3-R planar arm, and a closed, 1-D elastic network after adaptation to them. This manifold is smoothly contractible to a point since it does not llwrap around" any of the dimensions ofT3. This method requires data sample sizes exponential in the number of degrees of freedom of the manipulator and thus will not be adequate for large dof"snake" manipulators. However, practical industrial robots of 7-dof may be amenable to our technique, especially if, as is common, it is designed with a separable wrist and is thus composable into a 4-dof redundant positioner plus a 3-dof non-redundant orienter. This work can also be used to augment the differential methods of redundancy resolution. An approximate solution can be found extremely rapidly, and used to initialize gradient-based methods, which can then iterate to achieve a highly accurate solution. Global decisions such as choosing between multiple manifolds and identifying criteria for choosing locations on the manifold can now be made at run-time. Computation of an approximate direct inverse can then be made in constant time. Acknowledgements This work was supported in part by NSF Presidential Young Investigator award IRI9057631 and Fellowships from the California Space Institute and the McDonnell-Pew Center for Cognitive Neuroscience. The first author would like to thank the NIPS Foundation for providing student travel grants. References Daniel Baker (1990), "Some Topological Problems in Robotics", The Mathematical Intelligencer, Vol. 12, No. I, pp. 66-76. Joel Burdick (1988), "Kinematics and Design of Redundant Robot Manipulators", Stanford Ph.D. Thesis, Dept. of Mechanical Engineering. Joel Burdick (1991), "A Classification of 3R Regional Manipulator Singularities and Geometries", Proc. 19911£EE Inti. Con! Robotics & Automation, Sacramento. 262 DeMers and Kreutz-Delgado (,,;::================7-Il Joint 1 "ntl. - 2 Joi nt Z ",.,.1. 0 Joint ) .. ",1. 0 -2 -2 - 2 J---~------~-2 o Joint 1 h91e Figure 4: Two views of the same data points in one of the two Itself-motion" pre-image manifolds of a point in region B, and an elastic net after adaptation. It belongs to a different homotopy class than that of Fig 3 - it is not contractible to a point. John Craig (1986).Introduction to Robotics. David DeMers & Kenneth Kreutz-Delgado (1992). ''Learning Global Direct Inverse Kinematics'" in Moody, J. E .• Hanson. SJ. and Lippmann, R.P .• eds, Advances in Neural Information Processing Systems 4. 589-594. Michael 1. Jordan & David Rumelhart (1992), "Forward Models: Supervised Learning with a Distal Teacher", Cognitive Science 16,307-354. J. Kindermann & Alexander Linden (1990), "Inversion of Neural Networks by Gradient Descent". 1. Parallel Computing 14.277-286. Michael Kuperstein (1991), "INFANT Neural Controller for Adaptive Sensory-Motor Control". Neural Networks. Vol. 4, pp. 131-145. Thomas Martinetz, Helge Ritter. & Klaus Schulten (1990), ''Three-Dimensional Neural Networks for Learning Visuomotor Coordination of a Robot Arm", IEEE Trans. Neural Networks, Vol. 1, No. 1. Charles Nash & Siddhartha Sen (1983). Topology and Geometry for Physicists. Philippe Wenger (1992), "On the Kinematics of Manipulators with General Geometry: Application to the Feasibility Analysis of Continuous Trajectories", in M. Jamshidi, et al., eds, Robotics and Manufacturing: Recent Trends in Research, Education and Applications 4, 15-20 (ISRAM-92, Santa Fe).
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Object-Based Analog VLSI Vision Circuits Christof Koch Computation and Neural Systems California Institute of Technology Pasadena, CA John G. Harris Bimal Mathur, Shih-Chii Liu Rockwell International Science Center Thousand Oaks, CA MIT Artificial Intelligence Laboratory Cambridge, MA Jin Luo, Massimo Sivilotti Tanner Research, Inc. Pasadena, CA Abstract We describe two successfully working, analog VLSI vision circuits that move beyond pixel-based early vision algorithms. One circuit, implementing the dynamic wires model, provides for dedicated lines of communication among groups of pixels that share a common property. The chip uses the dynamic wires model to compute the arclength of visual contours. Another circuit labels all points inside a given contour with one voltage and all other with another voltage. Its behavior is very robust, since small breaks in contours are automatically sealed, providing for Figure-Ground segregation in a noisy environment. Both chips are implemented using networks of resistors and switches and represent a step towards object level processing since a single voltage value encodes the property of an ensemble of pixels. 1 CONTOUR-LENGTH CHIP Contour length computation is useful for further processing such as structural saliency (Shaashua and Ullman, 1988), which is thought to be an important stage before object recognition. This computation is impossible on an analog chip if we 828 Object-Based Analog VLSI Vision Circuits 829 CI ~ c 0 · ~ " > -CI · tI" 01 0 0 .... -" 0 > CI '0' ,1'4 L J o II • 01'4 , :JCI · 0 · .. 0 10 20 :so Contour Length Figure 1: Figure 1: Plot of measured voltage vs. contour length from 30 different contours scanned into the contour length chip. The voltage is a linear function of contour length. are restricted to pure pixel- or image-based operations. The dynamic wire methodology provides dedicated lines of communication among groups of pixels of an image which share common properties (Liu and Harris, 1992). In simple applications, object regions can be grouped together to compute the area or the center of mass of each object. Alternatively, object boundaries may be used to compute curvature or contour length. These ideas are not limited to sets of simple electrical wires; resistive networks can also be configured on the fly. The problem of smoothing object contours using resistive dynamic wires has been previously studied (Liu and Harris, 1992). In the contour-length application, pixels along image contours are electrically connected by a reconfigurable dynamic wire. The first step of processing requires that each contour choose an arbitrary but unique leader pixel. The top of Fig. 2 shows several examples of contours and indicates which pixels where chosen as leaders by the chip. The leader is responsible for connecting a shunting resistor between the shared dynamic wire and ground. If each pixel on the contour supplies a constant amount of current to the dynamic wire, all of the current must flow through the shunting resistor. Therefore, the voltage on the wire will encode the contour length. Fig. 1 shows the linear relationship between the measured voltage and the contour length. The bottom half of Fig. 2 shows the length of several example contours using an intensity coding. The brighter contours indicate a higher voltage and therefore a longer contour. The contour length chip was fabricated through MOSIS using 2J.Lm CMOS technology. The prototype 2x2 mm 2 chip contains an a 7x7 pixel array. 830 Koch, Mathur, Liu, Harris, Luo, and Sivilotti "'Q-l . .. Em a I I q i . =..c ..... a I I :.l;l.l;l.l;l.!;m:l:ll;l.l;l.l;l.l; ::::::::::::::::::::~ ~lli~r"" llii~l~ 11111111 lllllil .'.;.; ........ . II : .. :;!;i; .. I.;!.;i.:!.:; ... :.:.; .. :.;.:.·<:.;~ .. ' ... :.'.:~I.·.:i.:'.:.. ::i.I .. :I:;.:."::.I::.i.I::: .• fllt:it!lj ....... ~ .. Figure 2: Four binary contour images were scanned into the contour-length chip and are shown in the top figure. The highlighted pixel in each contour was chosen by the chip to be the leader. The bottom figure shows the measured voltages (indicated by intensity) from the contour-length chip for the four images are shown. Since the intensity of each pixel encodes its length, the longer contours are brighter. Object-Based Analog VLSI Vision Circuits 831 The most challenging aspect of the design of the contour-length chip is the circuitry to uniquely select a pixel from each contour to be the leader. The leader is selected by entering all the pixels along each contour in a competition. The winner of this competition will be the leader. This competition requires each node to charge up its own capacitor once a reset line has been triggered. The first node that charges its capacitor above the trip point of a digital inverter will pull down a global precharge wire which connects all the pixels along the contour. This wire will in turn latch the states of the winner and losers. One of the pixels will normally toggle first because of the inherent offsets and component mismatches in silicon. 2 FIGURE-GROUND CHIP Ullman (1984) proposed that a visual routine is used in human vision to determine if a specified point in the visual field is inside or outside of one (or more) closed visual contours. We describe such a chip that labels all points inside a givenpossibly incomplete and broken-contour. We assume that the presence of an edge in the image causes switches at the corresponding grid point within a rectangular resistive network to open (Fig. 3). A closed edge contour will then correspond to a series of open switches on this grid. We assume that the visual contour will always encompass the central grid point in the array. At this point, the resistive grid is connected to the battery V/ig, while the periphery of the array is grounded to Vgnd. If the voltage at all other grid points is left floating and the contour is complete, that is, the central grid point is completely isolated from the periphery of the chip by a series of open grid points, the voltage at all points inside the contour rises to V/ig, while the voltage at grid points outside the contour will settle to Vgnd . Thus, the figure will be labeled by one voltage level and ground by another. Contours in real images are frequently incomplete, but instead have broken segments of one or more pixels. This will enable the current to flow through these holes in the contour, smearing out the voltage level between inside and outside. We exploit a property of Mead's (1989) Hres circuit, used to implement the resistances, to achieve contour completion. While the current flowing through Hres is linear in the voltage gradient for small voltage differences, it saturates for large voltage gradients. At those locations where the contour is broken, the saturating resistances limit the current flow, preventing smoothing of the voltage profile to occur. Figure 4 shows the responses of the Figure-Ground chip to different input patterns collected with a fixed bias: V/ig = 3.5 V and Vgnd = 2 V. The two-dimensional data is presented as pairs of images. The input patterns are located on the left while the corresponding voltage outputs are presented next to the input on the right. The black-white patterns are used to represent the binary input data encoding object boundaries. Thus, at all locations marked in black, the associated switches shown in Fig. 1a are opened. The gray-scale on the right denotes output voltage levels, where the darkest value corresponds to V/ig and the brightest to Vgnd. The center pixel of the view field is always set to V/ig. Notice that at every node where a boundary input signal (in black) appears and the switches are opened, the output voltage at that node is tied to Vgnd . This can be seen best in (e; white outline). To evaluate the ability of our circuit to perform Figure-Ground segregation in the presence of breaks in the contour, more and wider breaks are introduced into a simple square 832 Koch, Mathur, Liu, Harris, Luo, and Sivilotti +~+ , ., Voltage tvr.-!" + +.¢. -+ ;:'. + -Vgnd .& (a) (b) Figure 3: (a) The Figure-Ground network is made up of resistors and switches. The input to the chip is a binary edge map. At every grid point in the rectangular array where edges have been found, four switches are opened, isolating that node from its four neighbors (the shaded edge contour corresponds to a series of isolated nodes). We assume that the central point in the array is always enclosed by the contour. This point is connected to a voltage source V/ig, while the periphery is connected to the voltage Vgnd. If the contour is unbroken, the voltage at each interior point will then rise to V/ig, while all outside grid points will settle to Vgnd. Thus, the object is rapidly segregated from the background. If the contour is not complete, the saturating resistors (indicated with simple resistors) will limit the current flowing through these holes in the contour and partially seal off the boundary. (b) represents a conceptual view of how an object (figure) is segregated from the background in the two-dimensional view field, in terms of two distinct voltage levels (V/ig labels the object and Vgnd labels the background). The circuit has 48 by 48 nodes on a 4.6 by 6.8 mm2 die size and was implemented using MOSIS 2 pm CMOS technology. n LJ I I I I I I I I II L-.J Object-Based Analog VLSI Vision Circuits 833 Figure 4 (a) The input consists of a completely enclosed box. The network is therefore broken into two isolated segments, the inside and the outside of the box and labeled by two very different voltage values, Vfig and Vgnd. (b) The object boundary has a break equal to one pixel at the center of the left and right edges. Due to the large Voltage difference across these two leaks, the saturated horizontal resistances, HRes, saturate, thereby helping to "seal" off these breaks using a very simple algorithm. (C) The width of the breaks in the contour increases to three pixels each. Yet HRes still acts to effectively seal the two holes and the "Figure" is segregated from the "Surround". (d) The width of the breaks increases to five pixels each. Due to the much smaller voltage gradient across this wider gap in the contour, the voltage spreads outside the figure. (e) A total of four breaks, each fi ve pixels wide, prevents the ''Figure'' from being segregated. The system can't decide whether a single object with wide breaks at its side or four separate objects are present. 834 Koch, Mathur, Liu, Harris, Luo, and Sivilotti • • • • • --_ .... __ ...... _-_. ( a ) ( b ) (c) Figure 5: The Figure-Ground response to a noisy and incomplete contour outlining a hand (the binary image shown in (a) is scanned in from off-chip). The output voltage is shown as intensity in (b) and as a 3-D plot in (c) . The center node is tied to 3.5 V and marked as black in (c). The shaded area labels all pixels whose voltage is above 2.4 V. Notice the voltage decay along the little finger, due to an incomplete contour at the finger tip. Object-Based Analog VLSI Vision Circuits 835 contour. The box and break points on the sides are center-row symmetrical and the breaks are respectively one, three and five pixels wide. In (e), two additional, five pixel wide breaks have been included. For small enough breaks, our circuit has an excellent boundary-completion capabilities. This is important for machine vision, since real images rarely have complete boundaries. The performance of the chip is illustrated in Fig. 4. If the contour is unbroken, the voltage inside the figure rises to V/ ig , segregating it from the surround. If a small gap appears in the contour, it can be partially sealed off by the action of the saturating resistance Hres, which limits the current flowing through this gap, inhibiting full voltage equalization from occurring. As the break in the contour becomes larger, the voltage gradient across the illusionary contour between the upper and the lower part of the figure becomes smaller and smaller. If Hres is set to a low conductance, the gradient becomes larger again (Fig. 5c); now, however, the chip fails to discriminate between very small and large gaps. Note that inside and outside are strictly defined only for a closed contour. Thus, it is somewhat arbitrary at what distance two edges are considered to be part of the same or separate contours (e.g., Fig. 5). If the output voltage is thresholded at 3.0 V (in the case of Fig. 5b), the contour with one or two pixel breaks would be considered a single Figure, while the two larger breaks would not be. 3 CONCLUSION Most analog vision chips are restricted to work either at the local, pixel-level or the global, image-level. The dynamic wire and figure-ground chips discussed in this paper allow data-dependent neighborhoods to form. With these configured neighborhoods, analog chips can now perform object-level processing. Acknowledgements This Work is supported by the National Science Foundation, the Office of Naval Research and Rockwell International Science Center. We thank MOSIS for all chip fabrication. JGH is supported by an NSF postdoctoral fellowship. References Liu, S. and Harris, J .G. (1992), Dynamic wires: an analog VLSI model for object processing, Internat. Journal of Compo Vision. 8: pp. 231-239. Luo, J., Koch, C. and Mathur, B. (1992), Figure-Ground segregation using an analog VLSI Chip, IEEE Micro, Vol. 12 46-57, 1992. Shaashua, A. and Ullman, S. (1988), Structural saliency: The detection of globally salient structures using a locally connected network. In Proceedings of the IEEE Computer Vision and Pattern Recognition Conference. Ullman, S. (1984), Visual routines, Cognition, Vol. 18, pp. 97-159, 1984.
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Hidden Markov Models in Molecular Biology: New Algorithms and Applications Pierre Baldi • Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109 Tim H lmkapiller Division of Biology California Institute of Technology Yves Chauvin t Net-ID, Inc. 8, Cathy Place Menlo Park, CA 94305 Marcella A. McClure Department of Evolutionary Biology University of California, Irvine Abstract Hidden Markov Models (HMMs) can be applied to several important problems in molecular biology. We introduce a new convergent learning algorithm for HMMs that, unlike the classical Baum-Welch algorithm is smooth and can be applied on-line or in batch mode, with or without the usual Viterbi most likely path approximation. Left-right HMMs with insertion and deletion states are then trained to represent several protein families including immunoglobulins and kinases. In all cases, the models derived capture all the important statistical properties of the families and can be used efficiently in a number of important tasks such as multiple alignment, motif detection, and classification. *and Division of Biology, California Institute of Technology. t and Department of Psychology, Stanford University. 747 748 Baldi, Chauvin, Hunkapiller, and McClure 1 INTRODUCTION Hidden Markov Models (e.g., Rabiner, 1989) and the more general EM algorithm in statistics can be applied to the modeling and analysis of biological primary sequence information (Churchill (1989), Lawrence and Reilly (1990), Baldi et al. (1992), Cardon and Storrr..o (1992), H~.usslel et al. (1992». Most not.ably, as in speech recognition applications, a family of evolutionarily related sequences can be viewed as consisting of different utterances of the same prototypical sequence resulting from a common under1ying HMM dynamics. A model trained from a family can then be used for a number of tasks including multiple alignments and classification. The multiple alignment is particularly important since it reveals the highly conserved regions of the molecules with functional and structural significance even in the absence of any tertiary information. The mUltiple alignment is also an essential tool for proper phylogenetic tree reconstruction and other important tasks. Good algorithms based on dynamic programming exist for the alignment of two sequences. However they scale exponentially with the number of sequences and the general multiple alignment problem is known to be NP-complete. Here, we briefly present a new algorithm and its variations for learning in HMMs and the results of some of the applications of this approach to new protein families. 2 HMMs FOR BIOLOGICAL PRIMARY SEQUENCES A HMM is characterized by a set of states, an alphabet of symbols, a probability transition matrix T = (tij) and a probability emission matrix eij. As in speech applications, we are going to consider left-right architectures: once a given state is left it can never be visited again. Common knowledge of evolutionary mechanisms suggests the choice of three types of states (in addition to the start and to the end state): the main states m}, ... , mN, the delete states d}, ... , dN+l and the insert states iI, ... , iN+l' N is the length of the model which is usually chosen equal to the average length of the sequences in the family and, if needed, can be adjusted in later stages. The details of a typical architecture are given in Figure 1. The alphabet has 4 letters in the case of DNA or RNA sequences, one symbol per nucleotide, and 20 letters in the case of proteins, one symbol per amino acid. Only the main and insert states emit letters, while the delete states are of course mute. The linear sequence of state transitions start ml m2 ... mN end is the backbone of the model and correponds to the path associated with the prototypical sequence in the family under consideration. Insertions and deletions are defined with respect to this backbone. Insertions and deletions are treated symmetrically except for the loops on the insert states needed to account for multiple insertions. The adjustable parameters of the HMM provide a natural way of incorporating variable gap penalties. A number of other architectures are also possible. 3 LEARNING ALGORITHMS Learning from examples in HMMs is typically accomplished using the Baum-Welch algorithm. In the Baum-Welch algorithm, the expected number nij (resp. mij) of i j transitions (resp. emissions of letter j from state i) induced by the data are calculated using the forward-backward procedure. The transition and emission Hidden Markov Models in Molecular Biology: New Algorithms and Applications 749 Figure 1: The basic left-right HMM architecture. Sand E are the start and end states. probabilities are then reset to the observed frequencies by t+ .. -_ nij and + mij eiJ· = -IJ ni mi (1) where ni = E j nij and m, = E j mij. It is clear that this algorithm can lead to abrupt jumps in parameter space and that the procedure cannot be used for online learning (after each training example). This is even more so if, in order to save some computations, the Viterbi approximation is used to estimate likelihoods and transition and emission statistics by computing only the most likely paths as opposed to the forward-bacward procedure where all possible paths are examined. A new algorithm for HMM learning which is smooth and can be used on-line or in batch mode, with or without the Viterbi approximation, can be defined as follows. First, we use a Boltzmann-Gibbs representation for the parameters. For each tij (resp. eij) we define a new parameter Wij (resp. Vii) by (2) Normalisation constraints are naturally enforced by this representation throughout learning with the added advantage that none of the parameters can reach the absorbing value O. After computing on-line or in batch mode the statistics nij and mii using the forward-backward procedure (or the usual Viterbi approximation), the update equations are particularly simple and given by n·· m·· ~Wij = 1}(....!L - tij) and ~Vij = 1}( ---2L - eii) (3) ni mi where 1} is the learning rate. In Baldi et al. (1992) a proof is given that this algorithm must converge to a maximum of the product of the likelihoods of the training sequences. In the case of an on-line Viterbi approximation, the optimal path associated with the current training sequence is first computed. The update equations are then given by ~Wij = 1}(t:i - tii) and ~vii = 1}(tii - eii) (4) 750 Baldi, Chauvin, Hunkapiller, and McClure Here, for a fixed state i, t:j and t1j are the target transition and emission values: t~j = 1 every time the transition Si -.. Sj is part of the Viterbi path of the corresponding training sequence sequence and 0 otherwise and similarly for tij' After training, the model derived can be used for a number of tasks. First, by computing for each sequence its most likely path through the model using the Viterbi algorithm, multiple sequences can be aligned to each other in time O(K N 2), linear in the number K of sequences. The model can also be used for classification and data base searches. The likelihood of any sequence (randomly generated or taken from any data base) can be calc°.llated and compared to the likelihood of the sequences in the family being modeled. Additional applications are discussed in Baldi et al. (1992). 4 EXPERIMENTS AND RESULTS The previous approach has been applied to a number of protein families including globins, immunoglobulins, kinases, aspartic acid proteases and G-coupled receptor proteins. The first application and alignment of the globin family using HMMs (trained with the Viterbi approximation of the Baum-Welch algorithm, and a number of additional heuristics) was given by Haussler et al. (1992). Here, we briefly describe some of our results on the immunoglobulin and the kinase families 1. 4.1 IMMUNOGLOBULINS Immunoglobulins or antibodies are proteins produced by B cells that bind with specificity to foreign antigens in order to neutralize them or target their destruction by other effector cells (e.g., Hunkapiller & Hood, 1989). The set of sequences used in our experiments consists of immunoglobulins V region sequences from the Protein Identification Resources (PIR) data base. It corresponds to 294 sequences, with minimum length 90, average length 117 and maximum length 254. The variation in length resulted from including any sequence with a V region, including those that also included signal or leader sequences, germline sequences that did not include the J segment, and some that contained the C region as well. Seventy seqences contained one or more special characters indicating an ambiguous amino acid determination and were removed. For the immunoglobulins variable V regions, we have trained a model of length 117 using a random subset of 150 sequences. Figure 2 displays the alignment corresponding to the first 20 sequences in this random subset. Letters emitted from the main states are upper case and letters emitted from insertion states are lower case. Dashes represent deletions or accomodate for insertions. As can be observed, the algorithm has been able to detect all the main regions of highly conserved residues. Most importantly, the cysteine residues towards the beginning and the end responsible for the disulphide bonds which holds the chains together are perfectly aligned and marked. The only exception is the fifth sequence from the bottom which has a serine residue in its terminal portion. It is also important to remark that some 1 Recently, Hausssler et al. have also independently applied their approach to the kinase family (Haussler, private communication). PHOI06 B27563 MHMS76 D28035 D24672 PHOIOO B27888 PL0160 E28833 D30539 C30560 AVMSX4 C30540 PL0123 H36005 PH0097 137267 A25114 D2HUWA A30539 Hidden Markov Models in Molecular Biology: New Algorithms and Applications 751 mklpvrllvlmfwipasssDvVMTQTPLSLpvSLGDQASISCRSSQSLVHSngnTYLNWYLQ--KAGQS----------------------LQQPGAELv-KPGASVKLSCKASGYTFTN---YWIHWVKQ--RPGRGL ------------------------ESGGGLv-QPGGSMKLSCVASGFTFSN---YWMNWVRQ--SPEKGL mefglswiflvailkgvqcEvRLVESGGDLv-EPGGSLRVSCEVSGFIFSK---AWMNWVRQ--APGKGL ---------------------------------------ISCKASGYTFTN---YGMNWVKQ--APGKGL -------------------LvQLQQSGPVLv-KPGTSMKISCKTSGYSFTG---YTMSWVRQ--SHGKSL -------------------EvMLVESGGGLa-KPGGSLKLSCTTSGFTFS1---HAMSWVRQ--TPEKRL -------------------QvQLQQSGPGLv-KPSQTLSLTCA1SGDSVSSns-AAWNW1RQ--SPSRGL -------------------DvVMTQTPLSLpvSLGDQASISCRSSQSLVRSngnTYLHWYLQ--KPGQP-------------------EvKLVESGGGLv-QSGGSLRLSCATSGFTFSD---FYMEWVRQ--PPGKSL -------------------QvHLQQSGAELv-KPGASVK1SCKASGYTFTS---YWMNWVKQ--RPGQGL -------------------EvKLLESGGGLv-QPGGSLKLSCAASGFDFSR---YWMSWVRQ--APGKGL -------------------EvKLVESGGGLv-QPGGSLRLSCATSGFTFSD---FYMEWVRQ--PPGKRL -------------------EvQLVESGGGLv-QPGGSLRLSCAASGFTFSS---YWMSWVRQ--APGKGL -------------------EvQLVESGGGLv-KPGGSLRLSCAASGFTFSN---AWMNWVRQ--APGKGL -------------------DvKLVESGGGLv-KPGGSLKLSCAASGFTFSS---Y1MSWVRQ--TPEKRL gsimg---------------vQLQQSGPELv-KPGASVKISCKTSGYTFTE---YTMHWVKQ--SHGKSL -------------------DvHLQESGPGLv-KPSQSLSLTCSVTGYS1TRg--YNWNW1RR--FPGNKL -------------------RIQLQESGPGLv-KPSETLSLTCIVSGGPIRRtg-YYWGWIRQ--PPGKGL -------------------EvKLVESGGGLv-QPGGSLRLSCATSGFTFSD---FYMEWVRQ--PPGKRL ......•............•....•............•............• * ...................•........ PHOI06 B27563 MHMS76 D28035 D24672 PHOIOO B27888 PL0160 E28833 D30539 C30560 AVMSX4 C30540 PL0123 H36005 PH0097 137267 A25114 D2HUWA A30539 PHOI06 B27563 MHMS76 D28035 D24672 PHOIOO B27888 PL0160 E28833 D30539 C30560 AVMSX4 C30540 PL0123 H36005 PH0097 137267 A25114 D2HUWA A30539 -p-KLLI-YKV---SNR-FSGVPDRFSGSG--SGTDFTLKI SRVEAEDLG IYFCSQ-------------E-W1GR1-DPNSGGTKY-NEKFKNKATLT1NKPSNTAYMQLSSLTSDDSAVYYCARGYDYSYY------E-WVAE1rLKSGYATHY-AESVKGRFT1SRDDSKSSVYLQMNNLRAEDTG1YYCTRPGV----------Q-WVGQ I kNKVDGGTIDYAAPVKGRF I ISRDDSKSTVYLQMNRLK1EDTAVYYCVGNYTGT--------K-WMGW1-NTYTGEPTY-ADDFKGRFAFSLETSASTAYLQ1NNLKNEDTATYFCARGSSYDYY------E-W1GL1-1PSNGGTNY-NQKFKDKASLTVDKSSSTAYMELLSLTSEDSAVYYCARPSYYGSRnyy---E-WVAA1-SSGGSYTFY-PDSVKGRFT1SRDNAKNTLYLQ1NSLRSEDTAIYYCAREEGLRLDdy----E-WLGRT-YYRSKWYNDYAVSVKSRITINPDTSKNQFSLQLNSVTPEDTAVYYCARELGDA---------p-KLLI-YKV---SNR-VSGVPDRFSGSG--SGTDFTLK1SRVEAEDLGVYFCSQSTHV---------E-WlAASrNEANDYTTEYSASVKGRFIVSRDTSQSILYLQM1ALRAEDTAIYYCSRDYYGSSYw-----E-W1GEI-DPSNSYTNN-NQKFKNKATLTVDKSSNTAYMQLSSLTSEDSAVYYCARWGTGSSWg-----E-W1GE1-NPDSST1NY-TPSLKDKFIISRDNAKNTLYLQMSKVRSEDTALYYCARLHYYGY-------E-WlAASrNKAHDYTTEYSASVKGRFIVSRDTSQSI LYLQMNALRAEDTA1YYCARDADYGSSshw---E-WVAN1-KQDGSEKYY-VDSVKGRFT1SRDNAKNSLYLQMNSLRAEDTAVYYCAR-------------E-WVGRlkSKTDGGTTDYAAPVKGRFT1SRDDSKNTLYLQMNSLKTEDTAVYYCTTDRGGSSQ------E-WVAT1-SSGGRYTYY-SDSVKGRFT1SRDNAKNTLYLQMSSLRSEDTAMYYSTASGDS---------E-W1GGI-NPNNGGTSY-NQKFKGKATLTVDKSSSTAYMELRSLTSEDSAVYYCARRGLTTVVaksy--E-WMGY1-NYDGS-NNY-NPSLKNR1SVTRDTSKNQFFLKMNSVTTEDTATYYCARL1PFSDGyyedyyE-W1GGV-YYTGS-1YY-NPSLRGRVT1SVDTSRNQFSLNLRSMSAADTAMYYCARGNPPPYYdigtgsd E-W1AAS rNKANDYTTEYSASVKGRF 1VSRDTSQS I LYLQMNALRAEDTA1YYCARDYYGSSYvw --------------------tthvpptfgggtkleikr-AMDYWGQGTSVTVSS--------------------PDYWGQGTTLTVSS--------------------VDYWGQGTLVTVSS-------------------AMDYWGQGTSVTVSS-------------------AMDYWGQGTSVTVSSak-----------------AMDYWGQGTSVTVS---------------------FD1WGQGTMVTVSS-------------------YFDVWGAGTTVTVSS-------------------WFAYWGQGTLVTVSA--------------------AAYWGQGTLVTVSAe------------------yFDVWGAGTTVTVSS--------------------GDYWGQGTLVTVSS--------------------FDYWGQGTTLTVSSak-----------------yFDYWGQGTTLTVSS-------------------AMDYWGQGT------------------------dG1DVWGQGTTVHVSS-------------------YFDVWGAGTTVTVSS------------------Figure 2: Immunoglobulin alignment. * 752 Baldi, Chauvin, HunkapiIler, and McClure of the sequences in the family have some sort of "header" (leader signal peptide) whereas the others do not. We did not remove the headers prior to training and used the sequences as they were given to us. The model was able to detect and accomodate these "headers" by treating them as initial inserts as can be seen from the alignment of two of the sequences. 4.2 KINASES Eukaryotic protein k~nases cO"lstitute a very large family of proteins that regulate the most basic of cellular processes through phosphorylation. They have been termed the "transistors" of the cell (Hunter (1987)). We have used the sequences available in the kinase data base maintained at the Salk Institute. Our basic set consists of 224 sequences, with minimum length 156, average length 287, and maximallength 569. Only one sequence containing a special symbol (X) was discarded. In one experiment, we trained a model of length 287 using a random subset of 150 kinase sequences. Figure 3 displays the corresponding alignment for a subset of 12 phylogenetically representative sequences. These include serine/threonine, tyrosine and dual specificity kinases from mammals, birds, fungi and retroviruses and herpes viruses. The percentage of identical residues within the kinase data sets ranges from 8-30%, suggesting that only those residues involved in catalysis are conserved among these highly divergent sequences. All the 12 characteristic catalytic domains or subdomains described in Hanks and Quinn (1991) are easily recognizable and marked. Additional highly conserved positions can also be observed consistent with previously constructed multiple alignments. For instance, the initial hydrophobic consensus Gly-X-Gly-XX-Gly together with the Lys located 15 or 20 residues downstream are part of the ATP /GTP binding site. The carboxyl terminus is characterized by the presence of an invariant Arg residue. Conserved residues in proximity to the acceptor amino acid are found in the VIb (Asp), VII (Asp-Phe-Gly) and VIII domains (Ala-Pro-Glu). In Figure 4, the entropy of the emission distribution of each main state is plotted: motifs are easily detectable and correspond to positions with very low entropy. 5 DISCUSSION HMMs are emerging as a powerful, adaptive, and modular tool for computational biology. Here, they have been used, together with a new learning algorithm, to model families of proteins. In all cases, the models derived capture all the important statistical properties of the families. Additional results and potential applications, such as phylogenetic tree reconstruction, classification, and superfamily modeling, are discussed in Baldi et al. (1992). References Baldi, P., Chauvin, Y., Hunkapiller, T. and McClure, M. A. (1992) Adaptive Algorithms for Modeling and Analysis of Biological Primary Sequence Information. Technical Report. Cardon, L. R. and Stormo, G. D. (1992) Expectation Maximization Algorithm for Identifying Protein-binding Sites with Variable Lengths from Unaligned DNA CD2a MLCK PSKH CAPK WEEl CSRC EGFR PDGF VFES RAFl CMOS HSVK Hidden Markov Models in Molecular Biology: New Algorithms and Applications 753 an~KR--LEKVGEGTYGVVYKALDLrpg--QGQRVVALK------KIRLESEDEGVPSTAIREISLLKEL-K-DDNIVRLYDIVH --FSMnsKEALGGGKFGAVCTCTEK-----STGLKLAAK---VI-KKQTPKDKE----MVMLEIEVMNQL-N-HRNLIQLYAAIE akYDI--KALIGRGSFSRVVRVEHR-----ATRQPYAIK---MIETKYREGRE-----VCESELRVLRRV-R-HANIIQLVEVFE dqFER--IKTLGTGSFGRVMLVKHM-----ETGNHYAMK---ILDKQKVVKLKQIE--HTLNEKRILQAV-N-FPFLVKLEFSFK trFRN--VTLLGSGEFSEVFQVEDPv----EKTLKYAVK---KL-KVKFSGPKERN--RLLQEVSIQRALkG-HDHIVELMDSWE esLRL--EVKLGQGCFGEVWMGTWN------GTTRVAIK---TLKPGNMSPE------AFLQEAQVMKKL-R-HEKLVQLYAVVS teFKK--IKVLGSGAFGTVYKGLWIpege-KVKIPVAIK---ELREATSPKANK----EILDEAYVMASV-D-NPHVCRLLGICL dqLVL--GRTLGSGAFGQVVEATAHglshsQATMKVAVK---MLKSTARSSEKQ----ALMSELY--GDL--v-DYLHRNKHTFL edLVL--GEQIGRGNFGEVFSGRLR-----ADNTLVAVK---SCRETLPPDIKA----KFLQEAKILKQY-S-HPNIVRLIGVCT seVML--STRIGSGSFGTVYKGKWH--------GDVAVK---ILKVVDPTPEQFQ---AFRNEVAVLRKT-R-HVNILLFMGYMT eqVCL--LQRLGAGGFGSVYKATYR-------GVPVAIKQvNKCTKNRLASRR-----SFWAELNV-ARL-R-HDNIVRVVAAST mgFTI--HGALTPGSEGCVFDSSHP-----DYFQRVIVK- -----AGWYT--------STSHEARLLRRL-D-HPAILPLLDLHV · ............... , ....•.. * .•................... • . .. 0 ••••••• • •••••••••••••••••••••••••••••••••• • .....•..•............. I . ......•.•....... . .... I I ...........•.••....... I I I .........•... IV ..... CD2E MLCK PSKH CAPK WEEl CSRC EGFR PDGF VFES RAFl CMOS HSVK SDAHk---------LY-L-V-FEFLDL-DLKRYMEGIpkd--------------------------------------------TPHE----------IV-L-F-KEYIEGGELFERIVDE-----------------------------------------------TQER----------VY-M-V-MELATGGELFDRIIAK-----------------------------------------------DNSN----------LY-M-V-MEYVPGGEMFSHLRRI-----------------------------------------------HGGF----------LY-M-Q-VELCENGSLDRFLEEQgql---------------------------------------------EEP----------IY-I-V-TEYMSKGSLLDFLKGE------------------------------------------------TST----------VQ-L-I-TQLMPFGCLLDYVREH------------------------------------------------QRHsnkhcppsaeLYs-n-a--LPVGFSLPSHLNLTgesdggymdmskdesidyvpmldmkgdikyadiespsymapydnyvps QKQP----------IY-I-V-MELVQGGDFLTFLRTE------------------------------------------------KDN----------LA-I-V-TQWCEGSSLYKHLHVQ-----------------------------------------------RTPAgsnsl-----GT-I-I-MEFGGNVTLHQVIYGAaghpegdaqephcrtg-------------------------------VSGV----------TC-L-V-LPKYQA-DLYTYLSRR-----------------------------------------------••••••••••••• •• •••••••••••••••••• V •••• • •••••••••••••••••••••••••••••••••••••••••••••••••••••• CD2a MLCK PSKH CAPK WEEl CSRC EGFR PDGF VFES RAFl CMOS HSVK ---------------QP-LGADIVKKFMMQ-LCKGIAYCHSHRILHRDLKPQNLL-INKDG---N-LKLGDFGLARAFGVPLRAY --------------DYH-LTEVDTMVFVRQ-ICDGILFMHKMRVLHLDLKPENILcVNTTG---H1VKIIDFGLARRYNPNEKL---------------GS-FTERDATRVLQM-VLDGVRYLHALGITHRDLKPENLL-YYHPGtdsK-IIITDFGLASARKKGDDCL ---------------GR-FSEPHARFYAAQ-IVLTFEYLHSLDLIYRDLKPENLL-IDQQG---Y-IQVTDFGFAKRVKGRT-----------------SR-LDEFRVWKILVE-VALGLQFIHHKNYVHLDLKPANVM-ITFEG---T-LKIGDFGMASVWPVPRG---------------MGKyLRLPQLVDMAAQ-IASGMAYVERMNYVHRDLRAANIL-VGENL---V-CKVADFGLARLIEDNEYTA --------------KDN-IGSQYLLNWCVQ-IAKGMNYLEDRRLVHRDLAARNVL-VKTPQ---H-VKITDFGLAKLLGAEEKEY apertyratlinds-PV-LSYTDLVGFSYQ-VANGMDFLASKNCVHRDLAARNVL-ICEGK---L-VKICDFGLARDIMRDSNYI --------------GAR-LRMKTLLQMVGD-AAAGMEYLESKCCIHRDLAARNCL-VTEKN---V-LKISDFGMSREAADGIYAA --------------ETK-FQMFQLIDIARQ-TAQGMDYLKAKNIIHRDMKSNNIF-LHEGL---T-VKIGDFGLATVKSRWSGSQ ---------------GQ-LSLGKCLKYSLD-VVNGLLFLHSQSIVHLDLKPANIL-ISEQD---V-CKISDFGCSEKLEDLLCFQ --------------LNP-LGRPQIAAVSRQ-LLSAVDYIHRQGIIHRDIKTENIF-INTPE---D-ICLGDFGAACFVQGSRSSP • ............................... . ....................... * •••• * .................. *. * •••••••••••• ..••........•...•.•... .. ...... . VIa ................... . ... VIb ...........•••.... VII ....•....... CD2a MLCK PSKH CAPK WEEl CSRC EGFR PDGF VFES RAFl CMOS HSVK ---THEIVTLWYRAPEVLLgGK---QYSTGVDTWSIGCIFAEMCNRKP---------------IFSGDSE-----IDQIFKIFRV ---KVNFGTPEFLSPEVVN-YD---QISDKTDMWSLGVITYMLLSGLS---------------PFLGDDD-----TETLNNVLSG M--KTTCGTPEYIAPEVLV-RK---PYTNSVDMWALGVIAYILLSGTM---------------PFEDDNR-----TRLYRQILRG ---WTLCGTPEYLAPEIIL-SK---GYNKAVDWWALGVLIYEMAAGYF---------------PFFADQP-----IQIYEKIVSG ---MEREGDCEYIAPEVLA-NH---LYDKPADIFSLGITVFEAAANIV--------------LPDNGQSW-----Q----KLRSG R--QGAKFPIKWTAPEAAL-YG---RFTIKSDVWSFGILLTELTTKGR--------------VPYPGMVN-----REVLDQVERG H-AEGGKVPIKWMALESIL-HR---IYTHQSDVWSYGVTVWELMTFGS--------------KPYDGIPA-----SEISSILEKG S-KGSTYLPLKWMAPESIF-NS---LYTTLSDVWSFGILLKEIFTLGG--------------TPYPELPM----NDQFYNAIKRG S-GGLRQVPVKWTAPEALN-YG---RYSSESDVWSFGILLKETFSLGA--------------SPYPNLSN-----QQTREFVEKG Q-VEQPTGSVLWMAPEVIR-MQdnnPFSFQSDVYSYGIVLYELMTGEL---------------PYS---R-----DQIIFMVGRG TpSYPLGGTYTHRAPELLK-GE---GVTPKADIYSFAITLWQMTTKQA---------------PYSGERQ-----HILYAVVAYD F-PYGIAGTIDTNAPEVLA-GD---PYTTTVDIWSAGLVIFETAVHNA------------------------------------· .......... . .......... . ** ...............•.... * ..... . ................. I •••••••••••••••••••••••• · ....••......•.•..... VIII •••........•.... IX ..........•••.••..•......... ' ............ X .•...... CD2a MLCK PSKH CAPK WEE 1 CSRC EGFR PDGF VFES RAFl CMOS HSVK ---LGTPNEAlwpdivylpdfkpsfpqwrrkdlsqvvpSLDPRGIDLLDKLLAYDPINRISARRAAIHPYFQES-------nwyFDEETFEA----------------------------VSDEAKDFVSNLIVKEQGARMSAAQCLAHPWLNNL-------kysYSGEPWPS----------------------------VSNLAKDFIDRLLTVDPGARMTALQALRHPWVVSM----------KVR-FPSH----------------------------FSSDLKDLLRNLLQVDLTKRFGNLKDGVNDIKNHK----------DLSDAPRLsstdngssltsssretpansii------GQGGLDRVVEWKLSPEPRNRPTIDQILATD--EVCWV--------YRMPCPPE----------------------------CPESLHDLMCQCWRRDPEERPTFEYLQAFLEDYFT----------ERLPQPPI----------------------------CTIDVYKIMVKCWKIDADSRPKFRELIIEFSKMAR----------YRMAQPAH----------------------------ASDEIYEIMQKCKEEKFETRPPFSQLVLLLERLLGEGykkky---GRLPCPEL----------------------------CPDAVFRLMEQCWAYEPGQRPSFSAIYQEL---------------YASPDLsKlykn------------------------CPKAMKRLVADCVKKVKEERPLFPQILSSIELLQH----------LRPSLSAAvfedsl----------------------PGQRLGDVIQRCWRPSAAQRPSARLLLVDLTSLKA-------· ..... ............................................................•.............. ......... ........• . .............. . .................. . .................•... . XI ......••.............. Figure 3: Kinase alignment of 12 representative sequences. 754 Baldi, Chauvin, Hunkapiller, and McClure , Main State Entropy Values 10 20 30 40 50 60 70 eo 90 100110120130140 150 160 170 1 eo 190 200 210 220 230 240 250 260 270 2eo Entropy Distribution , , , 0.0 0.5 2.5 3.0 Figure 4: Kinase emission entropy plot and distribution. Fragments. Journal of Molecular Biology, 223,159-170. Churchill, G. A. (1989) Stochastic Models for Heterogeneous DNA Sequences. Bulletin of Mathematical Biology, 51, 1, 79-94. Hanks, S. K., Quinn, A. M. (1991) Protein Kinase Catalytic Domain Sequences Database: Identification of Conserved Features of Primary Structure and Classification of Family Members. Methods in Enzymology, 200, 38-62. Haussler, D., Krogh, A., Mian, S. and Sjolander, K. (1992) Protein Modeling using Hidden Markov Models. Computer and Information Sciences Technical Report (UCSC-CRL-92-93), University of California, Santa Cruz. Hunkapiller, T. and Hood, L. (1989) Diversity of the Immunoglobulin Gene Superfamily. Advances in Immunology, 44, 1-63, Academic Press, Inc. Hunter, T. (1987) A Thousand and One Protein Kinases. Cell, 50, 823-829. Lawrence, C. E. and Reilly, A. A. (1990) An Expectation Maximization (EM) Algorithm for the Identification and Characterization of Common Sites in Unaligned Biopolymer Sequences. Proteins: Struct. Funct. Genet., 7, 41-51. Rabiner, L. R. (1989) A Tutorial on Hidden Markor Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77, 2, 257-286.
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Perceiving Complex Visual Scenes: An Oscillator Neural Network Model that Integrates Selective Attention, Perceptual Organisation, and Invariant Recognition Rainer Goebel Department of Psychology University of Braunschweig Spielmannstr. 19 W-3300 Braunschweig, Germany Abstract Which processes underly our ability to quickly recognize familiar objects within a complex visual input scene? In this paper an implemented neural network model is described that attempts to specify how selective visual attention, perceptual organisation, and invariance transformations might work together in order to segment, select, and recognize objects out of complex input scenes containing multiple, possibly overlapping objects. Retinotopically organized feature maps serve as input for two main processing routes: the 'wherepathway' dealing with location information and the 'what-pathway' computing the shape and attributes of objects. A location-based attention mechanism operates on an early stage of visual processing selecting a contigous region of the visual field for preferential processing. Additionally, location-based attention plays an important role for invariant object recognition controling appropriate normalization processes within the what-pathway. Object recognition is supported through the segmentation of the visual field into distinct entities. In order to represent different segmented entities at the same time, the model uses an oscillatory binding mechanism. Connections between the where-pathway and the what-pathway lead to a flexible cooperation between different functional subsystems producing an overall behavior which is consistent with a variety of psychophysical data. 903 904 Goebel 1 INTRODUCTION We are able to recognize a familiar object from many different viewpoints. Additionally, an object normally does not appear in isolation but in combination with other objects. These varying viewing conditions produce very different retinal neural representations. The task of the visual system can be considered as a transformation process forming high-level object representations which are invariant with respect to different viewing conditions. Selective attention and perceptual organisation seem to play an important. role in this transformation process. 1.1 LOCATION-BASED VS OBJECT-BASED ATTENTION N eisser (1967) assumed that visual processing is done in two stages: an early stage that operates in parallel across the entire visual field, and a later stage that can only process information from a limited part of the field at anyone time. Neisser (1967) proposed an object-based approach to selective attention: the first, 'preattentive', stage segments the whole field into seperate objects on the basis of Gestalt principles; the second stage, focal attention, selects one of these objects for detailed analysis. Other theories stress the location-based nature of visual attention: a limited contigous region is filtered for detailed analysis (e.g., Posner et al., 1980). There exists a number of models of location-based at.tention (e.g., Hinton & Lang, 1985; Mozer, 1991; Sandon, 1990) and a few models of object-based attention using whole object knowledge (e.g., Fukushima, 1986). Our model attempts to integrate both approaches: location-based attention - implemented as a 'spotlight' - operates on an early stage of visual processing selecting a contigous region for detailed processing. However, the position and the size of the attentional window is determined to a large extent from the results of a segmentation process operating at different levels within the system. 1.2 DYNAMIC BINDING The question of how groupings can be represented in a neural network is known as the binding problem. It occurs in many variations, e.g., as the problem of how to represent multiple objects simultaneously but sufficiently distinct that confusions ('illusory conjunctions') at later processing stages are avoided. An interesting solution of the binding problem is based on ideas proposed by Milner (1974) and von der Malsburg (1981). In contrast to most connectionist models assuming that only the average output activity of neurons encodes important information, they suggest that the exact timing of neuronal activity (the firing of individual neurons or the 'bursting' of cell groups) plays an important role for information processing in the brain. The central idea is that stimulated units do not respond with a constant output but with oscillatory bellavior which can be exploited to represent feature linkings. A possible solution for representing multiple objects might be that the parts of one object are bound together through synchronized (phase-locked) oscillations and separated from other objects through an uncorellated phase relation. Recent empirical findings (Eckhorn et al., 1988; Gray & Singer, 1989) provide some evidence that the brain may indeed use phase-locked oscillations as a means for representing global object properties. Perceiving Complex Visual Scenes: An Oscillator Neural Network Model 905 2 THE MODEL 2.1 SYSTEM DYNAMICS In order to establish dynamic binding via phase-locked oscillations the units of the model must be able to exhibit oscillatory behavior. Stimulated from the empirical findings mentioned earlier, a rapidly growing number of work has studied populations of oscillating units (e.g., Eckhorn et al., }990; Sompolinsky et al., }990). There exists also a number of models using phase-locked oscillations in order to simulate various aspects of perceptual organisation (e.g., Schillen & Konig, 1991; Mozer, Zemel, Behrmann & Williams, 1992). We defined computationally simple model neurons which allow to represent independently an activation value and a period value. Such a model neuron possesses two types of input areas: the activation gate (a-gate) and the period-gate (p-gate) which allow the model neurons to communicate via two types of connections (cf. Eckhorn et al., }990; they distinguish between 'feeding' and 'linking' connections). We make the following definitions: • wfj: weight from model neuron j to the a-gate of model neuron i. • wfj: weight from model neuron j to the p-gate of model neuron i. • ~i(t): internal time-keeper of unit i • T: globally defined period length • 7i (N): period length of unit i (Nth oscillation) Each model neuron possesses an internal time-keeper ~i(t) counting the number of bins elapsed since the last firing point. A model neuron is refractory until the timekeeper reaches the value Ii (e.g., Ii = T = 8). Then it may emit an activation value and resets the time-keeper. Depending on the stimulation received at the p-gate (see below) a model neuron fires either if ~ = T } or ~ = T. This variation of the individual period length Ii is the only possibility for a unit to change its phase relation to other units. The value of the globally defined period length T determines directly how many objects may be represented 'simultaneously'. The activation value aj at the internal time ~ is determined as follows: T. n neti(~ = Td = L L wfjaj(O (=lj=l if ~ = Ti otherwise (1) (2) where u(x) is the logistic (sigmoidal) function. If we consider an extreme case with T = 1 we obtain the following equations: n netj(t) = L W0 aj (3) j=l 906 Goebel (4) This derivation allows us to study the same network as a conventional connectionist network (T = 1) with a 'non-oscillatory' activation function to which we can add a dynamic binding mechanism by simply setting T > 1. In the latter case the input at the p-gate determines the length of the current period as either Ii = T - 1 or Ii = T. The decision to shift the phase relation to other neurons should be done in such a way that the 'belongingness constraints' imposed by the connectivity pattern of the p-weights wfj is maximized, e.g., if two units are positively p-coupled they should oscillate in phase, if they are negatively p-coupled they should oscillate out of phase. The decision whether a unit fires at T - 1 or T depends on two values, the stimulation received during the refractory period 1 < f. < Ii (N - 1) and on the stimulation received at the last firing point f. = Ti(N - 1). These values behave as two opposite forces gi determining the probability Pj< of shortening the next period: 1- 2r p.< = r + --~~~ z 1 + e(g~-g~) if f. = Ii if 1 < f. < Ti (5) (6) (7) If the value of gl-gl is large (e.g., there are many positively p-coupled units firing at the same time) it is unlikely that the unit shortens its next period length. If instead the value of gl-gl is large (e.g., there are many positively coupled neurons firing just before the considered unit) it is likely that the unit will shorten its next period. There exists also a small overall noise level r = 0.01 which allows for symmetry breaking (e.g., if two strongly negatively coupled neurons are accidentally phase-locked). 2.2 THE INPUT MODULE Figure 1 shows an overview of the architecture of the model, called HOTSPOT. An input is presented to the model by clamping on units at the model-retina consisting of two layers with 15x25 units. Each layer is meant to correspond to a different color-sensitive ganglion cell type. The retinal representation is then analyzed within different retinotopically organized feature maps (4 oriented line segments and 2 unoriented color blobs) as a simplified representation of an early visual processing stage (corresponding roughly to VI). A lateral connectivity pattern of p-weights within and between these feature maps computes initial feature linkings consistent with the findings of Eckhorn et al., (1988) and Gray and Singer (1989). Each feature map also projects to a second feature-specific layer. The weights between those layers compute the saliency at each position of a particular feature type. These saliency values are finally integrated within the saliency map. The retinotopic feature maps project to both the what pathway, corresponding roughly to the occipito-temporal processing stream and the where-pathway, corresponding to the occipita-parietal stream (e.g., Ungerleider & Mishkin, 1982). Perceiving Complex Visual Scenes: An Oscillator Neural Network Model 907 2.3 THE SPOTLIGHT-LAYER The spotlight-layer receives bottom-up input from the feature maps via the saliency map and top-down input from the spotlight-control module. Based on these sources of stimulation, the spotlight layer computes a circular region of activity representing the current focus of spatial attention. The spotlight-layer corresponds roughly to the pulvinar nucleus of the thalamus. The spotlight-layer gates the flow of information within the what-pathway. 2.4 THE WHAT-PATHWAY: FROM FEATURES TO OBJECTS Processing within the what-pathway includes spatial selection, invariance transformation, complex grouping, object-based selection and object recognition. 2.4.1 The Invariallce Module The task of the Invariance module is to retain the spatial arrangement of the features falling within the attentional spotlight while abstracting at the same time the absolute retinal position of the att.ended information. This goal is achieved in several stages along the what-pathway for each feature type. The basic idea is that each neuron connects to several neurons at the next layer. If a certain position is not attended its 'standard' way may be 'open'. If, however, a position is attended, the decision which way is currently gated for a neuron depends on the position and width of the attentional spotlight. Special control layers compute explicitly whether a certain absolute position falls within one of 5 horizontal and 5 vertical regions of the spotlight (e.g., the horizontal regions are 'far left', 'near left', 'center', 'near right', 'far right'). These layers gate the feedforward-synapses within the what-pathway. Finally, the selected information reaches the invariance-output layers which have a 7x7 resolution for each feature type. Recently Olshausen, Anderson and Van Essen (1992) proposed a strikingly similar approach for forming invariant representations. Despite invariance transformations the representation of an object at the invarianceoutput layers may not be exactly the same as in previous experiences. Therefore the model uses additional processes contributing to invariant object recognition, most importantly the extraction of global features and the exploitation of population codes for the length, position and orientation of features. This also establishes a limited kind of rotation invariance. The selection of information within the what-pathway is consistent with findings from Moran & Desimone (1985): unattended information is excluded from further processing only, if it would stimulate the same population of neurons at the next stage as the selected information. 2.4.2 The Object-Recogllition-Module The output of the Invariance Module, the perceptual-code stage, feeds to the objectrecognition layer and receives recurrent connections from that layer terminating both on the a-gate and the p-gate of its units. These connections are trained using the back-propagation learning rule (T = T = 1). The recurrent loop establishes an interactive recognition process allowing to recognize distorted patterns through the completion of missing informat.ion and the suppression of noise. At the perceptual-code stage perceptual organisation continues based on the initial feature linkings computed within the elementary feature maps. The p-weight pattern 908 Goebel normalisation \ complex grouplng what-pathway color maps ~ttellttve stage bIdlng. salienCy! onentatlon maps "&reeD/, ~7L. ~7 "blue:/ • -7L-'7 ./ "retina" Figure 1: The architecture of HOTSPOT Perceiving Complex Visual Scenes: An Oscillator Neural Network Model 909 within the perceptual-code stage implements a set of Gestalt principles such as spatial proximity, similarity and continuity of contour. In additon, acquired shape knowledge is another force acting on the perceptual-code stage in order to bind or separate global features. Object-based attention may select one of multiple oscillating objects. For determining a specific object it may use whole-object knowledge (e.g., 'select the letter H'), spatial cues (e.g., 'select the right object') or color cues (e.g., 'select the green object') as well as a combined cue. If the selected object does not use the whole resolution of the perceptual-code stage, commands are sent to the where-pathway in order to adjust the spotlight accordingly. 2.5 THE WHERE-PATHWAY The where-pathway consists of the saliency map, the spotlight-control module, the disengagement layer and the spatial-representation layer. The spotlight-control module performs relative movements and size changes of the attentional spotlight which are demanded by the saliency map, object-based selection or commands from a short-term store holding task instructions. If the current position of the spotlight is not changed for some time, the disengagement layer inhibits the corresponding position at the saliency map. The spatial-representation layer contains a coarsely tuned representation of all active retinal positions. If no position within the visual field is particularly salient, this layer determines possible target positions for spatial attention. If the model knows "where what is" this knowledge is transferred to the visual shortterm memory where a sequence of 'location-object couplings' can be stored. 3 CONCLUSION In this paper an oscillator neural network model was presented that integrates location-based attention, perceptual organisation, and invariance transformations. It was outlined how the cooperation between these mechanisms allow the model to segment, select and recognize objects within a complex input scene. The model was successfully applied to simulate a wide variety of psychophysical data including texture segregation, visual search, hierarchical segmentation and recognition. A typical 'processing cycle' of the model consists of an initial segmentation of the visual field with a broadly tuned spotlight. Then a segmented, but not necessarily recognizable, entity may be selected due to its saliency or by object-based attention. This selection in turn induces movements of the location-based attention mechanism until the selected entity is surrounded by the spotlight. Since in this case appropriate invariance transformations are computed the selected object is optimally recognized. Some predictions of the model concerning the object-based nature of selective attention are currently experimentally tested. HOTSPOT indicates a promising way towards a deeper understanding of complex visual processing by bringing together both neurobiological and psychophysical findings in a fruitful way. Acknowledgements I am grateful to Reinhard Eckhorn, Peter Konig, Michael Mozer, Werner X. Schneider, Wolf Singer and Dirk Vorberg for valuable discussions. 910 Goebel References Eckhorn, R, Bauer, R, Jordan, W., Brosch, M., Kruse, W., Munk, M. & Reitboeck, H.J. (1988) Coherent Oscillations: A mechanism of feature linking in the visual cortex? Biological Cybernetics, 60, 121-130 Eckhorn, R., Reitboeck, H. J., Arndt, M., & Dicke, P. (1990). Feature linking via synchronization among distributed assemblies: The simulation of results from cat visual cortex. Neural Computation, 2, 293-307. Fukushima, K. (1986). A neural network model for selective attention in visual pattern recognition. Biological Cybernetics, 55, 5-15. Gray, M. C. & Singer, W. (1989). Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. PNAS USA, 86, 1698-1702. Hinton, G.E, Lang, K.J. (1985). Shape Recognition and Illusory Conjunctions. Proceedings of the 9th IlCAI - Los Angeles, 1, 252-259. Milner, P.M. (1974). A model for visual shape recognition. Psych. Rev., 81,521-535. Moran, J. & Desimone, R. (1985). Selective attention gates visual processing in the extrastriate cortex. Science, 229,782-784. Mozer, M. C. (1991). The perception of multiple objects: a connectionist approach. MIT Press / Bradford Books. Mozer, M. C., Zemel, R. S., & Behrmann, M., Williams, C.K.l. (1992). Learning to segment images using dynamic feature binding. Neural Computation, 4, 650-665. Neisser, U. (1967) . Cognitive Psychology. New York: Appleton-Century-Crofts. Olshausen, B., Anderson, Ch., & Van Essen, D. (1992), A neural model of visual attention and invariant pattern recognition. CNS Memo 18, CalTech. Posner, M.I., Snyder, C. R. R., & Davidson, B.J. (1980). Attention and the detection of signals. Journal of Experimental Psychology: General, 109, 160-174. Sandon, P. (1990). Simulating visual attention. Journal of Cog. Neurosc., 2, 213-231. Schillen, Th. B. & Konig, P. (1991). Stimulus-dependent assembly formation of oscillatory responses: II. Desynchronization. Neural Computation, 3, 167-178. Sompolinsky, H., Golomb, D., & Kleinfeld, D. (1990). Global processing of visual stimuli in a neural network of coupled oscillators. Proc. Natl. Acad. Sci. USA, 87, 7200-7204. Ungerleider, L. G., & Mishkin, M. (1982). Two cortical visual systems. In D. J. Ingle, M. A. Goodale, & R. J. W. Mansfield (Eels.), Analysis of visual behavior. Cambridge, MA: MIT Press. Van Essen (1985). Functional organization of primate visual cortex. In A. Peters & E. G. Jones (Eds.)., Cerebral cortex, vol. 3. New York: Plenum Press. Von der Malsburg, C. (1981) The correlation theory of brain function. Internal Report 81-2, Dept. of Neurobiology, MPI for Biophysical Chemistry. PART XII COMPUTATIONAL AND THEORETICAL NEUROBIOLOGY
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Second order derivatives for network pruning: Optimal Brain Surgeon Babak Hassibi* and David G. Stork Ricoh California Research Center 2882 Sand Hill Road, Suite 115 Menlo Park, CA 94025-7022 stork@crc.ricoh.com and * Department of Electrical Engineering Stanford University Stanford, CA 94305 Abstract We investigate the use of information from all second order derivatives of the error function to perfonn network pruning (i.e., removing unimportant weights from a trained network) in order to improve generalization, simplify networks, reduce hardware or storage requirements, increase the speed of further training, and in some cases enable rule extraction. Our method, Optimal Brain Surgeon (OBS), is Significantly better than magnitude-based methods and Optimal Brain Damage [Le Cun, Denker and Sol1a, 1990], which often remove the wrong weights. OBS permits the pruning of more weights than other methods (for the same error on the training set), and thus yields better generalization on test data. Crucial to OBS is a recursion relation for calculating the inverse Hessian matrix H-I from training data and structural information of the net. OBS permits a 90%, a 76%, and a 62% reduction in weights over backpropagation with weighL decay on three benchmark MONK's problems [Thrun et aI., 1991]. Of OBS, Optimal Brain Damage, and magnitude-based methods, only OBS deletes the correct weights from a trained XOR network in every case. Finally, whereas Sejnowski and Rosenberg [1987J used 18,000 weights in their NETtalk network, we used OBS to prune a network to just 1560 weights, yielding better generalization. 1 Introduction A central problem in machine learning and pattern recognition is to minimize the system complexity (description length, VC-dimension, etc.) consistent with the training data. In neural networks this regularization problem is often cast as minimizing the number of connection weights. Without such weight elimination overfilting problems and thus poor generalization will result. Conversely, if there are too few weights, the network might not be able to learn the training data. If we begin with a trained network having too many weights, the questions then become: Which weights should be eliminated? How should the remaining weights be adjusted for best performance? How can such network pruning be done in a computationally efficient way? 164 Second order derivatives for network pruning: Optimal Brain Surgeon 165 Magnitude based methods [Hertz, Krogh and Palmer, 1991] eliminate weights that have the smallest magnitude. This simple and naively plausible idea unfortunately often leads to the elimination of the wrong weights small weights can be necessary for low error. Optimal Brain Damage [Le Cun, Denker and Solla, 1990] uses the criterion of minimal increase in training error for weight elimination. For computational simplicity, OBD assumes that the Hessian matrix is diagonal: in fact. however, Hessians for every problem we have considered are strongly non-diagonal, and this leads OBD to eliminate the wrong weights. The superiority of the method described here Optimal Brain Surgeon lies in great pan to the fact that it makes no restrictive assumptions about the form of the network's Hessian, and thereby eliminates the correct weights. Moreover, unlike other methods, OBS does not demand (typically slow) retraining after the pruning of a weight. 2 Optimal Brain Surgeon In deriving our method we begin, as do Le Cun, Denker and Solla [1990], by considering a network trained to a local minimum in error. The functional Taylor series of the error with respect to weights (or parameters, see below) is: (1) where H = ;]2 E/ aw2 is the Hessian matrix (containing all second order derivatives) and the superscript T denotes vector transpose. For a network trained to a local minimum in error, the first (linear) term vanishes: we also ignore the third and all higher order terms. Our goal is then to set one of the weights to zero (which we call wq) to minimize the increase in error given by Eq. l. Eliminating Wq is expressed as: owq+wq =0 ormoregenerally e~ ·OW+Wq =0 (2) where eq is the unit vector in weight space corresponding to (scalar) weight wq• Our goal is then to solve: Minq {Mint5w l! OW T . H . ow} such that e~. ow + W q = O} (3) To solve Eq. 3 we form a Lagrangian from Eqs. 1 and 2: L = 1-ow T . H . ow + A (e~ . ow + W q) (4) where A. is a Lagrange undetermined multiplier. We take functional derivatives, employ the constraints of Eq. 2, and use matrix inversion to find that the optimal weight change and resulting change in error are: w 1 w2 ow = q H-1 • e and L = q (5) [H-1] q q 2 [H-1] qq qq Note that neither H nor H·I need be diagonal (as is assumed by Le Cun et al.): moreover, our method recalculates the magnitude of all the weights in the network, by the left side of Eq. 5. We call Lq the "saliency" of weight q the increase in error that results when the weight is eliminated a definition more general than Le Cun et al. 's, and which includes theirs in the special case of diagonal H. Thus we have the following algorithm: Optimal Brain Surgeon procedure 1. Train a "reasonably large" network to minimum error. 2. Compute H·I . 3. Find the q that gives the smallest saliency Lq = Wq 2/(2[H·I ]qq). If this candidate error increase is much smaller than E, then the qth weight should be deleted, and we proceed to step 4; otherwise go to step 5. (Other stopping criteria can be used too.) 4. Use the q from step 3 to update all weights (Eq. 5). Go to step 2. 5. No more weights can be deleted without large increase in E. (At this point it may be desirable to retrain the network.) Figure 1 illustrates the basic idea. The relative magnitudes of the error after pruning (before retraining. if any) depend upon the particular problem, but to second order obey: E(mag) ~ E(OBD) ~ E(OBS). which is the key to the superiority of OBS. In this example OBS and OBD lead to the elimination of the same weight (weight 1). In many cases, however. OBS will eliminate different weights than those eliminated by OBD (cf. Sect. 6). We call our method Optimal Brain Surgeon because in addition to deleting weights, it 166 Hassibi and Stork calculates and changes the strengths of other weights without the need for gradient descent or other incremental retraining. Figure 1: Error as a function of two weights in a network. The (local) minimum occurs at weight w·, found by gradient descent or other learning method. In this illustration, a magnitude based pruning technique (mag) then removes the smallest weight, weight 2; Optimal Brain Damage before retraining (OBD) removes weight I. In contrast, our Optimal Brain Surgeon method (OBS) not only removes weight I, but also automatically adjusts the value of weight 2 to minimize the error, without retraining. The error surface here is general in that it has different curvatures (second derivatives) along different directions, a minimum at a non-special weight value, and a non-diagonal Hessian (i.e., principal axes are not parallel to the weight axes). We have found (to our surprise) that every problem we have investigated has strongly non-diagonal Hessians thereby explaining the improvment of our method over that of Le Cun et al. 3 Computing the inverse Hessian The difficulty appears to be step 2 in the OBS procedure, since inverting a matrix of thousands or millions of terms seems computationally intractable. In what follows we shall give a general derivation of the inverse Hessian for a fully trained neural network. It makes no difference whether it was trained by backpropagation, competitive learning, the Boltzmann algorithm, or any other method, so long as derivatives can be taken (see below). We shall show that the Hessian can be reduced to the sample covariance matrix associated with certain gradient vectors. Furthennore, the gradient vectors necessary for OBS are normally available at small computational cost; the covariance form of the Hessian yields a recursive formula for computing the inverse. Consider a general non-linear neural network that maps an input vector in of dimension nj into an output vector 0 of dimension no' according to the following: 0= F(w,in) (6) where w is an n dimensional vector representing the neural network's weights or other parameters. We shall refer to w as a weight vector below for simplicity and definiteness, but it must be stressed that w could represent any continuous parameters, such as those describing neural transfer function, weight sharing, and so on. The mean square error corresponding to the training set is dermed as: E = _1 i(t[k] _ o[k]{ (t[k] _ o[k]) 2P k=1 (7) where P is the number of training patterns, and tlk] and olk] are the desired response and network response for the kth training pattern. The first derivative with respect to w is: aE = _! i aF(w,in[k) (t[k) _ o[k]) (Jw P k=1 dw (8) and the second derivative or Hessian is: a2E 1 P aF(w in[k]) aF(w,in[k]{ H=--2 =- L[ , dw P k=1 (Jw (Jw :l2 • [k] (J F(w,m ) . (t[k) _ o[k)] (Jw2 (9) Second order derivatives for network pruning: Optimal Brain Surgeon 167 Next we consider a network fully trained to a local minimum in error at w*. Under this condition the network response O[k] will be close to the desired response t[k], and hence we neglect the tenn involving (t[k]- ork]). Even late in pruning, when this error is not small for a single pattern, this approximation can be justified (see next Section). This simplification yields: H =! f dF(w,in[k]). dF(w,in[k) T p k=1 dw dw (10) If out network has just a single output, we may define the n-dimensional data vector Xrk] of derivatives as: X[k) = dF(w,in[k]) (11) aw Thus Eq. 10 can be written as: H =! fX[k). X[k]T P k=1 If instead our network has mUltiple output units, then X will be an n x no matrix of the fonn: (12) X[k] = dF(w,in[k]) = (dF1(w,in[k]) dFno (W,in[k]» [kJ [k] aw aw ..... aw = (Xl •...• Xno) (13) where Fj is the ith component of F. Hence in this multiple output unit case Eq. 10 generalizes to: H =! f rx~k). X~k]T (14) P k=ll=1 Equations 12 and 14 show that H is the sample covariance matrix associated with the gradient variable X. Equation 12 also shows that for the single output case we can calculate the full Hessian by sequentially adding in successive "component" Hessians as: H - H + .!.X[m+I]. X[m+l]T with HO = aI and Hp = H (15) m+lm P But Optimal Brain Surgeon requires the inverse of H (Eq. 5). This inverse can be calculated using a standard matrix inversion fonnula [Kailath, 1980]: (A + 8 . C . 0)-1 = A-I - A-I. 8 . (C-I + D. A-I. 8)-1 . D . A-I (16) applied to each tenn in the analogous sequence in Eq. 16: H-1 . X[m+1) . X[m+1)T . H-I H-I - H-I m m with HOi = a-II and Hpl = H-I m+1 m p + x[m+I)T . H-I . X[m+lI m (17) and a (l0·8 S a S 10-4) a small constant needed to make HO•I meaningful, and to which our method is insensitive [Hassibi, Stork and Wolff, 1993b]. Actually, Eq. 17 leads to the calculation of the inverse of (H + ciI), and this corresponds to the introduction of a penalty term allliwll2 in Eq. 4. This has the benefit of penalizing large candidate jumps in weight space, and thus helping to insure that the neglecting of higher order Lenns in Eq. 1 is valid. Equation 17 permits the calculation of H·I using a single sequential pass through the training data 1 S m S P. It is also straightforward to generalize Eq. 18 to the multiple output case of Eq. 15: in this case Eq. 15 will have recursions on both the indices m and I giving: H - H 1 x[m) X[m)T m 1+1 ml + 1+1' 1+1 P H = H + ! x[m+1) . X[m+I]T m+11 milo P 1 I (18) To sequentially calculate U-I for the multiple output case, we use Eq. 16, as before. 4 The (t - 0) ~ 0 approximation The approximation used for Eq. 10 can be justified on computational and functional grounds, even late in pruning when the training error is not negligible. From the computational view, we note [rrst that nonnally H is degenerate especially before significant pruning has been done and its inverse not well defined. 168 Hassibi and Stork The approximation guarantees that there are no singularities in the calculation of H-1• It also keeps the computational complexity of calculating H-1 the same as that for calculating H O(p n2). In Statistics the approximation is the basis of Fisher's method of scoring and its goal is to replace the true Hessian with its expected value and guarantee that H is positive definite (thereby avoiding stability problems that can plague Gauss-Newton methods) [Seber and Wild, 1989]. Equally important are the functional justifications of the approximation. Consider a high capactiy network trained to small training error. We can consider the network structure as involving both signal and noise. As we prune, we hope to eliminate those weights that lead to "overfilting," i.e., learning the noise. If our pruning method did not employ the (t - 0) ~ 0 approximation, every pruning step (Eqs. 9 and 5) would inject the noise back into the system, by penalizing for noise tenns. A different way to think of the approximation is the following. After some pruning by OBS we have reached a new weight vector that is a local minimum of the error (cf. Fig. 1). Even if this error is not negligible, we want to stay as close to that value of the error as we can. Thus we imagine a new, effective teaching signal t*, that would keep the network near this new error minimum. It is then (t* - 0) that we in effect set to zero when using Eq. 10 instead of Eq. 9. 5 aBS and back propagation Using the standard tennino)ogy from backpropagation [Rumelhart, Hinton and Williams, 1986J and the single output network of Fig. 2, it is straightforward to show from Eq. 11 that the derivative vectors are: [k] - (x~]J X [k] (19) Xu where (20) refers to derivatives with respect to hidden-to-output weights Vj and [X~.t)]T = (f' (net[.t)f (net\.t)v\.t)o~!L .... f' (net[.t)f (net\.t)v~.t)o~~) .... , (21) f (net[.t)f' (net~.t)v~~)o\.t) ..... f (net(.t)f (net~.t.)V~.t.)o~~l) J J J J I refers to derivatives with respect to input-to-hidden weights uji' and where lexicographical ordering has been used. The neuron nonlinearity is f(·). 6 output hidden input i = n· 1 Figure 2: Backpropagation net with lli inputs and nj hidden units. The input-to-hidden weights are Uji and hidden-to-output weights Vj. The derivative ("data") vectors are Xv and Xu (Eqs. 20 and 21). Simulation results We applied OBS, Optimal Brain Damage, and a magnitude based pruning method to the 2-2-1 network with bias unit of Fig. 3, trained on all patterns of the XOR problem. The network was first trained to a local minimum, which had zero error, and then the three methods were used to prune one weight. As shown,the methods deleted different weights. We then trained the original XOR network from different initial conditions, thereby leading to a different local minima. Whereas there were some cases in which OBD or magnitude methods deleted the correct weight, only OBS deleted the correct weight in every case. Moreover, OBS changed the values of the remaining weights (Eq.5) to achieve perfect perfonnance without any retraining by the backpropagation algorithm. Figure 4 shows the Hessian of the trained but unpruned XOR network. output hidden input Second order derivatives for network pruning: Optimal Brain Surgeon 169 Figure 3: A nine weight XOR network trained to a local minimum. The thickness of the lines indicates the weight magnitudes, and inhibitory weights are shown dashed. Subsequent pruning using a magnitude based method (Mag) would delete weight v3; using Optimal Brain Damage (OBD) would delete U22. Even with retraining, the network pruned by those methods cannot learn the XOR problem. In contrast, Optimal Brain Surgeon (OBS) deletes U23 and furthennore changed all other weights (cf. Eq. 5) to achieve zero error on the XOR problem. Figure 4: The Hessian of the trained but unpruned XOR network, calculated by means of Eq. 12. White represents large values and black small magnitudes. The rows and columns are labeled by the weights shown in Fig. 3. As is to be expected, the hidden-to-output weights have significant Hessian components. Note especially that the Hessian is far from being diagonal. The Hessians for all problems we have investigated, including the MONK's problems (below), are far from being diagonal. VI v2 v3 UII Ul2 u13 u21 u22 U23 Figure 5 shows two-dimensional "slices" of the nine-dimensional error surface in the neighborhood of a local minimum at w· for the XOR network. The cuts compare the weight elimination of Magnitude methods (left) and OBD (right) with the elimination and weight adjustment given by OBS. E U23 o -1 -2 V3 u22 Figure 5: (Left) the XOR error surface as a function of weights V3 and U23 (cf. Fig. 4). A magnitude based pruning method would delete weight V3 whereas OBS deletes U23. (Right) The XOR error surface as a function of weights U22 and U23. Optimal Brain Damage would delete U22 whereas OBS deletes U23. For this minimum, only deleting U23 will allow the pruned network to solve the XOR problem. E 170 Hassibi and Stork After all network weights are updated by Eq. 5 the system is at zero error (not shown). It is especially noteworthy that in neither case of pruning by magnitude methods nor Optimal Brain Damage will further retraining by gradient descent reduce the training error to zero. In short, magnitude methods and Optimal Brain Damage delete the wrong weights, and their mistake cannot be overcome by further network training. Only Optimal Brain Surgeon deletes the correct weight. We also applied OBS to larger problems, three MONK's problems, and compared our results to those of Thrun et al. [1991], whose backpropagation network outperformed all other approaches (network and rulebased) on these benchmark problems in an extensive machine learning competition. Accuracy training testing # weights MONKl BPWD 100 100 58 aBS 100 100 14 MONK 2 BPWD 100 100 39 aBS 100 100 15 BPWD 93.4 97.2 39 aBS 93.4 97.2 4 MONK 3 Table 1: The accuracy and number of weights found by backpropagation with weight decay (BPWD) found by Thrun etal. [1991], and by OBS on three MONK's problems. Table I shows that for the same perfonnance, OBS (without retraining) required only 24%, 38% and 10% of the weights of the backpropagation network, which was already regularized with weight decay (Fig. 6). The error increaseL (Eq. 5) accompanying pruning by OBS negligibly affected accuracy. ", .' .... .... , _I. l..... .-'...... . .... Figure 6: Optimal networks found by Thrun using backpropagation with weight decay (Left) and by OBS (Right) on MONK I, which is based on logical rules. Solid (dashed) lines denote excitatory (inhibitory) connections; bias units are at left. The dramatic reduction in weights achieved by OBS yields a network that is simple enough that the logical rules that generated the data can be recovered from the pruned network, for instance by the methods of Towell and Shavlik [1992]. Hence OBS may help to address a criticism often levied at neural networks: the fact that they may be unintelligible. We applied OBS to a three-layer NETtalk network. While Sejnowski and Rosenberg [1987] used 18,000 weights, we began with just 5546 weights, which after backpropagation training had a test error of 5259. After pruning this net with OBS to 2438 weights, and then retraining and pruning again, we achieved a net with only 1560 weights and test error of only 4701 a significant improvement over the original, more complex network [Hassibi, Stork and Wolff, 1993a]. Thus OBS can be applied to real-world pattern recognition problems such as speech recognition and optical character recognition, which typically have several thousand parameters. 7 Analysis and conclusions Why is Optimal Brain Surgeon so successful at reducing excess degrees of freedom? Conversely, given this new standard in weight elimination, we can ask: Why are magnitude based methods so poor? Consider again Fig. 1. Starting from the local minimum at w·, a magnitude based method deletes the wrong weight, weight 2, and through retraining, weight 1 will increase. The final "solution" is weight 1 4 large, weight 2 = O. This is precisely the opposite of the solution found by OBS: weight 1 = 0, weight 2 4 large. Although the actual difference in error shown in Fig. 1 may be small, in large networks, differences from many incorrect weight elimination decisions can add up to a significant increase in error. Second order derivatives for network pruning: Optimal Brain Surgeon 171 But most importantly, it is simply wishful thinking to believe that after the elimination of many incorrect weights by magnitude methods the net can "sort it all out" through further training and reach a global optimum, especially if the network has already been pruned significantly (cf. XOR discussion, above). We have also seen how the approximation employed by Optimal Brain Damage that the diagonals of the Hessian are dominant does not hold for the problems we have investigated. There are typically many off-diagonal terms that are comparable to their diagonal counterparts. This explains why OBD often deletes the wrong weight, while OBS deletes the correct one. We note too that our method is quite general, and subsumes previous methods for weight elimination. In our terminology, magnitude based methods assume isotropic Hessian (H ex I); OBD assumes diagonal H: FARM [Kung and Hu, 1991] assumes linear f(net) and only updates the hidden-to-output weights. We have shown that none of those assumptions are valid nor sufficient for optimal weight elimination. We should also point out that our method is even more general than presented here [Hassibi, Stork and Wotff, 1993bl. For instance, rather than pruning a weight (parameter) by setting it to zero, one can instead reduce a degree of freedom by projecting onto an arbitrary plane, e.g., Wq = a constant, though such networks typically have a large description length [Rissanen, 1978]. The pruning constraint w q = 0 discussed throughout this paper makes retraining (if desired) particularly simple. Several weights can be deleted simultaneously; bias weights can be exempt from pruning, and so forth. A slight generalization of OBS employs cross-entropy or the Kullback-Leibler error measure, leading to Fisher Infonnation matrix rather than the Hessian (Hassibi, Stork and Wolff, 1993b). We note too that OBS does not by itself give a criterion for when to stop pruning, and thus OBS can be utilized with a wide variety of such criteria. Moreover, gradual methods such as weight decay during learning can be used in conjunction with OBS. Acknowledgements The first author was supported in part by grants AFOSR 91-0060 and DAAL03-91-C-OOlO to T. Kailath, who in tum provided constant encouragement Deep thanks go to Greg Wolff (Ricoh) for assistance with simulations and analysis, and Jerome Friedman (Stanford) for pointers to relevant statistics literature. REFERENCES Hassibi, B. Stork, D. G. and Wolff, G. (1993a). Optimal Brain Surgeon and general network pruning (submitted to ICNN, San Francisco) Hassibi, B. Stork, D. G. and Wolff, G. (1993b). Optimal Brain Surgeon, Information Theory and network capacity control (in preparation) Hertz, J., Krogh, A. and Palmer, R. G. (1991). Introduction to the Theory of Neural Computation Addison-Wesley. Kailath, T. (1980). Linear Systems Prentice-Hall. Kung, S. Y. and Hu, Y. H. (1991). A Frobenius approximation reduction method (FARM) for detennining the optimal number of hidden units, Proceedings of the IJCNN-9I Seattle, Washington. Le Cun, Y., Denker, J. S. and SoUa, S. A. (1990). Optimal Brain Damage, in Proceedings of the Neural Information Processing Systems-2, D. S. Touretzky (ed.) 598-605, Morgan-Kaufmann. Rissanen, J. (1978). Modelling by shortest data description, Aulomatica 14,465-471. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning Internal representations by error propagation, Chapter 8 (318-362) in Parallel Distributed Processing I D. E. Rumelhart and J. L. McClelland (eds.) MIT Press. Seber, G. A. F. and Wild, C. J. (1989). Nonlinear Regression 35-36 Wiley. Sejnowski, T. J., and Rosenberg, C. R. (1987). Parallel networks that learn to pronounce English text, Complex Syslems I, 145-168. Thrun, S. B. and 23 co-authors (1991). The MONK's Problems A perfonnance comparison of different learning algorithms, CMU-CS-91-197 Carnegie-Mellon U. Department of Computer ScienceTech Report. Towell, G. and Shavlik, J. W. (1992). Interpretation of artificial neural networks: Mapping knowledgebased neural networks into rules, in Proceedings of the Neural In/ormation Processing Systems-4, ]. E. Moody, D. S. Touretzky and R. P. Lippmann (eds.) 977-984, Morgan-Kaufmann.
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Automatic Learning Rate Maximization by On-Line Estimation of the Hessian's Eigenvectors Yann LeCun,l Patrice Y. Simard,l and Barak Pearlmutter2 1 AT&T Bell Laboratories 101 Crawfords Corner Rd, Holmdel, NJ 07733 2CS&E Dept. Oregon Grad. Inst., 19600 NW vonNeumann Dr, Beaverton, OR 97006 Abstract We propose a very simple, and well principled way of computing the optimal step size in gradient descent algorithms. The on-line version is very efficient computationally, and is applicable to large backpropagation networks trained on large data sets. The main ingredient is a technique for estimating the principal eigenvalue(s) and eigenvector(s) of the objective function's second derivative matrix (Hessian), which does not require to even calculate the Hessian. Several other applications of this technique are proposed for speeding up learning, or for eliminating useless parameters. 1 INTRODUCTION Choosing the appropriate learning rate, or step size, in a gradient descent procedure such as backpropagation, is simultaneously one of the most crucial and expertintensive part of neural-network learning. We propose a method for computing the best step size which is both well-principled, simple, very cheap computationally, and, most of all, applicable to on-line training with large networks and data sets. Learning algorithms that use Gradient Descent minimize an objective function E of the form p E(W) = ~EEP(W) p=O EP = E(W,XP) (1) where W is the vector of parameters (weights), P is the number of training patterns, and XP is the p-th training example (including the desired output if necessary). Two basic versions of gradient descent can be used to minimize E. In the first version, 156 Automatic Learning Rate Maximization by Estimation of Hessian's Eigenvectors 157 called the batch version, the exact gradient of E with respect to W is calculated, and the weights are updated by iterating the procedure W W - 1]VE(W) (2) where 1] is the learning rate or step size, and VE(W) is the gradient of E with respect to W. In the second version, called on-line, or Stochastic Gradient Descent, the weights are updated after each pattern presentation (3) Before going any further, we should emphasize that our main interest is in training large networks on large data sets. As many authors have shown, Stochastic Gradient Descent (SGD) is much faster on large problems than the "batch" version. In fact, on large problems, a carefully tuned SGD algorithm outperforms most accelerated or second-order batch techniques, including Conjugate Gradient. Although there have been attempts to "stochasticize" second-order algorithms (Becker and Le Cun, 1988) (Moller, 1992), most of the resulting procedures also rely on a global scaling parameter similar to 1]. Therefore, there is considerable interest in finding ways of optimizing 1]. 2 COMPUTING THE OPTIMAL LEARNING RATE: THE RECIPE In a somewhat unconventional way, we first give our simple "recipe" for computing the optimal learning rate 1]. In the subsequent sections, we sketch the theory behind the recipe. Here is the proposed procedure for estimating the optimal learning rate in a backpropagation network trained with Stochastic Gradient Descent. Equivalent procedures for other adaptive machines are strai~htforward. In the following, the notation N(V) designates the normalized vector V /11 VII. Let W be the N dimensional weight vector, 1. pick a normalized, N dimensional vector \If at random. Pick two small positive constants a and " say a = 0.01 and, = 0.01. 2. pick a training example (input and desired output) XP. Perform a regular forward prop and a backward prop. Store the resulting gradient vector G1 = VEP(W). 3. add aNew) to the current weight vector W, 4. perform a forward prop and a backward prop on the same pattern using the perturbed weight vector. Store the resulting gradient vector G2 ::: VEP(W + aN(w» 5. update vector W with the runmng average formula W (1 -,)w + ;( G2 - G.). 6. restore the weight vector to its original value W. 7. loop to step 2 until Ilwll stabilizes. 8. set the learning rate 1] to IIWII- 1, and go on to a regular training session. The constant a controls the size of the perturbation. A small a gives a better estimate, but is more likely to cause numerical errors. , controls the tradeoff between the convergence speed of wand the accuracy of the result. It is better to start with 158 LeCun, Simard, and Pearlmutter E(W) W z (a) W2 principal eigenvector ~--------------~~Wl (b) Figure 1: Gradient descent with optimal learning rate in (a) one dimension, and (b) two dimensions (contour plot). a relatively large 'Y (say 0.1) and progressively decrease it until the fluctuations on 1I\]i1l are less than say 10%. In our experience accurate estimates can be obtained with between one hundred and a few hundred pattern presentations: for a large problem, the cost is very small compared to a single learning epoch. 3 STEP SIZE, CURVATURE AND EIGENVALUES The procedure described in the previous section makes "\]ill converge to the largest positive eigenvalue of the second derivative matrix of the average obJective function. In this section we informally explain why the best learning rate is the inverse of this eigenvalue. More detailed analysis of gradient descent procedures can be found in Optimization, Statistical Estimation, or Adaptive Filtering textbooks (see for example (Widrow and Stearns, 1985». For didactical purposes, consider an objective function of the form E(w) = ~(w - z)2 + C where w is a scalar parameter (see fig l(a». Assuming w is the current value of the parameter, what is the optimal 1] that takes us to the minimum in one step? It is easy to visualize that, as it has been known since Newton, the optimal TJ is the inverse of the second derivative of E, i.e. 1/ h. Any smaller or slightly larger value will yield slower convergence. A value more then twice the optimal will cause divergence. In multidimension, things are more complicated. If the objective function is quadratic, the surfaces of equal cost are ellipsoids (or ellipses in 2D as shown on figure l(b». Intuitively, if the learning rate is set for optimal convergence along the direction of largest second derivative, then it will be small enough to ensure (slow) convergence along all the other directions. This corresponds to setting the learning rate to the inverse of the second derivative in the direction in which it is the largest. The largest learning rate that ensures convergence is twice that value. The actual optimal TJ is somewhere in between. Setting it to the inverse of the largest second derivative is both safe, and close enough to the optimal. The second derivative information is contained in the Hessian matrix of E(W): the symmetric matrix H whose (i,j) component is ()2 E(W)/OWiOWj. If the learning machine has N free parameters (weights), H is an N by N matrix. The Hessian can be decomposed (diagonalized) into a product of the form H = RART, where A is a diagonal matrix whose diagonal terms (the eigenvalues of H) are the second derivatives of E(W) Automatic Learning Rate Maximization by Estimation of Hessian's Eigenvectors 159 along the principal axes of the ellipsoids of equal cost, and R is a rotation matrix which defines the directions of these principal axes. The direction of largest second derivative is the principal eigenvector of H, and the largest second derivative is the corresponding eigenvalue (the largest one). In short, it can be shown that the optimal learning rate is the inverse of the largest eigenvalue of H: 1 1Jopt = ~ "max 4 COMPUTING THE HESSIAN'S LARGEST EIGENVALUE WITHOUT COMPUTING THE HESSIAN (4) This section derives the recipe given in section 2. Large learning machines, such as backpropagation networks can have several thousand free parameters. Computing, or even storing, the full Hessian matrix is often prohibitively expensive. So at first glance, finding its largest eigenvalue in a reasonable time seems rather hopeless. We are about to propose a shortcut based on three simple ideas: 1- the Taylor expansion, 2- the power method, 3- the running average. The method described here is general, and can be applied to any differentiable objective function that can be written as an average over "examples" (e.g. RBFs, or other statistical estimation techniques). Taylor expansion: Although it is often unrealistic to compute the Hessian H, there is a simple way to approximate the product of H by a vector of our choosing. Let \II be an N dimensional vector, and a a small real constant, the Taylor expansion of the gradient of E(W) around W along the direction \II gives us H\II = V'E(W + a\ll) - V'E(W) + O(a2 ) (5) a Assuming E is locally quadratic (i.e. ignoring the O(a2 ) term), the product of H by any vector W can be estimated by subtracting the gradient of E at point (W + a\ll) from the gradient at W. This is an O(N) process, compared to the O(N2) direct product. In the usual neural network context, this can be done with two forward propagations and two backward propagations. More accurate methods which do not use perturbations for computing H\II exist, but they are more complicated to implement than this one. (Pearlmutter, 1993). The power method: Let Amax be the largest eigenvalue! of H, and Vmax the corresponding normalized eigenvector (or a vector in the eigenspace if >'max is degenerate). If we pick a vector \II (say, at random) which is non-orthogonal to Vmax , then iterating the procedure \II .- H N(\II) (6) will make N(\II) converge to Vmax , and IIwll converge to I>'maxl. The procedure is slow if good accuracy is required, but a good estimate of the eigenvalue can be obtained with a very small number of iterations (typically about 10). The reason for introducing equation (5), is now clear: we can use it to compute the right hand side of (6), yielding \II .- 1. (V'E (W + aN(\II» - V'E(W» a (7) llargest in absolute value, not largest algebraically 160 LeCun, Simard, and Pearlmutter where W is the current estimate of the principal eigenvector of H, and a is a small constant. The "on-line" version: One iteration of the procedure (7) requires the computation of the gradient of E at two different points of the parameter space. This means that one iteration of (7) is roughly equivalent to two epochs of gradient descent learning (two passes through the entire training set). Since (7) needs to be iterated, say 10 times, the total cost of estimating Amax would be approximately equivalent to 20 epochs. This excessive cost can be drastically reduced with an "on-line" version of (7) which exploits the stationarity of the second-order information over large (and redundant) training sets. Essentially, the hidden "average over patterns" in VE can be replaced by a running average. The procedure becomes 1 \II <- (1 - ,)w + ,- (VE (W + aN(w» - VE(W» a (8) where , is a small constant which controls the tradeoff between the convergence speed and the accuracy 2. The "recipe" given in section 2 is a direct implementation of (8). Empirically, this procedure yields sufficiently accurate values in a very short time. In fact, in all the cases we have tried, it converged with only a few dozen pattern presentations: a fraction of the time of an entire learning pass through the training set (see the results section). It looks like the essential features of the Hessian can be extracted from only a few examples of the training set. In other words, the largest eigenvalue of the Hessian seems to be mainly determined by the network architecture and initial weights, and by short-term, low-order statistics of the input data. It should be noted that the on-line procedure can only find positive eigenvalues. 5 A FEW RESULTS Experiments will be described for two different network architectures trained on segmented handwritten digits taken from the NIST database. Inputs to the networks were 28x28 pixel images containing a centered and size-normalized image of the character. Network 1 was a 4-hidden layer, locally-connected network with shared weights similar to (Le Cun et al., 1990a) but with fewer feature maps. Each layer is only connected to the layer above. the input is 32x32 (there is a border around the 28x28 image), layer 1 is 2x28x28, with 5x5 convolutional (shared) connections. Layer 2 is 2x14x14 with 2x2 subsampled, averaging connections. Layer 3 is 4xl0xl0, with 2x5x5 convolutional connections. Layer 4 is 4x5x5 with 2x2 averaging connections, and the output layer is 10xlxl with 4x5x5 convolutional connections. The network has a total of 64,638 connections but only 1278 free parameters because of the weight sharing. Network 2 was a regular 784x30xlO fully-connected network (23860 weights). The sigmoid function used for all units in both nets was 1.7159 tanh(2/3x). Target outputs were set to +1 for the correct unit, and -1 for the others. To check the validity of our assumptions, we computed the full Hessian of Network 1 on 300 patterns (using finite differences on the gradient) and obtained the eigenvalues and eigenvectors using one of the EISPACK routines. We then computed 2the procedure (8) is not an unbiased estimator of (7). Large values of 'Yare likely to produce slightly underestimated eigenvalues, but this inaccuracy has no practical consequences. Automatic Learning Rate Maximization by Estimation of Hessian's Eigenvectors 161 80 70 II) 60 IS E 50 ;: 8 II) 40 '}'I=O.1 ::;, ii > 30 Ii '}'I=O.03 9 20 II) 10 '}'I=O.01 '}'I=O.003 o 0 60 100 150 200 250 300 350 400 Number of pattern presentations Figure 2: Convergence of the on-line eigenvalue estimation (Network 1) the principal eigenvector and eigenvalue using procedures (7), and (8). All three methods agreed within less than a percent on the eigenvalue. An example run of (8) on a 1000 pattern set is shown on figure 2. A 10% accurate estimate of the largest eigenvalue is obtained in less than 200 pattern presentations (one fifth of the database). As can be seen, the value is fairly stable over small portions of the set, which means that increasing the set size would not require more iterations of the estimation procedure. A second series of experiments were run to verify the accuracy of the learning rate prediction. Network 1 was trained on 1000 patterns, and network 2 on 300 patterns, both with SGD. Figure 3 shows the Mean Squared Error of the two networks after 1,2,3,4 and 5 passes through the training set as a function of the learning rate, for one particular initial weight vector. The constant I was set to 0.1 for the first 20 patterns, 0.03 for the next 60, 0.01 for the next 120, and 0.003 for the next 200 (400 total pattern presentations), but it was found that adequate values were obtained after only 100 to 200 pattern presentations. The vertical bar represents the value predicted by the method for that particular run. It is clear that the predicted optimal value is very close to the correct optimal learning rate. Other experiments with different training sets and initial weights gave similar results. Depending on the initial weights, the largest eigenvalue for Network 1 varied between 80 and 250, and for Network 2 between 250 and 400. Experiments tend to suggest that the optimal learning rate varies only slightly during the early phase of training. The learning rate may need to be decreased for long learning sessions, as SGD converts from the "getting near the minimum" mode to the "wobbling around" mode. There are many other method for adjusting the learning rate. Unfortunately, most of them are based on some measurement of the oscillations of the gradient (Jacobs, 1987). Therefore, they are difficult to apply to stochastic gradient descent. 6 MORE ON EIGENVALUES AND EIGENVECTORS We believe that computing the optimal learning rate is only one of many applications of our eigenvector estimation technique. The procedure can be adapted to serve many applications. 162 LeCun, Simard, and Pearl mutter 2 2 (a) (b) a:: g ~ ffi 1.5 ffi 1.5 c c ILl ILl a:: a:: c ~ ::> i3 1 0 1 CI) CI) z z ; ; 0.5 0.5 n n m A o 06-4 _____ -+-_ ...... __ ~~-~...o 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 3: Mean Squared Error after 1,2,3,4, and 5 epochs (from top to bottom) as a function of the ratio between the learning rate TJ and the learning rate predicted by the proposed method 1I'l111-1. (a) Network 1 trained on 1000 patterns, (b) Network 2 trained on 300 patterns. An important variation of the learning rate estimation is when, instead of update rule 3, we use a "scaled SGD" rule of the form W +- W - TJcI>V'EP(W), where cI> is a diagonal matrix (each weight has its own learning rate TJ4Jd. For example, each 4Ji can be the inverse of the corresponding diagonal term of the average Hessian, which can be computed efficiently as suggested in (Le Cun, 1987; Becker and Le Cun, 1988). Then procedure 8 must be changed to 'l1 +- (1 - ,)'l1 +, ! cI>~ (V'E (W + acI>~ N('l1)) - V'E(W)) (9) where the terms of cI>~ are the square root of the corresponding terms in cI>. More generally, the above formula applies to any transformation of the parameter space whose Jacobian is cI>~. The added cost is small since cI>~ is diagonal. Another extension of the procedure can compute the first J( principal eigenvectors and eigenvalues. The idea is to store J( ei~envector estimates 'l1k' k = 1 .. . J(, updated simultaneously with equation (8) tthis costs a factor J( over estimating only one). We must also ensure that the 'l1 k'S remain orthogonal to each other. This can be performed by projecting each 'l1 k onto the space orthogonal to the space sub tended by the 'l1l' I < k. This is an N J( process, which is relatively cheap if the network uses shared weights. A generalization of the acceleration method introduced in (Le Cun, Kanter and SoHa, 1991) can be implemented with this technique. The idea is to use a "Newton-like" weIght update formula of the type K W +- W - L II'l1kll-1 Pk k=1 where Pk, k = 1 ... J( 1 is the projection of V'E( W) onto 'l1 k, and PK is the projection of V'E(W) on the space orthogonal to the 'l1k' (k = 1 ... J( - 1). In theory, this procedure can accelerate the training by a factor 1I'l1111/II'l1KII, which is between 3 and 10 for J( = 5 in a typical backprop network. Results will be reported in a later publication. Interestingly, the method can be slightly modified to yield the smallest eigenvalues/eigenvectors. First, the largest eigenvalue Amax must be computed (or bounded Automatic Learning Rate Maximization by Estimation of Hessian's Eigenvectors 163 above). Then, by iterating W ~ (1 - ,)w + AmaxN(w) - ,.!. (VE (W + o:N(w» - VE(W» (10) a one can compute the eigenvector corresponding to the smallest (probably negative) eigenvalue of (H - AmaxI), which is the same as H's. This can be used to determine the direction(s) of displacement in parameter space that will cause the least increase of the objective function. There are obvious applications of this to weight elimination methods: a better version of OBD (Le Cun et al., 1990b) or a more efficient version of OBS (Hassibi and Stork, 1993). We have proposed efficient methods for (a) computing the product of the Hessian by any vector, and (b) estimating the few eigenvectors of largest or smallest eigenvalues. The methods were successfully applied the estimation of the optimal learning rate in Stochastic Gradient Descent learning We feel that we have only scratched the surface of the many applications of the proposed techniques. Acknowledgements Yann LeCun and Patrice Simard would like to thank the members of the Adaptive Systems Research dept for their support and comments. Barak Pearlmutter was partially supported by grants NSF ECS-9114333 and ONR N00014-92-J-4062 to John Moody. References Becker, S. and Le Cun, Y. (1988). Improving the Convergence of Back-Propagation Learning with Second-Order Methods. Technical Report CRG-TR-88-5, University of Toronto Connectionist Research Group. Hassibi, B. and Stork, D. (1993). Optimal Brain Surgeon. In Giles, L., Hanson, S., and Cowan, J., editors, Advances in Neural Information Processing Systems, volume 5, (Denver, 1992). Morgan Kaufman. Jacobs, R. A. (1987). Increased Rates of Convergence Through Learning Rate Adaptation. Department of Computer and Information Sciences COINS-TR87-117, University of Massachusetts, Amherst, Ma. Le Cun, Y. (1987). Modeles connexionnistes de l'apprentissage (connectionist learning models). PhD thesis, Universite P. et M. Curie (Paris 6). Le Cun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W., and Jackel, L. D. (1990a). Handwritten digit recognition with a backpropagation network. In Touretzky, D., editor, Advances in Neural Information Processing Systems 2 (NIPS *89} , Denver, CO. Morgan Kaufman. Le Cun, Y., Denker, J. S., SolI a, S., Howard, R. E., and Jackel, L. D. (1990b). Optimal Brain Damage. In Touretzky, D., editor, Advances in Neural Information Processing Systems 2 (NIPS*89), Denver, CO. Morgan Kaufman. Le Cun, Y., Kanter, I., and Solla, S. (1991). Eigenvalues of covariance matrices: application to neural-network learning. Physical Review Letters, 66(18):23962399. Moller, M. (1992). supervised learning on large redundant training sets. In Neural Networks for Signal Processing 2. IEEE press. Pearlmutter, B. (1993). Phd thesis, Carnegie-Mellon University. Pittsburgh PA. Widrow, B. and Stearns, S. D. (1985). Adaptive Signal Processing. Prentice-Hall.
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Computation of Heading Direction From Optic Flow in Visual Cortex Markus Lappe· JosefP. Rauschecker Laboratory of Neurophysiology, NIMH, Poolesville, MD, U.S.A. and Max-Planck-Institut fur Biologische Kybernetik, Tiibingen, Germany Abstract We have designed a neural network which detects the direction of egomotion from optic flow in the presence of eye movements (Lappe and Rauschecker, 1993). The performance of the network is consistent with human psychophysical data, and its output neurons show great similarity to "triple component" cells in area MSTd of monkey visual cortex. We now show that by using assumptions about the kind of eye movements that the obsenrer is likely to perform, our model can generate various other cell types found in MSTd as well. 1 INTRODUCTION Following the ideas of Gibson in the 1950's a number of studies in human psychophysics have demonstrated that optic flow can be used effectively for navigation in space (Rieger and Toet, 1985; Stone and Perrone, 1991; Warren et aI., 1988). In search for the neural basis of optic flow processing, an area in the cat's extrastriate visual cortex (PMLS) was described as having a centrifugal organization of neuronal direction preferences, which suggested an involvement of area PMLS in the processing of expanding flow fields (Rauschecker et a/., 1987; Brenner and Rauschecker, 1990). Recently, neurons in the dorsal part of the medial superior temporal area (MSTd) in monkeys have been described that respond to various combinations of large expanding/contracting, rotating, or shifting dot patterns (Duffy and Wurtz, 1991; Tanaka and Saito, 1989). Cells in MSTd show a continuum of response properties ranging from selectivity for only one movement pattern ("single ·Present address: Neurobiologie, ND7, Ruhr- Universitat Bochum, 4630 Bochum, Germany. 433 434 Lappe and Rauschecker component cells") to selectivity for one mode of each of the three movement types ("triple component cells"). An interesting property of many MSTd cells is their position invariance (Andersen et al., 1990). A sizable proportion of cells, however, do change their selectivity when the stimulus is displaced by several tens of degrees of visual angle, and their position dependence seems to be correlated with the type of movement selectivity (DuffY and Wurtz, 1991; Orban et al., 1992): It is most common for triple component cells and occurs least often in single component cells. Taken together, the wide range of directional tuning and the apparent lack of specificity for the spatial position of a stimulus seem to suggest, that MSTd cells do not possess the selectivity needed to explain the high accuracy of human observers in psychophysical experiments. Our simulation results, however, demonstrate that a population encoding can be used, in which individual neurons are rather broadly tuned while the whole network gives very accurate results. 2 THE NETWORK MODEL The major projections to area MST originate from the middle temporal area (M1). Area MT is a well known area of monkey cortex specialized for the processing of visual motion. It contains a retinotopic representation of local movement directions (AUman and Kaas, 1971; Maunsell and Van Essen, 1983). In our model we assume that area MT comprises a population encoding of the optic flow and that area MST uses this input from MT to extract the heading direction. Therefore, the network consists of two layers. In the first layer, 300 optic flow vectors at random locations within 50 degrees of eccentricity are represented. Each flow vector is encoded by a population of directionally selective neurons. It has been shown previously that a biologically plausible population encoding like this can also be modelled by a neural network (Wang et al., 1989). For simplicity we use only four neurons to represent an optic flow vector 8 i as 4 8, = 2: Sileil, 1=1 (1) with equally spaced preferred directions eil = (cos(1rk/2),sin(1rk/2»t. A neuron's response to a flow vector of direction 4>i and speed OJ is given by the tuning curve if COS(4)i -1rk/2) > 0 otherwise. The second layer contains a retinotopic grid of possible translational heading directions T j. Each direction is represented by a population of neurons, whose summed activities give the likelihood that Tj is the correct heading. The perceiVed direction is finally chosen to be the one that has the highest population activity. The calculation of this likelihood is based on the subspace algorithm by Heeger and Jepson (1992). It employs the minimization of a residual function over all possible heading directions. The neuronal populations in the second layer evaluate a related function that is maximal for the correct heading. The subspace algorithm works as follows: When an observer moves through a static environment all points in space share the same six motion parameters, the translation T = (Tx, Ty , Tz)t and the rotation n = (Ox, Oy, Oz)t. The optic flow 8(z, y) is the projection of the movement ofa 3D- point (X, Y, Z)t onto the retina, which, for simplicity, is modelled as an image plane. In a viewer centered coordinate Computation of Heading Direction From Optic Flow in Visual Cortex 435 system the optic flow can be written as: 1 8(z, y) = Z(z, y) A(z, y)T + B(z, y)fl with the matrices ( -f A(z,y) = 0 o Z) ( zy j f - f - z2 j f - f y and B(z, y) = f + y2 j f -zyj f (2) depending only on coordinates (z , y) in the image plane and on the "focal length" f (Heeger and Jepson, 1992). In trying to estimate T, given the optic flow 8, we first have to note that the unknowns Z (z, y) and T are multiplied together. They can thus not be determined independently so that the translation is considered a unit vector pointing in the direction of heading. Eq. (2) now contains six unknowns, Z (z , y), T and fl, but only two measurements {}z; and {}y. Therefore, flow vectors from m distinct image points are combined into the matrix equation S = C(T)q, (3) where S = (81, ... , 8m )t is a 2m-dimensional vector consisting of the components of the m image velocities, q = (ljZ(z1' yI), ... , 1jZ(zm, Ym), nz;, ny, nz)t an (m + 3)dimensional vector, and C(T) = (A(Zr)T ::~ (4) o a 2m x (m + 3) matrix. Heeger and Jepson (1992) show that the heading direction can be recovered by minimizing the residual function R(T) = IIStCl.(T)112. In this equation Cl.(T) is defined as follows: Provided that the columns of C(T) are linearly independent, they form a basis of an C m + 3 )-dimensional subspace of the R 2m, which is called therangeofC(T). Thematrix Cl.(T) spans the remaining (2m-( m+3))dimensional subspace which is called the orthogonal complement of C(T). Every vector in the orthogonal complement ofC(T) is orthogonal to every vector in the range ofC(T). In the network, the population of neurons representing a certain Tj shall be maximally excited when R(Tj) = O. Two steps are necessary to accomplish this. First an individual neuron evaluates part of the argument ofR(T j) by picking out one of the colwnn vectors of Cl.(Tj), denoted by Ct(Tj), and computing StCtCTj). This is done in the following way: m first layer populations are chosen to form the neuron's input receptive field The neuron's output is given by the sigmoid function m 4 Ujl = g(1: 1: Jij1:1Sik -1-'), (5) ;=1 k=1 in which Jij1:1 denotes the strength of the synaptic connection between the /-th output neuron in the second layer population representing heading direction T j and the k-th input neuron in the first layer population representing the optic flow vector 8;, I-' denotes the threshold For the synaptic strengths we require that: m 4 1:1: Jij1:1Si1: = StCt(Tj ). (6) i=1 1:=1 436 Lappe and Rauschecker At a single image location i this is: ~ J nt (Cl~2i-1 (Tj)) L..J ijlr:1Silr:=ui C1. .(T.) . lr:=1 1,21 , Substituting eq. (I) we find: ~ ~ t (Cl~2i-1(Tj)) L..J J,jlr:1 S ,lr: = L..J Silr:eilr: C1. .(T.) . lr:=1 lr:=1 1,21 , Therefore we set the synaptic strengths to: t (Cl~2i-1(Tj)) Jij1r:l = eilr: C1. .(T.) . 1,21 , Then. whenever T j is compatible with the measured optic flow, i.e. when 8 is in the range ofe(Tj), the neuron receives a net input of zero. In the second step, another neuron Ujl1 is constructed so that the sum of the activities of the two neurons is maximal in this situation. Bothneurons are connected to the same set of image locations but their connection strengths satisfY Jij k 11 = - Jij kl· In addition, the threshold JL is given a slightly negative value. Then both their sigmoid transfer functions overlap at zero input, and the sum has a single peak. Finally, the neurons in every second layer population are organized in such matched pairs so that each population j generates its maximal activity when R(T j) = O. In simulations, our network is able to compute the direction of heading with a mean error of less than one degree in agreement with human psychophysical data (see Lappe and Rauschecker, 1993). Like heading detection in human observers it functions over a wide range of speeds, it works with sparse flow fields and it needs depth in the visual environment when eye movements are performed. 3 DIFFERENT RESPONSE SELECTIVITIES For the remainder of this paper we will focus on the second layer neuron's response properties by carrying out simulations analogous to neurophysiological experiments (Andersen et al., 1990; DuffY and Wurtz, 1991; Orban et aI., 1992). A single neuron is constructed that receives input from 30 random image locations forming a 60 x 60 degree receptive field The receptive field occupies the lower left quadrant of the visual field and also includes the fovea (Fig. IA). The neuron is then presented with shifting, expanding/contracting and rotating optic flow patterns. The center (XC) Yc) of the expanding/contracting and rotating patterns is varied over the 100 x 100 degree visual field in order to test thepositiondependence of the neuron's responses. Directional tuning is assessed via the direction ~ of the shifting patterns. All patterns are obtained by choosing suitable translations and rotations in eq. (2). For instance, rotating patterns centered at (XC) Yc) are generated by T=O and 0= Jz1:~~+P (~). (7) In keeping with the most common experimental condition, all depth values Z(x,) Yi) are taken to be equal. Computation of Heading Direction From Optic Flow in Visual Cortex 437 A c E " -... .. . . m: • JI •• _Ja • • _.\ •• • • • Figure 1: Single Neuron Responding To All Three Types Of Optic Flow Stimuli (" Triple Component Cell") In the following we consider different assumptions about the observer's eye movements. These assumptions change the equations of the subspace algorithm. The rotational matrix B(.x, y) takes on different forms. We will show that these changes result in differwt cell types. First let us restrict the model to the biologically most important case: During locomotion in a static environment the eye movements of humans or higher animals are usually the product of intentional behavior. A very common situation is the fixation of a visible object during locomotion. A specific eye rotation is necessary to compensate for the translational body-movement and to keep the object fixed in the center (0, 0) of the visual field, so that its image velocity eq. (2) vanishes: 1 (-ITx) (-lOY) (0) 8(0,0) = ZF -ITy + +jOx = 0 . (8) ZF denotes the distance of the fixation point. We can easily calculate Ox and Oy from eq. (8) and chose Oz = O. The optic flow eq. (2) in the case of the fixation of a stationary object then is 1 1 8(.x,y) = Z(.x,y)A(.x,y)T + ZFB(.x,y)T, with ( 1+.x2 / f (.xy)/ f 0) B(.x, y) = (.xy)/ j 1+ y2 / f 0 . We would like to emphasize that another common situation, namely no eye movements at all, can be approximated by Z F -+ (X). We can now construct a new matrix o ~("" YllT ) B(.xm , Ym)T and form synaptic connections in the same way as described above. The resulting network is able to deal with the most common types of eye movements. The response properties of a 438 Lappe and Rauschecker .. y A C E ... •• • . " -... -oo· _10· .0· oo· ·50· -10· . . t . -.. -30 • •• • • • .. . • •• •• -.0 J'J' 1'1' B 0 F • Figure 2: Neuron Selective For Two Components ("Double Component Cell") single neuron from such a network are shown in Fig. 1. The neuron is selective for all three types of flow patterns. It exhibits broad directional tuning (Fig. IB) for upward shifting patterns (~ = 90 deg.). The responses to expanding (Fig. Ie), contracting (Fig. ID) and rotating (Fig. IE-F) patterns show large areas of position invariant selectivity. Inside the receptive field, which covers the second quadrant (see destribution of input locations in Fig. IA), the neuron favors upward shifts, contractions and counterclockwise rotations. It is thus compatible with a triple component cell in MSTd Also, lines are visible along which the selectivities reverse. This happens because the neuron's input is a linear function of the stimulus position (xc, Yc). For example. for rotational patterns we can calculate the input using eqs. (2), (6), and (7): ~~ J .. s· ±O ~ ~(x )Bt(x. .)(Cl~2i-1(T;)) ~~ 1.1 l1 ,k - v' 2 2 f2 ~ z c,Yc,f "y, Cl. .(T·) . ':=1 k=1 Xc + Yc + i=1 F 1,21 .1 As long as the threshold J.t is small, the neuron's output is halfway between its maximal and minimal values whenever its input is zero, i.e. when This is the equation of a line in the (xc, Yc) plane. The naJIon's selectivity for rotations reverses along this line. A similar equation holds expansion/contraction selectivity. Now, what would the neuron's selectivity look like, if we had not restricted the eye mov~ ments to the case of the fixation of an object. The responses of a neuron that is constructed following the unconstrained version of the algorithm, as described in section 2, is shown in Fig. 2. There is no selectivity for clockwise versus counterclockwise rotations at all, since both patterns elicit the same response everywhere in the visual field Inside the receptive field the neuron favors contractions and shifts towards the upper left (4) = 150 deg.). It can thus be regarded as a double component cell. To understand the absence of rotational selectivity we have to calculate the whole rotational optic flow pattern 0 rot by inserting T Computation of Heading Direction From Optic Flow in Visual Cortex 439 .. y A c E ... . " -... . .. • •• • -'0· ..... B • Figure 3: Predominantly Rotation Selective Neuron ("Single Component Cell") and n from eq. (7) into eq. (3). C(T) becomes ( 0 ... 0 C(O) = : ... : o ... 0 Denoting the three rightmost colwnn vectors of C (T) by B I• B 2, and B3 we:find ±n 8 rot = (ZeBI + Ye B 2 + /B3). Vz~ + y~ + /2 Comparison to C(T), eq. (4), shows that 8 rot can be written as a linear combination of colwnn vectorsofC(T). Thus 8 rot lies in therangeofC(T) and is orthogonal to Cl.(T), so that 8 rot ct (Tj) = 0 for all j and I. From eqs. (5) and (6) it follows, that the neuron's response to any rotationalpattem is always Ujl = g( -1-'). The last type of eye movements we want to consider is that of a general frontoparallel rotation, which is defined by nz = O. In addition to the fixation of a stationary object, frontoparallel rotations also include smooth pursuit eye movements necessary for the fixation of a moving object. Inserting nz = 0 into eq. (2) gives A 1 A (nx) 9(z,y) = Z(x,y)A(z,y)T+B(z,y) Oy with H(z y) ( zy/ / -(I + x2 / f) ) , / + y2 / / -zy/ / now being a 2 x 2 matrix, so that C(T), eq. (4), becomes a 2m x (m + 2) matrix C(T). A neuron that is constructed using C(T) can be seen in Fig. 3. It best responds to counterclockwise rotational patterns showing complete position invariance over the visual field The neuron is much less selective to expansions and unidirectional shifts, since 440 Lappe and Rauschecker the responses never reach saturation. It therefore resembles a single component rotation selective cell. The position invariant behavior can again be explained by looking at the rotational optic flow pattern. Using the same argument as above, one can show that the neuron's input is zero whenever Oz vanishes, i.e. when the rotational axis lies in the (X, Y)-plane. Then the flow pattern becomes ±O A A Brot = (XcBl + ycB 2), v'x~ + y~ + /2 and is an element of the range of C(T;). The (X, Y)-plane thus splits the space of all rotational axes into two half spaces, one in which the neuron's input is always positive and one in which it is always negative. Clockwise rotations are characterized by Oz > 0 and hence all lie in the same half space, while counterclockwise rotations lie in the other. As a result the neuron is exclusively excited by one mode of rotation in all of the visual field 4 Conclusion Our neural network model for the detection of ego-motion proposes a computational map of heading directions. A similar map could be contained in area MSTd of monkey visual cortex. Cells in MSTd exhibit a varying degree of selectivity for basic optic flow patterns, but often show a substantial indifference towards the spatial position of a stimulus. By using a population encoding of the heading directions, individual neurons in the model exhibit similar position invariant responses within large parts of the visual field Different neuronal selectivities found in MSTd can be modelled by assuming specializations pertaining to different types of eye movements. Consistent with experimental findings the position invariance of the model neurons is largest in the single component cells and less developed in the double and triple component cells. References Allman, J. M. and Kaas, 1. H. 1971. Brain Res. 31,85-105. Andersen, R, Graziano, M., and Snowden, R 1990. Soc. Neurosci. Abstr. 16, 7. Brenner, E. and Rauschecker, J. P. 1990. J. Physiol. 423,641-660. Duffy, C. J. and Wurtz, R H. 1991. J. Neurophysiol. 65(6), 1329-1359. Gibson, J. J. 1950. The Perception of the Visual World. HoughtonMi:fHin, Boston. Heeger, D. J. and Jepson, A. 1992. Int. J. Compo Vis. 7(2),95-117. Lappe, M. and Rauschecker, J. P. 1993. Neural Computation (in press). Maunsell,1. H. R and Van Essen, D. C. 1983. J. Neurophysiol. 49(5), 1127-1147. Orban, G. A., Lagae, L., Vern, A., Raiguel, S., Xiao, D., Maes, H., and Torre, V. 1992. Proc. Nat. Acad Sci. 89,2595-2599. Rauschecker, J. P., von Griinau, M. W., and Poulin, C. 1987. J. Neurosci. 7(4),943-958. Rieger, J. H. and Toe!, L. 1985. BioI. Cyb. 52,377-381. Stone, L. S. and Perrone, J. A. 1991. In Soc. Neurosci. Abstr. 17,857. Tanaka, K. and Saito, H.-A. 1989. J. Neurophysiol. 62(3),626--641. Wang, H. T., Mathur, B. P. and Koch, C. 1989. Neural Computation 1,92-103. Warren, W. H. Jr., and Hannon, D. J. 1988. Nature 336, 162-163.
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Statistical Modeling of Cell-Assemblies Activities in Associative Cortex of Behaving Monkeys Itay Gat and Naftali Tishby Institute of Computer Science and Center for Neural Computation Hebrew University, Jerusalem 91904, Israel * Abstract So far there has been no general method for relating extracellular electrophysiological measured activity of neurons in the associative cortex to underlying network or "cognitive" states. We propose to model such data using a multivariate Poisson Hidden Markov Model. We demonstrate the application of this approach for temporal segmentation of the firing patterns, and for characterization of the cortical responses to external stimuli. Using such a statistical model we can significantly discriminate two behavioral modes of the monkey, and characterize them by the different firing patterns, as well as by the level of coherency of their multi-unit firing activity. Our study utilized measurements carried out on behaving Rhesus monkeys by M. Abeles, E. Vaadia, and H. Bergman, of the Hadassa Medical School of the Hebrew University. 1 Introduction Hebb hypothesized in 1949 that the basic information processing unit in the cortex is a cell-assembly which may include thousands of cells in a highly interconnected network[l]. The cell-assembly hypothesis shifts the focus from the single cell to the * {itay,tishby }@cs.huji.ac.il 945 946 Gat and Tishby complete network activity. This view has led several laboratories to develop technology for simultaneous multi-cellular recording from a small region in the cortex[2, 3]. There remains, however, a large discrepancy between our ability to construct neuralnetwork models and their correspondence with such multi-cellular recordings. To some extent this is due to the difficulty in observing simultaneous activity of any significant number of individual cells in a living nerve tissue. Extracellular electrophysiological measurements have so far obtained simultaneous recordings from just a few randomly selected cells (about 10), a negligibly small number compared to the size of the hypothesized cell-assembly. It is quite remarkable therefore, that such local measurements in the associative cortex have yielded so much information, such as synfire chains [2], multi-cell firing correlation[6], and statistical correlation between cell activity and external behavior. However, such observations have so far relied mostly on the accumulated statistics of cell firing over a large number of repeated experiments, to obtain any statistically significant effect. This is due to the very low firing rates (about 10Hz) of individual cells in the associative cortex, as can be seen in figure 1. 30~--------------------------~~------------------------~ O~--------------------------~----------------------------r 111 1' II' "'1 I III I I I I I .1111 I I I t I I , I 1 '1 1, 1111 • I I I, I I I 1111." .. ,. I I • '1" I •• It' t ,', ,II , ••••• ,' "I " '" I I II "I ", I , I '"' , 1111 I . I III I '" 11111.' I I 1" II • ". , •• ,'" .,., '0' I I II I " II I • I II • I I I I I I , I It , " " I I " I I I " •• ,. I", I • 1" I I I I " • " I I I ". II II .,1 I 110'" I,,, " I,,, .. ,,, •• , " II •• II I 1,.,.' I" , •• , I I, I , I I I "'" ". "..11 " I II " I I • '" , •• I I I , I II "I' " I "." II I II ,. I I II ,.11 "."ltl III I .", ., • , 'I I • I I " ••• '" " I ., "'" • I I. I." •• " .. , "" ". , I ........ II , , , I ' I I I ,I 111 •• ,11' I ," I I I I I " I ., , " " •• , I '"' •• " I, I I I I II ,. I ,., I I tI, I ., " , I I 111"'" " • I , .. '1' I. II I I I " • """ " , " I I " I , I . , .,',., I, II , . , It, "" "" " I I I .', " I I. I '"1 " ••• III "." .. ',., II " •• '" "' " .", • I '" I " 1111.,1", "' ", III" .1 I • ., I •• I • I I' • ,. "' I, I I I. " • " I I .f! '" " • I '"~ It ., " ,. • I I '" , '" ••• ,' " " " I I , .. I • ""1 ' , .. , 'I' I' , , .,' I' I """ •• 11 I • • ,.... • ,.". ',.',',,"'1 " '" :" I', " • ',' " ,: ,.,. ",.' .,' I ,"\.'''' '." ,'" ,. " ., •• , , •• ",. .". II ,",:,: ::::,::: ;:;i:~:;,;> ,:,~:::: ",:,> :,~~i:',i: ::;'~', ,: ,.:, ::, ,::::: :\:~~:";;:i;.:'·~~>::/~;:',:,: ',~'::,.,:':;~::' ~;:C:'~' ::.: ~: "~;':::: ':, .. " : ,:~',"'," .. ,:: '~::," ",',::,:1 ,',,: ',"", :~' ".':'" :/:",:' ': ,:,:, ,: ':' ," ,', :':':;:~3,,-::-,:::::~::~: '::: :::; ,:':." ::': :::</'T: .. ~':,::"~::",:'::':: ',:'::' ~: ,'J I I • , 'I 01" '~. ~: "I " ' • -2000 o MiIliSec 2000 Figure 1: An example of firing times of a single unit. Shown are 48 repetitions of the same trial, aligned by the external stimulus marker, and drawn horizontally one on top of another. The accumulated histogram estimates the firing rate in 50msec bins, and exhibits a clear increase of activity right after the stimulus. Clearly, simultaneous measurements of the activity of 10 units contain more information than single unit firing and pairwise correlations. The goal of the present study is to develop and evaluate a statistical method which can better capture the multi- unit nature of this data, by treating it as a vector stochastic process. The firing train of each of these units is conventionally modeled as a Poisson process with a time-dependent average firing rate[2]. Estimating the firing rate parameter requires careful averaging over a sliding window. The size of this window should be long enough to include several spikes, and short enough to capture the variability. Modeling Cell-Assemblies Activities in Associative Cortex of Behaving Monkeys 947 Within such a window the process is characterized by a vector of average rates, and possibly higher order correlations between the units. The next step, in this framework, is to collect such vector-frames into statistically similar clusters, which should correspond to similar network activity, as reflected by the firing of these units. Furthermore, we can facilitate the well-established formulation of Hidden-Markov-Models[7] to estimate these "hidden" states of the network activity, similarly to the application of such models to other stochastic data, e.g. speech. The main advantage of this approach is its ability to characterize statistically the multi-unit process, in an unsupervised manner, thus allowing for finer discrimination of individual events. In this report we focus on the statistical discrimination of two behavioral modes, and demonstrate not only their distinct multi-unit firing patterns, but also the fact that the coherency level of the firing activity in these two modes is significantly different. 2 Origin of the data The data used for the present analysis was collected at the Hadassa Medical School, by recording from a Rhesus monkey Macaca Mulatta who was trained to perform a spatial delayed release task. In this task the monkey had to remember the location from which a stimulus was given and after a delay of 1-32 seconds, respond by touching that location. Correct responses were reinforced by a drop of juice. After completion of the training period, the monkey was anesthetized and prepared for recording of electrical activity in the frontal cortex. After the monkey recovered from the surgery the activity of the cortex was recorded, while the monkey was performing the previously learned routine. Thus the recording does not reflect the learning process, but rather the cortical activity of the well trained monkey while performing its task. During each of the recording sessions six microelectrodes were used simultaneously. With the aid of two pattern detectors and four windowinscriminates, the activity of up to 11 single units (neurons) was concomitantly recorded. The recorded data contains the firing times of these units, the behavioral events of the monkey, and the electro-occulogram (EOG)[5, 2,4]. 2.1 Behavioral modes To understand the results reported here it is important to focus on the behavioral aspect of these experiments. The monkey was trained to perform a spatial delayed response task during which he had to alternate between two behavioral modes. The monkey initiated the trial, by pressing a central key, and a fixation light was turned on in front of it. Then after 3-6 seconds a visual stimulus was given either from the left or from the right. The stimulus was presented for 100 millisec. After a delay the fixation light was dimmed and the monkey had to touch the key from which the visual stimulus came ("Go" mode), or keep his hand on the central key regardless of the external stimulus ("No-Go" mode). For the correct behavior the monkey was rewarded with a drop of juice. After 4 correct trials all the lights in front of the monkey blinked (this is called "switch" henceforth), signaling the monkey to change the behavioral mode - so that if started in the "Go" mode he now had to switch to "No-Go" mode, or vice versa. 948 Gat and Tishby There is a clear statistical indication, based on the accumulated firing histograms, that the firing patterns are different in these two modes. One of our main experimental results so far is a more quantitative analysis of this observation, both in terms of the firing patterns directly, and by using a new measure of the coherency level of the firing activity. 1 5 c5 5 c5 , .'.11. i.' .. , ' .. 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' .(11 ... 1 .. ~'.I .. I . , I I II ' .. - _ ... Svv'itch Figure 2: Multi-unit firing trains and their statistical segmentation by the model. Shown are 4 sec. of activity, in two trials, near the "switch". Estimated firing rates for each channel are also plotted on top of the firing spikes. The upper example is taken from the training data, while the lower is outside of the training set. Shown are also the association probabilities for each of the 8 states of the model. The monkey's cell-assembly clearly undergoes the state sequence "1", "5", "6", "5" in both cases. Similar sequence was observed near the same marker in many (but not all) other instances of the same event during that measurement day. 2.2 Method of analysis As was indicated before, most of the statistical analysis so far was done by accumulating the firing patterns from many trials, aligned by external markers. This supervised mode of analysis can be understood from figure 1, where 48 different Modeling Cell-Assemblies Activities in Associative Cortex of Behaving Monkeys 949 "Go" firing trains of a single unit are aligned by the marker. There is a clear increase in the accumulated firing rate following the marker, indicating a response of this unit to the stimulus. In contrast, we would like to obtain, in an unsupervised self organizing manner, a statistical characterization of the multi-unit firing activity around the marked stimuli, as well as in other unobserved cortical processes. We claim to achieve this goal through characteristic sequences of Markov states. 3 Multivariate Poisson Hidden Markov Model The following statistical assumptions underlie our model. Channel firing is distributed according to a Poisson distribution. The distances between spikes are distributed exponentially and their number in each frame, n, depends only on the mean firing rate A, through the distribution e-AAn PA(n) = I . n. (1) The estimation of the parameter A is performed in each channel, within a sliding window of 500ms length, every lOOms. These overlapping windows introduce correlations between the frames, but generate less noisy, smoother, firing curves. These curves are depicted on top of the spike trains for each unit in figure 2. The multivariate Poisson process is taken as a Maximum Entropy distribution with i.i.d. Poisson prior, subject to pairwise channel correlations as additional constraints, yielding the following parametric distribution d PA(nl,n2, ... ,nd) = II PA,(ni) exp[ - LAij(ni - Ad(nj - Aj) - AO]' (2) ij The Aij are additional Lagrange multipliers, determined by the observed pairwise correlation E[( ni - Ad( nj - Aj)), while AO ensures the normalization. In the analysis reported here the pairwise correlation term has not been implemented. The statistical distance between a frame and the cluster centers is determined by the probability that this frame is generated by the centroid distribution. This probability is asymptotically fixed by the empirical information divergence (KL distance) between the processes[8, 9]. For I-dimensional Poisson distributions the divergence is simply given by (3) The uncorrelated multi-unit divergence is simply the sum of divergences for all the units. Using this measure, we can train a multivariate Poisson Hidden Markov Model, where each state is characterized by such a vector Poisson process. This is a special case of a method called distributional clustering, recently developed in a more general setup[IO]. The clustering provides us with the desired statistical segmentation of the data into states. The probability of a frame, Xt, to belong to a given state, Sj, is determined by the probability that the vector firing pattern is generated by the state centroid's 950 Gat and Tishby distribution. Under our model assumptions this probability is a function solely of the empirical divergences, Eq.(3), and is given by (4) where f3 determines the "cluster-hardness". These state probability curves are plotted in figure 2 in correspondence with the spike trains. The most probable state at each instance determines the most likely segmentation of the data, and the frames are labeled by this most probable state number. These labels are also shown on top of the spike trains in figure 2. 4 Experimental results We used about 6000 seconds of recordings done during a single day. It is important to note that this was an exceptionally good day in terms of the measurement quality. During that period the monkey performed 60 repetitions of his trained routine, in sets of 4 trials of "Go" mode, followed by 4 trials in the "No-Go" mode. We selected the 8 most active recorded units for our modeling. The training of the models was done on the first 4000 seconds of recording, 2000 seconds for each mode, while the rest was used for testing. 4.1 The nature of the segmentation Any method can segment the data in some way, but the point is to obtain reliable predictions using this segmentation. As always, there is some arbitrariness in the choice of the number of states (or clusters), which ideally is determined by the data. Here we tested only 8 and 15 states, and in most cases 8 were sufficient for our purposes. Since we used "fuzzy", or "soft" clustering, each frame has some probability of belonging to any of the clusters. Although in most cases the most likely state is clearly defined, the complete picture is seen only from the complete association distribution. Notice, e.g., in the lower segment of figure 2, where a most likely state "7" "pops up" between states "6" and "5", but is clearly not significant, as seen from the corresponding probability curve. 4.2 Characterization of events by state-sequences The first test of the segmentation is whether it is correlated with the external markers in any way. Since the markers were not used in any way during the training of the model (clustering), such correlations is a valid test of consistency. Moreover, one would like this correspondence to the markers to hold also outside of the training data. An exhaustive statistical examination of this question has not been made, as yet, but we could easily find many instances of similar state sequences near the same external marker, both within and outside of the training data. In figure 2 we bring a typical example to this effect. The next step is to train smallieft-to-right Markov models to spot these events more reliably. .. Il .. .. ;:I U Modeling Cell-Assemblies Activities in Associative Cortex of Behaving Monkeys Go-Mode No Go-Mode 0 0 0 0 1::1 0 0 0 0 D t:I 0 0 0 0 0 0 0 0 0 CJ 0 0 0 0 0 0 t:I D 0 D 0 0 n t:I D 0 D 0 0 0 0 0 0 0 0 0 n ...... 0 0 0 0 0 0 ...... D 0 0 n 0 0 1::1 t:I 0 0 0 0 0 0 0 0 0 0 0 0 0 ,..., 0 0 D 0 0 0 0 0 D "" 0 n 0 0 0 0 0 0 D 0 0 0 0 ~ 0 0 0 0 Units Units Figure 3: Average firing rates for each unit in each state, for the "Go" and «NoGo" modes. Notice that while no single unit clearly discriminates the two modes, their overall statistical discrimination is big enough that on average 100 frames are enough to determine the correct mode, more than 95% of the time_ 4.3 Statistical Inference of "Go" and "No-Go" modes Next we examined the statistical difference between models trained on the "Go" vs. "No-Go" modes. Here we obtained a highly significant difference in the cluster centroid's distributions, as shown in figure 3. The average statistical divergence between different clusters within each mode were 9.18 and 9.52 (natural logarithm) ,in «Go" and "No-Go" respectively, while in between those modes the divergence was more than 35. 4.4 Behavioral mode and the network firing coherency In addition to the clearly different cluster centers in the two modes, there is another interesting and unexpected difference_ We would like to call this firing coherency level, and it characterize the spread of the data around the cluster centers. The average divergence between the frames and their most likely state is consistently much higher in the "No-Go" mode than in the "Go" mode (figure 4). This is in agreement with the assumption that correct performance of the «No-Go" paradigm requires little attention, and therefore the brain may engage in a variety of processes. Acknowledglnents Special thanks are due to Moshe Abeles for his continuous encouragement and support, and for his important comments on the manuscript. We would also like to thank Hagai Bergman, and Eilon Vaadia for sharing their data with us, and for numerous stimulating and encouraging discussions of our approach. This research was supported in part by a grant from the Unites States Israeli Binational Science Foundation (BSF). 951 952 Gat and Tishby ~ ~ is ~ 0g 0Q !! til 10.---~-------.-------,--------.-------.-------,----, 9.5 Go. NoGo .. 9 8.5 8 15 clusters 7.5 + + .. + + + 7 •••••••••••• + + + + + + • •••• • 8 clusters + + + + + + .. + .. + + + • •••••••• ••• ••• •• • 6.5~--~------~--____ ~L ________ ~ ______ ~ ______ ~ __ ~ 2 3 2 3 Trial Number Figure 4: Firing coherency in the two behavioral modes at different clustering trials. The "No-Go" average divergence to the cluster centers is systematically higher than in the "Go" mode. The effect is shown for both 8 and 15 states, and is even more profound with 8 states. References [1] D. O. Hebb, The Organization of Behavior, Wiley, New York (1949) [2] M. Abeles, Corticonics, (Cambridge University Press, 1991) [3] J. Kruger, Simultaneous Individual Recordings From Many Cerebral Neurons: Techniques and Results, Rev. Phys. Biochem. Pharmacol.: 98:pp. 177-233 (1983) [4] M. Abeles, E. Vaadia, H. Bergman, Firing patterns of single unit in the prefrontal cortex and neural-networks models., Network 1 (1990) [5] M. Abeles, H. Bergman, E. Margalit and E. Vaadia, Spatio Temporal Firing Patterns in the Frontal Cortex of Behaving Monkeys., Hebrew University preprint (1992) [6] E. Vaadia, E. Ahissar, H. Bergman, and Y. Lavner, Correlated activity of neurons: a neural code for higher brain functions in: J.Kruger (ed), Neural Cooperativity pp. 249-279, (Springer-Verlag 1991). [7] A. B. Poritz, Hidden Markov Models: A Guided tour,(ICASSP 1988 New York). [8] T.M. Cover and J.A. Thomas, Information Theory, (Wiley, 1991). [9] J. Ziv and N. Merhav, A Measure of Relative Entropy between Individual Sequences, Technion preprint (1992) [10J N. Tishby and F. Pereira, Distributional Clustering, Hebrew University preprint (1993).
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Harmonic Grammars for Formal Languages Paul Smolensky Department of Computer Science & Institute of Cognitive Science U ni versity of Colorado Boulder, Colorado 80309-0430 Abstract Basic connectionist principles imply that grammars should take the form of systems of parallel soft constraints defining an optimization problem the solutions to which are the well-formed structures in the language. Such Harmonic Grammars have been successfully applied to a number of problems in the theory of natural languages. Here it is shown that formal languages too can be specified by Harmonic Grammars, rather than by conventional serial re-write rule systems. 1 HARMONIC GRAMMARS In collaboration with Geraldine Legendre, Yoshiro Miyata, and Alan Prince, I have been studying how symbolic computation in human cognition can arise naturally as a higher-level virtual machine realized in appropriately designed lower-level connectionist networks. The basic computational principles of the approach are these: (1) a. \Vhell analyzed at the lower level, mental representations are distributed patterns of connectionist activity; when analyzed at a higher level, these same representations constitute symbolic structures. The particular symbolic structure s is characterized as a set of filler/role bindings {f d ri}, using a collection of structural roles {rd each of which may be occupied by a filler fi-a constituent symbolic struc847 848 Smolensky ture. The corresponding lower-level description is an activity vector s = Li fi0ri. These tensor product representations can be defined recursively: fillers which are themselves complex structures are represented by vectors which in turn are recursively defined as tensor product representations. (Smolensky, 1987; Smolensky, 1990). b. When analyzed at the lower level, mental processes are massively parallel numerical activation spreading; when analyzed at a higher level, these same processes constitute a form of symbol manipulation in which entire structures, possibly involving recursive embedding, are manipulated in parallel. (Dolan and Smolensky, 1989; Legendre et al., 1991a; Smolensky, 1990). c. When the lower-level description of the activation spreading processes satisfies certain mathematical properties, this process can be analyzed on a higher level as the construction of that symbolic structure including the given input structure which maximizes Harmony (equivalently, minimizes 'energy'. The Harmony can be computed either at the lower level as a particular mathematical function of the numbers comprising the activation pattern, or at the higher level as a function of the symbolic constituents comprising the structure. In the simplest cases, the core of the Harmony function can be written at the lower, connectionist level simply as the quadratic form H = aTWa, where a is the network's activation vector and W its connection weight matrix. At the higher level, H = LC1,C2 H C1 ; C2; each H C1 ; C2 is the Harmony ofhaving the two symbolic constituents Cl and C2 in the same structure (the Ci are constituents in particular structural roles, and may be the same). (Cohen and Grossberg, 1983; Golden, 1986; Golden j 1988; Hinton and Sejnowski, 1983; Hinton and Sejnowski, 1986; Hopfield, 1982; Hopfield, 1984; lIopfield, 1987; Legendre et al., 1990a; Smolensky, 1983; Smolensky, 1986). Once Harmony (connectionist well-formed ness) is identified with grammaticality (linguistic well-formedness), the following results (Ic) (Legendre et al., 1990a): (2) a. The explicit form of the Harmony function can be computed to be a sum of terms each of which measures the well-formedness arising from the coexistence, within a single structure, of a pair of constituents in their particular structural roles. b. A ( descriptive) grammar can thus be identified as a set of soft rules each of the form: If a linguistic structure S simultaneously contains constituent Cl in structural role rl and constituent C2 in structural role r2, then add to H(S), the Harmony value of S, the quantity H cl ,rl;c2,r2 (which may be positive or negative). A set of such soft rules (or "constraints," or "preferences") defines a Harmonic Grammar. c. The constituents in the soft rules include both those that are given in the input and the "hidden" constituents that are assigned to the input by the grammar. The problem for the parser (computational Harmonic Gt:ammars for Formal Languages 849 grammar) is to construct that structure S, containing both input and "hidden" constituents, with the highest overall Harmony H(S). Harmonic Grammar (IIG) is a formal development of conceptual ideas linking Harmony to linguistics which were first proposed in Lakoff's cognitive phonology (Lakoff, 1988; Lakoff, 1989) and Goldsmith's harmonic phonology (Goldsmith, 1990; Goldsmith, in press). For an application of HG to natural language syntax/semantics, see (Legendre et al., 1990a; Legendre et al., 1990b; Legendre et al., 1991b; Legendre et al., in press). Harmonic Grammar has more recently evolved into a non-numerical formalism called Optimality Theory which has been successfully applied to a range of problems in phonology (Prince and Smolensky, 1991; Prince and Smolensky, in preparation). For a comprehensive discussion of the overall research program see (Smolensky et al., 1992). 2 HGs FOR FORMAL LANGUAGES One means for assessing the expressive power of Harmonic Grammar is to apply it to the specification of formal languages. Can, e.g., any Context-Free Language (CFL) L be specified by an IIG? Can a set of soft rules of the form (2b) be given so that a string s E L iff the maximum-Harmony tree with s as terminals has, say, H ~ O? A crucial limitation of these soft rules is that each may only refer to a pair of constituents: in this sense, they are only second order. (It simplifies the exposition to describe as "pairs" those in which both constituents are the same; these actually correspond to first order soft rules, which also exist in HG.) For a CFL, a tree is well-formed iff all of its local trees are--where a local tree is just some node and all its children. Thus the HG rules need only refer to pairs of nodes which fall in a single local tree, i.e., parent-child pairs and/or sibling pairs. The II value of the entire tree is just the sum of all the numbers for each such pair of nodes given by the soft rules defining the I1G. It is clear that for a general context-free grammar (CFG), pairwise evaluation doesn't suffice. Consider, e.g., the following CFG fragment, Go A~B C, A~D E, F~B E, and the ill-formed local tree (A ; (B E)) (here, A is the parent, Band E the two children). Pairwise well-formedness checks fail to detect the ill-formed ness , since the first rule says B can be a left child of A, the second that E can be a right child of A, and the third that B can be a left sibling of E. The ill-formedness can be detected only by examining all three nodes simultaneously, and seeing that this triple is not licensed by any single rule. One possible approach would be to extend HG to rules higher than second order, involving more than two constituents; this corresponds to H functions of degree higher than 2. Such H functions go beyond standard connectionist networks with pairwise connectivity, requiring networks defined over hypergraphs rather than ordinary graphs. There is a natural alternative, however, that requires no change at all in I1G, but instead adopts a special kind of grammar for the CFL. The basic trick is a modification of an idea taken from Generalized Phrase Structure Grammar (Gazdar et al., 1985), a theory that adapts CFGs to the study of natural languages. It is useful to introduce a new normal form for CFGs, Harmonic Normal Form 850 Smolensky (HNF). In IINF, all rules of are three types: A[i]-B C, A-a, and A-A[i]; and there is the further requirement that there can be only one branching rule with a given left hand side-the unique branching condition. Here we use lowercase letters to denote terminal symbols, and have two sorts of non-terminals: general symbols like A and subcategorized symbols like A[I], A[2], ... , A[i]. To see that every CFL L does indeed have an HNF grammar, it suffices to first take a CFG for L in Chomsky Normal Form, and, for each (necessarily binary) branching rule A-B C, (i) replace the symbol A on the left hand side with A[i], using a different value of i for each branching rule with a given left hand side, and (ii) add the rule A-A[i]. Subcategorizing the general category A, which may have several legal branching expansions, into the specialized subcategories A[i], each of which has only one legal branching expansion, makes it possible to determine the well-formedness of an entire tree simply by examining each parent/child pair separately: an entire tree is wellformed iff every parent/child pair is. The unique branching condition enables us to evaluate the Harmony of a tree simply by adding up a collection of numbers (specified by the soft rules of an IIG), one for each node and one for each link of the tree. Now, any CFL L can be specified by a Harmonic Grammar. First, find an HNF grammar G H N F for L; from it, generate a set of soft rules defining a Harmonic Grammar GIl via the correspondences: a A A[i] start symbol S A-a (a = a or A[i)) A[i]-B C Ra: If a is at any node, add -1 to H RA: If A is at any node, add -2 to H RA[i]: If A[i] is at any node, add -3 to H Rroot : If S is at the root, add + 1 to H If a is a left child of A, add +2 to H If B is a left child of A[i], add +2 to H If C is a right child of A[i], add +2 to H The soft rules Ra , RA, RA[i] and Rroot are first-order and evaluate tree nodes; the remaining second-order soft rules are legal domination rules evaluating tree links. This IIG assigns H = 0 to any legal parse tree (with S at the root), and H < 0 for any other tree; thus s E L iff the maximal-Harmony completion of s to a tree has H ~ O. P1'OOf. 'Ve evaluate the Harmony of any tree by conceptually breaking up its nodes and links into pieces each of which contributes either + 1 or -1 to H. In legal trees, there will be complete cancellation of the positive and negative contributions; illegal trees will have uncancelled -Is leading to a total H < O. The decomposition of nodes and links proceeds as follows. Replace each (undirected) link in the tree with a pair of directed links, one pointing up to the parent, the other down to the child. If the link joins a lega.l parent/child pair, the corresponding legal domination rule will contribute +2 to H; break this +2 into two contributions of + 1, one for each of the directed links. We similarly break up the non-terminal nodes into sub-nodes. A non-terminal node labelled Harmonic Grammars for Formal Languages 851 A[i] has two children in legal trees, and we break such a node into three sub-nodes, one corresponding to each downward link to a child and one corresponding to the upward link to the parent of A[i]. According to soft rule RA[ij, the contribution of this node A[l1 to II is -3; this is distributed as three contributions of -1, one for each sub-node. Similarly, a non-terminal node labelled A has only one child in a legal tree, so we break it into two sub-nodes, one for the downward link to the only child, one for the upward link to the parent of A. The contribution of -2 dictated by soft rule RA is similarly decomposed into two contribution:) of -1, one for each sub-node. There is no need to break up terminal nodes, which in legal trees have only one outgoing link, upward to the parent; the contribution from Ra is already just -1. \Ve can evaluate the Harmony of any tree by examining each node, now decomposed into a set of sub-nodes, and determining the contribution to II made by the node and its outgoing directed links. We will not double-count link contributions this way; half the contribution of each original undirected link is counted at each of the nodes it connects. Consider first a non-terminal node n labelled by A[i]; if it has a legal parent, it will have an upward link to the parent that contributes +1, which cancels the -1 contributed by n's corresponding sub-node. If n has a legal left child, the downward link to it will contribute + 1, cancelling the -1 contributed by n's corresponding sub-node. Similarly for the right child. Thus the total contribution of this node will be 0 if it has a legal parent and two legal children. For each m,issing legal child or parent, the node contributes an uncancelled -1, so the contribution of this node n in the general case IS: (3) lIn = -(the number of missing legal children and parents of node n) The same result (3) holds of the non-branching non-terminals labelled A; the only difference is that now the only child that could be missing is a legal left child. If A happens to be a legal start symbol in root position, then the -1 of the sub-node corresponding to the upward link to a parent is cancelled not by a legal parent, as usual, but rather by the + 1 of the soft rule Rroot . The result (3) still holds even in this case, if we simply agree to count the root position itself as a legal parent for start symbols. And finally, (3) holds of a terminal node n labelled a; such a node can have no missing child, but might have a missing legal parent. Thus the total Harmony of a tree is II = Ln lIn, with lIn given by (3). That is, II is the minus the total number of missing legal children and parents for all nodes in the tree. Thus, II = 0 if each node has a legal parent and all its required legal children, otherwise H ~ O. Because the grammar is in Harmonic Normal Form, a parse tree is legal iff every every node has a legal parent and its required 852 Smolensky number of legal children, where "legal" parenti child dominations are defined only pairwise, in terms of the parent and one child, blind to any other children that might be present or absent. Thus we have established the desired result, that the maximum-Harmony parse of a string s has H > 0 iff s E L. We can also now see how to understand the soft rules of G H, and how to generalize beyond Context-Free Languages. The soft rules say that each node makes a negative contribution equal to its valence, while each link makes a positive contribution equal to its valence (2); where the "valence" of a node (or link) is just the number of links (or nodes) it is attached to in a legal tree. The negative contributions of the nodes are made any time the node is present; these are cancelled by positive contributions from the links only when the link constitutes a legal domination, sanctioned by the grammar. So in order to apply the same strategy to unrestricted grammars, we will simply set the magnitude of the (negative) contributions of nodes equal to their valence, as determined by the grammar. 0 We can illustrate the technique by showing how HNF solves the problem with the simple three-rule grammar fragment Go introduced early in this section. The corresponding HNF grammar fragment GHNF given by the above construction is A[l]~B C, A~A(1l, A[2]~D E, A~A[2l, F[l]~B E, F~F[l]. To avoid extraneous complications from adding a start node above and terminal nodes below, suppose that both A and F are valid start symbols and that B, C, D, E are terminal nodes. Then the corresponding HG GH assigns to the ill-formed tree (A ; (B E)) the Harmony -4, since, according to GHNF, Band E are both missing a legal parent and A is missing two legal children. Introducing a now-necessary subcategorized version of A helps, but not enough: (A ; (A[l] ; (B E))) and (A ; (A[2] ; (B E))) both have H = -2 since in each, one leaf node is missing a legal parent (E and B, respectively), and the A[i] node is missing the corresponding legal child. But the correct parse of the string B E, (F ; (F[l] ; (B E))), has H = O. This technique can be generalized from context-free to unrestricted (type 0) formal languages, which are equivalent to Turing Machines in the languages they generate (e.g., (Hopcroft and Ullman, 1979)). The ith production rule in an unrestricted grammar, Ri: ala2·· ·an• ~ i31i32·· ·i3m. is replaced by the two rules: R~ : ala2· .. ani -- r[i] and Ri' : r[i] ~ i31i32 ... i3mi' introducing new non-terminal symbols r[i]. The corresponding soft rules in the Harmonic Grammar are then: "If the kth parent of r[i] is ak, add +2 to H" and "If i3k is the kth child of r[il, add +2 to H"; there is also the rule Rr[;]: "If r[i] is at any node, add -ni - mi to H ." There are also soft rules Ra , RA , and Rroot , defined as in the context-free case. Acknowledgements I am grateful to Geraldine Legendre, Yoshiro Miyata, and Alan Prince for many helpful discussions. The research presented here has been supported in part by NSF grant BS-9209265 and by the University of Colorado at Boulder Council on Research and Creative Work. Harmonic Grammars for Formal Languages 853 References Cohen, M. A. and Grossberg, S. (1983). Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Transactions on Systems, Man, and Cybernetics, 13:815-825. Dolan, C. P. and Smolensky, P. (1989). Tensor Product Production System: A modular architecture and representation. Connection Science, 1:53-68. Gazdar, G., Klein, E., Pullum, G., and Sag, 1. (1985). Generalized Phrase Structure Grammar. Harvard University Press, Cambridge, MA. Golden, R. M. (1986). The "Brain-State-in-a-Box" neural model is a gradient descent algorithm. Mathematical Psychology, 30-31:73-80. Golden, R. M. (1988). A unified framework for connectionist systems. Biological Cybernetics, 59:109-120. Goldsmith, J. A. (1990). Autosegmental and lv/etrical Phonology. Basil Blackwell, Oxford. Goldsmith, J. A. (In press). Phonology as an intelligent system. In Napoli, D. J. and Kegl, J. A., editors, Bridges between Psychology and Linguistics: A Swarthmore Festschrift for Lila Gleitman. Cambridge University Press, Cambridge. Hinton, G. E. and Sejnowski, T. J. (1983). Analyzing cooperative computation. In Proceedings of the Fifth Annual Conference of the Cognitive Science Society, Rochester, NY. Erlbaum Associates. Hinton, G. E. and Sejnowski, T. J. (1986). Learning and relearning in Boltzmann machines. In Rumelhart, D. E., McClelland, J. L., and the PDP Research Group, editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Volume 1: Foundations, chapter 7, pages 282-317. MIT Press/Bradford Books, Cambridge, MA. Hopcroft, J. E. and Ullman, J. D. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading, MA. Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, USA, 79:2554-2558. Hopfield, J. J. (1984). Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences, USA, 81:3088-3092. IIopfield, J. J. (1987). Learning algorithms and probability distributions in feedforward and feed-back networks. Proceedings of the National Academy of Sciences, USA, 84:8429-8433. Lakoff, G. (1988). A suggestion for a linguistics with connectionist foundations. In Touretzky, D., Hinton, G. E., and Sejnowski, T. J., editors, Proceedings of the Connectionist Afodels Summer School, pages 301-314, San ~Iateo, CA. Morgan Kaufmann. Lakoff, G. (1989). Cognitive phonology. Paper presented at the UC-Bel'keley Workshop on Rules and Constraints. 854 Smolensky Legendre, G., Miyata, Y., and Smolensky, P. (1990a). Harmonic Grammar-A formal multi-level connectionist theory of linguistic well-formedness: Theoretical foundations. In Proceedings of the Twelfth Annual Conference of the Cognitive Science Society, pages 388-395, Cambridge, MA. Lawrence Erlbaum. Legendre, G., Miyata, Y., and Smolensky, P. (1990b). Harmonic Grammar-A formal multi-level connectionist theory of linguistic well-formedness: An application. In Proceedings of the Twelfth Annual Conference of the Cognitive Science Society, pages 884-891, Cambridge, MA. Lawrence Erlbaum. Legendre, G., Miyata, Y., and Smolensky, P. (1991a). Distributed recursive structure processing. In Touretzky, D. S. and Lippman, R., editors, Advances in Neural Information Processing Systems 3, pages 591-597, San Mateo, CA. Morgan Kaufmann. Slightly expanded version in Brian Mayoh, editor, Scandinavian Conference on Artificial Intelligence-g1, pages 47-53. lOS Press, Amsterdam. Legendre, G., Miyata, Y., and Smolensky, P. (1991b). Unifying syntactic and semantic approaches to unaccusativity: A connectionist approach. In Sutton, L. aud Johnson (with Ruth Shields), C., editors, Proceedings of the Seventeenth Annual Afeeting of the Berkeley Linguistics Society, pages 156-167, Berkeley, CA. Legendre, G., ~'1iyata, Y., and Smolensky, P. (In press). Can connectionism contribute to syntax? Harmonic Grammar, with an application. In Deaton, K., Noske, M., and Ziolkowski, M., editors, Proceedings of the 26th Afeeting of the Chicago Linguistic Society, Chicago, IL. Prince, A. and Smolensky, P. (1991). Notes on connectionism and Harmony Theory in linguistics. Technical report, Department of Computer Science, University of Colorado at Boulder. Technical Report CU-CS-533-91. Prince, A. and Smolensky, P. (In preparation). Optimality Theory: Constraint interaction in generative grammar. Smolensky, P. (1983). Schema selection and stochastic inference in modular environments. In Proceedings of the National Conference on Artificial Intelligence, pages 378-382, Washington, DC. Smolensky, P. (1986). Information processing in dynamical systems: Foundations of Harmony Theory. In Rumelhart, D. E., McClelland, J. L., and the PDP Research Group, editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations, chapter 6, pages 194-28l. MIT Press/Bradford Books, Cambridge, MA. Smolensky, P. (1987). On variable binding and the representation of symbolic structures in connectionist systems. Technical report, Department of Computer Science, University of Colorado at Boulder. Technical Report CU-CS-355-87. Smolensky, P. (1990). Tensor product variable binding and the representation of symbolic structures in connectionist networks. Artificial Intelligence, 46: 159216. Smolensky, P., Legendre, G., and Miyata, Y. (1992). Principles for an integrated connectionist/symbolic theory of higher cognition. Technical report, Department of Computer Science, University of Colorado at Boulder. Technical Report CU-CS-600-92.
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Silicon Auditory Processors as Computer Peripherals .T ollll Lmr,7.Hl'o . .T ollll Wawl'7.Ylwk CS Division UC B(~rk('ley Evans lIall Bcrl,plpy. Ct\ !H720 lazzaro~cs.berkeley.edu, johnw~cs.berkeley.edu M. Mahowald'" ~ Massimo Sivilottit, Dave Gillcspict Califol'lIia lnst,itult' of Technology Pasadena. CA !) 11:l!) Abstract Sever<tl resE'<lI'ch gl'Oups cue impl('lllt'lIt.ing allalog integrat.ed circuit. models of hiological audit.ory Pl"Occ'ssing. The outputs of these circuit models haV(~ takell sevel'al forms. includillg video [ormat. for monitor display, simple scanned Ollt.put [01' oscilloscope display anJ parallel analog out.put.s suitable ror dat.a-acquisition systems. In this pa.per, we describe an allel"llative out.put method for silicon auditory models, suit.able for din-'ct. interface to digital computers. * Present address: f\1. Mahowald, f\1H.C ,\natol1lical Ncmophamacology Unit, Mansfield TId, Oxfc)('d OXI :1'rl£ Ellgland . mam~vax.oxford.ac.uk t Present address: f\lass Siviloui. '1'(1111)(,1' H,csearrh, 180 Nort.h Vinedo Avenue, Pasadena, CA 9Il07. mass~tanner. corn :I: Present address: Dave Gill,>spiE', SYllapf,ics, :l()!)8 Orchard Parkway, San Jose CA, 95131. daveg~synaptics. com 820 Silicon Auditory Processors as Computer Peripherals 821 1. INTRODUCTION Several researchers have implemented comput.at.ional models of biological audit.ory processing, with the goal of incorporat.ing t.hese models into a speech recognition system (for a recent review, see (Jankowski, 1992)). These projects have shown the promise of the biological approach, someti Illes showing clear performance advantages over traditiona.l methods. The application of t.IH'se comput.('tional models is limited by t.heir large computation and communication I·eqllil·ement.::.;. A lIalog VLSI implementations of these neural models may relieve t.his cOlllput.at.ional burden; several VLSI research groups have effort.s in this area, and working int.egrated circuit models of nlany popular representat.ions present.ly exist .. A review of t.h('se models is present.ed in (Lazzaro, 1991). In this paper, we present. an interface met.hod (Ma.howald, 1992; Sivilott.i, 1991) that addresses the cOll1l1lllnicHt.ions issues between analog VLSI auditory implementations and digital processors. 2. COMMUNICATIONS IN NEURAL SYSTEMS Biological neurons COlli IllU lIicat.e 10llg dist.anc('s using a pulse represent.ation. Communications engineers have developed several schemes for communicat.ing on a wire using pulses as aLomic unit.s. In t.hese schemes, maximally using the communications bandwidth of a wire implies t.lw nlPan rat.e of pulses on t.he wire is a significant fract.ion of the maxil1lum pulse I'at.e allowed 011 the wire. Using this criterion, nemal syst.el\1s lise wir('s very illefficiently. Tn most. parts of the brain, most. of the wires arc esselltia.lly inactive 1Il0st of the time. If neural syst.ems are not organized t.o fully ut.ilii':E' t.lw available handwidt.h of each wire. what does neural communica.tioll opt.illli;!'c'? 8\'idclI(,(' sllggest.s that. f'llcrgy consel·vat.ion is an important isslle for neural syst.f'lI1s. A silnpl(' st rat.egy fOl' enel'gy conservat.ion is t.he reduction of t.he t.ot.al Ill11I1I)('1' of \>lIls('s ill t.he representat.ioll. ivlany possible coding st.rategies sat.isfy t.his elwrgy rC'qllil'('lIwnt.. The strat.egies observed ill lI(,lIl'al SYSt.('IIIS share anot her ('0111111011 propert.y. Neural systems oftE'1I implenwnt. H class of COIIlPlltilt.ions ill a. Il\allller t.ha.t. produces an energy-efficient. out.put. encoding as all addit.ional bypl'Oduct .. The energy-efficient coding is not perfOl'tnE'd simply for comll1unica.t.ioll and immediat.ely reversed upon receipt, but is an int.egral part. of t.h(' n('w r('IH'csent.at.ioll. In this way, energyefficient neural coding is int.rinsically diffnc'lIt. frolll engineering da.ta compression techniques. Temporal adaptat.ion. lat.('J'al inhihition, alld spike' colTelat.ions arc examples of neural processing methods tlIH1. perform illtercst.illg cOlllput.at.ions while producing an energy-efficient. OUt.PIlt. code. Thcst> repr('sent.a(.ional principles are t.he founda.tion of t.he neural computation and cOllll11unicat.ioll method we advocat.e in this paper, In this method, t.he out.put. units of a chip are spiking llelll'On circuit.s that use energy-efficient coding IlIet.hods. To COll1lllllllicat.f' t.his code off a c\rip, we liSE' a dist.inctly non-biological apPl'o(1eh. 822 Lazzaro, Wawrzynek, Mahowald, Sivilotti, and Gillespie 3. THE EVENT-ADDRESS PROTOCOL The unique characteristics of enel·gy-efficient. codes define the remaining off-chip communications problem. In the spiking nemon protocol, the height. and width of the spike carries no informat.ion; the neuron imparts new information only at the moment a spike begins. This moment occurs asynchronously; there is no global clock synchronizing the output units. One way of completely specifying the information in the output units is an event list., a tabulation of the precise time each output unit. begins a new spike. vVe can use this specification as a basis for an off-chip communicat.ions system, t.hat. sends an event.-list message off-chip at the moment. an output neuron begins a new spike. An eVl'nt.-list message includes the identificat.ion of the output unit., and the t.ime of firing . A pcrformance analysis of' this protocol can be found in (Lazzal'O et al., l!J92). Note that an explicit timest.amp for each clIt.l·y in t.he cvent list is not necessary, if communication lat.ency betwcen t.he scnding chip and the receiver is a constant. In this case, the sender simply com1l1I1nicRt,es, lIpon onset of a. spike from an output, the identit.y of the output. unit.; t.he l'eceivcl' can aPPclld a locally genera.ted timestamp to complete the event. If simplificd in t.his mallllcr, we refcl' to the event-list protocol as the event.-a.ddress protocol. ''''e have designed a \vol'king syst.em t.hat. comput.es a model of auditory nerve response, in rea] t.ime, usillg ana.log V LSI processing. This syst.em takes as input an analog sOllnd sOllrce. alld uses t.he eVPlIt.-list, I·('prl'sent.a.t.ion t.o communicat.e t.he model out.put to the host. computel·. Board Architecture CJ 0 1--------D Chip Architecture 1 I \\ttt~ \\\\\ I I I r-1 Spiking ArbIters 1 l3us DO Analog I 1 and I ~ Ou tPllt -i Paralld Processing Olll A,Sf1us cardU I En('odlll~ : I I Array ~,---' IPC 1 Tim.>r l " ."~ 1_- _______ J aa aaaaaaaaaa aaa aa aaaaaaaaaa aao DOD aa aaaaaaaaao aa aa aaaaaaaaa aao aa a aa Sound Input Figure 1. System block diagrnI1l. showing chip architecture, hoard a.rchit.ecture, a.nd the host. comput,(')' (SIlIl IPC). Silicon Auditory Processors as Computer Peripherals 823 4. SYSTEMS IMPLEMENTATION Figure 1 is a block diagl'am of this system. A single VLSI chip computes the auditory model response; an array of spiking neuron circuit.s is the final representation of the model. This chip also implements t.he event-address protocol, using asynchronous arbitration circuits. The chip produces a pa.rallel binary encoding of the model output, as an asynchronous stream of event addt·esses. These on-chip operations are shown inside the dashed recta.ngle in Figure 1, labelled Chip Architecture. Additional digital processing com~letes t.he custom hardwa.re in the system. This hardware transforms the event.-address prot.ocol int.o an event-list. protocol, by adding a time marker for each event (16 bit time markers with 20ps resolution). In addition, the hardware implemellt.s the bus interface to the host computer, in conjunction with a commercial int.erface board. The commercial interface board supports 10 MBytes/second asynchronous da.ta tl'ansfers between our custom hardware and the host computer, and includes 8 KByt.es of data buffers. Our display software produces a real-time graphical display of the audit.ory model response, using the X window syst.em . 5. VLSI CIRCUIT DETAILS Figure 2 shows a block diagram of t.he chip. The analog input signal connects to circuits t.hat. perform analog pl'Ocessing, t hat are fully described and referenced in (Lazzal'O et al., 1993). The outpllt. of this analog processing is represented by 150 spiking neurons, arranged in a ~O by 5 array. These are the output units of the chip; the event.-address prot.ocol commllnicates t.he a.ctivity of these units off chip. At the onset of a spike from an output unit, t.he array position of the spiking unit, encoded as a binary number, appears on the output bus. The asynchronous out.put bus is shown in FigurE' 2 as t.he dat.a signals marked Encoded X Output (column position) and Encoded Y Output (row position), and the acknowledge and request control signals Ae and Re. "Ve implemented the event.-address pl'Ot.ocol as an asynchl'Onous arbitration prot.ocol in two dimensions. ]ll t.his scheme, an out.put. unit. can access two request. lines, one a.<;socia.ted wit.h its row and one ;.)ssociat.ed with its column. Using a wire-OR signalling prot.ocol. any out.put. unit 011 a part.icular row or colllmn may assert the request line. Each request. lillE' is paired wit.h an acknowledge line, driven by the arbitration circuitt·y out.sidr. t.h0 array. Rowand column wires for acknowledge and request are explicit.ly showl! in Figll\'e 2. as t he lines that form a grid inside the output. unit. array. At the onset of a. spike, an out.put. unit. a~sert . s it.s row request. line, and wait.s for a reply on it.s \'Ow acknowlf>dge line. An aSYllchronous arbit.rat.ion syst.em, mar'ked in Figure 2 as Y Arhitration Tree. aSSl\res only one out.put row is acknowledged. Aft.er row acknowledgement., the output unit. assel't.s its column request line, and waits for a reply on it.s COlU11111 acknowledge line. The al·bit.ration system is shown in detail in Figure 2: fOllr two-input. arhit.f'r circuit.s, shown a.<; rectangles marked with the letter A, arc connect.ed as a t.ree t.o arhit.rat.e among t.he 5 column inputs. 824 Lazzaro, Wawrzynek, Mahowald, Sivilotti, and Gillespie Encoded Y Axis n A r-0 0 0 0 0 ~ I-rr~ f· · . · · · · · . · Q,) cd Q,) ..... v u H .-< ..... ..... bO = .-< a u 0 R t-=l fa ..... U 0 0 0 0 0 -r~ rr.-< cd 0 ..... A ..... r..... ~ ~ -0 ..... ..... u .... f"'> ;.::: a ~ ..... U ~ U') ~ · · . · · · · · . · f0 0 0 0 0 lfttr...... rSo und Input V I Control Loo'ic l") J I A I I AJ ~ AI II Ae Rc l ed X Output l A U Encod Figure 2. Block dia p;ralll or t Itt' chip , S('(' t('xt 1'0J'ddail:-;, Silicon Auditory Processors as Computer Peripherals 825 To Array n --..---H------t--- Ac (b) ----+--------+--+---- At A B l?y ,..,' (C) Ay /.' Figure 3. Diagrams of COllllllllllicatinn circllil:-; ill 1.1.(' citip. (a) Two-inpllt. arhit.er circuit.. (b) ('olltl'ol lo~ic to illt(·rl'a('(· arhil 1';,1 iOIl logic alld Olltpill. IInit array. (c) Ollt.put. Illlit. ('ircllit.. 826 Lazzaro, Wawrzynek, Mahowald, Sivilotti, and Gillespie Upon the an'ival of hot.h row and column acknowledgement.s, t.he output unit releases both row and column request. lines. St.a.t.ic lat.ches, shown in Figure 2 as the rectangles marked Control Logic:, ret.ain t.he stat.e of the row and column request lines. Binary encoders transfonl1 t.he row and column acknowledge lines int.o the output data bus. Another column encoder Sf'nses t.he acknowledgement of any column, and asserts the bus cont.rol OUt.PIlt. Re. \,"hell t.he ('xtcl'ncd device has secmed t.he data, it responds by a.'3sert.ing t.he Ae signal. The At: signal clears the st.at.ic latches in the Control Logic blocks and reset.s Re. When Ae is reset, the data transfer is complet.e, and the chip is reacly for t.he next communication event. Figure 3 shows the deta.ils of the communications circuit.s of Figure 2. Figure 3(a) shows the t.wo-input cHhit.er circuit used to create t.he binary arbitration trees in Figure 3. This digital circuit. t"kcs a~ inpllt t.wo request signals, RI and R2 , and produces the a<;sociated acknowledge sigll:1ls Al and A 2 • The acknowledgement of a request precludes the acknowledgement. of a second request. The cil·cuit. assert.s an acknowledge signal unt.il it.s clssocialed r'('qll!'st. is released. Ro is an auxiliary out.put. sigllal indicat.ing eit.her HI 01' R",1 has been asserted; Ao is an auxiliary input. sigllal t.ltal enahl('s t.ht' .. \ I rllld +~ out.put.s. Tlte auxiliary signals allow the two-input. 'Hhit<'r· t.o fllnctioll as an elcrllcnt. ill arhit.rat.ion trees, as shown in Figure 2; the Ro and AI) sip;nals of one I('vel of arbit.ra.t.ion cOllnect to the Rk and Ak signals at. t.he next level of cuhit.rat.ion. rn !.\Vo-input. operat.ioll, t.he Ro and Ao signals are connect.NI t.ogct.lwl', as sbon-II ill I he' root. arhit.er in Figure 2. Figure 3(b) shows t.lre circuit illlpl('lllellt.at.ioll of t.he Control Logic blocks ill Figure 2; this circuit is rcpC'at.cd for ea.ch row and col1lmn connect.ion. This circuit. interfaces t.he output. hilS cont.rol input. Ac wit II t.he c\l·hitrat.ion cil·cuitl'Y. If output. communicat.ion is not in progr('ss, Ac is at, ground, and Ac is at. \idd. The PFET transist.or marked a~ Load act.s as a st.at.ic pullllp to t.he array request line (R); out.put unit.s pull t.his lillt" low t.o as:'o:<Tt ,1 request.. The NOR gat.e invert.s the array request line, alld rout.es it. t.o (.\W arhitration t.ree. When a penuillg request is ackno",.ledged by tIl<' I.r<'<' ackllo\\'I(' dge linC', till' two NFET t.ransist.ors act. t.o latch the army request. lilw. The ass<'ltion of .... le releases t.he array request line and disa.bles t.he arhit.rat.ioll LI·ce request. inpllt.; t.hese actions reset all st.ate in the communicat.ions syste'm . '''llCn AI; is releas(~d, the syst.em is ready t.o communicate a. new event. Figure 3(c) shows t.he circuit. implernclltat.ion of a Hnit. in the output aITay. In t.his implcmentat.ion, each out.put ullit. is a t.wo-st.age lo\\,-powel· axon circuit. (Lazzaro, 1992). The first. axonal st.ag(' r('ceiV<'s t.he' cochlear input.; t.his axon st.age is not shown in Figure 3( c). The fil·sf. st.agf' couple'S illt.o the second st.age, shown in Figure 3(c), via t.he Sand F wires. To underst.and t.ht' opt'rat.ioll of' this circll it.. WI' C01ISid('r t I If' t.rallsrnis~ioll of a single spil-\t'. Init.ially, We' a~Sllllle the n'<jll('st. lilh's I?,. and ay are held high hy t.lte st.at.ic pl\lIup PFET t.ransist.ors shown ill Figlll·(' :J(h); ill addit.ion, we aSSIJIlle the a.cknowledge lines A.r and A" arp at. gWlllld, alld tilt' 1I0nillvert.ing hurrer input. voltage is at Silicon Auditory Processors as Computer Peripherals 827 ground. When the first. axonal st.age fires, the S signal changes from ground potential to Vdd. At this point the buffer input voJt.agf! begills t.o increase, at a rate determined by the analog cont.I·ol voltage P. Whcn t.he swit.ching threshold of t.he buffer is reached, the buffet output. volt.ag(' F swings \,0 \/dd ; capacit.ive feedback ensures a reliable switching transitioll. At this point, the output, unit. Plllls the request line Ry low, and the cOnllllllnirat.ions sequellre hegins. The Y arbitl'ation logic I'('plies 1.0 t 11<' Ry rf'qu('st. by asserting t.he Ay line. 'Vhen both F and Ay are asscJ'tcd, t.hc output ullit. pulls the reqllest. line R.r: low. The X arbitration logic replies to t.he R.r: I'equcst hy asserting t.he Ax line. The assertion of both Ax aBd Ay reset,s t.IH~ burrel' input voltagc t.o gl"Ound. As a result, the F line swings to ground potent.i<ll, t.hc out.put. unit. releases thc Rx a.nd Ry lines, and the first axon stage is enahled. At. t.his point., the l<lt.ch cil'cuit. of Figme 3(b) maintains the state of the Rx and 17,/ lines, IInt,il it is cleared by the off-chip acknowledge sign a. I. Acknowledgements Research and prot.ot.ypillg of t.he ('vc-nt-addr('ss illt('rface took place in Carver J\Jead's laborat.ory at. Calt,('('il: we arc' grateful for his illsights, CrtrOllragell1ellt., and support.. The Caltech-hased 1'('s<"Hrdl was funded by t.1t(' ONH, UP, and t.he Syst.ems Development. Foundat.ion. HCS('aJ'ch and prot,otyping of I.he allditory-nerve demollstrat.ion chip and syst.em took place a1 (1(' n(,l'kd(~y, alld was flll1(kd by t.he NSF (PYI award l\lIPS-895-8568), AT,,:T, al\d t.he ONH (UHI-l'\OOOl/1-02-.J-1672). Rcfel'cuees Jankowski. C. R. (1 m.l:2). "1\ ('olllparisoll of ,\ IIdit.ory l'\'lodcls for A lItomat.ir Speech Recognit.ion," S.13. TI1('~i:-;, ;\[1'1' Ikpt. of Electrical Ellgillc<.'\'ing and Comput.cl' Science. Lazz;uo, J. P. (IH!H). "Biologically-h(ls('d auditory signal pl'ocessing in analog V LSI," IEEE A.'iilo/lla1' ('ollje 1'( lin 011 Sl[jll(ll.~ . . '3ys/c1Il.'i. (/.lId Comp1liers. La7,zaro, J. P. (199:2). "l,o\\,-po\\,('J' ~ilicoll spiking I}(,\lron~ and :nons," IEEE 11Iterualiol/al S'ympo,<;iu1ll 01/ CiJ'CIt".~ (l1Id ,'1'.1/,1;/(1/1.,. Sail Di('go, CA, p. 2:22U-:22:2-1. Lazzaro, J., \\'awl'zYlWk, .1., ~Iaho\\'ald, ~I., Sivilot.f.i, ~L, and Gillespie, D. (1993). "Silicon audit.ory pro('(~ssorS as (,OlllplItcr peripIH'I'als," IEEE Trallsactiolls of Ne'IIT'a/ Nelworks. May (ill pr('ss). Mahowald, 1\I. (1992). Ph.n. Thesis, COlllplltat.ion and NCll\'al Syst.ems, California. Instittlt.r of Technology. Sivilot.t.i, 1\1. (1991), "Wirillg ('ollsi(krat.iolls ill analog VLSr syst.ems, wit.h applicat.iolls to fh'ld-progrnllll1lahlc' 11('\ works," (:0111 pllt.('r Sciel1r(' Technical Heport. (Ph. D. Thesis), Califol'llia 11lSit.II1.<' of'T('cllllology,
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Information, prediction, and query by committee Yoav Freund Computer and Information Sciences University of California, Santa Cruz yoavQcse.ucsc.edu Eli Shamir Institute of Computer Science Hebrew University, Jerusalem sharnirQcs.huji.ac.il H. Sebastian Seung AT &T Bell Laboratories Murray Hill, New Jersey seungQphysics.att.com N aft ali Tishby Institute of Computer Science and Center for Neural Computation Hebrew University, Jerusalem tishbyQcs.huji.ac.il Abstract We analyze the "query by committee" algorithm, a method for filtering informative queries from a random stream of inputs. We show that if the two-member committee algorithm achieves information gain with positive lower bound, then the prediction error decreases exponentially with the number of queries. We show that, in particular, this exponential decrease holds for query learning of thresholded smooth functions. 1 Introduction For the most part, research on supervised learning has utilized a random input paradigm, in which the learner is both trained and tested on examples drawn at random from the same distribution. In contrast, in the query paradigm, the learner is given the power to ask questions, rather than just passively accept examples. What does the learner gain from this additional power? Can it attain the same prediction performance with fewer examples? Most work on query learning has been in the constructive paradigm, in which the 483 484 Freund, Seung, Shamir, and Tishby learner constructs inputs on which to query the teacher. For some classes of boolean functions and finite automata that are not PAC learnable from random inputs, there are algorithms that can successfully PAC learn using "membership queries" [VaI84, Ang88]. Query algorithms are also known for neural network learning[Bau91]. The general relevance of these positive results is unclear, since each is specific to the learning of a particular concept class. Moreover, as shown by Eisenberg and Rivest in [ER90], constructed membership queries cannot be used to reduce the number of examples required for PAC learning. That is because random examples provide the learner with information not only about the correct mapping, but also about the distribution of future test inputs. This information is lacking if the learner must construct inputs. In the statistical literature, some attempt has been made towards a more fundamental understanding of query learning, there called "sequential design of experiments." 1. It has been suggested that the optimal experiment (query) is the one with maximal Shannon information[Lin56, Fed72, Mac92]. Similar suggestions have been made in the perceptron learning literature[KR90]. Although the use of an entropic measure seems sensible, its relationship with prediction error has remained unclear. Understanding this relationship is a main goal of the present work, and enables us to prove a positive result about the power of queries. Our work is derived within the query filtering paradigm, rather than the constructive paradigm. In this paradigm, proposed by [CAL90], the learner is given access to a stream of inputs drawn at random from a distribution. The learner sees every input, but chooses whether or not to query the teacher for the label. This paradigm is realistic in contexts where it is cheap to get unlabeled examples, but expensive to label them. It avoids the problems with the constructive paradigm described in [ER90] because it gives the learner free access to the input distribution. In [CAL90] there are several suggestions for query filters together with some empirical tests of their performance on simple problems. Seung et al.[SOS92] have suggested a filter called "query by committee," and analytically calculated its performance for some perceptron-type learning problems. For these problems, they found that the prediction error decreases exponentially fast in the number of queries. In this work we present a more complete and general analysis of query by committee, and show that such an exponential decrease is guaranteed for a general class of learning problems. We work in a Bayesian model of concept learning[HKS91] in which the target concept I is chosen from a concept class C according to some prior distribution P. The concept class consists of boolean-valued functions defined on some input space X. An example is an input x E X along with its label I = I( x). For any set of examples, we define the version space to be the set of all hypotheses in C that are consistent with the examples. As each example arrives, it eliminates inconsistent hypotheses, and the probability of the version space (with respect to P) is reduced. The instantaneous information gain (i.i.g.) is defined as the logarithm of the ratio IThe paradigm of (non-sequential) experimental design is analogous to what might be called "batch query learning," in which all of the inputs are chosen by the learner before a single label is received from the teacher Information, prediction, and query by committee 485 of version space probabilities before and after receiving the example. In this work, we study a particular kind oflearner, the Gibbs learner, which chooses a hypothesis at random from the version space. In Bayesian terms, it chooses from the posterior distribution on the concept class, which is the restriction of the prior distribution to the version space. If an unlabeled input x is provided, the expected i.i.g. of its label can be defined by taking the expectation with respect to the probabilities of the unknown label. The input x divides the version space into two parts, those hypotheses that label it as a positive example, and those that label it negative. Let the probability ratios of these two parts to the whole be X anti 1 - X. Then the expected i.i.g. is 1i(X) = -X logX - (1 - X) log(1- X) . (1) The goal of the learner is to minimize its prediction error, its probability of error on an input drawn from the input distribution V. In the ease of random input learning, every input x is drawn independently from V. Since the expected i.i.g. tends to zero (see [HKS91]), it seems that random input learning is inefficient. We will analyze query construction and filtering algorithms that are designed to achieve high information gain. The rest of the paper is organized as follows. In section 2 we exhibit query construction algorithms for the high-low game. The bisection algorithm for high-low illustrates that constructing queries with high information gain can improve prediction performance. But the failure of bisection for multi-dimensional high-low exposes a deficiency of the query construction paradigm. In section 3 we define the query filtering paradigm, and discuss the relation between information gain and prediction error for queries filtered by a committee of Gibbs learners. In section 4 lower bounds for information gain are proved for the learning of some nontrivial concept classes. Section 5 is a summary and discussion of open problems. 2 Query construction and the high-low game In this section, we give examples of query construction algorithms for the high-low game and its generalizations. In the high-low game, the concept class C consists of functions of the form { I w < x fw(x) = 0: w > x (2) where 0 ~ w, x ~ 1. Thus both X and C are naturally mapped to the interval [0,1]. Both P, the prior distribution for the parameter w, and V, the input distribution for x, are assumed to be uniform on [0,1]. Given any sequence of examples, the version space is [XL, XR] where XL is the largest negative example and XR is the smallest positive example. The posterior distribution is uniform in the interval [XL, XR] and vanishes outside. The prediction error of a Gibbs learner is Pr(fv(x) I- fw(x)) where x is chosen from V, and v and w from the posterior distribution. It is easy to show that Pr(fv (x) I- fw (x)) = (XR - xL)/3. Since the prediction error is proportional to the version space volume, always querying on the midpoint (XR + xL)/2 causes the prediction error after m queries to decrease like 2- m . This is in contrast to the case of random input learning, for which the prediction error decreases like l/m. 486 Freund, Seung, Shamir, and Tishby The strategy of bisection is clearly maximally informative, since it achieves 1i(lj2) = 1 bit per query, and can be applied to the learning of any concept class. Naive intuition suggests that it should lead to rapidly decreasing prediction error, but this is not necessarily so. Generalizing the high-low game to d dimensions provides a simple counterexample. The target concepts are functions of the form fw(i, x) = {6: Wi < X Wi> X (3) The prior distribution of 'Iii is uniform on the concept class C = [0, l]d. The inputs are pairs (i, x), where i takes on the values 1, ... , d with equal probability, and x is uniformly distributed on [0, 1]. Since this is basically d concurrent high-low games (one for each component of 'Iii), the version space is a product of subintervals of [0,1]. For d = 2, the concept class is the unit square, and the version space is a rectangle. The prediction error is proportional to the perimeter of the rectangle. A sequence of queries with i = 1 can bisect the rectangle along one dimension, yielding 1 bit per query, while the perimeter tends to a finite constant. Hence the prediction error tends to a finite constant, in spite of the maximal information gain. 3 The committee filter: information and prediction The dilemma of the previous section was that constructing queries with high information gain does not necessarily lead to rapidly decreasing prediction error. This is because the constructed query distribution may have nothing to do with the input distribution V. This deficiency can be avoided in a different paradigm in which the query distribution is created by filtering V. Suppose that the learner receives a stream of unlabeled inputs Xl, x2, ... drawn independently from the distribution V. After seeing each input Xi, the learner has the choice of whether or not to query the teacher for the correct label Ii = f( Xi). In [50592] it was suggested to filter queries that cause disagreement in a committee of Gibbs learners. In this paper we concentrate on committees with two members. The algorithm is: Query by a committee of two Repeat the following until n queries have been accepted 1. Draw an unlabeled input x E X at random from V. 2. Select two hypotheses hl' h2 from the posterior distribution. In other words, pick two hypotheses that are consistent with the labeled examples seen so far. 3. If hl(x) -:j:. h2(X) then query the teacher for the label of x, and add it to the training set. The committee filter tends to select examples that split the version space into two parts of comparable size, because if one of the parts contains most of the version space, then the probability that the two hypotheses will disagree is very small. More precisely, if x cuts the version space into parts of size X and 1 - X, then the probability of accepting x is 2x(1 - X). One can show that the i.i.g. of the queries is lower bounded by that obtained from random inputs. Information, prediction, and query by committee 487 In this section, we assume something stronger: that the expected i.i.g. of the committee has positive lower bound. Conditions under which this assumption holds will be discussed in the next section. The bound implies that the cumulative information gain increases linearly with the number of queries n. But the version space resulting from the queries alone must be larger than the version space that would result if the learner knew all of the labels. Hence the cumulative information gain from the queries is upper bounded by the cumulative information gain which would be obtained from the labels of all m inputs, which behaves like O(dlog r;;) for a concept class C with finite VC dimension d ([HKS91]). These O(n) and O(log m) behaviors are consistent only if the gap between consecutive queries increases exponentially fast. This argument is depicted in Fi!~ure 1. Cumulative Information Gain Cumulative Information of Queries Expected ~~;~~ r------- _r-----------Random Examples r ----Gap between example: _ _ accepted as queries x x x x Number of Random Examples Figure 1: Each tag on the x axis denotes a random example in a specific typical sequence. The symbol X under a tag denotes the fact that the example was chosen as a query. Recall that an input is accepted if it provokes disagreement between the two Gibbs learners that constitute the committee. Thus a large gap between consecutive queries is equivalent to a small probability of disagreement. But in our Bayesian framework the probability of disagreement between two Gibbs learners is equal to the probability of disagreement between a Gibbs learner and the teacher, which is the expected prediction error. Thus the prediction error is exponentially small as a function of the number of queries. The exact statement of the result is given below, a detailed proof of which will be published elsewhere. Theorem 1 Suppose that a concept class C has VC-dimension d < 00 and the expected information gained by the two member committee algorithm is bounded by c > 0, independent of the query number and of the previous queries. Then the probability that one of the two committee members makes a mistake on a randomly chosen example with respect to a randomly chosen fEe is bounded by (3+0(e-Cln»~exp (-2(d~ l)n) (4) for some constant Cl > 0, where n is the number of queries asked so far. 488 Freund, Seung, Shamir, and Tishby 4 Lower bounds on the information gain Theorem 1 is applicable to learning problems for which the committee achieves i.i.g. with positive lower bound. A simple case of this is the d-dimensional high-low game of section 2, for which the i.i.g. is 7 /(121n 2) R: 0.84, independent of dimension. This exact result is simple to derive because the high-low game is geometrically trivial: all version spaces are similar to each other. In general, the shape of the version space is more complex, and depends on the randomness of the examples. Nevertheless, the expected i.i.g. can be lower bounded even for some learning problems with nontrivial version space geometry. 4.1 The information gain for convex version spaces Define a class of functions f w by fw(x, t) = { ~: ... ... t w·x> , tij·x<t. (5) The vector tij E Rd is drawn at random from a prior distribution P, which is uniform over some convex body contained in the unit ball. The distribution of inputs (x, t) E Bd x [-1,1], is a product of any distribution over Bd (the unit ball centered at the origin) and the uniform distribution over [-1, +1]. Since each example defines a plane in the concept space, all version spaces for this problem are convex. We show that there is a uniform lower bound on the expected i.i.g. for any convex version space when a two member committee filters inputs drawn from V. In the next paragraphs we sketch our proof, the full details of which shall appear elsewhere. In fact, we prove a stronger statement, a bound on the expected i.i.g. for any fixed X. Fix x and define x( t) as the fraction of the version space volume for which x· w < t. Since the probability of filtering a query at t is proportional to 2X(t)[1 - X(t)], the expected i.i.g. is given by J~l 2X(t)[1 - X(t)]1i(X(t))dt I[X(t)] = (6) J~l 2X(t)[1 - X(t)]dt In the following, it is more convenient to define the expected i.i.g. as a functional of r(t) = d-VdX/dt, which is the radius function of the body of revolution with equivalent cross sectional area dX/dt. Using the Brunn-Minkowski inequality, it can be shown that any convex body has a concave radius function r(t). We have found a set of four transformations of r(t) which decrease I[r]. The only concave function that is a fixed point of these transformations is (up to volume preserving rescaling transformations): r*(t) = d-{/df2 (1 - It I) . This corresponds to the body constructed by placing two cones base to base with their axes pointing along x. We can calculate I[r*] explicitly for each dimension d. As the dimension of the space increases to infinity this value converges from above to a strictly positive value which is 1/9 + 7/(181n2) R: 0.672 bits, which is surprisingly close to the upper bound of 1 bit. Information, prediction, and query by committee 489 4.2 The information gain for thresholded continuous functions Consider a concept class consisting of functions of the form fw(x) = { ~: F(w,x)~O, F(w, i) <0 , (7) where i E R', w E Rd and F is a smooth function of both x and W. Random input learning of this type of concept class has been studied within the annealed approximation by [AFS92]. We assume that both V and P are described by density functions that are smooth and nonvanishing almost everywhere. Let the target concept be denoted by fwo(i). We now argue that in the small version space limit (reached in the limit of a large number of examples), the expected i.i.g. for query learning of this concept class has the same lower bound that was derived in section 4.1. This is because a linear expansion of F becomes a good approximation in the version space, F(w, X) = F(wa, i) + (w - wa) . 'V wF(wa, x) . (8) Consequently, the version space is a convex body containing wa, each boundary of which is a hyperplane perpendicular to 'VwF(wa, x) for some x in the training set. Because the prior density P is smooth and nonvanishing, the posterior becomes uniform on the version space. From Eq. (8) it follows that a small version space is only cut by hyperplanes corresponding to inputs i for which F(wa, X) is small. Such inputs can be parametrized by using coordinates on the decision boundary (the manifold in x space determined by F( wa, i) = 0), plus an additional coordinate for the direction normal to the decision boundary. Varying the normal coordinate of x changes the distance of the corresponding hyperplane from wa, but does not change its direction (to lowest order). Hence each normal average is governed by the lower bound of 0.672 bits that was derived in section 4.1 for planar cuts along a fixed axis of a convex version space. The expected i.i.g. is obtained by integrating the normal average over the rest of the coordinates, and therefore is governed by the same lower bound. 5 Summary and open questions In this work we have shown that the number of examples required for query learning behaves like the logarithm of the number required for random input learning. This result on the power of query filtering applies generally to concept classes for which the committee filter achieves information gain with positive lower bound, and in particular to concept classes consisting of thresholded smooth functions. A wide variety of learning architectures in common use fall in this group, including radial basis function networks and layered feedforward neural networks with smooth transfer functions. Our main unrealistic assumption is that the learned rule is assumed to be realizable and noiseless. Understanding how to filter queries for learning unrealizable or noisy concepts remains an important open problem. 490 Freund, Seung, Shamir, and Tishby Acknowledgment s Part of this research was done at the Hebrew University of Jerusalem. Freund, Shamir and Tishby would like to thank the US-Israel Binational Science Foundation (BSF) Grant no. 90-00189/2 for support of their work. We would also like to thank Yossi Azar and Manfred Opper for helpful discussions regarding this work. References [AFS92] S. Amari, N. Fujita, and S. Shinomoto. Four types of learning curves. Neural Comput., 4:605-618, 1992. [Ang88] D. Angluin. Queries and concept learning. Machine Learning, 2:319-342, 1988. [Bau91] E. Baum. Neural net algorithms that learn in polynomial time from examples and queries. IEEE Trans. Neural Networks, 2:5-19, 1991. [CAL90] D. Cohn, L. Atlas, and R. Ladner. Training connectionist networks with queries and selective sampling. Advances in Neural Information Processing Systems, 2:566-573, 1990. [ER90] B. Eisenberg and R. Rivest. On the sample complexity of PAC-learning using random and chosen examples. In M. Fulk and J. Case, editors, Proceedings of the Third Annual ACM Workshop on Computational Learning Theory, pages 154-162, San Mateo, Ca, 1990. Kaufmann. [Fed72] V. V. Fedorov. Theory of Optimal Experiments. Academic Press, New York, 1972. [HKS91] D. Haussler, M. Kearns, and R. Schapire. Bounds on the sample complexity of Bayesian learning using information theory and the VC dimension. In M. K. Warmuth and L. G. Valiant, editors, Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 61-74, San Mateo, CA, 1991. Kaufmann. [KR90] W. Kinzel and P. Rujan. Improving a network generalization ability by selecting examples. Europhys. Lett., 13:473-477, 1990. [Lin56] D. V. Lindley. On a measure of the information provided by an experiment. Ann. Math. Statist., 27:986-1005, 1956. [Mac92] D. J. C. MacKay. Bayesian methods for adaptive models. PhD thesis, California Institute of Technology, Pasadena, 1992. [SOS92] H. S. Seung, M. Opper, and H. Sompolinsky. Query by committee. In Proceedings of the fifth annual A CM workshop on computational learning theory, pages 287-294, New York, 1992. ACM. [VaI84] L. G. Valiant. A theory of the learnable. Comm. ACM, 27:1134-1142, 1984.
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Improving Convergence in Hierarchical Matching Networks for Object Recognition Joachim Utans* Gene Gindit Department of Electrical Engineering Yale University P. O. Box 2157 Yale Station New Haven, CT 06520 Abstract We are interested in the use of analog neural networks for recognizing visual objects. Objects are described by the set of parts they are composed of and their structural relationship. Structural models are stored in a database and the recognition problem reduces to matching data to models in a structurally consistent way. The object recognition problem is in general very difficult in that it involves coupled problems of grouping, segmentation and matching. We limit the problem here to the simultaneous labelling of the parts of a single object and the determination of analog parameters. This coupled problem reduces to a weighted match problem in which an optimizing neural network must minimize E(M, p) = LO'i MO'i WO'i(p), where the {MO'd are binary match variables for data parts i to model parts a and {Wai(P)} are weights dependent on parameters p . In this work we show that by first solving for estimates p without solving for Mai , we may obtain good initial parameter estimates that yield better solutions for M and p. *Current address: International Computer Science Institute, 1947 Center Street, Suite 600, Berkeley, CA 94704, utans@icsi.berkeley.edu tCurrent address: SUNY Stony Brook, Department of Electrical Engineering, Stony Brook, NY 11784 401 402 Utans and Gindi Figure 1: Stored Model for a 3-Level Compositional Hierarchy (compare Figure 3). 1 Recognition via Stochastic Forward Models The Frameville object recognition system introduced by Mjolsness et al [5, 6, 1] makes use of a compositional hierarchy to represent stored models. The recognition problem is formulated as the minimization of an objective function. Mjolsness [3,4] has proposed to derive the objective function describing the recognition problem in a principled way from a stochastic model that describes the objects the system is designed to recognize (stochastic visual grammar). The description mirrors the data representation as a compositional hierarchy, at each stage the description of the object becomes more detailed as parts are added. The stochastic model assigns a probability distribution at each stage of that process. Thus at each level of the hierarchy a more detailed description of parts in terms of their subparts is given by specifying a probability distribution for the coordinates of the subparts. Explicitly specifying these distributions allows for finer control over individual part descriptions than the rather general parameter error terms used before [1, 8]. The goal is to derive a joint probability distribution for an instance of an object and its parts as it appears in the scene. This gives the probability of observing such an object prior to the arrival of the data. Given an observed image, the recognition problem can be stated as a Bayesian inference problem that the neural network solves. 1.1 3-Level Stochastic Model For example, consider the model shown in Figure 1 and 3. The object and its parts are represented as line segments (sticks), the parameters were p = (x, y, I, ())T with x , y denoting position, I the length of a stick and () its orientation. The model considers only a rigid translation of an object in the image. Only one model is stored. From a central position p = (x, y, I, ()), itself chosen from a uniform density, the N{3 parts at the first level are placed. Their structural relationships is stored as coordinates u{3 in an object-centered coordinate frame, i.e. relative to p. While placing the parts, Gaussian distributed noise with mean 0 and is added to the position coordinates to capture the notion of natural variation of the object's shape. The variance is coordinate specific, but we assume the same distribution for the x and y coordinates, O"'ix; O"'~, is the variance for the length Improving Convergence in Hierarchical Matching Networks for Object Recognition 403 component and UI9 for the relative angle. In addition, here we assume for simplicity that all parts are independently distributed. Each of the parts {3 is composed of subparts. For simplicity of notation, we assume that each part {3 is composed from the same number of subparts Nm (note that the index 'Y in Figure 2 here corresponds to the double index {3m to keep track of which part {3 subpart {3m belongs to on the model side, i.e. the index (3m denotes the mth sub-part of part (3). The next step models the unordering of parts in the image via a permutation matrix M, chosen with probability P(M), by which their identity is lost. If this step were omitted, the recognition problem would reduce to the problem of estimating part parameters because the parts would already be labeled. From the grammar we compute the final joint probability distribution (all constant terms are collected in a constant C): P(M, {P,3m}, {PtJ}, p) = 1.2 Frameville Architecture for Part Labelling within a single Object The stochastic forward model for the part labelling problem with only a single object present in the scene translates into a reduced Frameville architecture as depicted in Figure 2. The compositional hierarchy parallels the steps in the stochastic model as parts are added at each level. Match variables appear only at the lowest level, corresponding to the permutation step of the grammar. Parts in the image must be matched to model parts and parts found to belong to the stored object must be grouped together. The single match neuron Mai at the highest level can be set to unity since we assume we know the object's identity and only a single object is present. Similarly, all terms inaij from the first to the second level can be set to unity for the correct grouping since the grouping is known at this point from the forward model description. In addition, at the intermediate (second) level, we may set all M,3j = 1 for {3 = j and MtJj = 0 otherwise with no loss of generality. These mid-level frames may be matched ahead of time, but their parameters must be computed from data. Introducing a part permutation at the intermediate levels thus is redundant. Given this, an additional simplification ina grouping variables at the lowest (third) level is possible. Since parts are pre-matched at all but the lowest level, inaj k can be expressed in terms of the part match M"{k as inajk = M"{k1NA"{tJM,3j and explicitly representing inaj k as variables is not necessary. The input to the system are the {pk}, recognition involves finding the parameters 404 Utans and Gindi Model <l rame • x • y • 9 • I Data Figure 2: Frameville Architecture for the Stochastic Model. The 3-level grammar leads to a reduced "Frameville" style network architecture: a single model is stored on the model side and only one instance of the model is present in the input data. The ovals on the model side represent the object, its parts and subparts (compare Figure 1); the arcs INA represent their structural relationship. On the data side, the triangles represent parameter vectors (or frames) describing an instance of the object in the scene. At the lowest level the Pk represent the input data, parameters at higher levels in the hierarchy must be computed by the network (represented as bold triangles) . ina represents the grouping of parts on the data side (see text) . The horizontal lines represent assignments from frames on the data side to nodes on the model side. At the intermediate level, frames are prematched to the corresponding parts on the model side; match variables are necessary only at the lowest level (represented as bold lines with circles). P and {Pi} as well as the labelling of parts M. Thus, from Bayes Theorem P( {pdIM, p, {Pi} )P(M, p, {Pi}) P({Pk} ) ex: P(M, p, {Pi}, {pd) (2) and recognition reduces to finding the most probable values for p, {Pi} and M given the data: arg max P(M, p, {Pi}, {pd) M,P,{Pi} (3) Solving the inference problem involves finding the MAP estimate and is is equivalent to minimizing the exponent in equation (1) with respect to M, P and {Pi}. 2 Bootstrap: Coarse Scale Hints to Initialize the Network 2.1 Compositional Hierarchy and Scale Space In some labelling approaches found in the vision literature, an object is first labelled at the coarse, low resolution, level and approximate parameters are found . In this top-down approach the information at the higher, more abstract, levels is used Improving Convergence in Hierarchical Matching Networks for Object Recognition 405 spatial scale II III im---:.+-Human 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 L ______ I r---------' 1 1-- -, 1 iVt Am i (I) ~ i 1 1 ______ _ ___ J abstraction Figure 3: Compositional Hierarchy vs. Scale Space Hierarchy_ A compositional hierarchy can represent a scale space hierarchy. At successive levels in the hierarchy, more and more detail is added to the object_ to select initial values for the parts at the next lower level of abstraction. The segmentation and labelling at this next lowest level is thus not done blindly; rather it is strongly influenced contextually by the results at the level above. In fact, in very general terms such a scheme was described by Marr and Nishihara [2]. They advocate in essence a hierarchical model base in which a shape is first matched to the highest levels, and defaults in terms of relative object-based parameters of parts at the next level are recalled from memory. These defaults then serve as initial values in an unspecified segmentation algorithm that derives part parameters; this step is repeated recursively until the lowest level is reached. Note that the highest level of abstractions correspond to the coarsest levels of spatial scale. There is nothing in the design of the model base that demands this, but invariably, elements at the top of a compositional hierarchy are of coarser scale since they must both include the many subparts below, and summarize this inclusion with relatively few parameters. Figure 3 illustrates the correspondence between these representations. In this sense, the compositional hierarchy as applied to shapes includes a notion of scale, but there is no "scale-space" operation of intentionally blurring data. The notion of Scale Space as utilized here thus differs from the application of the method to low-level computations in the visual domain where auxiliary coarse scale representations are computed explicitly. The object representations in the Frameville system as described earlier combines both, bottom-up and top-down elements. If the top-down aspects of the scheme described by Marr and Nishihara [2] could be incorporated into the Frameville architecture, then our previous simulation results [8] suggest that much better performance can be expected from the neural network. Two problems must be addressed: (1) How do we obtain, from the observed raw data alone, a coarse estimate of the slot parameters at the highest level and (2) given these crude estimates how do we utilize them to recall default settings for the segmentation one level below? 406 Utans and Gindi 0, Bootstrap y Model Data Figure 4: Bootstrap computation for a network from a 3- level grammar. Analog frame variables at the top and intermediate level are initialized from data by a bootstrap computation (bold lines indicate the flow of information) 2.2 Initialization of Coarse Scale Parameters We propose to aid convergence by supplying initial values for the analog variables p and {Pi}; these must be computed from data without making use of the labelling. In general, it is not possible to solve for the analog parameters without knowledge of the correct permutation matrix M. However, for the purpose of obtaining an approximation f> one can derive a new objective function that does not depend on M and the parameters {Pi} by integrating over the {Pi} and summing over all possible permutation matrices M: P(p,{pk}) = L J d{pj}P(P,{Pi},{pd,M) {M}IM is a permutation (4) This formulation leads to an Elastic Net type network [9, 7]. However, this implementation of a separate network for the bootstrap computations is expensive. Here we use simpler computation where the coarse scale parameters are estimated by computing sample averages, corresponding to finding the solution for the Elastic Net in the high temperature limit [7]. For the position x we find, after integrating over the {xi}, x 1 L M{3mkXk L{3m 1/(O"~xO"~mx) 13m k O"~xO"~mx _ 1 L U{3x L{3 1/ O"~x (3 O"~x (5) and similarly for y. Since the assignment M{3m k of subparts k on the data side to subparts fJm on the model side is not known at this point, the first term in equations (5) cannot be evaluated. After approximating the actual variance with Improving Convergence in Hierarchical Matching Networks for Object Recognition 407 an average variance, these equations reduce to 1 1 1 x N N L Xk N N L uf3mx - N L uf3x 13 m k 13 m 13m 13 13 (6) In terms of the objective function this translates into assuming that here the error terms for all parts are weighted equally. Since these weights would depend on the actual part match, this just corresponds to our ignorance regarding identity of the parts. This approximation assumes that the variances do not differ by a large amount, otherwise the approximation p will not be close to the true values. Since the model can be designed such that the part primitives used at the lowest level of the grammar are not highly specialized as would be the case for abstractions at higher levels of the model, the approximation proved sufficient for the problems studied here. The neural network can be used to perform the calculation. The Elastic Net formulation assigns approximately equal weights to all possible assignments at high temperatures. Thus, this behavior can be expressed in the original network with match variables by choosing Mf3mk = l/{Nf3Nm) V i,j. This leads to the following two-pass bootstrap computation. Using this specific choice for M only the analog variables need to be updated to compute the coarse scale estimates. The network with constant M is just the neural network implementation for computing x from equation (6). After these have converged, x can be used to compute Xj = x + uf3. Thus, the parameters for intermediate levels can by hypothesized from the coarse scale estimate x by adding the known transformation (recall that for intermediate levels, the part identity is preserved and no permutation steps takes place (see Figure 2)). Then the network is restarted with random values for the match variables to compute the correct labelling and the correct parameters. 2.3 Simulation Results The bootstrap procedure has been implemented for a 3-level hierarchical model. The model describes a "gingerbread man" as shown in Figure 3. The incorrect solutions observed did not, in the vast majority of cases, violate the permutation matrix constraint, i.e. the assignment was unique. However, even though the assignment is unique, parts where not always assigned correctly. Most commonly, the identity of neighboring parts was interchanged, in particular for cases with large variance. The advantage of using the bootstrap initialization is clear from Figure 5. For the simulation, cr~ = 2crt; the noise variance was identical for all parts. The network computed the solution reliably for large noise variances. In such cases the performance of the network without initialization deteriorates rapidly. Only one set of 10 experiments was used for the graph but in all simulations performed, the network with initialization consistently outperformed the network without initialization. Figure 5(right) shows the time measured in the number of iterations necessary for the network to converge; it is almost unaffected by the increase in the noise variance. This is because the initial values derived from data are still close to the final solution. While in some cases, the random starting point happens to be close to the correct solution and the network without initialization converges rapidly, Figure 5 reflect the typical behavior and demonstrate the advantage of computing approximate initial values. 408 Utans and Gindi Success Rate 100 80 80 ~ .0 20 'h.o 0.2 0 .• 0.8 0.8 ott. 2 (122 I 0 '" -= o ..oj ., 300 ... 11)200 .oj ..... o ... II) ~ 100 -= o 0.0 Convergence Speed 0. 2 0.4 0.8 0.8 1.0 a"1' 2 CT22 Figure 5: Results Comparing the Network without and with Initialization (solid line). Left : The success rate indicates the rate at which the network converged to the correct solutions. /1~ denotes the noise variance at the intermediate level of the model and /1~ the noise variance at the lowest level. Only one set of 10 experiments was used for the graph but in all simulations performed, the network with initialization consistently outperformed the network without initialization. Right: The graph shows the average time it takes for the network to converge (as measured by the number of iterations) averaged over 10 experiments. Only simulations where the network converged to the correct solution are used to compute the average time for convergence. The stopping criterion used required all the match neurons to assume values M'j > 0.95 or M'J < 0.05. The error bars denote the standard deviation. Acknowledgements This work was supported in part by AFOSR grant AFOSR 90-0224. Vie thank E. Mjolsness and A. Rangarajan for many helpful discussions. References [1] G. Gindi, E. Mj~lsness, and P. Anandan. Neural networks for model based recognition. In Neural Networks: Concepts, Applications and Implementations, pages 144-173. Prentice-Hall, 1991. [2] David Marr. Vision. W. H. Freeman and Co., New York, 1982. [3] E. Mjolsness. Bay~sian inference on visual grammars by neural nets that optimize. Technical Report YALEU-DCS-TR-854, Yale University, Dept. of Computer Science, 1991. [4] E. Mj~lsness. Visual grammars and their neural nets. In R.P. Lippmann J.E. Moody, S.J. Hanson, editor, Advances in Neural Information Processing Systems 4. Morgan Kaufmann Publishers, San Mateo, CA, 1992. [5] Eric Mjolsness, Gene Gindi, and P. Anandan. Optimization in model matching and perceptual organization: A first look. Research report yaleu/dcs/rr-634, Yale University, Department of Computer Science, 1988. [6] Eric Mjolsness, Gene R. Gindi, and P. Anandan. Optimization in model matching and perceptual organization. Neural Computation, vol. 1, no. 2, 1989. [7] Joachim Utans. Neural Networks for Object Recognition within Compositional Hierarchies. PhD thesis, Department of Electrical Engineering, Yale University, New Haven, CT 06520, 1992. [8] Joachim Utans, Gene R. Gindi, Eric Mjolsness, and P. Anandan. Neural networks for object recognition within compositional hierarchies: Initial experiments. Technical report 8903, Yale University, Center for Systems Science, Department Electrical Engineering, 1989. [9] A. L. Yuille. Generalized deformable models, statistical physics, and matching problems. Neural Computation, 2(2):1-24, 1990.
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A dynamical model of priming and repetition blindness Daphne Bavelier Laboratory of Neuropsychology The Salk Institute La J oHa, CA 92037 Michael I. Jordan Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge MA 02139 Abstract We describe a model of visual word recognition that accounts for several aspects of the temporal processing of sequences of briefly presented words. The model utilizes a new representation for written words, based on dynamic time warping and multidimensional scaling. The visual input passes through cascaded perceptual, comparison, and detection stages. We describe how these dynamical processes can account for several aspects of word recognition, including repetition priming and repetition blindness. 1 INTRODUCTION Several psychological phenomena show that the construction of organized and meaningful representations of the visual environment requires establishing separate representations (termed episodic representations) for the different objects viewed. Three phenomena in the word recognition literature suggest that the segregation of the visual flow into separate episodic representations can be characterized in terms of specific temporal constraints. We developed a model to explore the nature of these constraints. 2 DESCRIPTION OF THE BEHAVIORAL DATA In a typical priming experiment, subjects are presented with a first word, termed the "prime," and then asked to name or make a judgment to a second word, termed 879 880 Bavelier and Jordan the "target." The performance of subjects is compared in conditions in which the target and prime are related versus conditions in which they are unrelated. When the prime is presented fast enough so that it cannot be identified (about 40 ms), subjects' performance on the target is facilitated when the prime and the target are identical compared to the case in which they are unrelated. This effect, known as "lnasked priming," is very short lasting, appearing only within trials, and lasting on the order of 100 ms (Humphreys, Evett, Quinlan & Besner, 1987). If the prime, however, is presented for a period such that it is just identifiable (about 100 ms), subjects' performance on the target is hindered when prime and target are identical (Kanwisher, 1987; Humphreys et al., 1987). This effect, known as "repetition blindness," is conditional on the conscious identification of the prime. The size of the effect decreases as the duration between the two items increases. Repetition blindness is observed only within trials and vanishes for inter-stimulus durations on the order of 500 ms. When the prime is presented long enough to be easily identifiable (about 250 ms or more), subjects' performance on the target is once again facilitated when prime and target are identical (Salasoo, Shiffrin & Feustel, 1985). This effect, known as "classical repetition priming," is long lasting, being observed not only within trials, but between trials and even between sessions. In certain experimental conditions, it has been observed to last up to a year. These results implicate two factors influencing word recognition: the time of presentation and whether or not the prime has been identified. We have developed a model that captures the rather non-intuitive result that as the time of presentation of the prime increases, recall of the target is first facilitated, then inhibited and then facilitated again. The two main features of the model are the dynamical properties of the word representations and the dependence of the detection processes for each word on previous conscious identification of that word. 3 REPRESENTATION The representation that we developed for our model is a vector space representation that allows each word to be represented by a fixed-length vector, even though the words are of different length. We developed an algorithmic method for finding the word representations that avoids some of the difficulties with earlier proposals (cf. Pinker & Prince, 1988). The algorithm proceeds in three stages. First, dynamic programming (Bellman, 1957) is used to compute an inter-word similarity matrix. The transition costs in the dynamic programming procedure were based on empirically-determined values of visual similarity between individual letters (Townsend, 1971). Interestingly, we found that dynamic programming solutions naturally capture several factors that are known to be important in human sensitivity to orthographic similarity (for example, orthographic priming increases as a function of the number of letters shared between the prime and the target in a nonlinear manner, shared end-letters are more important than shared middle-letters, and relative letter position determines orthographic similarity (Humphreys et al., 1987)). A dynamical model of priming and repetition blindness 881 After the dynamic programming stage, multidimensional scaling (Torgerson, 19.58) is used to convert the inter-word similarity ma.trix into a vector space representation in which distance correlates with similarity. Next, word vectors are normalized by projecting them onto a semi-hypersphere. This gives the origin of the vector space a meaning, allowing us to use vector magnitude to represent signal energy. This representation also yielded natural choices for the "blank" stimulus and the "mask" stimulus. The "blank" was taken to be the origin of the space and the "mask" was taken to be a vector on the far side ofthe hypersphere. In the dynamical model that we describe below, vectors that are far apart have maximally disruptive effects on each other. A distant stimulus causes the state to move rapidly away from a particular word vector, thus interfering maximally with its processing. 4 PROCESSING 4.1 FORMALIZATION OF THE PROBLEM AS A SIGNAL DETECTION PROBLEM We formalize the problem of visual word recognition as a problem of detecting significant fluctuations of a multidimensional signal embedded in noise. This can be viewed as a maximum likelihood detection problem in which the onsets and durations of the signal are not known a priori. Our model has two main levels of processing: a perceptual stage and a detection stage. Perceptual Stage The perceptual stage is a bank of noisy linear filters. Let Wi denote the ndimensional word vector presented at time t, with components Wi,k. The word vector is corrupted with white noise crt] to form the input Uk [t]: udt] = VVi ,k + crt], and this input is filtered: rdt] = -aork[t - 1] - alrk[t - 2] + budt] + 7][t], in the presence of additional white noise 7][t]. Detection Stages The first detection stage in the model is a linear filter whose inverted impulse response is matched to the impulse response of the perceptual filter: sdt] = -cosdt - 1] - clsdt - 2] + drk[t]. Such a filter is known as a matched filter, and is known to have optimality properties that make it an effective preprocessor for a system that utilizes thresholds for making decisions (van Trees, 1968). The output of the matched filter is projected onto each of the words in the lexicon to form scalar "word activation" signals xdt] that can be compared to thresholds: n Xi[t] = L Wi ,ksdt]. k=l 882 Bavelier and Jordan • A !i 0 Input ., ~ Perceptual ,.. <0, 6" ." 1000 TlME(n.) Stage !" 2 Per09ptual Filter f G 11 . ,,, <0, "" .00 "00 lltjE(ITW) Matched Filter ~ ~ Ii: Detection f iI ~ Stag .. !i l< Word Activation '00 .. , .00 ,ao ,,00 TIME(ITW) 0 [=-=- n \ , ' , ~ Word deleaion SI '00 '00 .00 .00 10'" TIMEtlTW) Figure 1: The processing stages of the model. The figures on the right show the signals in the model projected onto the vector for the word bring. Bring was presented for 100 ms, followed by a 300 ms blank, followed by a second presentation of bring for 300 ms. The decision process is a simple binary decision based on a variable baseline pdt] and a variable threshold edt]: { I if xdt] - pdt] > edt] Yi = a otherwise 4.2 DETECTION DYNAMICS The problem of detecting signals that may overlap in time and have unknown onsets and unknown durations requires the system to focus on fluctuations rather than the absolute heights of the activation curves. Moreover, the test for significance of a fluctuation must be dependent on the state of the detection mechanism and the state of the filters. Our significance test utilizes two time-varying quantities to capture this state-dependence: the baseline p and the threshold o. The baseline Pi [t] varies as follows. On time steps for which the fluctuations are subthreshold (ydt] = 0, for all i), each baseline simply tracks the most recent A dynamical model of priming and repetition blindness 883 minimum value of the corresponding word activation signal: I/. • [t] _ { fl,i [t - 1] rt Xi [t] if Xi[t] > fl,i[t] otherwise When a fluctuation passes threshold (ydt] = 1, for some i), the word i is "detected," and the baselines of all words are increased: fl,dt] = fl,dt - 1] + ~c/>(i, k), where c/>(i, k) is the angle between Wi and Wk and ~ is a positive scaling parameter. This rule prevents multiple detections during a single presentation and it prevents the neighbors of a detected word from being detected due to their overlap with the detected word. The threshold (}i is subject to first-order dynamics that serve to increase or decrease the threshold as a function of the recent activation history of the word (a rudimentary form of adaptation): Odt] = a(}dt -1] + (1- a)()~ - ,B(Xi[t] - fl,dt]) + , where a and ,B are positive numbers. This rule has the effect of decreasing the threshold if the activation of the word is currently above its baseline, and increasing the threshold toward its nominal value ()? otherwise. 4.3 PARAMETERS The parameters in the model were determined from the behavioral data and from the structural assumptions of the model in the following manner. The dynamics of the perceptual filter were determined by the time constants of masked priming, as given by the behavioral data. This choice also fixed the dynamics of the matched filter, since the matched filter was tied to the dynamics of the perceptual filter. The dynamics of the baseline fl, (i.e., the value 0 were determined by the constraint that a long presentation of a word not lead to multiple detections of the word. Finally, the dynamics of the threshold 0 were determined by the dynamics of classical repetition priming as given by the behavioral data. Note that the behavioral data on repetition blindness were not used in adjusting the parameters of the model. 5 ACCOUNTS OF THE THREE BASIC PHENOMENA 5.1 MASKED PRIMING The facilitation observed in masked priming is due to temporal superposition in the perceptual filter and the matched filter. At the time scale at which masked priming is observed, the activation due to the first critical word (Cl) overlaps with the activation due to the second critical word (C2) (see Figure 2A), leading to a larger word activation value when C1 and C2 are identical than when they are different. 5.2 REPETITION BLINDNESS The temporal superposition that leads to masked priming is also responsible for repetition blindness (see Figure 3). The temporal overlap from the filtering dynamics 884 Bavelier and Jordan 2A 28 ~ 0 ... . ~ 0 (') . .., u 0 c( N 0 ~ 0 o 200 400 600 800 1000 o 200 400 600 800 1000 TIME(ms) TIME(ms) Figure 2: Activation curves at the perceptual level (A) and the matched filter level (B) for the word bring during the presentation of the sequence bring, character, bring. Each word was presented for 40 ms. o It) o 1-=- 1 o 200 400 600 800 1000 TIME(ms) Figure 3: Activation curves for the word bring during the presentation of the sequence bring, character, bring. Each word was presented for 100 ms. will prevent the baseline I-' from getting reset to a sufficiently small value to allow a second detection. That is, repetition blindness arises because the fluctuation due to the brief presentation of C2 is not judged significant against the background of the recent detection of the word. Note that such a failure to detect the second occurrence will happen only when C 1 has been correctly detected, because only then will the baseline be increased. This dependence of repetition blindness on explicit detection of the first occurrence also characterizes the behavioral data (Kanwisher, 1987). 5.3 CLASSICAL REPETITION PRIMING The facilitation observed in classical repetition priming is due to the dynamics of the threshold (). The value of () decreases during significant increases in the activation of a word; hence a smaller fluctuation in activation is needed for the next occurrence A dynamical model of priming and repetition blindness 885 ~ ----..... " I ItIR_,.... 8 / 0 ~ ~f\ II) 0 a 200 400 600 800 1000 nME(ma) Figure 4: Activation curves for the word bring during the presentation of the word bring for 300ms, followed by a 300ms blank, followed by bring again for lOOms. to be detected (see Figure 4). 6 OTHER DATA ACCOUNTED FOR BY THE MODEL The model captures most of the specific characteristics of the three basic phenomena that we have reviewed. For example, it accounts for the finding of masked priming between orthographic neighbors (Humphreys, et al., 1987). This effect arises in the model because a distributed representation is used for the words. The model also captures the finding that the size of repetition blindness decreases as the interval between the critical stimuli increases (this is due to the fact that the baseline is reset to increasingly lower values as the inter-stimulus interval increases), as well as the fact that the size of repetition blindness decreases as the duration of presentation of C2 increases (because the activation for C2 continues to increase while the baseline remains fixed). Similarly, the model accounts for the finding that. the manifestation of repetition blindness is dependent on the conscious identification of the first occurrence, as well the finding of repetition blindness between orthographic neighbors (Kanwisher, 1987). Specifics of classical repetition priming, such as the finding that priming is restricted to a word identity, and the fact that its size increases with the number of repetitions and diminishes as the lag between repetitions increases (Salasoo, Shiffrin & Feustel, 1985), are also captured by the model. The model also accounts for other behavioral phenomena described in the literature on word recognition. Our vector space representation allows us t.o account naturally for the fact that the final words in a list are recalled better than the middle words in the list (the "recency" effect). This occurs because dissimilar words tend to have large angle between them (and therefore "inhibit" each other dynamically). whereas the "blank" is at the origin of the space and is relatively "close" to an of the words. The residual activation for a presented word therefore tends to be stronger if followed by a blank than by a dissimilar word. The model also captures certain of the effects of pattern masks on word recognition. For example, "forward" masking, a condition in which the mask precedes the word to be detected, is known t.o be less disruptive than "backward" masking, a condition in which the mask follows the word to be detected. This occurs in the model because of the dynamics of the 886 Bavelier and Jordan baselines: preceding a word with a mask tends to reset its baseline to lower values and therefore renders the test for significance relatively more sensitive. 7 CONCLUSIONS From the point of view of the current model, the fact that the detection of repeated items is enhanced, then suppressed, then once again enhanced as the duration of the items is increased finds a natural explanation in the nature of the signal processing task that the word recognition system must solve. The signals that arrive in temporal sequence for perceptual processing have unknown onset times, unknown durations, and are corrupted by noise. The fact that signals have unknown onset times and can superimpose implies that the system must detect fluctuations in signal strength rather than absolute values of signal strength. The presence of noise, inevitable given the neural hardware and the complex multidimensional nature of the signal, implies that the system must detect significant fluctuations and must incorporate information about recent events into its significance tests. The real-time constraints of this detection task and the need to guard against errors imply that certain of the fluctuations will be missed, a fact that will result in "blindness" to repeated items at certain time scales. Acknowledgments This research was funded by the McDonnell-Pew Centers for Cognitive Neuroscience at UCSD and MIT, by a grant from the McDonnell-Pew Foundation to Michael!. Jordan, and by NIDCD Grant 5R01-DC-00128 to Helen Neville. References Humphreys, G. W., Evett, L. J., Quinlan, P. T., & Besner, D. (1987). Orthographic priming. In M. Coltheart (Ed.), Attention and Performance XII (pp. 105-125). Hillsdale, NJ: Erlbaum. Kanwisher, N. (1987). Repetition blindness: Type recognition without token individuation. Cognition, 27, 117-143. Pinker, S. & Prince, A. (1988). On language and connectionism: Analysis of a parallel distributed processing model of language acquisition. Cognition, 28, 73193. Salasoo, A., Shiffrin, R. M., & Feustel, T. C. (1985). Building Permanent Memory Codes: Codification and Repetition Effects in Word Identification. Journal of Experimental Psychology: General, 114, 50-77. Torgerson, W. S. (1958). Theory and Methods of Scaling. J. Wiley & Sons: New York. Townsend, J. T. (1971). Theoritical analysis of an alphabetic confusion matrix. Perception and Psychophysics, 9, 40-50. (see also 449-454). Vall Trees, F. (1968). Detection, Estimation and Modulation Theory, Part 1. New York: Wiley.
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Statistical and Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons John K. Douglass and Frank Moss Department of Biology and Department of Physics University of Missouri at St. Louis St. Louis, MO 63121 Andre Longtin Department of Physics University of Ottawa Ottawa, Ontario, Canada KIN 6N5 Abstract We interpret the time interval data obtained from periodically stimulated sensory neurons in terms of two simple dynamical systems driven by noise with an embedded weak periodic function called the signal: 1) a bistable system defined by two potential wells separated by a barrier, and 2) a FitzHugh-Nagumo system. The implementation is by analog simulation: electronic circuits which mimic the dynamics. For a given signal frequency, our simulators have only two adjustable parameters, the signal and noise intensities. We show that experimental data obtained from the periodically stimulated mechanoreceptor in the crayfish tail fan can be accurately approximated by these simulations. Finally, we discuss stochastic resonance in the two models. 1 INTRODUcnON It is well known that sensory information is transmitted to the brain using a code which must be based on the time intervals between neural firing events or the mean firing rate. However, in any collection of such data, and even when the sensory system is stimulated with a periodic signal, statistical analyses have shown that a significant fraction of the intervals are random, having no coherent relationship to the stimulus. We call this component the ''noise". It is clear 993 994 Douglass, Moss, and Longtin that coherent and incoherent subsets of such data must be separated. Moreover, the noise intensity depends upon the stimulus intensity in a nonlinear manner through, for example, efferent connections in the visual system (Kaplan and Barlow, 1980) and is often much larger (sometimes several orders of magnitude larger!) than can be accounted for by equilibrium statistical mechanics (Denk and Webb, 1992). Evidence that the noise in networks of neurons can dynamically alter the properties of the membrane potential and time constants has also been accumulated (Kaplan and Barlow, 1976; Treutlein and Schulten, 1985; Bemander, Koch and Douglas, 1992). Recently, based on comparisons of interspike interval histograms (ISIH's) obtained from passive analog simulations of simple bistable systems, with those from auditory neurons, it was suggested that the noise intensity may play a critical role in the ability of the living system to sense the stimulus intensity (Longtin, Bulsara and Moss, 1991). In this work, it is shown that in the simulations, ISIH9s are reproduced provided that noise is added to a weak signal, i.e. one that cannot cause firing by itself. All of these processes are essentially nonlinear, and they indicate the ultimate futility of simply measuring the 'background spontaneous rate" and later subtracting it from spike rates obtained with a stimulus applied. Indeed, they raise serious doubts regarding the applicability of any linear transform theory to neural problems. In this paper, we investigate the possibility that the noise can enhance the ability of a sensory neuron to transmit information about periodic stimuli. The present study relies on two objects, the ISIH and the power spectrum, both familiar measurements in electrophysiology. These are obtained from analog simulations of two simple dynamical systems, 1) the overdamped motion of a particle in a bistable, quartic potential; and 2) the FitzHugh-Nagumo model. The results of these simulations are compared with those from experiments on the mechanoreceptor in the tailfan of the crayfish Procambarus clarkii. 2 THE ANALOG SIMULATOR Previously, we made detailed comparisons of ISIH's obtained from a variety of sensory modalities (Longtin, Bulsara and Moss, 1991) with those measured on the bistable system, .:t = x - x3 + ~(t) + f sin(wt) (1) where f is the stimulus intensity, and ~ is a quasi white, Gaussian noise, defined by (~(t)~(s» = (DI r)exp( - It-sll r) with D the noise intensity and r a (dimensionless) noise correlation time. Quasi white means that the actual noise correlation time is at least one order of magnitude smaller than the integrator Statistical & Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons 995 time constant (the "clock" by which the simulator measures time). It was shown that the neurophysiological data could be satisfactorily matched by data from the simulation by adjusting either the noise intensity or the stimulus intensity provided that the other quantity had a value not very different from the height of the potential barrier. Moreover, bistable dynamical systems of the type represented by Eq. (1) (and many others as well) have been frequently used to demonstrate stochastic resonance (SR), an essentially nonlinear process whereby the signal-to-noise ratio (SNR) of a weak signal can be enhanced by the noise. Below we show that SR can be demonstrated in a typical excitable system of the type often used to model sensory neurons. This raises a tantalizing question: can SR be discovered as a naturally occurring phenomenon in living systems? More information can be found in a recent review and workshop proceedings (Moss, 1993; Chialvo and Apkarian, 1993; Longtin, 1993). There is, however, a significant difference between the dynamics represented by Eq. (1) and the more usual neuron models which are excitable systems. A simple example of the latter is the FitzHugh-Nagumo (FN) model, the ISIH's of which have recently been studied (Longtin, 1993). The FN model is an excitable system controlled by a bifurcation parameter. When the voltage variable is perturbed past a certain boundary, a large excursion, identified with a neural firing event, occurs. Thus a detenninistic refractory period is built into the model as the time required for the execution of a single firing event. By contrast, in the bistable system, a firing event is represented by the transition from well A to well B. Before another firing can occur, the system must be reset by a reverse transition from B to A, which is essentially stochastic. The bistable system thus exhibits a statistical distribution of refractory periods. The FN system is not bistable, but, depending on the value of the bifurcation parameter, it can be either periodically firing (oscillating) or residing on a fixed point. The FN model used here is defined by (Longtin, 1993), v = \-(v - 0.5)(1 - v) - w + ~(t), w = v- w - [b + fsin(wt)], (2) (3) where v is the fast variable (action potential) to which the noise ~ is added, W is the recovery variable to which the signal is added, and b is the bifurcation parameter. The range of behaviors is given by: b >0.65, fixed point and b ~ 0.65, oscillating. We operate far into the fixed point regime at b = 0.9, so that bursts of sustained oscillations do not occur. Thus single spikes at more-orless random times but with some coherence with the signal are generated. A schematic diagram of the analog simulator is shown in Fig. 1. The simulator is constructed of standard electronic chips: voltage multipliers (X) and operational 996 Douglass, Moss, and Longtin Output, P(T), P(w) vet) wet) PC Asyst softwo.re v-w Action potential: fast variable v - v(v-O.5)(t-v) w + Sit) S (,:.Lt. _---, + n Noise gen. v(l-v)(f +0.5) + g (t) n Recovery: slow variable w = v-w-b- E sinwt b=0.1 to 1.0 v-w-b Signal gen. Esinwt Fig. 1 Analog simulator of FitzHugh-Nagumo model. The characteristic response times are determined by the integrator time constants as shown. The noise correlation time was Tn = 10-5 S. amplifier summers (+). The fast variable, vCt), was digitized and analyzed for the ISIH and the power spectrum by the PC shown. Note that the noise correlation time, 10-5 s, is equal to the fast variable integrator time constant and is much larger than the slow variable time constant. This noise is, therefore, colored. Analog simulator designs, nonlinear experiments and colored noise have recently been reviewed (Moss and McClintock, 1989). Below we compare data from this simulator with electrophysiological data from the crayfish. Statistical & Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons 997 3 EXPERIMENTS WITH CRAYFISH MECHANORECEPTOR CELLS Single hair mechanoreceptor cells of the crayfish tailfan represent a simple and robust system lacking known efferents. A simple system is necessary, since we are searching for a specific dynamical behavior which might be masked in a more complex physiology. In this system, small motions of the hairs (as small as a few tens of nanometers) are transduced into spike trains which travel up the sensory neuron to the caudal ganglion. These neurons show a range of spontaneous firing rates (internal noise). In this experiment, a neuron with a relatively high internal noise was chosen. Other experiments and more details are described elsewhere (Bulsara, Douglass and Moss, 1993). The preparation consisted of a piece of the tailfan from which the sensory nerve bundle and ganglion were exposed surgically. This appendage was sinusoidally moved through the saline solution by an electromagnetic transducer. Extracellular recordings from an identified hair cell were made using standard methods. The preparations typically persisted in good physiological condition for 8 to 12 hours. An example ISIH is shown in the upper panel of Fig. 2. The stimulus period was, To = 14 ms. Note the peak sequence at the integer multiples of To (Longtin, et ai, 1991). This ISIH was measured in about 15 minutes for which about 8K spikes were obtained. An ISIH obtained from the FN simulator in the same time and including about the same number of spikes is shown in the lower panel. The similarity demonstrates that neurophysiological ISIH's can easily be mimicked with FN models as well as with bistable models. Our model is also able to reproduce non renewal effects (data not shown) which occur at high frequency and! or low stimulus or noise intensity, and for which the first peak in the ISIH is not the one of maximum amplitude. We turn now to the question of whether SR, based on the power spectrum, can be demonstrated in such excitable systems. The power spectrum typically shows a sharp peak due to the signal at frequency wo, riding on a broad noise background. An example, measured on the FN simulator, is shown in the left panel in Fig. 3. This spectrum was obtained for a constant signal intensity set just above threshold and for the stated external noise intensity. The SNR, in decibels, is defined as the ratio of the strength S(w) of the signal feature to the noise amplitude, N(w), measured at the base of the signal feature: SNR = 10 10glOS! N. The panel on the right of Fig. 3 shows the SNR's obtained from a large number of such power spectra, each measured for a different noise intensity. Clearly there is an optimal noise intensity which maximizes the SNR. This is, to our knowledge, the first demonstration of SR based on the power spectra in an excitable system. Just as for the bistable systems (Moss, 1993), when the external noise intensity is too low, the signal is not "sampled" frequently enough and the SNR is low. By contrast, when the noise intensity is too 998 Douglass, Moss, and Longtin Iii 27.0 --..... .::! 24.0 ~ 21.0 ..... :g 18.0 C1J C1 15.0 >:!:! 12.0 ...... ;; 9.00 It! .g 6.00 to. 3.00 16.0 32.0 48.0 64.0 80 .0 96 .0 112. 128 . 144.xEb60. Time Imsl Iii 27.0 --..... .::::! 24.0 ~ 21.0 .... :g 18.0 C1J C1 15.0 >:!:! 12.0 .... E 9.00 It! .g 6.00 to. 3.00 , 16.0 32.0 48.0 64.0 80.0 96.0 112. 128. 144'xEb60. Time Ims) Fig. 2. ISIH's obtained from the crayfish stimulated at 68.6 Hz (upper); and the FN simulator driven at the same frequency with b = 0.9, Vnoise = 0.022 V nns' and Vsig = 0.53 V nns (lower). high, the signal becomes randomized. The occurrence of a maximum in the SNR is thus motivated. SR has also been studied using well residence time probability densities, which are analogous to the physiological ISIH's (Longtin, el ai, 1991), and was further studied in the FN system (Longtin, 1993). In these cases, it is observed that the individual peak heights pass through maxima as the noise intensity is varied, thus demonstrating SR, similar to that shown in Fig. 3, based on the ISIH (or residence time probability density). 4 DISCUSSION We have shown that physiological measurements such as the familiar ISIH patterns obtained from periodically stimulated sensory neurons can be easily mimicked by analog simulations of simple noisy systems, in particular bistable sysStatistical & Dynamica1lnterpretation of ISIH Data from Periodically Stimulated Sensory Neurons 999 >~ 7 33 en • c: ~ ~:D~ c... 2.20 .... ~ 1.47 C1 tn t .733 x o n. .050 .100 .150 .200 .250 .300 .350 .400 .450xEIfiOO Frequency (Kllz) 1200 ,-----------, 1000 ~66t:r:.~~~ ~ ~ ~~ 8.00 ~ ~ ~ ~ 600 : ~ '" 4.00 - ~ 2.00 OO~OO I~~~~~. UO NOISE VOI.TACE (toIV.msl Fig. 3. A power spectrum from the FN simulator stimulated by a 20 Hz signal for b = 0.9, f = 0.25 V and V,.oise = 0.021 V nns (left). The SNR's versus noise voltage measured in the FN system showing SR at V,.olse ~ 10 mV nns (right). Similar SR results based on the ISIH have been obtained by Longtin (1993) and by Chialvo and Apkarian (1993). tems for which the refractory period is strictly stochastic and excitable systems for which the refractory period is deterministic. Further, we have shown that SR, based on SNR's obtained from the power spectrum, can be demonstrated for the FitzHugh-Nagumo model. It is worth emphasizing that these results are possible only because the systems are inherently nonlinear. The signal alone is too weak to cause firing events in either the bistable or the excitable models. Thus these results suggest that biological systems may be able to detect weak stimuli in the presence of background noise which they could not otherwise detect. Careful behavioral studies will be necessary to decide this question, however, a recent and interesting psychophysics experiment using human interpretations of ambiguous figures, presented in sequences with both coherent and random components points directly to this possibility (Chialvo and Apkarian, 1993). Acknowledgements This work was supported by the Office of Naval Research grant NOOOI4-92-J1235 and by NSERC (Canada). References Bemander, 0, Koch, C. and Douglas R. (1992) Network activity determines spatio-temporal integration in single cells, in Advances in Neural Information Processing Systems 3; R. Lippman, J. Moody and D. Touretzky, editors; Morgan Kaufmann, San Mateo, CA. 43-50 1000 Douglass, Moss, and Longtin Bulsara, A., Douglass, J. and Moss, F. (1993) Nonlinear Resonance: Noiseassisted information processing in physical and neurophysiological systems. Nav. Res. Rev. in press. Chialvo, D. and Apkarian, V. (1993) Modulated noisy biological dynamics: three examples; in Proceedings of the NATO ARW on Stochastic Resonance in Physics and Biology, edited by F. Moss, A. Bulsara, and M. F. Shlesinger, 1. Stat. Phys. 70, forthcoming Oenk, W. and Webb, W. (1992) Forward and reverse transduction at the limit of sensitivity studied by correlating electrical and mechanical fluctuations in frog saccular hair cells. Hear. Res. 60, 89-102. Kaplan, E. and Barlow, R. (1976) Energy, quanta and Limulus vision. Vision Res. 16, 745-751 Kaplan, E. and Barlow, R. (1980) Circadian clock in Limulus brain increases response and decreases noise of retinal photoreceptors. Nature 286, 393 Longtin, A. (1993) Stochastic resonance in neuron models, in Proceedings of the NATO ARWon Stochastic Resonance in Physics and Biology, edited by F. Moss, A. Bulsara, and M. F. Shlesinger, J. Stat. Phys. 70, forthcoming Longtin, A, Bulsara, A and Moss F. (1991) Time interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons. Phys. Rev. Lett. 67, 656-659 Moss, F. (1993) Stochastic resonance: from the ice ages to the monkey's ear; in, Some Problems in Statistical Physics, edited by G. H. Weiss, SIAM, Philadelphia, in press Moss, F. and McClintock, P.V.E. editors (1989) Noise in Nonlinear Dynamical Systems, Vols. 1 - 3, Cambridge University Press. Treutlein, H. and Schulten, K. (1985) Noise induced limit cycles of the Bonhoeffer-Van der Pol model of neural pulses. Ber. Bunsenges. Phys. Chern. 89, 710.
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Unsupervised Discrimination of Clustered Data via Optimization of Binary Information Gain Nicol N. Schraudolph Computer Science & Engr. Dept. University of California, San Diego La Jolla, CA 92093-0114 nici@cs.ucsd.edu Terrence J. Sejnowski Computational Neurobiology Laboratory The Salk Institute for Biological Studies San Diego, CA 92186-5800 tsejnowski@ucsd.edu Abstract We present the information-theoretic derivation of a learning algorithm that clusters unlabelled data with linear discriminants. In contrast to methods that try to preserve information about the input patterns, we maximize the information gained from observing the output of robust binary discriminators implemented with sigmoid nodes. We deri ve a local weight adaptation rule via gradient ascent in this objective, demonstrate its dynamics on some simple data sets, relate our approach to previous work and suggest directions in which it may be extended. 1 INTRODUCTION Unsupervised learning algorithms may perform useful preprocessing functions by preserving some aspects of their input while discarding others. This can be quantified as maximization of the information the network's output carries about those aspects of the input that are deemed important. (Linsker, 1988) suggests maximal preservation of information about all aspects of the input. This In/omax principle provides for optimal reconstruction of the input in the face of noise and resource limitations. The I-max algorithm (Becker and Hinton, 1992), by contrast, focusses on coherent aspects of the input, which are extracted by maximizing the mutual information between networks looking at different patches of input. Our work aims at recoding clustered data with adaptive discriminants that selectively emphasize gaps between clusters while collapsing patterns within a cluster onto near499 500 Schraudolph and Sejnowski identical output representations. We achieve this by maximizing in/ormation gain the information gained through observation of the network's outputs under a probabilistic in terpretati on. 2 STRATEGY Consider a node that performs a weighted summation on its inputs i and squashes the resulting net input y through a sigmoid function f : z = f(y), where f(y) = 1 +le_Y and y = tV · i . (1) Such a sigmoid node can be regarded as a "soft" discriminant: with a large enough weight vector, the output will essentially be binary, but smaller weights allow for the expression of varying degrees of confidence in the discrimination. To make this notion more precise, consider y a random variable with bimodal distribution, namely an even mixture of two Gaussian distributions. Then if their means equal ± half their variance, z is the posterior probability for discriminating between the two source distributions (Anderson, 1972). This probabilitstic interpretation of z can be used to design a learning algorithm that seeks such bimodal projections of the input data. In particular, we search for highly informative discriminants by maximizing the information gained about the binary discrimination through observation of z. This binary in/ormation gain is given by dH(z) = H(i) - H(z), (2) where H (z) is the entropy of z under the above interpretation, and i is an estimate of z based on prior knowledge. 3 RESULTS 3.1 THE ALGORITHM In the Appendix, we present the derivation of a learning algorithm that maximizes binary information gain by gradient ascent. The resulting weight update rule is dw 0( f'(y) i (y - fI), (3) where fI, the estimated net input, must meet certain conditions1 (see Appendix). The weight change dictated by (3) is thus proportional to the product of three factors: • the derivative of the Sigmoid squashing function, • the presynaptic input i, and • the difference between actual and anticipated net input. 1 In what follows, we have successfully used estimators that merely approximate these conditions. Unsupervised Discrimination of Clustered Data via Optimization of Binary Information Gain 501 Ay = iy AH(z) l.00-----+------t-------j---------jf-------I----0.50 ==::f;~~~i~~~~$~~~~,~~~~±::= 0.00-___ ._ -0.50 -1.00 -----+------t-------j---------jf-------I-----4.00 -2.00 0.00 2.00 4.00 y Figure I: Phase plot of ll.y against net input y for y = {-3, -2, ... 3}. See text for details. 3.2 SINGLE NODE DYNAMICS For a single, isolated node, we use (y), the average net input over a batch of input patterns, as estimator for y. The behavior of our algorithm in this setting is best understood from a phase plot as shown in Figure I, where the change in net input resulting from a weight change according to (3) is graphed against the net input that causes this weight change. Curves are plotted for seven different values of y. The central curve (y = 0) is identical to that of the straightforward Hebb rule for sigmoid nodes: both positive and negative net inputs are equally amplified until they reach saturation. For non-zero values of y, however, the curves become asymmetric: positive y favor negative changes ll.y and vice versa. For y = (y), it is easy to see that this will have the effect of centering net inputs around zero. The node will therefore converge to a state where its output is one for half of the input patterns, and zero for the other half. Note that this can be achieved by any sufficiently large weight vector, regardless of its direction! However, since simple gradient ascent is both greedy and local in weight space, starting it from small random initial weights is equivalent to a bias towards discriminations that can be made confidently with smaller weight vectors. To illustrate this effect, we have tested a single node running our algorithm on a set of vowel formant frequency data due to (Peterson and Barney, 1952). The most prominent feature of this data is a central gap that separates front from back vowels; however, this feature is near-orthogonal to the principal component of the data and thus escapes detection by standard Hebbian learning rules. Figure 2 shows the initial, intermediate and final phase of this experiment, using a visualization technique suggested by (Munro, 1992). Each plot shows the pre-image of zero net input superimposed on a scatter plot of the data set in input space. The two flanking lines delineate the "active region" where the sigmoid is not saturated, and thus provide an indication of weight vector size. As demonstrated in this figure, our algorithm is capable of proceeding smoothly from a small initial weight vector that responds in principal component direction to a solution which uses a large weight vector in near-orthogonal direction to successfully discriminate between the two data clusters. 502 Schraudolph and Sejnowski 1 i. ; .. ... . .. , ' " . . ' .,. ., .. ;#~~~~}t"" . ,:' .... ~~.~ r·· ... <:/0' Figure 2: Single node discovers distinction between front and back vowels in unlabelled data set of 1514 multi-speaker vowel utterances (Peterson and Barney, 1952). Superimposed on a scatter plot of the data are the pre-images of Y = 0 (solid center line) and Y = ±1,31696 (flanking lines) in input space. Discovered feature is far from principal component direction. 3.3 EXTENSION TO A LAYER OF NODES A learning algorithm for a single sigmoid node has of course only limited utility. When extending it to a layer of such nodes, some form oflateral interaction is needed to ensure that each node makes a different binary discrimination. The common technique of introducing lateral competition for activity or weight changes would achieve this only at the cost of severely distorting the behavior of our algorithm. Fortunately our framework is flexible enough to accommodate lateral differentiation in a less intrusive manner: by picking an estimator that uses the activity of every other node in the layer to make its prediction, we force each node to maximize its information gain with respect to the entire layer. To demonstrate this technique we use the linear second-order estimator Yi = (Yi) + L (Yj - (Yj)) (}ij j#-i (4) to predict the net input Yi of the ith node in the layer, where the (.) operator denotes averaging over a batch of input patterns, and {}ij is the empirical correlation coefficient (5) Figure 3 shows a layer of three such nodes adapting to a mixture of three Gaussian distributions, with each node initially picking a different Gaussian to separate from the other two. After some time, all three discriminants rotate in concert so as to further maximize information gain by splitting the input data evenly. Note that throughout this process, the nodes always remain well-differentiated from each other. For most initial conditions, however, the course of this experiment is that depicted in Figure 4: two nodes discover a more efficient way to discriminate between the three input clusters, to the detriment of the third. The latecomer repeatedly tries to settle into one of the gaps in the data, but this would result in a high degree of predictability. Thus the node with the shortest weight vector and hence most volatile discriminant is weakened further, its weight vector all but eliminated in an effective demonstration of Occam's razor. Unsupervised Discrimination of Clustered Data via Optimization of Binary Information Gain 503 Figure 3: Layer of three nodes adapts to a mixture of three Gaussian distributions. In the final state, each node splits the input data evenly. ; / Figure 4: Most initial conditions, however, lead to a minimal solution involving only two nodes. The weakest node is "crowded out" by Occam's razor, its weight vector reduced to near-zero length. 4 DISCUSSION 4.1 RELATED WORK By maximizing the difference of actual from anticipated response, our algorithm makes binary discriminations that are highly informative with respect to clusters in the input. The weight change in proportion to a difference in acti vity is reminiscent of the covariance rule (Sejnowski, 1977) but generalizes it in two important respects: • it explicitly incorporates a sigmoid nonlinearity, and • fj need not necessarily be the average net input. Both of these are critical improvements: the first allows the node to respond only to inputs in its non-saturated region, and hence to learn local features in projections other than along the principal component direction. The second provides a convenient mechanism for extending the algorithm by incorporating additional information in the estimator. We share the goal of seeking highly informative, bimodal projections of the input with the Bienenstock-Cooper-Munro (BCM) algorithm (Bienenstock et al., 1982; Intrator, 1992). A critical difference, however, is that BCM uses a complex, asymmetric nonlinearity that increases the selectivity of nodes and hence produces a localized, l-of-n recoding of the input, whereas our algorithm makes symmetric, robust and independent binary discriminations. 504 Schraudolph and Sejnowski 4.2 FUTURE DIRECTIONS Since the learning algorithm described here has demonstrated flexibility and efficiency in our initial experiments, we plan to scale it up to address high-dimensional, real-world problems. The algorithm itself is likely to be further extended and improved as its applications grow more demanding. For instance, although the size of the weight vector represents commitment to a discriminant in our framework, it is not explicitly controlled. The dynamics of weight adaptation happen to implement a reasonable bias in this case, but further refinements may be possible. Other priors implicit in our approach such as the preference for splitting the data evenly could be similarly relaxed or modified. Another attractive generalization of this learning rule would be to implement nonlinear discriminants by backpropagating weight derivatives through hidden units. The dynamic stability of our algorithm is a Significant asset for its expansion into an efficient unsupervised multi-layer network. In such a network, linear estimators are no longer sufficient to fully remove redundancy between nodes. In his closely related predictability minimization architecture, (Schmidhuber, 1992) uses backpropagation networks as nonlinear estimators for this purpose with some success. Since the notion of estimator in our framework is completely general, it may combine evidence from multiple, disparate sources. Thus a network running our algorithm can be trained to complement a heterogeneous mix of pattern recognition methods by maximizing information gain relative to an estimator that utilizes all such available sources of information. This flexibility should greatly aid the integration of binary information gain optimization into existing techniques. APPENDIX: MATHEMATICAL DERIVATION We derive a straightforward batch learning algorithm that performs gradient ascent in the binary information gain objective. On-line approximations may be obtained by using exponential traces in place of the batch averages denoted by the (.) operator. CONDITIONS ON THE ESTIMATOR To eliminate the deri vati ve term from ( 11 d) below we require that the estimator i be • unbiased: (i) = (z),and • honest: tz i = tz (i) . The honesty condition ensures that the estimator has access to the estimated variable only on the slow timescale of batch averaging, thus eliminating trivial "solutions" such as i = z. For an unbiased and honest estimator, oi 0 0 (oz) oz = oz (i) = oz (z) = oz = 1. (6) Unsupervised Discrimination of Clustered Data via Optimization of Binary Information Gain 505 BINARY ENTROPY AND ITS DERIVATIVE The enuopy of a binary random variable X as a function of z = Pr( X = 1) is given by H(z) = -zlogz - (1- z)log(l- z); (7) its derivative with respect to z is o oz H(z) = log(l - z) -log z. (8) Since z in our case is produced by the sigmoid function f given in (I), this conveniently simplifies to o -H(z) = -yo oz GRADIENT ASCENT IN INFORMATION GAIN The information dH gained from observing the output z of the discriminator is dH(z) = H(i) - H(z), (9) (10) where z is an estimate of z based on prior knowledge. We maximize dH(z) by batched gradient ascent in weight space: dill ex (o~ dH(Z») (Ila) ( 0: . ~ [H(i) - H(Z»)) ow oz (lIb) ( z (1 - z) :~ [~ ~ . :i H (i) - :Z H (z ) 1 ) (llc) ( z (1 - z) i (Y - ~! . y) ) , (lId) where estimation of the node's output z has been replaced by that of its net input y. Substitution of (6) into (lId) yields the binary information gain optimization rule dill ex (z (1 - z) i(y - y»). (12) • Acknowledgements We would like to thank Steve Nowlan, Peter Dayan and Rich Zemel for stimulating and helpful discussions. This work was supported by the Office of Naval Research and the McDonnell-Pew Center for Cognitive Neuroscience at San Diego. 506 Schraudolph and Sejnowski References Anderson, 1. (1972). Logistic discrimination. Biometrika, 59:19-35. Anderson, 1. and Rosenfeld, E., editors (1988). Neurocomputing: Foundations of Research. MIT Press, Cambridge. Becker, S. and Hinton, G. E. (1992). A self-organizing neural network that discovers surfaces in random-dot stereograms. Nature, 355: 161-163. Bienenstock, E., Cooper, L., and Munro, P. (1982). Theory for the development of neuron selectivity: Orientation specificity and binocular interaction in visual cortex. Journal of Neuroscience, 2. Reprinted in (Anderson and Rosenfeld, 1988). Intrator, N. (1992). Feature extraction using an unsupervised neural network. Neural Computation, 4:98-107. Linsker, R. (1988). Self-organization in a perceptual network. Computer, pages 105-117. Munro, P. W. (1992). Visualizations of 2-d hidden unit space. In International Joint Conference on Neural Networks, volume 3, pages 468-473, Baltimore 1992. IEEE. Peterson, G. E. and Barney, H. L. (1952). Control methods used in a study of the vowels. Journal of the Acoustical Society of America, 24: 175-184. Schmidhuber, 1. (1992). Learning factorial codes by predictability minimization. Neural Computation, 4:863-879. Sejnowski, T. 1. (1977). Storing covariance with nonlinearly interacting neurons. Journal Of Mathematical Biology, 4:303-321.
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On-Line Estimation of the Optimal Value Function: HJB-Estimators James K. Peterson Department of Mathematical Sciences Martin Hall Box 341907 Clemson University Clemson, SC 29634-1907 email: petersonOmath. clemson. edu Abstract In this paper, we discuss on-line estimation strategies that model the optimal value function of a typical optimal control problem. We present a general strategy that uses local corridor solutions obtained via dynamic programming to provide local optimal control sequence training data for a neural architecture model of the optimal value function. ION-LINE ESTIMATORS In this paper, the problems of adaptive control using neural architectures are explored in the setting of general on-line estimators. 'Ve will try to pay close attention to the underlying mathematical structure that arises in the on-line estimation process. The complete effect of a control action Uk at a given time step t/.; is clouded by the fact that the state history depends on the control actions taken after time step tk' So the effect of a control action over all future time must be monitored. Hence, choice of control must inevitably involve knowledge of the future history of the state trajectory. In other words, the optimal control sequence can not be determined until after the fact. Of course, standard optimal control theory supplies an optimal control sequence to this problem for a variety of performance criteria. Roughly, there are two approaches of interest: solving the two-point boundary value 319 320 Peterson problem arising from the solution of Pontryagin 's maximum or minimum principle or solving the Hamilton-J acobi-Bellman (HJB) partial differential equation. However, the computational burdens associated with these schemes may be too high for realtime use. Is it possible to essentially use on-line estimation to build a solution to either of these two classical techniques at a lower cost? In other words, if TJ samples are taken of the system from some initial point under some initial sequence of control actions, can this time series be use to obtain information about the true optimal sequence of controls that should be used in the next TJ time steps? We will focus here on algorithm designs for on-line estimation of the optimal control law that are implement able in a control step time of 20 milliseconds or less. vVe will use local learning methods such as CMAC (Cerebellar Model Articulated Controllers) architectures (Albus, 1 and W. Miller, 7), and estimators for characterizations of the optimal value function via solutions of the Hamilton-Jacobi-Bellman equation, (adaptive critic type methods), (Barto, 2; Werbos, 12). 2 CLASSICAL CONTROL STRATEGIES In order to discuss on-line estimation schemes based on the Hamilton- JacobiBellman equation, we now introduce a common sample problem: where J(x, u, t) Subject to: y'(s) y(t) y(s) u(s) mm J(x, u, t) uEU i t! dist(y(tf), r) + t L(y(s), u(s), s) ds f(y(s), u(s), s), t::; s ::; tf x E y ( s) ~ RN , t::; s ::; t f E U ( s) ~ RM, t::; s ::; t f (1) (2) (3) (4) (5) (6) Here y and u are the state vector and control vector of the system, respectively; U is the space of functions that the control must be chosen from during the minimization process and ( 4) - ( 6) give the initialization and constraint conditions that the state and control must satisfy. The set r represents a target constraint set and dist(y(tf), r) indicates the distance from the final state y(tf) to the constraint set r. The optimal value of this problem for t.he initial state x and time t will be denoted by J(x, t) where J(x, t) minJ(x,u,t). u On-Line Estimation of the Optimal Value Function: HJB-Estimators 321 It is well known that the optimal value function J(x, t) satisfies a generalized partial differential equation known as the Hamilton-J acobi-Bellman (HJB) equation. aJ(x, t) at J(x,t,) . {( ) aJ(x, t) ( )} m~n L x, u, t + ax I x, u, t dist(x, f) In the case that J is indeed differentiable with respect to both the state and time arguments, this equation is interpreted in the usual way. However, there are many problems where the optimal value function is not differentiable, even though it is bounded and continuous. In these cases, the optimal value function J can be interpreted as a viscosity solution of the HJB equation and the partial derivatives of J are replaced by the sub and superdifferentials of J (Crandall, 5). In general, once the HJB equation is solved, the optimal control from state x and time t is then given by the minimum condition . { aJ(x,t) ( )} U E argm~n L(x,u,t)+ ax I x,u,t If the underlying state and time space are discretized using a state mesh of resolution r and a time mesh of resolution s, the HJB equation can be rewritten into the form of the standard Bellman Principle of Optimality (BPO): where X(Xi, u) indicates the new state achieved by using control u over time interval [tj,tj+d from initial state Xi. In practice, this equat.ion is solved by successive iterations of the form: where T denotes the iteration cycle and the process is started by initializing J~~ (Xi, tj) in a suitable manner. Generally, the iterations continue until the values J;tl(Xi,tj) and J;tl(Xi,tj) differ by negligible amounts. This iterative process is usually referred to as dynamic programming (DP). Once this iterative process converges, let Jr~(Xi,tj) = limT->ooJ:~, and consider linl(r,s)->(O,O) Jrs(xi,tj), where (xi, tj) indicates that the discrete grid points depend on the resolution (r, s). In many situations, this limit gives the viscosity solution J(x, t) to the HJB equation. Now consider the problem of finding J(x,O). The Pontrya.gin minimum principle gives first order necessary conditions that the optimal state x and costate p variables must satisfy. Letting fl(x, u, p, t) = L(x, u, t) + pT I(x, u, t) and defining 322 Peterson H(x,p, t) min H(x, u, p, t), u (7) the optimal state and costate then must satisfy the following two-point boundary value problem (TPBVP): '(t) oH(x,p,t) x op , x(O) = x, p'(t) = _ aH~;p,t) p(tj) = 0 (8) and the optimal control is obtained from ( 7) once the optimal state and costate are determined. Note that ( 7) can not necessarily be solved for the control u in terms of x and p, i.e. a feedback law may not be possible. If the TPBVP can not be solved, then we set J(x,O) = 00. In conclusion, in this problem, we are led inevitably to an optimal value function that can be poorly behaved; hence, we can easily imagine that at many (x, t), ~; is not available and hence J will not satisfy the HJB equation in the usual sense. So if we estimate J directly using some form of on-line estimation, how can we hope to back out the control law if ~; is not available? 3 HJB ESTIMATORS A potential on-line estimation technique can be based on approximations of the optimal value function. Since the optimal value function should satisfy the HJB equation, these methods will be grouped under the broad classification HJD estimators. Assume that there is a given initial state Xo with start time O. Consider a local patch, or local corridor, of the state space around the initial state xo, denoted by n(xo). The exact size ofO(xo) will depend on the nature of the state dynamics and the starting state. If O( xo) is then discretized using a coarse grid of resolution r and the time domain is discretized using resolution s, an approximat.e dynamic programming problem can be formulated and solved using the BPa equations. Since the new states obtained via integration of the plant dynamics will in general not land on coarse grid lines, some sort of interpolation must be used to assign the integrated new state value an appropriate coarse grid value. This can be done using the coarse encoding implied by the grid resolution r of O(xo). In addition, multiple grid resolutions may be used with coarse and fine grid approximations interacting with one another as in multigrid schemes (Briggs, 3). The optimal value function so obtained will be denoted by Jr~(Zi,tj) for any discrete grid point Zi E O(xo) and time point t j. This approximate solution also supplies an estimate of the optimal control sequence (u*)£j-l = (u*)'j-l(Zi,tj)' Some papers on approximate dynamic programming are (Peterson, 8; (Sutton, 10; Luus, 6). It is also possible to obtain estimates of the optimal control sequences, states and costates using an 7J step lookahead and the Pontryagin minimum principle. The associated two point boundary value problem is solved and the controls computed via Ui E arg minu H(x;, u, pi, ti) where (x*)ri and (P*)ri are the calculated optimal state and costate sequences respectively. This approach is developed in (Peterson, 9) and implemelltated for On-Line Estimation of the Optimal Value Function: HJB-Estimators 323 vibration suppression in a large space structure, by (Carlson, Rothermel and Lee, 4) For any Zi E n(xo), let (u){j-1 (u)J- 1(Zi' tj) be a control sequence used from initial state Zi and time point tj. Thus Uij is the control used on time interval [tj,tj+1] from start point Zi. Define zl/1 = Z(Zi,Uij,tj), the state obtained by integrating the plant dynamics one time step using control Uij and initial state Zi" Then Ui,j+1 is the control used on time interval [tj+1, tj+2] from start point zl/l and the new state is zl/2 = z(zl/l, Ui,j+l, ij+d; in general, Ui,j+k is the control used on time interval [tj+k, tj+k+1] from start point zl/k and the new state is j+k+1 (j+k t) h jZij = Z Zij ,Ui,j+k, j+k , were Zij = Zi· Let's now assume that optimal control information Uij (we will dispense with the superscript * labeling for expositional cleanness) is available at each of the discrete grid points (Zi, tj) E n(xo). Let <Prs(Zi, tj) denote the value of a neural architecture (CMAC, feedforward, associative etc.) which is trained as follows using this optimal information for 0 ~ k < T} j - 1 (the equation below holds for the converged value of the network's parameters and the actual dependence of the network on those parameters is notationally suppressed): ·+k "+k+l "+k <Prs (zfj ,tj+k) = e<Prs (zfj ,tj+k+d + (~(zfj ,Ui,j+k) (9) where 0 < e, ( ~ 1 and we define a typical reinforcement function ~ by if j ~ k < T} - j - 1 if k = T} - 1 (10) (11) For notational convenience, we will now drop the notational dependence on the time grid points and simply refer to the reinforcement by ~(zf/k, Ui,j+k) Then applying ( 9) repeatedly, for any 0 ~ p ~ '1] - i, p-1 e <Prs (zf/P, tj+p ) + ( E e 3i(zf/k, Ui,j+k) (12) k=O Thus, the function wr .• can be defined by 324 Peterson where the term uif7 will be interpreted as Uj,1}-1. It follows then that since Uij is optimal, Clearly, the function <Prs(Zi, tj) = Wrs(Zi' tj, 1, 1) estimates the optimal value Jrs (Zi, tj) itself. (See, Q-Learning (Watkins, 11». An alternate approach that does not model J indirectly, as is done above, is to train a neural model <Prs(Zi,tj) directly on the data J(Zi,tj) that is computed in each local corridor calculation. In either case, the above observations lead to the following algorithm: Initialization: Here, the iteration count is r = O. For given starting state Xo and local look ahead of 7J time steps, form the local corridor O(xo) and solve the associated approximate BPO equation for Jrs(Zi, tj). Compute the associated optimal control sequences for each (Zi,tj) pair, (u*){j-1 = (u*)1- 1(Zi,tj)' Initialize the neural architecture for the optimal value estimate using cI>~8(Zi' tj) = J r 8 (Zi , t i)' Estilnate of New Optimal Control Sequence: For the next TJ time steps, an estimate must be made of the next optimal control action in time interval [t f7+k, t f7+k+1]' The initial state is any Zi in O( xf7) (xf7 is one such choice) and the initial time is tf7' For the time interval [tf7, t f7+1], if the model <P~8 (Zi, tj) is differentiable, the new control can be estimated by { L(zf7,u,tf7)(tf7+1 -tTl) } Uf7+1 E arg ~in + a:.:. (zf7' tf7) f(zf7,u,t f7 )(t1}+l -tf7) For ease of notation, let Zf7+1 denote the new state obtained using the control Uf7+1 on the interval [tf7' t f7H]' Then choose the next control via Clearly, if Zf7+k denote the new state obtained using the control ttf7+k-1 on the interval [t,/+k, t f7+k+1], the next control is chosen to satisfy E On-Line Estimation of the Optimal Value Function: HJB-Estimators 325 Alternately, if the neural architecture is not differentiable (that is 0:;, is not availa.ble), the new control action can be computed via E Update of the Neural Estimator: The new starting point for the dynamics is now x1/ and there is a new associateclloca.l corridor n( x1/). The neural estimator is then updated using either the HJB or the BPa equations over the local corridor n(x1/). Using the BPa equations, for all Zi E n(x1/) the updates are: where (it )1- 1 indicates the optimal control estimates obtained in the previous algorithm step. Finally, using the HJB equation, for all Zi E n(x1/) the updates are: { L( Zi, u, t1/+1) (t77+1 +1 - t'I+1) } ~~s (Zi, t 77+1+1) + mJn + a:;, (Zi, t77+i) !(Zi,u,t77+i)(t77+i+1 -t77+i) Comparison to BPO optimal control sequence: Now solve the associated approximate BPa equation for each Zi in the local corridor n(x1/) for Jrs(Zi' t77+j). Compute the new approximate optimal control sequences for each (Zi' t77+j) pair, (u* )~~j 1 = (u* )~~j 1 (Zi, t77+i) and compare them to the estimated sequences (it )~~j 1. If the discrepancy is out of tolerance (this is a design decision) initialize the neural architecture for the optimal value estimate using ~~s(Zi,t'1+i) = Jrs(Zi,t 77+i). If the discrepancy is acceptable, terminate the BPa approximation calculations for M future iterations and use the neural architectures alone for on-line estimation. The determination of the stability and convergence properties of anyon-line approximation procedure of this sort is intimately connected with the the optimal value function which solves the generalized HJB equation. We conjecture the following limit converges to a viscosity solution of the HJB equation for the given optimal control problem: J(x, t) Further, there are stability questions and there are interesting issues relating to the use of multiple state resolutions rl and r2 and the corresponding different approximations to J, leading to the use of multigrid like methods on the HJ B equation (see, for example, Briggs, 3). Also note that there is an advantage to using CMAC 326 Peterson architectures for the approximation of the optimal value function J j since J need not be smooth, the CMAC's lack of differentiability wit.h respect to its inputs is not a problem and in fact is a virtue. Acknowledgements We acknowledge the partial support of NASA grant NAG 3-1311 from the Lewis Research Center. References 1. Albus, J. 1975. "A New Approach to Manipulator Control: The Cerebellar Model Articulation Controller (CMAC)." J. Dynamic Systems, Measurement and Control, 220 - 227. 2. Barto, A., R. Sutton, C. Anderson. 1983 "Neuronlike Adaptive Elements That Can Solve Difficult Learning Control Problems." IEEE Trans. Systems, Man Cybernetics, Vol. SMC-13, No.5, September/October, 834 846. 3. Briggs, W. 1987. A Multigrid Tutorial, SIAM, Philadelphia, PA. 4. Carlson, R., C. Lee and K. Rothermel. 1992. "Real Time Neural Control of an Active Structure", Artificial Neural Networks in Engineering 2, 623 - 628. 5. Crandall, M. and P. Lions. 1983. "Viscosity solutions of Hamilton-Jacobi Equations." Trans. American Math. Soc., Vol. 277, No.1, 1 - 42. 6. Luus, R. 1990. " Optimal Control by Dynamic Programming Using Systematic Reduction of Grid Size", Int. J. Control, Vol. 51, No.5, 995 - 1013. 7. Miller, W. 1987. "Sensor-Based Control of Robotic Manipulators Using as General Learning Algorithm." IEEE J. Robot. Automat., Vol RA-3, No.2, 157 - 165 8. Peterson, J. 1992. "Neural Network Approaches to Estimating Directional Cost Information and Path Planning in Analog Valued Obstacle Fields", HEURISTICS: The Journal of Knowledge Engineering, Special Issue on Artificial Neural Networks, Vol. 5, No.2, Summer, 50 - 61. 9. Peterson, J. 1992. "On-Line Estimation of Optimal Control Sequences: Pontryagin Estimators", Artificial Neural Networks in Engineering 2, ed. Dagli et. al., 579 - 584. 10. Sutton, R. 1991. " Planning by Incremental Dynamic Programming", Proceedings of the Ninth International Workshop on Machine Learning, 353 357. 11. Watkins, C. 1989. Learning From Delayed Rewards, Ph. D. Dissertation, King's College. 12. Werbos, P. 1990. "A Menu of Designs for Reinforcement Learning Over Time". In Neural Networks for Control, Ed. Miller, W. R. 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Using Aperiodic Reinforcement for Directed Self-Organization During Development PR Montague P Dayan SJ Nowlan A Pouget TJ Sejnowski CNL, The Salk Institute 10010 North Torrey Pines Rd. La Jolla, CA 92037, USA read~helmholtz.sdsc.edu Abstract We present a local learning rule in which Hebbian learning is conditional on an incorrect prediction of a reinforcement signal. We propose a biological interpretation of such a framework and display its utility through examples in which the reinforcement signal is cast as the delivery of a neuromodulator to its target. Three exam pIes are presented which illustrate how this framework can be applied to the development of the oculomotor system. 1 INTRODUCTION Activity-dependent accounts of the self-organization of the vertebrate brain have relied ubiquitously on correlational (mainly Hebbian) rules to drive synaptic learning. In the brain, a major problem for any such unsupervised rule is that many different kinds of correlations exist at approximately the same time scales and each is effectively noise to the next. For example, relationships within and between the retinae among variables such as color, motion, and topography may mask one another and disrupt their appropriate segregation at the level of the thalamus or cortex. It is known, however, that many of these variables can be segregrated both within and between cortical areas suggesting that certain sets of correlated inputs are somehow separated from the temporal noise of other inputs. Some form of supervised learning appears to be required. Unfortunately, detailed supervision and 969 970 Montague, Dayan, Nowlan, Pouget, and Sejnowski selection in a brain region is not a feasible mechanism for the vertebrate brain. The question thus arises: What kind of biological mechanism or signal could selectively bias synaptic learning toward a particular subset of correlations? One answer lies in the possible role played by diffuse neuromodulatory systems. It is known that multiple diffuse modulatory systems are involved in the selforganization of cortical structures (eg Bear and Singer, 1986) and some of them a ppear to deliver reward and/or salience signals to the cortex and other structures to influence learning in the adult. Recent data (Ljunberg, et al, 1992) suggest that this latter influence is qualitatively similar to that predicted by Sutton and Ba.rto's (1981,1987) classical conditioning theory. These systems innervate large expanses of cortical and subcortical turf through extensive axonal projections that originate in midbrain and basal forebrain nuclei and deliver such compounds as dopamine, serotonin, norepinephrine, and acetylcholine to their targets. The small number of neurons comprising these subcortical nuclei relative to the extent of the territory their axons innervate suggests that the nuclei are reporting scalar signals to their target structures. In this paper, these facts are synthesized into a single framework which relates the development of brain structures and conditioning in adult brains. We postulate a modification to Hebbian accounts of self-organization: Hebbian learning is conditional on a incorrect prediction of future delivered reinforcement from a diffuse neuromodulatory system. This reinforcement signal can be derived both from externally driven contingencies such as proprioception from eye movements as well as from internal pathways leading from cortical areas to subcortical nuclei. The next section presents our framework and proposes a specific model for how predictions about future reinforcement could be made in the vertebrate brain utilizing the firing in a diffuse neuromodulatory system (figure 1). Using this model we illustrate the framework with three examples suggesting how mappings in the oculomotor system may develop. The first example shows how eye movement commands could become appropriately calibrated in the absence of visual experience (figure 3). The second example demonstrates the development of a mapping from a selected visual target to an eye movement which acquires the target. The third example describes how our framework could permit the development and alignment of multimodal maps (visual and auditory) in the superior colliculus. In this example, the transformation of auditory signals from head-centered to eyecentered coordinates results implicitly from the development of the mapping from parietal cortex onto the colliculus. 2 THEORY We consider two classes of reinforcement learning (RL) rule: static and dynamic. 2.1 Static reinforcement learning The simplest learning rule that incorporates a reinforcement signal is: (1) U sing Aperiodic Reinforcement for Directed Self-Organization During Development 971 where, all at times t, Wt is a connection weight, Xt an input measure, Yt an output measure, 1't a reinforcement measure, and ex. is the learning rate. In this case, l' can be driven by either external events in the world or by cortical projections (internal events) and it picks out those correlations between x and Y about which the system learns. Learning is shut down if nothing occurs that is independently judged to be significant, i.e. events for which l' is O. 2.2 Dynamic Reinforcement learning -learning driven by prediction error A more informative way to utilize reinforcement signals is to incorporate some form of prediction. The predictive form of RL, called temporal difference learning (TD, Sutton and Barto, 1981,1987), specifies weight changes according to: (2) where 1't+ 1 is the reward delivered in the next instant in time t + 1. V is called a value function and its value at any time t is an estimate of the future reward. This framework is closely related to dynamiC programming (Barto et aI, 1989) and a body of theory has been built around it. The prediction error [(1"t+l + Vt+ J) - Vt], measures the degree to which the prediction of future reward Vt is higher or lower than the combination of the actual future reward 1't+ 1 and the expectation of reward from time t + 1 onward (Vt+1). To place dynamic RL in a biological context, we start with a simple Hebbian rule but make learning contingent on this prediction error. Learning therefore slows as the predictions about future rewards get better. In contrast with static RL, in a TD account the value of l' per se is not important, only whether the system is able to predict or anticipate the the future value of r. Therefore the weight changes are: ~Wt = ex.xtlJt[(1't+l + Vt+l) - Vtl (3) including a measure of post-synaptic response, 1)t. 3 MAKING PREDICTIONS IN THE BRAIN In our account of RL in the brain, the cortex is the structure tha t makes predictions of future reinforcement. This reinforcement is envisioned as the output of subcortical nuclei which deliver various neuromodulators to the cortex that permit Hebbian learning. Experiments have shown that various of these nuclei, which have access to cortical representations of complex sensory input, are necessary for instrumental and classical conditioning to occur (Ljunberg et ai., 1992). Figure 1 shows one TD scenario in which a pattern of activity in a region of cortex makes a prediction about future expected reinforcement. At time t, the prediction of future reward Vt is viewed as an excitatory drive from the cortex onto one or more subcortical nuclei (pathway B). The high degree of convergence in B ensures that this drive predicts only a scalar output of the nucleus R. Consider a pattern of activity onto layer II which provides excitatory drive to R and concomitantly causes some output, say a movement, at time t + 1. This movement provides a separate source of excitatory drive rt+ 1 to the same nucleus through independent 972 Montague, Dayan, Nowlan, Pouget, and Sejnowski Layer I A Layer II B , External __ C_~I R II-___ D ______ --.. contingencies I I Figure 1: Making predictions about future reinforcement. Layer I is an array of units that projects topographically onto layer II. (A) Weights from I onto II develop according to equation 3 and represent the value function V t. (B) The weights from II onto R are fixed. The prediction of future reward by the weights onto II is a scalar because the highly convergent excitatory drive from II to the reinforcement nucleus (R) effectively sums the input. (C) External events in the world provide independent eXcitatory drive to the reinforcement nucleus. (D) Scalar signal which results from the output firing of R and is broadcast throughout layer II. This activity delivers to layer II the neuromodulator required for Hebbian learning. The output firing of R is controlled by temporal changes in its excitatory input and habituates to constant or slowly varying input. This makes for learning in layer II according to equation 3 (see text). connections conveying information from sensory structures such as stretch receptors (pathway C). Hence, at time t + 1, the excitatory input to R is the sum of the 'immediate reward' Tt+ 1 and the new prediction of future reward Vt+ I. If the reinforcement nucleus is driven primarily by changes in its input over some time window, then the difference between the excitatory drive at time t and t + 1, ie [(Tt+1 + Vt+d - Vt] is what its output reflects. The output is distributed throughout a region of cortex (pathway D) and permits Hebbian weight changes at the individual connections which determine the value function Vt. The example hinges on two assumptions: 1) Hebbian learning in the cortex is contingent upon delivery of the neuromodulator, and 2) the reinforcement nucleus is sensitive to temporal changes in its input and otherwise habituates to constant or slowly varying input. Initially, before the system is capable of predicting future delivery of reinforcement correctly, the arrival of TH 1 causes a large learning signal because the prediction error [(Tt+1 + Vt+1) - Vtl is large. This error drives weight changes at synaptic connections with correlated pre- and postsynaptic elements until the predictions come to a pproximate the actual future delivered reinforcement. Once these predictions become accurate, learning abates. At that point, the system has learned about whatever contingencies are currently controlling reinforcement delivery. For the case in which the delivery of reinforcement is not controlled by any predictable contingencies, Hebbian learning can still occur if the fluctuations of the prediction error have a positive mean. Using Aperiodic Reinforcement for Directed Self~Organization During Development 973 Drive to _ =:::::~~~~~~~~I~~~~~~~~"6. rnotoneuron~ 64x64 L ~ D R Motoneurons 4x4 EYE MUSCLE (up) E Figure 2: Upper layer is a 64 by 64 input array with 3 by 3 center-surround filters at each position which projects topographically onto the middle layer. The middle layer projects randomly to four 4 X 4 motoneuron layers which code for an equilibrium eye position signal, for example, through setting equilibrium muscle tensions in the 4 muscles. Reinforcement signals originate from either eye movement (muscle' stretch') or foveation. The eye is moved according to h = (T - t)g. " = (u - d)g where r,l,u,d are respectively the average activities on the right, left, up, down motoneuron layers and 9 is a fixed gain parameter. hand" are linearly combined to give the eye position. In the presence of multiple statistically independent sources of control of the reinforcement signal (pathways onto R), the system can separately 'learn away' the contingencies for each of these sources. This passage of control of reinforcement delivery can allow the development of connections in a region to be staged. Hence, control of reinforcement can be passed between contingencies without supervision. In this manner, a few nuclei can be used to deliver information globally about many different circumstances. We illustrate this point below with development of a sensorimotor mapping. 4 EXAMPLES 4.1 Learning to calibrate without sensory experience Figure 2 illustra tes the architecture for the next two exam pIes. Briefly, cortical layers drive four 'motor' layers of units which each provide an equilibrium command to one of four extraocular muscles. The mapping from the cortical layers onto these four layers is random and sparse (15%-35% connectivity) and is plastic according to the learning rule described above. Two external events control the delivery of reinforcement: eye movement and foveation of high contrast objects in the visual input. The minimum eye movement necessary to cause a reinforcement is a change of two pixels in any direction (see figure 3). We begin by demonstrating how an unbalanced mapping onto the motoneuron 974 Montague, Dayan, Nowlan, Pouget, and Sejnowski y x Figure 3: Learning to calibrate eye movement commands. This example illustrates how a reinforcement signal could help to organize an appropriate balance in the sensorimotor mapping before visual experience. The dark bounding box represents the 64x64 pixel working area over which an 8x8 fovea can move. A Foveal position during the first 400 cycles of learning. The architecture is as in figure 2, but the weights onto the right/left and up/down pairs are not balanced. Random activity in the layer providing the drive to the motoneurons initially drives the eye to an extreme position at the upper right. From this position, no movement of the eye can occur and thus no reinforcement can be delivered from the proprioceptive feedback causing all the weights to begin to decrease. With time, the weights onto the motoneurons become balanced and the eye moves. B Foveal position after 400 cycles of learning and after increasing the gain 9 to 10 times its initial value. After the weights onto antagonistic muscles become balanced, the net excursions of the eye are small thus requiring an increase in 9 in order to allow the eye to explore its working range. C Size of foveal region relative the working range of the eye. The fovea covered an 8x8 region of the working area of the eye and the learning rate ex was varied from 0.08 to 0.25 without changing the result. layers can be automatically calibrated in the absence of visual experience. Imagine that the weights onto the right/left and up/down pairs are initially unbalanced, as might happen if one or more muscles are weak or the effective drives to each muscle are unequal. Figure 3, which shows the position of the fovea during learning, indicates that the initially unbalanced weights cause the eye to move immediately to an extreme position (figure 3, A). Since the reinforcement is controlled only by eye movement and foveation and neither is occurring in this state, Tt+ 1 is roughly O. This is despite the (randomly generated) activity in the motoneurons continually making predictions that reinforcement from eye-movement should be being delivered. Therefore all the weights begin to decrease, with those mediating the unbalanced condition decreasing the fastest, until balance is achieved (see path A). Once the eye reaches equilibrium, further random noise will cause no mean net eye movement since the mappings onto each of the four motoneuron layers are balanced. The larger amplitude eye movements shown in the center of figure 3 (labeled B) are the result of increasing the gain g (figure 2). Using Aperiodic Reinforcement for Directed Self-Organization During Development 975 Figure 4: Development of foveation map. The map after 2000 learning cycles shows the approximate eye movement vector from stimulation of each position in the visual field. Lengths were normalized to the size of the largest movement. The undisplayed quadrants were qualitatively similar. Note that this scheme does not account for activity or contrast differences in the input and assumes that these have already been normalized. Learning rate = 0.12. Connectivity from the middle layer to the motoneurons was 35% and was randomized. Unlike the previous example, the weights onto the four layers of motoneurons were initially balanced. I II I 7 / / / 1/ I ,/ V"1-i------o 4.2 Learning a foveation map with sensory experience Although reinforcement would be delivered by foveation as well as successful eye-movements, the former would be expected to be a comparatively rare event. Once equilibrium is achieved, however, the reinforcement that comes from eye movements is fully predicted by the prior activity of the motoneurons, and so other contingencies, in this case foveation, grab control of the delivery of reinforcement. The resulting TD signals now provide information about the link between visual input on the top layer of figure 2 and the resulting command, and the system learns how to foveate correctly. Figure 4 shows the motor map that has developed after 2000 learning cycles. In the current example, the weights onto the four layers of motoneurons initially were balancedand the gain g was 10 times larger than before calibration (see figure 3). This learning currently assumes that some cortical area selects the salient targets. 4.3 Learning to align separate mappings In the primate superior colliculus, it is known that cells can respond to multiple modalities including auditory input which defines a head centered coordinate system. Auditory receptive fields shift their position in the colliculus with changing eye position suggesting the existence of a mechanism which maintains the registration between auditory and visual maps (Jay and Sparks, 1984). Our framework suggests a developmental explanation of these findings in terms of an activitydependent self-organizing principle. Consider an intermediate layer, modeling the parietal cortex, which receives signals representing eye position (proprioception), retinal position of a visual target (selected visual input), and head position of an auditory target and which projects onto the superior colliculus. This can be visualized using figure 2 with parietal cortex as the top layer and the ,colliculus as the drive to the motoneurons. As before (figure 2), assume that foveation of a target, whether auditory or visual, delivers reinforcement and that learning in this layer and the colliculus follows equation 3. In a manner analagous to the example in figure 4, those combinations of retinal, eye position, and head centered signals in this parietal layer which predict a foveating eye movement are selected by this learning rule. Hence, as before, the weights from this layer onto the colliculus make predictions about future reinforcement. In figure 4, a foveation map develops which codes for eye movements in absolute coordinates relative to some equilibrium position of the eye. In the current example, such a foveation map would be inappropriate since it requires persistent activity in the collicular layer to maintain a fixed eye position. Instead, the collicular to motoneuron mapping must represent changes in the balance between antagonistic muscles with some other system coding for current eye position. 976 Montague, Dayan, Nowlan, Pouget, and Sejnowski Why would such an initial architecture, acting under the aegis of the learning rule expressed in equation 3, develop the collicular mappings observed in experiments? Those combinations of signals in the parietal layer that correctly predict foveation have their connections onto the collicular layer stabilized. In the current representation, foveation of a target will occur if the correct change in firing between antagonistic motoneurons occurs. After learning slows, the parietal layer is left with cells whose visual and auditory responses are modulated by eye position signals. In the collicular layer, the visual responses of a cell are not modulated by eye position signals while the head-centered auditory responses are modulated by eye position. The reasons for these differences in thecolliculus layer and parietal layer are implicit in the new motoneuron model and the way the equation 3 polices learning. The collicular layer is driven by combinations of the three signals and the learning rule enforces a common frame of reference for these combinations because foveation of the target is the only source of reinforcement. Consider, for example, a visual target on a region of retina for two different eye positions. The change in the balance between right and left muscles required to foveate such a retinal target is the same for each eye position hence the projection from the parietal to collicular layer develops so that the influence of eye position for a fixed retinal target is eliminated. The influence of eye position for an auditory target remains, however, because successful foveation of an auditory target requires different regions of the collicular map to be active as a function of eye position. These examples illustrate how diffuse modulatory systems in the midbrain and basal forebrain can be employed in single framework to guide activity-dependent map development in the vertebrate brain. This framework gives a natural role to such diffuse system for both development and conditioning in the adult brain and illustrates how external contingencies can be incorporated into cortical representations through these crude scalar signals. References [1] Barto, AG, Sutton, RS & Watkins, CJCH (1989). Learning and Sequential Decision Making. Technical Report 89-95, Computer and Information Science, University of Massachusetts, Amherst, MA. [2] Bear, MF & Singer, W (1986). Modulation of visual cortical plasticity by acetylcholine and noradrenaline. Nature, 320, 172-176. [3] Jay, MF & Sparks, DL (1984). Auditory receptive fields in primate superior colliculus shift with changes in eye position. Nature, 309, 345-347. [4] Ljunberg, T, Apicella, P & Schultz, W (1992). Responses of monkey dopamine neurons during learning of behavioral reactions. Journal of Neurophysiology, 67(1), 145-163. [5] Sutton, RS (1988). Learning to predict by the methods of temporal difference. Machine Learning, 3, pp 9-44. [6] Sutton, RS & Barto, AG (1981). Toward a modern theory of adaptive networks: Expectation and prediction. Psychological Review, 882, pp 135-170. [7] Sutton, RS & Barto, AG (1987). A temporal-difference model of classical conditioning. Proceedings of the Ninth Annual Conference of the Cognitive Science Society. Seattle, WA.
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A Connectionist Symbol Manipulator That Discovers the Structure of Context-Free Languages Michael C. Mozer and Sreerupa Das Department of Computer Science & Institute of Cognitive Science University of Colorado Boulder, CO 80309-0430 Abstract We present a neural net architecture that can discover hierarchical and recursive structure in symbol strings. To detect structure at multiple levels, the architecture has the capability of reducing symbols substrings to single symbols, and makes use of an external stack memory. In terms of formal languages, the architecture can learn to parse strings in an LR(O) contextfree grammar. Given training sets of positive and negative exemplars, the architecture has been trained to recognize many different grammars. The architecture has only one layer of modifiable weights, allowing for a straightforward interpretation of its behavior. Many cognitive domains involve complex sequences that contain hierarchical or recursive structure, e.g., music, natural language parsing, event perception. To illustrate, "the spider that ate the hairy fly" is a noun phrase containing the embedded noun phrase "the hairy fly." Understanding such multilevel structures requires forming reduced descriptions (Hinton, 1988) in which a string of symbols or states ("the hairy fly") is reduced to a single symbolic entity (a noun phrase). We present a neural net architecture that learns to encode the structure of symbol strings via such red uction transformations. The difficult problem of extracting multilevel structure from complex, extended sequences has been studied by Mozer (1992), Ring (1993), Rohwer (1990), and Schmidhuber (1992), among others. While these previous efforts have made some 863 864 Mozer and Das input queue d~ft demon units stack pop push push Figure 1: The demon model. progress, no one has claimed victory over the problem. Our approach is based on a new perspective-one of symbolic reduction transformations-which affords a fresh attack on the problem. 1 A BLACKBOARD ARCHITECTURE Our inspiration is a blackboard style architecture that works as follows. The input, a sequence of symbols, is copied onto a blackboard-a scratch pad memory-one symbol at a time. A set of demon, watch over the blackboard, each looking for a specific pattern of symbols. When a demon observes its pattern, it fire" causing the pattern to be replaced by a symbol associated with that demon, which we'll call its identity. This process continues until the entire input string has been read or no demon can fire. The sequence of demon firings and the final blackboard contents specify the structure of the input. The model we present is a simplified version of this blackboard architecture. The blackboard is implemented as a stack. Consequently, the demons have no control over where they write or read a symbol; they simply push and pop symbols from the stack. The other simplification is that the demon firing is based on template matching, rather than a more sophisticated form of pattern matching. The demon model is sketched in Figure 1. An input queue holds the input string to be parsed, which is gradually transferred to the stack. The top k stack symbols are encoded in a set of dack unit&; in the current implementation, k = 2. Each demon is embodied by a special processing unit which receives input from the stack units. The weights of each demon unit specify a pair of symbols, which the demon unit matches against the two stack symbols. If there is a match, the demon unit pops the top two stack symbols and pushes its identity. If no demon unit matches, an additional unit, called the default unit, becomes active. The default unit is responsible for transferring a symbol from the input queue onto the stack. Connectionist Symbol Manipulator Discovers Structure of Context-Free Languages 865 S -+ a b S -+ a X X -+ S b S /\ a X /\ S b /\ a b Figure 2: The rewrite rules defining a grammar that generates strings of the form anbn and a parse tree for the string aabb. 2 PARSING CONTEXT-FREE LANGUAGES Each demon unit reduces a pair of symbols to a single symbol. We can express the operation of a demon as a rewrite rule of the form X --+ a b, where the lower case letters denote symbols in the input string and upper case letters denote the demon identities, also symbols in their own right. The above rule specifies that when the symbols a and b appear on the top of the stack, in that order, the X demon unit should fire, erasing those two symbols and replacing them with an X. Demon units can respond to internal symbols (demon identities) instead of input symbols, allowing internal symbols on the right hand side of the rule. Demon units can also respond to individual input symbols, achieving rules of the form X --+ a. Multiple demon units can have the same identity, leading to rewrite rules of a more general form, e.g., X --+ a b lYe I d Z I a. This class of rewrite rules can express a subset of context-free grammars. Figure 2 shows a sample grammar that generates strings of the form anbn and a parse tree for the input string aabb. The demon model essentially constructs such parse trees via the sequence of reduction operations. That each rule has only one or two symbols on the right hand side imposes no limitation on the class of grammars that can be recognized. However, the demon model does require certain knowledge about the grammars to be identified. First, the maximum number of rewrite rules and the maximum number of rules having the same left-hand side must be specified in advance. This is because the units have to be allocated prior to learning. Second, the LR-class of the grammar must be given. To explain, any context-free grammar can be characterized as LR( n), which indicates that the strings of the grammar can be parsed from left to right with n symbols of look ahead on the input queue. The demon model requires that n be specified in advance. In the present work, we examine only LR(O) grammars, but the architecture can readily be generalized to arbitrary n. Giles et al. (1990), Sun et al. (1990), and Das, Giles, and Sun (1992) have previously explored the learning of context-free grammars in a neural net. Their approach was based on the automaton perspective of a recognizer, where the primary interest was to learn the dynamics of a pushdown automaton. There has also been significant work in context-free grammar inference using symbolic approaches. In general, these approaches require a significant amount of prior information about the grammar and, although theoretically sound, have not proven terribly useful in practice. A promising exception is the recent proposal of Stolcke (1993). . .. 866 Mozer and Das 3 CONTINUOUS DYNAMICS So far, we have described the model in a discrete way: demon firing is all-ornone and mutually exclusive, corresponding to the demon units achieving a unary representation. This may be the desired behavior following learning, but neural net learning algorithms like back propagation require exploration in continuous state and weight spaces and therefore need to allow partial activity of demon units. The continuous activation dynamics follow. Demon unit i computes the distance between its weights, Wi, and the input, x: di.ti = bi IWi - xl 2 , where bi is an adjustable bias associated with the unit. The activity of unit i, denoted .i, is computed via a normalized exponential transform (Bridle, 1990j Rumelhart, in press), e-di,ti ·i = L:i e-didj , which enforces a competition among the units. A special unit, called the default unit, is designed to respond when none of the demons fire strongly. Its activity, .del, is computed like that of any demon unit with di.tdel = bdel' 4 CONTINUOUS STACK Because demon units can be partially active, stack operations need to be performed partially. This can be accomplished with a continuou.s .stack (Giles et al., 1990). Unlike a discrete stack where an item is either present or absent, items can be present to varying degrees. Each item on the stack has an associated thickneu, a scalar in the interval [0,1] indicating what fraction of the item is present (Figure 3). To understand how the thickness plays a role in processing, we digress briefly and explain the encoding of symbols. Both on the stack and in the network, symbols are represented by numerical vectors that have one component per symbol. The vector representation of some symbol X, denoted rx, has value 1 for the component corresponding to X and 0 for all other components. H the symbol has thickness t, the vector representation is trX' Although items on the stack have different thicknesses, the network is presented with compo.site .ymbol.s having thickness 1.0. Composite symbols are formed by combining stack items. For example, in Figure 3, composite symbol 1 is defined as the vector .2rX + .5rz + .3rv. The input to the demon network consists of the top two composite symbols on the stack. The advantages of a continuous stack are twofold. First, it is required for network learningj if a discrete stack were used, a small change in weights could result in a big (discrete) change in the stack. This was the motivation underlying the continuous stack used by Giles et ale Second, the continuous stack is differentiable and hence allows us to back propagate error through the stack during learning. While we have summarized this point in one sentence, the reader must appreciate the fact that it is no small feat! Giles et ale did not consider back propagation through the stack. Each time step, the network performs two operations on the stack: Connectionist Symbol Manipulator Discovers Structure of Context-Free Languages 867 top of stack thickness x .2 composite Z .5 symbol! V .4 composite X .7 symbol 2 y .4 Figure 3: A continuous stack. The symbols indicate the contentsj the height of a stack entry indicates its thickness, also given by the number to the right. The top composite symbol on the stack is a combination of the items forming a total thickness of 1.0j the next composite symbol is a combination of the items making up the next 1.0 units of thickness. Pop. IT a demon unit fires, the top two composite symbols should be popped from the stack (to be replaced by the demon's identity). If no demon unit fires, in which case the default unit becomes active, the stack should remain unchanged. These behaviors, as well as interpolated behaviors, are achieved by multiplying by 6deJ the thickness of any portion of a stack item contributing to the top two composite symbols. Remember that BdeJ is 0 when one or more demon units are strongly active, and is 1 when the default unit is fully active. Push. The symbol written onto the stack is the composite symbol formed by summing the identity vectors of the demon units, weighted by their activities: L:i 8iri, where ri is the vector representing demon i's identity. Included in this summation is the default unit, where rdeJ is defined to be the composite symbol over thickness 'deJ of the input queue. (After a thickness of BdcJ is read from the input queue, it is removed from the queue.) 5 TRAINING METHODOLOGY The system is trained on positive and negative examples of a context-free grammar. Its task is to classify each input string as grammatical or not. Because the grammars can always be written such that the root of the parse tree is the symbol S (e.g., Figure 2), the stack should contain just S upon completion of processing ofa positive example. For a negative example, the stack should contain anything but s. These criteria can be translated into an objective function as follows. If one assumes a Gaussian noise distribution over outputs, the probability that the top of the stack contains the symbol S following presentation of example i is pioot <X e- 1c,-rsI2 , where Ci is the vector representing the top composite symbol on the stackj and the probability that the total thickness of the stack is 1 (i.e., the stack contains exactly one item) is 868 Mozer and Das where n is the total thickness of the stack and ~ is a constant. For a positive example, the objective function should be greatest when there is a high probability of S being on the stack and a high probability of it being the sole item on the stackj for a negative example, the objective function should be greatest when either event has a low probability. We thus obtain a likelihood objective function whose logarithm the learning procedure attempts to maximize: L= IT IT iEpos example iEneg example Training sets were generated by hand, with a preference for shorter strings. Positive examples were generated from the grammarj negative examples were either randomly generated or were formed by perturbing a grammatical string. In most training sets, there were roughly 3-5 times as many negative examples as positive. One might validly be concerned that we introduced some bias in our selection of examples. Ifso, it was not deliberate. In the initial experiments reported below, our goal was primarily to demonstrate that under some conditions, the network could actually induce the grammar. In the next phase of our research, we plan a systematic investigation of the number and nature of examples required for successful learning. The total number of demon units and the (fixed) identity of each was specified in advance of learning. For the grammar in Figure 2, we provided at least two S demons and one X demon. Any number of demons beyond the minimum did not affect performance. The initial weights {Wij} were selected from a uniform distribution over the interval [.45, .55]. The bi were initialized to 1.0. Before an example is presented, the stack is reset to contain only a single symbol, the null symbol with vector representation 0 and infinite thickness. The example string is placed in the input queue. The network is then allowed to run for 21-1 time steps, which is exactly the number of steps required to process any grammatical string of length I. One can intuit this fact by considering that it takes two operations to process each symbol, one to transfer the symbol from the input queue to the stack, and another to reduce the symbol. The derivative of the objective function is computed with respect to the weight parameters using a form of back propagation through time (Rumelhart, Hinton, & Williams, 1986). This involves "unfolding" the architecture in time and back propagating through the stack. Weights are then updated to perform gradient ascent in the log likelihood function. 6 RESULTS AND DISCUSSION We have successfully trained the architecture on a variety of grammars, including those shown in Table 1. In each case, the network discriminates positive and negative examples perfectly on the training set. For the first three grammars, additional (longer) strings were used to test network generalization performance. In each case, generalization performance was 100%. Connectionist Symbol Manipulator Discovers Structure of Context-Free Languages 869 s s x a a x Figure 4: Sample weights for anbn • Weights are organized by demon unit, whose identities appear above the rectangles. The top and bottom halves of the rectangle represents connections from composite symbols 1 and 2, respectively. The darker the shading is of a symbol in a rectangle, the larger the connection strength is from the input unit representing that symbol to the demon unit. The weights clearly indicate the three rewrite rules of the grammar. Table 1: Grammars successfully learned by the demon model I grammar name I rewrite rulel I anbn S--+ablaX X--+Sb parenthesis balancing S --+ (J) 11 X T S S X--+S postfix s--+Yxlsx x--+Y+ls+ Y--+alb pseudo natural language S --+ NP VP NP --+ d NP2 I NP2 NP2 --+ n I an VP --+ v NP Due to the simplicity of the architecture-the fact that there is only one layer of modifiable weights-the learned weights can often be interpreted as symbolic rewrite rules (Figure 4). It is a remarkable achievement that the numerical optimization framework of neural net learning can be used to discover symbolic rules (see also Mozer &. Bachrach, 1991). The first three grammars were successfully learned by the model of Giles et al. (1990), although the analysis required to interpret the weights is generally more cumbersome and tentative. The last grammar could not be learned by their model (Das et al., 1992). When more demon units are provided to the model than are required for the domain, the weights tend to be less interpretable, but generalization performance is just as good. (Of course, this result can hold for only a limited range of network sizes.) The model also does well with very small training sets (e.g., three positive, three negative examples for anbn). This is no doubt because the architecture imposes strong biases on the learning process. We performed some preliminary experiments with staged training in which the length of strings in the training set was increased gradually, allowing the model to first learn simple cases and then move on to more difficult cases. This substantially improved the training time and robustness. 870 Mozer and Das Although the current version of the model is designed for LR(O) context-free grammars, it can be extended to LR(n) by including connections from the first n composite symbols in the input queue to the demon units. However, our focus is not necessarily on building the theoretically most powerful formal language recognizer and learning systemj rather, our primary interest has been on integrating symbol manipulation capabilities into a neural network architecture. In this regard, the model makes a clear contribution. It has the ability represent a string of symbols with a single symbol, and to do so iteratively, allowing for the formation of hierarchical and recursive structures. This is the essence of symbolic information processing, and, in our view, a key ingredient necessary for structure learning. Acknowledgements This research was supported by NSF Presidential Young Investigator award IRI9058450 and grant 90-21 from the James S. McDonnell Foundation. Our thanks to Paul Smolensky, Lee Giles, and J urgen Schmidhuber for helpful comments regarding this work. References Bridle, J. (1990). Training stochastic model recognition algorithms as networks can lead to maximum mutual information estimation of parameters. In D. S. Touretzk,. (Ed.), Adllancu in neural infor. mation procelling .ydem. J (pp. 211-217). San Mateo, CA: Morgan Kaufmann. Das, S., Giles, C. L., &t: Sun, G. Z. (1992). Learning context-free grammars: Capabilities and limitations of neural network with an external stack memorJ. In Proceeding. of the Fourteenth Annual Conference of the Cognitille Science (pp. 791-795). Hillsdale, NJ: Erlbaum. Giles, C. L., Sun, G. Z., Chen, H. H., Lee, Y. C., &t: Chen, D. (1990). Higher order recurrent networks and grammatical inference. In D. S. Tourebk,. (Ed.), Adllancu in neural information procelling .y.tem. J (pp. 380-387). San Mateo, CA: Morgan Kaufmann. Hinton, G. E. (1988). Representing part-whole hierarchies in connectionist networks. Proceeding. of the Eighth Annual Conference of the Cognitille Science Society. Mozer, M. C. (1992). The induction of multiscale temporal structure. In J. E. Mood,., S. J. Hanson, &. R. P. Lippman (Eds.), Adllancu in neural information procelling .y.tem. IV (pp. 275-282). San Mateo, CA: Morgan Kaufmann. Mozer, M. C., &t: Bachrach, J. (1991). SLUG: A connectionist architecture for inferring the structure of finite-state environments. Machine Learning, 7, 139-160. Ring, M. (1993). Learning sequential tasks b,. incrementall,. adding higher orders. Thi. 1I0lume. Rohwer, R. (1990). The 'moving targets' training algorithm. In D. S. Touretzk,. (Ed.), Adllance. in neural informa.tion procelling .y.tem. J (pp. 558-565). San Mateo, CA: Morgan Kaufmann. Rumelhart, D. E., Hinton, G. E., &t: Williams, R. J. (1986). Learning internal representations by error propagation. In D. E. Rumelhart &t: J. L. McClelland (Eds.), Pa.rallel di.tributed procelling: E:z:ploration. in the microdructure of cognition. Volume I: Foundation. (pp. 318-362). Cambridge, MA: MIT Press/Bradford Books. Rumelhart, D. E. (in press). Connectionist processing and learning as statistical inference. In Y. Chauvin &t: D. E. Rumelhart (Ed •. ), Backpropagation: Theory, architecturu, and application •. Hillsdale, NJ: Erlbaum. Schmidhuber, J. (1992). Learning unambiguous reduced sequence descriptions. In J. E. Moody, S. J. Hanson, &t: R. P. Lippman (Eds.), Adllancu in neural information proceuing .y.tem. IV (pp. 291-298). San Mateo, CA: Morgan Kaufmann. Stolcke, A., &t: Omohundro, S. (1993). Hidden markov model induction b,. Ba,.esian model merging. Thi. 1Iolume. Sun, G. Z., Chen, H. H., Giles, C. L., Lee, Y. C., &t: Chen, D. (1990). Connectionist pushdown automata that learn context-Cree grammars. In Proceeding. of the International Joint Conference on Neural Network, (pp. 1-577). Hillsdale, NJ: Erlbaum Associates.
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Analogy--Watershed or Waterloo? Structural alignment and the development of connectionist models of analogy Dedre Gentner Department of Psychology Northwestern University 2029 Sheridan Rd. Evanston, IL 60208 Arthur B. Markman Department of Psychology Northwestern University 2029 Sheridan Rd. Evanston, IL 60208 ABSTRACT Neural network models have been criticized for their inability to make use of compositional representations. In this paper, we describe a series of psychological phenomena that demonstrate the role of structured representations in cognition. These findings suggest that people compare relational representations via a process of structural alignment. This process will have to be captured by any model of cognition, symbolic or subsymbolic. 1.0 INTRODUCTION Pattern recognition is central to cognition. At the perceptual level, we notice key features of the world (like symmetry), recognize objects in front of us and identify the letters on a printed page. At a higher level, we recognize problems we have solved before and determine similarities-including analogical similarities-between new situations and old ones. Neural network models have been successful at capturing sensory pattern recognition (e.g., Sabourin & Mitiche, 1992). In contrast, the determination of higher level similarities has been well modeled by symbolic processes (Falkenhainer, Forbus, & Gentner, 1989). An important question is whether neural networks can be extended to high-level similarity and pattern recognition. In this paper, we will summarize the constraints on cognitive representations suggested by the psychological study of similarity and analogy. We focus on three themes: (1) structural alignment; (2) structural projection; and (3) flexibility. 855 856 Gentner and Markman 2.0 STRUCTURAL ALIGNMENT IN SIMILARITY Extensive psychological research has examined the way people compare pairs of items to determine their similarity. Mounting evidence suggests that the similarity of two complex items depends on the degree of match between their component objects (common and distinctive attributes) and on the degree of match between the relations among the component objects. Specifically, there is evidence that (I) similarity involves structured pattern matching, (2) similarity involves structured pattern completion, (3) comparing the same item with different things can highlight different aspects of the item and (4) even comparisons of a single pair of items may yield multiple interpretations. We will examine these four claims in the following sections. 2.1 SIMILARITY INVOLVES STRUCTURED PATTERN MATCHING The central idea underlying structured pattern matching is that similarity involves an alignment of relational structure. For example, in Figure la, configuration A is clearly more similar to the top configuration than configuration B, because A has similar objects taking part in the same relation (above), while B has similar objects taking part in a different relation (next-to). This determination can be made regardless of whether the objects taking part in the relations are similar. For example, in Figure Ib configuration A is also more similar to the top configuration than is configuration B, because A shares a relation with the top configuration, while B does not. As a check on this intuition, 10 subjects were asked to tell us which configuration (A or B) went best with the top configuration for the triads in Figures la and lb. Al1lO subjects chose configuration A for both triads. This example demonstrates that relations (such as the common above relation) are important in similarity processing. £ A B A B (A) (B) Figure 1. Examples of structural alignment in perception. The importance of relations was also demonstrated by Palmer (1978) who asked subjects to rate the similarity of pairs of configurations like those in Figure 2. The pair in Figure 2a shares the global property that both are open figures, while the pair in Figure 2b does not. As would be expected if Subjects attend to relations when determining similarity, Structural alignment & the development of connectionist models of analogy 857 higher similarity ratings were given to pairs like the one in Figure 2a than to pairs like the one in Figure 2b. This finding can only be explained by appealing to structural similarity, because both pairs of configurations share the same number of local line segments. Consistent with this result, Palmer also found that subjects were faster to say that the items in Figure 2b are different than that the items in Figure 2a are different. A similar result was obtained by Lockhead and King (1977). (A) (B) Figure 2. Structured pattern matching in a study by Palmer (1978). Further research suggests that common bindings between relations and the items they relate are also central to similarity. For example, Clement and Gentner (1991) presented subjects with pairs of analogous stories. One story described organisms called Tams that ate rocks, while the other described robots that collected data on a planet. In each story, one matching fact also had a matching causal antecedent. For example, the Tams' exhausting the minerals on the rock CAUSED them to move to another rock, while the robots' exhausting the data on a planet CAUSED them to move to another planet. A second matching fact did not have a matching causal antecedent. For example, the Tams' underbelly could not function on a new rock and the robots' probe could not function on a new planet, but the causes of these facts did not match. Subjects were asked which of the two pairs of key facts (shown in bold) was more important to the stories. Subjects selected the pair that had the matching causal antecedent, suggesting that their mappings preserved the relational connections in the stories. 2.2 STRUCTURED PATTERN COMPLETION Pattern completion has long been a central feature of neural network models (Anderson, Silverstein, Ritz, & Jones, 1977; Hopfield, 1982). For example, in the BSB model of Anderson et aI., vectors in which some units are below their maximum value are filled in by completing a pattern based on the vector similarities of the current activation pattern to previously learned patterns. The key issue here is the kind of information that guides pattern completion in humans. Data from psychological studies suggests that subjects' pattern matching ability is controlled by structural similarities rather than by geometric similarities. For example, Medin and Goldstone presented subjects with pairs of objects like those in Figure 3 (Medin, Goldstone & Gentner, in press). The left-hand figure in both pairs is somewhat ambiguous, but the right-hand figure is not. Subjects who were asked to list the commonalities of the pair in Figure 3a said that both figures had three prongs, while subjects who were asked to list the commonalities of the pair in Figure 3b said that both figures had four prongs. This finding was obtained for 15 of 16 triads tested, and suggests 858 Gentner and Markman that subjects were mapping the structure from the unambiguous figure onto the ambiguous one. Of course, in order for the mapping to take place, the underlying structure of the figures has to be readily alignable, and there must be ambiguity in the target figure. In the pair in Figure 3c, the left hand item cannot be viewed as having four prongs, and so this mapping is not made. (A) (B) Itb.m ~ (C) Figure 3: Example of structured pattern completion. Structured pattern completion also occurs in conceptual structures. Clement and Gentner (1991) extended the study described above by deleting the key matching facts from one of the stories (e.g., the bold facts from the robot story). Subjects read both stories, then predicted one new fact about the robot story. Subjects were free to predict anything at all, but 50% of the subjects predicted the fact with the matching causal antecedent, while only 28% of the subjects predicted the fact with no matching causal antecedent. By comparison, a control group that made predictions about the target story without seeing the base predicted both facts at the same rate (about 5%). This finding underlines the importance of connectivity in pattern completion. People's predictions were determined not just by the local information, but by whether it was connected to matching information. Thus, pattern completion is structure-sensitive. 2.3 DIFFERENT COMPARISONS-DIFFERENT INTERPRETATIONS Comparison is flexible. When an item takes part in many comparisons, it may be interpreted differently in each comparison. For example, in Figure 3a, the left figure is interpreted as having 3 prongs, while in Figure 3b, it is interpreted as having 4 prongs. Similarly, the comparison 'My surgeon is a butcher' conveys a clumsy surgeon, but 'Genghis Khan was a butcher' conveys a ruthless killer (Glucksberg and Keysar, 1992). This type of flexibility is also evident in an example presented by Spellman and Holyoak (1992). They pointed out that some politicians likened the Gulf War to World War II, implying that the United States was acting as the world's policeman to stop a tyrant. Other politicians compared Operation Desert Storm to Vietnam, implying that the United States entered into a potentially endless conflict between two other nations. Clearly, different comparisons highlighted different features of the Gulf War. Structural alignment & the development of connectionist models of analogy 859 2.4 SAME COMPARISON-DIFFERENT INTERPRETATIONS Even a single comparison can yield more than one distinct interpretation. This situation may arise when the items are richly represented, with many different clusters of knowledge. It can also arise when the comparison permits more than one alignment, as when the similarities of the objects in an item suggest different correspondences than do the relational similarities (i.e. components are cross-mapped (Gentner & Toupin, 1986». Markman and Gentner (in press) presented subjects with pairs of scenes like those depicting the perceptual higher order relation monotonic increase in size shown in Figure 4. In Figure 4, the circle with the arrow over it in the left-hand figure is the largest circle in the array. It is cross-mapped, since it is the same size as the middle circle in the righthand figure, but plays the same relational role as the left (largest) circle. Subjects were given a mapping task in which they were asked to point to the object in the right-hand figure that went with with the cross-mapped circle in the left-hand figure. In this task, subjects chose the circle that looked most similar 91 % of the time. However, a second group of subjects, who rated the similarity of the pair before doing this mapping, selected the object playing the same relational role 61 % of the time. In both tasks, when subjects were asked whether there were any other good choices, they generally described the other possible mapping. These results show that the same comparison can be aligned in different ways, and that similarity comparisons promote structural alignment. ,. ~ " """ .:. ' : ~ A "'-.; .... Figure 4: Stimuli with a cross-mapping from Markman and Gentner (m press). Goldstone (personal communication) has demonstrated that, not only are comparisons flexible, but subjects can attend to attribute and relation matches selectively. He presented subjects with triads like the one in Figure 5. Subjects were asked to choose either the bottom figure with the most attribute similarity to the top one, or the bottom figure with the most relational similarity to the top one. In this study, and other pilot studies, subjects were highly accurate at both task, suggesting flexibility to attend to different kinds of similarity. DD DO 00 A B Figure 5: Sample stimuli from study by Goldstone. 860 Gentner and Markman Similar flexibility can also be found in stimuli with conceptual relations. Gentner (1988) presented children with double metaphors that can have two meanings, one based on attribute similarities and a second based on relational similarities. For example, the metaphor 'Plant stems are like drinking straws' can mean that both are round and skinny, or that both transport fluids from low places to high places. Gentner found that young children (age 5-6) made the attribute-based interpretation, while older children (age 9-10) and adults could make either interpretation (but preferred the relation-based interpretation). There are limits to this flexibility. People prefer to make structurally consistent mappings (Gentner, 1983). For example, Spellman and Holyoak (1992) told subjects to map Operation Desert Storm onto World War II. When they asked subjects to find a correspondence for George Bush given that Saddam Hussein corresponded to Hitler, subjects generally chose either FDR or Churchill. Then, subjects were asked to make a mapping for the United States in 1991. Interestingly, subjects who mapped Bush to FDR almost always mapped the US in 1991 to the US during World War II. In contrast, subjects who mapped Bush to Churchill almost always mapped the US in 1991 to Britain during World War II. Thus, subjects maintained structurally consistent mappings. This type of flexibility adds significant complexity to the comparison process, because a system cannot simply be trained to search for relational correspondences or be taught to prefer only attribute matches. Rather, the comparison process must determine both attribute and relation matches and must be able to keep different mappings distinct from each other. 2.5 SUMMARY OF EMPIRICAL EVIDENCE These findings suggest that comparisons of both perceptual and conceptual materials involve structural alignment. Further, structural alignment promotes structure sensitive pattern completion. Finally, comparisons allow for multiple interpretations of a single item in different comparisons or multiple interpretations of a single comparison. Any model of human cognition that involves comparison must exhibit these properties. 3.0 IMPLICATIONS FOR COGNITIVE MODELS Many of the questions concerning the adequacy of connectionist models and neural networks for high-level cognitive tasks have centered on linguistic processing and the crucial role of compositional relational structures in sentence comprehension (Fodor & McLaughlin, 1990; Fodor & Pylyshyn, 1988). Recent work has addressed this problem by examining ways to represent hierarchical structure in connectionist models, implementing stacks and binary trees to model variable binding and recursive sentence processing (e.g., Elman, 1990; Pollack, 1990; Smolensky, 1990; see also Quinlan, 1991 for a review). It is too soon to tell how successful these methods will be, or whether they can be extended to the general case of structural alignment. The results summarized here underline the need for representations that permit structural alignment. How should this be done? As van Gelder (1990) discusses, symbolic systems traditionally use concatenative representation, in which symbol names are concatenated to build a compositional representation. For example, a circle above a triangle could be Structural alignment & the development of connectionist models of analogy 861 represented by the assertion above(circle,triangle). Such symbolic representations have been used to model the analogy and similarity phenomena described here with some success (Falkenhainer, et al., 1989). Van Gelder (1990) suggests a weaker criterion of Junctional compositionality. In functionally compositional representations, tokens for the symbols are not directly present in the representation, but they can be extracted from the representation via some process. Van Gelder suggests that the natural representation used by neural networks is functionally compositional. Analogously, the question of whether connectionist models can model the phenomena described here should be couched in terms of Junctional alignability: whether the representations can be decomposed and aligned, rather than whether the structure is transparently present. Along this track, an intriguing question is whether the surface form of functionally compositional representations will be similar to the degree that the structures they represent are similar. If so, the alignment process could take place simply by comparing activation vectors. As yet, there are no networks that exhibit this behavior. Further, given the evidence that geometric representations are insufficient to model human similarity comparisons (see Tversky (1977) for a review), we are pessimistic about the prospects that this type of model will be developed. In conclusion, substantial psychological evidence suggests that determining the similarity of two items requires a flexible alignment of structured representations. We suspect that connectionist models of cognitive processes that involve comparisons will have to exhibit concatenative compositionality in order to capture the flexibility inherent in comparisons. However, we leave open the possibility that systems exhibiting functional alignability will be successful. Acknowledgments This research was sponsored by ONR grant BNS-87-20301. We thank Jon Handler, Ed Wisniewski, Phil Wolff and the whole Similarity and Analogy group for comments on this work. We also thank Laura Kotovsky, Catherine Kreiser and Russ Poldrack for running the pilot studies described above. References Anderson, J. A., Silverstein, 1. W., Ritz, S. A., & Jones, R. S. (1977). Distinctive features, categorical perception and probability learning: Some applications of a neural model. Psychological Review, 81,413-451. Clement, C. A., & Gentner, D. (1991). Systematicity as a selection constraint in analogical mapping. Cognitive Science, l5., 89-132. Elman, J.L. (1990). Finding structure in time. Cognitive Science, M(2), 179-212. Falkenhainer, B., Forbus, K. D., & Gentner, D. (1989). The structure-mapping engine: Algorithm and examples. Artificial Intelligence, 41(1), 1-63. Fodor, J., & McLaughlin, B. (1990). Connectionism and the problem of systematicity: Why Smolensky's solution doesn't work. Cognition,.3.5., 183-204. 862 Gentner and Markman Fodor, J. A., & Pylyshyn, Z. W. (1988). Connectionism and cognitive architecture: A critical analysis. Cognition, 28, 3-71. Gentner, D. (1983). Structure mapping: A theoretical framework for analogy. Cognitive Science, 1, 155-170. Gentner, D. (1988). Metaphor as structure mapping: The relational shift. Qllk:l Development,,52,47-59. Gentner, D., & Toupin, C. (1986). Systematicity and surface similarity in the development of analogy. Cognitive Science,.ill, 277-300. Glucksberg, S. & Keysar, B. (1990). Understanding metaphorical comparisons: Beyond similarity. PsychQlogical Review, 21(1), 3-18. Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, 'fl., 25542558. Lockhead, G. R., & King, M. C. (1977). Classifying integral stimuli. Journal of Experimental PsycholQgy: Human Perception and Performance, .3.(3), 436-443. Markman, A. B., & Gentner, D. (in press). Structural alignment during similarity comparisons. Cognitive PsycholQgy. Medin, D. L., Goldstone, R. L., & Gentner, D. (in press). Respects for similarity. Psychological Review. Palmer, S. E. (1978). Structural aspects of visual similarity. MemQry and Cognition, Q(2),91-97. Pollack, J. B. (1990). Recursive distributed representations. Artificial Intelligence, 46(12), 77-106. Quinlan, P.T. (1991). CQnnectionism and PsycholQgy: A psycholQgical perspective on new connectionist research. Chicago: The University of Chicago Press. Sabourin, M. & Mitiche, A. (1992). Optical character recognition by a neural network. Neural Networks, ~(5), 843-852. Smolensky, P. (1990). Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artificial Intelligence,~, 159-216. Spellman, B. A., & Holyoak, K. 1. (1992). If Saddam is Hitler then who is George Bush? Analogical mapping between systems of social roles. Journal of Personality and Social Psychology, Q1(6), 913-933. Tversky, A. (1977). Features of similarity. Psychological Review, 84(4),327-352. van Gelder, T. (1990). Compositionality: A connectionist variation on a classical theme. Cognitive Science, 14(3), 355-384.
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A Recurrent Neural Network for Generation of Ocular Saccades Lina L.E. Massone Department of Physiology Department of Electrical Engineering and Computer Scienc~ Northwestern University 303 E. Chicago Avenue, Chicago, 1160611 Abstract This paper presents a neural network able to control saccadic movements. The input to the network is a specification of a stimulation site on the collicular motor map. The output is the time course of the eye position in the orbit (horizontal and vertical angles). The units in the network exhibit a one-to-one correspondance with neurons in the intermediate layer of the superior colliculus (collicular motor map), in the brainstem and with oculomotor neurons. Simulations carried out with this network demonstrate its ability to reproduce in a straightforward fashion many experimental observations. 1. INTRODUCTION It is known that the superior colliculus (SC) plays an important role in the control of eye movements (Schiller et a1. 1980). Electrophysiological studies (Cynader and Berman 1972, Robinson 1972) showed that the intermediate layer of SC is topographically organized into a motor map. The location of active neurons in this area was found to be related to the oculomotor error (Le. how far the eyes are from the target) and their firing rate to saccade velocity (Roher et al. 1987, Berthoz et al. 1987). Neurons in the rostral area of the motor map, the so-called fixation neurons, tend to become active when the eyes are on target (Munoz and Wurtz 1992) and they can provide a gating mechanism to 1014 A Recurrent Neural Network for Generation of Ocular Saccades 1015 arrest the movement (Guitton 1992). SC sends signals to the brainstem whose circuitry translates them into commands to the oculomotor neurons that innervate the eye muscles (Robinson 1981). This paper presents a recurrent neural network that performs a spatio-temporal transformation from a stimulation site on the collicular motor map and an eye movement. The units in the network correspond to neurons in the intermediate layer of the colliculus, neurons in the brainstem and to oculomotor neurons. Caudal (right) Medial (up) : . : . Lateral (down) Caudal (left) Figure 1: An array of units that represents the collicular motor map. The dark square represents the fixation area. The units in the array project to four units that represent burst cells devoted to process rightward, leftward, upward and downward saccades. The network was built entirely on anatomical and physiological observations. Specifically, the following assumptions were used: (1) The activity on the collicular motor map shifts towards the fixation area during movement (Munoz et aI. 1991, Droulez and Berthoz 1991). (2) The output of the superior colliculus is a vectorial velocity signal 1016 Massone that is the sum of the contributions from each active collicular neuron. (3) Such signal is decomposed into horizontal velocity and vertical velocity by a topographic and graded connectivity pattern from SC to the burst cells in the brainstem. (4) The computation performed from the burst-cells level down to the actual eye movement is carried out according to the push-pull arrangement proposed by Robinson (1981). (5) The activity on the collicular motor map is shifted by signals that represent the eye velocity. Efferent copies of the horizontal and vertical eye velocities are fed back onto the collicular map in order to implement the activity shift. (a) (b) (c) (d) Figure 2: The topographic and graded pattern of connectivity from the collicular array to the four burst cells. Black means no connection, brighter colors represent larger weight values. (a) To the right cell. (b) To the left cell. (c) To the up cell. (d) To the down cell. Simulations conducted with such a system (Massone submitted) demonstrated the network's ability to reproduce a number of experimental observations. Namely the network can: (1) Spontaneously produce oblique saccades whose curvature varies with the ratio between the horizontal and vertical components of the motor error.(2) Automatically hold the eye position in the orbit at the end of a saccade by exploiting the internal dynamic of the network. (3) Continuously produce efferent copies of the movements A Recurrent Neural Network for Generation of Ocular Saccades 1017 without the need for reset signals. (4) Account for the outcome of the lidocaine experiment (Lee et al. 1988) without assuming a population averaging mechanism. Section 2 describes the network architecture. A more detailed description of the network, it mechanisms and physiological ground as well as a number of simulation results can be found in Massone (submitted). 2. THE NETWORK The network input layer is a bidimensional array of linear units that represent neurons in the collicular motor map. The array is topographically arranged as shown in Figure 1. Activity along the caudal axis produces horizontal saccades in a contralateral fashion, activity along the medio-Iateral axis produces vertical saccades, activity in the rest of the array produces oblique saccades. The dark square in the center (rostral area) represents the fixation area. The units in this array project to four logistic units that represent two pairs of burst cells, one pair devoted to control horizontal movements, one pair devoted to control vertical movements. The pattern of connectivity between the collicular array and the units that represent the burst cells is qualitatively shown in Figure 2. The value of the weights of such connections increases exponentially when one moves from the center towards the periphery of the array. The fixation area projects to four other units that represent the so-called omnipause neurons. These units send a gating signal to the burstcells units and are responsible for arresting the movement when the eyes are on target. i.e. when the activity in the input array reaches the center. Each pair of burst-cells units project to the network shown in Figure 3. This network is a computational version of the push-pull arrangement proposed by Robinson (1981). The bottom part of the network represents the oculomotor plant, the top part represents the brainstem circuitry and the oculomotor neurons. The weights in the bottom part of the network were derived by splitting into two equations the differential equation proposed by Robinson (1981) to describe the behavior of the oculomotor plant under a combined motorneuron input R. d91 R1 =k91 +r<l R 1 and R2 are the firing rates of the agonist and antagonist motorneurons, 91 and 92 are the components of the eye position due to motions in opposite directions (e.g. left and right), k is the eye stiffness and r is the eye viscosity. The weights in the top part of the network were analytically computed from the weights in the bottom part of the network by imposing the following constraints: (1) The difference between 91 and 92 must produce the correct 9. (2) The output of the neural integrators must be an efferent copy of the eye movement. (3) The output of the motorneurons must hold the eye at the current orbital position when the burst-cells units are shut off by the gating action of the omnipause cells. Efferent copies of the horizontal and vertical eye velocities were computed by differentiating the output of the neural 1018 Massone from fixation neurons from fixation neurons \1/ from collicular array \1/ a Rl .1t1r .1t/r R2 ,--, l-k.1t1r + 1 -1 a1 a J--------, "":YE_ I Figure 3: The recurrent network used to control eye movements in one direction, e.g. horizontal. An identical network is required to control vertical movements. OPN: omnipause neurons. Bel, BC2: burst cells. NIl, NI2: neural integrators. MNI, MN2: motor neurons. The architecture is based on Robinson'S push-pull arrangement. k=4.0, r=O.95, a=O.5, L\t=l msec. A Recurrent Neural Network for Generation of Ocular Saccades 1019 integrators. These signals were recurrently fed back onto the input array and made the activity in the array shift towards the fixation area. This architecture assumes that the output of the collicular array represents saccade velocity. The network is started by selecting one unit in the input array, i.e. a "stimulation" site. When the unit is selected, a square area centered at that unit becomes active with a gaussian activity profile (Ottes et a1. 1986. Munoz and Guitton 1991). At the time the input units are activated the eye starts moving and, as a consequence of the velocity feedback the activity on the input array starts shifting. The movement is arrested when the fixation area becomes activated. The activity of all units i~l the network represents neurons firing rates and is expressed in spikes/second. Figure 4 shows the response of the network when the collicular array is stimulated at two sites sequentially. Each site causes an oblique saccade with unequal components. Stimulation number 1 brings the eye up and to the right. stimulation number 2 brings the eye back to the initial position. Fixation is maintained for a while inbetween stimulations and at the end of the two movements. The reSUlting trajectories in the movement plane (vertical angle versus horizontal angle) demonstrate the ability of the network to (i) maintain the eye position in the orbit when the burst cells activation is set to zero by the gating action of the omnipause neurons. (ii) produce curved trajectories with opposite curvatures when the eye moves back and forth between the same two angular positions. None of the units in the network is ever reset between saccades; because of the push-pull arrangement, when the activity of one neural integrator increases, the activity of the antagonist integrator decreases. This mechanism ensures that their activity does not grow indefinetely. 3. CONCLUSIONS In this paper I presented an anatomically and physiologically inspired network able to control saccadic movements and to reproduce the outcome of some experimental observations. The results of simulations carried out with this network can be found in Massone (submitted). This work is currently being extended to (i) modeling the activity shift phenomenon as the relaxation of a dynamical system to its equilibrium configuration rather than as a feedback-driven mechanism, (ii) studying the role of the collicular output signals in the calibration and accuracy of arm movements (Massone 1992). Acknowledgements This work was supported by the National Science Foundation, grant BCS-9113455 to the author. References Berthoz A., Grantyn A., Droulez J. (1987) Some collicular neurons code saccadic eye velocity, Neuroscience Letters, 72.289-294. BC_JtP& BC}eft BC_ .. BC_ ..... ~.~. C. E=. ~. [A. L.~. TJwta PhI M c ~~ ........... Figure 4: The response of the network to two sequential stimulations that produce two oblique saccades with unequal components. "'"' o N o ~ I\) VI VI o ::3 ~ A Recurrent Neural Network for Generation of Ocular Saccades 1021 Cynader M., Berman N. (1972) Receptive-field organization of monkey superior colliculus, Journal of Neurophysiology, 35, 187-201. Droulez J., Berthoz A. (1991) The concept of dynamic memory in sensorimotor control, in Motor Control Concepts and Issues, Humphrey D.R. and Freund H.J. Eds., 1. Whiley and Sons, 137-161. GuiUon D. (1992) Control of eye-head coordination during orienting gaze shifts, Trends i1l Neuroscience, 15(5),174-179. Lee C., Roher W.H., Sparks D.L. (1988) Population coding of saccadic eye movements by neurons in the superior colliculus. Nature, 332, 357-360. Massone L. E. (1992) A biologically-inspired architecture for reactive motor control, in Neural Networks for Control, G. Beckey and K. Goldberg Eds., Kluwer Academic Publishers, 1992. Massone L.E. (submitted) A velocity-based model for control of ocular saccades, Neural Computation. Munoz D.P., Pellisson D., Guitton D. (1991) Movement of Neural Activity on the Superior Colliculus Motor Map during Gaze Shifts, Science, 251. 1358-1360. Munoz D.P., Guitton D. (1991) Gaze control by the tecto-reticulo-spinal system in the head-free cat. II. Sustained discharges coding gaze position error, Journal of Neurophysiology, 66, 1624-1641. Munoz D.P., Wurtz R.H. (1992) Role of the rostral superior colliculus in active visual fixation and execution of express saccades, Journal of Neurophysiology, 67, 1000-1002. Ottes F.P., Van Gisbergen J.A.M., Eggermont J.J. (1986) Visuomotor fields of the superior colliculus: a quantitative model, Vision Research, 26, 857-873. Robinson D.A. (1972) Eye movements evoked by collicular stimulation in the alert monkey, Vision Research, 12, 1795-1808. Robinson D.A. (1981) Control of eye movements, in Handbook of Physiology - The Nervous System fl, V.B. Brooks Ed., 1275-1320. Roher W.H., White J.M., Sparks D.L. (1987) Saccade-related burst cells in the superior colliculus: relationship of activity with saccade velocity. Society of Neuroscience Abstracts, 13, 1092.
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Network Structuring And Training Using Rule-based Knowledge Volker Tresp Siemens AG Central Research Otto-Hahn-Ring 6 8000 Munchen 83, Germany Jiirgen Hollatz* Institut fur Informatik TV Munchen ArcisstraBe 21 8000 Munchen 2, Germany Subutai Ahmad Siemens AG Central Research Otto-Hahn-Ring 6 8000 Munchen 83, Germany Abstract We demonstrate in this paper how certain forms of rule-based knowledge can be used to prestructure a neural network of normalized basis functions and give a probabilistic interpretation of the network architecture. We describe several ways to assure that rule-based knowledge is preserved during training and present a method for complexity reduction that tries to minimize the number of rules and the number of conjuncts. After training the refined rules are extracted and analyzed. 1 INTRODUCTION Training a network to model a high dimensional input/output mapping with only a small amount of training data is only possible if the underlying map is of low complexity and the network, therefore, can be oflow complexity as well. With increasing *Mail address: Siemens AG, Central Research, Otto-Hahn-Ring 6, 8000 Miinchen 83. 871 872 Tresp, Hollatz, and Ahmad network complexity, parameter variance increases and. the network prediction becomes less reliable. This predicament can be solved if we manage to incorporate prior knowledge to bias the network as it was done by Roscheisen, Hofmann and Tresp (1992). There, prior knowledge was available. in the form of an algorithm which summarized the engineering knowledge accumulated over many years. Here, we consider the case that prior knowledge is available in the form of a set of rules which specify knowledge about the input/output mapping that the network has to learn. This is a very common occurrence in industrial and medical applications where rules can be either given by experts or where rules can be extracted from the existing solution to the problem. The inclusion of prior knowledge has the additional advantage that if the network is required to extrapolate into regions of the input space where it has not seen any training data, it can rely on this prior knowledge. Furthermore, in many on-line control applications, the network is required to make reasonable predictions right from the beginning. Before it has seen sufficient training data it has to rely primarily on prior knowledge. This situation is also typical for human learning. If we learn a new skill such as driving a car or riding a bicycle, it would be disastrous to start without prior knowledge about the problem. Typically, we are told some basic rules, which we try to follow in the beginning, but which are then refined and altered through experience. The better our initial knowledge about a problem, the faster we can achieve good performance and the less training is required (Towel, Shavlik and Noordewier, 1990). 2 FROM KNOWLEDGE TO NETWORKS We consider a neural network y = N N(x) which makes a prediction about the state of y E ~ given the state of its input x E ~n. We assume that an expert provides information about the same mapping in terms of a set of rules. The premise of a rule specifies the conditions on x under which the conclusion can be applied. This region of the input space is formally described by a basis function bi(x). Instead of allowing only binary values for a basis function (1: premise is valid, 0: premise is not valid), we permit continuous positive values which represent the certainty or weight of a rule given the input. We assume that the conclusion of the rule can be described in form of a mathematical expression, such as conc/usioni: the output is equal to Wi(X) where Wi(X) is a function of the input (or a subset of the input) and can be a constant, a polynomial or even another neural network. Since several rules can be active for a given state of the input, we define the output of the network to be a weighted average of the conclusions of the active rules where the weighting factor is proportional to the activity of the basis function given the input ( ) = NN( ) = Li Wj(x) bj(x) (1) y x x Lj bj(x) . This is a very general concept since we still have complete freedom to specify the form of the basis function bi(x) and the conclusion Wj(x). If bi(x) and Wi(X) are Network Structuring And Training Using Rule-based Knowledge 873 described by neural networks themselves, there is a close relationship with the adaptive mixtures of local experts (Jacobs, Jordan, Nowlan and Hinton, 1991). On the other hand, if we assume that the basis function can be approximated by a multivariate Gaussian 1 ~ (x· - W·)2 bi(x) = Ki exp[-2 LJ u? 'J ], j 'J (2) and if the Wi are constants, we obtain the network of normalized basis functions which were previously described by Moody and Darken (1989) and Specht (1990). In some cases the expert might want to formulate the premise as simple logical expressions. As an example, the rule IF [((Xl:::::: a) AND (X4:::::: b)] OR (X2:::::: c) THEN y = d X x2 is encoded as premzsei : b( ) [1(xl-a)2+(x4- b)2] [1(x2-c)2] i x = exp -2 u2 + exp -2 u 2 conclusioni : Wi(X) = d X x 2 . This formulation is related to the fuzzy logic approach of Tagaki and Sugeno (1992). 3 PRESERVING THE RULE-BASED KNOWLEDGE Equation 1 can be implemented as a network of normalized basis functions N Ninit which describes the rule-based knowledge and which can be used for prediction. Actual training data can be used to improve network performance. We consider four different ways to ensure that the expert knowledge is preserved during training. Forget. We use the data to adapt N Ninit with gradient descent (we typically adapt all parameters in the network). The sooner we stop training, the more of the initial expert knowledge is preserved. Freeze. We freeze the parameters in the initial network and introduce a new basis function whenever prediction and data show a large deviation. In this way the network learns an additive correction to the initial network. Correct. Whereas normal weight decay penalizes the deviation of a parameter from zero, we penalize a parameter if it deviates from its initial value q}nit E 1 ~( init)2 p = 20:"j L- qj - qj j (3) w here the qj is a generic network parameter. Internal teacher. We formulate a penalty in terms of the mapping rather than in terms of the parameters Ep = ~O:" j(NNinit(x) - NN(x»2dx. This has the advantage that we do not have to specify priors on relatively unintuitive network parameters. Instead, the prior directly reflects the certainty that we 874 Tresp, Hollatz, and Ahmad associate with the mapping of the initialized network which can often be estimated. Roscheisen, Hofmann and Tresp (1992) estimated this certainty from problem specific knowledge. We can approximate the integral in Equation 3 numerically by Monte-Carlo integration which leads to a training procedure where we adapt the network with a .~ixture of measured training data and training data artificially generated by JV JV'"&t(x) at randomly chosen inputs. The mixing proportion directly relates to the weight of the penalty, a (Roscheisen, Hofmann and Tresp, 1992). 4 COMPLEXITY REDUCTION After training the rules can be extracted again from the network but we have to ensure that the set of rules is as concise as possible, otherwise the value of the extracted rules is limited. We would like to find the smallest number of rules that can still describe the knowledge sufficiently. Also, the network should be encouraged to find rules with the smallest number of conjuncts, which in this case means that a basis function is only dependent on a small number of input dimensions. We suggest the following pruning strategy for Gaussian basis functions. 1. Prune basis functions. Evaluate the relative weight of each basis function at its center Wi = bi(J-Ldl 2:j bj (J-Li) which is a measure of its importance in the network. Remove the unit with the smallest Wi. Figure 1 illustrates the pruning of basis functions. 2. Prune conjuncts. Successively, set the largest (J'ij equal to infinity, effectively removing input j from basis function i. Sequentially remove basis functions and conjuncts until the error increases above a threshold. Retrain after a unit or a conjunct is removed. 5 A PROBABILISTIC INTERPRETATION One of the advantages of our approach is that there is a probabilistic interpretation of the system. In addition, if the expert formulates his or her knowledge in terms of probability distributions then a number of useful properties can be derived (it is natural here to interpret probability as a subjective degree of belief in an event.). We assume that the system can be in a number of states Si which are unobservable. Formally, each of those hidden states corresponds to a rule. The prior probability that the system is in state Sj is equal to P(Si). Assuming that the system is in state Si there is a probability distribution P(x, ylSi) that we measure an input vector x and an output y and For every rule the expert specifies the probability distributions in the last sum. Let's consider the case that P(x, ylsd = P(XISi) P(ylsd and that P(XISi) and P(yISi) can be approximated by Gaussians. In this case Equation 4 describes a Gaussian mixture model. For every rule, the expert has to specify Network Structuring And Training Using Rule-based Knowledge 875 20 units 5 units Figure 1: 80 values of a noisy sinusoid (A) are presented as training data to a network of20 (Cauchy) basis functions, (bi(x) = Ki [1+ Lj (Xj - P.ij )2/O'fj]-2). (B) shows how this network also tries to approximate the noise in the data. (D) shows the basis functions bi(x) and (F) the normalized basis functions bi(x)/ Lj bj(x). Pruning reduces the network architecture to 5 units placed at the extrema of the sinusoid (basis functions: E, normalized basis functions: G). The network output is shown in (C). y • • . .... ,. • ~ I," _ .. "' .. : .......... -: .. • ,.:,.. • ~ " '1. • • • . --.c '41' • . .,.--. '.' . . • " . . .... . . -....-.---.. .~- .. ~.j :." .... . • -; ..... f# ... ~. • ~! ':!-.~.!.-.-'. "' ... . . . . • x y i i E(xty) , • ! • i .... · i - • . ~. ~. ~.:: ..... ""'. . . . ,: ,. .. . .. • •• " --c : : # • .' ......... . .t '-..... t... • • • • ··-;'-'·f •• . ..... . '.. . ~.. .. .. . .t ':, .# ..--\. z:t .~.. • '.a •• : • ~ i ' • • x Figure 2: Left: The two rectangles indicate centers and standard deviations of two Gaussians that approximate a density. Right: the figure shows the expected values f(Ylx) (continuous line) and f(xly) (dotted line). 876 Tresp, Hollatz, and Ahmad • P (Si), the probability of the occurrence of state Si (the overall weight of the rule), • P(xlsd = Nj(x; J-Lj, Ed, the probability that an input vector x occurs, given that the system is in state Sj, and • P(ylsd = Nl (y; Wi, un, the probability of output y given state Si. The evidence for a state given an input x becomes P(sdx) = P(XISi)P(Si) Lj P(xlsj )P(Sj) and the expected value of the output E(ylx) = Li J y P(ylx, Si) dy P(xlsdP(sj), Lj P(xlsj)P(sj) (5) where, P(xlsj) = J P(x, ylSi) dy. If we substitute bi(x) = P(XISi)P(sd and Wi(X) = J y P(ylx, bi) dy we can calculate the expected value of y using the same architecture as described in Equation 1. Subsequent training data can be employed to improve the model. The likelihood of the data {xk, y"} becomes L = IIEp(xk,ykISd P(sd k i which can be maximized using gradient descent or EM. These adaptation rules are more complicated than supervised learning since according to our model the data generating process also makes assumptions about the distributions of the data in the input space. Equation 4 gives an approximation of the joint probability density of input and output. Input and output are formally equivalent (Figure 2) and, in the case of Gaussian mixtures, we can easily calculate the optimal output given just a subset of inputs (Ahmad and Tresp, 1993). A number of authors used clustering and Gaussian approximation on the input space alone and resources were distributed according to the complexity of the input space. In this method, resources are distributed according to the complexity of both input and output space. 1 6 CLASSIFICATION A conclusion now specifies the correct class. Let {bikli = l. .. Nd denote the set of basis functions whose conclusion specifies c1assk. We set wt = Dkj, where wt is the weight from basis function bij to the kth output and Dkj is the Kronecker symbol. The kth output of the network ( ) _ NN ( ) - Lij wtbij(X) _ Li bik(X) Yk X k X . Lim b,m(X) Lim blm(x) (6) 1 Note, that a probabilistic interpretation is only possible if the integral over a basis function is finite, i.e. all variances are finite. Network Structuring And Training Using Rule-based Knowledge 877 specifies the certainty of classk, given the input. During training, we do not adapt the output weights wt. Therefore, the outputs of the network are always positive and sum to one. A probabilistic interpretation can be found if we assume that P(xlclassk)P(classk) ~ L:i bik(X). We obtain, P( I I) P(xlclassk)P(classk) c assk x = L:, P(xlclass,)P(class,) and recover Equation 6. If the basis functions are Gausssians, again we obtain a Gaussian mixture learning problem and, as a special case (one unit per class), a Gaussian classifier. 7 APPLICATIONS We have validated our approach on a number of applications including a network that learned how to control a bicycle and an application in the legal sciences (Hollatz and Tresp, 1992). Here we present results for a well known data set, the Boston housing data (Breiman et al., 1981), and demonstrate pruning and rule extraction. The task is to predict the housing price in a Boston neighborhood as a function of 13 potentially relevant input features. We started with 20 Gaussian basis functions which were adapted using gradient descent. We achieved a generalization error of 0.074. We then pruned units and conjuncts according to the procedure described in Section 4. We achieved the best generalization error (0.058) using 4 units (this is approximately 10% better than the result reported for CART in Breiman et al., 1981). With only two basis functions and 3 conjuncts, we still achieved reasonable prediction accuracy (generalization error of 0.12; simply predicting the mean results in a generalization error of 0.28). Table 1 describes the final network. Interestingly, our network was left with the input features which CART also considered the most relevant. The network was trained with normalized inputs. If we translate them back into real world values, we obtain the rules: Rule14: IF the number of rooms (RM) is approximately 5.4 (0.62 corresponds to 5.4 rooms which is smaller than the average of 6.3) AND the pupil/teacher value is approximately 20.2 (0.85 corresponds to 20.2 pupils/teacher which is higher than the average of 18.4) THEN the value of the home is approximately $14000 (0.528 corresponds to $14000 which is lower than the average of $22500). Table 1: Network structure after pruning. conclusion feature j CART rating center: J-I.ij width: l1'ij Unit#: i = 14 Wi = 0.528 RM second 0.62 0.21 Ki = 0.17 PIT third 0.85 0.35 Unit#: i = 20 Wi = 1.6 LSTAT most important 0.06 0.24 Ki = 0.83 878 Tresp, Hollatz, and Ahmad Rule2o: IF the percentage of lower-status population (LSTAT) is approximately 2.5% (0.06 corresponds to 2.5% which is lower than the average of 12.65%), THEN the value of the home is approximately $34000 (1.6 corresponds to $34000 which is higher than the average of $22500). 8 CONCLUSION We demonstrated how rule-based knowledge can be incorporated into the structuring and training of a neural network. Training with experimental data allows for rule refinement. Rule extraction provides a quantitative interpretation of what is "going on" in the network, although, in general, it is difficult to define the domain where a given rule "dominates" the network response and along which boundaries the rules partition the input space. Acknowledgements We acknowledge valuable discussions with Ralph N euneier and his support in the Boston housing data application. V.T. was supported in part by a grant from the Bundesminister fiir Forschung und Technologie and J. H. by a fellowship from Siemens AG. References S. Ahmad and V. Tresp. Some solutions to the missing feature problem in vision. This volume, 1993. L. Breiman et al.. Classification and regression trees. Wadsworth and Brooks, 1981. R. A. Jacobs, M. I. Jordan, S. J. Nowlan and G. E. Hinton. Adaptive mixtures of local experts. Neural Computation, Vol. 3, pp. 79-87, 1991. J. Hollatz and V. Tresp. A Rule-based network architecture. Artificial Neural Networks II, I. Aleksander, J. Taylor, eds., Elsevier, Amsterdam, 1992. J. Moody and C. Darken. Fast learning in networks of locally-tuned processing units. Neural Computation, Vol. 1, pp. 281-294, 1989. M. Roscheisen, R. Hofmann and V. Tresp. Neural control for rolling mills: incorporating domain theories to overcome data deficiency. In: Advances in Neural Information Processing Systems 4, 1992. D. F. Specht. Probabilistic neural networks. Neural Networks, Vol. 3, pp. 109-117, 1990. T. Takagi and M. Sugeno. Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics, Vol. 15, No.1, pp. 116-132, 1985. G. G. Towell, J. W. Shavlik and M. O. Noordewier. Refinement of approximately correct domain theories by knowledge-based neural networks. In Proceedings of the Eights National Conference on Artificial Intelligence, pp. 861-866, MA, 1990.
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Hybrid Circuits of Interacting Computer Model and Biological Neurons Sylvie Renaud-LeMassonDepartment of Physics Brandeis University Waltham. MA 02254 Eve Marder Department of Biology Brandeis University Waltham. MA 02254 Abstract Gwendal LeMasson' Department of Biology Brandeis University Waltham. MA 02254 L.F. Abbott Department of Physics Brandeis University Waltham. MA 02254 We demonstrate the use of a digital signal processing board to construct hybrid networks consisting of computer model neurons connected to a biological neural network. This system operates in real time. and the synaptic connections are realistic effective conductances. Therefore. the synapses made from the computer model neuron are integrated correctly by the postsynaptic biological neuron. This method provides us with the ability to add additional. completely known elements to a biological network and study their effect on network activity. Moreover. by changing the parameters of the model neuron. it is possible to assess the role of individual conductances in the activity of the neuron. and in the network in which it participates. "Present address. lXL. Universit6 de Bordeaux l-Enserb. CNRSURA 846. 351 crs de la Liberation. 33405 Talence Cedex. France. 'Present address. LNPC. CNRS. Universite de Bordeaux 1. Place de Dr. Peyneau. 33120 Arcachon. France 813 814 Renaud-LeMasson, LeMasson, Marder, and Abbott 1 INTRODUCTION A primary goal in neuroscience is to understand how the electrical properties of individual neurons contribute to the complex behavior of the networks in which they are found. However, the experimentalist wishing to assess the contribution of a given neuron or synapse in network function is hampered by lack of adeq.uate tools. For example, although pharmacological agents are often used to block synaptic connections within a network (Eisen and Marder, 1982), or individual currents within a neuron (Tierney and Harris-Warrick, 1992), it is rarely possible to do precise pharmacological dissections of network function. Computational models of neurons and networks (Koch and Segev, 1989) allow the investigator the control over parameters not possible with pharmacology. However, because realistic computer models are always based on inadequate biophysical data, the investigator must always be concerned that the simulated system may differ from biological reality in a critical way. We have developed a system that allows us to construct hybrid networks consisting of a model neuron interacting with a biological neural network. This allows us to work with a real biological system while retaining complete control over the parameters of the model neuron. 2 THE MODEL NEURON Biophysical data describing the ionic currents of the Lateral Pyloric (LP) neuron of the crab stomatogastric ganglion (STG) (Golowasch and Marder, 1992) were used to construct an isopotential model LP neuron using MAXIM. MAXIM is a software package that runs on MacIntosh systems and provides a graphical modeling tool for neurons and small neural networks (LeMasson, 1993). The model LP neuron used uses Hodgkin-Huxley type equations and contains a fast Na+ conductance, a Ca+ conductance, a delayed rectifier K+ conductance, a transient outward current (iJ and a hyperpolarization-activated current OJ, as well as a leak conductance. This model is similar to that reported in Buchholtz et al. (1992) but because the raw data were refit using MAXIM, details are slightly different. 3 ARTIFICIAL SYNAPSES Artificial chemical synapses are produced by the same method used in Sharp et at. (1993). An axoclamp in discontinuous current clamp (DCC) mode is used to record the membrane potential and inject current into the biological neurons (Fig. 1). The presynaptic membrane potential is used to control current injection into the postsynaptic neuron simulating a conductance change (rather than an injected current as in Yarom et al.). The synaptic current injected into the postsynaptic neuron depends on the programmed synaptic conductance and an investigator-determined reversal potential. The investigator also specifies the threshold and the function relating "transmitter release" to presynaptic membrane potential, as well as the time course of the synaptic conductance. Hybrid Circuits of Interacting Computer Model and Biological Neurons 815 parameters Mac IIfx DSP data ,...------, V m & Is .... Vl--m ----4 Axoclamp t--~ D. C. C. STG Figure 1: Schematic diagram of the system used to establish hybrid circuits. 4 HARDWARE Our system uses a Digital Signal Processor (DSP) board with built-in AID and DI A 16 bit-precision converters (Spectral Innovations MacDSP256KNI), with DSP32C (AT&T) mounted in a Macintosh II fx (MAC) computer. A block diagram of the system is shown Fig. 1. The parameters describing the membrane and synaptic conductances of the model neuron are stored in the MAC and are transferred to the DSP board RAM (256x32K) through the standard NuBus interface. The DSP translates the parameter files into look-up tables via a polynomial fitting procedure. The differential equations of the LP model and the artificial synapses are integrated by the DSP board, taking advantage of its optimized arithmetic functions and data access. In this system, the computational model runs on the DSP board, and the Mac IIfx functions to store and display data on the screen. The computational speed of this system depends on the integration time step and the complexity of the model (the number of differential equations implemented in the model). For the results shown here, the integration time step was 0.7 msec, and under the conditions described below, 10-15 differential equations were used. The current system is limited to two real neurons and one model neuron because the DSP board has only two input and two output channels. A later generation system with more input and output channels and additional speed will increase the number of neurons and connections that can be created. During anyone time step, the membrane potential of the model neuron is computed, the synaptic currents are determined, and a voltage command is exported to the Axoclamp instructing it to inject the appropriate current into the biological neuron (typically a few nA). During each time step the Axoclamp is used to measure the membrane potential of the biological neurons (typically between -80mV and OmV) used to compute the value of the synaptic inputs to the model neuron. The computed and measured membrane potentials are periodically (every 500 time steps) sent to the computer main memory to be displayed and recorded. To make this system run in real time, it is necessary to maintain perfect timing among all the components. Therefore in every experiment we first determine the minimum time step needed to do the integration depending on the complexity of the model being implemented. For complex models we used the internal clock of the MacII to drive the board. Under some conditions it was preferable to drive the board with an external clock. It is critical that the Axoclamp sampling rate be more than twice the board time step if the two are not synchronized. In our experiments, the Axoclamp switching 816 Renaud-LeMasson, LeMasson, Marder, and Abbott A B C: increaseofglh Model LP ~ real synapses • ..,~ artificial synapses 0.5 sec ISOmV Figure 2: Hybrid network consisting of a model LP neuron connected to a PD neuron of a biological stomatogastric ganglion. A: Simplified connectivity diagram of the pyloric circuit of the stomatogastric ganglion. The AB/PD group consists of one AB neuron electrically coupled to two PD neurons. All chemical synapses are inhibitory. B: Simultaneous intracellular recordings from two biological neurons (PD and LP) and a plot of the membrane potential of the model LP neuron connected to the circuit. The parameters of the synaptic connections and the model LP neuron were adjusted so that the model LP neuron fired in the same time in the pyloric cycle as the biological LP neuron. C: Same recording configuration as Bt but maximal conductance of ih in the model neuron was increased. Hybrid Circuits of Interacting Computer Model and Biological Neurons 817 A B py Model LP ~ real synapses ~ artificial synapses 0.5 sec 110 mV 150 mV 110 mV Figure 3: Hybrid network in which the model LP neuron is connected to. two diffe~ent biological neurons. A: Connectivity diagram showing the pattern of synaptic connections shown in part B. B: Simultaneous recordings from the biological AB neuron, the model LP neuron, and a biological PY neuron. circuit was running about three times faster than the board time step. However, if experimental conditions force a slower Axoclamp sampling rate, then it will be important to synchronize the Axoclamp clock with the board. S RESULTS The STG of the lobster, Panulirus interruptus contains one LP neuron, two Pyloric Dilator (PO) neurons, one Anterior Burster (AB), and eight Pyloric (PY) neurons (Eisen and Marder, 1982; Harris-Warrick et al., 1992). The connectivity among these neurons is known, and is shown in Figure 2A. The PD and LP neurons fire in alternation, because of the reciprocal inhibitory connections between them. Figure 2B shows a model LP neuron connected with reciprocal inhibitory synapses to a biological PD neuron. The parameters controlling the threshold, activation curve, time course, and reversal potential of the model neuron were adjusted until the model neuron fired at the same time within the rhythmic pyloric cycle as the biological LP neuron (Fig. 2B). Once these parameters were set, it was then possible to ask what effect changing the membrane properties of the model neuron had on emergent network activity. Figure 2C shows the result of increasing the maximal conductance of one of the currents in the model LP neuron, it,. Note that increasing this current 818 Renaud-LeMasson, LeMasson, Marder, and Abbott increased the number of LP action potentials per burst. The increased activity in the LP neuron delayed the onset of the next burst in the PO neurons because of the inhibitory synapse between the model LP neuron and the biological PO neuron, and the cycle period therefore also increased. Another effect seen in this example, is that the increased conductance of ~ in the LP neuron delayed the onset of the model LP neuron's firing relative to that of the biological LP neuron. In the experiment shown in Figure 3 we created reciprocal inhibitory connections between the model LP neuron and two biological neurons, the AB and a PY (Fig. 3A). (The action potentials in the AB neuron are higbly attenuated by the cable properties of this neuron). This example shows clearly the unitary inhibitory postsynaptic potentials (IPSPs) in the biological neurons resulting from the model LP's action potentials. During each burst of LP action potentials the IPSPs in the AB neuron increase considerably in amplitude, although the AB neuron's membrane potential is moving towards the reversal potential of the IPSPs. This occurs presumably because the conductance of the AB neuron is higher right at the end of its burst, and decreases as it hyperpolarizes. The same burst of LP action potentials evokes IPSPs in the PY neuron that increase in amplitude, here presumably because the PY neuron is depolarizing and increasing the driving force on the artificial chemical synapse. These recordings demonstrate that although the same function is controlling the synaptic -release- properties in the model LP neuron, the actual change in membrane potential evoked by action potentials in the LP neuron is affected by the total conductance of the biological neurons. 6 CONCLUSIONS The ability to connect a realistic model neuron to a biological network offers a unique opportunity to study the effects of individual currents on network activity. It also provides realistic, two-way interactions between biological and computer-based networks. As well as providing an ·important new tool for neuroscience, this represents an exciting new direction in biologically-based computing. 7 ACKNOWLEDGMENTS We thank Ms. Joan McCarthy for help with manuscript preparation. Research supported by MH 46742, the Human Science Frontier Program, and NSF DMS9208206. 8 REFERENCES Buchholtz, F., Golowasch, J., Epstein, I.R., and Marder, E. (1992) Mathematical model of an identified stomatogastric ganglion neuron. 1. Neurophysiology 67:332340. Eisen, J.S., and Marder, E. (1982) Mechanisms underlying pattern generation in lobster stomatogastric ganglion as determined by selective inactivation of identified neurons. III. Synaptic connections of electrically coupled pyloric neurons. 1. Neurophysiology 48: 1392-1415. Golowasch, J. and Marder, E. (1992) Ionic currents of the lateral pyloric neuron of Hybrid Circuits of Interacting Computer Model and Biological Neurons 819 the stomatogastric ganglion of the crab. J. Neurophysiology 67:318-331. Harris-Warrick. R.M .• Marder, E .• Selverston, A.I.. and Maurice, M .• eds. (1992) Dynamic Biological Networks. Cambridge. MA: MIT Press. Koch. C .• and Segev. I.. eds. (1989) Methods in Neuronal Modeling. Cambridge, MA: MIT press. LeMasson, G. (1993) Maxim: A software system for simulating single neurons and neural networks, in preparat~on. Sharp, A.A .• O'Neil, M.B., Abbott, L.F. and Marder. E. (1993) The dynamic clamp: Computer-generated conductances in real neurons. J. Neurophysiology, in press. Yarom, Y. (1992) Rhythmogenesis in a hybrid system interconnecting an olivary neuron to an network of coupled oscillators. Neuroscience 44:263-275.
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A Knowledge-Based Model of Geometry Learning Geoffrey Towell Siemens Corporate Research 755 College Road East Princeton, NJ 08540 towe ll@ learning. siemens. com Abstract Richard Lehrer Educational Psychology University of Wisconsin 1025 West Johnson St. Madison, WI 53706 lehrer@vms.macc. wisc.edu We propose a model of the development of geometric reasoning in children that explicitly involves learning. The model uses a neural network that is initialized with an understanding of geometry similar to that of second-grade children. Through the presentation of a series of examples, the model is shown to develop an understanding of geometry similar to that of fifth-grade children who were trained using similar materials. 1 Introduction One of the principal problems in instructing children is to develop sequences of examples that help children acquire useful concepts. In this endeavor it is often useful to have a model of how children learn the material, for a good model can guide an instructor towards particularly effective examples. In short, good models of learning help a teacher maximize the utility of the example presented. The particular problem with which we are concerned is learning about conventional concepts in geometry, like those involved in identifying, and recognizing similarities and differences among, shapes. This is a difficult subject to teach because children (and adults) have a complex set of informal rules for geometry (that are often at odds with conventional rules). Hence, instruction must supplant this informal geometry with a common formalism. To be efficient in their instruction, teachers need a model of geometric learning which, at the very least: 1. can represent children's understanding of geometry prior to instruction, 2. can describe how understanding changes as a result of instruction, 3. can predict the effect of differing instructional sequences. In this paper we describe a neural network based model that has these properties. 887 888 Towell and Lehrer An extant model of geometry learning, the "van Hiele model" [6] represents children's understanding as purely perceptual -- appearances dominate reasoning. However, our research suggests that children's reasoning is better characterized as a mix of perception and rules. Moreover, unlike the model we propose, the van Hiele model can neither be used to test the effectiveness of instruction prior to trying that instruction on children nor can it be used to describe how understanding changes as a result of a specific type of instruction. Briefly, our model uses a set of rules derived from interviews with first and second grade children [1, 2], to produce a stereotypical informal conception of geometry. These rules, described in more detail in Section 2.1, give our model an explicit representation of pre-instructional geometry understanding. The rules are then translated into a neural network using the KBANN algorithm [3]. As a neural network, our model can test the effect of differing instructional sequences by simply training two instances with different sets of examples. The experiments in Section 3 take advantage of this ability of our model; they show that it is able to accurately model the effect of two different sets of instruction. 2 ANew Model This section describes the initial state of our model and its implementation as a neural network. The initial state of the model is intended to reproduce the decision processes of a typical child prior to instruction. The methodology used to derive this information and a brief description of this information both are in the first subsection. In addition, this subsection contains a small experiment that shows the accuracy of the initial state of the model. In the next subsection, we briefly describe the translation of those rules into a neural network. 2.1 The initial state of the model Our model is based upon interviews with children in first and second grade [1, 2]. In these interviews, children were presented with sets of three figures such as the triad in Figure 1. They were asked which pair of the three figures is the most similar and why they made their decision. These interviews revealed that, prior to instruction, children base judgments of similarity upon the seven attributes in Table 1. For the triad discrimination task, children find ways in which a pair is similar that is not shared by the other two pairs. For instance, Band C in Figure 1.2 are both pointy but A is not. As a result, the modal response of children prior to instruction is that {B C} is the most similar pair. This decision making process is described by the rules in Table 2. In addition to the rules in Table 2, we include in our initial model a set of rules that describe templates for standard geometric shapes. This addition is based upon interviews with children which suggest that they know the names of shapes such as triangles and squares, and that they associate with each name a small set of templates. Initially, children treat these shape names as having no more importance than any of the attributes in Table 1. So, our model initial treats shape names exactly as one of those attributes. Over time children learn that the names of shapes are very important because they are diagnostic (the name indicates properties). Our hope was that the model would make a similar transition so that the shape names would become sufficient for similarity determination. Note that the rules in Table 2 do not always yield a unique decision. Rather, there are A Knowledge-Based Model of Geometry Learning 889 Table 1: Attributes used by children prior to instruction. Attribute name Possible values Attribute name Possible values Tilt 0, 10,20,30,40 Slant yes, no Area small, medium, large Shape skinny, medium, fat Pointy yes, no Direction +-,-,j,l 2 long & short yes, no Table 2: Rules for similarity judgment in the triad discrimination task. 1. IF fig-val(figl?, att?) = fig-val(fig2?, att?) THEN same-att-value(figl?, fig2?, att?). 2. IF not (same-att-value(figl?, fig3?, att?)) AND figl? * fig3? AND fig2? * fig3? THEN unq-sim(figl?, fig2?, att?). 3. IF c(unq-sim(figl?, fig2?, att?)) > c(unq-sim(figl?, fig2?, att?)) AND c(unq-sim(figl?, fig3?, att?)) > c(unq-sim(fig2?, fig3?, att?)) AND figl? * fig3? AND fig2?* fig3? THEN 1 2 3 4 5 most-similar(figl?, fig2?). A D D 0 Labels followed by a '?' indicate variables. fig-val(fig?, att?) returns the value of att? in fig? cO counts the number of instances. B C A 0 <> 6 D A ~ 7 c==-0 b 8 ~ C7 D ~ 9 0 ~ ~ t. ~ Figure 1: Triads used to test learning. B C D ~ ~ ~ C> ~ /\ ~ triads for which these rules cannot decide which pair is most similar. This is not often the case for a particular child, who usually finds one attribute more salient than another. Yet, frequently when the rules cannot uniquely identify the most similar pair, a classroom of children is equally divided. Hence, the model may not accurately predict an individual response, but is it usually correct at identifying the modal responses. To verify the accuracy of the initial state of our model, we used the set of nine testing triads shown in Figure 1 which were developed for the interviews with children. As shown in Table 3, the model matches very nicely responses obtained from a separate sample of 48 second grade children. Thus, we believe that we have a valid point from which to start. 2.2 The translation of rule sets into a neural network We translate rules sets into neural networks using the KBANN algorithm [3] which uses a set of hierarchically-structured rules in the form of propositional Horn clauses to set the topology and initial weights of an artificial neural network. Because the rules in Table 2 are 890 Towell and Lehrer Table 3: Initial responses by the model. Triad Number 1 2 3 4 5 6 7 8 9 Initial Model BC BC AC AC BC ABIBC AC ADIBC ACIBC Second Grade Children BC BC AC AC BC ABIBC AC AD ACIBC Answers in the "initial model" row indicate the responses generated by the initial rules. More than response in a column indicates that the rules could not differentiate among two pairs. Answers in the "second grade" row are the modal responses of second grade children. More than one answer in a column indicates that equal numbers of children judged the pairs most similar. Table 4: Properties used to describe figures. Property name values Property name values Convex Yes No # Pairs Equal Opposite Angles 01234 # Sides 34568 # Pairs Opposite Sides Equal 01234 # Angles 34568 # Pairs Parallel Sides 01234 All Sides Equal Yes No Adj acent Angles = 180 Yes No # Right Angles 01234 # Lines of Symmetry 01234568 All Angles Equal Yes No # Equal Sides 0234568 # Equal Angles 0234568 not in propositional form, they must be expanded before they can be accepted by KBANN. The expansion turns a simple set of three rules into an ugly set of approximately 100 rules. Figure 2 is a high-level view of the structure of the neural network that results from the rules. In this implementation we present all three figures at the same time and all decisions are made in parallel. Hence, the rules described above must be repeated at least three times. In the neural network that results from the rule translation, these repeated rules are not independent. Rather they are linked so that modifications of the rules are shared across every pairing. Thus, the network cannot learn a rule which applies only to one pair. Finally, the model begins with the set of 13 properties listed in Table 4 in addition to the attributes of Table 1. (Note that we use "attribute" to refer to the informal, visual features in Table 1 and "property" to refer to the symbolic features in Table 4.) As a result, each figure is described to the model as a 74 position vector (18 positions encode the attributes; the remaining 56 positions encode the properties). 3 An Experiment Using the Model One of the points we made in the introduction is that a useful model of geometry learning should be able to predict the effect of instruction. The experiment reported in this section tests this facet of our model. Briefly, this experiment trains two instances of our model using different sets of data. We then compare the instances to children who have been trained using a set of problems similar to one of those used to train the model. Our results show that the two instances learn quite different things. Moreover, the instance trained witn material similar to the children predicts the children's responses on test problems with a high level of accuracy. A Knowledge-Based Model of Geometry Learning 891 Unique Same ? AB Boxes indicate one or more units. Dashed boxes indicate units associated with duplicated rules Dashed lines indicate one or more negatively weighted links. Solid lines indicate one or more positively weighted links. rll"Bci~''''''~ i most similar § .... . ....... .. .. .. .. ... .. .. Figure 2: The structure of the neural network for our model. 3.1 Training the model For this experiment, we developed two sets of training shapes. One set contains every polygon in a fifth-grade math textbook [4] (Figure 3). The other set consists of 81 items which might be produced by a child using a modified version of LOGO (Figure 4). Here we assume that one of the effects of learning geometry with a tool like LOGO is simply to increase the extent and range of possible examples. A collection of 33 triads were selected from each set to train the model. 1 Training consisted of repeated presentations of each of the 33 triads until the network correctly identified the most similar pair for each triad. 3.2 Tests of the model In this section, we test the ability of the model to accurately predict the effects of instruction. We do this by comparing the two trained instances of the model to the modal responses of fifth graders who had used LOGO for two weeks. In those two weeks, the children had generates many (but not all) of the figures in Figure 4. Hence, we expected that the instance 1 In choosing the same number of triads for each training set, we are being very generous to the textbook. In reality, not only do children see more figures when using LOGO, they are also able to make many more contrasts between figures. Hence, it might be more accurate to make the LOGO training set much larger than the textbook training set. 892 Towell and Lehrer Figure 3: Representative textbook shapes. Figure 4: Representative shapes encountered using a modified version of LOGo. of the model trained using triads drawn from Figure 4 would better predict the responses of these children than the other instance of the model. Clearly, the results in Table 5 verify our expectations. The LOGo-trained model agrees with the modal responses of children on an average of six examples while the textbook-trained model agrees on an average of three examples. The respective binomial probabilities of six and three matches is 0.024 and 0.249. These probabilities suggest that the match between the LOGo-trained model and the children is unlikely to have occurred by chance. On the other hand, the instance of the model trained by the textbook examples has the most probable outcome from simply random guessing. Thus, we conclude that the LOGo-trained model is a good predictor of children's learning when using LOGO. In addition, whereas the textbook-trained model was no better than chance at estimating the conventional response, the LOGo-trained model matched convention on an average of seven triads. Interestingly, on both triads where the LOGo-trained model did not match convention, it could not due to lack of appropriate information. For triad 3, convention matches the trapezoid with the parallelogram rather than either of these with the quadrilateral because the trapezoid and the parallelogram both have some pairs of parallel lines. The model, however, has only information about the number of pairs of parallel lines. On the basis of this feature, the three figures are equally dissimilar. For triad 7, the other triad for which the LOGo-trained model did not match convention, the conventional paring matches two obtuse triangles. However, the model has no information about angles other than number and number of right angles. Hence, it could not possibly get this triad correct (at least not for the right reason). We expect that correcting these minor weaknesses will improve the model's ability to make the conventional response. Table 5: Responses after learning by trained instances of the model and children. Triad Number 1 2 3 4 5 6 7 8 9 Textbook Trained ABIBC BC AC AC BC AB AC AB AC LOGO Trained AB/BC AB ?? BC AB AB AB AB BC Fifth Grade Children ABIBC AB AC/AB BC AB ABIBC AC ABIBC BC Convention AB AB AB BC AB AB BC AB BC Responses by the model are the modal responses over 500 trials. ?? indicates that the model was unable to select among the pairings. The success of our model in the prediction experiment lead us to investigate the reasons unoerlying the answers generated by its two instances. In so doing we hoped to gain an understanding of the networks' reasoning processes. Such an understanding would A Knowledge-Based Model of Geometry Learning 893 be invaluable in the design of instruction for it would allow the selection of examples that fill specific learning deficits. Unfortunately, trained neural networks are often nearly impossible to comprehend. However, using tools such as those described by Towell and Shavlik [5], we believe that we developed a reasonably clear understanding of the effects of each set of training examples. The LOGo-trained model made comprehensive adjustments of its initial conditions. Of the eight attributes, it attends to only size and 2 long & short after training. \Vhile learning to ignore most of the attributes, the model also learned to pay attention to several of the properties. In particular, number of angles, number of sides, all angles, equal, all sides equal, and number of pairs of opposite sides parallel, all were important to the network after training. Thus, the LOGo-trained instance of the model made a significant transition in its basis for geometric reasoning. Sadly, in making this transition, the declarative clarity of the initial rules was lost. Hence, it is impossible to precisely state the rules that the trained model used to make its final decisions. By contrast, the textbook-trained instance of the model failed to learn that most of the attributes were unimportant. Instead, the model simply learned that several of the properties were also important. As a result, reasons for answers on the test set often seemed schizophrenic. For instance, in responding Be on test triad 2, the network attributed the decision to similarities in: area, pointiness, point-direction, number of sides, number of angles, number of right angle and all angles equal. Given this combination, it is not surprising that the example is answered incorrectly. This result suggests that typical textbooks may accentuate the importance of conventional properties, but they provide little grist for abandoning the mill of informal attributes. 3.3 Discussion This experiment demonstrated the utility of our model in several ways. First, it showed that the model is sensitive to differences in training set. Of itself, this is neither a surprising nor interesting conclusion. What is important about the difference in learning is that the model trained in a manner similar to a classroom of fifth grade children made responses to the test set that we quite similar to those of fifth grade children. In addition to making different responses to the test set, the two trained instances of the model appeared to learn different things. In particular, the LOGO-trained instance essentially replaced its initial knowledge with something much more like the formal geometry. On the other hand, the textbook-trained instance simply added several concepts from formal geometry to the informal concept with which it was initialized. An improved transition from informal to formal geometry is one of the advantages claimed for LOGO based instruction [2]. Hence, the difference between the two instances of the model agrees with observation of children. This result suggests that our model is able to predict the effect of differing instructional sequences. A further experiment of this hypothesis would be to use our model to design a set of instruction materials. This could be done by starting with an apparently good set of materials, training the model, examining its deficiencies and revising the training materials appropriately. Our hypothesis is that a set of materials so constructed would be superior to the materials normally used in classrooms. Testing of this hypothesis is one of our major directions for future research. 894 Towell and Lehrer 4 Conclusions In this paper we have described a model of the initial stages of geometry learning by elementary school children. This model is initialized using a set of rules based upon interviews with first and second grade children. This set of rules is shown to accurately predict the responses of second grade children on a hard set of similarity determination problems. Given that we have a valid starting point for our model, we test it by training those rules, after re-representing them in a neural network, with two different sets of training materials. Each instance of the model is analyzed in two ways. First, they are compared, on an independent set of testing examples, to fifth grade children who had been trained using materials similar to one of the model's training sets. This comparison showed that the model trained with materials similar to the children accurately reproduced the responses of the children. The second analysis involved examining the model after training to determine what it had learned. Both instances of the model learned to attend to the properties that were not mentioned in the initial rules. The model trained with the richer (LOGo-based) training set also learned that the informal attributes were relatively unimportant. Conversely, the model trained with the textbook-based training examples merely added information about properties to the pre-existing information. Therefore, we believe that the model we have described is has the potential to become a valuable tool for teachers. References [1] R. Lehrer, W. Knight, M. Love, and L. Sancilio. Software to link action and description in pre-proof geometry. Presented at the Annual Meeting of the American Educational Research Association, 1989. [2] R. Lehrer, L. Randle, and L. Sancilio. Learning preproof geometry with LOGO. Cognition and Instruction, 6:159--184, 1989. [3] M. O. Noordewier, G. G. Towell, and J. W. Shavlik. Training knowledge-based neural networks to recognize genes in DNA sequences. In Advances in Neural Infonnation Processing Systems, volume 3, pages 530--536, Denver, CO, 1991. Morgan Kaufmann. [4] M. A. Sobel, editor. Mathematics. McGraw-Hill, New York, 1987. [5] G. G. Towell and J. W. Shavlik. Interpretation of artificial neural networks: Mapping knowledgebased neural networks into rules. In Advances in Neural Infonnation Processing Systems, volume 4, pages 977--984, Denver, CO, 1991. Morgan Kaufmann. [6] P. M. van HieJe. Structure and Insight. Academic Press, New York, 1986.
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Integration of Visual and Somatosensory Information for Preshaping Hand in Grasping Movements Yoji Uno ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan Naohiro Fukumura* Faculty of Engineering University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan Mitsuo Kawato Ryoji Suzuki Faculty of Engineering University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan Abstract The primate brain must solve two important problems in grasping movements. The first problem concerns the recognition of grasped objects: specifically, how does the brain integrate visual and motor information on a grasped object? The second problem concerns hand shape planning: specifically, how does the brain design the hand configuration suited to the shape of the object and the manipulation task? A neural network model that solves these problems has been developed. The operations of the network are divided into a learning phase and an optimization phase. In the learning phase, internal representations, which depend on the grasped objects and the task, are acquired by integrating visual and somatosensory information. In the optimization phase, the most suitable hand shape for grasping an object is determined by using a relaxation computation of the network. * Present Address: Parallel Distributed Processing Research Dept., Sony Corporation, 6-7-35 Kitashinagawa, Shinagawa-ku, Tokyo 141, Japan 311 312 Uno, Fukumura, Suzuki, and Kawato 1 INTRODUCTION It has previously been established that, while reaching out to grasp an object, the human hand preshapes according to the shape of the object and the planned manipulation (Jeannerod, 1984; Arbib et al., 1985). The preshaping of the human hand suggests that prior to grasping an object the 3-dimensional form of the object is recognized and the most suitable hand configuration is preset depending on the manipulation task. It is supposed that the human recognizes objects using not only visual information but also somatosensory information when the hand grasps them. Visual information is made from the 2-dimensional image in the visual system of the brain. Somatosensory information is closely related to motor information, because it depends on the prehensile hand shape (Le., finger configuration). We hypothesize that an internal representation of a grasped object is formed in the brain by integrating visual and somatosensory information. Some physiological studies support our hypothesis. For example, Taira et al. (1990) found that the activity of hand-movement-related neurons in the posterior parietal association cortex were highly selective to the shape and/or the orientation of manipulated switches. How can the neural network integrate different kinds of information? Merely uniting visual image with somatosensory information does not lead to any interesting representation. Our basic idea is that information compression is applied to integrating different kinds of information. It is useful to extract the essential information by compressing the visual and somatosensory information. lrie & Kawato (1991) pointed out that multi-layered perceptrons have the ability to extract features from the input signals by compressing the information from input signals. Katayama & Kawato (1990) proposed a learning schema in which an internal representation of the grasped object was acquired using information compression. Developing the schema of Katayama et al., we have devised a neural network model for recognizing objects and planning hand shapes (e.g., Fukumura et al. 1991). This neural network consists of five layers of neurons with only forward connections as shown in Figure 1. The input layer (1 st layer) and the output layer (5th layer) of the network have the same structure. There are fewer neurons in the 3rd layer than in the 1st and 5th layers. The operations of the network are divided into the learning phase, which is discussed in section 2 and the optimization phase, which is discussed in section 3. 2 INTEGRATION OF VISUAL AND SOMATOSENSORY INFORMATION USING NETWORK LEARNING In the learning phase, the neural network learns the relation between the visual information (Le.,visual image) and the somatosensory information which, in this paper, is regarded as information on the prehensile hand configuration (Le., finger configuration). Both vector x representing the visual image of an object and vector y representing the prehensile hand configuration to grasp it are fed into the 1 st layer (the input layer). The synaptic weights of the network are repeatedly adjusted so that the 5th layer outputs the same vectors x and y as are fed into the 1st layer. In other words, the network comes to realize the identity map between the 1st layer and the 5th layer through a learning process. The most important point of the neural network model is that the number of neurons in the 3rd layer is smaller than the number of neurons in the 1st layer (which is equal to the number of neurons in the 5th layer). Therefore, the information from x and y is compressed between Visual & Somatosensory Information for Preshaping Hand in Grasping Movements 313 y (hand) Figure 1: A neural network model for integrating visual image x and prehensile hand configuration y. The internal representation z of a grasped object is acquired in the third layer. the 1st layer and the 3rd layer, and restored between the 3rd layer and the 5th layer. Once the network learning process is complete, visual image x and prehensile hand configuration yare integrated in the network. Consequently, the internal representation z of the grasped object, which should include enough information to reproduce x and y, is formed in the 3rd layer. Prehensile hand configuration in grasping movements were measured and the learning of the network was simulated by a computer. In behavioral experiments, three kinds of wooden objects were prepared: five circular cylinders whose diameters were 3 cm, 4 cm, 5 cm, 6 cm and 7 cm; four quadrangular prisms whose side lengths were 3 cm, 4 cm, 5 cm and 6 cm; and three spheres whose diameters were 3 cm, 4 cm and 5 cm. Data input to the network was comprised of visual image x and prehensile hand configuration y. Visual images of objects are formed through complicated processes in the visual system of the brain. For simplicity, however, projections of objects onto a side plane and/or a bottom plane were used instead of real visual images. The area of each pixel of the ,e-rojected image was fed into the network as an element of visual image x. A DataGloveT (V P L) was used to measure finger configurations in grasping movements. We attached sixteen optical fibers, whose outputs were roughly inversely proportional to finger joint-angles, to the DataGlove. The subject was instructed to grasp the objects on the table tightly with the palm and all the fingers. The subject grasped twelve objects thirty times each, which produced 360 prehensile patterns for use as training data for network learning. In the computer simulation, six neurons were set in the 3rd layer. The baCk-propagation learning method was applied in order to modify the synaptic weights in the network. Figure 2 shows the activity of neurons in the 3rd layer after the learning had sufficiently been performed. Some interesting features of the internal representations were found in Figure 2. The first is that the level of neuron activity in the 3rd layer increased monotonically as the size of the object increased. The second is that, except for the magnitude, the neuron activation patterns for the same kinds of objects were almost the same. Furthermore, the activation patterns were similar for circular cylinders and quadrangular prisms, but were quite different for spheres. In other words, similar representations were acquired for similarly shaped objects. We concluded that the internal representations were formed in the 3rd 314 Uno, Fukumura, Suzuki, and Kawato Neuron activity Neuron activity 1.00 p, Q' \' 0 ~ \ 0--, Diameter ,../ ~ \ \, --3cm 1.00 £? 0 1 \ I \ P 10 tJ \ \ 1 '\ \ '0/ b/\ \ YI -o--4cm \6 Q, b -Q-Scm 0.00 / I P, V\' -{)-6cm o. d ,( \ '-O '7cm o.J /\ \0 ~\~~ 1~ j\"-o ~ , \ J ' Neuron activity Side Length -3cm -Q'-4cm -D-Scm 1.00 -D-6cm 0.00 Diameter ..... -4cm -i:rScm -/:r6cm -I.OO-l--~1 ~2 ~3 ----;=4 -5 ::"":;:"'6 ~ Neuron index of the 3rd layer -1.00 123456 Neuron index of the 3rd layer -1.00-l--~1 -2......-13~4~5~6~ Neuron index of the 3rd layer a) Circular cylinder b) Quadrangular prism c) Sphere Figure 2: Internal representations of grasped objects. Graph a) shows the neuron activation patterns for five circular cylinders whose diameters were 3 cm, 4 cm, 5 cm, 6 cm and 7 cm. Graph b) shows the neuron activation patterns for four quadrangular prisms whose side lengths were 3 cm, 4 cm, 5 cm and 6 cm. Finally, Graph c) shows the neuron activation patterns for three spheres whose diameters were 3 cm, 4 cm and 5 cm. The abscissa represents the index of the six neurons in the 3rd layer, while the ordinate represents their activity. These values were normalized from -1 to + 1. layer and changed topologically according to the shapes and sizes of the grasped objects. 3 DESIGN OF PREHENSILE HAND SHAPES The neural network that has completed the learning can design hand shapes to grasp any objects in the optimization phase. Determining prehensile hand shape (i.e., finger configuration) is an ill-posed problem, because there are many ways to grasp any given object. In other words, prehensile hand configuration cannot be determined uniquely for anyone object. In order to solve this indeterminacy, a criterion, a measure of performance for any possible prehensile configuration is introduced. The criterion should normally be defined based on the dynamics of the human hand and the manipulation task. However, for simplicity, the criterion is defined based only on the static configuration of the fingers, which is represented by vector y. We assumed that the central nervous system adopts a stable hand configuration to grasp an object, which corresponds to flexing the fingers as much as possible. The output of the DataGlove sensor decreases as finger flexion increases. Therefore, the criterion C l (y) is defined as follows: (1) where Yi represents the ith output of the sixteen DataGlove sensors. Minimizing the criterion Cl (y) requires as much finger flexing as possible. Finding values of Yi (i = 1,2, ... , 16) so as to minimize Cl (y) is an optimization problem Visual & Somatosensory Information for Preshaping Hand in Grasping Movements 315 with constraints. In the optimization phase, the neural network can solve this optimization problem using a relaxation computation as follows. When an object is specified, the visual image x* of the object is input to the 1 st layer as an input signal and given to the 5th layer as a reference signal. We call neurons in the 1st and the 5th layers which represent visual image x image neurons, and call neurons in the 1st and the 5th layers which represent finger configuration y hand neurons. Let us define the following energy function of the network. 1 ~ * I 2 1 ~ I 2 1 ~_2 E(y) = 2 ~(Xi Xi) + 2 ~(Yj - Yj) + A' 2: ~ Yj' i j j (2) Here, xi is the ith element of the image x* which is fed into the ith image neuron in the 1 st layer, and x~ is the output of the ith image neuron in the 5th layer. Yj is the activity of the jth hand neuron in the 1st layer, and yj is the output of the jth hand neuron in the 5th layer. A is a positive regularization parameter which decreases gradually during the relaxation computation. The first term and the second term of equation (2) require that the network realizes the identity map between the input layer and the output layer as well as in the learning phase. This requirement guarantees that a hand whose configuration is specified by vector y can grasp an object whose visual image is x*. The third term of equation (2) represents the criterion C1 (y). In the optimization phase, the values of the synaptic weights are fixed. Instead. the hand neuron changes its state autonomously while obeying the following differential equation: dYk aE c ds = - aYk' k = 1,2, ... ,16. (3) Here. s is the relaxation time required for the state change of the hand neuron, and c is a positive time constant. The right-hand side of equation (3) can be transformed as follows: _ aE = ~(x; _ x~) ax~ + ~(yj _ y',) ay} + (Yk _ yk) (ay~ - 1) - AYk. (4) 8Yk ~ 8Yk ~ J 8Yk 8Yk I J It is straightforward to show that the first three terms of equation (4) are the error signals at the kth hand neuron, which can be calculated backward from the output layer to the input layer. The fourth term of equation (4) is a suppressive signal which is given to the hand neuron by itself. When the state of the hand neuron obeys the differential equation (3), the time change E can be expressed as : dE = L dYk aE = -c L(dyk )2 < O. (5) ds k ds aYk k ds Therefore, the energy function E always decreases and the network comes to the equilibrium state that is the (local) minimum energy state. The outputs of the hand neurons in the equilibrium state represent the solution of the optimization problem which corresponds to the most suitable finger configuration. The relaxation computation of the neural network was simulated. For example, when given the image of a circular cylinder whose diameter was 5 cm, the prehensile finger configuration was computed. After a hundred-thousand iterations for the relaxation computation, we had the results shown in Figure 3. The left sied shows the hand shape that had the minimum value of the criterion of all the training data recorded when the subject grasped a circular cylinder whose diameter was 5 cm. The right side shows the hand shape produced by relaxation computation. These two hand shapes were very similar, which indicated that the network reproduced hand shape by using relaxation computation. 316 Uno, Fukumura, Suzuki, and Kawato Trainnig Data Results of relaxation Figure 3: Prehesile hand shapes for grasping a circular cylinder whose diameter was 5cm. 4 VARIOUS TYPES OF PREHENSIONS In the sections above, the subject was instructed to grasp objects using only one type of prehension. It is, however, thought that a human chooses different types of prehensions depending on the manipulation tasks. In order to investigate the dependence of the internal representation on the type of prehension, the second behavioral experiment was conducted. In this experiment, five circular cylinders and three spheres which were the same size as those in the first experiment were prepared. The subject was first instructed to grasp the objects tightly with his palm and all of his fingers, and then to grasp the same objects with only his fingertips. Iberall et al.(1988) referred to the first prehension and the second prehension as palm opposition and pad opposition, respectively. The subject grasped eight objects in two different types of prehensions twenty times each, which produced 320 prehensile patterns. Four neurons were set in the 3rd layer of the network and the network learning was simulated using these prehensile patterns as training data. Figure 4 shows the neuronal activation patterns formed in the 3rd layer after the network learning. Even if the grasped objects were the same, the neuron activation pattern for palm opposition was quite different from that for palm opposition. The neural network can reproduce different prehension, by introducing different criteria. Cl (y) is definded corresponding to palm opposition. Furthermore, we defind another criterion C2(Y), corresponding to pad opposition. i{MP,CM idP C 2(y) = 2: yf + 2:(1.0 - Yj)2. (6) Minimizing the criterion C2(y) demands that the MP joints (metacarpophalangeal joints) of the four fingers and the eM joint (carpometacarpal joint) of the thumb be flexed as much as possible and that the IP joints (interphalangeal joints) of all five fingers be stretched as much as possible. The relaxation computation of the neural network was simulated, when given the image of a sphere whose diameter was 5 cm. The results of the relaxation computation are shown in Figure 5. Adopting the different criteria, the neural network reproduced different prehensile hand configurations which corresponded to a) palm opposition and b) pad opposition. Visual & Somatosensory Information for Preshaping Hand in Grasping Movements 317 Neuron activity Neuron activity 1.00 €1 q 1.00 e.-fl-, 9 ~ , \. 'Q\ ~ \. ,', \ ~ \\~/f\~ Diameter \ / ~ laD 0 . , -0- 4cm V -0- Scm 0,00 000 --0-6cm -0' 7cm ,100 '1.oo_~..--~~ ,1234 1234 Neuron index of the 3rt! layer Neuron index of the 3rd layer a) Palm Opposition b) Pad Oppositon Circular cylinder Neuron activity 100 {> I I I I 0.00 A I I 'A I I ~\ ~ Neuron activity 100 ,1.00+--___ ..--. Diameter -- 4cm 1234 1234 Neuron index of the 3rd layer Neuron index of the 3rd layer C) Palm Opposition d) Pad Oppositon Sphere Figure 4: Internal representations of grasped objects formed in the 3rd layer of the network_ Graphs a), b), c) and d) show the activation patterns of neurons for palm oppositions when grasping 5 circular cylinders, for pad oppositions when grasping 5 circular cylinders, for palm oppositions when grasping 3 spheres and for pad oppositions when grasping 3 spheres, respectively. See Figure 2 legend for description. 5 DISCUSSION In view of the function of neurons in the posterior parietal association cortex, we have devised a neural network model for integrating visual and motor information. The proposed neural network model is an active sensing model, as it learns only when an object is successfully grasped. In this paper, tactile information is not treated, as the materials of the grasped objects are ,not considered for simplicity. We know that tactile information plays an important role in the recognition of grasped objects. The neural network model shown in Figure 1 can easily be developed so as to integrate visual, motor and tactile information. However, it is not clear how the internal representations of grasped objects is changed by adding tactile information. The critical problem in our neural network model is how many neurons should be set in the 3rd layer to represent the shapes of grasped objects. If there are too few neurons in the 3rd layer, the 3rd layer cannot represent enough information to restor x and y between the 3rd layer and the 5th layer; that is, the network cannot learn to realize the identity map between the input layer and the output layer. If there are too many neurons in the 3rd layer, the network cannot obtain useful representations of the grasped objects in the 3rd layer and the relaxation computation sometimes fails. In the present stage, we have no method to decide an adequate number of neurons for the 3rd layer. This is an important task for the future. Acknowledgements The main part ofthis study was done while the first author (Y.U.) was working at University of Tokyo. Y. Uno, N. Fukumura and R. Suzuki was supported by Japanese Ministry of 318 Uno, Fukumura, Suzuki, and Kawato Trainnig Data Result of relaxation a) Plam Opposition Training Data Result of relaxation b) Pad opposition Figure 5: Prehensile hand configuration a) for palm opposition and prehensile hand configuration b) for pad opposition when grasping a sphere whose diameter was 5 cm. The left sides show the hand shapes with the minimum values of the criterions for all training data recorded when the subject grasped a sphere whose diameter was 5 cm. The right sides show the hand shapes made by the relaxation computation. Education, Science and Culture Grants, NO.03251102 and No.03650338. M. Kawato was supported by Human Frontier Science Project Grant. References M. Jeannerod. (1984) The timing of natural prehension movements, J. Motor Behavior, 16: 235-254. M.A. Arbib, T. Iberall and D. Lyons. (1985) Coordinated control programs for movements of the hand. Hand Function and the Neocortex. Experimental Brain Research, suppl.l0, 111-129. N. Fukumura, Y. Uno, R. Suzuki andK. Kawato (1991) A neural network model which recognizes shape of a grasped object and decides hand configuration. Japan IEICE Technical Report, NC90-104: 213-218 (in Japanese). Katayama and M. Kawato (1990) Neural network model integrating visual and somatic information. J. Robotics Society of Japan, 8: 117-125 (in Japanese). T. Iberall (1998) A neural network for planning hand shapes in human prehension. proc. Automation and controls Con/.: 2288-2293. B. Irie and Kawato (1991) "Acquisition of Internal Representation by Multilayered Perceptrons." Electronics and Communications in Japan, Part 3, 74: 112-118. M. Taira, S. Mine, A.P. Georgopoulos, A. Murata and S. Sakata. (1990) Parietal cortex neurons of the monkey related to the visual guidance of hand movement. Exp. Brain Res., 83: 29-36.
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Spiral Waves in Integrate-and-Fire Neural Networks John G. Milton Department of Neurology The University of Chicago Chicago, IL 60637 Po Hsiang Chu Department of Computer Science DePaul University Chicago, IL 60614 Jack D. Cowan Department of Mathematics The University of Chicago Chicago, IL 60637 Abstract The formation of propagating spiral waves is studied in a randomly connected neural network composed of integrate-and-fire neurons with recovery period and excitatory connections using computer simulations. Network activity is initiated by periodic stimulation at a single point. The results suggest that spiral waves can arise in such a network via a sub-critical Hopf bifurcation. 1 Introduction In neural networks activity propagates through populations, or layers, of neurons. This propagation can be monitored as an evolution of spatial patterns of activity. Thirty years ago, computer simulations on the spread of activity through 2-D randomly connected networks demonstrated that a variety of complex spatio-temporal patterns can be generated including target waves and spirals (Beurle, 1956, 1962; Farley and Clark, 1961; Farley, 1965). The networks studied by these investigators correspond to inhomogeneous excitable media in which the probability of interneuronal connectivity decreases exponentially with distance. Although travelling spiral waves can readily be formed in excitable media by the introduction of non-uniform 1001 1002 Milton, Chu, and Cowan initial conditions (e.g. Winfree, 1987), this approach is not suitable for the study and classification of the dynamics associated with the onset of spiral wave formation. Here we show that spiral waves can "spontaneously" arise from target waves in a neural network in which activity is initiated by periodic stimulation at a single point. In particular, the onset of spiral wave formation appears to occur via a sub-critical Hopf bifurcation. 2 Methods Computer simulations were used to simulate the propagation of activity from a centrally placed source in a neural network containing 100 x 100 neurons arranged on a square lattice with excitatory interactions. At t = 0 all neurons were at rest except the source. There were free boundary conditions and all simulations were performed on a SUN SPARC 1+ computer. The network was constructed by assuming that the probability, A, of interneuronal connectivity was an exponential decreasing function of distance, i.e. A = (3 exp( -air!) where a = 0.6, {3 = 1.5 are constants and Ir I is the euclidean interneuronal distance (on average each neuron makes 24 connections and '" 1.3 connections per neuron, i.e. multiple connections occur). Once the connectivity was determined it remained fixed throughout the simulation. The dynamics of each neuron were represented by an integrate-and-fire model possessing a "leaky" membrane potential and an absolute (1 time step) and relative refractory or recovery period as described previously (Beurle, 1962; Farley, 1965; Farley and Clark, 1961): the membrane and threshold decay constants were, respectively, km = 0.3 msec- 1, ko = 0.03 msec- 1• The time step of the network was taken as 1 msec and it was assumed that during this time a neuron transmits excitation to all other neurons connected to it. 3 Results We illustrate the dynamics of a particular network as a function of the magnitude of the excitatory interneuronal excitation, E, when all other parameters are fixed. When E < 0.2 no activity propagates from the central source. For 0.2 < E < 0.58 target waves regularly emanate from the centrally placed source (Figure 1a). For E ~ 0.58 the activity patterns, once established, persisted even when the source was turned off. Complex spiral waves occurred when 0.58 < E < 0.63 (Figures 1b-ld). In these cases spiral meandering, spiral tip break-up and the formation of new spirals (some with multiple arms) occur continuously. Eventually the spirals tend to migrate out of the network. For E ~ 0.63 only disorganized spatial patterns occurred without clearly distinguishable wave fronts, except initially (Figures ie-f). Spiral Waves in Integrate-and-Fire Neural Networks 1003 Figure 1: Representative examples of the spatial pattern of neural activity as a function of E:(a) E = 0.45, (b - e) E = 0.58 and (f) E = 0.72. Color code: gray = quiescent, white = activated, black = relatively refractory. See text for details. .1 D5 c.:I 0 Z ~ .14 i:L CIl z 0 .07 ~ :::> U.I z z 0 0 ~ u .2 < ~ u.. o ,~~~~~~~WL o 5 ITERATION x 102 b c 10 Figure 2: Plot of the fraction of neurons firing per unit time for different values of E: (a) 0.45, (b) 0.58, and (c) 0.72. At t = 0 all neurons except the central source are quiescent. At t = 500 (indicated by .J-) the source is shut off. The region indicated by (*) corresponds to an epoch in which spiral tip breakup occurs. 1004 Milton, Chu, and Cowan The temporal dynamics of the network can be examined by plotting the fraction F of neurons that fire as a function of time. As E is increased through target waves (Figure 2a) to spiral waves (Figure 2b) to disorganized patterns (Figure 2c), the fluctuations in F become less regular, the mean value increases and the amplitude decreases. On closer inspection it can be seen that during spiral wave propagation (Figure 2b) the time series for F undergoes amplitude modulation as reported previously (Farley, 1965). The interval of low amplitUde, very irregular fluctuations in F (* in Figure 2b) corresponds to a period of spiral tip breakup (Figure lc). The appearance of spiral waves is typically preceded by 20-30 target waves. The formation of a spiral wave appears to occur in two steps. First there is an increase in the minimum value of F which begins at t '" 420 and more abruptly occurs at t '" 460 (Figure 2b). The target waves first become asymmetric and then activity propagates from the source region without the more centrally located neurons first entering the quisecent state (Figure 3c). At this time the spatially coherent wave front of the target waves becomes replaced by a disordered noncoherent distribution of active and refractory neurons. Secondly, the dispersed network activity begins to coalesce (Figures 3c and 3d) until at t '" 536 the first identifiable spiral occurs (Figure 3e). Figure 3: The fraction of neurons firing per unit time, for differing values of generation time t: (a) 175, (b) 345, (c) 465, (d) 503 (e) 536, and (f) 749. At t = 0 all neurons except the central source are quiescent. It was found that only 4 out of 20 networks constructed with the same 0, j3 produced spiral waves for E = 0.58 with periodic central point stimulation (simulations, in some cases, ran up to 50,000 generations). However, for all 20 networks, spiral waves could be obtained by the use of non-uniform initial conditions. Moreover, for those networks in which spiral waves occurred, the generation at which they formed differed. These observations emphasize that small fluctuations in the local connectivity of neurons likely play a major role in governing the dynamics of the network. Spiral Waves in Integrate-and-Fire Neural Networks 1005 4 Discussion Self-maintaining spiral waves can a..rise in an inhomogeneous neural network with uniform initial conditions. Initially well-formed target waves emanate periodically from the centrally placed source. Eventually, provided that E is in a critical range (Figures 1 & 3), the target waves may break up and be replaced by spiral waves. The necessary conditions for spiral wave formation are that: 1) the network be sufficiently tightly connected (Farley, 1965; Farley and Clark, 1961) and 2) the probability of interneuronal connectivity should decrease with distance (unpublished observations). As the network is made more tightly connected the probability that self-maintained activity arises increases provided that E is in the appropriate range (unpublished observations). These criteria are not sufficient to ensure that selfmaintained activity, including spiral waves, will form in a given realization of the neural network. It has previously been shown that partially formed spiral-like waves can arise from periodic point stimulation in a model excitable media in which the inhomogeneity arises from a dispersion of refractory times, k;l (Kaplan, et al, 1988). Integrate-and-fire neural networks have two stable states: a state in which all neurons are at rest, another associated with spiral waves. Target waves represent a transient response to perturbations away from the stable rest state. Since the neurons have memory (i.e. there is a relative refractory state with ko « k m ), the mean threshold and membrane potential of the network evolve with time. As a consequence the mean fraction of firing neurons slowly increases (Figure 2b). Our simulations suggest that at some point, provided that the connectivity of the network is suitable, the rest state suddenly becomes unstable and is replaced by a stable spiral wave. This exchange of stability is typical of a sub-critical Hopf bifurcation. Although complex, but organized, spatio-temporal patterns of spreading activity can readily be generated by a randomly connected neural network, the significance of these phenomena, if any, is not presently clear. On the one hand it is not difficult to imagine that these spatio-temporal dynamics could be related to phenomena ranging from the generation of the EEG, to the spread of epileptic and migraine related activity and the transmission of visual images in the cortex to the formation of patterns and learning by artificial neural networks. On the other hand, the occurence of such phenomena in artificial neural nets could conceivably hinder efficient learning, for example, by slowing convergence. Continued study of the properties of these networks will clearly be necessary before these issues can be resolved. Acknowledgements The authors acknowledge useful discussions with Drs. G. B. Ermentrout, L. Glass and D. Kaplan and financial support from the National Institutes of Health (JM), the Brain Research Foundation (JDC, JM), and the Office of Naval Research (JDC). References R. L. Beurle. (1956) Properties of a mass of cells capable of regenerating pulses. Phil. Trans. Roy. Soc. Lond. 240 B, 55-94. 1006 Milton, Chu, and Cowan R. L. BeUl-Ie. (1962) FUnctional organization in random networks. In Principles of Self-Organization, H. v. Foerster and G. W. Zopf, eds., pp 291-314. New York, Pergamon Press. B. G. Farley_ (1965) A neuronal network model and the "slow potentials" of electrophysiology. Compo in Biomed_ Res. 2, 265-294. B. G. Farley & W. A. Clark. (1961) Activity in networks of neuron-like elements. In Information Theory, C. Cherry, ed., pp 242-251. Washington, Butterworths. D. T. Kaplan, J. M.Smith,B. E. H. Saxberg & R. J. Cohen. (1988) Nonlinear dynamics in cardiac conduction. Math. Biosci. 90, 19-48. A. T. Winfree. (1987) When Time Breaks Down, Princeton University Press, Princeton, N.J.
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Kohonen Feature Maps and Growing Cell Structures a Performance Comparison Bernd Fritzke International Computer Science Institute 1947 Center Street, Suite 600 Berkeley, CA 94704-1105, USA Abstract A performance comparison of two self-organizing networks, the Kohonen Feature Map and the recently proposed Growing Cell Structures is made. For this purpose several performance criteria for self-organizing networks are proposed and motivated. The models are tested with three example problems of increasing difficulty. The Kohonen Feature Map demonstrates slightly superior results only for the simplest problem. For the other more difficult and also more realistic problems the Growing Cell Structures exhibit significantly better performance by every criterion. Additional advantages of the new model are that all parameters are constant over time and that size as well as structure of the network are determined automatically. 1 INTRODUCTION Self-organizing networks are able to generate interesting low-dimensional representations of high-dimensional input data. The most well-known of these models is the Kohonen Feature Map (Kohonen [1982)). So far it has been applied to a large variety of problems including vector quantization (Schweizer et al. [1991)), biological modelling (Obermayer, Ritter & Schulten [1990)), combinatorial optimization (Favata & Walker [1991]) and also processing of symbolic information(Ritter & Kohonen [1989)) . 123 124 Fritzke It has been reported by a number of researchers, that one disadvantage of Kohonen's model is the fact, that the network structure had to be specified in advance. This is generally not possible in an optimal way since a necessary piece of information, the probability distribution of the input signals, is usually not available. The choice of an unsuitable network structure, however, can badly degrade network performance. Recently we have proposed a new self-organizing network model - the Growing Cell Structures - which can automatically determine a problem specific network structure (Fritzke [1992]). By now the model has been successfully applied to clustering (Fritzke [1991]) and combinatorial optimization (Fritzke & Wilke [1991]). In this contribution we directly compare our model to that of Kohonen. We first review some general properties of self-organizing networks and several performance criteria for these networks are proposed and motivated. The new model is then briefly described. Simulation results are presented and allow a comparison of both models with respect to the proposed criteria. 2 SELF-ORGANIZING NETWORKS 2.1 CHARACTERlSTICS A self-organizing network consists of a set of neurons arranged in some topological structure which induces neighborhood relations among the neurons. An ndimensional reference vector is attached to every neuron. This vector determines the specific n-dimensional input signal to which the neuron is maximally sensitive. By assigning to every input signal the neuron with the nearest reference vector (according to a suitable norm), a mapping is defined from the space of all possible input signals onto the neural structure. A given set of reference vectors thus divides the input vector space into regions with a common nearest reference vector. These regions are commonly denoted as Voronoi regions and the corresponding partition of the input vector space is denoted Voronoi partition. Self-organizing networks learn (change internal parameters) in an unsupervised manner from a stream of input signals. These input signals obey a generally unknown probability distribution. For each input signal the neuron with the nearest reference vector is determined, the so-called "best matching unit" (bmu). The reference vectors of the bmu and of a number of its topological neighbors are moved towards the input signal. The adaptation of topological neighbors distinguishes self-organization ("winner take most") from competitive learning where only the bmu is adapted ("winner take all"). 2.2 PERFORMANCE CRlTERlA One can identify three main criteria for self-organizing networks. The importance of each criterion may vary from application to application. Topology Preservation. This denotes two properties of the mapping defined by the network. We call the mapping topology-preserving if Kohonen Feature Maps and Growing Cell Structures-a Performance Comparison 125 a) similar input vectors are mapped onto identical or closely neighboring neurons and b) neighboring neurons have similar reference vectors. Property a) ensures, that small changes of the input vector cause correspondingly small changes in the position of the bmu. The mapping is robust against distortions of the input, a very important property for applications dealing with real, noisy data. Property b) ensures robustness of the inverse mapping. The topology preservation is especially interesting when the dimension of the input vectors is higher than the network dimension. Then the mapping reduces the data dimension but usually preserves important similarity relations among the input data. Modelling of Probability Distribution. A set of reference vectors is said to model the probability distribution, ifthe local density of reference vectors in the input vector space approaches the probability density of the input vector distribution. This property is desirable for two reasons. First, we get an implicit model of the unknown probability distribution underlying the input signals. Second, the network becomes fault-tolerant against damage, since every neuron is only "responsible" for a small fraction of all input vectors. If neurons are destroyed for some reason the mapping ability of the network degrades only proportionally to the number of the destroyed neurons (soft fail) . This is a very desirable property for technical (as well as natural) systems. Minimization of Quantization Error. The quantization error for a given input signal is the distance between this signal and the reference vector of the bmu. We call a set of reference vectors error minimizing for a given probability distribution if the mean quantization error is minimized. This property is important, if the original signals have to be reconstructed from the reference vectors which is a very common situation in vector quantization. The quantization error in this case limits the accuracy of the reconstruction. One should note that the optimal distribution of reference vectors for error minimization is generally different from the optimal distribution for distribution modelling. 3 THE GROWING CELL STRUCTURES The Growing Cell Structures are a self-organizing network an important feature of which is the ability to automatically find a problem specific network structure through a growth process. Basic building blocks are k-dimensional hypertetrahedrons: lines for k = 1, triangles for k = 2, tetrahedrons for k = 3 etc. The vertices of the hypertetrahedrons are the neurons and the edges denote neighborhood relations. By insertion and deletion of neurons the structure is modified. This is done during a self-organization process which is similar to that in Kohonen's model. Input signals cause adaptation of the bmu and its topological neighbors. In contrast to Kohonen's model all parameters are constant including the width of the neighborhood around 126 Fritzke the bmu where adaptation takes place. Only direct neighbors and the bmu itself are being adapted. 3.1 INSERTION OF NEURONS To determine the positions where new neurons should be inserted the concept of a resource is introduced. Every neuron has a local resource variable and new neurons are always inserted near the neuron with the highest resource value. New neurons get part of the resource of their neighbors so that in the long run the resource is distributed evenly among all neurons. Every input signal causes an increase of the resource variable of the best matching unit. Choices for the resource examined so far are • the summed quantization error caused by the neuron • the number of input signals received by the neuron Always after a constant number of adaptation steps (e.g. 100) a new neuron is inserted. For this purpose the neuron with the highest resource is determined and the edge connecting it to the neighbor with the most different reference vector is "split" by inserting the new neuron. Further edges are added to rebuild a structure consisting only of k-dimensional hypertetrahedrons. The reference vector of the new neuron is interpolated from the reference vectors belonging to the ending points of the split edge. The resource variable of the new neuron is initialized by subtracting some resource from its neighbors) the amount of which is determined by the reduction of their Voronoi regions through the insertion. 3.2 DELETION OF NEURONS By comparing the fraction of all input signals which a specific neuron has received and the volume of its Voronoi region one can derive a local estimate of the probability density of the input vectors. Those neurons) whose reference vectors fall into regions of the input vector space with a very low probability density) are regarded as "superfluous)) and are removed. The result are problem-specific network structures potentially consisting of several separate sub networks and accurately modelling a given probability distribution. 4 SIMULATION RESULTS A number of tests have been performed to evaluate the performance of the new model. One series is described in the following. Three methods have been compared. a) Kohonen Feature Maps (KFM) b) Growing Cell Structures with quantization error as resource (GCS-l) c) Growing Cell Structures with number of input signals as resource (GCS-2) Kohonen Feature Maps and Growing Cell Structures-a Performance Comparison 127 Distribution A: The probability density is uniform in the unit square o Distribution B: The probability density is uniform in the lOx 10-field, by a factor 100 higher in the 1 x I-field and zero elsewhere [J [J c [J c [J [J Distribution C: The probability density is uniform inside the seven lower squares, by a factor 10 higher in the two upper squares and zero elsewhere. Figure 1: Three different probability distributions used for a performance comparison. Distribution A is very simple and has a form ideally suited for the Kohonen Feature Map which uses a square grid of neurons. Distribution B was chosen to show the effects of a highly varying probability density. Distribution C is the most realistic with a number of separate regions some of which have also different probability densities. These models were applied to the probability distributions shown in fig. 1. The Kohonen model was used with a 10 x 10-grid of neurons. The Growing Cell Structures were used to build up a two dimensional cell structure of the same size. This was achieved by stopping the growth process when the number of neurons had reached 100. At the end of the simulation the proposed criteria were measured as follows: • The topology preservation requires two properties. Property a) was measured by the topographical product recently proposed by Bauer e.a. for this purpose (Bauer & Pawelzik [1992]). Property b) was measured by computing the mean edge length in the input space, i.e. the mean difference between reference vectors of directly neighboring neurons. • The distribution modelling was measured by generating 5000 test signals according to the specific probability distribution and counting for every neuron the number of test signals it has been bmu for. The standard deviation of all counter values was computed and divided by the mean value of the counters to get a normalized measure, the distribution error, for the modelling of the probability distribution. • The error minimization was measured by computing the mean square quantization error of the test signals. The numerical results of the simulations are shown in fig. 2. Typical examples of the final network structures can be seen in fig. 3. It can be seen from fig. 2 that the 128 Fritzke model A B C model A B C KFM 0.022 0.048 KFM 0.092 0.110 GCS-1 0.014 0.044 GCS-1 0.11 10.056 1 0.015 GCS-2 10.0111 10.019 1 GCS-2 0.11 0.071 10.0131 a) topographical product b) mean edge length model A B C model A B C KFM 0.84 0.90 KFM 0.0020 0.00077 0.00086 GCS-1 §I] 10.591 GCS-1 0.0019 0.00089 0.00010 GCS-2 0.26 1.57 0.73 GCS-2 0.0019 10.000551 10.000041 c) distribution error d) quantization error Figure 2: Simulation results of the performance comparison. The model of Kohonen(KFM) and two versions of the Growing Cell Structures have been compared with respect to different criteria. All criteria are such, that smaller values are better values. The best (smallest) value in each column is enclosed in a box. Simulations were performed with the probability distributions A, Band C from fig. 1. model of Kohonen has superior values only for distribution A, which is very regular and formed exactly like the chosen network structure (a square). Since generally the probability distribution is unknown and irregular, the distributions Band C are by far more realistic. For these distributions the Growing Cell Structures have the best values. The modelling of the distribution and the minimization of the quantization error are generally concurring objectives. One has to decide which objective is more important for the current application. Then the appropriate version ofthe Growing Cell Structures can optimize with respect to that objective. For the complicated distribution C, however, either version of the Growing Cell Structures performs for every criterion better than Kohonen's model. Especially notable is the low quantization error for distribution C and the error minimizing version (GCS-2) of the Growing Cell Structures (see fig. 2d). This value indicates a good potential for vector quantization. 5 DISCUSSION Our investigations indicate that - w.r.t the proposed criteria - the Growing Cell Structures are superior to Kohonen's model for all but very carefully chosen trivial examples. Although we used small examples for the sake of clarity, our experiments lead us to conjecture, that the difference will further increase with the difficulty and size of the problem. There are some other important advantages of our approach. First, all parameters are constant. This eliminates the difficult choice of a "cooling schedule" which is necessary in Kohonen's model. Second, the network size does not have to be specified in advance. Instead the growth process can be continued until an arbitrary performance criterion is met. To meet a specific criterion with Kohonen's model, one generally has to try different network sizes. To start always with a very large Kohonen Feature Maps and Growing Cell Structures-a Performance Comparison 129 Distribution A Distribution B Distribution C a) f-I-v -" b) c) Figure 3: Typical simulation results for the model of Kohonen and the two versions of the Growing Cell Structures. The network size is 100 in every case. The probability distributions are described in fig. 1. a) Kohonen Feature Map (KFM). For distributions Band C the fixed network structure leads to long connections and neurons in regions with zero probability density. b) Growing Cell Structures, distribution modelling variant (GCS-1). The growth process combined with occasional removal of "superfluous" neurons has led to several sub networks for distributions Band C. For distribution B roughly half of the neurons are used to model either of the squares. This corresponds well to the underlying probability density. c) Growing Cell Structures, error minimizing variant (GCS-2). The difference to the previous variant can be seen best for distribution B, where only a few neurons are used to cover the small square. 130 Fritzke network is not a good solution to this problem, since the computational effort grows faster than quadratically with the network size. Currently applications of variants of the new method to image compression and robot control are being investigated. Furthermore a new type of radial basis function network related to (Moody & Darken [1989]) is being explored, which is based on the Growing Cell Structures. REFERENCES Bauer, H.-U. & K. Pawelzik [1992}, "Quantifying the neighborhood preservation of self-organizing feature maps," IEEE Transactions on Neural Networks 3, 570-579. Favata, F. & R. Walker [1991]' "A study of the application of Kohonen-type neural networks to the travelling Salesman Problem," Biological Cybernetics 64, 463-468. Fritzke, B. [1991], "Unsupervised clustering with growing cell structures," Proc. of IJCNN-91, Seattle, 531-536 (Vol. II). Fritzke, B. [1992], "Growing cell structures - a self-organizing network in k dimensions," in Artificial Neural Networks II, I. Aleksander & J. Taylor, eds., North-Holland, Amsterdam, 1051-1056. Fritzke, B. & P. Wilke [19911, "FLEXMAP - A neural network with linear time and space complexity for the traveling salesman problem," Proc. of IJCNN-91, Singapore, 929-934. Kohonen, T. [19821, "Self-organized formation of topologically correct feature maps," Biological Cybernetics 43, 59-69. Moody, J. & C. Darken [19891, "Fast Learning in Networks of Locally-Tuned Processing Units," Neural Computation 1, 281-294. Obermayer, K., H. Ritter & K. Schulten [1990J, "Large-scale simulations of selforganizing neural networks on parallel computers: application to biological modeling," Parallel Computing 14,381-404. Ritter, H.J. & T. Kohonen [1989], "Self-Organizing Semantic Maps," Biological Cybernetics 61,241-254. Schweizer, L., G. Parladori, G.L. Sicuranza & S. Marsi [1991}, "A fully neural approach to image compression," in Artificial Neural Networks, T. Kohonen, K. Miikisara, O. Simula & J. Kangas, eds., North-Holland, Amsterdam, 815-820.
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A Neural Model of Descending Gain Control in the Electrosensory System Mark E. Nelson Beckman Institute University of Illinois 405 N. Mathews Urbana, IL 61801 Abstract In the electrosensory system of weakly electric fish, descending pathways to a first-order sensory nucleus have been shown to influence the gain of its output neurons. The underlying neural mechanisms that subserve this descending gain control capability are not yet fully understood. We suggest that one possible gain control mechanism could involve the regulation of total membrane conductance of the output neurons. In this paper, a neural model based on this idea is used to demonstrate how activity levels on descending pathways could control both the gain and baseline excitation of a target neuron. 1 INTRODUCTION Certain species of freshwater tropical fish, known as weakly electric fish, possess an active electric sense that allows them to detect and discriminate objects in their environment using a self-generated electric field (Bullock and Heiligenberg, 1986). They detect objects by sensing small perturbations in this electric field using an array of specialized receptors, known as electroreceptors, that cover their body surface. Weakly electric fish often live in turbid water and tend to be nocturnal. These conditions, which hinder visual perception, do not adversely affect the electric sense. Hence the electrosensory system allows these fish to navigate and capture prey in total darkness in much the same way as the sonar system of echolocating bats allows them to do the same. A fundamental difference between bat echolocation and fish 921 922 Nelson "electrolocation" is that the propagation of the electric field emitted by the fish is essentially instantaneous when considered on the time scales that characterize nervous system function. Thus rather than processing echo delays as bats do, electric fish extract information from instantaneous amplitude and phase modulations of their emitted signals. The electric sense must cope with a wide range of stimulus intensities because the magnitude of electric field perturbations varies considerably depending on the size, distance and impedance of the object that gives rise to them (Bastian, 1981a). The range of intensities that the system experiences is also affected by the conductivity of the surrounding water, which undergoes significant seasonal variation. In the electrosensory system, there are no peripheral mechanisms to compensate for variations in stimulus intensity. Unlike the visual system, which can regulate the intensity of light arriving at photoreceptors by adjusting pupil diameter, the electrosensory system has no equivalent means for directly regulating the overall stimulus strength experienced by the electroreceptors, 1 and unlike the auditory system, there are no efferent projections to the sensory periphery to control the gain of the receptors themselves. The first opportunity for the electrosensory system to make gain adjustments occurs in a first-order sensory nucleus known as the electrosensory lateral line lobe (ELL). In the ELL, primary afferent axons from peripheral electroreceptors terminate on the basal dendrites of a class of pyramidal cells referred to as E-cells (Maler et al., 1981; Bastian, 1981b), which represent a subset of the output neurons for the nucleus. These pyramidal cells also receive descending inputs from higher brain centers on their apical dendrites (Maler et al., 1981). One noteworthy feature is that the descending inputs are massive, far outnumbering the afferent inputs in total number of synapses. Experiments have shown that the E-cells, unlike peripheral electroreceptors, maintain a relatively constant response amplitude to electrosensory stimuli when the overall electric field normalization is experimentally altered. This automatic gain control capability is lost, however, when descending input to the ELL is blocked (Bastian, 1986ab). The underlying neural mechanisms that subserve this descending gain control capability are not yet fully understood, although it is known that GABAergic inhibition plays a role (Shumway & Maler, 1989). We suggest that one possible gain control mechanism could involve the regulation of total membrane conductance of the pyramidal cells. In the next section we present a model based on this idea and show how activity levels on descending pathways could regulate both the gain and baseline excitation of a target neuron. 2 NEURAL CIRCUITRY FOR DESCENDING GAIN CONTROL Figure 1 shows a schematic diagram of neural circuitry that could provide the basis for a descending gain control mechanism. This circuitry is inspired by the circuitry found in the ELL, but has been greatly simplified to retain only the aspects that lIn principle, this could be achieved by regulating the strength of the fish's own electric discharge. However, these fish maintain a remarkably stable discharge amplitude and such a mechanism has never been observed. (CONTROL) (INPUT) A Neural Model of Descending Gain Control in the Electrosensory System 923 descending_~ ____ ..... excitatory descending~ inhibitory ~ inhibitory interneuron primary afferent --->!)o----~O o excitatory synapse • inhibitory synapse pyramidal cell (OUTPUT) Figure 1: Neural circuitry for descending gain control. The gain of the pyramidal cell response to an input signal arriving on its basilar dendrite can be controlled by adjusting the tonic levels of activity on two descending pathways. A descending excitatory pathway makes excitatory synapses (open circles) directly on the pyramidal cell. A descending inhibitory pathway acts through an inhibitory interneuron (shown in gray) to activate inhibitory synapses (filled circles) on the pyramidal cell. are essential for the proposed gain control mechanism. The pyramidal cell receives afferent input on a basal dendrite and control inputs from two descending pathways. One descending pathway makes excitatory synaptic connections directly on the apical dendrite of the pyramidal cell, while a second pathway exerts a net inhibitory effect on the pyramidal cell by acting through an inhibitory interneuron. We will show that under appropriate conditions, the gain of the pyramidal cell's response to an input signal arriving on its basal dendrite can be controlled by adjusting the tonic levels of activity on the two descending pathways. At this point it is worth pointing out that the spatial segregation of input and control pathways onto different parts of the dendritic tree is not an essential feature of the proposed gain control mechanism. However, by allowing independent experimental manipulation of these two pathways, this segregation has played a key role in the discovery and subsequent characterization of the gain control function in this system (Bastian, 1986ab ). The gain control function of the neural circuitry show in Figure 1 can best be understood by considering an electrical equivalent circuit for the pyramidal cell. The equivalent circuit shown in Figure 2 includes only the features that are necessary to understand the gain control function and does not reflect the true complexity of ELL pyramidal cells, which are known to contain many different types of voltagedependent channels (Mathieson & Maler, 1988). The passive electrical properties of the circuit in Figure 2 are described by a membrane capacitance em, a leakage conductance gleak, and an associated reversal potential E leak . The excitatory descending pathway directly activates excitatory synapses on the pyramidal cell, giving rise to an excitatory synaptic conductance gex with a reversal potential Eex. 924 Nelson g leak em E leak Figure 2: Electrical equivalent circuit for the pyramidal cell in the gain control circuit. The excitatory and inhibitory conductances, gex and 9inh, are shown are variable resistances to indicate that their steady-state values are dependent on the activity levels of the descending pathways. The inhibitory descending pathway acts by exciting a class of inhibitory interneurons which in turn activate inhibitory synapses on the pyramidal cell with inhibitory conductance 9inh and reversal potential Einh. In this model, the excitatory and inhibitory conductances gex and 9inh represent the population conductances of all the individual excitatory and inhibitory synapses associated with the descending pathways. Although individual synaptic events give rise to a time-dependent conductance change (which is often modeled by an a function), we consider the regime in which the activity levels on the descending pathways, the number of synapses involved, and the synaptic time constants are such that the summed effect can be well described by a single time-invariant conductance value for each pathway. The input signal (the one under the influence of the gain control mechanism) is modeled in a general form as a time-dependent current I( t). This current can represent either the synaptic current arising from activation of synapses in the primary afferent pathway, or it can represent direct current injection into the cell, such as might occur in an intracellular recording experiment. The behavior of the membrane potential Vet) for this model system is described by d~r(t) Cm~ + 9leak(V(t) - Eleak ) + gex(V(t) - Eex) + 9inh(V(t) - Einh) = let) (1) In the absence of an input signal (I = 0), the system will reach a steady-state (dV/dt = 0) membrane potential ~8 given by ~s (I = 0) = 9leak E leak + gex E ex + 9inhEinh gleak + gex + 9inh (2) A Neural Model of Descending Gain Control in the Electrosensory System 925 If we consider the input I(t) to give rise to fluctuations in membrane potential U(t) about this steady state value U(t) = V(t) 11;;8 (3) then (1) can be rewritten as dU(t) Cm~ + 9totU(t) = I(t) (4) where 9tot is the total membrane conductance 9tot = 9leak + gex + 9inh (5) Equation (4) describes a first-order low-pass filter with a transfer function 0(8), from input current to output voltage change, given by 0(8) = Rtot 'TS + 1 (6) where 8 is the complex frequency (8 = iw), Rtot is the total membrane resistance (Rtot = 1/ 9tot), and 'T is the RC time constant (7) The frequency dependence of the response gain 10(iw)1 is shown in Figure 3. For low frequency components of the input signal (W'T « 1), the gain is inversely proportional to the total membrane conductance 9toh while at high frequencies (W'T» 1), the gain is independent of 9tot. This is due to the fact that the impedance of the RC circuit shown in Figure 2 is dominated by the resistive components at low frequencies and by the capacitive component at high frequencies. Note that the RC time constant 'T, which characterizes the low-pass filter cutoff frequency, varies inversely with 9tot. For components of the input signal below the cutoff frequency, gain control can be accomplished by regulating the total membrane conductance. In electrophysiological terms, this mechanism can be thought of in terms of regulating the input resistance of the neuron. As the total membrane conductance is increased, the input resistance is decreased, meaning that a fixed amount of current injection will cause a smaller change in membrane potential. Hence increasing the total membrane conductance decreases the response gain. In our model, we propose that regulation of total membrane conductance occurs via activity on descending pathways that activate excitatory and inhibitory synaptic conductances. For this proposed mechanism to be effective, these synaptic conductances must make a significant contribution to the total membrane conductance of the pyramidal cell. Whether this condition actually holds for ELL pyramidal cells has not yet been experimentally tested. However, it is not an unreasonable assumption to make, considering recent reports that synaptic background activity can have 926 Nelson 20 0 gtot= g leak: 1:= 100 msec ,-..., gtot= 10 gl == -20 k -e '-' c: gtot= 100 gleak: 1: = 1 msec .-40 ~ ~ -60 -80~~~~~~~~~~~~m-~~~~~~~~ 101 1& 1d 1~ 1d 1<f (0 (rad/sec) Figure 3: Gain as a function of frequency for three different values oftotal membrane conductance gtot. At low frequencies, gain is inversely proportional to gtot. Note that the time constant T, which characterizes the low-pass cutoff frequency, also varies inversely with gtot . Gain is normalized to the maximum gain: Gmax = _1_; 9teak Gain(dB) = 20 10glO( G~aJ. a significant influence on the total membrane conductance of cortical pyramidal cells (Bernander et aI., 1991) and cerebellar Purkinje cells (Rapp et aI., 1992). 3 CONTROL OF BASELINE EXCITATION If the only functional goal was to achieve regulation of total membrane conductance, then synaptic background activity on a single descending pat.hway would be sufficient and there would be no need for the paired excitatory and inhibitory control pathways shown in Figure 1. However, the goal of gain control is regulate the total membrane conductance while holding the baseline level of excitation constant. In other words, we would like to be able to change the sensitivity of a neuron's response without changing its spontaneous level of activity (or steady-state resting potential if it is below spiking threshold). By having paired excitatory and inhibitory control pathways, as shown in Figure 1, we gain the extra degree-of-freedom necessary to achieve this goal. Equation (2) provided us with a relationship between the synaptic conductances in our model and the steady-state membrane potential. In order to change the gain of a neuron, without changing its baseline level of excitation, the excitatory and inhibitory conductances must be adjusted in a way that achieves the desired total membrane conductance gtot and maintains a constant V3S • Solving equations (2) and (5) simultaneously for gex and ginh, we find A Neural Model of Descending Gain Control in the Electrosensory System 927 (8) (9) For example, consider a case where the reversal potentials are El eak -70 m V, Eex = 0 m V, and Einh = -90 m V. Assume want to find values of the steady-state conductances, gex and ginh, that would result in a total membrane conductance that is twice the leakage conductance (i.e. gtot = 2gleak) and would produce a steady-state depolarization of 10 mV (i.e. ~s = -60 mY). Using (8) and (9) we find the required synaptic conductance levels are gex = ~ gleak and ginh = ~ gleak. 4 DISCUSSION The ability to regulate a target neuron's gain using descending control signals would provide the nervous system with a powerful means for implementing adaptive signal processing algorithms in sensory processing pathways as well as other parts of the brain. The simple gain control mechanism proposed here, involving the regulation of total membrane conductance, may find widespread use in the nervous system. Determining whether or not this is the case, of course, requires experimental verification. Even in the electrosensory system, which provided the inspiration for this model, definitive experimental tests of the proposed mechanism have yet to be carried out. Fortunately the model provides a straightforward experimentally testable prediction, namely that if activity levels on the presumed control pathways are changed, then the input resistance of the target neuron will reflect those changes. In the case of the ELL, the prediction is that if descending pathways were silenced while monitoring the input resistance of an E-type pyramidal cell, one would observe an increase in input resistance corresponding to the elimination of the descending contributions to the total membrane conductance. We have mentioned that the gain control circuitry of Figure 1 was inspired by the neural circuitry of the ELL. For those familiar with this circuitry, it is interesting to speculate on the identity of the interneuron in the inhibitory control pathway. In the gymnotid ELL, there are at least six identified classes of inhibitory interneurons. For the proposed gain control mechanism, we are interested in the identifying those that receive descending input and which make inhibitory synapses onto pyramidal cells. Four of the six classes meet these criteria: granule cell type 2 (GC2), polymorphic, stellate, and ventral molecular layer neurons. While all four classes may participate to some extent in the gain control mechanism, one would predict, based on cell number and synapse location, that GC2 (as suggested by Shumway & Maler, 1989) and polymorphic cells would make the dominant contribution. The morphology of GC2 and polymorphic neurons differs somewhat from that shown in Figure 1. In addition to the apical dendrite, which is shown in the figure, these neurons also have a basal dendrite that receives primary afferent input. GC2 and polymorphic neurons are excited by primary afferent input and thus provide additional inhibition to pyramidal cells when afferent activity levels increase. This can be viewed as providing a feedforward component to the automatic gain control mechanism. 928 Nelson In this paper, we have confined our analysis to the effects of tonic changes in descending activity. While this may be a reasonable approximantion for certain experimental manipulations, it is unlikely to be a good representation of the dynamic patterns that occur under natural conditions, particularly since the descending pathways form part of a feedback loop that includes the ELL output neurons. The full story in the electrosensory system will un doubt ably be much more complex. For example, there is already experimental evidence demonstrating that, in addition to gain control, descending pathways influence the spatial and temporal filtering properties of ELL output neurons (Bastian, 1986ab; Shumway & Maler, 1989). Acknowledgements This work was supported by NIMH 1-R29-MH49242-01. Thanks to Joe Bastian and Lenny Maler for many enlightening discussions on descending control in the ELL. References Bastian, J. (1981a) Electrolocation I: An analysis of the effects of moving objects and other electrical stimuli on the electroreceptor activity of Apteronotus alhi/rons. J. Compo Physiol. 144, 465-479. Bastian, J. (1981b) Electrolocation II: The effects of moving objects and other electrical stimuli on the activities of two categories of posterior lateral line lobe cells in Apteronotus alhi/rons. J. Compo Physiol. 144, 481-494. Bastian, J. (1986a) Gain control in the electrosensory system mediated by descending inputs to the electrosensory lateral line lobe. J. N eurosci. 6, 553-562. Bastian, J. (1986b) Gain control in the electrosensory system: a role for the descending projections to the electrosensory lateral line lobe. J. Compo Physiol. 158, 505-515. Bernander, 0., Douglas, R.J., Martin, K.A.C. & Koch, C. (1991) Synaptic background activity influences spatiotemporal integration in single pyramidal cells. Proc. Natl. Acad. Sci. USA 88, 11569-11573. Bullock, T.H. & Heiligenberg, W., eds. (1986) Electro reception. Wiley, New York. Maler, 1., Sas, E. and Rogers, J. (1981) The cytology of the posterior lateral line lobe of high frequency weakly electric fish (Gymnotidei): Dendritic differentiation and synaptic specificity in a simple cortex. J. Compo Neurol. 195,87-140. Mathieson, W.B. & Maler, 1. (1988) Morphological and electrophysiological properties of a novel in vitro preparation: the electrosensory lateral line lobe brain slice. J. Compo Physiol. 163, 489-506. Rapp, M., Yarom, Y. & Segev, I. (1992) The impact of parallel fiber background activity on the cable properties of cerebellar Purkinje Cells. Neural Compo 4, 518533. Shumway, C.A. & Maler, L.M. (1989) GABAergic inhibition shapes temporal and spatial response properties of pyramidal cells in the electrosensory lateral line lobe of gymnotiform fish J. Compo Physiol. 164, 391-407.
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Memory-based Reinforcement Learning: Efficient Computation with Prioritized Sweeping Andrew W. Moore awm@ai.mit.edu NE43-759 MIT AI Lab. 545 Technology Square Cambridge MA 02139 Christopher G. At:iteson Abstract cga@ai.mit.edu NE43-771 MIT AI Lab. 545 Technology Square Cambridge MA 02139 We present a new algorithm, Prioritized Sweeping, for efficient prediction and control of stochastic Markov systems. Incremental learning methods such as Temporal Differencing and Q-Iearning have fast real time performance. Classical methods are slower, but more accurate, because they make full use of the observations. Prioritized Sweeping aims for the best of both worlds. It uses all previous experiences both to prioritize important dynamic programming sweeps and to guide the exploration of statespace. We compare Prioritized Sweeping with other reinforcement learning schemes for a number of different stochastic optimal control problems. It successfully solves large state-space real time problems with which other methods have difficulty. 1 STOCHASTIC PREDICTION The paper introduces a memory-based technique, prioritized 6weeping, which is used both for stochastic prediction and reinforcement learning. A fuller version of this paper is in preparation [Moore and Atkeson, 1992]. Consider the 500 state Markov system depicted in Figure 1. The system has sixteen absorbing states, depicted by white and black circles. The prediction problem is to estimate, for every state, the long-term probability that it will terminate in a white, rather than black, circle. The data available to the learner is a sequence of observed state transitions. Let us consider two existing methods along with prioritized sweeping. 263 264 Moore and Atkeson Figure 1: A 500-state Markov system. Each state has a random number (mean 5) of random successors chosen within the local neighborhood. Temporal Differencing (TD) is an elegant incremental algorithm [Sutton, 1988] which has recently had success with a very large problem [Tesauro, 1991]. The classical method proceeds by building a maximum likelihood model of the state transitions. qij (the transition probability from i to i) is estimated by ANum ber of observations i ~ i llij = Number of occasions in state i (1) After t + 1 observations the new absorption probability estimates are computed to satisfy, for each terminal state k, the linear system (2) jeSucC8(i)nNONTERMS where the i'iJ& [t]'s are the absorption probabilities we are trying to learn, where succs(i) is the set of all states which have been observed as immediate successors of i and NONTERMS is the set of non-terminal states. This set of equations is solved after each transition is observed. It is solved using Gauss-Seidel-an iterative method. What initial estimates should be used to start the iteration? An excellent answer is to use the previous absorption probability estimates i'iJ& [t]. Prioritized sweeping is designed to combine the advantages of the classical method with the advantages of TD. It is described in the next section, but let us first examine performance on the original 500-state example of Figure 1. Figure 2 shows the result. TD certainly learns: by 100,000 observations it is estimating the terminal-white probability to an RMS accuracy of 0.1. However, the performance of the classical method appears considerably better than TD: the same error of 0.1 is obtained after only 3000 observations. Figure 3 indicates why temporal differencing may nevertheless often be more useful. TD requires far less computation per observation, and so can obtain more data in real time. Thus, after 300 seconds, TD has had 250,000 observations and is down Memory-based Reinforcement Learning: Efficient Computation with Prioritized Sweeping 265 Mean ± Standard Dev'n TD Classical Pri. Sweep After 100,000 observations 0.40 ± 0.077 0.024 ± 0.0063 0.024 ± 0.0061 After 300 seconds 0.079 ± 0.067 0.23 ± 0.038 0.021 ± 0.0080 Table 1: RMS prediction error: mean and standard deviation for ten experiments. 1D -----1D -----1.5 0.5 .. us Classical .. us Classical ---= Pri. Sweep e Pri.Sweep .. I.. 0.4 .. .. u 1.35 u 0.35 ----J:I J:I --.............. oS 1.3 -.. .. = 0.3 .. ~ ~ ...... 1 " "1.25 , 0.25 ] , 'tI \ " 1.2 , u 1.2 " \ IS. 1.15 Do 0.15 \ \ rIJ 0.1 \ fI.l o.t ~ \ ~ =c ... 5 =c 0.15 o o o 100 300 le3 3e3 ... 3e. ..5 o 0.3 1 3 10 30 100 300 No. observations (log scale) Real time, seconds (log scale) Figure 2: RMS prediction against obFigure 3: RMS prediction against real servation during three learning algatime rithms. to an error of 0.05, whereas even after 300 seconds the classical method has only 1000 observations and a much cruder estimate. In the same figures we see the motivation behind prioritized sweeping. Its performance relative to observations is almost as good as the classical method, while its performance relative to real time is even better than TD. The graphs in Figures 2 and 3 were based on only one learning experiment each. Ten further experiments, each with a different random 500 state problem, were run. The results are given in Table 1. 2 PRIORITIZED SWEEPING A longer paper [Moore and Atkeson, 1992] will describe the algorithm in detail. Here we summarize the essential insights, and then simply present the algorithm in Figure 4. The closest relation to prioritized sweeping is the search scheduling technique of the A* algorithm [Nilsson, 1971]. Closely related research is being performed by [Peng and Williams, 1992] into a similar algorithm to prioritized sweeping, which they call Dyna-Q-queue . • The memory requirements oflearning a N, x N, matrix, where N, is the number of states, may initially appear prohibitive, especially since we intend to operate with more than 10,000 states. However, we need only allocate memory for the 266 Moore and Atkeson 1. Promote state irecent (the source of the most recent transition) to top of priority queue. 2. While we are allowed further processing and priority queue not empty 2.1 Remove the top state from the priority queue. Call it i 2.2 amax = 0 2.3 for each Ie E TERMS Pnew = qil& + L q,j ijl& j esuccs(i)nNONTERMS a:= I Pnew i"il& I iil& : = Pnew amax := max(amax, a) 2.4 for each i' E preds(i) P := qi1iamax if i' not on queue, or P exceeds the current priority of i', then promote i' to new priority P. Figure 4: The prioritized sweeping algorithm. This sequence of operations is executed each time a transition is observed. experiences the system actually has, and for a wide class of physical systems there is not enough time in the lifetime of the physical system to run out of memory . • We keep a record of all predecessors of each state. When the eventual absorption probabilities of a state are updated, its predecessors are alerted that they may need to change. A priority value is assigned to each predecessor according to how large this change could be possibly be, and it is placed in a priority queue. • After each real-world observation i ~ j, the transition probability estimate qij is updated along with the probabilities of transition to all other previously observed successors of i. Then state i is promoted to the top of the priority queue so that its absorption probabilities are updated immediately. Next, we continue to process further states from the top of the queue. Each state that is processed may result in the addition or promotion of its predecessors within the queue. This loop continues for a preset number of processing steps or until the queue empties. If a real world observation is interesting, all its predecessors and their earlier ancestors quickly find themselves near the top of the priority queue. On the other hand, if the real world observation is unsurprising, then the processing immediately proceeds to other, more important areas of state-space which had been under consideration on the previous time step. These other areas may be different nom those in which the system currently finds itself. Memory-based Reinforcement Learning: Efficient Computation with Prioritized Sweeping 267 Dyna-PI+ Dyna-OPT PriSweep 15 States 400 300 150 117 States > 500 900 1200 605 States > 36000 21000 6000 4528 States > > 500000 245000 59000 Table 2: Number of observations before 98% of decisions were subsequently optimal. Dyna and Prioritized Sweeping were each allowed to process ten states per real-world observation. 3 LEARNING CONTROL FROM REINFORCEMENT Prioritized sweeping is also directly applicable to stochastic control problems. Remembering all previous transitions allows an additional advantage for controlexploration can be guided towards areas of state space in which we predict we are ignorant. This is achieved using the exploration philosophy of [Kaelbling, 1990] and [Sutton, 1990]: optimism in the face of uncertainty. 4 RESULTS Results of some maze problems of significant size are shown in Table 2. Each state has four actions: one for each direction. Blocked actions do not move. One goal state (the star in subsequent figures) gives 100 units of reward, all others give no reward, and there is a discount factor of 0.99. Trials start in the bottom left corner. The system is reset to the start state whenever the goal state has been visited ten times since the last reset. The reset is outside the learning task: it is not observed as a state transition. Prioritized sweeping is tested against a highly tuned Q-learner [Watkins, 1989] and a highly tuned Dyna [Sutton, 1990]. The optimistic experimentation method (described in the full paper) can be applied to other algorithms, and so the results of optimistic Dyna-learning is also included. The same mazes were also run as a stochastic problem in which requested actions were randomly corrupted 50% of the time. The gap between Dyna-OPT and Prioritized Sweeping was reduced in these cases. For example, on a stochastic 4528-state maze Dyna-OPT took 310,000 steps and Prioritized sweeping took 200,000. We also have results for a five state bench-mark problem described in [Sato et al., 1988, Barto and Singh, 1990]. Convergence time is reduced by a factor of twenty over the incremental methods. 268 Moore and Atkeson Experiences to converge Real time to converge Q never Dyna-PI+ never Optimistic Dyna 55,000 1500 secs Prioritized Sweeping 14,000 330 secs Table 3: Performance on the deterministic rod-in-maze task. Both Dynas and prioritized sweeping were allowed 100 backups per experience. Finally we consider a task with a 3-d state space quantized into 15,000 potential discrete states (not all reachable). The task is shown in Figure 5 and involves finding the shortest path for a rod which can be rotated and translated. Q, Dyna-PI+, Optimistic Dyna and prioritized sweeping were all tested. The results are in Table 3. Q and Dyna-PI+ did not even travel a quarter of the way to the goal, let alone discover an optimal path, within 200,000 experiences. Optimistic Dyna and prioritized sweeping both eventually converged, with the latter requiring a third the experiences and a fifth the real time. When 2000 backups per experience were permitted, instead of 100, then both optimistic Dyna and prioritized sweeping required fewer experiences to converge. Optimistic Dyna took 21,000 experiences instead of 55,000 but took 2,900 secondsalmost twice the real time. Prioritized sweeping took 13,500 instead of 14,000 experiences-very little improvement, but it used no extra time. This indicates that for prioritized sweeping, 100 backups per observation is sufficient to make almost complete use of its observations, so that all the long term reward (J,) estimates are very close to the estimates which would be globally consistent with the transition probability estimates ('if';). Thus, we conjecture that even full dynamic programming after each experience (which would take days of real time) would do little better. Figme 5: A three-DOF problem, and the optimal solution path. Memory-based Reinforcement Learning: Efficient Computation with Prioritized Sweeping 269 f--- [3 f--rr 1--,.Y f-I" lI ....L. Figure 6: Dotted states are all those visFigure 7: A kd.-tree tessellation of state ited when the Manhattan heuristic was space of a sparse mase used 5 DISCUSSION Our investigation shows that Prioritized Sweeping can solve large state-space realtime problems with which other methods have difficulty. An important extension allows heuristics to constrain exploration decisions. For example, in finding an optimal path through a maze, many states need not be considered at all. Figure 6 shows the areas explored using a Manhattan heuristic when finding the optimal path from the lower left to the center. For some tasks we may be even satisfied to cease exploration when we have obtained a solution known to be, say, within 50% of the optimal solution. This can be achieved by using a heuristic which lies: it tells us that the best possible reward-ta-go is that of a path which is twice the length of the true shortest possible path. Furthermore, another promising avenue is prioritized sweeping in conjunction with kd-tree tessellations of state space to concentrate prioritizing sweeping on the important regions [Moore, 1991]. Other benefits of the memory-based approach, described in [Moore, 1992], allow us to control forgetting in changing environments and automatic scaling of state variables. Acknowledgements Thanks to Mary Soon Lee, Satinder Singh and Rich Sutton for useful comments on an early draft. Andrew W. Moore is supported by a Postdoctoral Fellowship from SERC/NATO. Support was also provided under Air Force Office of Scientific Research grant AFOSR-89-0500, an Alfred P. Sloan Fellowship, the W. M. Keck Foundation Associate Professorship in Biomedical Engineering, Siemens Corporation, and a National Science Foundation Presidential Young Investigator Award to Christopher G. Atkeson. 270 Moore and Atkeson References [Barto and Singh, 1990] A. G. Barto and S. P. Singh. On the Computational Economics of Reinforcement Learning. In D. S. Touretzky, editor, Connectioni.t Mode": Proceeding. of the 1990 Summer School. Morgan Kaufmann, 1990. [Kaelbling, 1990] L. P. Kaelbling. Learning in Embedded Systems. PhD. Thesisj Technical Report No. TR-90-04, Stanford University, Department of Computer Science, June 1990. [Moore and Atkeson, 1992] A. V;!. !-Ioore and C. G. Atkeson. Memory-based Reinforcement Learning: CO:lverging with Less Data and Less Real Time. In preparation, 1992. [Moore, 1991] A. W. Moore. Variable Resolution Dynamic Programming: Efficiently Learning Action Maps in Multivariate Real-valued State-spaces. In L. Birnbaum and G. Collins, editors, Machine Learning: Proceeding. of the Eighth International Work.thop. Morgan Kaufman, June 1991. [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, Advance. in Neural Information Proceuing Sydem6 4. Morgan Kaufmann, April 1992. [Nilsson, 1971] N. J. Nilsson. Problem-60lving Method6 in Artificial Intelligence. McGraw Hill, 1971. [Peng and Williams, 1992] J. Peng and R. J. Williams. Efficient Search Control in Dyna. College of Computer Science, Northeastern University, March 1992. [Sato et al., 1988] M. Sato, K. Abe, and H. Takeda. Learning Control of Finite Markov Chains with an Explicit Trade-off Between Estimation and Control. IEEE Tran6. on SY6tem6, Man, and Cybernetic.t, 18{5}:667-684, 1988. [Sutton, 1988] R. S. Sutton. Learning to Predict by the Methods of Temporal Differences. Machine Learning, 3:9-44, 1988. [Sutton, 1990] R. S. Sutton. Integrated Architecture for Learning, Planning, and Reacting Based on Approximating Dynamic Programming. In Proceeding. of the 7th International Conference on Machine Learning. Morgan Kaufman, June 1990. [Tesauro, 1991] G. J. Tesauro. Practical Issues in Temporal Difference Learning. RC 17223 (76307), IBM T. J. Watson Research Center, NY, 1991. [Watkins, 1989] C. J. C. H. Watkins. Learning from Delayed Rewards. PhD. Thesis, King's College, University of Cambridge, May 1989.
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Learning Fuzzy Rule-Based Neural Networks for Control Charles M. Higgins and Rodney M. Goodman Department of Electrical Engineering, 116-81 California Institute of Technology Pasadena, CA 91125 Abstract A three-step method for function approximation with a fuzzy system is proposed. First, the membership functions and an initial rule representation are learned; second, the rules are compressed as much as possible using information theory; and finally, a computational network is constructed to compute the function value. This system is applied to two control examples: learning the truck and trailer backer-upper control system, and learning a cruise control system for a radio-controlled model car. 1 Introduction Function approximation is the problem of estimating a function from a set of examples of its independent variables and function value. If there is prior knowledge of the type of function being learned, a mathematical model of the function can be constructed and the parameters perturbed until the best match is achieved. However, if there is no prior knowledge of the function, a model-free system such as a neural network or a fuzzy system may be employed to approximate an arbitrary nonlinear function. A neural network's inherent parallel computation is efficient for speed; however, the information learned is expressed only in the weights of the network. The advantage of fuzzy systems over neural networks is that the information learned is expressed in terms of linguistic rules. In this paper, we propose a method for learning a complete fuzzy system to approximate example data. The membership functions and a minimal set of rules are constructed automatically from the example data, and in addition the final system is expressed as a computational 350 Learning Fuzzy Rule-Based Neural Networks for Control 351 -1.0 0 1.0 Variable Value Figure 1: Membership function example Pos 5.0 (neural) network for efficient parallel computation of the function value, combining the advantages of neural networks and fuzzy systems. The proposed learning algorithm can be used to construct a fuzzy control system from examples of an existing control system's actions. Hereafter, we will refer to the function value as the output variable, and the independent variables of the function as the input variables. 2 Fuzzy Systems In a fuzzy system, a function is expressed in terms of membership functions and rules. Each variable has membership functions which partition its range into overlapping classes (see figure 1). Given these membership functions for each variable, a function may be expressed by making rules from the input space to the output space and smoothly varying between them. In order to simplify the learning of membership functions, we will specify a number of their properties beforehand. First, we will use piecewise linear membership functions. We will also specify that membership functions are fully overlapping; that is, at any given value of the variable the total membership sums to one. Given these two properties of the membership functions, we need only specify the positions of the peaks of the membership functions to completely describe them. We define a fuzzy rule as if y then X, where y (the condition side) is a conjunction in which each clause specifies an input variable and one of the membership functions associated with it, and X (the conclusion side) specifies an output variable membership function. 3 Learning a Fuzzy System from Example Data There are three steps in our method for constructing a fuzzy system: first, learn the membership functions and an initial rule representation; second, simplify (compress) the rules as much as possible using information theory; and finally, construct a computational network with the rules and membership functions to calculate the function value given the independent variables. 352 Higgins and Goodman 3.1 Learning the Membership Functions Before learning, two parameters must be specified. First, the maximum allowable RMS error of the approximation from the example data; second, the maximum number of membership functions for each variable. The system will not exceed this number of membership functions, but may use fewer if the error is reduced sufficiently before the maximum number is reached. 3.1.1 Learning by Successive Approximation to the Target Function The following procedure is performed to construct membership functions and a set of rules to approximate the given data set. All of the rules in this step are eel/based, that is, they have a condition for every input variable; there is a rule for every combination of input variables (eeIQ. We begin with input membership functions at input extrema. The closest example point to each "corner" of the input space is found and a membership function for the output is added at its value at the corner point. The initial rule set contains a rule for each corner, specifying the closest output membership function to the actual value at that corner. We now find the example point with the greatest RMS error from the current model and add membership functions in eaeh variable at that point. Next, we construct a new set of rules to approximate the function. Constructing rules simply means determining the output membership function to associate with each cell. While constructing this rule set, we also add any output membership functions which are needed. The best rule for a given cell is found by finding the closest example point to the rule (recall each rule specifies a point in the input space). If the output value at this point is "too far" from the closest output membership function value, this output value is added as a new output membership. After this addition has been made, if necessary, the closest output membership function to the value at the closest point is used as the conclusion of the rule. At this point, if the error threshold has been reached or all membership functions are full, we exit. Otherwise, we go back to find the point with the greatest error from the model and iterate again. 3.2 Simplifying the Rules In order to have as simple a fuzzy system as possible, we would like to use the minimum possible number of rules. The initial cell-based rule set can be "compressed" into a minimal set of rules; we propose the use of an information-theoretic algorithm for induction of rules from a discrete data set [1] for this purpose. The key to the use of this method is the interpretation of each of the original rules as a discrete example. The rule set becomes a discrete data set which is input to a rule-learning algorithm. This algorithm learns the best rules to describe the data set. There are two components of the rule-learning scheme. First, we need a way to tell which of two candidate rules is the best. Second, we need a way to search the space of all possible rules in order to find the best rules without simply checking every rule in the search space. Learning Fuzzy Rule-Based Neural Networks for Control 353 3.2.1 Ranking Rules Smyth and Goodman[2] have developed an information-theoretic measure of rule value with respect to a given discrete data set. This measure is known as the j-measure; defining a rule as if y then X, the j-measure can be expressed as follows: . p(Xly) p(Xly) J(Xly) = p(Xly) log2( p(X) ) + p(Xly) log2( p(X) ) [2] also suggests a modified rule measure, the J-measure: J(Xly) = p(y)j(Xly) This measure discounts rules which are not as useful in the data set in order to remove the effects of "noise" or randomness. The probabilities in both measures are computed from relative frequencies counted in the given discrete data set. Using the j-measure, examples wilt be combined only when no error is caused in the prediction ofthe data set. The J-measure, on the other hand, will combine examples even if some prediction ability of the data is lost. If we simply use the j-measure to compress our original rule set, we don't get significant compression. However, we can only tolerate a certain margin of error in prediction of our original rule set and maintain the same control performance. In order to obtain compression, we wish to allow some error, but not so much as the J-measure will create. We thus propose the following measure, which allows a gradual variation of the amount of noise tolerance: I -ax -e L(Xly) = f(p(y),a)j(XIY) where !(x,a) = 1- e- a The parameter a may be set at 0+ to obtain the J-measure since !(x,O+) = x or at 00 to obtain thej-measure, since f(x, 00) = 1 (x> 0). Any value ofa between o and 00 will result in an amount of compression between that of the J-measure and the j-measure; thus if we are able to tolerate some error in the prediction of the original rule set, we can obtain more compression than the j-measure could give us, but not as much as the J-measure would require. We show an example of the variation of a for the truck backer-upper control system in section 4.1. 3.2.2 Searching for the Best Rules In [1], we presented an efficient method for searching the space of all possible rules to find the most representative ones for discrete data sets. The basic idea is that each example is a very specific (and quite perfect) rule. However, this rule is applicable to only one example. We wish to generalize this very specific rule to cover as many examples as possible, while at the same time keeping it as correct as possible. The goodness-measures shown above are just the tool for doing this. If we calculate the "goodness" of all the rules generated by removing a single input variable from the very specific rule, then we will be able to tell if any of the slightly more general rules generated from this rule are better. If so, we take the best and continue in this manner until no more general rule with a higher "goodness" exists. When we have performed this procedure on the very specific rule generated from each example (and removed duplicates), we will have a set of rules which represents the data set. 354 Higgins and Goodman Lateral inhibitory connecti~ns Input Membership Functions Rules Defuzzification Output Membership Functions Figure 2: Computational network constructed from fuzzy system 3.3 Constructing a Network Constructing a computational network to represent a given fuzzy system can be accomplished as shown in figure 2. From input to output, layers represent input membership functions, rules, output membership functions, and finally defuzzification. A novel feature of our network is the lateral links shown in figure 2 between the outputs of various rules. These links allow inference with dependent rules. 3.3.1 The Layers of the Network The first layer contains a node for every input membership function used in the rule set. Each of these nodes responds with a value between zero and one to a certain region of the input variable range, implementing a single membership function. The second layer contains a node for each rule - each of these nodes represents a fuzzy AND, implemented as a product. The third layer contains a node for every output membership function. Each of these nodes sums the outputs from each rule that concludes that output fuzzy set. The final node simply takes the output memberships collected in the previous layer and performs a defuzzification to produce the final crisp output by normalizing the weights from each output node and performing a convex combination with the peaks of the output membership functions. 3.3.2 The Problem with Dependent Rules and a Solution There is a problem with the standard fuzzy inference techniques when used with dependent rules. Consider a rule whose conditions are all contained in a more specific rule (i.e. one with more conditions) which contradicts its conclusion. Using standard fuzzy techniques, the more general rule will drive the output to an intermediate value between the two conclusions. What we really want is that a more general rule dependent on a more specific rule should only be allowed to fire to the degree that the more specific rule is not firing. Thus the degree of firing of the more specific rule should gate the maximum firing allowed for the more general rule. This is expressed in network form in the links between the rule layer and the output membership functions layer. The lateral arrows are inhibitory connections which take the value at their input, invert it (subtract it from one), and multiply it by the value at their output. Learning Fuzzy Rule-Based Neural Networks for Control 355 '---Truck and Trailer Cab Angle --'---- '---- '-----)-Truck '---Angle '---- '---Loading t Dock Y position (of truck rear) Figure 3: The truck and trailer backer-upper problem 4 Experimental Results In this section, we show the results of two experiments: first, a truck backer-upper in simulation; and second, a simple cruise controller for a radio-controlled model car constructed in our laboratory. 4.1 Truck and Trailer Backer-Upper Jenkins and Yuhas [3] have developed by hand a very efficient neural network for solving the problem of backing up a truck and trailer to a loading dock. The truck and trailer backer-upper problem is parameterized in figure 3. The function approximator system was trained on 225 example runs of the Yuhas controller, with initial positions distributed symmetrically about the field in which the truck operates. In order to show the effect of varying the number of membership functions, we have fixed the maximum number of membership functions for the y position and cab angle at 5 and set the maximum allowable error to zero, thus guaranteeing that the system will fill out all of the allowed membership functions. We varied the maximum number of truck angle membership functions from 3 to 9. The effects of this are shown in figure 4. Note that the error decreases sharply and then holds constant, reaching its minimum at 5 membership functions. The Yuhas network performance is shown as a horizontal line. At its best, the fuzzy system performs slightly better than the system it is approximating. For this experiment, we set a goal of 33% rule compression. We varied the parameter a in the L-measure for each rule set to get the desired compression. Note in figure 4 the performance of the system with compressed rules. The performance is in every case almost identical to that of the original rule sets. The number of rules and the amount of rule compression obtained can be seen in table 1. 4.2 Cruise Controller In this section, we describe the learning of a cruise controller to keep a radio controlled model car driving at a constant speed in a circle. We designed a simple PD controller to perform this task, and then learned a fuzzy system to perform the same task. This example is not intended to suggest that a fuzzy system should replace a simple PD controller, since the fuzzy system may represent far more complex 356 Higgins and Goodman NO> , .. MO> 1\ I~ ~ \ I"-PO \ \F IIIVS .. ... \ \ \. \ , .. ... ... 1\ c;jj:jj ~ '1'.:'" \ \ \ 1"'",1 III \\ \ DO> ,0> .0> 1\ \ \ YuI iooSy\ . ... ~ Yuba Sv ... ,0> -~~---.. S , J • .. 5 , 7 I ~w~W«~Fm=~~~_ Numw ofwct qIo m=bcnhip tunCllona a) Cmtrol error: final y positim b) Cmtrol error: fmal truck angle Figure 4: Results of experiments with the truck backer-upper N umber of truck angle membership functions 3 4 5 6 7 8 9 N umber of Rules Cell-Based 75 100 125 150 175 200 225 Compressed 48 67 86 100 114 138 154 CompressIOn 36% 33% 31% 33% 35% 31% 32% Table 1: Number of rules and compression figures for learned TBU systems functions, but rather to show that the fuzzy system can learn from real control data and operate in real-time. The fuzzy system was trained on 6 runs of the PD controller which included runs going forward and backward, and conditions in which the car's speed was perturbed momentarily by blocking the car or pushing it. Figure 5 shows the error trajectory of both the hand-crafted PD and learned fuzzy control systems from rest. The car builds speed until it reaches the desired set point with a well-damped response, then holds speed for a while. At a later time, an obstacle was placed in the path of the car to stop it and then removed; figure 5 shows the similar recovery responses of both systems. It can be seen from the numerical results in table 2 that the fuzzy system performs as well as the original PD controller. No compression was attempted because the rule sets are already very small. PD Controller Learned Fuzzy System Time from 90% error to 10% error (s) 0.9 0.7 RMS error at steady state (uncal) 59 45 Time to correct after obstacle (s) 6.2 6.2 Table 2: Analysis of cruise control performance Learning Fuzzy Rule-Based Neural Networks for Control 357 • 00 .. ~ n ... " 11'1 I~ ~ u... DO I/Y -'",y' , .r V ..... · ~V' '" V'W -.00 -I -... J -. ~ 000 too 1000 1500 :3(01) 21.C1O 1)0) g<o soo 10(0 1500 lOCO 2100 :1100 Timo(o) Timo(o) a) PD COII.trol SYl1lem b) Fuzzy COII.troi System Figure 5: Performance of PD controller vs. learned fuzzy system 5 Summary and Conclusions We have presented a method which, given examples of a function and its independent variables, can construct a computational network based on fuzzy logic to predict the function given the independent variables. The user must only specify the maximum number of membership functions for each variable and the maximum RMS error from the example data. The final fuzzy system's actions can be explicitly explained in terms of rule firings. If a system designer does not like some aspect of the learned system's performance, he can simply change the rule set and the membership functions to his liking. This is in direct contrast to a neural network system, in which he would have no recourse but another round of training. Acknowledgements This work was supported in part by Pacific Bell, and in part by DARPA and ONR under grant no. NOOOI4-92-J-1860. References [1] C. Higgins and R. Goodman, "Incremental Learning using Rule-Based Neural Networks," Proceedings of the International Joint Conference on Neural Networks, vol. 1, 875-880, July 1991. [2] R. Goodman, C. Higgins, J. Miller, P. Smyth, "Rule-Based Networks for Classification and Probability Estimation," Neural Computation 4(6),781-804, November 1992. [3] R. Jenkins and B. Yuhas, "A Simplified Neural-Network Solution through Problem Decomposition: The Case of the Truck Backer-Upper," Neural Computation 4(5), 647-9, September 1992. PART IV VISUAL PROCESSING
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Neural Network On-Line Learning Control of Spacecraft Smart Structures Dr. Christopher Bowman Ball Aerospace Systems Group P.O. Box 1062 Boulder. CO 80306 Abstract The overall goal is to reduce spacecraft weight. volume, and cost by online adaptive non-linear control of flexible structural components. The objective of this effort is to develop an adaptive Neural Network (NN) controller for the Ball C-Side 1m x 3m antenna with embedded actuators and the RAMS sensor system. A traditional optimal controller for the major modes is provided perturbations by the NN to compensate for unknown residual modes. On-line training of recurrent and feed-forward NN architectures have achieved adaptive vibration control with unknown modal variations and noisy measurements. On-line training feedback to each actuator NN output is computed via Newton's method to reduce the difference between desired and achieved antenna positions. 1 ADAPTIVE CONTROL BACKGROUND The two traditional approaches to adaptive control are 1) direct control (such as perfonned in direct model reference adaptive controllers) and 2) indirect control (such as performed by explicit self-tuning regulators). Direct control techniques (e.g. model-reference adaptive cootrul) provide good stability however are susceptible to noise. Whereas indirect control techn;'q~es (e.g. explicit self-tuning regulators) have low noise susceptibility and good convergence rate. However they require more control effort and have worse stability and are less roblistto mismodeling. NNs synergistically augment traditional adaptive control techniques by providing improved mismodeling robustness both adaptively on-line for time-varying dynamics as well as in a learned control mode at a slower rate. The NN control approaches which correspond to direct and indirect adaptive control are commonly known as inverse and forward modeling. respectively. More specifically, aNN which maps the plant state and its desired perfonnance to the control command is called an inverse model, a NN mapping both the current plant state and control to the next state and its performance is called the forward model. When given a desired performwce and the current state. the inverse model generates the control. see Figure 1. The actual perfonnance is observed and is used to train/update the inverse model. A significant problem occurs when the desired and achieved perfonnance differ greatly since the model near the desired slate is not changed. This condition is corrected by adding random noise to the control outputs so as to extend the state space 303 304 Bowman being explored. However, this correction has the effect of slowing the learning and reducing broadband stability. easurements Trainin , .. ---=.,:-:-=.....n..:=; __ ~-----' "" n;o:uu;u.;k ,." lnV_~;""1 Con 1 Nonlinear r~ s .... ""' 1 L.!!.:.~-- ents Nel!Pll'Controller ~-~ Structures x Filters Y ... " Previous controls and stale measurements Figure 1: Direct Adaptive Control Using Inverse Modeling Neural Network Controller Trainin " " Feedback " " II~F~=fI-..1 Current and Provisions state "" pr;.::ev.;.;l;.;;,O.::.;Us;..;c;.;o;;.;n.;;.tro~ ___ .....;;meas=;.;;urements I':"" -lnv-I-~-'M"'I""ode-I" Control Net.Tl!JtControlier u I' 1 N_ ~i -I ~M=eas=ur=e=m=en=ts=~ Structures x Fillers y Previous controls and stale measurements ements y Figure 2: Dual (Indirect and Direct) Adaptive Control Using Forward Modeling Neural Network State Predictor To Aid Inverse Model Convergence For forward modeling the map from the current control and state to the resulting state and performance is learned, see Figure 2. For cases where the performance is evaluated at a future time (i.e. distal in time), a predictive critic [Barto and Sutton, 1989] NN model is learned. In both cases the Jacobian of this performance can be computed to iteratively generate the next control action. However, this differentiating of the critic NN for backpropagation training of the controller network is very slow and in some cases steers the searching the wrong direction due to initial erroneous forward model estimates. As the NN adapts itself the performance flattens which results in the slow halting of learning at an Neural Network On-Line Learning Control of Spacecraft Smart Structures 305 unacceptable solution. Adding noise to the controller's output [Jordan and Jacobs, 1990] breaks the redundancy but forces the critic to predict the effects of future noise. This problem has been solved by using a separately trained intermediate plant model to predict the next state from the prior state and control while having an independent predictor model generate the performance evaluation from the plant model predicted state [Werbos, 1990] and [Brody, 1991]. The result is a 50-100 fold learning speed improvement over reinforcement training of the forward model controller NN. However, this method still relies on a "good" forward model to incrementally train the inverse model. These incremental changes can still lead to undesirable solutions. For control systems which follow the stage 1,2 or 3 models given in [Narendra, 1991) the control can be analytically computed from a forward-only model. For the most general, non-linear (stage 4) systems, an alternative is the memory-based forward model [Moore, 1992]. Using only a forward NN model, a direct hill-climbing or Newton's method search of candidate actions can be applied until a control decision is reached. The resulting state and its performance are used for on-line training of the forward model. Judicial random control actions are applied to improve behavior only where the forward model error is predicted to be large (e.g. via cross-validation). Also using robust regression, experiences can be deweighted according to their quality and their age. The high computational burden of these cross-validation techniques can be reduced by parallel on-line processing providing the "policy" parameters for fast on-line NN control. For control problems which are distal in time and space, a hybrid of these two forwardmodeling approaches can be used. Namely, a NN plant model is added which is trained off-line in real-time and updated as necessary at a slower rate than the on-line forward model which predicts performance based upon the current plant model. This slower rate trained forward-model NN supports learned control (e.g. via numerical inversion) whereas the on-line forward model provides the faster response adaptive control. Other NN control techniques such as using a Hopfield net to solve the optimal-control quadraticprogramming problem or the supervised training of ART II off-line with adaptive vigilance for on-line pole placement have been proposed. However, their on-line robustness appears limited due to their sensitivity to a priori parameter assumptions. A forward model NN which augments a traditional controller for unmodeled modes and unforeseen situations is presented in the following section. Performance results for both feed-faward and current learning versions are compared in Section 3. 2 RESIDUAL FORWARD MODEL NEURAL NETWORK (RFM-NN) CONTROLLER A type of forward model NN which acts as a residual mode mter to support a reduced-order model (ROM) traditional optimal state controller has been evaluated. see Figure 3. The ROM determines the control based upon its model coordinate approximate representation of the structure. Model coordinates are obtained by a transformation using known primary vibration modes, [Young, 1990]. The transformation operator is a set of eigenvectors (mode shapes) generated by finite element modeling. The ROM controller is traditionally augmented by a residual-mode mter (RMF). Ball's RFM-NN Ball's RFM-NN replaces the RMF in order to better capture the mismodeled. unmodeled and changing modes. The objective of the RFM-NN is to provide ROM controller with ROM derivative state perturbations, so that the ROM controls the structure as desired by the user. The RFMNN is trained on-line using scored supervised feedback to generate these desired ROM state perturbations. The scored supervised training provides a score for each perturbation output based upon the measured position of the structure. The measured deviations, Y*(t), from the desired structure position are converted to errors in the estimated ROM state using the ROM. transformation. Specifically, the training score, S(t), for each ROM derivative state XN (t) is expressed in the following discrete equation: .. S(t) = BN Y * (t) - xN(t) where *N(t) = [AN + BNGN - KNCN]XN(t -1) + KN Y(t -1) 306 Bowman " ;" ," .. :,::..-:.;" .. :: . , . . ; .} ::: ': .. : ", Figure 3: Residual Forward Model Neural Network Adaptive Controller Replaces Traditional Residual Mode Filter Newton's method is then applied to find lbe 0* (1) ROM state ~ations which zero the score. First, the score is smoothed, Set) = ~S(t -1) + (1- o)S(t) and the neural network output is smoothed similarly. Second, Newton's method computes the adjusbnents needed to zero the scores, ~(O*N(t» = -S(t)(8iN(t) - 8iN(t -1» I [S(t) - Set -1)] = -EXN(t) (if either difference = 0) Third, the NN is trained, 8*T(t + 1) = ~(8iN(t» + 8iN(t) with the appropriate learning rate, a (e.g. approximation to inverse of largest eigenvalue of the Hessian weight matrix). 3 RFM-NN ON-LINE LEARNING RESULTS Both feed-forward and recurrent RFM-NNs have been incorporated into an interactive simulation of Ball's Control-Structure Interaction Demonstration Experiment (C-SIDE) see Figure 4. This 1m x 3m lightweight antenna facesheet has 8 embedded actuators plus three auxiliary input actuators and uses 8 remote angular measurement sensors (RAMS) plus 4 displacement and 3 velocity auxiliary sensors. In order to evaluate the on-line performance of the RFM-NNs the ROM controller was given insufficient and partially incorrect modes. The ROM without the RFM-NN grew unstable (i.e. greater than 10 millimeter C-SIDE displacements) in 13 seconds. The initial feed-forward RFM-NN used 8 sensor and 6 ROM state feedback estimate inputs as well as 5 hidden units and 3 ROM velocity state perturbation outputs. This RFM-NN had random initial weights, logistic Neural Network On-Line Learning Control of Spacecraft Smart Structures 307 activation functions. and back-propagation training using one sixth the learning rate for the output layers (e.g .. 06 and .01). Newton RFM-NN training search used a step size of one with smoothing factor of one tenth. Figure 4: 1m x 3m C-SIDE Antenna Facesheet With Embedded Actuators. This RFM-NN learned on-line to stabilized and reduce vibration to less than ±Imm within 20 seconds, see Figure 5. A five Newton force applied a few seconds later is compensated for within nine seconds, see Figure 6. This is accomplished with learning off as well as when on. To test the necessity of the RFM-NN the ROM was given the scored supervised training (Le. Newton's search estimates) directly instead of the RFMNN outputs. This caused immediate unstable behavior. To test the RFM-NN sensitivity to measurement accuracy a unifonn error of ±5% was added. Starting from the same random weight start the RFM-NN required 25 seconds to learn to stabilize the antenna, see Figure 7. The best stability was achieved when the product of the Newton and BPN steps was approximately .01. This feed-forward NN was compared to an Elman-type recurrent NN (i.e. hidden layer feedback to itself with one-step BP training). The recurrent RFM-NN on-line learning stability was much less sensitive to initial weights. The recurrent RFM-NN stabilized C-SIDE with up to 10% - 20% measurement noise versus 5% - 10% limit for feed-forward RFM-NN. 4 SUMMARY AND RECOMMENDATIONS Adaptive smart sbUctures promise to reduce spacecraft weight and dependence on extensive ground monitoring. A recurrent forward model NN is used as a residual mode fllter to augment a traditional reduced-order model (ROM) controller. It was more robust than the feed-forward NN and the traditional-only controller in the presence of unmodeled modes and noisy measurements. Further analyses and hardware implementations will be perfonned to better quantify this robustness including the sensitivity to the ROM controller mode fidelity, number of output modes. learning rates, measurement-to-state errors, and time quantization effects. To improve robustness to ROM mode changes a comparison to the dual forward/inverse NN control approach is recommended. The forward model will adjust the search used to train an inverse model which provides control augmentations to the ROM controller. This will enable control searches to occur both off-line faster than real-time using the forward model (Le. imagination) and on-line using direct search trials with varying noise levels. The forward model will adapt using quality experiences (e.g. via cross validation) which improves inverse models searches. The inverse model reliance on forward model will reduce until forward model prediction errors increase. Future challenges. include solving 308 Bowman (-SIll APtU'lcial "8UNl ttetuaI'k Reai411&1 no .. 1 Cantrall ... ROft State Esti .... tea ROft State Ed inat. AcIjud_nts /) /"f'Xi r\ I I I· • \ X \i i \, '0xYJ Figure 5: RFM-NN On-Line Learning To Achieve Stable Control (-Sill APtltlcial "lW'al Hetwal'k Reai411&1 I10MI Cantrall ... DiapllClIII8IIt lteuvennta (+.(-18I111d '/\X~, . , \; \ . , Figure 5: RFM-NN On-Line Learning To Achieve Stable Control (concluded) the temporal credit assignment problem, partitioning to restricted chip sizes, combining with incomplete a priori knowledge, and balancing adaptivity of response with long-term learning. The goal is to extend stabiJity-dominated, fixed-goal traditional control with adaptive robotic-type neural control to enable better autonomous control where fullyjustified fixed models and complete system knowledge are not required. The resultant robust autonomous control will capitalize on the speed of massively parallel analog neural-like computations (e.g. with NN pulse stream chips). Neural Network On-Line Learning Control of Spacecraft Smart Structures 309 C-SIJI APtificial tteuNl IIetuaI'k Resiaul IIoUI Cantrall ... lb.,: 36.4& Paus. - Hit to conti.... lletworll: Figure 6: 5 Newton Force Vibration Removed Using RFM-NN Learned Forward Model C-SIJlI APtificial ltauPal tIIriwarIc Reaiwl twal Cantrall ... fl,.: 25.28 Pause.t - Hit to contb... network: . ~,~ .. ... ~.; ' . .. \i r~ .\ " .. ~. l \. ~. \,: \l "-. / ,J... R~ Est inat. Acljustaents : -'. --.~ ",. ~~- . ":::=<I>~-::::. :::::.:::;::::::::::::::-=:-==--=~O:::= ' =.::..-.:.-= ..... ~==::::.::::: ..... :::_== / ~ .. , "t·"'" ..: ~ --... I ~; ~~~ __ ~ ______ .~ __ ~ __ ~.~ - .-~ . ~.---~.-~-\, ... ---/./ Figure 7: RFM·NN Learning to Remove Vibrations in C·SIDE With ±15% Noisy Displacement Measurements 310 Bowman 5 REFERENCES Barto, A.G., Sutton, R.S .• and Watkins, CJ.C.H., Learning and Sequential Decision Making. Univ. of Mass. at Amherst COINS Technical Report 89-95, September 1989 Bowman, C.L., Adaptive Neural Networks Applied to Signal Recognition, 3rd TriService Data fusion Symposium, May 1989 Brody, Carlos, Fast Learning With Predictive Forward Models. Neural Information Processing Systems 4 (NIPS4), 1992 Jorden, M.I., and Jacobs, R.A., Learning to Control and Unstable System with Forward Modeling, in D.S. Touretzky, ed., Advances in NIPS 2, Morgan Kaufmann 1990. Moore, A.W., Fast. Robust Adaptive Control by Learning Only Forward Models. NIPS 4, 1992 Mukhopadhyay S. and Narendra, D.S .• Disturbance Rejection in Nonlinear Systems Using Neural Networks Yale University Report No. 9114 December 1991 Werbos, P., Architectures For Reinforcement Learning, in Miller, Sutton and Werbos. ed., Neural Networks for Control, MIT Press 1990 Young, D.O., Distributed Finite-Element Modeling and Control Approach for Large Flexible Structures, J. of Guidance, Control and Dynamics, Vol. 13 (4),703-713,1990
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Hidden Markov Model Induction by Bayesian Model Merging Andreas Stolcke*'** *Computer Science Division University of California Berkeley, CA 94720 stolcke@icsi.berkeley.edu Stephen Omohundro" **International Computer Science Institute 1947 Center Street, Suite 600 Berkeley, CA 94704 om@icsi.berkeley.edu Abstract This paper describes a technique for learning both the number of states and the topology of Hidden Markov Models from examples. The induction process starts with the most specific model consistent with the training data and generalizes by successively merging states. Both the choice of states to merge and the stopping criterion are guided by the Bayesian posterior probability. We compare our algorithm with the Baum-Welch method of estimating fixed-size models, and find that it can induce minimal HMMs from data in cases where fixed estimation does not converge or requires redundant parameters to converge. 1 INTRODUCTION AND OVERVIEW Hidden Markov Models (HMMs) are a well-studied approach to the modelling of sequence data. HMMs can be viewed as a stochastic generalization of finite-state automata, where both the transitions between states and the generation of output symbols are governed by probability distributions. HMMs have been important in speech recognition (Rabiner & Juang, 1986), cryptography, and more recently in other areas such as protein classification and alignment (Haussler, Krogh, Mian & SjOlander, 1992; Baldi, Chauvin, Hunkapiller & McClure, 1993). Practitioners have typically chosen the HMM topology by hand, so that learning the HMM from sample data means estimating only a fixed number of model parameters. The standard approach is to find a maximum likelihood (ML) or maximum a posteriori probability (MAP) estimate of the HMM parameters. The Baum-Welch algorithm uses dynamic programming 11 12 Stokke and Omohundro to approximate these estimates (Baum, Petrie, Soules & Weiss, 1970). A more general problem is to additionally find the best HMM topology. This includes both the number of states and the connectivity (the non-zero transitions and emissions). One could exhaustively search the model space using the Baum-Welch algorithm on fully connected models of varying sizes, picking the model size and topology with the highest posterior probability. (Maximum likelihood estimation is not useful for this comparison since larger models usually fit the data better.) This approach is very costly and BaumWelch may get stuck at sub-optimal local maxima. Our comparative results later in the paper show that this often occurs in practice. The problem can be somewhat alleviated by sampling from several initial conditions, but at a further increase in computational cost. The HMM induction method proposed in this paper tackles the structure learning problem in an incremental way. Rather than estimating a fixed-size model from scratch for various sizes, the model size is adjusted as new evidence arrives. There are two opposing tendencies in adjusting the model size and structure. Initially new data adds to the model size, because the HMM has to be augmented to accommodate the new samples. If enough data of a similar structure is available, however, the algorithm collapses the shared structure, decreasing the model size. The merging of structure is also what drives generalization, Le., creates HMMs that generate data not seen during training. Beyond being incremental, our algorithm is data-driven, in that the samples themselves completely determine the initial model shape. Baum-Welch estimation, by comparison, uses an initially random set of parameters for a given-sized HMM and iteratively updates them until a point is found at which the sample likelihood is locally maximal. What seems intuitively troublesome with this approach is that the initial model is completely uninformed by the data. The sample data directs the model formation process only in an indirect manner as the model approaches a meaningful shape. 2 HIDDEN MARKOV MODELS For lack of space we cannot give a full introduction to HMMs here; see Rabiner & Juang (1986) for details. Briefly, an HMM consists of states and transitions like a Markov chain. In the discrete version considered here, it generates strings by performing random walks between an initial and a final state, outputting symbols at every state in between. The probability P(xlM) that a model M generates a string x is determined by the conditional probabilities of making a transition from one state to another and the probability of emitting each symbol from each state. Once these are given, the probability of a particular path through the model generating the string can be computed as the product of all transition and emission probabilities along the path. The probability of a string x is the sum of the probabilities of all paths generating x. For example, the model M3 in Figure 1 generates the strings ab, abab, ababab, ... with b b·l·· 2 2 2 . I pro a I lUes 3' 3!' 3!' ... , respective y. 3 HMM INDUCTION BY STATE MERGING 3.1 MODEL MERGING Omohundro (1992) has proposed an approach to statistical model inference in which initial Hidden Markov Model Induction by Bayesian Model Merging 13 models simply replicate the data and generalize by similarity. As more data is received, component models are fit from more complex model spaces. This allows the formation of arbitrarily complex models without overfitting along the way. The elementary step used in modifying the overall model is a merging of sub-models, collapsing the sample sets for the corresponding sample regions. The search for sub-models to merge is guided by an attempt to sacrifice as little of the sample likelihood as possible as a result of the merging process. This search can be done very efficiently if (a) a greedy search strategy can be used, and (b) likelihood computations can be done locally for each sub-model and don't require global recomputation on each model update. 3.2 STATE MERGING IN HMMS We have applied this general approach to the HMM learning task. We describe the algorithm here mostly by presenting an example. The details are available in Stolcke & Omohundro (1993). To obtain an initial model from the data, we first construct an HMM which produces exactly the input strings. The start state has as many outgoing transitions as there are strings and each string is represented by a unique path with one state per sample symbol. The probability of entering these paths from the start state is uniformly distributed. Within each path there is a unique transition arc whose probability is 1. The emission probabilities are 1 for each state to produce the corresponding symbol. As an example, consider the regular language (abt and two samples drawn from it, the strings ab and abab. The algorithm constructs the initial model Mo depicted in Figure 1. This is the most specific model accounting for the observed data. It assigns each sample a probability equal to its relative frequency, and is therefore a maximum likelihood model for the data. Learning from the sample data means generalizing from it. This implies trading off model likelihood against some sort of bias towards 'simpler' models, expressed by a prior probability distribution over HMMs. Bayesian analysis provides a formal basis for this tradeoff. Bayes' rule tells us that the posterior model probability P(Mlx) is proportional to the product of the model prior P(M) and the likelihood of the data P(xlM). Smaller or simpler models will have a higher prior and this can outweigh the drop in likelihood as long as the generalization is conservative and keeps the model close to the data. The choice of model priors is discussed in the next section. The fundamental idea exploited here is that the initial model Mo can be gradually transformed into the generating model by repeatedly merging states. The intuition for this heuristic comes from the fact that if we take the paths that generate the samples in an actual generating HMM M and 'unroll' them to make them completely disjoint, we obtain Mo. The iterative merging process, then, is an attempt to undo the unrolling, tracing a search through the model space back to the generating model. Merging two states q] and q2 in this context means replacing q] and q2 by a new state r with a transition distribution that is a weighted mixture of the transition probabilities of q], q2, and with a similar mixture distribution for the emissions. Transition probabilities into q] or q2 are added up and redirected to r. The weights used in forming the mixture distributions are the relative frequencies with which q] and q2 are visited in the current model. Repeatedly performing such merging operations yields a sequence of models Mo, MJ , 14 Stokke and Omohundro Mo: a b log L(xIMo) = - 1. 39 log L(xIMJ) = log L(xIMo) a b Figure I: Sequence of models obtained by merging samples {ab, abab}. All transitions without special annotations have probability 1; Output symbols appear above their respective states and also carry an implicit probability of 1. For each model the log likelihood is given. M2 •... , along which we can search for the MAP model. To make the search for M efficient, we use a greedy strategy: given Mi. choose a pair of states for merging that maximizes P(Mi+llX)· Continuing with the previous example, we find that states 1 and 3 in Mo can be merged without penalizing the likelihood. This is because they have identical outputs and the loss due to merging the outgoing transitions is compensated by the merging of the incoming transitions. The .5/.5 split is simply transferred to outgoing transitions of the merged state. The same situation obtains for states 2 and 4 once 1 and 3 are merged. From these two first merges we get model M. in Figure 1. By convention we reuse the smaller of two state indices to denote the merged state. At this point the best merge turns out to be between states 2 and 6, giving model M2. However, there is a penalty in likelihood, which decreases to about .59 of its previous value. Under all the reasonable priors we considered (see below), the posterior model probability still increases due to an increase in the prior. Note that the transition probability ratio at state 2 is now 2/1, since two samples make use of the first transition, whereas only one takes the second transition. Finally, states 1 and 5 can be merged without penalty to give M3, the minimal model that generates (ab)+. Further merging at this point would reduce the likelihood by three orders of magnitude. The resulting decrease in the posterior probability tells the algorithm to stop Hidden Markov Model Induction by Bayesian Model Merging 15 at this point. 3.3 MODEL PRIORS As noted previously, the likelihoods P(XIMj ) along the sequence of models considered by the algorithm is monotonically decreasing. The prior P(M) must account for an overall increase in posterior probability, and is therefore the driving force behind generalization. As in the work on Bayesian learning of classification trees by Buntine (1992), we can split the prior P(M) into a term accounting for the model structure, P(Ms), and a term for the adjustable parameters in a fixed structure P(MpIMs). We initially relied on the structural prior only, incorporating an explicit bias towards smaller models. Size here is some function of the number of states and/or transitions, IMI. Such a prior can be obtained by making P(Ms) <X e- 1M1 , and can be viewed as a description length prior that penalizes models according to their coding length (Rissanen, 1983; Wallace & Freeman, 1987). The constants in this "MOL" term had to be adjusted by hand from examples of 'desirable' generalization. For the parameter prior P(MpIMs), it is standard practice to apply some sort of smoothing or regularizing prior to avoid overfitting the model parameters. Since both the transition and the emission probabilities are given by multinomial distributions it is natural to use a Dirichlet conjugate prior in this case (Berger, 1985). The effect of this prior is equivalent to having a number of 'virtual' samples for each of the possible transitions and emissions which are added to the actual samples when it comes to estimating the most likely parameter settings. In our case, the virtual samples made equal use of all potential transitions and emissions, adding bias towards uniform transition and emission probabilities. We found that the Dirichlet priors by themselves produce an implicit bias towards smaller models, a phenomenon that can be explained as follows. The prior alone results in a model with uniform, flat distributions. Adding actual samples has the effect of putting bumps into the posterior distributions, so as to fit the data. The more samples are available, the more peaked the posteriors will get around the maximum likelihood estimates of the parameters, increasing the MAP value. In estimating HMM parameters, what counts is not the total number of samples, but the number of samples per state, since transition and emission distributions are local to each state. As we merge states, the available evidence gets shared by fewer states, thus allowing the remaining states to produce a better fit to the data. This phenomenon is similar, but not identical, to the Bayesian 'Occam factors' that prefer models with fewer parameter (MacKay, 1992). Occam factors are a result of integrating the posterior over the parameter space, something which we do not do because of the computational complications it introduces in HMMs (see below). 3.4 APPROXIMATIONS At each iteration step, our algorithm evaluates the posterior resulting from every possible merge in the current HMM. To keep this procedure feasible, a number of approximations are incorporated in the implementation that don't seem to affect its qualitative properties. • For the purpose of likelihood computation, we consider only the most likely path through the model for a given sample string (the Viterbi path). This allows us to 16 Stokke and Omohundro express the likelihood in product form, computable from sufficient statistics for each transition and emission. • We assume the Viterbi paths are preserved by the merging operation, that is, the paths previously passing through the merged states now go through the resulting new state. This allows us to update the sufficient statistics incrementally, and means only O(number of states) likelihood terms need to be recomputed. • The posterior probability of the model structure is approximated by the posterior of the MAP estimates for the model parameters. Rigorously integrating over all parameter values is not feasible since varying even a single parameter could change the paths of all samples through the HMM. • Finally, it has to be kept in mind that our search procedure along the sequence of merged models finds only local optima, since we stop as soon as the posterior starts to decrease. A full search of the space would be much more costly. However, we found a best-first look-ahead strategy to be sufficient in rare cases where a local maximum caused a problem. In those cases we continue merging along the best-first path for a fixed number of steps (typically one) to check whether the posterior has undergone just a temporary decrease. 4 EXPERIMENTS We have used various artificial finite-state languages to test our algorithm and compare its performance to the standard Baum-Welch algorithm. Table 1 summarizes the results on the two sample languages ac· a u bc· band a+ b+ a+ b+. The first of these contains a contingency between initial and final symbols that can be hard for learning algorithms to uncover. We used no explicit model size prior in our experiments after we found that the Dirichlet prior was very robust in giving just the the right amount of bias toward smaller models.! Summarizing the results, we found that merging very reliably found the generating model structure from a very small number of samples. The parameter values are determined by the sample set statistics. The Baum-Welch algorithm, much like a backpropagation network, may be sensitive to its random initial parameter settings. We therefore sampled from a number of initial conditions. Interestingly, we found that Baum-Welch has a good chance of settling into a suboptimal HMM structure, especially if the number of states is the minimal number required for the target language. It proved much easier to estimate correct language models when extra states were provided. Also, increasing the sample size helped it converge to the target model. 5 RELATED WORK Our approach is related to several other approaches in the literature. The concept of state merging is implicit in the notion of state equivalence classes, which is fundamental to much of automata theory (Hopcroft & Ullman, 1979) and has been applied I The number of 'virtual' samples per transition/emission was held constant at 0.1 throughout. Hidden Markov Model Induction by Bayesian Model Merging 17 (a) Method Sample Entropy Cross-entropy Language n Merging 8 m.p. 2.295 2.188 ± .020 ac'a v bc'b 6 Merging 20 random 2.087 2.158 ± .033 ac'a v bc'b 6 Baum-Welch 8 m.p. 2.087 2.894 ± .023 (best) (a v b)c'(a v b) 6 (10 trials) 2.773 4.291 ± .228 (worst) (a v b)c'(a v b) 6 Baum-Welch 20 random 2.087 2.105 ± .031 (best) ac'a v bc'b 6 (l0 trials) 2.775 2.825 ± .031 (worst) (a v b)c'(a v b) 6 Baum-Welch 8 m.p. 2.384 3.914 ± .271 ac'a v bc'b 10 Baum-Welch 20 random 2.085 2.155 ± .032 ac'a v bc'b 10 (b) Method Sample Entropy Cross-entropy Language n Merging 5m.p. 2.163 7.678 ± .158 a+b+a+b+ 4 Baum-Welch 5m.p. 3.545 8.963 ± .161 (best) (a+b+t 4 (3 trials) 3.287 59.663 ± .007 (worst) (a+b+t 4 Merging 10 random 5.009 5.623 ± .074 a+b+a+b+ 4 Baum-We1ch 10 random 5.009 5.688 ± .076 (best) a+b+a+b+ 4 (3 trials) 6.109 8.395 ± .137 (worst) (a+b+t 4 Table 1: Results for merging and Baum-Welch on two regular languages: (a) ac'a v bc'b and (b) a+b+a+b+. Samples were either the top most probable (m.p.) ones from the target language, or a set of randomly generated ones. 'Entropy' is the average negative log probability on the training set, whereas 'cross-entropy' refers to the empirical cross-entropy between the induced model and the generating model (the lower, the better generalization). n denotes the final number of model states for merging, or the fixed model size for Baum-Welch. For Baum-Welch, both best and worst performance over several initial conditions is listed. to automata learning as well (Angluin & Smith, 1983). Tomita (1982) is an example of finite-state model space search guided by a (nonprobabilistic) goodness measure. Horning (1969) describes a Bayesian grammar induction procedure that searches the model space exhaustively for the MAP model. The procedure provably finds the globally optimal grammar in finite time, but is infeasible in practice because of its enumerative character. The incremental augmentation of the HMM by merging in new samples has some of the flavor of the algorithm used by Porat & Feldman (1991) to induce a finite-state model from positive-only, ordered examples. Haussler et al. (1992) use limited HMM 'surgery' (insertions and deletions in a linear HMM) to adjust the model size to the data, while keeping the topology unchanged. 6 FURTHER RESEARCH We are investigating several real-world applications for our method. One task is the construction of unified multiple-pronunciation word models for speech recognition. This is currently being carried out in collaboration with Chuck Wooters at ICSI, and it appears that our merging algorithm is able to produce linguistically adequate phonetic models. Another direction involves an extension of the model space to stochastic context-free grammars, for which a standard estimation method analogous to Baum-Welch exists (Lari 18 Stokke and Omohundro & Young, 1990). The notions of sample incorporation and merging carry over to this domain (with merging now involving the non-terminals of the CFO), but need to be complemented with a mechanism that adds new non-terminals to create hierarchical structure (which we call chunking). Acknowledgements We would like to thank Peter Cheeseman, Wray Buntine, David Stoutamire, and Jerry Feldman for helpful discussions of the issues in this paper. References Angluin, D. & Smith, C. H. (1983), 'Inductive inference: Theory and methods', ACM Computing Surveys 15(3), 237-269. Baldi, P., Chauvin, Y., Hunkapiller, T. & McClure, M. A. (1993), 'Hidden Markov Models in molecular biology: New algorithms and applications' , this volume. Baum, L. E., Petrie, T., Soules, G. & Weiss, N. (1970), 'A maximization technique occuring in the statistical analysis of probabilistic functions in Markov chains', The Annals of Mathematical Statistics 41( 1), 164-17l. Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis, Springer Verlag, New York. Buntine, W. (1992), Learning classification trees, in D. J. Hand, ed., 'Artificial Intelligence Frontiers in Statistics: AI and Statistics III' , Chapman & Hall. Haussler, D., Krogh, A., Mian, 1. S. & Sjolander, K. (1992), Protein modeling using hidden Markov models: Analysis of globins, Technical Report UCSC-CRL-92-23, Computer and Information Sciences, University of California, Santa Cruz, Ca. Revised Sept. 1992. Hopcroft, J. E. & Ullman, J. D. (1979), Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, Mass. Horning, J. J. (1969), A study of grammatical inference, Technical Report CS 139, Computer Science Department, Stanford University, Stanford, Ca. Lari, K. & Young, S. J. (1990), 'The estimation of stochastic context-free grammars using the Inside-Outside algorithm' , Computer Speech and Language 4, 35-56. MacKay, D. J. C. (1992), 'Bayesian interpolation', Neural Computation 4,415-447. Omohundro, S. M. (1992), Best-first model merging for dynamic learning and recognition, Technical Report TR-92-004, International Computer Science Institute, Berkeley, Ca. Porat, S. & Feldman, 1. A. (1991), 'Learning automata from ordered examples', Machine Learning 7, 109-138. Rabiner, L. R. & Juang, B. H. (1986), 'An introduction to Hidden Markov Models', IEEE ASSP Magazine 3(1), 4-16. Rissanen, J. (1983), 'A universal prior for integers and estimation by minimum description length', The Annals of Statistics 11(2), 416-431 . Stolcke, A. & Omohundro, S. (1993), Best-first model merging for Hidden Markov Model induction, Technical Report TR-93-003, International Computer Science Institute, Berkeley, Ca. Tomita, M. (1982), Dynamic construction of finite automata from examples using hill-climbing, in 'Proceedings of the 4th Annual Conference of the Cognitive Science Society', Ann Arbor, Mich., pp. 105-108. Wallace, C. S. & Freeman, P. R. (1987), 'Estimation and inference by compact coding', Journal of the Royal Statistical Society, Series B 49(3),240-265.
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Statistical Mechanics of Learning Large Committee Machine Holm Schwarze CONNECT, The Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark John Hertz· Nordita Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Abstract • In a We use statistical mechanics to study generalization in large committee machines. For an architecture with nonoverlapping receptive fields a replica calculation yields the generalization error in the limit of a large number of hidden units. For continuous weights the generalization error falls off asymptotically inversely proportional to Q, the number of training examples per weight. For binary weights we find a discontinuous transition from poor to perfect generalization followed by a wide region of metastability. Broken replica symmetry is found within this region at low temperatures. For a fully connected architecture the generalization error is calculated within the annealed approximation. For both binary and continuous weights we find transitions from a symmetric state to one with specialized hidden units, accompanied by discontinuous drops in the generalization error. 1 Introduction There has been a good deal of theoretical work on calcula.ting the generalization ability of neural networks within the fra.mework of statistical mechanics (for a review • Address in 1993: Laboratory of Neuropsychology, NIMH, Bethesda, MD 20892, USA 523 524 Schwarze and Hertz see e.g. Watkin et.al., 1992; Seung et.al., 1992). This approach has mostly been applied to single-layer nets (e.g. Gyorgyi and Tishby, 1990; Seung et.al., 1992). Extensions to networks with a hidden layer include a model with small hidden receptive fields (Sompolinskyand Tishby, 1990), some general results on networks whose outputs are continuous functions of their inputs (Seung et.al., 1992; Krogh and Hertz, 1992), and calculations for a so-called committee machine (Nilsson, 1965), a two-layer Boolean network, which implements a majority decision of the hidden units (Schwarze et.al., 1992; Schwarze and Hertz, 1992; Mato and Parga, 1992; Barkai et.al., 1992; Engel et.al., 1992). This model has previomlly been studied when learning a function which could be implemented by a simple perceptron (i.e. one with no hidden units) in the high-temperature (i.e. high-noise) limit (Schwarze et.al., 1992). In most practical applications, however, the function to be learnt is not linearly separable. Therefore, we consider here a committee machine trained on a rule which itself is defined by another committee machine (the 'teacher' network) and hence not linearly separable. We calculate the generalization error, the probability of misclassifying an arbitrary new input, as a function of 0, the ratio of the number of training examples P to the number of adjustable weights in the network. First we present results for the 'tree' committee machine, a restricted version of the model in which the receptive fields of the hidden units do not overlap. In section 3 we study a fully connected architecture allowing for correlations between different hidden units in the student network. In both cases we study a large-net limit in which the total number of inputs (N) and the number of hidden units (K) both go to infinity, but with K «: N. 2 Committee machine with nonoverlapping receptive fields In this model each hidden unit receives its input from N I K input units, subject to the restriction that different hidden units do not share common inputs. Therefore there is only one path from each input unit to the output. The hidden-output weights are all fixed to +1 as to implement a majority decision of the hidden units. The overall network output for inputs 5, E R N/K, 1 = 1, ... , K, to the K branches is given by 0"( {S,}) = sign ( ~ t. 0", (5,») , (1) where 0"1 is the output of the lth hidden unit, given by 0".(5,) = sign ( 1ft w, . 5.) . (2) Here W, is the N I K -dimensional weight vector connecting the input with the Ith hidden unit. The training examples ({~#-' ,}, r( {~#-' ,}), j.£ = I, ... , P, are generated by another committee machine with weight vectors 11, and an overall output r({~#-'I})' defined analogously to (1). There are N adjustable weights in the network, and therefore we have 0 = PIN. Statistical Mechanics of Learning in a Large Committee Machine 525 As in the corresponding calculations for simple perceptrons (Gardner and Derrida, 1988; Gyorgyi and Tishby, 1990; Seung et.al., 1992), we consider a stochastic learning algorithm which for long training times yields a Gibbs distribution of networks. The statistical mechanics approach starts out from the partition function Z = jdpo({W,})e-13E({W ,}), an integral over weight space with a priori measure Po({W,}), weighted with a thermal factor e-13E({w,l), where E is the total error on the training examples p E({W,}) = I:e[-u({{IL,}) .r({{ILI})]. (3) 1L=1 The formal temperature T = 1/ f3 defines the level of noise during the training process. For T = 0 this procedure corresponds to simply minimizing the training error E. From this the average free energy F = -T ((lnZ)), averaged over all possible sets of training examples can be calculated using the replica method (for details see Schwarze and Hertz, 1992). Like the calculations for simple perceptrons, our theory has two sets of order parameters: 0.13 _ K Wo. WI3 q, - N-I ·-1 a. K a. RI = N WI ·V,. Note that these are the only order parameters in this model. Due to the tree structure no correlations between different hidden units exist. Assuming both replica symmetry and 'translational symmetry' we are left with two parameters: q, the pattern average of the square of the average input-hidden weight vector, and R, the average overlap between this weight vector and a corresponding one for the teacher. We then obtain expressions for the replica-symmetric free energy of the form G(q, R, tI, R) = 0 G1(q, R) + G 2(q, R, tI, R), where the 'entropy' terms G 2 for the continuous- and binary-weight cases are exactly the same as in the simple perceptron (Gyorgyi and Tishby, 1990, Seung et.al., 1992). In the large-K limit another simplification similar to the zero-temperature capacity calculation (Barkai et.al., 1992) is found in the tree model. The 'energy' term G 1 is the same as the corresponding term in the calculation for the simple perceptron, except that the order parameters have to be replaced by f(q) = (2/1r) sin- 1 q and f(R) = (2/1r) sin- 1 R. The generalization error 1 €g = - arccos If(R)] 7r (4) can then be obtained from the value of R at the saddle point of the free energy. For a network with continuous weights, the solution of the saddle point equations yields an algebraically decreasing generalization error. There is no phase transition at any value of 0 or T. For T = 0 the asymptotic form of the generalization error in powers of 1/0 can be easily obtained as 1.25/0 + ('1/0 2), twice the €g found for the simple perceptron in this limit. 526 Schwarze and Henz 0.50 0.400.30 '" \I) 0.20 0.10 0.00 0 2 3 4 Figure 1: Learning curve for the large-K tree committee (solid line) with binary weights at T = 1. The phase transition occurs at Oc = 1.98, and the spinodal point is at 0, = 3.56. The analytic results are compared with Monte Carlo simulations with K = 9, N = 75 and T = I, averaged over 10 runs. In each simulation the number of training examples is gradually increased (dotted line) and decreased (dashed line), respectively. The broken line shows the generalization error for the simple perceptron. In contrast, the model with binary weights exhibits a phase transition at all temperatures from poor to perfect generalization. The corresponding generalization error as a function of 0 is shown in figure 1. At small values of 0 the free energy has two saddle points, one at R < 1 and the other at R = 1. Initially the solution with R < 1 and poor generalization ability has the lower free energy and therefore corresponds to the equilibrium state. When the load parameter is increased to a critical value Oc, the situation changes and the solution at R = 1 becomes the global minimum of the free energy. The system exhibits a first order phase transition to the state of perfect generalization. In the region Oc < 0 < 0, the R < 1 solution remains metastable and disappears at the spinodal point 0,. We find the same qualitative picture at all temperatures, and the complete replica symmetric phase diagram is shown in figure 2. The solid line corresponds to the phase transition to perfect generalization, and in the region between the solid and the dashed lines the R < 1 state of poor generalization is metastable. Below the dotted line, the replica-symmetric solution yields a negative entropy for the metastable state. This is unphysical in a binary system and replica symmetry has to be broken in this region, indicating the existence of many different metastable states. The simple perceptron without hidden units corresponds to the case K = 1 in our model. A comparison of the generalization properties with the large-K limit shows that both limits exhibit qualitatively similar behavior. The locations of the thermodynamic transitions and the spinodal line, however, are different and the generalization error of the R < 1 state in the large-K committee machine is higher than in the simple perceptron. The case of general finite K is rather more involved, but the annealed approximation Statistical Mechanics of Learning in a Large Committee Machine 527 1.0 0.8 R<1 metastability 0.6 .· · ··· · ·~ ·>·;.·r 0.4 ~ ~ R=1 ~ / I 0.2 RSB I 0.0 1.0 1.5 2.0 2.5 3.0 0/ Figure 2: Replica-symmetric phase diagram ofthe large-K tree committee machine with binary weights. The solid line shows the locations of the phase transition, and the spinodal line is shown dashed. Below the the dotted line the replica-symmetric solution is incorrect. for finite K indicates a rather smooth K -dependence for 1 < K < 00 (Mato and Parga, 1992). We performed Monte-Carlo simulations to check the validity of the assumptions made in our calculation and found good agreements with our analytic results. Figure 1 compares the analytic predictions for large K with Monte Carlo simulations for K = 9. The simulations were performed for a slowly increasing and decreasing training set size, respectively, yielding a hysteresis loop around the location of the phase transition. 3 Fully connected committee machine In contrast to the previous model the hidden units in the fully connected committee machine receive inputs from the entire input layer. Their output for a given Ndimensional input vector 5 is given by 0',(5) = sign (.Jw W, . 5), (5) while the overall output is again of the form (1). Note that the weight vectors W, are now N-dimensional, and the load parameter is given by a = P / (K N). For this model we solved the annealed approximation, which replaces ((In Z)) by In ((Z)). This approximation becomes exact at high temperatures (high noise level during training). For learnable target rules, as in the present problem, previous work indicates that the annealed approximation yields qualitatively correct results and correctly predicts the shape of the learning curves even at low temperatures (Seung et.al., 1992). Performing the average over all possible training sets again leads to two sets of order parameters: the overlaps between the student and teacher weight 528 Schwarze and Hertz vectors, RlIe = N- 1 W, . V An and the mutual overlaps in the student network CUe = N -1 W,, Wk' The weight vectors of the target rule are assumed to be un correlated and normalized, N- 1 L . V k = O,k. As in the previous model we make symmetry assumptions for the order parameters. In the fully connected architecture we have to allow for correlations between different hidden units (RlIe, ClIe :f! 0 for l =f. Ie) but also include the possibility of a specialization of individual units (Rll =f. RlIe). This is necessary because the ground state of the system with vanishing generalization error is achieved for the choice R'k = C'k = O,k. Therefore we make the ansatz R'k = R + 1101111, C'k = C + (1 - C)O/k (6) and evaluate the annealed free energy of the system using the saddle point method (details will be reported elsewhere). The values of the order parameters at the minimum of the free energy finally yield the average generalization error fg as a function of o. For a network with continuous weights and small 0 the global minimum of the free energy occurs at 11 = 0 and R '" qK- 3 / 4 ). Hence, for small training sets each hidden unit in the student network has a small symmetric overlap to all the hidden units in the teacher network. The information obtained from the training examples is not sufficient for a specialization of hidden units, and the generalization error approaches a plateau. To order 1/VK, this approach is given by €g = fO + ~ + 0(1/ K), fO = ~ arccos ( )2/71") ~ 0.206, (7) with 'Y({3) = )71"/2 - 1 [(1 e-~)-1 - foJ/(471"). Figure 3 shows the generalization error as a function of 0, including 1/VK-corrections for different values of K. 0.50 0.40 D' 0.30 ...... (x, (Xc \U 0.20!:" ~'~'~" "':'''':'''.:::.:::':: :':'-",:,·::·::·..:r.:.-.: - -'''''r'' t. i 0.10 ·'·'·-·-.L.~ 0.00 t.......o~ ......................... ......L.....~ ........... ~~L........o.-.........:l o 5 10 15 a=P/KN 20 25 Figure 3: Generalization error for continuous weights and T = 0.5. The approach to the residual error is shown including 1/VJ(-corrections for K=5 (solid line), K=ll (dotted line), and K=100 (dashed line). The broken line corresponds to the solution with nonvanishing 11. When the training set size is increased to a critical value 0, of the load parameter, Statistical Mechanics of Learning in a Large Committee Machine 529 a second minimum of the free energy appears at a finite value of /:::,. close to 1. For a larger value Oc > 0, this becomes the global minimum of the free energy and the system exhibits a first order phase transition. The generalization error of the specialized solution decays smoothly with an asymptotic behavior inversely proportional to o. However, the poorly-generalizing symmetric state remains metastable for all ° > Oc. Therefore, a stochastic learning procedure starting with /:::,. = 0 will first settle into the metastable state. For large N it will take an exponentially long time to cross the free energy barrier to the global minimum of the free energy. In a network with binary weights and for large K we find the same initial approach to a finite generalization error as in (7) for continuous weights. In the large-K limit the discreteness of the weights does not influence the behavior for small training sets. However, while a perfect match of the student to the teacher network (Rue = e'k = Olk) cannot happen for ° < 00 in the continuous model, such a 'freezing' is possible in a discrete system. The free energy of the binary model always has a local minimum at R'k = e'k = Olk. When the load parameter is increased to a critical value, this minimum becomes the global minimum of the free energy, and a discontinuous transition into this perfectly generalizing state occurs, just as in the binary-weight simple perceptron and the tree described in section 2. As in the case of continuous weights, the symmetric solution remains metastable here even for large values of o. Figure 4 shows the generalization error for binary weights, including 1/v'K-corrections for K = 5. The predictions of the large-K theory are compared with Monte Carlo simulations. Although we cannot expect a good quantitative agreement for such a small committee, the simulations support our qualitative results. Note that the leading order correction to €o in eqn. (7) is only small for ° ~ 11K. However, we have obtained a different solution, which is valid for ° "-' (111 K). The corresponding generalization error is shown as a dotted line in figure 4. 0.50 0.40 •.... 0.30 '" UJ • 1M 1M lI( • 1M 0.20 ~ t t 0.10 t t ~ 0.00 0 5 10 15 20 25 30 ex = P/KN Figure 4: Generalization error for binary weights at T = 5. The large-K theory for different regions of ° is compared with simulations for K = 5 and N = 45 averaged over all simulations (+) and simulations, in which no freezing occurred (*), respectively. The solid line shows the finite-o results including II v'K -corrections. The dotted line shows the small-o solution. 530 Schwarze and Hertz Compared to the tree model the fully connected committee machine shows a qualitatively different behavior. This difference is particularly pronounced in the continuous model. While the generalization error of the tree architecture decays smoothly for all values of a, the fully connected model exhibits a discontinuous phase transition. Compared to the tree model, the fully connected architecture has an additional symmetry, because each permutation of hidden units in the student network yields the same output for a given input (Barkai et.al., 1992). This additional degree of freedom causes the poor generalization ability for small training sets. Only if the training set size is sufficiently large can the hidden units specialize on one of the hidden units in the teacher network and achieve good generalization. However, the poorly generalizing states remain metastable even for arbitrarily large a. A similar phenomenon has also been found in a different architecture with only 2 hidden units performing a parity operation (Hansel et.al., 1992). Acknowledgements H. Schwarze acknowledges support from the EC under the SCIENCE programme and by the Danish Natural Science Council and the Danish Technical Research Council through CONNECT. References E. Barkai, D. Hansel, and H. Sompolinsky (1992), Phys.Rev. A 45, 4146. A. Engel, H.M. Kohler, F. Tschepke, H. Vollmayr, and A. Zippelius (1992), Phys.Rev. A 45, 7590. E. Gardner, B. Derrida (1989), J.Phys. A 21, 271. G. Gyorgyi and N. Tishby (1990) in Neural Networks and Spin Glasses, edited K. Thuemann and R. Koberle (World Scientific, Singapore). D. Hansel, G. Mato, and C. Meunier (1992), Europhys.Lett. 20, 471. A. Krogh, l. Hertz (1992), Advances in Neural Information Processing Systems IV, edited by l.E. Moody, S.l. Hanson, and R.P. Lippmann, (Morgan Kaufmann, San Mateo). G. Mato, N. Parga (1992), J.Phys. A 25, 5047. N.J. Nilsson (1965) Learning Machines, (McGraw-Hill, New York). H. Schwarze, M. Opper, and W. Kinzel (1992), Phys.Rev. A 45, R6185. H. Schwarze, J. Hertz (1992), Europhys.Lett. 20,375. H.S. Seung, H. Sompolinsky, and N. Tishby (1992), Phys.Rev. A 45, 6056. H. Sompolinsky, N. Tishby (1990), Europhys.Lett. 13,567. T. Watkin, A. Rau, and M. Biehl (1992), to be published in Review of Modern Physics.
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An Information-Theoretic Approach to Deciphering the Hippocampal Code William E. Skaggs Bruce L. McNaughton Katalin M. Gothard Etan J. Markus Center for Neural Systems, Memory, and Aging 344 Life Sciences North University of Arizona Tucson AZ 85724 bill@nsma.arizona.edu Abstract Information theory is used to derive a simple formula for the amount of information conveyed by the firing rate of a neuron about any experimentally measured variable or combination of variables (e.g. running speed, head direction, location of the animal, etc.). The derivation treats the cell as a communication channel whose input is the measured variable and whose output is the cell's spike train. Applying the formula, we find systematic differences in the information content of hippocampal "place cells" in different experimental conditions. 1 INTRODUCTION Almost any neuron will respond to some manipulation or other by changing its firing rate, and this change in firing can convey information to downstream neurons. The aim of this article is to introduce a very simple formula for the average rate at which a cell conveys information in this way, and to show how the formula is helpful in the study of the firing properties of cells in the rat hippocampus. This is by no means the first application of information theory to the study of neural coding; see especially Richmond and Optican (1990). The thing that particularly distinguishes 1030 An Information-Theoretic Approach to Deciphering the Hippocampal Code 1031 our approach is its simplemindedness. To get the basic idea, imagine we are recording the activity of a neuron in the brain of a rat, while the rat is wandering around randomly on a circular platform. Suppose we observe that the cell fires only when the rat is on the left half of the platform, and that it fires at a constant rate everywhere on the left half; and suppose that on the whole the rat spends half of its time on the left half of the platform. In this case, if we are prevented from seeing where the rat is, but are informed that the neuron has just this very moment fired a spike, we obtain thereby one bit of information about the current location of the rat. Suppose we have a second cell, which fires only in the southwest quarter of the platform; in this case a spike would give us two bits of information. If there were in addition a small amount of background firing, the information would be slightly less than two bits. And so on. Going back to the cell that fires everywhere on the left half of the platform, suppose that when it is active, it fires at a mean rate of 10 spikes per second. Since it is active half the time, it fires at an overall mean rate of 5 spikes per second. Since a spike conveys one bit of information about the rat's location, the cell's spike train conveys information at an average rate of 5 bits per second. This does not mean that if the cell is observed for one second, on average 5 bits will be obtained-rather it means that if the cell is observed for a sufficiently short time interval dt, on average 5dt bits will be obtained. In 20 milliseconds, for example, the expected information conveyed by the cell about the location of the rat will be very nearly 0.1 bits. The longer the time interval over which the cell is observed, the more redundancy in the spike train, and hence the farther below 5dt the total information falls. The formula that leads to these numbers is 1= l.\(X) log2 .\~) p(x)dx, (1) where I is the information rate of the cell in bits per second, x is spatial location, p( x) is the probability density for the rat being at location x, .\( x) is the mean firing rate when the rat is at location x, and .\ = Jz .\(x)p(x)dx is the overall mean firing rate of the cell. The derivation of this formula appears in the final section. (To our knowledge the formula, though very simple, has not previously been published.) Note that, as far as the formula is concerned, there is nothing special about spatial location: the formula can equally well be used to define the rate at which a cell conveys information about any aspect of the rat's state, or any combination of aspects. The only mathematical requirement1 is that the rat's state x and the spike train of the cell both be stationary random variables, so that the probability density p( x) and the expected firing rate .\( x) are well-defined. The information rate given by formula (1) is measured in bits per second. If it is divided by the overall mean firing rate.\ of the cell (expressed in spikes per second), then a different kind of information rate is obtained, in units of bits per spike-let us call it the information per spike. This is a measure of the specificity of the cell: the more "grandmotherish" the cell, the more information per spike. For a population lOther than obvious requirements of integrability that are sure to be fulfilled in natural situations. 1032 Skaggs, McNaughton, Gothard, and Markus of cells, then, a highly distributed representation equates to little information per spike. . .' ' .. :. .. ,'. . . . • .11 ••• • • ' • .. ':,'. . J .. ' ': . I, • .. ••• I I ", 'I: .' ,I • • .: •••• . ..'.. '. .. "I •• II'~ ,'I' .. II ' •• -.. .', •• ::.' '1 ...' IJ I '. II:. .. ~.' "_ .' ••••• 1 ___ • L_ ," .". II .,' •••• II:'. II ~'.' "': I I I.' •. • • .. ILl. II. I. • ,', .' .' I • • I. • • ' •• II • '. • ••••• I • " •• /" I • .... ' •• ' •• • I. I' .1 '.. • • :.. ••• .1 '.' . I.' I ... • •• II. . ' ' . . • • • • • • II • • ••• . •.... .1 .': . ,.'. .. .. I • • : .1 ••• I .. II • • '.' .... ".' ......:.. • • • I ",1' .,'1,' ,'.' •••••••••• I. I '. • '. • •• ',.' ," ••• • • •• I' · .:' .... ", I····· ·' •. 1.·· ....•.... ... ••• • ? ...1. .\ I • I • • I' "1 I.' I. • I ." •••• • II • ,\... • ..... · II .. ..... .' . . I...: '. '. I ," I.' '.' ':". . • • '.' .1, .~:. '. I.". I. '.. .· .... 1._.·.. . .' ..... .. '. \ '. I· I'. •••••• I • ••••• _ ', •• ' ." I ":.' I. I .. 1 •••• I' .' '. '. •• I :.: I... I '. ..,... . . ..... ..... '-• '. I" II. I • • ." • I I • I'" • I • I J I I • II. ..' I.. I I I I..... t .. 1 _ ' ••• :1.' . • • ~I II.'. • ••• ..1 • . I I • I • • I CO' '. '.:" .'. . .. I... . '. .1. • I .' .... . • I. '1 ••• I. •••••••• • '.l!)' • • ... • • I • I.' .' • II I. • • • • • • '. '. I.. .; " I:. :~. . I,.~. .1' • • I I.' .. I 1.1 • ........... '1 •• -..; :- • I'·~ ~~~~\H~,,-.... ~g.!~lt£1~ ... --;~~~ Figure 1: "Spot plot" of the activity of a single pyramidal cell in the hippocampus of a rat, recorded while the rat foraged for food pellets inside a small cylinder. The dots show locations visited by the rat, and the circles show points where the cell fired-large circles mean that several spikes occurred within a short time. The lines indicate which direction the rat was facing when the cell fired. The plot represents 29 minutes of data, during which the cell fired at an overall mean rate of 1.319 Hz. Consider, as an example, a typical "place cell" (actually an especially nice place cell) from the CAl layer of the hippocampus of a rat-Figure I shows a "spot plot" of the activity of the cell as the rat moves around inside a 76 cm diameter cylinder with high, opaque walls, foraging for randomly scattered food pellets. This cell, like most pyramidal cells in CAl, fires at a relatively high rate (above 10 Hz) when the rat is in a specific small portion of the environment-the "place field" of the cellbut at a much lower rate elsewhere. Different cells have place fields in different locations; there are no systematic rules for their arrangement, except that there may be a tendency for neighboring cells to have nearby place fields. The activity of place cells is known to be related to more than just place: in some circumstances it is sensitive to the direction the rat is facing, and it can also be modulated by running speed, alertness, or other aspects of behavioral state. The dependence on An Information-Theoretic Approach to Deciphering the Hippocampal Code 1033 head direction has given rise to a certain amount of controversy, because in some types of environment it is very strong, while in others it is virtually absent. Table 1 gives statistics for the amount of information conveyed by this cell about spatial location, head direction, running speed, and combinations of these variables. Note that the information conveyed about spatial location and head direction is hardly more than the information conveyed about spatial location alone-the difference is well within the error bounds of the calculation. Thus this cell has no detectable directionality. This seems to be typical of cells recorded in unstructured environments. Table 1: Information conveyed by the cell whose activity is plotted in Figure 1. VARIABLES Location Head Direction Running Speed Location and Head Direction Location and Running Speed INFO 2.40 bits/sec 0.48 bits/sec 0.03 bits/sec 2.53 bits/sec 2.36 bits/sec INFO PER SPIKE 1.82 bits 0.37 bits 0.02 bits 1.92 bits 1.79 bits The information-rate measure may be helpful in understanding the computations performed by neural populations. Consider an example. Cells in the CA3 and CAl regions of the rat hippocampal formation have long been known to convey information about a rat's spatial location (this is discussed in more detail below). Data from our lab suggest that, in a given environment, an average CA3 cell conveys something in the neighborhood of 0.1 bits per second about the rat's position-some cells convey a good deal more information than this, but many are virtually silent. Cells in CAl receive most of their input from cells in CA3; each gets on the order of 10,000 such inputs. Question: How long must the integration time of a CAl cell be in order for it to form a good estimate of the rat's location? Answer: With 10,000 inputs, each conveying on average 0.1 bits per second, the cell receives information at a rate of 1000 bits per second, or 1 bit per millisecond, so in 5-10 msec the cell receives enough information to form a moderately precise estimate of location. 2 APPLICATIONS We now very briefly describe two experimental studies that have found differences in the spatial information content of rat hippocampal activity under different conditions. The methods used for recording the cells are described in detail in McNaughton et al (1989)-to summarize, the cells were recorded with stereotrodes, which are twisted pairs of electrodes, separated by about 15 microns at the tips, that pick up the extracellular electric fields generated when cells fire. A single stereotrode can detect the activity of as many as six or seven distinct hippocampal cells; spikes from different cells can be separated on the basis of their amplitudes on the two electrodes, as well as other differences in wave shape. The location of the rat was tracked using arrays of LEDs attached to their heads and a video camera on the ceiling. Spatial firing rate maps for each cell were constructed using an adaptive binning technique designed to minimize error (Skaggs and McNaughton, submitted), 1034 Skaggs, McNaughton, Gothard, and Markus and information rates were calculated using these firing rate maps. As a control, the spike train was randomly time-shifted relative to the sequence of locations; this was done 100 times, and the cell was deemed to have significant spatial dependence if its information rate was more than 2.29 standard deviations above the mean of the 100 control information rates. 2.1 EXPERIMENT: PROXIMAL VERSUS DISTAL VISUAL CUES In this study (a preliminary account of which appears in Gothard et al (1992», the activity of place cells was recorded successively in two environments, the first a 76 em diameter cylinder with four patterned cue-cards on the high, opaque gray wall, the second a cylinder of the same shape, but with a low, transparent plexiglass wall and four patterned cue-cards on the distant black walls of the recording room. The two environments thus had the same shape, and from any given point were visually quite similar; the difference is that in one all of the visual cues were proximal to the rat, while in the other many of them were distal. DISTAL CUES PROXIMAL CUES ... . .. .:.,,: :.:; ;:;: . .. .. .; . .. : : : . . " . . :. : Figure 2: Firing rate maps of four simultaneously recorded cells, in the distal cue environment (top) and proximal cue environment (bottom). The scale is identical for all plots; black ~ 5 Hz. Fifty cells were recorded with robust place-dependent firing in one or the other cylinder. There was no discernable relationship between place fields in the two environments-a cell having a place field in the proximal cue environment might be nearly silent in the distal cue environment, and even if it did fire, its place field would be in a different location. (Figure 2 shows firing rate maps for four of the cells.) A substantially higher fraction of the cells had place fields in the proximal cue environment, and overall the average information per second was almost 50% higher An Information-Theoretic Approach to Deciphering the Hippocampal Code 1035 in the proximal cue environment. For the cells possessing fields, the information per spike was significantly higher in the proximal cue environment, meaning that place fields were more compact. These results indicate that in the proximal cue environment, spatial location is represented by the hippocampal population more precisely, and by a larger pool of cells, than in the distal cue environment. The most likely explanation is that, at least in the absence of local cues, the configuration of visual landmarks controls the activity of the place cell population. 2.2 EXPERIMENT: LIGHT VERSUS DARK Visual cues have a great deal of influence on place fields, but they are not the only important factor; in fact, some hippocampal cells maintain place fields even in complete darkness (McNaughton et a/., 1989b; Quirk et a/., 1990). This experiment (Markus et a/., 1992) was designed to examine how lack of visual cues changes the properties of place fields. Rats traversed an eight-arm radial maze for chocolate milk reward, with the room lights being turned on and oft' on alternate trials. (A trial consisted of one visit to each of the eight arms of the maze.) Figure 3 shows firing rate maps for four simultaneously recorded cells. LIGHT .tt:·: :/::;. .. : .... : '::::.::;.: DARK Figure 3: Firing rate maps of four simultaneously recorded cells, with room lights turned on (top) and off (bottom). The scale is identical for all plots; black ~ 5 Hz. (The loops at the ends of the arms are caused by the rat turning around there.) The most salient effect was that a much larger fraction of cells showed significant spatially selective firing in the light than in the dark: 35% as opposed to 20%. However, the average information per second decreased only by 15% in the dark as compared to the light, from 0.326 bits per second in the light to 0.278 bits per 1036 Skaggs, McNaughton, Gothard, and Markus second in the dark. (These are overestimates of the population averages, because cells silent in both light and dark were not included in the sample.) Interestingly, the drop in information content from light to dark seemed to be much smaller than the drop from proximal cues to distal cues in the previous experiment. A major difference between the two experime:nts is that, in the eight-arm maze, tactile cues potentially give a great deal of information about spatial location, but in a cylinder they serve only to distinguish the center from the wall. While it is dangerous to compare the two experiments, which differed methodologically in several ways, the results suggest that tactile cues can have a very strong influence on hippocampal firing, at least when visual cues are absent. 3 THEORY The information-rate formula (1) is derived by considering a neuron as a "channel" (in the information-theoretic sense) whose input is the spatial location of the rat, and whose output is the spike train. During a sufficiently short time interval the spike train is effectively a binary random variable (Le. the only possibilities are to spike once or not at all), and the probability of spiking is determined by the spatial location. The event of spiking may be indicated by a random variable S whose value is 1 if the cell spikes and 0 otherwise. If the environment is partitioned into a set of nonoverlapping bins, then spatial location may be represented by an integer-valued random variable X giving the index of the currently occupied bin. In information theory, the information conveyed by a discrete random variable X about another discrete random variable Y, which is identical to the mutual information of X and Y, is given by where Z i and Yi are the possible values of X and Y, and pO is probability. If Aj is the mean firing rate when the rat is in bin j, then the probability of a spike during a brief time interval tl.t is P(S= 11X =j) = Ajtl.t. Also, the overall probability of a spike is P(S=1) = Atl.t, where with Pi = P(X =j). After these expressions are plugged in to the equation for I(Y IX) above, it is a matter of straightforward algebra, using power series expansions of logarithms and keeping only lower order terms, to derive a discrete approximation of equation (1). An Information-Theoretic Approach to Deciphering the Hippocampal Code 1037 4 DISCUSSION In many situations, neurons must decide whether to fire on the basis of relatively brief samples of input, often 100 milliseconds or less. A cell cannot get much information from a single input in such a short time, so to achieve precision it needs to integrate many inputs. Formula (1) provides a measure of how much information a single input conveys about a given variable in such a brief time interval. The formula can be applied to any type of cell that uses firing rate to convey information. The only requirement is to have enough data to get good, stable estimates of firing rates. In practice, for a hippocampal cell having a mean firing rate of around 0.5 Hz in an environment, twenty minutes of data is adequate for measuring position-dependence; and for a "theta cell" (an interneuron, firing at a considerably higher rate), very clean measurements are possible. We have used the measure in the study of hippocampal place cells, but it might actually work better for some other types. The problem with place cells is that they fire at low overall rates, so it is time-consuming to get an adequate sample. Cortical pyramidal cells often have mean rates at least ten times faster, so it ought to be easier to get accurate numbers for them. The information measure might naturally be applied to study, for example, the changes in information content of visual cortical cells as a visual stimulus is blurred or dimmed. Supported by NIMH grant MH46823 References Gothard, K. M., Skaggs, W. E., McNaughton, B. L., Barnes, C. A., and Youngs, S. P. (1992). Place field specificity depends on proximity of visual cues. Soc Neurosci Abstr, 18:1216. 508.10. Markus, E. J., Barnes, C. A., McNaughton, B. L., Gladden, V., Abel, T. W., and Skaggs, W. E. (1992). Decrease in the information content of hippocampal cal cell spatial firing patterns in the dark. Soc Neuroscience Abstr, 18:1216. 508.12. McNaughton, B. L., Leonard, B., and Chen, L. (1989b). Cortical-hippocampal interactions and cognitive mapping: A hypothesis based on reintegration of the parietal and inferotemporal pathways for visual processing. Psychobiology, 17:230-235. McNaughton, B. L., Barnes, C. A., Meltzer, J., and Sutherland, R. J. (1989a). Hippocampal granule cells are necessary for normal spatial learning but not for spatially selective pyramidal cell discharge. Exp Brain Res, 76:485-496. Quirk, G. J., Muller, R. U., and Kubie, J. L. (1990). The firing of hippocampal place cells in the dark depends on the rat's previous experience. J N eurosci, 10:2008-2017. Richmond, B. J. and Optican, L. M. (1990). Temporal encoding of two-dimensional patterns by single units in primate primary visual cortex: Ii information transmission. J Neurophysiol, 64:370-380.
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Using hippocampal 'place cells' for navigation, exploiting phase coding Neil Burgess, John O'Keefe and Michael Recce Department of Anatomy, University College London, London WC1E 6BT, England. (e-mail: n.burgess<Ducl.ac . uk) Abstract A model of the hippocampus as a central element in rat navigation is presented. Simulations show both the behaviour of single cells and the resultant navigation of the rat. These are compared with single unit recordings and behavioural data. The firing of CAl place cells is simulated as the (artificial) rat moves in an environment. This is the input for a neuronal network whose output, at each theta (0) cycle, is the next direction of travel for the rat. Cells are characterised by the number of spikes fired and the time of firing with respect to hippocampal 0 rhythm. 'Learning' occurs in 'on-off' synapses that are switched on by simultaneous pre- and post-synaptic activity. The simulated rat navigates successfully to goals encountered one or more times during exploration in open fields. One minute of random exploration of a 1m2 environment allows navigation to a newly-presented goal from novel starting positions. A limited number of obstacles can be successfully avoided. 1 Background Experiments have shown the hippocampus to be crucial to the spatial memory and navigational ability of the rat (O'Keefe & Nadel, 1978). Single unit recordings in freely moving rats have revealed 'place cells' in fields CA3 and CAl of the hippocampus whose firing is restricted to small portions of the rat's environment (the corresponding 'place fields') (O'Keefe & Dostrovsky, 1971), see Fig. 1a. In addition cells have been found in the dorsal pre-subiculum whose primary behavioural 929 930 Burgess, O'Keefe, and Reece a b A 360· I II • IIII II Phase B 1 Theta [mV) ·1 Time [s] Figure 1: a) A typical CAl place field, max. rate (over 18) is 13.6 spikes/so b) One second of the EEG () rhythm is shown in C, as the rat runs through a place field. A shows the times of firing of the place cell. Vertical ticks immediately above and below the EEG mark the positive to negative zero-crossings of the EEG, which we define as 00 (or 3600 ) of phase. B shows the phase of () at which each spike was fired (O'Keefe & Recce, 1992). correlate is 'head-direction' (Taube et aI., 1990). Both are suggestive of navigation. Temporal as well as spatial aspects of the electrophysiology of the hippocampal region are significant for a model. The hippocampal EEG '() rhythm' is best characterised as a sinusoid of frequency 7 - 12H z and occurs whenever the rat is making displacement movements. Recently place cell firing has been found to have a systematic phase relationship to the local EEG (O'Keefe & Recce, 1992), see §3.1 and Fig. lb. Finally, the () rhythm has been found to modulate long-term potentiation of synapses in the hippocampus (Pavlides et al., 1988). 2 Introduction We are designing a model that is consistent with both the data from single unit recording and the behavioural data that are relevant to spatial memory and navigation in the rat. As a first step this paper examines a simple navigational strategy that could be implemented in a physiologically plausible way to enable navigation to previously encountered reward sites from novel starting positions. We assume the firing properties of CAl place cells, which form the input for our system. The simplest map-based strategies (as opposed to route-following ones) are based on defining a surface over the whole environment, on which gradient ascent leads to the goal (e.g. delayed reinforcement or temporal difference learning). These tend to have the problem that, to build up this surface, the goal must be reached many times, from different points in the environment (by which time the rat has died of old age). Further, a new surface must be computed if the goal is moved. Specific problems are raised by the properties of rats' navigation: (i) the position of CAl place fields is independent of goal position (Speakman & O'Keefe, 1990); (ii) high firing rates in place cells are restricted to limited portions of the environment; (iii) rats are able to navigate after a brief exploration of the environment, and (iv) can take novel short-cuts or detours (Tolman, 1948). Using hippocampal 'place cells' for navigation, exploiting phase coding 931 To overcome these problems we propose that a more diffuse representation of position is rapidly built up downstream of CAl, by cells with larger firing fields than in CAL The patterns of activation of this group of cells, at two different locations in the environment, have a correlation that decreases with the separation of the two locations (but never reaches zero, as is the case with small place fields). Thus the overlap between t.he pattern of activity at any moment and the pattern of activity at the goal location would be a measure of nearness to the goal. We refer to these cells as 'subicular' cells because the subiculum seems a likely site for them, given single unit recordings (Barnes et al., 1990) showing spatially consistent firing over large parts of the environment. We show that the output of these subicular cells is sufficient to enable navigation in our model. In addition the model requires: (i) 'goal' cells (see Fig. 4a) that fire when a goal is encountered, allowing synaptic connections from subicular cells to be switched on, (ii) phase-coded place cell firing, (iii) 'head-direction' cells, and (iv) synaptic change that is modulated by the phase of the EEG. The relative firing rates of groups of goal cells code for the direction of objects encountered during exploration, in the same way that cens in primate motor cortex code for the direction of arm movements (Georgopoulos et al., 1988). 3 The model In our simulation a rat is in constant motion (speed 30cm/ s) in a square environment of size L x L (L ~ 150cm). Food or obstacles can be placed in the environment at any time. The rat is aware of any objects within 6cm (whisker length) of its position. It bounces off any obstacles (or the edge of the environment) with which it collides. The f) frequency is taken to be 10Hz (period O.ls) and we model each f) cycle as having 5 different phases. Thus the smallest timestep (at which synaptic connections and cell firing rates are updated) is 0.02s. The rat is either 'exploring' (its current direction is a random variable within 30 0 of its previous direction), or 'searching' (its current direction is determined by the goal cells, see below). Synaptic and cell update rules are the same during searching or exploring. 3.1 The phase of CAl place cell firing When a rat on a linear track runs through a place field, the place cell fires at successively earlier phases of the EEG f) rhythm. A cell that fires at phase 3600 when the rat enters the place field may fire as much as 355 0 earlier in the f) cycle when exiting the field (O'Keefe & Recce, 1992), see Fig. lb. Simulations below involve 484 CAl place cells with place field centres spread evenly on a grid over the whole environment. The place fields are circular, with diameters 0.25L, 0.35L or Oo4L (as place fields appear to scale with the size of an environment; Muller & Kubie, 1987). The fraction of cells active during any O.ls interval is thus 7r(0.1252 + 0.1752 + 0.2 2)/3 = 9%. When the rat is in a cell's place field it fires 1 to 3 spikes depending on its distance from the field centre, see Fig. 2b. When the (simulated) rat first enters a place field the cell fires 1 spike at phase 3600 of the f) rhythm; as the rat moves through the place field, its phase of firing shifts backwards by 72 0 every time the number of spikes fired by the cell changes 932 Burgess, O'Keefe, and Reece a c EJlElm •• 360· 288· 216· 144· 72· 0.0 0.2 0.4 0.6 0.8 Figure 2: a) Firing rate map of a typical place cell in the model (max. rate 11.6 spikes/s); b) Model of a place field; the numbers indicate the number of spikes fired by the place cell when the rat is in each ring. c) The phase at which spikes would be fired during all possible straight trajectories of the rat through the place field from left to right. d) The total number of spikes fired in the model of CAl versus time, the phase of firing of one place cell (as the rat runs through the centre of the field) is indicated be vertical ticks above the graph. (i.e. each time it crosses a line in Fig. 2b). Thus each theta cycle is divided into 5 timesteps. No shift results from passing through the edge of the field, whereas a shift of 288 0 (0.08s) results from passing through the middle of the field, see Fig. 2c. The consequences for the model in terms of which place cells fire at different phases within one () cycle are shown in Fig. 3. The cells that are active at phase 3600 have place fields centred ahead of the position of the rat (i.e. place fields that the rat is entering), those active at phase 00 have place fields centred behind the rat. If the rat is simultaneously leaving field A and entering field B then cell A fires before cell B, having shifted backwards by up to 0.08s. The total number of spikes fired at each phase as the rat moves about implies that the envelope of all the spikes fired in CAl oscillates with the () frequency. Fig. 2d shows the shift in the firing of one cell compared to the envelope (cf. Fig. 1b). 3.2 Subicular cells We simulate 6 groups of 80 cells (480 in total); each subicular cell receives one synaptic connection from a random 5% of the CAl cells. These connections are either on or off (1 or 0). At each timestep (0.02s) the 10 cells in each group with the greatest excitatory input from CAl fire between 1 and 5 spikes (depending on their relative excitation). Fig. 3c shows a typical subicular firing rate map. The consequences of phase coding in CAl (Figs. 3a and b) remain in these subicular cells as they are driven by CAl: the net firing field of all cells active at phase 360 0 of () is peaked ahead of the rat. Using hippocampal 'place cells' for navigation, exploiting phase coding 933 a b Figure 3: Net firing rate map of all the place cells that were active at the 3600 ( a) and 72 0 (b) phases of e as the rat ran through the centre of the environment from left to right. c) Firing rate map of a typical 'subicular' cell in the model; max. rate (over LOs) is 46.4 spikes/so Barnes et al. (1990) found max. firing rates (over O.ls) of 80 spikes/s (mean 7 spikes/s) in the subiculum. N SEW a o 0 0 0 Goal cells b Subicular cells <? q q 9 6x80 (480) ....•.... onloff synapses 5% connectivity Place cells 000000000000 22x22 (484) Figure 4: a) Connections and units in the model; interneurons shown between the subicular cells indicate competitive dynamics, but are not simulated explicitly. b) The trajectory of 90 seconds of 'exploration' in the central 126 x 126cm2 of the environment. The rat is shown in the bottom left hand corner, to scale. 3.2.1 Learning The connections are initialised such that each subicular cell receives on average one 'on' connection. Subsequently a synaptic connection can be switched on only during phases 1800 to 3600 of e. A synapse becomes switched on if the pre-synaptic cell is active, and the post-synaptic cell is above a threshold activity (4 spikes), in the same timestep (0.02s). Hence a subicular firing field is rapidly built up during exploration, as a superposition of CAl place fields, see Fig 3c. 3.3 Goal cells The correlation between the patterns of activity of the subicular cells at two different locations in the environment decreases with the separation of the two locations. Thus if synaptic connections to a goal cell were switched on when the rat encountered food then a firing rate map of the goal cell would resemble a cone covering the entire environment, peaked at the food site, i.e. the firing rate would indicate 934 Burgess, O'Keefe, and Reece a_ c Figure 5: Goal cell firing fields, a) West, b) East, of 'food' encountered at the centre of the environment. c) Trajectories to a goal from 8 novel starting positions. All figures refer to encountering food immediately after the exploration in Fig. 4b. Notice that much of the environment was never visited during exploration. the closeness of the food during subsequent movement of the rat. The scheme we actually use involves groups of goal cells continuously estimating the distance to 4 points displaced from the goal site in 4 different directions. Notice that when a freely moving rat encounters an interesting object a fair amount of 'local investigation' takes place (sniffing, rearing, looking around and local exploration). During the local investigation of a small object the rat crosses the location of the object in many different directions. We postulate groups of goal cells that become excited strongly enough to induce synaptic change in connections from subicular cells whenever the rat encounters a specific piece of food and is heading in a particular direction. This supposes the joint action of an object classifier and of head-direction cells; head-direction cells corresponding to different directions being connected to different goal cells. Since synaptic change occurs only at the 180 0 to 3600 phases of 8, and the net firing rate map of all the subicular cells that are active at phase 3600 during any 8 cycle is peaked ahead of the rat, goal cells have firing fields that are peaked a little bit away from the goal position. For example, goal cells whose subicular connections are changed when the rat is heading east have firing rate fields that are peaked to the east of the goal location, see Fig. 5. Local investigation of a food site is modelled by the rat moving 12cm to the north, south, east and west and occurs whenever food is encountered. Navigation is restricted to the central 126 x 126cm2 portion of the 150 x 150cm2 environment (over which firing rate maps are shown) to leave room for this. There are 4 goal cells for every piece of food found in the environment, (GC-Ilorth, GC.-South, GC_east, GC_west), see Fig. 4a. Initially the connections from all subicular cells are off; they are switched on if the subicular cell is active and the rat is at the particular piece of food, travelling in the right direction. When the rat is searching, goal cells simply fire a number of spikes (in each 0.025 timestep) that is proportional to their net excitatory input from the subicular cells. 3.4 Maps and navigation When the rat is to the north of the food, GC-Ilorth fires at a higher rate than GC..south. We take the firing rate of GC_north to be a 'vote' that the rat is north a Using hippocampal 'place cells' for navigation, exploiting phase coding 935 b c ",., ; .. --_ .. _ .. --- ...... , :' ~* f ~~ : + -- ". Figure 6: a) Trajectory of rat with alternating goals. b) an obstacle is interposed; the rat collides with the obstacle on the first run, but learns to avoid the collision site in the 2 subsequent runs. c) Successive predictions of goal (box) and obstacle (cross) positions generated as the rat ran from one goal site to the other; the predicted positions get more accurate as the rat gets closer to the object in question. of the goal. Similarly the firing rate of GCJlouth is a vote that the rat is south of the goal: the resultant direction (the vector sum of directions north, south, east and west, weighted by the firing rates of the corresponding cells) is an estimate of the direction of the rat from the food (cf.Georgopoulos et al., 1988). Since the firing rate maps of the 4 goal cells are peaked quite close to the food location, their net firing rate increases as the food is approached, i.e. it is an estimation of how close the food is. Thus the firing rates of the 4 goal cells associated with a piece of food can be used to predict its approximate position relative to the rat (e.g. 70cm northeast), as the rat moves about the environment (see Fig. 6c). We use groups of goal cells to code for the locations at which the rat encountered any objects (obstacles or food), as described above. A new group of goal cells is recruited every time the rat encounters a new object, or a new (6cm) part of an extended object. The output of the system acts as a map for the rat, telling it where everything is relative to itself, as it moves around. The process of navigation is to decide which way to go, given the information in the map. When there are no obstacles in the environment, navigation corresponds to moving in the direction indicated by the group of goal cells corresponding to a particular piece of food. When the environment includes many obstacles the task of navigation is much harder, and there is not enough clear behavioural data to guide modelling. We do not model navigation at a neuronal level, although we wish to examine the navigation that would result from a simple reading of the 'map' provided by our model. The rules used to direct the simulated rat are as follows: (i) every 0.18 the direction and distance to the goal (one of the pieces of food) are estimated; (ii) the direction and distance to all locations at which an obstacle was encountered are estimated; (iii) obstacle locations are classified as 'in-the-way' if (a) estimated to be within 45° of the goal direction, (b) closer than the goal and (c) less than L/2 away; (iv) the current direction of the rat becomes the vector sum of the goal direction (weighted by the net firing rate of the corresponding 4 goal cells) minus the directions to any in-the-way obstacles (weighted by the net firing rate of the '0 bstacle cells' and by the similarity of the obstacle and goal directions). 936 Burgess, O'Keefe, and Reece 4 Performance The model achieves latent learning (i.e. the map is constructed independently of knowledge of the goal, see e.g. Tolman, 1948). A piece of food encountered only once, after exploration, can be returned to, see Fig. 5c. Notice that a large part of the environment was never visited during exploration (Fig. 4b). Navigation is equally good after exploration in an environment containing food/obstacles from the beginning. If the food is encountered only during the earliest stages of exploration (before a stable subicular representation is built up) then performance is worse. Multiple goals and a small number of obstacles can be accommodated, see Fig. 6. Notice that searching also acts as exploration, and that synaptic connections can be switched at any time: all learning is incremental, but saturates when all the relevant synapses have been switched on. Performance does not depend crucially on the parameter values, used although it is worse with fewer cells, and smaller environments require less exploration before reliable navigation is possible (e.g. 60s for a 1m2 box). Quantitative analysis will appear in a longer paper. References Barnes C A, McNaughton B L, Mizumori S J Y, Leonard B W & Lin L-H (1990) 'Comparison of spatial and temporal characteristics of neuronal activity in sequential stages of hippocampal processing', Progreu in Brain Re6earch 83 287-300. Georgopoulos A P, Kettner R E & Schwartz A B (1988) 'Primate motor cortex and free arm movements to visual targets in three-dimensional space. II. Coding of the direction of movement by a neuronal population', J. Neur06d. 8 2928-2937. Muller R U & Kubie J L (1987) 'The effects of changes in the environment on the spatial firing of hippocampal complex-spike cells', J. Neur06d. 7 1951-1968. O'Keefe J & Dostrovsky J (1971) 'The hippocampus as a spatial map: preliminary evidence from unit activity in the freely moving rat', BrainRe6. 34 171-175. O'Keefe J & Nadel L (1978) The hippocampu6 a6 a cognitive map, Clarendon Press, Oxford. O'Keefe J & Reece M (1992) 'Phase relationship between hippocampal place units and the EEG theta rhythm', Hippocampu!, to be published. Pavlides C, Greenstein Y J, Grudman M & Winson J (1988) 'Long-term potentiation in the dentate gyrus is induced preferentially on the positive phase of O-rhythm', Brain Re6. 439 383-387. Speakman A S & O'Keefe J (1990) 'Hippocampal complex spike cells do not change their place fields if the goal is moved within a cue controlled environment', European Journal of Neuro!cience 2 544-555. Taube J S, Muller R U & Ranck J B Jr (1990) 'Head-direction cells recorded from the postsubiculum in freely moving rats. I. Description & quantitative analysis', J. Neur06ci. 10 420-435. Tolman E C (1948) 'Cognitive Maps in rats and men', P6ychological Review 55 189-208.
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Weight Space Probability Densities in Stochastic Learning: I. Dynamics and Equilibria Todd K. Leen and John E. Moody Department of Computer Science and Engineering Oregon Graduate Institute of Science & Technology 19600 N.W. von Neumann Dr. Beaverton, OR 97006-1999 Abstract The ensemble dynamics of stochastic learning algorithms can be studied using theoretical techniques from statistical physics. We develop the equations of motion for the weight space probability densities for stochastic learning algorithms. We discuss equilibria in the diffusion approximation and provide expressions for special cases of the LMS algorithm. The equilibrium densities are not in general thermal (Gibbs) distributions in the objective function being minimized, but rather depend upon an effective potential that includes diffusion effects. Finally we present an exact analytical expression for the time evolution of the density for a learning algorithm with weight updates proportional to the sign of the gradient. 1 Introduction: Theoretical Framework Stochastic learning algorithms involve weight updates of the form w(n+1) = w(n) + /-l(n)H[w(n),x(n)] (1) where w E 7£m is the vector of m weights, /-l is the learning rate, H[.] E 7£m is the update function, and x(n) is the exemplar (input or input/target pair) presented 451 452 Leen and Moody to the network at the nth iteration of the learning rule. Often the update function is based on the gradient of a cost function H(w,x) = -a£{w,x) law. We assume that the exemplars are Li.d. with underlying probability density p{x). We are interested in studying the time evolution and steady state behavior of the weight space probability density P(w, n) for ensembles of networks trained by stochastic learning. Stochastic process theory and classical statistical mechanics provide tools for doing this. As we shall see, the ensemble behavior of stochastic learning algorithms is similar to that of diffusion processes in physical systems, although significant differences do exist. 1.1 Dynamics of the Weight Space Probability Density Equation (1) defines a Markov process on the weight space. Given the particular input x, the single time-step transition probability density for this process is a Dirac delta function whose arguments satisfy the weight update (1): W ( w' ~ w I x) = 8 ( w - w' - J-t H[ w' , x]) . (2) From this conditional transition probability, we calculate the total single time-step transition probability (Leen and Orr 1992, Ritter and Schulten 1988) W(w' ~ w) = ( 8( w - w' - J-tH[w',x]) }z (3) where ( ... }z denotes integration over the measure on the random variable x. The time evolution of the density is given by the Kolmogorov equation P(w, n + 1) = J dw' P(w', n) W(w' ~ w) , (4) which forms the basis for our dynamical description of the weight space probability density 1. Stationary, or equilibrium, probability distributions are eigenfunctions of the transition probability Ps(w) = J dw' Ps(w') W(w' ~ w). (5) It is particularly interesting to note that for problems in which there exists an optimal weight w,. such that H(w,.,x) = 0, "Ix, one stationary solution is a delta function at w = w,.. An important class of such examples are noise-free mapping problems for which weight values exist that realize the desired mapping over all possible input/target pairs. For such problems, the ensemble can settle into a sharp distribution at the optimal weights (for examples see Leen and Orr 1992, Orr and Leen 1993). Although the Kolmogorov equation can be integrated numerically, we would like to make further analytic progress. Towards this end we convert the Kolmogorov 1 An alternative is to base the time evolution on a suitable master equation. Both approaches give the same results. Weight Space Probability Densities in Stochastic Learning: I. Dynamics and Equilibria 453 equ~,tion into a differential· difference equation by expanding (3) as a power series in J.l. Since the transition probability is defined in the sense of generalized functions (i.e. distributions), the proper way to proceed is to smear (4) with a smooth test function of compact support f(w) to obtain J dw f{w) P(w, n + 1) = J dw dw' f(w) P(w', n) W(w' -t w). (6) Next we use the transition probability (3) to perform the integration over wand expand the resulting expression as a power series in J.l. Finally, we integrate by part5 to take derivatives off f, dropping the surface terms. This results in a discrete time version of the classic Kramers·Moyal expansion (llisken 1989) P(w,n+1) P(w,n) = where Hja denotes the ja th component of the m-component vector H. In section 3, we present an algorithm for which the Kramers-Moyal expansion can be explicitly summed. In general the full expansion is not analytically tractable, and to make further analytic progress we will truncate it at second order to obtain the Fokker-Planck equation. 1.2 The Fokker-Planck (Diffusion) Approximation For small enough 1J.l HI, the Kramers-Moyal expansion (7) can be truncated to second order to obtain a Fokker-Planck equation:2 P(w, n + 1) P(w, n) = {) -J.l {)Wi [ Ai(W) P(w, n) ] (8) In (8), and throughout the remainder of the paper, repeated indices are summed over. In the Fokker-Planck approximation, only two coefficients appear: Ai ( w) = (Hi)z, called the drift vector, and Bij(W) = (Hi Hj)z' called the diffusion matrix. The drift vector is simply the average update applied at w. Since the diffusion coefficients can be strongly dependent on the position in weight space, the equilibrium densities will, in general, not be thermal (Gibbs) distributions in the potential corresponding to (H( w, x) ) z' This is exemplified in our discussion of equilibrium densities for the LMS algorithm in section 2.1 below3 • 2Radons et al. (1990) independently derived a Fokker-Planck equation for backpropagation. Earlier, Ritter and Schulten (1988) derived a Fokker-Planck equation (for Kohonen's self-ordering feature map) that is valid in the neighborhood of a local optimum. 3See (Leen and Orr 1992, Orr and Leen 1993) for further examples. 454 Leen and Moody 2 Equilibrium Densities in the Fokker-Planck Approximation In equilibrium the probability density is stationary, P(w, n+1) = P(w, n) = Ps(w), so the Fokker-Planck equation (8) becomes 0= - a:i Ji(w) == - a:i (11. Ai(W) P8(W) ~2 a:j [Bij(W) P8(W)] ) (9) Here, we have implicitly defined the probability density current J(w). In equilibrium, its divergence is zero. If the drift and diffusion coefficients satisfy potential conditions, then the equilibrium current itself is zero and detailed balance is obtained. The potential conditions are (Gardiner, 1990) OZk OZ, [J-l 0 1 OWl - OWk = 0, where Zk(W) = Bk/(w) 2" ow; Bi;(W) - Ai(W) (10) Under these conditions the solution to (9) for the equilibrium density is: Ps(w) = !... e-2:F(w)/~, F(w) = 1 dWk Zk(W) J( w (11) where J( is a normalization constant and F( w) is called the effective potential. In general, the potential conditions are not satisfied for stochastic learning algorithms in multiple dimensions.4 In this respect, stochastic learning differs from most physical diffusion processes. However for LMS with inputs whose correlation matrix is isotropic, the conditions are satisfied and the equilibrium density can be reduced to the quadrature in (11). 2.1 Equilibrium Density for the LMS Algorithm The best known on-line learning system is the LMS adaptive filter. For the LMS algorithm, the training examples consist of input/target pairs x(n) = {s(n),t(n)}, the model output is u(n) = W· s(n), and the cost function is the squared error: 1 1 £(w,x(n)) = 2 [t(n)-u(n)]2 = 2 [t(n)-w·s(n)]2 The resulting update equations (for constant learning rate J-l) are w(n+l) = w(n) + J-l[t(n)-w.s(n)]s(n). (12) (13) We assume that the training data are generated according to a "signal plus noise" model: t(n) = w • . s(n) + €(n) , (14) where w. is the "true" weight vector and €( n) is LLd. noise with mean zero and variance (12. We denote the correlation matrix of the inputs s( n) by R and the 4For one-dimensional algorithms, the potential conditions are trivially satisfied. Weight Space Probability Densities in Stochastic Learning: I. Dynamics and Equilibria 455 fourth order correlation tensor of the inputs by S. It is convenient to shift the origin of coordinates in weight space and define the weight error vector v = w w •. In terms of v, the weight update is v(n+l) = v(n) JJ[s(n).v(n)]s(n) + JJf(n)s(n). The drift vector and diffusion matrix are given by Ai=-(SiSj}s Vj = -RijVj and (15) Bij = (Si Sj Sle SI Vie VI + f2 Sj Sj ) s ( = Sijlel Vie VI + (72 Rij (16) , respectively. Notice that the diffusion matrix is quadratic in v. Thus as we move away from the global minimum at v = 0, diffusive spreading of the probability density is enhanced. Notice also that, in general, both terms of the diffusion matrix contribute an anisotropy. We further assume that the inputs are drawn from a zero-mean Gaussian process. This assumption allows us to appeal to the Gaussian moment factoring theorem (Haykin, 1991, p318) to express the fourth-order correlation S in terms of R Sijlel = Rij Rlcl + Rile Rjl + Ril Rjle . The diffusion matrix reduces to (17) To compute the effective potential (10 and 11) the diffusion matrix is inverted using the Sherman-Morrison formula (Press, 1987, p67). As a final simplification, we assume that the input distribution is spherically symmetric. Thus R = rI , where I denotes the identity matrix. Together these assumptions insure detailed balance, and we can integrate (11) in closed form. In figure 1, we compare the effective potential F(v) (for 1-D LMS) with the potential corresponding to the quadratic cost function. v Fig.l: Effective potential (dashed curve) and cost function (solid curve) for I-D LMS. The spatial dependence of the the diffusion coefficient forces the effective potential to soften relative to the cost function for large Ivl. This accentuates the tails of the distribution relative to a gaussian. 456 Leen and Moody The equilibrium density is 1 [ 3r ] -( ~+m ) Ps{v) = K 1 + u21vl2 , (18) where, as before, m and J( denote the dimension of the weight vector and the normalization constant for the density respectively. For a l-D filter, the equilibrium density can be found in closed form without assuming Gaussian input data. We find (19) With gaussian inputs (for which S = 3r2 ) (19) properly reduces to (18) with m = 1. The equilibrium densities (18) and (19) are clearly not gaussian, however in the limit of very small J.lr they reduce to gaussian distributions with variance J.lu2/2. Figure 2 shows a comparison between the theoretical result and a histogram of 200,000 values of v generated by simulation with J.l = 0.005, and u 2 = 1.0. The input data were drawn from a zero-mean Gaussian distribution with r = 4.0. I I i I I -0.2 -0.1 0.0 0.1 0.2 v Fig.2: Equilibrium density for 1-D LMS 3 An Exactly Summable Model As in the case of LMS learning above, stochastic gradient descent algorithms update weights based on an instantaneous estimate of the gradient of some average cost function £(w) = {£(w, x) }z. That is, the update is given by o Hi(W,X) = --0 £(w,x). Wi An alternative is to increment or decrement each weight by a fixed amount depending only on the sign of O£/OWi. We formulated this alternative update rule because it avoids a common problem for sigmoidal networks, getting stuck on "flat spots" or "plateaus". The standard gradient descent update rule yields very slow movement on plateaus, while second order methods such as gauss-newton can be unstable. The sign-of-gradient update rule suffers from neither of these problems.s 5The use of the sign of the gradient has been suggested previously in the stochastic approximation literature by Fabian (1960) and in the neural network literature by Derthick (1984). Weight Space Probability Densities in Stochastic Learning: I. Dynamics and Equilibria 457 If at each iteration one chooses a weight at random for updating, then the KramersMoyal expansion can be exactly summed. Thus at each iteration we 1) choose a weight Wi and an exemplar x at random, and 2) update Wi with H .( ) _ . (8£(w,x(n))) I w,x -Sign 8 Wi (20) With this update rule, Hj = ±1 or 0 and Hi Hj = lSij (or 0). All of the coefficients (HiHj Hk ... ) z in the Kramers-Moyal expansion (7) vanish unless i = j = k = .... The remaining series can be summed by breaking it into odd and even parts. This leav\!s P(w,n+l) P(w,n) = m 1 L { P(w + Ilj,n) Aj(w + Ilj) - P(w -Ilh n) Aj(w -Ilj) } 2m j=1 m + 1 L { P(w + Ilj, n) Bjj(w + Ilj) - 2P(w, n) Bjj(w) 2m j=1 + P(w -Ilj, n) Bjj(w -Ilj) } (21) where /-tj denotes a displacement along Wj a distance /-t, Aj(w) = (Hj(w, x) )z' and Bjj(w) = (H;(w,x)z' Note that Bjj(w) = 1 unless H(w,x) = 0, for all x, in which case Bjj(w) = O. Although exact, (21) curiously has the form of a second order finite difference approximation to the Fokker-Planck equation with diagonal diffusion matrix. This form is understandable, since the dynamics (20) restrict the weight values W to a hypercubic lattice with cell length /-t and generate only nearest neighbor interactions. L: -0.5 -0.5 !J!: k!~ 0.5 1 1.5 2 2.5 -0.5 0.5 1 1.5 2 2.5 v v n=5oo -0.5 n =5000 0.5 1 1.5 2 2.5 v Fig.3: Sequence of densities for the XOR problem As an example, figure 3 shows the cost function evaluated along a 1-D slice through the weight space for the XOR problem. Along this line are local and global minima at v = 1 and v = 0 respectively. Also shown is the probability density (vertical lines). The sequence shows the spreading of the density from its initialization at the local minimum, and its eventual collection at the global minimum. 458 Leen and Moody 4 Discussion A theoretical approach that focuses on the dynamics of the weight space probability density, as we do here, provides powerful tools to extend understanding of stochastic search. Both transient and equilibrium behavior can be studied using these tools. We expect that knowledge of equilibrium weight space distributions can be used in conjunction with theories of generalization (e.g. Moody, 1992) to assess the influence of stochastic search on prediction error. Characterization of transient phenomena should facilitate the design and evaluation of search strategies such as data batching and adaptive learning rate schedules. Transient phenomena are treated in greater depth in the companion paper in this volume (Orr and Leen, 1993). Acknowledgements T. Leen was supported under grants N00014-91-J-1482 and N00014-90-J-1349 from ONR. J. Moody was supported under grants 89-0478 from AFOSR, ECS-9114333 from NSF, and N00014-89-J-1228 and N00014-92-J-4062 from ONR. References Todd K. Leen and Genevieve B. Orr (1992), Weight-space probability densities and convergence times for stochastic learning. In International Joint Conference on Neural Networks, pages IV 158-164. IEEE, June. H. Ritter and K. Schulten (1988), Convergence properties of Kohonen's topology conserving maps: Fluctuations, stability and dimension selection, Bioi. Cybern., 60, 59-71. Genevieve B. Orr and Todd K. Leen (1993), Probability densities in stochastic learning; II. Transients and Basin Hopping Times. In Giles, C.L., Hanson, S.J., and Cowan, J.D. (eds.), Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann Publishers. H. Risken (1989), The Fokker-Planck Equation Springer-Verlag, Berlin. G. Radons, H.G. Schuster and D. Werner (1990), Fokker-Planck description oflearning in backpropagation networks, International Neural Network Conference - INNC 90, Paris, II 993-996, Kluwer Academic Publishers. C.W. Gardiner (1990), Handbook of Stochastic Methods, 2nd Ed. Springer-Verlag, Berlin. Simon Haykin (1991), Adaptive Filter Theory, 2nd edition. Prentice Hall, Englewood Cliffs, N.J. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling (1987) Numerical Recipes - the Art of Scientific Computing. Cambridge University Press, Cambridge I New York. V. Fabian (1960), Stochastic approximation methods. Czechoslovak Math J., 10, 123-159. Mark Derthick (1984), Variations on the Boltzmann machine learning algorithm. Technical Report CMU-CS-84-120, Department of Computer Science, Carnegie-Mellon University, Pittsburgh, PA, August. John E. Moody (1992), The effective number of parameters: An analysis of generalization and regularization in nonlinear learning systems. In J .E. Moody, S.J. Hanson, and R.P. Lipmann, editors, Advances in Neural Information Processing Systems 4. Morgan Kaufmann Publishers, San Mateo, CA.
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Adaptive Stimulus Representations: A Computational Theory of Hippocampal-Region Function Mark A. Gluck Catherine E. Myers Center for Molecular and Behavioral Neuroscience Rutgers University. Newark. NJ 07102 g IlIck@pOl·/OI·.I'lI(gers.edll mycrs@p(/\-Iol'.rl/(gers.edll Abstract We present a theory of cortico-hippocampal interaction in discrimination learning. The hippocampal region is presumed to form new stimulus representations which facilitate learning by enhancing the discriminability of predictive stimuli and compressing stimulus-stimulus redundancies. The cortical and cerebellar regions, which are the sites of long-term memory. may acquire these new representations but are not assumed to be capable of forming new representations themselves. Instantiated as a connectionist model. this theory accounts for a wide range of trial-level classical conditioning phenomena in normal (intact) and hippocampal-Iesioned animals. It also makes several novel predictions which remain to be investigated empirically. The theory implies that the hippocampal region is involved in even the simplest learning tasks; although hippocampal-Iesioned animals may be able to use other strategies to learn these tasks. the theory predicts that they will show consistently different patterns of transfer and generalization when the task demands change. 1 INTRODUCTION It has long been known that the hippocampal region (including the entorhinal cortex. subicular complex. hippocampus and dentate gyrus) plays a role in leaming and memory. For example. the hippocampus has been implicated in human declarative memory (Scoville & Millner. 1957: Squire. 1987) while hippocampal damage in animals impairs such seemingly disparate abilities as spatial mapping (O'Keefe & Nadel. 1978). contextual sensitivity (Hirsh. 1974; Winocur. Rawlins & Gray. 1987; Nadel & Willner. 1980). temporal processing (Buszaki. 1989; Akase. Aikon & Disterhoft. 1989). configural association (Sutherland & Rudy. 1989) and the tlexible use of representations in novel situations (Eichenbaum & Buckingham. 1991). Several theories have characterized hippocampal function in terms of one or more of these abilities. However. a theory which can predict the full range of deficits after hippocampal lesion has been elusive. This paper attempts to provide a functional interpretation of a hippocampal-region role in associative learning. We propose that one function of the hippocampal region is to construct new representations which facilitate discrimination learning. We argue that this 937 938 Gluck and Myers representational function is sufficielll to derive and unify a wide range of trial-level conditioned effects observable in the illloct and lesioned animal. 2 A THEORY OF CORTICO-HIPPOCAMPAL INTERACTION Psychological theories have often found it useful to characterize stimuli as occupying points in an internal representation space (c.f. Shepard. 1958: Nosofsky. 1984). Connectionist theories can be interpreted in a similar geometric framework. For example. in a connectionist network (see Figure IA) a stimulus input such as a tone is recoded in the network's internal layer as a pallern of activations. A light input will activate a different pallern of activations in the internal layer nodes (Figure I B). These internal layer activations can be viewed as a representation of the stimulus inputs. and can be plotted in multi-dimensional internal representation space (Figure IC). Learning to classify stimulus inputs corresponds to finding an appropriate partition of representation space. In the connectionist model. the lower layer of network weights determine the representation while the upper layer of network weights detennine the classification. Our basic premise is that the hippocampal region has the ability to modify stimulus representations to facilitate classification. and that its representations are biased by two constraints. The first constraint. predictive differentiation. is a bias to differentiate the representations of stimuli which are to be classified differently. Predictive differentiation increases the representational resources (i.e .. hidden units) devoted to representing stimulus features which are especially predictive of how a stimulus is to be classified. For example. if red stimuli alone should evoke a response. then many represelllational (A) (C) Internal Representation Space (8) Response 1.0n one 0.8 R 0.6 : 2 ..... :. Classification • Tone Light Context 0.4 ~ Light 0.2 0.0 +--...---..---.---.--...., 0.0 0.2 0.4 O. 0.8 1.0 R, Tone Light Context Figure 1: Stimulus representations. The activations of the internal layer nodes in a connectionist network constitute a representation of the network's stimulus inputs. (A) Internal representation for an example tone stimulus. (B) Internal representation for an example light stimulus. (C) Translation of these representations into points in an internal representation space. with one dimension encoding the activation level of each internal node. Classifying stimuli corresponds 10 partitioning representation space so that representations of stimuli which ought to be classified together lie in the same partition. Classification is easier if the representations of stimuli to be classified together are clustered while representations of stimuli to be classified differently are widely separated in this space. Adaptive Stimulus Representations: Computational Theory of Hippocampal-Region Function 939 resources should be devoted to encoding color. The second constraint, redundancv compression. reduces the resources allocated to represent features which are redundant or irrelevant in predicting the desired response . These IWO constraints are by nature complementary. given a finite amount of representational resources. Compressing redundant features frees resources 10 encode more predictive features. Conversely. increasing the resources alloC:lIed to predictive features forces compression of the remaining (less predictive) features. This proposed hippocampal-region function may be modelled by a predictive autoencoder (on the right in Figure 2). An autoencoder (Hinton, 1989) learns to map from stimulus inputs. through :m internal layer, to an output which is a reproduction of those inputs. This is also known as stimulus-stimulus learning. To do this. the network must have access to some multi-layer learning algorithm such as error backpropagation (Rumelhart. Hinton & Williams. 1986). When the internal layer is narrower than the input and output layers. the system develops a recoding in the internal layer which takes advantage of redundancies in the inputs. A predictive autoencoder has the further constraint that it must also output a classification response to the inputs. This is also known as stimulusresponse learning. The internal layer recoding must therefore also emphasize stimulus features which are especially predictive of this classification. Therefore, a predictive autoencoder learns to fonn internal representations constrained by both predictive differentiation and redundancy compression. and is thus an example of a mechanism for implementing the two representational biases described above. The cerebral and cerebellar cortices form the sites of long term memory in this theory. but are not themselves directly able to form new representations. They can. however. acquire new representations formed in the hippocampal region. A simplified model of one such cerebellar region is shown on the left in Figure 2. This network does not have access to multi-layer learning which would allow it to independently form new internal representations by itself. Instead, the two layers of weights in this network evolve independently. The bollom layer of weights is trained so that the current input pallern generates an internal representation equivalent to that developed in the hippocampal model. Independently and simultaneously. weights in the cortical network top layer are trained to map from this evolving representation to the classification response. Because the cortical networks are not creating new representations. but only learning two independent single-layer mappings. they can use a much simpler learning rule than the hippocampal model. One such algorithm is the LMS learning rule (Widrow & Hoff. 1960), which can instantiate the Rescorla-Wagner (1972) model of classical conditioning. Cortical (Cerebellar) Network Hippocampal-System Model Single-layer I learning + ." .... s,:nv Input Sensory Input (training signal) --Multi-layer learning Figure 2. The cortico-hippocampal model: new representations developed in the hippocampal model can be acquired by cortical networks which are incapable. of developing such representations by themselves. 940 Gluck and Myers 3 MODELLING HIPPOCAMPAL INVOLVEMENT IN CLASSICAL CONDITIONING A popular experimental paradigm for the study of associative learning in :lI1imals is classical conditioning of the rabbit eyeblink response (see Gormezano. Kehoe & Marshall. 1983. for review). A puff of air delivered to the eye elicits a blink response in the rabbit. If a previously neutral stimulus. such as a lOne or light (called the conditioned stimulus). is repeatedly presented just before the airpuff. the animal will develop a blink response to this stimulus -- and time the response so that the lid is maximally closed just when the airpuff is scheduled to arrive. Ignoring Ihe many lemporal factors -- such as the interval between stimuli or precise timing of the response -- this reduces to a classification problem: learning which stimuli accuralely predict the airpuff and should therefore evoke a response. During a training trial. both the hippocampal and cortical networks receive the same input pallern. This pallem represents the presence or absence of all stimulus cues -- both conditioned stimuli and background contextual cues. Contextual cues are always present. but may change slowly over time. The hippocampus is trained incrementally to predict the current values of all cues -- including the US. The evolving hippoocampal internal layer representation is provided to the cortical network. which concurrently learns to reproduce this representation and to associate this evolving internal representation with a prediction of the US. This cortical network prediction is interpreted as the system's response. The complete (intact) cortico-hippocampal model of Figure 2 can be shown to produce conditioned behavior comparable to that of nonnal (intact) animals. Hippocampal lesions can be simulated by disabling the hippocampal model. This eliminates the training signal which the cortical model would otherwise use to construct internal layer representations. As a result. the lower layer of cortical network weights remains fixed. The lesioned model's cortical network can still modify its upper layer of weights to learn new discriminations for which its current (now fixed) internal representation is sufficient. 4 BEHAVIORAL RESULTS A stimulus discrimination task involves learning that one stimulus A predicts the airpuff but a second stimulus B does not. The notation <A+. B-> is used to indicate a series of training trials intermixing A+ (A preceeds the airpuff). B- (B does not preceed the airpuff) and context-alone presentations. Figure 3A shows the appropriate development of responses to A but nOI to B during this task. Both the intact and lesioned systems can acquire this discrimination. In fact. the lesioned system leams somewhat faster: it is only learning a classification. since its representation is fixed and (for this simple task) generally sufficient. In the intact system. by conlrast. the hippocampal model is developing a new representation and transferring it to the cortical network The cortical network must then learn classifications based on this changing representation. This will be slower than learning based on a fixed representation. This paradox of discrimination facilitation after hippocampal lesion has often been reported in the animal literature (Schmaltz & Theios. 1972: Eichenbaum. Fagan. Mathews & Cohen. 1988): one previous interpretation has been to suggest that the hippocampal region is somehow "unncccessary for" or even "inhibitory to" simple discrimination learning. Our model suggests a different interpretation: the intact system learns more slowly because it is actually learning more than the lesioned system. The intact system is learning not only how to map from stimuli to responses. it is also developing new stimulus representations which enhance the differentiation among representations of predictive stimulus features while compressing the representations of redundant and irrelevant stimulus features. Adaptive Stimulus Representations: Computational Theory of Hippocampal-Region Function 941 The benefit of this re-representation can most readily be seen when the task demands suddenly change. For example. suppose the task valences shift from <A+. B-> to <A-. B+>. The representation developed during the first training phase. which maximally differentiated features distinguishing stimulus A from B. will still be useful in the second training phase. Only the classification needs to be relearned. Figure 3B shows that the intact system can learn the reversed task slightly more quickly than it learned the original task. Successive reversals are expected to be even more facilitated. as the representations of A and B grow ever more distinct (see Sutherland & Mackintosh. 1971. for a review of the relevant empirical data). In contrast. the lesioned system is severely impaired in the reversal task (Figure 3 B). In the lesioned system. with a fixed representation. all the information is contained in the upper classificatory layer of weights. This information must be unlearned before the reversal task can be learned. Consistent with the model's behavior. empirical studies of hippocarnpal-Iesioned animals show strong impairment at reversal learning (Berger & Orr. 1983). The simplest evidence for redundancy compression likewise occurs during a transfer task. During unreinforced pre-exposure to a stimulus cue A. the presence or absence of A is irrelevant in terms of predicting US arrival (since a US never comes). Our theory expects that the representation of A will therefore become compressed with the representations of of the background contextual cues. In a subsequent training phase in which A does predict the US. the system must learn to respond to a feature it previously learned to ignore. The representation of A must now be re-differentiated from the context. Our theory therefore expects that learning to respond to A will be slowed. relative to learning (A) Response 1 0.8 0.6 0.4 0.2 o (C) Response 1 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 Training Trials AA+ A+ only I I I I I I I I I I I 50 0 100 Training Trials (8) Trials to Learn 400 300 200 100 O~----------~-----------, A+. B· A-. B+ (0) Response 1 AA+ 0.8 , _ ;%.C( ~" 0.6 '11' .... " 01' 0.4 l' Ii 0.2 0 0 50 0 100 Training Trials Figure 3. Behavioral results. Solid line = intact system. dashed line = lesioned system (A) Discrimination learning <A+. B-> in intact and lesioned models: lesioned model learns slightly faster. (B) Discrimination reversal «A+. B-.> then <A-. B+» Intact system shows facilitation on successive reversals, lesioned system is severely impaired. (C) Latent inhibition (A- impairs A+) in the intact model: (0) No latent inhibition in the lesioned model. All results shown are consistent with empirical data (see text for references). 942 Gluck and Myers without pre-exposure to A (Figure 3C). This effect occurs in animals and is known as latent inhibition (Lubow. 1973). In this theory. latent inhibition arises from hippocampal-dependent recodings. In the lesioned system. there is no stimulus-stimulus learning during the pre-exposure phase. and no redundancy compression in the (fixed) internal representation. Therefore. unreinforced pre-exposure does not slow the learning of a response to A (Figure 3D). Empirical studies have shown that hippocampal lesions also eliminate latent inhibition in animals (Solomon & Moore. 1975). Incidentally. a standard feedforward backpropagation network. with the same architecture as the cortical network. but with access to a multi-layer learning algorithm. fails to show latent inhibition. Such a network can fonn representations in its internal layer. but unlike the hippocampal model it does not perform stimulus-stimulus learning. Therefore. there is no effect of unreinfored pre-exposure of a stimulus. and no latent inhibition effect (simulations not shown). This cortico-hippocampal theory can account for many other effects of hippocampal lesions (see Gluck & Myers. 1992. 1993 / in press): including increased stimulus generalization and elimination of sensory preconditioning. It also provides all interpretation of the observation that hippocampal disruption can damage learning more than complete hippocampal removal (Solomon. Solomon. van der Schaaf & Perry. 1983): if the training signals from the hippocampus are "noisy". the cortical network will acquire a distorted and continuously changing internal representation. In general. this will make classification learning harder than in the lesioned system where the illlernal representation is simply fixed. The theory also makes several novel and testable predictions. For example. in the intact animal. training 10 discriminate two highly similar stimuli is facilitated by pre-training on an easier version of the same task -- even if the hard task is a reversal of the easy task (Mackintosh & Little. 1970). The theory predicts that this effect arises from predictive differentiation during the pre-training phase. and therefore should be eliminated after hippocampal lesion. Another effect observed in intact animals is compound preconditioning: discrimination of two stimuli A and B is impaired by pre-exposure to the compound AB (Lubow. Rifkin & Alek. 1976). The theory attributes this effect 10 redundancy compression in the pre-exposure phase. and therefore again predicts that the effect should disappear in the hippocampal-Iesioned animal. 5 CONCLUSIONS There are many hippocampal-dependent phenomena which the model. in its present form. does not address. For example. the model does not consider real-time tempor • .tl effects. or operant choice behavior. Because it is a trial-level model. it does not address the issue of a consolidation period during which memories gradually become independent of the hippocampus. We have also not considered here the physiological mechanisms or structures within the hippocampal region which might implement the proposed hippocampal function. Finally. the model would require extensions before it could apply to such high-level behaviors as spatial navigation. human declarative memory. and working memory -- all of which are known to be disrupted by hippocampal lesions. Despite the theory's restricted scope. it provides a simple and unified account of a wide range of trial-level conditioning data. It also makes several novel predictions which remain to be investigated in lesioned animals. The theory suggests that the effects of hippocampal damage may be especially informative in studies of two-phase transfer tasks. In these paradigms. both intact and hippocampal-Iesioned animals are expected to behave similarly on a simple initial learning task. but exhibit different behaviors on a subsequent transfer or generalization task. Adaptive Stimulus Representations: Computational Theory of Hippocampal-Region Function 943 REFERENCES Akase, E., Alkon, D., & Disterhoft. J. (1989). Hippocampal lesions impair memory of shorl-delay conditioned eye blink in rabbits. Behavioral Neuroscience, 103(5}, 935-943. Berger. T. W .. & Orr. W. B. (1983). Hippocampectomy seleclively disrupts discrimination reversal learning of the rabbit nictitating membrane response. Behavioral Brain Research, a, 49-68. Buszaki, G. (1989). Two-stage model of memory trace formation: A role for "noisy" brain states. Neuroscience, 31(3). 551-570. Eichenbaum, H., & Buckingham, J. (1991). Studies on hippocampal processing: Experiment. theory, and model. In M. Gabriel & J. Moore (Eds.), Neurocomputation and learning: Foundations of adaptive networks Cambridge. MA: M.l.T. Press. Eichenbaum, H .. Fagan. A.. Mathews, P .. & Cohen, N. (1988). Hippocampal system dysfunction and odor discrimination learning in rats: lmpainnent or facilitation depending on representational demands. Behavioral Neuroscience, 102(3).331-339. Gluck, M. & Myers, C. (1992). Hippocampal-system function in stimulus representation and generalization: A computational theory. Proceedin~s 14th Annual Conference of the Co~nitive Science Society. Bloomington. IN, 390-395. Gluck, M .. & Myers, C. (1993 / in press). Hippocampal mediation of stimulus representation: A computational theory, Hippocampus. Gormezano, I.. Kehoe, E. K .. & MarshaL B. S. (1983). Twenty years of classical conditioning research with the rabbit. Progress in Psychobiology and Physiological Psychology, lQ. 197-275. Hinton, G. E. (1989). Connectionist learning procedures. Artificial Intelligence, 4Q. 185234. Hirsh, R. (1974). The hippocampus and contextual retrieval of information from memory: A theory. Behavioral Biology. il. 421-444. Lubow, R. E. (1973). Latent inhibition. Psychological Bylletin.l!l... 398-407. Lubow, R, Ritkin, B .. & Alek, M. (1976). The context effect: The relationship between stimulus pre-exposure and environmental pre-exposure determines subsequent learning. Journal of Experimental Psychology: Animal Behavior Processes, 2(1). 38-47. Mackintosh, N. & Little. L. (1970). An analysis of transfer along a continuum. Canad. J. Psychol.1 Rev. Canad. Psychol.. 24(5).362-369. Nadel. L.. & Willner. J. (1980). Context and conditioning: A place for space. Physiological Psychology. a. 218-228. Nosofsky. R. M. (1974). Choice. similarity. and the context theory of classification. Joyrnal of Experimental Psychol0I:Y: Learning. Memo!), and Cognition. lQ, 104-114. O'Keefe, L & NadeL L. (1978). The Hippocampus as a Cognitive Map. Oxford. UK: Claredon University Press. 944 Gluck and Myers Rescorla. R. A .. & Wagner. A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and non-reinforcement. In A. H. Black & W. F. Prokasy (Eds.). Classical Conditioning II: Current Research and Theory New York: Appleton-Century-Crofts. Rumelhart. D. Eo, Hilllon. G. E .. & Williams. R. J. (1986). Learning internal representations by error propagation. In D. Rumelhart & J. McClelland (Eds.). Parallel DistribUled Processing: Explorations in the MicrOS!Tucture of Cognition (Vol. I: Foundations) (pp. 318-362). Cambridge. MA: MIT Press. Schmaltz. L. W .. & Theios. J. (1972). Acquisition and extinction of a classically conditioned response in hippocarnpectomized rabbits (Oryctolagus cuniculus). Journal of Comparative and Physiological Psychology. 79. 328-333 . Scoville. W. B .. & Milner. B. (1957). Loss of recent memory after bilateral hippocampal lesions. journal of Neurology. Neurosurgery. & Psychiatry, 2.0. 11-21. Shepard. R. N. (1958). Stimulus and response generalization: Deduction of the generalization gradient from a trace model. PSychological Review.~. 242-256. Solomon. P. R .. & Moore. J. W. (1975). Latent inhibition and stimulus generalization of the classically conditioned nictitating membrane response in rabbits (Oryctolagus cuniculus) following dorsal hippocampal ablation. Journal of Comparative and Physiological Psychology. 82. 1192-1203. Solomon. P .. Solomon. S .. van der Schaaf. E. & Perry, H. (1983). Altered activity in the hippocampus is more detrimental to classical conditioning than removing the structure. Science. 220. 329-331. Squire. L. R. (1987). Memory and brain. New York: Oxford University Press. Sutherland. N. & Mackintosh, N. (1971). Mechanisms of Animal Discrimination Learning. New York: Academic Press. Sutherland. R. J.. & Rudy. J. W. (1989). Configural association theory: The role of the hippocampal formation in learning. memory. and amnesia. Psychobiology. 17 (2), 129144. Widrow. B .. & Hoff. M. (1960). Adaptive switching circuits. Institute of Radio Engineers. Western Electronic Show and Convention. Convention Record.~. 96-194. Winocur. G .. Rawlins. 1. & Gray. J. R. (1987). The hippocampus and conditioning lO contextual cues. Behavioral Neuroscience.lQl. 617 -625.
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Discriminability-Based Transfer between Neural Networks L. Y. Pratt Department of Mathematical and Computer Sciences Colorado School of Mines Golden, CO 80401 lpratt@mines.colorado.edu Abstract Previously, we have introduced the idea of neural network transfer, where learning on a target problem is sped up by using the weights obtained from a network trained for a related source task. Here, we present a new algorithm. called Discriminability-Based Transfer (DBT), which uses an information measure to estimate the utility of hyperplanes defined by source weights in the target network, and rescales transferred weight magnitudes accordingly. Several experiments demonstrate that target networks initialized via DBT learn significantly faster than networks initialized randomly. 1 INTRODUCTION Neural networks are usually trained from scratch, relying only on the training data for guidance. However, as more and more networks are trained for various tasks, it becomes reasonable to seek out methods that. avoid "'reinventing the wheel" , and instead are able to build on previously trained networks' results. For example, consider a speech recognition network that was only trained on American English speakers. However, for a new application, speakers might have a British accent. Since these tasks are sub-distributions of the same larger distribution (English speakers), they may be related in a way that. can be exploited to speed up learning on the British network, compared to when weights are randomly initialized. We have previously introduced the question of how trained neural networks can be 204 Discriminability-Based Transfer between Neural Networks 205 "recycled' in this way [Pratt et al., 1991]; we've called this the transfer problem. The idea of transfer has strong roots in psychology (as discussed in [Sharkey and Sharkey, 1992]), and is a standard paradigm in neurobiology, where synapses almost always come "pre-wired". There are many ways to formulate the transfer problem. Retaining performance on the source task mayor may not be important. When it is, the problem has been called sequential learning, and has been explored by several authors (cf. [McCloskey and Cohen, 1989]). Our paradigm assumes that source task performance is not important, though when thl~ source task training data is a subset of the target training data, our method may be viewed as addressing sequential learning as well. Transfer knowledge can also be inserted into several different entry points in a back-propagation network (see [Pratt, 1993al). We focus on changing a network's initial weights; other studies change other aspects, such as the objective function (cf. [Thrun and Mitchell, 1993, Naik et al., 1992]). Transfer methods mayor may not use back-propagation for target task training. Our formulation does, because this allows it to degrade, in the worst case of no source task relevance, to back-propagation training on the tar~et task with randomly initialized weights. An alternative approach is described by lAgarwal et al., 1992]. Several studies have explored literal transfer in back-propagation networks, where the final weights from training on a source task are used as the initial conditions for target training (cf. [Martin, 1988]). However, these studies have shown that often networks will demonstrate worse performance after literal transfer than if they had been randomly initialized. This paper describes the Discriminability-Based Transfer (DBT) algorithm, which overcomes problems with literal transfer. DBT achieves the same asymptotic accuracy as randomly initialized networks, and requires substantially fewer training updates. It is also superior to literal transfer, and to just using the source network on the target task. 2 ANALYSIS OF LITERAL TRANSFER As mentioned above, several studies have shown that networks initialized via literal transfer give worse asymptotic perlormance than randomly initialized networks. To understand why. consider the situation when only a subset. of the source network input-to-hidden (IH) layer hyperplanes are relevant to the target problem. as illustrated in Figure 1. We've observed that some hyperplanes initialized by source network training don't shift out of their initial positions, despite the fact that they don't help to separate the target training data. The weights defining such hyperplanes often have high magnitudes [Dewan and Sontag, 1990]. Figure 2 (a) shows a simulation of such a situation, where a hyperplane that has a high magnitude, as if it came from a source network, causes learning to be slowed down. 1 Analysis of the back-propagation weight update equations reveals that high source weight magnitudes retard back-propagation learning on the target task because this 1 Neural network visualization will be explored more thoroughly in an upcoming pape ... An X-based animator is available from the author via anonymous ftp. Type "archie ha". 206 Pratt Source training data Target training data 0.9 : 0 ! 0 ....•....•.... .Q~ •... ; .•.....•...•...•.•....•.......•.•...•.••.......•.•..•........ Q ....••••...••.•..•..••.. o f 1./ 0 ~ Hyperplanes 0 1 0 o i 1 ./ 0 should be 0 1 0 ! 1! f retained 1 ·············if···f··'O································· ......................... 0-•...... 0 ............ . ;...j.~ _ Hyperplanes need to move 0 0.1 0.1 0.5 0.9 Feature 1 Figure 1: Problem Illustrating the need for DBT. The source and target tasks are identical, except that the target task has been shifted along one axis, as represented by the training data shown. Because of this shift, two of the source hyperplanes are helpful in separating class-O from class-l data in the target task, and two are not. equation is not scaled relative to weight magnitudes. Also, the weight update equation contains the factor y( 1 y) (where y is a unit's activation). which is small for large weights. Considering this analysis, it might at first appear that a simple solution to the problem with literal transfer is to uniformly lower all weight magnitudes. However, we have also observed that hyperplanes in separating positions will move unless they are given high weight magnitudes. To address both of these problems, we must rescale hyperplanes so that useful ones are defined by high-magnitude weights and less useful hyperplanes receive low magnitudes. To implement such a method, we need a metric for evaluating hyperplane utility. 3 EVALUATING CLASSIFIER COMPONENTS We borrow the 1M metric for evaluating hyperplanes from decision tree induction [Quinlan, 1983]. Given a set of training data and a hyperplane that crosses through it, the 1M function returns a value between 0 and 1, indicating the amount that the hyperplane helps to separate the data into different classes. The formula for 1M, for a decision surface in a multi-class problem, is: 1M ~ ( L L xij logxij - L Xi. log Xi. - L X.j logx.j + Nlog N) [Mingers, 1989). Here, N is the number of patterns, i is either 0 or 1, depending on the side of a hyperplane on which a pattern falls, j indexes over all classes, Xij is the count of class j patterns on side i of the hyperplane, ~. i is the count of all patterns on side i, and x.j is the total number of patterns in class j. 4 THE DBT ALGORITHM The DBT algorithm is shown in Figure 3. It inputs the target training data and weights from the source network, along with two parameters C and S (see below). DBT outputs a modified set of weights, for initializing training on the target task. Figure 2 (b) shows how the problem of Figure 2 (a) was repaired via DBT. DBT modifies the weights defining each source hyperplane to be proportional to the Discriminability-Based Transfer between Neural Networks 207 (a) Literal '" ~ 1 ....... , EpodI1 \ 0 l ! 0 f· · ·· · ··· · · ·~ ............. , 0 I 0 . , .. I.L. C! \ ~ 0: C! '" e J ~ C! C! '" e i ~ Lt C! 0.2 0.2 0.2 0.2 D.2 1· ... . o 1 0.4 0.6 0.1 F .. u.l 100 o ... , ............... \\ .~ ... . ..., , III 0.4 0.8 0.' F....".., 300 \ 0 ······1··....... \ 0 ········\.;.·······m \ III 0.4 0.8 0.8 FHU.1 400 o " 0 '. "'''',' ·····m \ III 0.4 0.8 0.1 F .. u.1 3100 1 \ 0 0·········· ... 1. \ I ... .. .\ 0 III .\ ......... ..., \ 0.4 0.8 0.8 FHU.1 (b) OBT o EpodI1 \ \ i 0 I· ...... r.. 1 l : . . .......................... \ 0.2 1 o 1 0.4 0.8 F .. ture 1 EDOdI100 \ \ \ \ :J 0.8 o o 0.2 0.4 0.8 0.8 F .. ture 1 HIf'2O '. . . . EpodI 300 \ 0 l f ·····1........ \ 0 .. .. .. \. ," .. .... ··tlJ , 0.2 0.4 0.8 0.8 FHtur.1 '" Cl I ....... ~ ......... :400 \ 0 tlJr e 0 f ... -- ....... s;: ...... O' . .. +tJ 1 0 '\ ~ . 0.2 0.4 0.8 0.' 0.2 0.4 1 ...... '0" 1 FHture 1 E:~ 0.8 0.8 F .. tI.n 1 3100 \ 0 ttl . • • \ • • ••• • • I)- •• • +tJ , , 0.2 0.4 0.6 0.8 F .. ture1 Figure 2: Hyperplane 110vement speed in Literal Transfer. Compared to DBT. Each image in this figure shows the hyperplanes implemented by IH weights at a different epoch of training. Hidden unit l's hyperplane is a solid line; HU2's is a dotted line, and HU3's hyperplane is shown as a dashed line. In (a) note how HUl seems fixed in place. Its high magnitude causes learning to be slow (taking about 3100 epochs to converge). In (b) note how DBT has given HUl a small magnitude, allowing it to be flexible, so that the training data is separated by epoch 390. A randomly initialized network on this problem takes about 600 epochs. 208 Pratt Input: Source network weights Target training data Parameters: C (cutoff (actor), S (scaleup (actor) Output: Initial weigbts (or target network, assuming same topology as source network Method: For eacb source network bidden unit i Compare tbe byperplane defined by incoming weights to i to tbe target training data, calculating [Mt ; (f [0,1]) Rescale [Mfa values so that largest has value S. Put. result in s, . For [Mt ; 's tbat are less tban C I( higbest magnitude ratio between weights defining hyperplane i is > 100.0, reset weights for that hyperplane randomly Else uniformly scale down byperplane to bave low-valued weigbts (maximum magnitude of 0.5), but to be in the same position. For eacb remaining IH hidden unit i For eacb weight wj; defining hyperplane i in target network L t · h • et Wj. = source weJg t Wj; X Si Set hidden-to-output target network weights randomly in [-0.5,0.5] Figure 3: The Discriminability-Based Transfer (DBT) Algorithm. 1M value, according to an input parameter, S. DBT is based on the idea that the best initial magnitude Aft for a target hyperplane is Mt = S X M. x I M t , where S ("scaleup") is a constant of proportionality, At. is the magnitude of a source network hyperplane, and I Mt is the discriminability of the source hyperplane on the target training data. We assume that this simple relationship holds over some range of I M t values. A second parameter, C, determines a cut-off in this relationship - source hyperplanes with I Mt < C receive very low magnitudes, so that the hyperplanes are effectively equivalent to those in a randomly initialized network. The use of the C parameter was motivated by empirical experiments that indicated that the multiplicative scaling via S was not adequate. To determine S and C for a particular source and target task, we ran DBT several times for a small number of epochs with different Sand C values. We chose the S and C values that yielded the best average TSS (total sum of squared errors) after a few epochs. We used local hill climbing in average TSS space to decide how to move in S, C space. DBT randomizes the weights in the network's hidden-to-output (HO) layer. See [Sharkey and Sharkey, 1992) for an extension to this work showing that literal transfer of HO weights might also be effective. 5 EMPIRICAL RESULTS DBT was evaluated on seven tasks: female-to-male speaker transfer on a lO-vowel recognition task (PB), a 3-class subset of the PB task (PB123). transfer from all females to a single male in the PB task (Onemale), transfer for a heart disease diagnosis problem from Hungarian to Swiss patients (Heart-HS). transfer for the same task from patients in California to Swiss patients (Heart-VAS). transfer from a subset of DNA pattern recognition exanlples to a superset (DNA). and transfer Discriminability-Based Transfer between Neural Networks 209 from a subset of chess endgame problems to a superset (Chess). Note that the DNA and chess tasks effectively address the sequential learning problem; as long as the source data is a subset of the target data, the target network can build on the previous results. DBT was compared to randomly initialized networks on the target task. We measured generalization performance in both conditions by using 10-way crossvalidation on 10 different initial conditions for each t.arget task, resulting in 100 different runs for each of the two conditions, and for each of the seven tasks. Our empirical methodology controlled carefully for initial conditions, hidden unit count, back-propagation paranleter:> '1 (learning rate) and Q" (momentum), and DBT parameters S and C. 5.1 SCENARIOS FOR EVALUATION There are at least two different practical situations in which we may want to speed up learning. First, we may have a limited amount of computer time. all of which will be used because we have no way of detecting when a network's performance has reached some criterion. In this case. if our speed-up method (i.e. DBT) is significantly superior to a baseline for a large proportion of epochs during training. then the probability that we'll have to stop during that period of significant superiority is high If we do stop at an epoch when our method is significantly better, then this justifies it over the baseline, because the resulting network has better petfonnance. A second situation is when we have some way of detecting when petformance is "good enough" for an application. In contrast to the above situation. here a DBT network may be run for a shorter time than a baseline network, because it reaches this criterion faster. In this case, the number of epochs of DBT significant superiority is less important than the speed with which it achieves the criterion. 5.2 RESULTS To evaluate networks according to the first scenario, we tested for statistical significance at the 99.0% level between the 100 DBT and the 100 randomly initialized networks at each training epoch. We found (1) that asymptotic DBT petformance scores were the same as for random networks and (2), that DBT was superior for much of the training period. Figure 4 (a) shows the number of weight updates for which a significant difference was found for the seven tasks. For the second scenario, we also found (3) that DBT networks required many fewer epochs to reach a criterion performance score. For this test, we found the last significantly different epoch between the two methods. Then we measured the number of epochs required to reach 98%, 95o/c" and 66%, of that level. The number of weight updates required for DBT and randomly initialized networks to reach the 98% criterion are shown in Figure 4 (b). Note that the y axis is logarithmic, so, for example. over 30 million weight updates were saved by using DBT instead of random initialization in the PB123 problem. Results for the 95% and 66% criteria also showed DBT to be at least as fast as random initialization for every task. Using the same tests described for DBT above, we also tested literal networks on the seven transfer tasks. We found that, unlike DBT, literal networks reached sig210 Pratt 00 o (a) Time for sig. epoch difference: OBT vs. random (b) Time required to train to 98% criterion o o g,~ o PB PB1230nema1eHeart· HeartDNA aPB PB123 0nemaIe HeartHeart- aHS VAS HS VAS Task Task Figure 4: Summary of DBT Empirical Results. nificantly worse asymptotic performance scores than randomly initialized networks. Literal networks also learned slower for some tasks. These results justify the use of the more complicated D BT method oyer literal transfer. \\Te also evaluated the source networks directly on the target tasks, without any back-propagation training on the target training data. Scores were significantly and substantially worse than random networks. This result indicates that the transfer scenarios we chose for evaluation were nontrivial. 6 CONCLUSION We have described the DBT algorithm for transfer between neural networks. 2 DBT demonstrated substantial and significant learning speed improvement over randomly initialized networks in 6 out of 7 tasks studied (and the same learning speed in the other task). DBT never displayed worse asymptotic performance than a randomly initialized network. We have also shown that DBT is superior to literal transfer, and to simply using the source network on the target task. Acknowledgements The author is indebted to John Smith. Gale MartinI and Anshu Agarwal for their valuable comments 011 this paper, and to Jack Mostow and Haym Hirsh for their contribution to this research program. 2See [Pratt, 1993b] for more details. Discriminability-Based Transfer between Neural Networks 211 References [Agarwal et al., 19921 A. Agarwal, R. J. Mammone, and D. K. Naik. An on-line training algorithm to overcome catastrophic forgetting. In Intelligence Engineering Systems through Artificial Neural Networks. volume 2, pages 239-244. The American Society of Mechanical Engineers, AS~IE Press, 1992. [Dewan and Sontag, 1990) Hasanat M. Dewan and Eduardo Sontag. Using extrapolation to speed up the backpropagation algorithm. In Proceedings oj the International Joint Conjerence on Neural Networks, Washington, DC, volume 1, pages 613-616. IEEE Pub:ications, Inc., January 1990. [Martin, 1988] Gale Martin. The effects of old learning on new in Hopfield and Backpropagation nets. Technical Report ACA-HI-0l9. Microelectronics and Computer Technology Corporation (MCC), 1988. [McCloskey and Cohen, 1989J Michael McCloskey and Neal J. Cohen. Catastrophic interference in connectionist networks: the sequential learning problem. The psychology oj learning and motivation, 24, 1989. [Mingers. 1989J John Mingers. An empirical comparison of selection measures for decision- tree induction. Machine Learning, 3( 4):319-342, 1989. [Naik et al., 1992] D. K. Naik, R. J. Mammone. and A. Agarwal. Meta-neural network approach to learning by learning. In Intelligence Engineering Systems through Artificial Neural Networks, volume 2. pages 245-252. The American Society of Mechanical Engineers, AS ME Press. 1992. [Pratt et al., 19911 Lorien Y. Pratt, Jack Mostow. and Candace A. Kamm. Direct transfer of learned information among neural networks. In Proceedings oj the Ninth National Conjerence on Artificial Intelligence (AAAI-91), pages 584-589, Anaheim, CA, 1991. [Pratt, 1993aJ Lorien Y. Pratt. Experiments in the transfer of knowledge between neural networks. In S. Hanson, G. Drastal, and R. Rivest, editors, Computational Learning Theory and Natural Learning Systems. Constraints and Prospects, chapter 4.1. MIT Press, 1993. To appear. [Pratt, 1993b] Lorien Y. Pratt. Non-literal transfer of informat.ion among inductive learners. In R.J.Mammone and Y. Y. Zeevi. editors. Neural Networks: Theory and Applications II. Academic Press, 1993. To appear. [Quinlan, 1983] J. R. Quinlan. Learning efficient classification procedures and their application to chess end games. In Machine Learning, pages 463-482. Palo Alto, CA: Tioga Publishing Company. 1983. [Sharkey and Sharkey, 19921 Noel E. Sharkey and Amanda J. C. Sharkey. Adaptive generalisation and the transfer of knowledge. 1992. \Vorking paper, Center for Connection Science, University of Exeter, 1992. [Thrun and Mitchell, 1993J Sebastian B. Thrun and Tom M. Mitchell. Integrating inductive neural network learning and explanation-based learning. In C.L. Giles, S. J. Hanson, and J . D. Cowan, editors. Advances In Neural Injormahon. Processing Systems 5. Morgan Kaufmann Publishers. San Mateo, CA, 1993.
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Learning Sequential Tasks by Incrementally Adding Higher Orders Mark Ring Department of Computer Sciences, Taylor 2.124 University of Texas at Austin Austin, Texas 78712 (ring@cs. utexas.edu) Abstract An incremental, higher-order, non-recurrent network combines two properties found to be useful for learning sequential tasks: higherorder connections and incremental introduction of new units. The network adds higher orders when needed by adding new units that dynamically modify connection weights. Since the new units modify the weights at the next time-step with information from the previous step, temporal tasks can be learned without the use of feedback, thereby greatly simplifying training. Furthermore, a theoretically unlimited number of units can be added to reach into the arbitrarily distant past. Experiments with the Reber grammar have demonstrated speedups of two orders of magnitude over recurrent networks. 1 INTRODUCTION Second-order recurrent networks have proven to be very powerful [8], especially when trained using complete back propagation through time [1, 6, 14]. It has also been demonstrated by Fahlman that a recurrent network that incrementally adds nodes during training-his Recurrent Cascade-Correlation algorithm [5]-can be superior to non-incremental, recurrent networks [2,4, 11, 12, 15]. The incremental, higher-order network presented here combines advantages of both of these approaches in a non-recurrent network. This network (a simplified, con115 116 Ring tinuous version of that introduced in [9]), adds higher orders when they are needed by the system to solve its task. This is done by adding new units that dynamically modify connection weights. The new units modify the weights at the next time-step with information from the last, which allows temporal tasks to be learned without the use of feedback. 2 GENERAL FORMULATION Each unit (U) in the network is either an input (I), output (0), or high-level (L) unit. Ui(t) Ii(t) Oi(t) ret) L~y(t) value of ith unit at time t. Ui(t) where i is an input unit. Ui(t) where i is an output unit. Target value for Oi (t) at time t. Ui(t) where i is the higher-order unit that modifies weight wxy at time t. 1 The output and high-level units are collectively referred to as non-input (N) units: { Oi (t) if Ui = Oi. L~y(t) if Ui = L~y. In a given time-step, the output and high-level units receive a summed input from the input units. Ni(t) == L Ij (t)g(i, j, t). (1) j g is a gating function representing the weight of a particular connection at a particular time-step. If there is a higher-order unit assigned to that connection, then the input value of that unit is added to the connection's weight at that time-step.2 ( .. t) _ { Wij(t) + Lij(t - 1) If Lij exists g Z,), Wij (t) Otherwise (2) At each time-step, the values of the output units are calculated from the input units and the weights (possibly modified by the activations of the high-level units from the previous time-step). The values of the high-level units are calculated at the same time in the same way. The output units generate the output of the network. The high-level units simply alter the weights at the next time-step. All unit activations can be computed simultaneously since the activations of the L units are not required 1 A connection may be modified by at most one L unit. Therefore Li , Lzy , and L~y are identical but used as appropriate for notational convenience. 21t can be seen that this is a higher-order connection in the usual sense if one substitutes the right-hand side of equation 1 for L'0 in equation 2 and then replaces g in equation 1 with the result. In fact, as the network increases in height, ever higher orders are introduced, while lower orders are preserved. Learning Sequential Tasks by Incrementally Adding Higher Orders 117 until the following time-step. The network is arranged hierarchically in that every higher-order units is always higher in the hierarchy than the units on either side of the weight it affects. Since higher-order units have no outgoing connections, the network is not recurrent. It is therefore impossible for a high-level unit to affect, directly or indirectly, its own input. There are no hidden units in the traditional sense, and all units have a linear activation function. (This does not imply that non-linear functions cannot be represented, since non-linearities do result from the multiplication of higher-level and input units in equations 1 and 2.) Learning is done through gradient descent to reduce the sum-squared error. E(t) ! L:(Ti(t) - Oi(t»2 2 . , (3) where 1] is the learning rate. Since it may take several time-steps for the value of a weight to affect the network's output and therefore the error, equation 3 can be rewritten as: BE(t) ~Wij(t) = B ( ") , Wij t - r' (4) where { 0 if Ui = Oi ri = 1 + rX if Ui = L~y The value ri is constant for any given unit i and specifies how "high" in the hierarchy unit i is. It therefore also specifies how many time-steps it takes for a change in unit i's activation to affect the network's output. Due to space limitations, the derivation of the gradient is not shown, but is given elsewhere [10]. The resulting weight change rule, however, is: ~ .. (t) - Ii (t _ i) { Ti(t) - Oi(t) If u~ = O~ w') r ~W (t) If U' = LI xy xy (5) The weights are changed after error values for the output units have been collected. Since each high-level unit is higher in the hierarchy than the units on either side of the weight it affects, weight changes are made bottom up, and the ~Wxy(t) in equation 5 will already have been calculated at the time ~Wij(t) is computed. The intuition behind the learning rule is that each high-level unit learns to utilize the context from the previous time-step for adjusting the connection it influences at the next time-step so that it can minimize the connection's error in that context. Therefore, if the information necessary to decide the correct value of a connection at one time-step is available at the previous time-step, then that information is used by the higher-order unit assigned to that connection. If the needed information is not available at the previous time-step, then new units may be built to look for the information at still earlier steps. This method concentrating on unexpected events is similar to the "hierarchy of decisions" of Dawkins [3], and the "history compression" of Schmidhuber [13]. 118 Ring 3 WHEN TO ADD NEW UNITS A unit is added whenever a weight is being pulled strongly in opposite directions (i.e. when learning is forcing the weight to increase and to decrease at the same time) . The unit is created to determine the contexts in which the weight is pulled in each direction. This is done in the following way: Two long-term averages are kept for each connection. The first of these records the average change made to the weight, ~Wij(t) = O'~Wij(t) + (1 O')~Wij(t - 1); 0 S; 0' S; l. The second is the long-term mean absolute deviation, given by: The parameter, 0', specifies the duration of the long-term average. A lower value of 0' means that the average is kept for a longer period of time. When ~Wij(t) is small, but I~Wij(t)1 is large, then the weight is being pulled strongly in conflicting directions, and a new unit is built. if I~Wij(t)1 c + I~Wij(t)1 >8 then build unit L~ +1, where c is a small constant that keeps the denominator from being zero, 8 is a threshold value, and N is the number of units in the network. A related method for adding new units in feed-forward networks was introduced by Wynne-Jones [16]. When a new unit is added, its incoming weights are initially zero. It has no output weights but simply learns to anticipate and reduce the error at each time-step of the weight it modifies. In order to keep the number of new units low, whenever a unit, Lij is created, the statistics for all connections into the destination unit (Ui ) are reset: I~Wij(t)1 ~ 0.0 and ~Wij(t) ~ 1.0. 4 RESULTS The Reber grammar is a small finite-state grammar of the following form: sO X y. .. . B ~ E ... ... ~ /v • ... TO Transitions from one node to the next are made by way of the labeled arcs. The task of the network is: given as input the label of the arc just traversed, predict Learning Sequential Tasks by Incrementally Adding Higher Orders 119 Elman Recurrent Incremental Network RTRL Cascade Higher-Order Correlation Network Sequences Seen: Mean 25,000 206 Best 20,000 19,000 176 "Hidden" Units 15 2 2-3 40 Table 1: The incremental higher-order network is compared against recurrent networks on the Reber grammar. The results for the recurrent networks are quoted from other sources [2, 5]. The mean and/or best performance is shown when available. RTRL is the real-time recurrent learning algorithm [15]. the arc that will be traversed next. A training sequence, or string, is generated by starting with a B transition and then randomly choosing an arc leading away from the current state until the final state is reached. Both inputs and outputs are encoded locally, so that there are seven output units (one each for B, T, S, X, V, P, and E) and eight input units (the same seven plus one bias unit). The network is considered correct if its highest activated outputs correspond to the arcs that can be traversed from the current state. Note that the current state cannot be determined from the current input alone. An Elman-type recurrent network was able to learn this task after 20,000 string presentations using 15 hidden units [2]. (The correctness criteria for the Elman net was slightly more stringent than that described in the previous paragraph.) Recurrent Cascade-Correlation (RCC) was able to learn this task using only two or three hidden units in an average of 25,000 string presentations [5]. The incremental, higher-order network was trained on a continuous stream of input: the network was not reset before beginning a new string. Training was considered to be complete only after the network had correctly classified 100 strings in a row. Using this criterion, the network completed training after an average of 206.3 string presentations with a standard deviation of 16.7. It achieved perfect generalization on test sets of 128 randomly generated strings in all ten runs. Because the Reber grammar is stochastic, a ceiling of 40 higher-order units was imposed on the network to prevent it from continually creating new units in an attempt to outguess the random number generator. Complete results for the network on the Reber grammar task are given in table 1. The parameter settings were: TJ = 0.04, (J" = 0.08, e = 1.0, f = 0.1 and Bias = 0.0. (The network seemed to perform better with no bias unit.) The network has also been tested on the "variable gap" tasks introduced by Mozer [7], as shown in figure 1. These tasks were intended to test performance of networks over long time-delays. Two sequences are alternately presented to the network. Each sequence begins with an X or a Y and is followed by a fixed string of characters with an X or a Y inserted some number of time-steps from the beginning. In figure 1 the number of time-steps, or "gap", is 2. The only difference between the two sequences is that the first begins with an X and repeats the X after the gap, while the second begins with a Y and repeats the Y after the gap. The network must learn to predict the next item in the sequence given the current item as input 120 Ring Time-step: 0 Sequence 1: X Sequence 2: Y 1 2 a b a b 3 4 X c Y c 5 6 d e d e 789 f g h f g h 10 11 J J 12 k k Figure 1: An example of a "variable gap" training sequence [7]. One item is presented to the network at each time-step. The target is the next item in the sequence. Here the "gap" is two, because there are two items in the sequence between the first X or Y and the second X or Y. In order to correctly predict the second X or Y, the network must remember how the sequence began. (where all inputs are locally encoded). In order for the network to predict the second occurrence of the X or Y, it must remember how the sequence began. The length of the gap can be increased in order to create tasks of greater difficulty. Results of the "gap" tasks are given in table 2. The values for the standard recurrent network and for Mozer's own variation are quoted from Mozer's paper [7]. The incremental higher-order net had no difficulty with gaps up to 24, which was the largest gap I tested. The same string was used for all tasks (except for the position of the second X or V), and had no repeated characters (again with the exception of the X and Y). The network continued to scale linearly with every gap size both in terms of units and epochs required for training. Because these tasks are not stochastic, the network always stopped building units as soon as it had created those necessary to solve each task. The parameter settings were: TJ = 1.5, (j = 0.2, e = 1.0, f = 0.1 and Bias = 0.0. The network was considered to have correctly predicted an element in the sequence if the most strongly activated output unit was the unit representing the correct prediction. The sequence was considered correctly predicted if all elements (other than the initial X or Y) were correctly predicted. Mean number of Training sets required by: Gap Standard Mozer Incremental Umts Recurrent Net Network Higher-Order Net Created 2 468 328 4 10 4 7406 584 6 15 6 9830 992 8 19 8 > 10000 1312 10 23 10 > 10000 1630 12 27 24 26 49 Table 2: A comparison on the "gap" tasks of a standard recurrent-network and a network devised specifically for long time-delays (quoted from Mozer [7], who reported results for gaps up to ten) against an incremental higher-order network. The last column is the number of units created by the incremental higher-order net. Learning Sequential Tasks by Incrementally Adding Higher Orders 121 5 CONCLUSIONS The incremental higher-order network performed much better than the networks that it was compared against on these tiny tests. A few caveats are in order, however. First, the parameters given for the tasks above were customized for those tasks. Second, the network may add a large number of new units if it contains many context-dependent events or if it is inherently stochastic. Third, though the network in principle can build an ever larger hierarchy that searches further and further back in time for a context that will predict what a connection's weight should be, many units may be needed to bridge a long time-gap. Finally, once a bridge across a time-delay is created, it does not generalize to other time-delays. On the other hand, the network learns very fast due to its simple structure that adds high-level units only when needed. Since there is no feedback (i.e. no unit ever produces a signal that will ever feed back to itself), learning can be Qone without back propagation through time. Also, since the outputs and high-level units have a fan-in equal to the number of inputs only, the number of connections in the system is much smaller than the number of connections in a traditional network with the same number of hidden units. Finally, the network can be thought of as a system of continuous-valued conditionaction rules that are inserted or removed depending on another set of such rules that are in turn inserted or removed depending on another set, etc. When new rules (new units) are added, they are initially invisible to the system, (i.e., they have no effect), but only gradually learn to have an effect as the opportunity to decrease error presents itself. Acknowledgements This work was supported by NASA Johnson Space Center Graduate Student Researchers Program training grant, NGT 50594. I would like to thank Eric Hartman, Kadir Liano, and my advisor Robert Simmons for useful discussions and helpful comments on drafts of this paper. I would also like to thank Pavilion Technologies, Inc. for their generous contribution of computer time and office space required to complete much of this work. References [1] Jonathan Richard Bachrach. Connectionist Modeling and Control of Finite State Environments. PhD thesis, Department of Computer and Information Sciences, University of Massachusetts, February 1992. [2] Axel Cleeremans, David Servan-Schreiber, and James L. McClelland. Finite state automata and simple recurrent networks. Neural Computation, 1(3):372381, 1989. [3] Richard Dawkins. Hierarchical organisation: a candidate principle for ethology. In P. P. G. Bateson and R. A. Hinde, editors, Growing Points in Ethology, pages 7-54, Cambridge, 1976. Cambridge University Press. [4] Jeffrey L. Elman. Finding structure in time. CRL Technical Report 8801, University of California, San Diego, Center for Research in Language, April 1988. 122 Ring [5] Scott E. Fahlman. The recurrent cascade-correlation architecture. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 190-196, San Mateo, California, 1991. Morgan Kaufmann Publishers. [6] C. L. Giles, C. B. Miller, D. Chen, G. Z. Sun, H. H. Chen, and Y. C. Lee. Extracting and learning an unknown grammar with recurrent neural networks. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4, pages 317-324, San Mateo, California, 1992. Morgan Kaufmann Publishers. [7] Michael C. Mozer. Induction of multiscale temporal structure. In John E. Moody, Steven J. Hanson, and Richard P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 275-282, San Mateo, California, 1992. Morgan Kaufmann Publishers. [8] Jordan B. Pollack. The induction of dynamical recognizers. Machine Learning, 7:227-252, 1991. [9] Mark B. Ring. Incremental development of complex behaviors through automatic construction of sensory-motor hierarchies. In Lawrence A. Birnbaum and Gregg C. Collins, editors, Machine Learning: Proceedings of the Eighth International Workshop (ML91), pages 343-347. Morgan Kaufmann Publishers, June 1991. [10] Mark B. Ring. Sequence learning with incremental higher-order neural networks. Technical Report AI 93-193, Artificial Intelligence Laboratory, University of Texas at Austin, January 1993. [11] A. J. Robinson and F. Fallside. The utility driven dynamic error propagation network. Technical Report CUED/F-INFENG/TR.l, Cambridge University Engineering Department, 1987. [12] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland, editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition. V1: Foundations. MIT Press, 1986. [13] Jiirgen Schmidhuber. Learning unambiguous reduced sequence descriptions. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4, pages 291-298, San Mateo, California, 1992. Morgan Kaufmann Publishers. [14] Raymond L. Watrous and Gary M. Kuhn. Induction of finite-state languages using second-order recurrent networks. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4, pages 309-316, San Mateo, California, 1992. Morgan Kaufmann Publishers. [15] Ronald J. Williams and David Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural Computation, 1(2):270-280, 1989. [16] Mike Wynn-Jones. Node splitting: A constructive algorithm for feed-forward neural networks. Neural Computing and Applications, 1(1):17-22, 1993.
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Topography and Ocular Dominance with Positive Correlations Geoffrey J. Goodhill University of Edinburgh Centre for Cognitive Science 2 Buccleuch Place Edinburgh EH8 9LW SCOTLAND Abstract A new computational model that addresses the formation of both topography and ocular dominance is presented. This is motivated by experimental evidence that these phenomena may be subserved by the same mechanisms. An important aspect of this model is that ocular dominance segregation can occur when input activity is both distributed, and positively correlated between the eyes. This allows investigation of the dependence of the pattern of ocular dominance stripes on the degree of correlation between the eyes: it is found that increasing correlation leads to narrower stripes. Experiments are suggested to test whether such behaviour occurs in the natural system. 1 INTRODUCTION The development of topographic and interdigitated mappings in the nervous system has been much studied experimentally, especially in the visual system (e.g. [8, 15]). Here, each eye projects in a topographic manner to more central brain structures: i.e. neighbouring points in the eye map to neighbouring points in the brain. In addition, when fibres from the two eyes invade the same target structure, a competitive interaction often appears to take place such that eventually postsynaptic cells receive inputs from only one eye or the other, in a pattern of interdigitating “ocular dominance” stripes. These phenomena have received a great deal of theoretical attention: several models have been proposed, each based on a different variant of a Hebb-type rule (e.g. [16, 17, 14, 13]). However, there are two aspects of the experimental data which previous models have not satisfactorily accounted for. Firstly, experimental manipulations in the frog and goldfish have shown that when fibres from a second eye invade a region of brain which is normally innervated by only one eye, ocular dominance stripes can be formed (e.g. [2]). This suggests that ocular dominance may be a byproduct of the expression of the rules for topographic map formation, and does not require additional mechanisms [1]. However, previous models of topography have required additional implausible assumptions to account for ocular dominance (e.g. [16, 11], [17, 10]), while previous models of ocular dominance (e.g. [14]) have not simultaneously addressed the development of topography. Secondly, the simulationresults presented for most previous models of ocular dominance have used only localized rather than distributed patterns of input activity (e.g. [11]), or zero or negative correlations in activity between the two eyes (e.g. [3]). It is clear that, in reality, between-eye correlations have a minimum value of zero (which might be achieved for instance in the case of strabismus), and in general these correlations will be positive after eye-opening. In the cat for instance, the majority of ocular dominance segregation occurs anatomically three to six weeks after birth, whereas eye opening occurs at postnatal day 7-10. Here I present a new model that accounts for (a) both topography and ocular dominance with the same mechanisms, and (b) ocular dominance segregation and receptive field refinement for input patterns which are both distributed, and positively correlated between the eyes. 2 OUTLINE OF THE MODEL The model is formulated at a general enough level to be applicable to both the retinocortical and the retinotectal systems. It consists of two two dimensional sheets of input units (indexed by ) connected to one two-dimensional sheet of output units (indexed by ) by fibres with variable synaptic weights . It is assumed in the retinocortical case that the topography of the retina is essentially unchanged by the lateral geniculate nucleus (LGN) on its way to the cortex, and in addition, the effects of retinal and LGN processing are taken together. Thus for simplicity we refer to the input layers of the model as being retinae and the output layer as being the cortex. An earlier version of the model appeared in [4], and a fuller description can be found in [5]. Both retina and cortex are arranged in square arrays. All weights and unit activities are positive. Lateral interactions exist in the cortical sheet of a circular centersurround excitation/inhibition form, although these are not modeled explicitly. Initially there is total connectivity of random strengths between retinal and cortical units, apart from a small bias that specifies an orientation for the map. At each time step, a pattern of activity is presented by setting the activities of retinal units. Each cortical unit calculates its total input according to a linear summation rule: We use assumptions similar to [9] concerning the effect of inhibitory lateral connections, to obtain the learning rule: is a small positive constant, is the cortical unit with the largest input , and is the function that specifies how the activities of units near to decrease with distance from . We assume to be a gaussian function of the Euclidean distance between units in the cortical sheet, with standard deviation . Inputs to the model are random dot patterns with short range spatial correlation introduced by convolution with a blurring function. Locally correlated patterns of activity were generated by assigning the value 0 or 1 to each pixel in each eye with a probability of 50%, and then convolving each eye with a gaussian function of standard deviation . Between-eye correlations were produced in the following way. Once each retina has been convolved individually with a gaussian function, activity of each unit in each retina is replaced with , where is the activity of the corresponding unit to in the other eye, and specifies the similarity between the two eyes. Thus by varying it is possible to vary the degree of correlation between the eyes: if they are uncorrelated, and if they are perfectly correlated (i.e. the pattern of activity is identical in the two eyes). The correlations existing in the biological system will clearly be more complicated than this. However, the simple correlational structure described above aims to capture the key features of the biological system: on average, cells in each retina are correlated to an extent that decreases with distance between cells, and (after eye opening) corresponding positionsin the two eyes are also on the average somewhat correlated. The sum of the weights for each postsynaptic unit is maintained at a constant fixed value. However, whereas this constraint is most usually enforced by dividing each weight by the sum of the weights for that postsynaptic unit ("divisive" normalization), it is enforced in this model by subtracting a constant amount from each weight ("subtractive" normalization), as in [13]. 3 RESULTS Typical results for the case of two positively correlated eyes are shown in figure 1. Gradually receptive fields refine over the course of development, and cortical units eventually lose connections from one or the other eye (figure 1(a-c)). After a large number of input patterns have been presented, cortical units are almost entirely monocular, and units dominant for the left and right eyes are laid out in a pattern of alternating stripes (figure 1(c)). In addition, maps from the two eyes are in register and topographic (figure 1(d-f)). The map of cortical receptive fields (a) (b) (c) (d) (e) (f) Figure 1: Typical results for two eyes. (a-c) show the ocular dominance of cortical units after 0, 50,000, and 350,000 iterations respectively. Each cortical unit is represented by a square with colour showing the eye for which it is dominant (black for right eye, white for left eye), and size showing the degree to which it is dominant. (d-f) represent cortical topography. Here the centre of mass of weights for each cortical unit is averaged over both eyes, imagining the retinae to be lying atop one another, and neighbouring units are connected by lines to form a grid. This type of picture reveals where the map is folded to take into account that the cortex must represent both eyes. It can be seen that discontinuities in terms of folds tend to follow stripe boundaries: first particular positions in one eye are represented, and then the cortex “doubles back” as its ocularity changes in order to represent corresponding positions in the other eye. Figure 2: The receptive fields of cortical units, showing topography and eye preference. Units are coloured white if they are strongly dominant for the left eye, black if they are strongly dominant for the right eye, and grey if they are primarily binocular. “Strongly dominant” is taken to mean that at least 80% of the total weight available to a cortical unit is concentrated in one eye. Within each unit is a representation of its receptive field: there is a 16 by 16 grid within each cortical unit with each grid point representing a retinal unit, and the size of the box at each grid point encodes the strength of the connection between each retinal unit and the cortical unit. For binocular (grey) units, the larger of the two corresponding weights in the two eyes is drawn at each position, coloured white or black according to which eye that weight belongs. It can be seen that neighbouring positions in each eye tend to be represented by neighbouring cortical units, apart from discontinuities across stripe boundaries. For instance, the bottom right corner of the right retina is represented by the bottom right cortical unit, but the bottom left corner of the right retina is represented by cortical unit (3,3) (counting along and up from the bottom left corner of the cortex), since unit (1,1) represents the left retina. (figure 2) confirms that, as described for the natural system [8], there is a smooth progression of retinal position represented across a stripe, followed by a doubling back at stripe boundaries for the cortex to “pick up where it left off” in the other eye. An important aspect of the model is that the effect on stripe width of the strength of correlation between the two eyes can be investigated, which has not been done in previous models. Figure 3 shows a series of results for the model, from which it can be seen that stronger between-eye correlations lead to narrower stripes. It is interesting to note that a similar relationship is seen in the elastic net model of topography and ocular dominance [6], even though this is formulated on a rather different mathematical basis to the model presented here. 4 DISCUSSION It has sometimes been argued that it is not necessary to also consider the development of topography in models for ocular dominance, since in the cat for instance, topography develops first, and is established before ocular dominance segregation occurs. However, non-simultaneity of development does not imply different mechanisms. As a theoretical example, in the elastic net model [6], minimisation by gradient descent of a particular objective function produces two clear stages of development: first topography formation, then ocular dominance segregation. A similar (though less marked) effect can be seen with the present model in figure 1: the rough form of the map is established before ocular dominance segregation is complete. Interest in the contribution to map formation and eye-specific segregation of previsual activity in the retina has been re-awakened recently by the finding that spontaneous retinal activity takes the form of waves sweeping across the retina in a random direction [12]. Although this finding has in turn generated a wave of theoretical activity, it is important to note that the theoretical principles of how correlated activity can guide map formation have been fairly well worked out since the 1970’s [16, 11]. Discovery of the precise form that these correlations take does not invalidate earlier modelling studies. Finally, the results for the model presented in figure 3 raise the question of whether stronger between-eye correlations lead to narrower stripes in the natural system. Perhaps the simplestway to test this experimentally would be to look for changes in stripe width in the cat after artificially induced strabismus, which severely reduces the correlations between the two eyes. Although the effect of strabismus on the degree of monocularity of cortical cells has been extensively investigated (e.g. [7]), the effect on stripe width has not been examined. Such experiments would shed light on the extent to which the periodicity of ocular dominance stripes is determined by environmental as opposed to innate factors. Acknowledgements This work was funded by an SERC postgraduate studentship, and an MRC/JCI postdoctoral training fellowship. I thank Harry Barrow for advice relating to this (a) (b) (c) Figure 3: Effect on stripe width of the degree of correlation between the two eyes. Shown from left to right are the cortical topography averaged over both eyes, the stripe pattern, and the power spectrum of the fourier transform for each case. (a) (b) (c) . Note that stripe width tends to decrease as increases, and also that the topography becomes smoother. work, and David Willshaw and David Price for helpful comments on an earlier draft of this paper. References [1] Constantine-Paton, M. (1983). Position and proximity in the development of maps and stripes. Trends Neurosci. 6, 32-36. [2] Constantine-Paton, M. & Law, M.I. (1978). Eye-specific termination bands in tecta of three-eyed frogs. Science, 202, 639-641. [3] Cowan, J.D. & Friedman, A.E. (1991). Studies of a model for the development and regeneration of eye-brain maps. In D.S. Touretzky, ed, Advances in Neural Information Processing Systems, III, 3-10. [4] Goodhill, G.J. (1991). Topography and ocular dominance can arise from distributed patterns of activity. International Joint Conference on Neural Networks, Seattle, July 1991, II, 623-627. [5] Goodhill, G.J. (1991). Correlations, Competition and Optimality: Modelling the Development of Topography and Ocular Dominance. PhD Thesis, Sussex University. [6] Goodhill, G.J. & Willshaw, D.J. (1990). Application of the elastic net algorithm to the formation of ocular dominance stripes. Network, 1, 41-59. [7] Hubel, D.H. & Wiesel, T.N. (1965). Binocular interaction in striate cortex of kittens reared with artificial squint. Journal of Neurophysiology, 28, 1041-1059. [8] Hubel, D.H. & Wiesel, T.N. (1977). Functional architecture of the macaque monkey visual cortex. Proc. R. Soc. Lond. B, 198, 1-59. [9] Kohonen, T. (1988). Self-organization and associative memory (3rd Edition). Springer, Berlin. [10] Malsburg,C. von der (1979). Development ofocularity domains and growth behaviour of axon terminals. Biol. Cybern., 32, 49-62. [11] Malsburg, C. von der & Willshaw, D.J. (1976). A mechanism for producing continuous neural mappings: ocularity dominance stripes and ordered retino-tectal projections. Exp. Brain. Res. Supplementum 1, 463-469. [12] Meister, M., Wong, R.O.L., Baylor, D.A. & Shatz, C.J. (1991). Synchronous bursts of action potentials in ganglion cells of the developing mammalian retina. Science, 252, 939-943. [13] Miller, K.D., Keller, J.B. & Stryker, M.P. (1989). Ocular dominance column development: Analysis and simulation. Science, 245, 605-615. [14] Swindale, N.V. (1980). A model for the formation of ocular dominance stripes. Proc. R. Soc. Lond. B, 208, 243-264. [15] Udin, S.B. & Fawcett, J.W. (1988). Formation of topographic maps. Ann. Rev. Neurosci., 11, 289-327. [16] Willshaw, D.J. & Malsburg, C. von der (1976). How patterned neural connections can be set up by self-organization. Proc. R. Soc. Lond. B, 194, 431-445. [17] Willshaw, D.J. & Malsburg, C. von der (1979). A marker induction mechanism for the establishment of ordered neural mappings: its application to the retinotectal problem. Phil. Trans. Roy. Soc. B, 287, 203-243.
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A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition G. Zavaliagkos Northeastern University Boston MA 02115 R. Schwartz BBN Systems and Technologies Cambridge, MA 02138 Y. Zhao BBN Systems and Technologies Cambridge, MA 02138 J. Makhoul BBN Systems and Technologies Cambridge, MA 02138 Abstract Untill recently, state-of-the-art, large-vocabulary, continuous speech recognition (CSR) has employed Hidden Markov Modeling (HMM) to model speech sounds. In an attempt to improve over HMM we developed a hybrid system that integrates HMM technology with neural networks. We present the concept of a "Segmental Neural Net" (SNN) for phonetic modeling in CSR. By taking into account all the frames of a phonetic segment simultaneously, the SNN overcomes the well-known conditional-independence limitation of HMMs. In several speaker-independent experiments with the DARPA Resource Management corpus, the hybrid system showed a consistent improvement in performance over the baseline HMM system. 1 INTRODUCTION The current state of the art in continuous speech recognition (CSR) is based on the use of hidden Markov models (HMM) to model phonemes in context. Two main reasons for the popularity of HMMs are their high performance, in terms of recognition accuracy, and their computational efficiency However, the limitations of HMMs in modeling the speech signal have been known for some time. Two such limitations are (a) the conditional-independence assumption, which prevents a HMM from taking full advan704 A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition 705 tage of the correlation that exists among the frames of a phonetic segment, and (b) the awkwardness with which segmental features can be incorporated into .HM:M systems. We have developed the concept of Segmental Neural Nets (SNN) to overcome the two .HM:M limitations just mentioned for phonetic modeling in speech. A segmental neural net is a neural network that attempts to recognize a complete phonetic segment as a single unit, rather than a sequence of conditionally independent frames. Neural nets are known to require a large amount of computation, especially for training. Also, there is no known efficient search technique for finding the best scoring segmentation with neural nets in continuous speech. Therefore, we have developed a hybrid SNN/HM:M system that is designed to take full advantage of the good properties of both methods. The two methods are integrated through a novel use of the N-best (multiple hypotheses) paradigm developed in conjunction with the BYBLOS system at BBN [1]. 2 SEGMENTAL NEURAL NET MODELING There have been several recent approaches to the use of neural nets in CSR. The SNN differs from these approaches in that it attempts to recognize each phoneme by using all the frames in a phonetic segment simultaneously to perform the recognition. By looking at a whole phonetic segment at once, we are able to take advantage of the correlation that exists among frames of a phonetic segments, thus ameliorating the limitations of .HM:Ms . warping phonetic .egment • core neural network Figure 1: The SNN model samples the frames and produces a single segment score. The structure of a typical SNN is shown in Figure 1. The input to the network is a fixed length representation of the speech segment. This input is scored by the network. If the network was trained to minimize a mean square error (MSE) or a relative entropy distortion measure, the output of the network will be an estimate of the posterior probability P(CI:z:) of the phonetic class C given the segment :z: [2, 3]. This property of the SNN allows a natural extension to CSR: We segment the utterance into phonetic segments, and score each one of them seperately. The score of the utterance is simply the product of the scores of the individual segments. 706 Zavaliagkos, Zhao, Schwartz, and Makhoul The procedure described above requires the availability of some form of phonetic segmentation of the speech. We describe in Section 3 how we use the HMM to obtain likely candidate segmentations. Here, we shall assume that a phonetic segmentation has been made available and each segment is represented by a sequence of frames of speech features. The actual number of such frames in a phonetic segment is variable. However, for input to the neural network, we need a fixed length representation. Therefore, we have to convert the variable number of frames in each segment to a fixed number of frames. We have considered two approaches to cope with this problem: time sampling and Oiscrete Cosine Transfonn (ocr). In the first approach, the requisite time warping is performed by a quasi-linear sampling of the feature vectors comprising the segment to a fixed number of frames (5 in our system). For example, in a 17-frame phonetic segment, we use frames 1, 5, 9, 13, and 17 as input to the SNN. The second approach uses the Discrete Cosine Transfonn (OCT). The ocr can be used to represent the frame sequence of a segment as follows. Consider the sequence of cepstral features across a segment as a time sequence and take its ocr. For an m frame segment, this transfonn will result in a set of m OCT coefficients for each feature. Truncate this sequence to its first few coefficients (the more coefficients , the more precise the representation). To keep the number of features the same as in the quasi-linear sampling, we use only five coefficients. If the input segment has less than five frames, we initially interpolate in time so that a five-point ocr is possible. Compared to the quasi-linear sampling, OCT has the advantage of using information from all input frames. Duration: Because of the time warping function, the SNN score for a segment is independent of the duration of the segment. In order to provide duration infonnation to the SNN, we constructed a simple durational model. For each phoneme, a histogram was made of segment durations in the training data. This histogram was then smoothed by convolving with a triangular window, and probabilities falling below a floor level were reset to that level. The duration score was multiplied by the neural net score to give an overall segment score. 3 THE N-BEST RESCORING PARADIGM Our hybrid system is based on the N-best rescoring paradigm [1], which allows us to design and test the SNN with little regard to the usual problem of searching for the segmentation when dealing with a large vocabulary speech recognition system. Figure 2 illusrates the hybrid system. Each utterance is decoded using the BBN BYBLOS system [4]. The decoding is done in two steps: First the N-best recognition is performed, producing a list of the candidate N best-scoring sentence hypotheses. In this stage, a relatively simple HMM: is used for computation pUIposes. The length of the N-best list is chosen to be long enough to almost always include the correct answer. The second step is the HMM: rescoring, where a more sophisticated HMM is used. The recognition process may stop at this stage, selecting the top scoring utterance of the list (HMM I-best output). To incOlporate the SNN in the N -best paradigm, we use the HMM system to generate a segmentation for each N-best hypothesis, and the SNN to generate a score for the hypothesis USing the HMM: segmentation. The N-best list may be reordered based on A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition 707 SNN scores alone. In this case the recognition process stops by selecting the top scoring utterance of the rescored list (NN I-best output). N-&e.tHMM Recognition N-Best ~ r List HMM Rescoring HMM 1-best HMM Scores Speech N-be8t Labels and Segmentation + + III. Combine SCores and Reorder List • Hybrid SNNIHMM Top Choice Segmental Neural Net Rescoring SNN Scor .. SNN 1-bMt Figure 2: Schematic diagram of the hybrid SNN/HMM system The last stage in the hybrid system is to combine several scores for each hypothesis, such as SNN score, HMM: score, grammar score, and the hypothesized number of words and phonemes. (The number of words and phonemes are included because they serve the same pUIpose as word and phoneme insertion penalties in a HMM: CSR system.) We form a composite score by taking a linear combination of the individual scores. The linear combination is determined by selecting the weights that give the best performance over a development test set. These weights can be chosen automatically [5]. After we have rescored the N-Best list, we can reorder it according to the new composite scores. If the CSR system is required to output just a single hypothesis, the highest scoring hypothesis is chosen (hybrid SNN/HMM top choice in Figure 2). 4 SNN TRAINING The training of the phonetic SNNs is done in two steps. In the first training step, we segment all of the training utterances into phonetic segments using the HMM: models and 708 Zavaliagkos, Zhao, Schwartz, and Makhoul the utterance transcriptions. Each segment then serves as a positive example of the SNN output corresponding to the phonetic label of the segment and as a negative example for all the other phonetic SNN outputs (we are using a total of 53 phonetic outputs). We call this training method i-best training. The SNN is trained using the log-error distortion measure [6], which is an extension of the relative entropy measure to an M -class problem. To ensure that the outputs are in fact probabilities, we use a sigmoidal nonlinearity to restrict their range in [0, 1] and an output normalization layer to make them sum to one. The models are initialized by removing the sigmoids and using the MSE measure. Then we reinstate th~ sigmoids and proceed with four iterations of a quasi-Newton [7] error minimization algorithm. For the adopted error measure, when the neural net non-linearity is the usual sigmoid function, there exists a unique minimum for single-layer nets [6]. The I-best training described has one drawback: the training does not cover all the cases that the network will be required to encounter in the N-best rescoring paradigm. With 1best training, given the correct segmentation, we train the network to discriminate between correct and incorrect labeling. However, the network will also be used to score N-best hypotheses with incorrect segmentation. Therefore, it is important to train based on the N-best lists in what we call N-best training. During N-best training, we produce the Nbest lists for all of the training sentences, and we then train positively with all the correct hypotheses and negatively on the "misrecognized" parts of the incorrect hypothesis. 4.1 Context Modelling Some of the largest gains in accuracy for HMM CSR systems have been obtained with the use of context (i.e., phonetic identity of neighbOring segments). Consequently, we implemented a version of the SNN that provided a simple model of left-context. In addition to the SNN previously described, which only models a segment's phonetic identity and makes no reference to context, we trained 53 additional left-context networks. Each of these 53 networks were identical in structure to the context-independent SNN. In the recognition process, the segment score is obtained by combining the output of the context-independent SNN with the corresponding output of the SNN that models the left-context of the segment. This combination is a weighted average of the two network values, where the weights are determined by the number of occurrences of the phoneme in the training data and the number of times the phoneme has its present context in the training data. 4.1.1 Regularization Techniques for Context Models During neural net training of context models, a decrease of the distortion on the training set often causes an increase of the distortion on the test set. This problem is called overtraining, and it typically occurs when the number of training samples is on the order of the number of the model parameters. Regularization provides a class of smoothing techniques to ameliorate the overtraining problem. Instead of minimizing the distortion measure alone, we are minimizing the following objective function: (1) A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition 709 where Wo is the set of weights corresponding to the context-independent model, Nd is the number of data points, and >'1, >'2, 711, 712 are smoothing parameters. The first regularization tenn is used to control the excursion of the weights in general and the other to control the degree to which the context-dependent model is allowed to deviate from the corresponding context-independent model (to achieve this first we initialize the contextdependent models with the context-independent model). In our initial experiments, we used values of >'1 = >'2 = 1.0, 711 = I, 712 = 2. When there are very few training data for a particular context model, the regularization tenns in (!) p:'evail, Cflnstraining the model parameters to remain close to their initial estimates. The regularization tenn is gradually turned off with the presence of more data. What we accomplish in this way is an automatic mechanism that controls overtraining. 4.2 Elliptical Basis Functions Our efforts to use multi-layer structures has been rather unsuccessful so far. The best improvement we got was a mere 5% reduction in error rate over the single-layer performance, but with a 10-fold increase in both number of parameters and computation time. We suspect that our training is getting trapped in bad local minima. Due to the above considerations, we considered an alternative multi-layer structure, the Elliptical Basis Function (EBF) network. EBFs are natural extensions of Radial Basis Functions, where a full covariance matrix is introduced in the basis functions. As many researchers have suggested, EBF networks provide modelling capabilities that are as powerful as multilayer perceptrons. An advantage of EBF is that there exist well established techniques for estimating the elliptical basis layer. As a consequence, the problem of training an EBF network can be reduced to a one-layer problem, i.e., training the second layer only. Our approach with EBF is to initialize them with Maximum Likelihood (ML). ML training allows us to use very detailed context models, such as triphones. The next step, which is not yet implemented, is to either proceed with discriminative NN training, or use a nonlinearity at the outout layer and treat the second layer as a single-layer feedforward model, or both. 5 EXPERIMENTAL CONDITIONS AND RESULTS Experiments to test the performance of the hybrid system were performed on the speakerindependent (SI) portion of the DARPA 1000-word Resource Management speech corpus. The training set consisted of utterances from 109 speakers, 2830 utterances from male speakers and 1160 utterances from female speakers. We have tested our system with 5 different test sets. The Feb '89 set was used as a cross-validation set for the SNN system. Feb '89 and Oct '89 were used as development sets whenever the weights for the combination of two or more models were to be estimated. Feb '91 and the two Sep '92 sets were used as independent test sets. Both the NN and the HMM systems had 3 separate models made from male, female, and combined data. During recognition all 3 models were used to score the utterances, and the recognition answer was decided by a 3-way gender selection: For each utterance, the model that produced the highest score was selected. The HMM used was the February '91 version of the BBN BYBLOS system. 710 Zavaliagkos, Zhao, Schwartz, and Makhoul In the experiments, we used SNNs with 53 outputs, each representing one of the phonemes in our system. The SNN was used to rescore N-best lists of length N = 20. The input to the net is a fixed number of frames of speech features (5 frames in our system). The features in each to-ms frame consist of 16 scalar values: power, power difference, and 14 mel-warped cepstral coefficients. For the EBF, the differences of the cepstral parameters were used also. Table 1: SNN development on February '89 test set ~--------------------~~~~--~-~ Word EITor (%) Original SSN (MSE) + Log-Error Criterion + N-Best training + Left Context + Regularization + word,phoneme penalties EBF 13.7 11.6 9.0 7.4 6.6 5.7 4.9 Table I shows the word error rates at the various stages of development. All the experiments mentioned below used the Feb '89 test set. The original I-layer SNN was trained using the I-best training algorithm and the MSE criterion, and gave a word error rate of 13.7%. The incorporation of the duration and the adoption of the log-error training criterion both resulted in some improvement, bringing the error rate down to 11.6%. With N-best training the error rate dropped to 9.0%; adding left context models reduced the word error rate down to 7.4%. When the the context models were trained with the regularization criterion the error rate dropped to 6.6%. All of the above results were obtained using the mean NN score (NN score divided by the number of segments). When we used word and phone penalties, the perfonnance was even better, a 5.7% word error rate. For the same conditions, the perfonnance for the EBF system was 4.9% word error rate. We should mention that the implementation of training with regularization was not complete at the time the hybrid system was tested on the September 92 test, so we will exclude it from the NN results presented below. The final hybrid system included the HM:M, the SNN and EBF models, and Table 2 summarizes its perfonnance (in this table, NN stands for the combination of SNN and EBF). We notice that with the exception of of the Sep '92 test sets the word error of the mfM was roughly around 3.5%(3.8, 3.7 and 3.4%). For the same test sets, the NN had a word error slightly higher than 4.0%, and the hybrid NN/HMM system a word error rate of 2.7%. We are very happy to see the perfonnance of our neural net approaching the perfonnance of the HMM. It is also worthwhile to mention that the perfonnance of the hybrid system for Feb '89, Oct '89 and Feb '91 is the best perfonnance reported so far for these sets. Special mention has to be made for the Sep '92 test sets. These test sets proved to be radically different than the previous released RM tests, resulting in almost a doubling of the HM:M word error rate. The deterioration in perfonnance of the hybrid system was bigger, and the improvement due to the hybrid system was less than 10% (compared to an improvement of :::::: 25% for the other 3 sets). We have all been baffled by these unexpected results, and although we are continuously looking for an explanation of this A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition 711 System HMM: NN NN+HMM: Feb '89 3.7 4.0 2.7 Word Error % Oct '89 Feb '91 3.8 3.4 4.2 4.1 2.7 2.7 Sep '92 6.0 7.4 5.5 Table 2: Hybrid Neural Net/HM1vf system. strange behaviour our efforts have not yet been successful. 6 CONCLUSIONS We have presented the Segmental Neural Net as a method for phonetic modeling in large vocabulary CSR systems and have demonstrated that, when combined with a conventional HMM, the SNN gives a significant improvement over the perfonnance of a state-of-theart HMM CSR system. The hybrid system is based on the N-best rescoring paradigm which, by providing the HMM segmentation, drastically reduces the computation for our segmental models and provides a simple way of combining the best aspects of two systems. The improvements achieved from the use of a hybrid system vary from less than 10% to about 25 % reduction in word error rate, depending on the test set used. References [1] R. Schwartz and S. Austin, "A Comparison of Several Approximate Algorithms for Finding Multiple (N-Best) Sentence Hypotheses," IEEE Int. Con[ Acoustics, Speech and Signal Processing, Toronto, Canada, May 1991, pp. 701-704. [2] A. Barron, "Statistical properties of artificial neural networks," IEEE Conf. Decision and Control, Tampa, FL, pp. 280-285, 1989. [3] H. Gish, "A probabilistic approach to the understanding and training of neural network classifiers," IEEE Int. ConfAcoust., Speech, Signal Processing, April 1990. [4] M. Bates et. all, "The BBN/HARC Spoken Language Understanding System" IEEE Int. Con[ Acoust., Speech,Signal Processing, Apr 1992, Minneapolis, MI, Apr. 1993 [5] M. Ostendorf et. all, "Integration of Diverse Recognition Methodologies Through Reevaluation of N-Best Sentence Hypotheses," Proc. DARPA Speech and Natural Language Workshop, Pacific Grove, CA, Morgan Kaufmann Publishers, February 1991. [6] A. El-Jaroudi and J. Makhoul, "A New Error Criterion for Posterior Probability Estimation with Neural Nets," International Joint Conference on Neural Networks, San Diego, CA, June 1990, Vol III, pp. 185-192. [7] D. Luenberger, Linear and Nonlinear Programming, Addison-Wesley, Massachusetts, 1984. [8] R. Schwartz et. all, "Improved Hidden Markov Modeling of Phonemes for Continuous Speech Recognition," IEEE Int. Con[ Acoustics, Speech and Signal Processing, San Diego, CA, March 1984, pp. 35.6.1-35.6.4.
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A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results. Christiane Linster David Marsan ESPCI, Laboratoire d'Electronique 10, Rue Vauquelin Claudine Masson Laboratoire de Neurobiologie Comparee des Invertebrees INRA/CNRS (URA 1190) 75005 Paris, France Michel Kerszberg Institut Pasteur CNRS (URA 1284) Neurobiologie Moleculaire 25, Rue du Dr. Roux 75015 Paris, France 91140 Bures sur Yvette, France Gerard Dreyfus Leon Personnaz ESPCI, Laboratoire d'Electronique 10, Rue Vauquelin 75005 Paris, France Abstract It is known from biological data that the response patterns of interneurons in the olfactory macroglomerulus (MGC) of insects are of central importance for the coding of the olfactory signal. We propose an analytically tractable model of the MGC which allows us to relate the distribution of response patterns to the architecture of the network. 1. Introduction The processing of pheromone odors in the antennallobe of several insect species relies on a number of response patterns of the antennallobe neurons in reaction to stimulation with pheromone components and blends. Antennallobe interneurons receive input from different receptor types, and relay this input to antennal lobe projection neurons via excitatory as well as inhibitory synapses. The diversity of the responses of the interneurons and projection neurons as well the long response latencies of these neurons to pheromone stimulation or electrical stimulation of the antenna, suggest a polysynaptic pathway 1022 A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1023 between the receptor neurons and these projection neurons (for a review see (Kaissling, 1990; Masson and Mustaparta, 1990)). I. Pf-EROMONE ce..ERALlSTS A. Carnot Discrinilate Single Odors AN) Camot COde Ter1lXlI'aI Olanges ,. Excited Type Stml'S BAL C1S Blend 2. Wtited Type Stint'S BAL C15 B1erd Resooose • Response n. PJ-EROMO\E SPECtAUSTS ill' i. I 'I i ' :IIIIIIIM~IIIH~11 l III : ... '-I:.!.! tU.i)! D II .. ua .~.~ _ . ..-._,,' .. I I I I I .111.J1IL._-_ '_'_'J1J~UlUlJ . . . . . A. Can Oiscrini1ate Singe Odors BUT Camot Code Terrporal Olanges Stinhs BAl C15 Blend Response (1) 00 (2) + 0 0 • • • I,L • • • I B. can Discriri1ate 5ilgIe Odors f>K) Can Code T~a1 Olanges stmt§ BAL C15 Blend 8espoose (1) 00 (2) + • -/./-/./Figure 1: With courtesy of John Hildebrand, by permission from Oxford University Press, from: Christensen, Mustaparta and Hildebrand: Discrimination of sex pheromone blends in the olfactory system of the moth, Chemical Senses, Vol 14, no 3, pp 463-477, 1989. 1024 Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz In the MOC of Manduca sexta, antennal lobe interneurons respond in various ways to antennal stimulation with single pheromone components or the blend: pheromone generalists respond by either excitation or inhibition to both components and the blend: they cannot discriminate the components; pheromone specialists respond (i) to one component but not to the other by either excitation or inhibition, (ii) with different response patterns to the presence of the single components or the blend, namely with excitation to one component, with inhibition to the other component and with a mixed response to the blend. These neurons can also follow pulsed stimulation up to a cut-off frequency (Figure 1). A model of the MOC (Linster et aI, 1993), based on biological data (anatomical and physiological) has demonstrated that the full diversity of response patterns can be reproduced with a random architecture using very simple ingredients such as spiking neurons governed by a first order differential equation, and synapses modeled as simple delay lines. In a model with uniform distributions of afferent, inhibitory and excitatory synapses, the distribution of the response patterns depends on the following network parameters: the percentage of afferent, inhibitory and excitatory synapses, the ratio of the average excitation of any interneuron to its spiking threshold, and the amount of feedback in the network. In the present paper, we show that the behavior of such a model can be described by a statistical approach, allowing us to search through parameter space and to make predictions about the biological system without exhaustive simulations. We compare the results obtained with simulation of the network model to the results obtained analytically by the statistical approach, and we show that the approximations made for the statistical descriptions are valid. 2. Simulations and comparison to biological data In (Linster et aI, 1993), we have used a simple neuron model: all neurons are spiking neurons, governed by a first order differential equation, with a membrane time constant and a probabilistic threshold 9. The time constant represents the decay time of the membrane potential of the neuron. The output of each neuron consists of an all-or-none action potential with unit amplitude that is generated when the membrane potential of the cell crosses a threshold, whose cumulative distribution function is a continuous and bounded probabilistic function of the membrane potential. All sources of delay and signal transformation from the presynaptic neuron to its postsynaptic site are modeled by a synaptic time delay. These delays are chosen in a random distribution (gaussian), with a longer mean value for inhibitory synapses than for excitatory synapses. We model two main populations of olfactory neurons: receptor neurons which are sensitive to the main pheromone component (called A) or to the minor pheromone component (called B) project uniformly onto the network of interneurons; two types of interneurons (excitatory and inhibitory) exist: each interneuron is allowed to make one synapse with any other interneuron. The model exhibits several behaviors that agree with biological data, and it allows us to state several predictive hypotheses about the processing of the pheromone blend. We observe two broad classes of intemeurons: selective (to one odor component) and nonselective neurons (in comparison to Figure 1). Selective neurons and non-selective neurons exhibit a variety of response patterns, which fall into three classes: inhibitory, excitatory and mixed (Figure 2). Such a classification has indeed been proposed for olfactory antennal A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1025 lobe neurons (local interneurons and projection neurons) in the specialist olfactory system in Manduca (Christensen and Hildebrand, 1987) and for the cockroach (Burrows et al, 1982; Boeckh and Ernst, 1987). Action potenUals Membrane potential Inhibitory response Excitatory response Simple mixed response I,!r"'"'' ",1""""""",,1', II II I."='.,_we." dlllI, •• ',IIIIIIII'''QII,'', "III, I Stimulus A ,r-------., Stimulus B ,,------"""', ,,....-----"""'\ '~----"""'\ Mixed responses ....-......... -------.~ Ibll ••• , ,111M "" IU! ,'d, ", ", "b I, ,." " .. , 11" 111111 , ,""I j 111," 1".' I.h IgUIUIIi. I dil'"'' I I rI ........ 500 ms ,,-----------, Oscillatory responses 4 ~ ,,---------.., ,,---------.., ~"I II."I.! ,., I. II." II II" , •• I.h •• I.! I.,!" '" ,I,U'!, !., ,.! .".," ,,',,! II" II ",! V\fVrfl.]"'" ''tMN\ ",,,, -\JWtJV'''' "'J"" , I ,,-------, ,,-------.., ,,-----"""', Figure 2: Response patterns of interneurons in the model presented, in response to stimulation with single components A and B, and with a blend with equal component concentrations. Receptor neurons fIre at maximum frequency during the stimulations. The interneuron in the upper row is inhibited by stimulus A, excited by stimulus B, and has a mixed response (excitation followed by inhibition) to the blend: in reference to Figure 1, this is a pheromone specialist receiving mixed input from both types of receptor neurons. These types of simple and mixed responses can be observed in the model at low connectivity, where the average excitation received by an interneuron is low compared to its spiking threshold. The neuron in the middle row responds with similar mixed responses to stimuli A, Band A+B. The neuron in the lower row responds to all stimuli with the same oscillatory response, here the average excitation received by an interneuron approaches or exceeds the spiking threshold of the neurons. Network parameters: 15 receptor neurons; 35 interneurons; 40% excitatory interneurons; 60% inhibitory interneurons; afferent connectivity 10%; membrane time constant 25 ms; mean inhibitory synaptic delays 100 ms; mean excitatory synaptic delays 25 ms, spiking threshold 4.0, synaptic weights + 1 and -1. 1026 Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz In our model, as well as in biological systems (Christensen and Hildebrand 1988, Christensen et ai., 1989) we observe a number of local interneurons that cannot follow pulsed stimulation beyond a neuron-specific cut-off frequency. This frequency depends on the neuron response pattern and on the duration of the interstimulus interval. Therefore, the type of response pattern is of central importance for the coding of the olfactory signal. Thus, in order to be able to relate the coding capabilities of a (model or biological) network to its architecture, we have investigated the distribution of response patterns both analytically and by simulations. 3. Analytical approach In order to investigate these questions in a more rigorous way, some of us (C.L., D.M., G.D., L.P.) have designed a simplified, analytically tractable model. We define two layers of interneurons: those which receive direct afferent input from the receptor neurons (layer 1), and those which receive only input from other interneurons (layer 2). In order to predict the response pattern of any interneuron as a function of the network parameters, we make the following assumptions: (i) statistically, all interneurons within a given layer receive the same synaptic input, (ii) the effect of feedback loops from layer 2 can be neglected, (iii) the response patterns have the same distribution for stimulations either by the blend or by pure components. Assumption (i) is correct because of the uniform distribution of synapses in the network of interneurons. Assumption (ii) is valid at low connectivity: if the average amount of excitation received by an interneuron is low as compared to its spiking threshold, its firing probability is low; therefore, the effect of the excitation from the receptors is vanishingly small beyond two interneurons: we thus neglect the effect of signals sent from layer 2. Thus, feedback is present within layer 1, and layer 2 receives only feed forward connections. Assumption (iii) is plausible if we suppose that the natural pheromone blend is more relevant for the system than the single components of the blend. We further assume in the analytical approach (as in the simulations) that the synaptic delays are longer on the average for inhibitory synapses than for excitatory synapses . An interneuron can thus respond with four types of patterns: non-response, which means that it does not have a presynaptic neuron (this response pattern can only occur in layer 2, at low connectivity); excitation, meaning that an interneuron receives only afferent input from receptor neurons or from excitatory interneurons; inhibition, meaning that an interneuron receives only input from inhibitory interneurons (this can occur in layer 2 only); and mixed responses, covering all other combinations of presynaptic input. We consider a network of N + Nr neurons, N (number of interneurons) and Nr (number of receptor neurons) being random variables, N + Nr being fixed. We define the probability ni that a neuron is an inhibitory interneuron, and the probability ne that it is an excitatory interneuron. Any interneuron has a probability c to make one synapse (with synaptic weight +1 or -1) with any other interneuron and a probability (1 - c) not to make a synapse with this interneuron; Cr is the afferent connectivity: any receptor neuron has a probability Cr to connect once to any interneuron, and a probability (1 - cr) not to connect to this interneuron. Then na = 1 - (1 - cr)Nr is the probability that an interneuron belongs to layer 1, and the number of interneurons in layer I obeys a binomial distribution with expectation value N nQ and variance N na (1 - na). In the following, the fixed number of interneurons in layer 1 will be taken equal to its expectation value. Similarly, the number of interneurons in layer 2 is taken to be N (1 - na). A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1027 Because of the assumptions made above, in both layers, we take into account for each interneuron the N na c synapses from presynaptic neurons of layerl. In layer 1, these neurons respond with excitatory or mixed responses. P 1 = nena N C is the probability that an interneuron in layer 1 responds with an excitation, and p~= 1 - neflaN c is the probability that an interneuron in layer 1 receives mixed synaptic input. In layer 2, we have to consider two cases: (i) at low connectivity, if N c na < 1, P6 = 1 - N c na is the probability that an interneuron of layer 2 does not receive a synapse, thus does not respond to stimulation, P; = N c nane is the probability that a neuron in layer 2 responds with excitation, p? = N c nam is the probability that an interneuron responds with inhibition; (ii) at higher connectivity, N c na > 1, P6 = 0, P; = ne naN c and pl = m naN c. In both cases (i) and (ii), the probability that an interneuron in layer 2 has a mixed response pattern is P; = 1 - P6 -Pe - Pl. Thus, an interneuron in the model responds with excitation with probability P e = na P; + (1 - na) P;, with inhibition with probability Pi = na p/ + (1 - na) p? and has a mixed response with probability Pm =na P ~ + (1 - na) p;. P 0 .80 Layer 1 0 .60 0 .40 0 .20 0 .10 0.20 0 .30 0 .40 0.50 0 .60 0 .70 C P 0.80 0 .60 Layer 2 0 .40 0.20 0.20 0.30 0.40 0 .50 0 .60 0 .70 C 0 .80 0.60 Layers 1 & 2 0.40 0.20 0.10 0.20 0 .30 0 .40 0 .50 0.60 0 .70 C Figure 4: Analytically derived distribution of the response patterns in a typical network (35 interneurons, 15 receptor neurons, 40% excitation, 60% inhibition, spiking threshold 4.0); the curves show the percentage of interneurons in the model which respond with a given pattern, as a function of the connectivity c. In this case, the average excitation an interneuron receives from other interneurons is 3.15 at c=O.3. Figure 4 shows the distribution of the response patterns computed analytically for a typical set of parameters. In order to perform comparisons between computed pattern distributions and pattern distributions obtained from simulations with the model, we designed an automatic classifier for the response patterns, based on the perceptron learning rule and the pocket algorithm (Gallant 1986). The classifier is trained to classify the responses of 1028 Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz individual interneurons, based on their membrane potential, into 5 typical response classes: non-response, pure excitation, pure inhibition, simple mixed response and oscillatory responses. Figure 5 shows the simulation results for the same set of parameters as for Figure 4. The agreement between the two curves shows that the approximations which we have made in order to describe the analytical model are valid. Figure 6 shows how the mixed responses in the simulations divide into simple mixed and oscillatory responses. When the validity limit of the approximations made in the analytical approach is reached, all neurons fire at maximum frequency and the network oscillates. Therefore, the analytical model describes satisfactorily the whole range of connectivity in which the pattern distribution does not reduce to oscillations. The oscillation frequency is determined by the mean synaptic delays and by the membrane time constants; more detailed results on the oscillatory behavior will be published in a future paper. P 80 60 40 20 P 80 60 40 20 80 60 40 20 ~ ................ . ../'4- Mixed ..-* Layer 1 0.3 0.4 o~ c Layer 2 Layers 1 & 2 0.4 Figure 5: Distribution of the response patterns obtained from simulations of the model with the set of parameters described above. The curves show the percentages of interneurons that respond with a given pattern, as a function of connectivity c. For each value of c, 100 simulation runs with three different stimulation inputs have been averaged. pr-------------------7-=-=--------==~-----------------=------80 60 40 20 Layers 1 & 2 o .~ o.b o. c Figure 6: Distribution of simple mixed and oscillatory responses in the simulation model. With the set of parameters chosen, condition ne c :::: e is satisfied for c::::O.3. A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1029 4. Conclusion In the olfactory system of insects and mammals, a number of response patterns are observed, which are of central importance for the coding of the olfactory signal. In the present paper, we show that, under some constraints, an analytical model can predict the existence and the distribution of these response patterns. We further show that the transition between non-oscillatory and oscillatory regimes is governed by a single parameter (ne c / E». It is thus possible, to explore the parameter space without exhaustive simulations, and to relate the coding capabilities of a model or biological network to its architecture. Acknowledgements This work was supported in part by a grant from Ministere de la Recherche et de la Technologie (Sciences de la Cognition). C. Linster has been supported by a research grant (BFR91/051) from the Ministere des Affaires Culturelles, Grand-Duche de Luxembourg. References Boeckh, J. and Ernst, K.D. (1987). Contribution of single unit analysis in insects to an understanding of olfactory function. 1. Compo Physiolo. AI61:549-565. Burrows, M., Boeckh, J., Esslen, J. (1982). Physiological and Morphological Properties of Interneurons in the Deutocerebrum of Male Cockroaches which respond to Female Pheromone. 1. Compo Physiolo. 145:447-457. Christensen, T.A., Hildebrand, J.G. (1987). Functions, Organization, and Physiology of the Olfactory Pathways in the Lepidoteran Brain. In Arthropod Brain: its Evolution, Development, Structure and Functions, A.P. GuPta, (ed), John Wiley & Sons. Christensen, T.A., Hildebrand, J.G. (1988). Frequency coding by central olfactory neurons in the spinx moth Manduca sexta. Chemical Senses 13 (1): 123-130. Christensen, T.A., Mustaparta, H., Hildebrand, J.G. (1989). Discrimination of sex pheromone blends in the olfactory system of the moth. Chemical Senses 14 (3):463-477. Kaissling, K-E., Kramer, E. (1990). Sensory basis of pheromone-mediated orientation in moths. Verh. Dtsch. Zoolo. Ges. 83:109-131. Linster, C., Masson, C., Kerszberg, M., Personnaz, L., Dreyfus, G. (1993). Computational Diversity in a Formal Model of the Insect Olfactory Macroglomerulus. Neural Computation 5:239-252. Masson, C., Mustaparta, H. (1990). Chemical Information Processing in the Olfactory System of Insects. Physiol. Reviews 70 (1): 199-245.
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Using Prior Knowledge in a NNPDA to Learn Context-Free Languages Guo-Zheng SUD Sreerupa Das Dept. of Compo Sc. & Inst. of Cognitive Sc. University of Colorado Boulder, CO 80309 c. Lee Giles· NEC Research Inst. 4 Independence Way Princeton, NJ 08540 "'lnst. for Adv. Compo Studies University of Maryland College Park, MD 20742 Abstract Although considerable interest has been shown in language inference and automata induction using recurrent neural networks, success of these models has mostly been limited to regular languages. We have previously demonstrated that Neural Network Pushdown Automaton (NNPDA) model is capable of learning deterministic context-free languages (e.g., anbn and parenthesis languages) from examples. However, the learning task is computationally intensive. In this paper we discus some ways in which a priori knowledge about the task and data could be used for efficient learning. We also observe that such knowledge is often an experimental prerequisite for learning nontrivial languages (eg. anbncbmam ). 1 INTRODUCTION Language inference and automata induction using recurrent neural networks has gained considerable interest in the recent years. Nevertheless, success of these models has been mostly limited to regular languages. Additional information in form of a priori knowledge has proved important and at times necessary for learning complex languages (Abu-Mostafa 1990; AI-Mashouq and Reed, 1991; Omlin and Giles, 1992; Towell, 1990). They have demonstrated that partial information incorporated in a connectionist model guides the learning process through constraints for efficient learning and better generalization. 'Ve have previously shown that the NNPDA model can learn Deterministic Context 65 66 Das, Giles, and Sun Output Top-or-stack push ... 11:0 t pop or no-op Action External State(t+l) ~ stack 00 0 0 \ \ .~~ ~ t ~ " alphabets on the It;;:::::::.. "hig.her order stack ::;;;;::.. weights '::~::::'. ='000 00 00 copy State Neurons Input Neurons Read Neurons it it 11' State(t) Input(t) Top-of-stack(t) Figure 1: The figure shows the architecture of a third-order NNPDA. Each weight relates the product of Input(t), State(t) and Top-of-Stack information to the State(t+1). Depending on the activation of the Action Neuron, stack action (namely, push, pop or no operation) is taken and the Top-of-Stack (i.e. value of Read Neurons) is updated. Free Languages (DCFLs) from a finite set of examples. However, the learning task requires considerable amount of time and computational resources. In this paper we discuss methods in which a priori knowledge, may be incorporated in a N eum! network Pushdown Automaton (NNPDA) described in (Das, Giles and Sun, 1992; Giles et aI, 1990; Sun et aI, 1990). 2 THE NEURAL NETWORK PUSHDOWN AUTOMATA 2.1 ARCHITECTURE The description of the network architecture is necessarily brief, for further details see the references above. The network consists of a set of recurrent units, called state neurons and an external stack memory. One state neuron is designated as the output neuron. The state neurons get input (at every time step) from three sources: from their own recurrent connections, from the input neurons and from the read neurons. The input neurons register external inputs which consist of strings of characters presented one at a time. The read neurons keep track of the symbol(s) on top of the stack. One non-recurrent state neuron, called the action neuron, indicates the stack action (push, pop or no-op) at any instance. The architecture is shown in Figure 1. The stack used in this model is continuous. Unlike an usual discrete stack where an element is either present or absent, elements in a continuous stack may be present in varying degrees (values between [0, 1]). A continuous stack is essential in order Using Prior Knowledge in a NNPDA to Learn Context-Free Languages 67 to permit the use of a continuous optimization method during learning. The stack is manipulated by the continuous valued action neuron. A detailed discussion on the operations may be found in (Das, Giles and Sun, 1992). 2.2 LEARNABLE CLASS OF LANGUAGES The class of language learnable by the NNPDA is a proper subset of deterministic context-free languages. A formal description of a Pushdown Automaton (PDA) requires two distinct sets of symbols - one is the input symbol set and the other is the stack symbol set!. We have reduced the complexity of this PDA model in the following ways: First, we use the same set of symbols for the input and the stack. Second, when a push operation is performed the symbol pushed on the stack is the one that is available as the current input. Third, no epsilon transitions are allowed in the NNPDA. Epsilon transition is one that performs state transition and stack action without reading in a new input symbol. Unlike a deterministic finite state automata, a deterministic PDA can make epsilon transitions under certain restrictions!. Although these simplifications reduce the language class learnable by NNPDA, nevertheless the languages in this class retain essential properties of eFLs and is therefore more complex than any regular language. 2.3 TRAINING The activation of the state neurons s at time step t + 1 may be formulated as follows (we will only consider third order NNPDA in this paper): (1) where g(x) = frac1/1 + exp( -x), i is the activation of the input neurons and r is the activation of the read neuron and W is the weight matrix of the network. We use a localized representation for the input and the read symbols. During training, input sequences are presented one at a time and activations are allowed to propagate until the end of the string is reached. Once the end is reached the activation of the output neuron is matched with the target (which is 1.0 for positive string and 0.0 for a negative string) The learning rule used in the NNPDA is a significantly enhanced extension to Real Time Recurrent Learning (\Villiams and Zipser, 1989). 2.4 OBJECTIVE FUNCTION The objective function used to train the network consists of two error terms: one for positive strings and the other for negative strings. For positive strings we require (a) the NNPDA must reach a final state and (b) the stack must be empty. This criterion can be reached by minimizing the error function: (2) where So(l) is the activation of an output neuron and L(I) is the stack length, after a string of length I has been presented as input a character at a time. For negative 1 For details refer to (Hopcroft, 1979). 68 Das, Giles, and Sun avg of total parenthesis postfix anbn presentations w IL wjo IL w IL wjo IL w IL wjo IL # of strings 2671 5644 8326 15912 108200 >200000 # of character 10628 29552 31171 82002 358750 >700000 Table 1: Effect of Incremental Learning (IL) is displayed in this table. The number of strings and characters required for learning the languages are provided here. parenthesis anbn w SSP wjo SSP w SSP wjo SSP epochs 50-80 50-80 150-250 150-250 generalization 100% 100% 100% 98.97% number of units 1+1 2 1+1 2 anbncbmam an+mbncm w SSP wjo SSP w SSP wjo SSP epochs 150 "''''''' 150-250 *** generalization 96.02% *** 100% *** number of units 1+1 *** 1+1 *** Table 2: This table provides some statistics on epochs, generalization and number of hidden units required for learning with and without selective string presentation (SSP). strings, the error function is modified as: E - { so(1) - L(l) rror 0 if (so(1) - L(l)) > 0.0 else (3) Equation (2) reflects the criterion that, for a negative pattern we require either the final state so(l) = 0.0 or the stack length L(1) to be greater than 1.0 (only when so(l) = 1.0 and the stack length L(l) is close to zero, the error is high). 3 BUILDING IN PRIOR KNOWLEDGE In practical inference tasks it may be possible to obtain prior knowledge about the problem domain. In such cases it often helps to build in knowledge into the system under study. There could be at least two different types of knowledge available to a model (a) knowledge that depends on the training data with absolutely no knowledge about the automaton, and (b) partial knowledge about the automaton being inferred. Some of ways in which knowledge can be provided to the model are discussed below. 3.1 KNOWLEDGE FROM THE DATA 3.1.1 Incremental Learning Incremental Learning has been suggested by many (Elman, 1991; Giles et aI, 1990, Sun et aI, 1990), where the training examples are presented in order of increasing Using Prior Knowledge in a NNPDA to Learn Context-Free Languages 69 0.9 0.8 0.7 0.6 ., ., ..., ... 0.5 0 ... ... .. 0.4 0.3 0.2 50 100 150 epochs with SSP without SSP --. 200 250 Figure 2: Faster convergence using selective string presentation (SSP) for parenthesis language task. length. This model of learning starts with a training set containing short simple strings. Longer strings are added to the training set as learning proceeds. We believe that incremental learning is very useful when (a) the data presented contains structure, and (b) the strings learned earlier embody simpler versions of the task being learned. Both these conditions are valid for context-free languages. Table 1 provides some results obtained when incremental learning was used. The figures are averages over several pairs of simulations, each of which were initialized with the same initial random weights. 3.1.2 Selective Input Presentation Our training data contained both positive and negative examples. One problem with training on incorrect strings is that, once a symbol in the string is reached that makes it negative, no further information is gained by processing the rest of the string. For example, the fifth a in the string aaaaba ... makes the string a negative example of the language a"b", irrespective of what follows it. In order to incorporate this idea we have introduced the concept of a dead state. During training, we assume that there is a teacher or an oracle who has knowledge of the grammar and is able to identify the first (leftmost) occurrence of incorrect sequence of symbols in a negative string. When such a point is reached in the input string, further processing of the string is stopped and the network is trained so that one designated state neuron called the dead state neuron is active. To accommodate the idea of a dead state in the learning rule, the following change is made: if the network is being trained on negative strings that end in a dead state then the length L(l) in the error function in equation (1) is ignored and it simply becomes 70 Das, Giles, and Sun 0.5 0 . 45 0 . 4 0.35 ... 0 0 . 3 ... ... .. 0. 0 . 25 ~ 0.2 0.15 0 . 1 0.05 a W/o IW wlth 1 IW --_. wlth 2 IW ' .. -wlth 3 IW --1000 2000 3000 4000 5000 6000 7000 8000 9000 No . of gtrlngg Figure 3: Learning curves when none, one or more initial weights (IW) were set for postfix language learning task Error = ~(1 - Sdead{l))2. Since such strings have an negative subsequence, they cannot be a prefix to any positive string. Therefore at this point we do not care about the length of the stack. For strings that are either positive or negative but do not go to a dead state (an example would be a prefix of a positive string); the objective function remains the same as described earlier in Equations 1 and 2. Such additional information provided during training resulted in efficient learning, helped in learning of exact pushdown automata and led to better generalization for the trained network. Information in this form was often a prerequisite for successfully learning certain languages. Figure 2 shows a typical plot of improvement in learning when such knowledge is used. Table 2 shows improvements in the statistics for generalization, number of units needed and number of epochs required for learning. The numbers in the tables were averages over several simulations; changing the initial conditions resulted in values of similar orders of magnitude. 3.2 KNOWLEDGE ABOUT THE TASK 3.2.1 Knowledge About The Target PDA's Dynamics One way in which knowledge about the target PDA can be built into a system is by biasing the initial conditions of the network. This may be done by assigning predetermined initial values to a selected set of weights (or biases). For example a third order NNPDA has a dynamics that maps well onto the theoretical model of a PDA. Both allow a three to two mapping of a similar kind. This is because in the third order NNPDA, the product of the activations of the input neurons, the read neurons and the state neurons determine the next state and the next action to be Using Prior Knowledge in a NNPDA to Learn Context-Free Languages 71 Start 9 [.5 .5] b -* , , Dead [.9 *] [.1 .9] a/-/push (a) PDA for parenthesis End [* .9] Start [.5 .5] (b) PDA for a~n [.0.6] e/-/* O End [* .9] Figure 4: The figure shows some of the PDAs inferred by the NNPDA. In the figure the nodes in the graph represent states inferred by the NNPDA and the numbers in "[]" indicates the state representations. Every transition is indicated by an arrow and is labeled as "x/y /z" where "x" corresponds to the current input symbol, "y" corresponds to the symbol on top of the stack and "z" corresponds to the action taken. taken. It may be possible to determine some of the weights in a third order network if certain information about the automaton in known. Typical improvement in learning is shown in Figure 3 for a postfix language learning task. 3.2.2 U sing Structured Examples Structured examples from a grammar are a set of strings where the order of letter generation is indicated by brackets. An example would be the string (( ab)c) generated by the rules S ---+ Xc; X ---+ abo Under the current dynamics and limitations of the model, this information could be interpreted as providing the stack actions (push and pop) to the NNPDA. Learning the palindrome language is a hard task because it necessitates remembering a precise history over a long period of time. The NNPDA was able to learn the palindrome language for two symbols when structured examples were presented. 4 AUTOMATON EXTRACTION FROM NNPDA Once the network performs well on the training set, the transition rules in the inferred PDA can then be deduced. Since the languages learned by the NNPDA so far corresponded to PDAs with few states, the state representations in the induced PDA could be inferred by looking at the state neuron activations when presented with all possible character sequences. For larger PDAs clustering techniques could be used to infer the state representations. Various clustering techniques for similar tasks have been discussed in (Das and Das, 1992; Giles et al., 1992). Figure 4 shows some of the PDAs inferred by the NNPDA. 72 Das, Giles, and Sun 5 CONCLUSION This paper has described some of the ways in which prior knowledge could be used to learn DCFGs in an NNPDA. Such knowledge is valuable to the learning process in two ways. It may reduce the solution space, and as a consequence may speed up the learning process. Having the right restrictions on a given representation can make learning simple: which reconfirms an old truism in Artificial Intelligence. References Y.S. Abu-Mostafa. (1990) Learning from hints in neural networks. Journal of Complexity, 6:192-198. K.A. AI-Mashouq and I.S. Reed. (1991) Including hints in training neural networks. Neural Computation, 3(3):418-427. S. Das and R. Das. (1992) Induction of discrete state-machine by stabilizing a continuous recurrent network using clustering. To appear in CSI Journal of Computer Science and Informatics. Special Issue on Neural Computing. S. Das, C.L. Giles, and G.Z. Sun. (1992) Learning context free grammars: capabilities and limitations of neural network with an external stack memory. Proc of the Fourteenth Annual Conf of the Cognitive Science Society, pp. 791-795. Morgan Kaufmann, San Mateo, Ca. J .L. Elman. (1991) Incremental learning, or the importance of starting small. CRL Tech Report 9101, Center for Research in Language, UCSD, La Jolla, CA. C.L. Giles, G.Z. Sun, H.H. Chen, Y.C. Lee and D. Chen, (1990) Higher Order Recurrent Networks & Grammatical Inference, Advances in Neural Information Processing Systems 2, pp. 380-387, ed. D.S. Touretzky, Morgan Kaufmann, San Mateo, CA. C.L. Giles, C.B. Miller, H.H. Chen, G.Z. Sun, and Y.C. Lee. (1992) Learning and extracting finite state automata with second-order recurrent neural networks. Neural Computation, 4(3):393-405. J .E. Hopfcroft and J.D. Ullman. (1979) Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading, MA. C.W. Omlin and C.L. Giles. (1992) Training second-order recurrent neural networks using hints. Proceedings of the Ninth Int Conf on Machine Learning, pp. 363-368. D. Sleeman and P. Edwards (eds). Morgan Kaufmann, San Mateo, Ca. G.Z. Sun, H.H. Chen, C.L. Giles, Y.C. Lee and D. Chen. (1991) Neural networks with external memory stack that learn context-free grammars from examples. Proc of the Conf on Information Science and Systems, Princeton U., Vol. II, pp. 649-653. G.G. Towell, J.W. Shavlik and M.O Noordewier. (1990) Refinement of approximately correct domain theories by knowledge-based neural-networks. In Proc of the Eighth National Conf on Artificial Intelligence, Boston, MA. pp. 861. R.J. Williams and D. Zipser. (1989) A learning algorithm for continually running fully recurrent neural networks. Neural Computation 1(2):270-280.
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A Method for Learning from Hints Yaser s. Abu-Mostafa Departments of Electrical Engineering, Computer Science, and Computation and Neural Systems California Institute of Technology Pasadena, CA 91125 e-mail: yaser@caltech.edu Abstract We address the problem of learning an unknown function by pu tting together several pieces of information (hints) that we know about the function. We introduce a method that generalizes learning from examples to learning from hints. A canonical representation of hints is defined and illustrated for new types of hints. All the hints are represented to the learning process by examples, and examples of the function are treated on equal footing with the rest of the hints. During learning, examples from different hints are selected for processing according to a given schedule. We present two types of schedules; fixed schedules that specify the relative emphasis of each hint, and adaptive schedules that are based on how well each hint has been learned so far. Our learning method is compatible with any descent technique that we may choose to use. 1 INTRODUCTION The use of hints is coming to the surface in a number of research communities dealing with learning and adaptive systems. In the learning-from-examples paradigm, one often has access not only to examples of the function, but also to a number of hints (prior knowledge, or side information) about the function. The most common difficulty in taking advantage of these hints is that they are heterogeneous and cannot be easily integrated into the learning process. This paper is written with the specific goal of addressing this problem. The paper develops a systematic method 73 74 Abu-Mostafa for incorporating different hints in the usuallearning-from-examples process. Without such a systematic method, one can still take advantage of certain types of hints. For instance, one can implement an invariance hint by preprocessing the input to achieve the invariance through normalization. Alternatively, one can structure the learning model in a way that directly implements the invariance (Minsky and Papert, 1969). Whenever direct implementation is feasible, the full benefit of the hint is realized. This paper does not attempt to offer a superior alternative to direct implementation. However, when direct implementation is not an option, we prescribe a systematic method lor incorporating practically any hint in any descent technique lor learning. The goal is to automate the use of hints in learning to a degree where we can effectively utilize a large number of different hints that may be available in a practical situation. As the use of hints becomes routine, we are encouraged to exploit even the simplest observations that we may have about the function we are trying to learn. The notion of hints is quite general and it is worthwhile to formalize what we mean by a hint as far as our method is concerned. Let I be the function that we are trying to learn. A hint is a property that I is known to have. Thus, all that is needed to qualify as a hint is to have a litmus test that I passes and that can be applied to different functions. Formally, a hint is a given subset of functions that includes I. We start by introducing the basic nomenclature and notation. The environment X is the set on which the function I is defined. The points in the environment are distributed according to some probability distribution P. I takes on values from some set Y I:X-+Y Often, Y is just {O, I} or the interval [0, 1]. The learning process takes pieces of information about (the otherwise unknown) I as input and produces a hypothesis g g:X-+Y that attempts to approximate f. The degree to which a hypothesis g is considered an approximation of I is measured by a distance or 'error' E(g, !) The error E is based on the disagreement between g and I as seen through the eyes of the probability distribution P. Two popular forms of the error measure are E = Pr[g(x) =F f(x)] and E = £[(g(x) - l(x))2] where Pr[.] denotes the probability of an event, and £[.J denotes the expected value of a random variable. The underlying probability distribution is P. E will always be a non-negative quantity, and we will take E(g,!) = ° t.o mean that 9 and I are identical for all intents and purposes. We will also assume that when the set of hypotheses is parameterized by real-valued parameters (e.g., the weights in the case of a neural network), E will be well-behaved as a function of the parameters A Method for Learning from Hints 75 (in order to allow for derivative-based descent techniques). We make the same assumptions about the error measures that will be introduced in section 2 for the hints. In this paper, the 'pieces of information' about f that are input to the learning process are more general than in the learning-from-examples paradigm. In that paradigm, a number of points Xl, ... , X N are picked from X (usually independently according to the probability distribution P) and the values of f on these points are provided. Thus, the input to the learning process is the set of examples (Xl, f(XI)),"', (XN' f(XN)) and these examples are used to guide the search for a good hypothesis. We will consider the set of examples of f as only one of the available hints and denote it by Ho. The other hints HI,' .. ,HM will be additional known facts about f, such as invariance properties for instance. The paper is organized as follows. Section 2 develops a canonical way for representing different hints. This is the first step in dealing with any hint that we encounter in a practical situation. Section 3 develops the basis for learning from hints and describes our method, including specific learning schedules. 2 REPRESENTATION OF HINTS As we discussed before, a hint Hm is defined by a litmus test that f satisfies and that can be applied to the set of hypotheses. This definition of Hm can be extended to a definition of 'approximation of Hm' in several ways. For instance, 9 can be considered to approximate Hm within f. if there is a function h that strictly satisfies H~ for which E(g, h) ::; f.. In the context of learning, it is essential to have a notion of approximation since exact learning is seldom achievable. Our definitions for approximating different hints will be part of the scheme for representing those hints. The first step in representing Hm is to choose a way of generating 'examples' of the hint. For illustration, suppose that Hm asserts that f: [-1,+1]- [-1,+1] is an odd function. An example of Hm would have the form f(-x) = -f(x) for a particular X E [-1, +1]. To generate N examples of this hint, we generate Xl,'" ,XN and assert for each Xn that f( -xn) = - f(xn). Suppose that we are in the middle of a learning process, and that the current hypothesis is 9 when the example f( -x) = - f(x) is presented. We wish to measure how much 9 disagrees with this example. This leads to the second component of the representation, the error measure em. For the oddness hint, em can be defined as em = (g(x) + g( _x))2 so that em = 0 reflects total agreement with the example (i.e., g( -x) = -g(x)). Once the disagreement between 9 and an example of Hm has been quantified 76 Abu-Mostafa through em, the disagreement between 9 and Hm as a whole is automatically quantified through Em, where Em = £(em) The expected value is taken w.r.t. the probability rule for picking the examples. Therefore, Em can be estimated by a.veraging em over a number of examples that are independently picked. The choice of representation of H m is not unique, and Em will depend on the form of examples, the probability rule for picking the examples, and the error measure em. A minimum requirement on Em is that it should be zero when E = O. This requirement guarantees that a hypothesis for which E = 0 (perfect hypothesis) will not be excluded by the condition Em = O. Let us illustrate how to represent different types of hints. Perhaps the most common type of hint is the invariance hint. This hint asserts that I(x) = I(x') for certain pairs x, x'. For instance, "I is shift-invariant" is formalized by the pairs x, x' that are shifted versions of each other. To represent the invariance hint, an invariant pair (x, x') is picked as an example. The error associated with this example is em = (g(x) - g(x'))2 Another related type of hint is the monotonicity hint (or inequality hint). The hint asserts for certain pairs x, x' that I(x) :S I(x') . For instance, "I is monotonically nondecreasing in x" is formalized by all pairs x, x' such that x < x'. To represent the monotonicity hint, an example (x, x') is picked, and the error associated with this example is given by _ {(g(x) - g(X'»2 em 0 if g(x) > g(x') if g(x) :S g(x' ) The third type of hint we discuss here is the approximation hint. The hint asserts for certain points x E X that I(x) E [ax, bx]. In other words, the value of 1 at x is known only approximately. The error associated with an example x of the approximation hint is if g(x) < ax if g(x) > bx if g(x) E [ax,bx] Another type of hints arises when the learning model allows non-binary values for 9 where 1 itself is known to be binary. This gives rise to the binary hint. Let X ~ X be the set where 1 is known to be binary (for Boolean functions, X is the set of binary input vectors). The binary hint is represented by examples of the form x, where x E X. The error function associated with an example x (assuming 0/1 binary convention, and assuming g( x) E [0, 1]) is em = g(x)(l- g(x» This choice of em forces it to be zero when g(x) is either 0 or 1, while it would be positive if g( x) is between 0 and 1. A Method for Learning from Hints 77 It is worth noting that the set of examples of f can be formally treated as a hint, too. Given (Xl, f(xt)},···, (XN' f(XN )), the examples hint asserts that these are the correct values of f at those particular points. Now, to generate an 'example' of this hint, we pick a number n from I to N and use the corresponding (xn, f(xn)). The error associated with this example is eo (we fix the convention that m = 0 for the examples hint) eo = (g(xn ) f(:c n ))2 Assuming that the probability rule for picking n is uniform over {I,··· ,N}, 1 N Eo = E(eo) = N I)g(xn) - f(xn))2 n=l In this case, Eo is also the best estimator of E = E[(g(x) - f(x))2] given Xl,··· ,xN that are independently picked according to the original probability distribution P. This way of looking at the examples of f justifies their treatment exactly as one of the hints, and underlines the distinction between E and Eo. In a practical situation, we try to infer as many hints about f as the situation will allow. Next, we represent each hint according to the scheme discussed in this section. This leads to a list Ho, H l ,··· ,HM of hints that are ready to produce examples upon the request of the learning algorithm. We now address how the algorithm should pick and choose between these examples as it moves along. 3 LEARNING SCHEDULES If the learning algorithm had complete information about f, it would search for a hypothesis g for which E(g, f) = o. However, f being unknown means that the point E = 0 cannot be directly identified. The most any learning algorithm can do given the hints Ho, HI,··· ,HM is to reach a hypothesis g for which all the error measures Eo, El, · ·· , EM are Zeros. Indeed, we have required that E = 0 implies that Em = 0 for all m. If that point is reached, regardless of how it is reached, the job is done. However, it is seldom the case that we can reach the zero-error point because either (1) it does not exist (i.e., no hypothesis can satisfy all the hints simultaneously, which implies that no hypothesis can replicate f exactly), or (2) it is difficult to reach (Le., the computing resources do not allow us to exhaustively search the space of hypotheses looking for that point). In either case, we will have to settle for a point where the Em's are 'as small as possible'. How small should each Em be? A balance has to be struck, otherwise some Em's may become very small at the expense of the others. This situation would mean that some hints are over-learned while the others are under-learned. We will discuss learning schedules that use different criteria for balancing between the hints. The schedules are used by the learning algorithm to simultaneously minimize the Em's. Let us start by exploring how simultaneous minimization of a number of quantities is done in general. Perhaps the most common approach is that of penalty functions (Wismer and Chat78 Abu-Mostafa tel'gy, 1978). In order to minimize Eo, E 1 ,· •• ,EM, we minimize the penalty function M L am Em m=O where each am is a non-negative number that may be constant (exact penalty function) or variable (sequential penalty function). Any descent technique can be employed to minimize the penalty function once the am's are selected. The am's are weights that reflect the relative emphasis or 'importance' of the corresponding Em's. The choice of the weights is usually crucial to the quality of the solution. Even if the am's are determined, we still do not have the explicit values of the Em's in our case (recall that Em is the expected value of the error em on an example of the hint). Instead, we will estimate Em by drawing several examples and averaging their error. Suppose that we draw Nm examples of Hm. The estimate for Em would then be 1 Nm _ ~ e(n) Nm Lm n=l where e~) is the error on the nth example. Consider a batch of examples consisting of No examples of Ho, Nl examples of HI, ... , and NM examples of HM. The total error of this batch is m=O n=l If we take Nm ex: am, this total error will be a proportional estimate of the penalty function M L am Em m=O In effect, we translated the weights into a schedule, where different hints are emphasized, not by magnifying their error, but by representing them with more examples. A batch of examples can be either a uniform batch that consist of N examples of one hint at a time, or, more generally, a mixed batch where examples of different hints are allowed within the same batch. If the descent technique is linear and the learning rate is small, a schedule that uses mixed batches is equivalent to a schedule that alternates between uniform batches (wit.h frequency equal to the frequency of examples in the mixed batch). If we are using a nonlinear descent technique, it is generally more difficult to ascertain a direct translation from mixed batches to uniform batches, but there may be compelling heuristic correspondences. All schedules discussed here are expressed in terms of uniform batches for simplicity. The implementation of a given schedule goes as follows: (1) The algorithm decides which hint (which m for m = 0,1,···, M) to work on next, according to some criterion; (2) The algorithm then requests a batch of examples of this hint; (3) It performs its descent on this batch; and (4) When it is done, it goes back to step (1). We make a distinction between fixed schedules, where the criterion for selecting the hint can be 'evaluated' ahead of time (albeit time-invariant or time-varying, A Method for Learning from Hints 79 deterministic or stochastic), and adaptive schedules, where the criterion depends on what happens as the algorithm runs. Here are some fixed and adaptive schedules: Simple Rotation: This is the simplest possible schedule that tries to balance between the hints. It is a fixed schedule that rotates between Ho, HI'···' HM. Thus, at step k, a batch of N examples of Hm is processed, where m = k mod (M + 1). This simple-minded algorithm tends to do well in situations where the Em's are somewhat similar. Weighted Rotation: This is the next step in fixed schedules that tries to give different emphasis to different Em's. The schedule rotates between the hints, visiting Hm with frequency Vm. The choice of the vm's can achieve balance by emphasizing the hints that are more important or harder to learn. Maximum Error: This is the simplest adaptive schedule that tries to achieve the same type of balance as simple rotation. At each step k, the algorithm processes the hint with the largest error Em. The algorithm uses estimates of the Em's to make its selection. Maximum Weighted Error: This is the adaptive counterpart to weighted rotation. It selects the hint with the largest value of vmEm. The choice of the vm's can achieve balance by making up for disparities between the numerical ranges of the Em's. Again, the algorithm uses estimates of the Em's. Adaptive schedules attempt to answer the question: Given a set of values for the Em's, which hint is the most under-learned? The above schedules answer the question by comparing the individual Em's. Althongh this works well in simple cases, it does not take into consideration the correlation between different hints. As we deal with more and more hints, the correlation between the Em's becomes more significant. This leads us to the final schedule that achieves the balance between the Em's through their relation to the actual error E. Adaptive Minimization: Given the estimates of Eo, EI , ... , EM, make M + 1 estimates of E, each based on all but one of the hints: E(., Ell E2,···, EM) E(Eo,., E2,···, EM) E(Eo, EI,.,···, EM) E (Eo, EI, E2, ... , • ) and choose the hint for which the corresponding estimate is the smallest. In other words, E becomes the common thread between the Em's. Knowing that we are really trying to minimize E, and that the Em's are merely a vehicle to this end, the criterion for balancing the Em's should be based on what is happening to E as far as we can tell. 80 Abu-Mostafa CONCLUSION This paper developed a systematic method for using different hints as input to the learning process, generalizing the case of invariance hints (Abu-Mostafa, 1990). The method treats all hints on equal footing, including the examples of the function. Hints are represented in a canonical way that is compatible with the common learning-from-examples paradigm. No restrictions are made on the learning model or the descent technique to be used. The hints are captured by the error measures Eo, El,"', EM, and the learning algorithm attempts to simultaneously minimize these quantities. The simultaneous minimization of the Em's gives rise to the idea of balancing between the different hints. A number of algorithms that minimize the Em's while maintaining this balance were discussed in the paper. Adaptive schedules in particular are worth noting because they automatically compensate against many artifacts of the learning process. It is worthwhile to distinguish between the quality of the hints and the quality of the learning algorithm that uses these hints. The quality of the hints is determined by how reliably one can predict that the actual error E will be close to zero for a given hypothesis based on the fact that Eo, E1 , ••• , EM are close to zero for that hypothesis. The quality of the algorithm is determined by how likely it is that the Em's will become nearly as small as they can be within a reasonable time. Acknowledgements The author would like to thank Ms. Zehra Kok for her valuable input. This work was supported by the AFOSR under grant number F49620-92-J-0398. References Abu-Mostafa, Y. S. (1990), Learning from hints in neural networks, Journal of Complexity 6, 192-198. AI-Mashouq, K. and Reed, 1. (1991), Including hints in training neural networks, Neural Computation 3, 418-427. Minsky, M. L. and Papert, S. A. (1969), "Perceptrons," MIT Press. Omlin, C. and Giles, C. L. (1992), Training second-order recurrent neural networks using hints, Machine Learning: Proceedings of the Ninth International Conference (ML-92), D. Sleeman and P. Edwards (ed.), Morgan Kaufmann. Suddarth, S. and Holden, A. (1991), Symbolic neural systems and the use of hints for developing complex systems, International Journal of Machine Studies 35, p. 291. Wismer, D. A. and Chattergy, R. (1978), "Introduction to Nonlinear Optimization," North Holland.
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Holographic Recurrent Networks Tony A. Plate Department of Computer Science University of Toronto Toronto, M5S lA4 Canada Abstract Holographic Recurrent Networks (HRNs) are recurrent networks which incorporate associative memory techniques for storing sequential structure. HRNs can be easily and quickly trained using gradient descent techniques to generate sequences of discrete outputs and trajectories through continuous spaee. The performance of HRNs is found to be superior to that of ordinary recurrent networks on these sequence generation tasks. 1 INTRODUCTION The representation and processing of data with complex structure in neural networks remains a challenge. In a previous paper [Plate, 1991b] I described Holographic Reduced Representations (HRRs) which use circular-convolution associative-memory to embody sequential and recursive structure in fixed-width distributed representations. This paper introduces Holographic Recurrent Networks (HRNs), which are recurrent nets that incorporate these techniques for generating sequences of symbols or trajectories through continuous space. The recurrent component of these networks uses convolution operations rather than the logistic-of-matrix-vectorproduct traditionally used in simple recurrent networks (SRNs) [Elman, 1991, Cleeremans et a/., 1991]. The goals ofthis work are threefold: (1) to investigate the use of circular-convolution associative memory techniques in networks trained by gradient descent; (2) to see whether adapting representations can improve the capacity of HRRs; and (3) to compare performance of HRNs with SRNs. 34 Holographic Recurrent Networks 35 1.1 RECURRENT NETWORKS & SEQUENTIAL PROCESSING SRNs have been used successfully to process sequential input and induce finite state grammars [Elman, 1991, Cleeremans et a/., 1991]. However, training times were extremely long, even for very simple grammars. This appeared to be due to the difficulty of findin& a recurrent operation that preserved sufficient context [Maskara and Noetzel, 1992J. In the work reported in this paper the task is reversed to be one of generating sequential output. Furthermore, in order to focus on the context retention aspect, no grammar induction is required. 1.2 CIRCULAR CONVOLUTION Circular convolution is an associative memory operator. The role of convolution in holographic memories is analogous to the role of the outer product operation in matrix style associative memories (e.g., Hopfield nets). Circular convolution can be viewed as a vector multiplication operator which maps pairs of vectors to a vector (just as matrix multiplication maps pairs of matrices to a matrix). It is defined as z = x@y : Zj = I:~:~ YkXj-k, where @ denotes circular eonvolution, x, y, and z are vectors of dimension n , Xi etc. are their elements, and subscripts are modulo-n (so that X-2 = Xn -2). Circular convolution can be computed in O(nlogn) using Fast Fourier Transforms (FFTs). Algebraically, convolution behaves like scalar multiplication: it is commutative, associative, and distributes over addition. The identity vector for convolution (I) is the "impulse" vector: its zero'th element is 1 and all other elements are zero. Most vectors have an inverse under convolution, i.e., for most vectors x there exists a unique vector y (=x- 1) such that x@y = I. For vectors with identically and independently distributed zero mean elements and an expected Euclidean length of 1 there is a numerically stable and simply derived approximate inverse. The approximate inverse of x is denoted by x· and is defined by the relation x; = Xn-j. Vector pairs can be associated by circular convolution. Multiple associations can be summed. The result can be decoded by convolving with the exact inverse or approximate inverse, though the latter generally gives more stable results. Holographie Reduced Representations [Plate, 1991a, Plate, 1991b] use c.ircular convolution for associating elements of a structure in a way that can embody hierarchical structure. The key property of circular convolution that makes it useful for representing hierarchical structure is that the circular convolution of two vectors is another vector of the same dimension, which can be used in further associations. Among assoeiative memories, holographic memories have been regarded as inferior beeause they produee very noisy results and have poor error correcting properties. However, when used in Holographic Reduced Representations the noisy results can be cleaned up with conventional error correcting associative memories. This gives the best of both worlds - the ability to represent sequential and recursive structure and clean output vectors. 2 TRAJECTORY-ASSOCIATION A simple method for storing sequences using circular convolution is to associate elements of the sequence with points along a predetermined trajectory. This is akin 36 Plate to the memory aid called the method of loci which instructs us to remember a list of items by associating each term with a distinctive location along a familiar path. 2.1 STORING SEQUENCES BY TRAJECTORY-ASSOCIATION Elements of the sequence and loci (points) on the trajectory are all represented by n-dimensional vectors. The loci are derived from a single vector k - they are its suc,cessive convolutive powers: kO, kl, k 2, etc. The convolutive power is defined in the obvious way: kO is the identity vector and k i+1 = ki@k. The vector k must be c,hosen so that it does not blow up or disappear when raised to high powers, i.e., so that IlkP II = 1 'V p. The dass of vec.tors which satisfy this constraint is easily identified in the frequency domain (the range of the discrete Fourier transform). They are the vectors for which the magnitude of the power of each frequenc.y component is equal to one. This class of vectors is identic,al to the class for which the approximate inverse is equal to the exact inverse. Thus, the trajectory-association representation for the sequence "abc" is Sabc. = a + b@k + c@k2. 2.2 DECODING TRAJECTORY-ASSOCIATED SEQUENCES Trajectory-associated sequences can be decoded by repeatedly convolving with the inverse of the vector that generated the encoding loci. The results of dec,oding summed convolution products are very noisy. Consequently, to decode trajec.tory associated sequences, we must have all the possible sequenc,e elements stored in an error c,orrecting associative memory. I call this memory the "clean up" memory. For example, to retrieve the third element of the sequence Sabc we convolve twice with k- 1 , which expands to a@k- 2 + b@k- 1 + c. The two terms involving powers of k are unlikely to be correlated with anything in the clean up memory. The most similar item in clean up memory will probably be c. The clean up memory should recognize this and output the dean version of c. 2.3 CAPACITY OF TRAJECTORY-ASSOCIATION In [Plate, 1991a] the capacity of circular-convolution based assoc.iative memory was c,alculated. It was assumed that the elements of all vectors (dimension n) were c,hosen randomly from a gaussian distribution with mean zero and variance lin (giving an expec.ted Eudidean length of 1.0). Quite high dimensional vec.tors were required to ensure a low probability of error in decoding. For example, with .512 element vec.tors and 1000 items in the clean up memory, 5 pairs can be stored with a 1 % chance of an error in deeoding. The scaling is nearly linear in n: with 1024 element vectors 10 pairs can be stored with about a 1% chance of error. This works out to a information c,apac.ity of about 0.1 bits per element. The elements are real numbers, but high precision is not required. These capacity calculations are roughly applicable to the trajectory-association method. They slightly underestimate its capacity because the restriction that the encoding loci have unity power in all frequencies results in lower decoding noise. Nonetheless this figure provides a useful benchmark against which to compare the capacity of HRNs which adapt vec.tors using gradient descent. Holographic Recurrent Networks 37 3 TRAJECTORY ASSOCIATION & RECURRENT NETS HRNs incorporate the trajectory-association scheme in recurrent networks. HRNs are very similar to SRNs, sueh as those used by [Elman, 1991] and [Cleeremans et al. , 1991]. However, the task used in this paper is different: the generation of target sequences at the output units, with inputs that do not vary in time. In order to understand the relationship between HRNs and SRNs both were tested on the sequence generation task. Several different unit activation functions were tried for the SRN: symmetric (tanh) and non-symmetric sigmoid (1/(1 + e- X )) for the hidden units, and soft max and normalized RBF for the output units. The best combination was symmetric sigmoid with softmax outputs. 3.1 ARCHITECTURE The H RN and the SRN used in the experiments described here are shown in Figure I. In the H RN the key layer y contains the generator for the inverse loci (corresponding to k- 1 in Section 2). The hidden to output nodes implement the dean-up memory: the output representation is local and the weights on the links to an output unit form the vector that represents the symbol corresponding to that unit. The softmax function serves to give maximum activation to the output unit whose weights are most similar to the activation at the hidden layer. The input representation is also loeal, and input activations do not ehange during the generation of one sequence. Thus the weights from a single input unit determine the acti vations at the code layer. Nets are reset at the beginning of each seq lIenee. The HRN computes the following functions. Time superscripts are omitted where all are the same. See Figure 1 for symbols. The parameter 9 is an adaptable input gain shared by all output units. Code units: Hidden units: Context units: Output units: (first time step) (subsequent steps) (total input) (output) (h = p@y) (softmax) In the SRN the only differenee is in the reeurrence operation, i.e., the computation of the activations of the hidden units whieh is, where bj is a bias: hj = tanh(cj + Ek wjkPk + bj). The objective function of the network is the asymmetric divergence between the activations of the output units (or) and the targets (tr) summed over eases sand timesteps t, plus two weight penalty terms (n is the number of hidden units): ( """ st lor) 0.0001 (""" r """ c) """ (1 """ 0 2) 2 E = ~ tj og t;; + n ~ Wjk + ~ Wjk + ~ - L.J Wjk stJ J J k J k J k The first weight penalty term is a standard weight cost designed to penalize large 38 Plate Output 0 Output 0 HRN SRN (;ontext p Input i Figure 1: Holographic. Recurrent Network (HRN) and Simple Recurrent Network (SRN). The backwards curved arrows denote a copy of activations to the next time step. In the HRN the c.ode layer is active only at the first time step and the c.ontext layer is active only after the first time step. The hidden, code, context, and key layers all have the same number of units. Some input units are used only during training, others only during testing. weights. The sec.ond weight penalty term was designed to force the Eudidean length of the weight vector on each output unit to be one. This penalty term helped the HRN c.onsiderably but did not noticeably improve the performance of the SRN. The partial derivatives for the activations were c.omputed by the unfolding in time method [Rumelhart et ai., 1986]. The partial derivatives for the activations of the context units in the HRN are: DE DE a-: = L 81 . Yk-j (= 'lpE = 'lh@Y*) PJ k ~J When there are a large number of hidden units it is more efficient to compute this derivative via FFTs as the convolution expression on the right. On all sequenc.es the net was cycled for as many time steps as required to produc.e the target sequence. The outputs did not indic.ate when the net had reached the end of the sequence, however, other experiments have shown that it is a simple matter to add an output to indic.ate this. 3.2 TRAINING AND GENERATIVE CAPACITY RESULTS One of the motivations for this work was to find recurrent networks with high generative capacity, i.e., networks whic.h after training on just a few sequences c.ould generate many other sequences without further modification of recurrent or output weights. The only thing in the network that changes to produce a different sequence is the activation on the codes units. To have high generative capacity the function of the output weights and recurrent weights (if they exist) must generalize to the production of novel sequenc.es. At each step the recurrent operation must update and retain information about the current position in the sequence. It was Holographic Recurrent Networks 39 expected that this would be a difficult task for SRNs, given the reported difficulties with getting SRNs to retain context, and Simard and LeCun's [1992] report of being unable to train a type of recurrent network to generate more than one trajectory through c.ontinuous space. However, it turned out that HRNs, and to a lesser extent SRNs, c.ould be easily trained to perform the sequence generation task well. The generative capacity of HRNs and SRNs was tested using randomly chosen sequences over :3 symbols (a, b, and c). The training data was (in all but one case) 12 sequences of length 4, e.g., "abac", and "bacb". Networks were trained on this data using the conjugate gradient method until all sequences were correctly generated. A symbol was judged to be correct when the activation of the correct output unit exceeded 0.5 and exceeded twice any other output unit activation. After the network had been trained, all the weights and parameters were frozen, except for the weights on the input to c.ode links. Then the network was trained on a test set of novel sequences of lengths 3 to 16 (32 sequences of each length). This training could be done one sequence at a time since the generation of each sequence involved an exclusive set of modifiable weights, as only one input unit was active for any sequence. The search for code weights for the test sequences was a c.onjugate gradient search limited to 100 iterations. 100% 80% 'x HRN 64 --060% HRN 32 -+HRN 16 ~ 40% A HRN 8 . X· )( N 4 'L:!.' • 20% A x. 0% 4 6 8 10 12 14 16 4 6 8 10 12 14 16 Figure 2: Percentage of novel sequences that can be generated versus length. The graph on the left in Figure 2 shows how the performance varies with sequence length for various networks with 16 hidden units. The points on this graph are the average of 5 runs; each run began with a randomization of all weights. The worst performance was produced by the SRN. The HRN gave the best performance: it was able to produce around 90% of all sequences up to length 12. Interestingly, a SRN (SRNZ in Figure 2) with frozen random recurrent weights from a suitable distribution performed significantly better than the unconstrained SRN. To some extent, the poor performance of the SRN was due to overtraining. This was verified by training a SRN on 48 sequences oflength 8 (8 times as much data). The performance improved greatly (SRN+ in Figure 2), but was still not as good that of the HRN trained on the lesser amount of data. This suggests that the extra parameters provided by the recurrent links in the SRN serve little useful purpose: the net does well with fixed random values for those parameters and a HRN does better without modifying any parameters in this operation. It appears that all that 40 Plate is required in the recurrent operation is some stable random map. The scaling performance of the HRN with respect to the number of hidden units is good. ThE" graph on the right in Figure 2 shows the performance of HRNs with R output units and varying numbers of hidden units (averages of 5 runs). As the number of hidden units increases from 4 to 64 the generative capaeity increases steadily. The sealing of sequence length with number of outputs (not shown) is also good: it is over 1 bit per hidden unit. This compares very will with the 0.1 bit per element aehieved by random vector eircular-c.onvolution (Section 2.3). The training times for both the HRNs and the SRNs were very short. Both required around 30 passes through the training data to train the output and recurrent weights. Finding a c.ode for test sequence of length 8 took the HRN an average of 14 passes. The SRN took an average of .57 passes (44 with frozen weights). The SRN trained on more data took mueh longer for the initial training (average 281 passes) but the c.ode searc.h was shorter (average 31 passes). 4 TRAJECTORIES IN CONTINUOUS SPACE HRNs ean also be used to generate trajectories through c.ontinuous spaee. Only two modifieations need be made: (a) ehange the function on the output units to sigmoid and add biases, and (b) use a fractional power for the key vector. A fractional power vector f can be generated by taking a random unity-power vector k and multiplying the phase angle of each frequency component by some fraction (\', i.e., f = kC/. The result is that fi is similar to fi when the difference between i and j is less than 1/ (\', and the similarity is greater for closer i and j. The output at the hidden layer will be similar at successive time steps. If desired, the speed at which the trajectory is traversed can be altered by changing (\'. target X target Y net Y Figure 3: Targets and outputs of a HRN trained to generate trajectories through c.ontinuous space. X and Yare plotted against time. A trajectory generating HRN with 16 hidden units and a key veetor k O.06 was trained to produce pen trajectories (100 steps) for 20 instances of handwritten digits (two of each). This is the same task that Simard and Le Cun [1992] used. The target trajectories and the output of the network for one instance are shown in Figure 3. 5 DISCUSSION One issue in processing sequential data with neural networks is how to present the inputs to the network. One approach has been to use a fixed window on the sequence, e.g., as in NETtaik [Sejnowski and Rosenberg, 1986]. A disadvantage of this is any fixed size of window may not be large enough in some situations. Another approach is to use a recurrent net to retain information about previous Holographic Recurrent Networks 41 inputs. A disadvantage of this is the difficulty that recurrent nets have in retaining information over many time steps. Generative networks offer another approach: use the codes that generate a sequence as input rather than the raw sequence. This would allow a fixed size network to take sequences of variable length as inputs (as long as they were finite), without having to use multiple input blocks or windows. The main attraction of circular convolution as an associative memory operator is its affordance of the representation of hierarchical structure. A hierarchical HRN, which takes advantage of this to represent sequences in chunks, has been built. However, it remains to be seen if it can be trained by gradient descent. 6 CONCLUSION The c.ircular convolution operation can be effectively incorporated into recurrent nets and the resulting nets (HRNs) can be easily trained using gradient descent to generate sequences and trajectories. HRNs appear to be more suited to this task than SRNs, though SRNs did surprisingly well. The relatively high generative capacity of HRNs shows that the capacity of circular convolution associative memory tplate, 1991a] can be greatly improved by adapting representations of vectors. References [Cleeremans et al., 1991] A. Cleeremans, D. Servan-Schreiber, and J. 1. McClelland. Graded state machines: The representation of temporal contingencies in simple recurrent networks. Machine Learning, 7(2/3):161-194, 1991. [Elman, 1991] J. Elman. Distributed representations, simple recurrent networks and grammatical structure. Machine Learning, 7(2/3):195-226, 1991. [Maskara and Noetzel, 1992] Arun Maskara and Andrew Noetzel. Forcing simple recurrent neural networks to encode context. In Proceedings of the 1992 Long Island Conference on Artificial Intelligence and Computer Graphics, 1992. [Plate, 1991a] T. A. Plate. Holographic Reduced Representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto, 1991. [Plate, 1991 b] T. A. Plate. Holographic Reduced Representations: Convolution algebra for compositional distributed representations. In Proceedings of the 12th International Joint Conference on Artificial Intelligence, pages 30-35, Sydney, Australia, 1991. [Rumelhart et al., 1986] D. E. Rumelhart, G. E. Hinton, and Williams R. J. Learning internal representations by error propagation. In Parallel distributed processing: Explorations in the microstructure of cognition, volume 1, chapter 8, pages 318-362. Bradford Books, Cambridge, MA, 1986. [Sejnowski and Rosenberg, 1986] T. J. Sejnowski and C. R. Rosenberg. NETtalk: A parallel network that learns to read aloud. Technical report 86-01, Department of Electrical Engineering and Computer Science, Johns Hopkins University, Baltimore, MD., 1986. [Simard and LeCun, 1992] P. Simard and Y. LeCun. Reverse TDNN: an architecture for trajectory generation. In J. M. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4 (NIPS*91), Denver, CO, 1992. Morgan Kaufman.
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Generalization Abilities of Cascade Network Architectures E. Littmann* Department of Information Science Bielefeld University D-4800 Bielefeld, FRG littmann@techfak.uni-bielefeld.de H. Ritter Department of Information Science Bielefeld University D-4800 Bielefeld, FRG helge@techfak.uni-bielefeld.de Abstract In [5], a new incremental cascade network architecture has been presented. This paper discusses the properties of such cascade networks and investigates their generalization abilities under the particular constraint of small data sets. The evaluation is done for cascade networks consisting of local linear maps using the MackeyGlass time series prediction task as a benchmark. Our results indicate that to bring the potential of large networks to bear on the problem of extracting information from small data sets without running the risk of overjitting, deeply cascaded network architectures are more favorable than shallow broad architectures that contain the same number of nodes. 1 Introduction For many real-world applications, a major constraint for the successful learning from examples is the limited number of examples available. Thus, methods are required, that can learn from small data sets. This constraint makes the problem of generalization particularly hard. If the number of adjustable parameters in a * to whom correspondence should be sent 188 Generalization Abilities of Cascade Network Architectures 189 network approaches the number of training examples, the problem of overfitting occurs and generalization becomes very poor. This severely limits the size of networks applicable to a learning task with a small data set. To achieve good generalization also in these cases, particular attention must be paid to a proper architecture chosen for the network. The better the architecture matches the structure of the problem at hand, the better is the chance to achieve good results even with small data sets and small numbers of units. In the present paper, we address this issue for the class of so called Cascade Network Architectures [5, 6] on the basis of an empirical approach, where we use the MackeyGlass time series prediction as a benchmark problem. In our experiments we want to exploit the potential of large networks to bear on the problem of extracting information from small data sets without running the risk of overfitting. Our results indicate that it is more favorable to use deeply cascaded network architectures than shallow broad architectures, provided the same number of nodes is used in both cases. The width of each individual layer is essentially determined by the size of the training data set. The cascade depth is then matched to the total number of nodes available. 2 Cascade Architecture So far, mainly architectures with few layers containing many units have been considered, while there has been very little research on narrow, but deeply cascaded networks. One of the few exceptions is the work of Fahlman [1], who proposed networks trained by the cascade-correlation algorithm. In his original approach, training is strictly feed-forward and the nonlinearity is achieved by incrementally adding percept ron units trained to maximize the covariance with the residual error. 2.1 Construction Algorithm In [5] we presented a new incremental cascade network architecture based 011 error minimization instead of covariance maximization. This leads to all architecture that differs significantly from Fahlman's proposal and allows an inversion of the construction process of the network. Thus, at each stage of the construction of the network all cascaded modules provide an approximation of the target function t(e), albeit corresponding to different states of convergence (Fig. 1). The algorithm starts with the training of a neural module with output yeo) to approximate a target function t(e), yielding (1) the superscript (0) indicating the cascade level. After an arbitrary number of training epochs, the weight vector w(O) becomes "frozen". Now we add the output y(O) of this module as a virtual input unit and train another neural module as new output 190 Littmann and Ritter Output Cascade laye (Output) Cascade laye (Output) Input { Bias Neural Module r2 Neural Module r 1 Neural Module Xn X2 X, 1 Figure 1: Cascade Network Architecture unit y(1) with (2) where x(1)(~) = {x(O)(~),y(O)(~)} denotes the extended input. This procedure can be iterated arbitrarily and generates a network sttucture as shown in Fig. 1. 2.2 Cascade Modules The details and advantages of this approach are discussed in [5, 6]. In particular, this architecture can be applied to any arbitrary nonlinear module. It does not rely on the availability of a procedure for error backpropagation. Therefore, it is also applicable to (and has been extensively tested with) pure feed-forward approaches like simple perceptrons [5] and vector quantization or "Local linear maps" ("LLM networks") [6, 7]. 2.3 Local Linear Maps LLM networks have been introduced earlier ((Fig. 2); for details, d. [11, 12]) and are related to the GRBF -approach [10] and the self-organizing maps [2, 3, 11]. They consist of N units r = 1, ... ,N, with an input weight vector w~in) E }RL, an output weight vector w~out) E }RM and a MxL-matrix Ar for each unit r. Generalization Abilities of Cascade Network Architectures 191 Output space Input space Figure 2: LLM Network Architecture The output y(net) of a single LLM-network for an input feature vector x E }RL is the "winner" node s determined by the minimality condition This leads to the learning steps for a training sample (x(o), yeo»): f (x(o) _ W(in») 1 8' (3) (4) (5) (6) (7) applied for T samples (x(o),y(o»),a = 1,2, ... T, and 0 < fi « 1, i = 1,2,3 denote learning step sizes. The additional term in (6), not given in [11, 121, leads to a better decoupling of the effects of (5) and (6,7). 3 Experiments In order to evaluate the generalization performance of this architecture, we consider the problem of time series prediction based on the Mackey-Glass differential equation, for which results of other networks already have been reported in the literature. 192 Littmann and Ritter 1.25 ] := 1 t 0.75 0.50 o 100 200 300 400 Time (t) Figure 3: Mackey-Glass function 3.1 Time Series Prediction Lapedes and Farber [4] introduced the prediction of chaotic time series as a benchmark problem. The data is based on the Mackey-Glass differential equation [8]: x(t) = -bx(t) + (ax(t - r))J(l + xlO(t - r)). (8) With the parameters a = 0.2, b = 0.1, and r = 17, this equation produces a chaotic time series with a strange attractor of fractal dimension d ~ 2.1 (Fig. 3). The input data is a vector x(t) = {x(t),x(t ~),x(t 2~),x(t 3~)}T. The learning task is defined to predict the value x(t + P). To facilitate comparison, we adopt the standard choice ~ = 6 and P = 85. Results with these parameters have been reported in [4, 9, 13]. The data was generated by integration with 30 steps per time unit. We performed different numbers of training epochs with samples randomly chosen from training sets consisting of 500 (5000 resp.) samples. The performance was measured on an independent test set of 5000 samples. All results are averages over ten runs. The error measure is the normalized root mean squ.are error (NRMSE), i.e. predicting the average value yields an error value of 1. 4 Results and Discussion The training of the single LLM networks was performed without extensive parameter tuning. If fine tuning for each cascade unit would be necessary, the training would be unattractively expensive. The first results were achieved with cascade networks consisting of LLM units after 30 training epochs per layer on a learning set of 500 samples. Figs. 4 and 5 represent the performance of such LLM cascade networks on the independent test set for different numbers of cascaded layers as a function of the number of nodes per layer Generalization Abilities of Cascade Network Architectures 193 NRMSE E3 1 layer B 2 layers B 3 layers B 4 layers E3 5 layers 0.5 '~----"'---"---"-'-- '--~"--'---r~"-- I -_··\·_····--1 0.4 ··.. ., .... ····... .·.,·. ····r .. 1 ····· 0.3 , . I ...... L ....• ~~-~~j ••••• ~.- _.~·····: i •• --. .. ,J.... . . ...... ... ~ • ••••••••••• ~ •••••••••••• <t •••••••••••• ; •.••••••••• J : : , ! ': ! O~------------------------------~· 0.2 0.1 .................................... . 10 20 30 40 50 60 70 80 90 100 # nodes per layer Figure 4: Iso-Layer-Dependence ----+--- ---0.5 0 •• D. l lO),8I"S Nodes 90 ID 100 Figure 5: Error Landscape ("iso-layer-curves"). The graphs indicate that there is an optimal number N~!~ of nodes for which the performance of the single layer network has a best value p!~:. Within the single layer architecture, additional nodes lead to a decrease of performance due to overfitting. This can only be avoided if the training set is enlarged, since N~!~ grows with the number of available training examples. However, Figs. 4 and 5 show that adding more units in the form of an additional, cascaded layer allows to increase performance significantly beyond p!~:. Similarly, the optimal performance of the resulting two-layer network cannot be improved beyond an optimal value p!;~ by arbitrarily increasing the number of nodes in the two-layer system. However, adding a third cascaded layer again allows to make use of more nodes to improve performance further, although this time the relative gain is smaller than for the first cascade step. The same situation repeats for larger numbers of cascaded layers. This suggests that the cascade architecture is very suitable to exploit the computational capabilities of large numbers of nodes for the task of building networks that generalize well from small data sets without running into the problem of overfitting when many nodes are used. A second way of comparing the benefits of shallow and broad versus narrow and deep architectures is to compare the performance achieveable by distributing a fixed number N of nodes over different numbers L of cascaded layers. Fig. 6 shows the result for the same benchmark problem as in Fig. 4, each graph belonging to one of the values N = 40,60,120,240 nodes and representing the NRMSE for distributing the N nodes among L layers of N / L nodes each t, L ranging from 1 to 10 layers ("iso-nodes-curves" ). 1 rounding to the nearest integral, whenever N / L is nonintegral. 194 Littmann and Ritter NRMSE E3 240 E3 120 B 60 B 40 Nodes 0.5 ........................ \ .............. ................. ..... ... ...................... ; .. .... : 0.4 0.3 ,l,:.' •••• ! •••• ., .: :. ... ,'..: .... . ' .. ~...... : : · ' ~.~ .. _~ . ___ ~~~:r.'= 0.1 .......... t.~ .~.~.~~ .~ .. ~ . . ··········1············: ···········, : j 0.2 o~----~--------------------~ 1 2 3 4 5 6 7 8 9 10 L = # cascaded layers Figure 6: Iso-Nodes-Dependence The results show that ----t------D.5 -1:------_ G •• G.3 0.2 '.1 I 2 J • , l"' .... Figure 7: Nodes-Layer-Dependence (i) the optimal number of layers increases monotonously with and is roughly proportional to the number of nodes to be used. (ii) if for each number of nodes the optimal number of layers is used, performance increases monotonously with the number of available nodes, and thus, as a consequence of (i), with the number of cascaded layers. These results are not restricted to small data sets only. The application of the cascade algorithm is also useful if larger training sets are available. Fig. 7 represents the performance of LLM cascade networks on the test set after 300 training epochs overall on a learning set consisting of 5000 samples. As could be expected, there is still no sign of overfitting, even using LLM networks with 100 nodes per layer. But regardless of the size of the single LLM unit, network performance is improved by the cascade process at least in a zone involving a total of some 300 nodes in the whole cascade. 5 Conclusions Summarizing, we find that Cascade Network Architectures allow to use the benefits of large numbers of nodes even for small training data sets, and still bypass the problem of overfitting. To achieve this, the "width" of each layer must be matched to the size of the training set. The "depth" of the cascade then is determined by the total number of nodes available. Generalization Abilities of Cascade Network Architectures 195 Acknowledgements This work was supported by the German Ministry of Research and Technology (BMFT), Grant No. ITN9104AO. Any responsibility for the contents of this publication is with the authors. References [1] Fahlman, S.E., and Lebiere, C. (1989), "The Cascade-Correlation Learning Architecture", in Advances in Neural Information Processing Systems II, ed. D.S. Touretzky, pp. 524-532. [2] Kohonen, T. (1984), Self-Organization and Associative Memory, Springer Series in Information Sciences 8, Springer, Heidelberg. [3] Kohonen, T. (1990), "The Self-Organizing Map", in Proc. IEEE 78, pp. 14641480. [4] Lapedes, A., and Farber, R. (1987), "Nonlinear signal processing using neural networks; Prediction and system modeling", TR LA-UR-87-2662 [5] Littmann, E., Ritter, H. (1992), "Cascade Network Architectures", in Proc. Intern. Joint Conference On Neural Networks, pp. II/398-404, Baltimore. [6] Littmann, E., Ritter, H. (1992), "Cascade LLM Networks", in Artificial Neural Networks II, eds. I. Aleksander, J. Taylor, pp. 253-257, Elsevier Science Publishers (North Holland). [7] Littmann, E., Meyering, A., Ritter, H. (1992), "Cascaded and Parallel Neural Network Architectures for Machine Vision A Case Study", in Proc. 14. DAGM-Symposium 1992, Dresden, ed. S. Fuchs, pp. 81-87, Springer, Heidelberg. [8] Mackey, M., and Glass, 1. (1977), "Oscillations and chaos in physiological control systems", in Science, pp. 287-289. [9] Moody, J., Darken, C. (1988). "Learning with Localized Receptive Fields", in Proc. of the 1988 Connectionist Models Summer School, Pittsburg, pp. 133143, Morgan Kaufman Publishers, San Mateo, CA. [10] Poggio, T., Edelman, S. (1990), "A network that learns to recognize threedimensional objects", in Nature 343, pp. 263-266. [11] Ritter, H. (1991), "Learning with the Self-organizing Map", in Artificial Neural Networks 1, eds. T. Kohonen, K. Makisara, O. Simula, J. Kangas, pp. 357-364, Elsevier Science Publishers (North-Holland). [12] Ritter, H., Martinetz, T., Schulten, K. (1992). Neural Computation and Selforganizing Maps, Addison-Wesley, Reading, MA. [13] Walter, J., Ritter, H., Schulten, K. (1990). "Non-linear prediction with selforganizing maps", in Proc. Intern. Joint Conference On Neural Networks, San Diego, Vol.1, pp. 587-592.
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Explanation-Based Neural Network Learning for Robot Control Tom M. Mitchell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 E-mail: mitchell@cs.cmu.edu Sebastian B. Thrun University of Bonn Institut fUr Infonnatik III ROmerstr. 164, D-5300 Bonn, Germany thrnn@uran.informatik.uni-bonn.de Abstract How can artificial neural nets generalize better from fewer examples? In order to generalize successfully, neural network learning methods typically require large training data sets. We introduce a neural network learning method that generalizes rationally from many fewer data points, relying instead on prior knowledge encoded in previously learned neural networks. For example, in robot control learning tasks reported here, previously learned networks that model the effects of robot actions are used to guide subsequent learning of robot control functions. For each observed training example of the target function (e.g. the robot control policy), the learner explains the observed example in terms of its prior knowledge, then analyzes this explanation to infer additional information about the shape, or slope, of the target function. This shape knowledge is used to bias generalization when learning the target function. Results are presented applying this approach to a simulated robot task based on reinforcement learning. 1 Introduction Neural network learning methods generalize from observed training data to new cases based on an inductive bias that is similar to smoothly interpolating between observed training points. Theoretical results [Valiant, 1984], [Baum and Haussler, 1989] on learnability, as well as practical experience, show that such purely inductive methods require significantly larger training data sets to learn functions of increasing complexity. This paper introduces explanation-based neural network learning (EBNN), a method that generalizes successfully from fewer training examples, relying instead on prior knowledge encoded in previously learned neural networks. EBNN is a neural network analogue to symbolic explanation-based learning methods (EBL) [DeJong and Mooney, 1986], [Mitchell et al., 19861 Symbolic EBL methods generalize based upon pre-specified domain knowledge represented by collections of symbolic rules. 287 288 Mitchell and Thrun For example. in the task of learning general rules for robot control EBL can use prior know ledge about the effects of robot actions to analytically generalize from specific training examples of successful control actions. This is achieved by a. observing a sequence of states and actions leading to some goal. b. explaining (i.e .• post-facto predicting) the outcome of this sequence using the domain theory. then c. analyzing this explanation in order to determine which features of the initial state are relevant to achieving the goal of the sequence. and which are nol In previous approaches to EBL. the initial domain knowledge has been represented symbolically. typically by propositional rules or hom clauses. and has typically been assumed to be complete and correct 2 EBNN: Integrating inductive and analytical learning EBNN extends explanation-based learning to cover situations in which prior knowledge (also called the domain theory) is approximate and is itself1earned from scratch. In EBNN, this domain theory is represented by real-valued neural networks. By using neural network representations. it becomes possible to learn the domain theory using training algorithms such as the Backpropagation algorithm [Rumelhart et al., 19861 In the robot domains addressed in this paper. such domain theory networks correspond to action models. i.e., networks that model the effect of actions on the state of the world M:s x a -+ Sf (here a denotes an action, s a state. and Sf the successor state). This domain theory is used by EBNN to bias the learning of the 'robot control function. Because the action models may be only approximately correct. we require that EBNN be robust with respect to severe errors in the domain theory. The remainder of this section describes the EBNN learning algorithm. Assume that the robot agent's action space is discrete. and that its domain knowledge is represented by a collection of pre-trained action models Mi:S -+ Sf. one for each discrete action i. The learning task of the robot is to learn a policy for action selection that maximizes the reward, denoted by R. which defines the task. More specifically. the agent has to learn an evaluation function Q(s, a). which measures the cumulativefuture expected reward when action a is executed at state s. Once learned. the function Q(s, a) allows the agent to select actions that maximize the reward R (greedy policy). Hence learning control reduces to learning the evaluation function Q.1 How can the agent use its previously learned action models to focus its learning of Q? To illustrate. consider the episode shown in Figure 1. The EBNN learning algorithm for learning the target function Q consists of two components, an inductive learning component and an analytical learning component. 2.1 The inductive component of EBNN The observed episode is used by the agent to construct training examples, denoted by Q, for the evaluation function Q: Q(sl,ad:= R Q(s2,a2):= R Q(s3,a3):= R Q could for example be realized by a monolithic neural network, or by a collection of networks trained with the Backpropagation training procedure. As observed training episodes are accumulated, Q will become increasingly accurate. Such pure inductive learning typIThis approach to learning a policy is adopted from recent research on reinforcenuml learning [Barto et al., 1991]. Explanation-Based Neural Network Learning for Robot Control 289 _-----~~ reward: R (goal state) Figure 1: Episode: Starting with the initial state SI. the action sequence aI, az, a3 was observed to produce the final reward R. The domain knowledge represented by neural network action models is used to post-facto predict and analyze each step of the observed episode. ically requires large amounts of training data (which will be costly in the case of robot learning). 2.2 The analytical component or EBNN In EBNN, the agent exploits its domain lrnowledge to extract additional shape lrnowledge about the target function Q. to speed convexgence and reduce the number of training examples required. This shape lrnowledge. represented by the estimated slope of the target function Q. is then used to guide the generalization process. More specifically. EBNN combines the above inductive learning component with an analytical learning component that performs the following three steps for each observed training episode: 1. Explain: Post-facto predict the obsexved episode (states and final reward), using the action models Mi (c.f. Fig. 1). Note that thexe may be a deviation between predicted and observed states. since the domain lrnowledge is only approximately correct. 2. Analyze: Analyze the explanation to estimate the slope of the target function for each observed state-action pair (81:, a1:) (k = 1..3). i.e .• extract the derivative of the final reward R with respect to the features of the states 81:. according to the action models Mi. For instance. consider the explanation of the episode shown in Fig. 1. The domain theory networks Mi represent differentiable functions. Therefore it is possible to extract the derivative of the final reward R with respect to the preceding state 83. denoted by "V '3R. Using the chain rule of differentiation. the derivatives of the final reward R with respect to all states 81: can be extracted. These derivatives "V,,, R describe the dependence of the final reward upon features of the previous states. They provide the target slopes. denoted by "V,,, Q. for the target function Q: ( 8M43 (S3) "V '3 Q 83, a3) "V '3 R 0 83 oM4,(83) OM42 (82) OS3 082 "V'2R OM43(83) 8M42 (82) 8M41 (81) 883 082 881 3. Learn: Update the learned target function to better fit both the target values and target slopes. Fig. 2 illustrates training information extracted by both the inductive (values) and the analytical (slopes) components ofEBNN. Assume that the "true" Q-function 290 Mitchell and Thrun Figure2: Fitting slopes: Let/bea target function for which tbreeexampies (Xl, I(xt)}. (X2, I(X2)). and (X3, 1 (X3)) are known. Based on these points the learner might generate the hypothesis g. If the slopes are also known. the learner can do much better: h. is shown in Fig. 2a, and that three training instances at Xl, X2 and X3 are given. When only values are used for learning, i.e., as in standard inductive learning, the learner might conclude the hypothesis g depicted in Fig. 2b. If the slopes are known as well, the learner can better estimate the target function (Fig. 2c). From this example it is clear that the analysis in EBNN may reduce the need for training data, provided that the estimated slopes extracted from the explanations are sufficiently accurate. In EBNN, the function Q is learned by a real-valued function approximator that fits both the target values and target slopes. If this approximator is a neural network, an extended version of the Backpropagation algorithm can be employed to fit these slope constraints as well, as originally shown by [Simard et al., 19921 Their algorithm "Tangent Prop" extends the Backpropagation error function by a second term measuring the mean square error of the slopes. Gradient descent in slope space is then combined with Backpropagation to minimize both error functions. In the experiments reported here, however, we used an instance-based function approximation technique described in Sect. 3. 2.3 Accommodating imperfect domain theories Notice that the slopes extracted from explanations will be only approximately correct, since they are derived from the approximate action models Mi. If this domain knowledge is weak, the slopes can be arbitrarily poor, which may mislead generalization. EBNN reduces this undesired effect by estimating the accuracy of the extracted slopes and weighting the analytical component of learning by these estimated slope accuracies. Generally speaking, the accuracy of slopes is estimated by the prediction accuracy of the explanation (this heuristic has been named LOB *). More specifically, each time the domain theory is used to post-facto predict a state sk+1, its prediction st~icted may deviate from the observed state sr+ied• Hence the I-step prediction accuracy at state Sk, denoted by Cl (i), is defined as 1 minus the normalized prediction error: ( .) 1 _ II st~cted - skb+red II Cl Z := max..prediction...error For a given episode we define the n-step accuracy cn(i) as the product of the I-step accuracies in the next n steps. The n-step accuracy, which measures the accuracy of the deri ved slopes n steps away from the end of the episode, posseses three desireable properties: a. It is I if the learned domain theory is perfectly correct, b. it decreases monotonically as the length of the chain of inferences increases, and c. it is bounded below by O. The n-step accuracy is used to determine the ratio with which the analytical and inductive components Explanation-Based Neural Network Learning for Robot Control 291 are weighted when learning the target concept. If an observation is n steps away from the end of the episode. the analytically derived training information (slopes) is weighted by the n-step accuracy times the weight of the inductive component (values). Although the experimental results reported in section 3 are promising. the generality of this approach is an open question. due to the heuristic nature of the assumption LOB *. 2.4 EBNN and Reinforcement Learning To make EBNN applicable to robot learning, we extend it here to a more sophisticated scheme for learning the evaluation function Q. namely Watkins' Q-Learning [Watkins, 1989] combined with Sutton's temporal difference methods [Sutton, 19881 The reason for doing so is the problem 0/ suboptimal action choices in robot learning: Robots must explore their environment. i.e., they must select non-optimal actions. Such non-optimal actions can have a negative impact on the final reward of an episode which results in both underestimating target values and misleading slope estimates. Watkins' Q-Learning [Watkins, 1989] permits non-optimal actions during the course of learning Q. In his algorithm targets for Q are constructed recursively, based on the maximum possible Q-value at the next state:2 ...... { R if k is the final step and R final reward Q(Sk, ak) = , m~ Q(Sk+l, a) otherwise a acuon Here , (O~,~I) is a discount/actor that discounts reward over time, which is commonly used for minimizing the number of actions. Sutton's TD(A) [Sutton, 1988] can be used to combine both Watkins' Q-Learning and the non-recursive Q-estimation scheme underlying the previous section. Here the parameter A (0 ~ A ~ 1) determines the ratio between recursive and non-recursive components: ...... { R if k final step ) Q(sk,ak) = (I-A),max a Q(sk+l,a) + A,Q(sk+l,ak+d otherwise (1 Eq. (1) describes the extended inductive component of the EBNN learning algoriLhm. The extension of the analytical component in EBNN is straightforward. Slopes are extracted via the derivative of Eq. (1), which is computed via the derivative of both the models !IIi and the derivative of Q. if k last step otherwise 3 Experimental results EBNN has been evaluated in a simulated robot navigation domain. The world and the action space are depicted in Fig. 3a&b. The learning task is to find a Q function, for which the greedy policy navigates the agent to its goal location (circle) from arbitrary starting locations, while avoiding collisions with the walls or the obstacle (square). States are 210 order to simplify the notation. we assume that reward is only received at the end of the episode, and is also modeled by the action models. The extension to more general cases is straightforward. 292 Mitchell and Thrun (a) (b) 0'*' robot (c) error 0.2 0.15 0.1 0.05 0 number of training examples Figure 3: a. The simulated robot world. b. Actions. c. The squared generalization error of the domain theory networks decreases monotonically as the amount of training data increases. These nine alternative domain theories were used in the experiments. described by the local view of the agent. in terms of distances and angles to the center of the goal and to the center of the obstacle. Note that the world is deterministic in these experiments, and that there is no sensor noise. We applied Watkins' Q-Learning and TD(~) as described in the previous section with A=0.7 and a discount factor ;=0.8. Each of the five actions was modeled by a separate neural network (12 hidden units) and each had a separate Q evaluation function. The latter functions were represented by a instance-based local approximation technique. In a nutshell, this technique memorizes all training instances and their slopes explicitly, and fits a local quadratic model over the [=3 nearest neighbors to the query point, fitting both target values and target slopes. We found empirically that this technique outperformed Tangent Prop in the domain at hand.3 We also applied an experience replay technique proposed by Lin [Lin, 19911 in order to optimally exploit the information given by the observed training episodes. Fig. 4 shows average performance curves for EBNN using nine different domain theories (action models) trained to different accuracies, with (Fig. 4a) and without (Fig. 4b) taking the n-step accuracy of the slopes into account. Fig. 4a shows the main result. It shows clearly that (1) EBNN outperfonns purely inductive learning, (2) more accurate domain theories yield better performance than less accurate theories, and (3) EBNN learning degrades gracefully as the accuracy of the domain theory decreases, eventually matching the performance of purely inductive learning. In the limit, as the size of the training data set grows, we expect all methods to converge to the same asymptotic performance. 4 Conclusion Explanation-based neural network learning. compared to purely inductive learning, generalizes more accurately from less training data. It replaces the need for large training data sets by relying instead on a previously learned domain theory. represented by neural networks. In this paper, EBNN has been described and evaluated in terms of robot learning tasks. Because the learned action models Mi are independent of the particular control task (reward function), this knowledge acquired during one task transfers directly to other tasks. 3Note that in a second experiment not reported here, we applied EBNN using neural network representation for Q and Tangent Prop successfully in a real robot domain. (a) Prob(success) 1 0 . 8 0.6 0 . 4 0.2 (b) Prob(success) 1 0.8 0 . 6 0.4 0.2 0 20 Explanation-Based Neural Network Learning for Robot Control 293 " ...... .. ~. 40 60 " ~ 80 n\lI1lber of 100 epiSOdes ,"",_10 nU!Jlberof 100 episodes Figure 4: How does domain knowledge improve generalization? a. Averaged results for EBNN domain theories of differing accuracies, pre-trained with from 5 to 8 192 training examples for each action model network. In contrast, the bold grey line reflects the learning curve for pure inductive learning, i.e., Q-Leaming and TD(A). b. Same experiments, but without weighting the analytical component of EBNN by its accuracy, illustrating the importance of the WB* heuristic. All curves are averaged over 3 runs and are also locally window-averaged. The perfonnance (vertical axis) is measured on an independent test set of starting positions. EBNN differs from other approaches to knowledge-based neural network learning, such as Shavlik/fowell's KBANNs [Shavlik and Towell, 1989]. in that the domain knowledge and the target function are strictly separated, and that both are learned from scratch. A major difference from other model-based approaches to robot learning, such as Sutton's DYNA architecture [Sutton, 1990] or Jordan!Rumelhart's distal teacher method [Jordan and Rumelhart, 1990], is the ability of EBNN to operate across the spectrum of strong to weak domain theories (using LOB*). EBNN has been found to degrade gracefully as the accuracy of the domain theory decreases. We have demonstrated the ability of EBNN to transfer knowledge among robot learning tasks. However, there are several open questions which will drive future research, the most significant of which are: a. Can EBNN be extended to real-valued, parameterized 294 Mitchell and Thrun action spaces? So far we assume discrete actions. b. Can EBNN be extended to handle first-order predicate logic. which is common in symbolic approaches to EBL? c. How will EBNN perform in highly stochastic domains? d. Can knowledge other than slopes (such as higher order derivatives) be extracted via explanations? e. Is it feasible to automatically partition/modularize the domain theory as well as the target function, as this is the case with symbolic EBL methods? More research on these issues is warranted. Acknowledgments We thank Ryusuke Masuoka, Long-Ji Lin. the CMU Robot Learning GrouP. Jude Shavlik, and Mike Jordan for invaluable discussions and suggestions. This research was sponsored in part by the Avionics Lab, Wright Research and Development Center. Aeronautical Systems Division (APSC). U. S. Air Force. Wright-Patterson AFB. OH 45433-6543 under Contract F33615-90-C-1465.Arpa Order No. 7597 and by a grant from Siemens Corporation. References [Barto et aI., 1991] Andy G. Barto, StevenJ. Bradtke, and Satinder P. Singh. Real-time learning and control using asynchronous dynamic programming. Technical Report COINS 91-57, Department of Computer Science, University of Massachusetts, MA, August 1991. [Baum and Haussler, 1989] Eric Baum and David Haussler. What size net gives valid generalization? Neural Computation, 1(1):151-160,1989. [Dejong and Mooney, 1986] Gerald DeJong and Raymond Mooney. Explanation-based learning: An alternative view. Machine Learning, 1(2):145-176, 1986. [Jordan and Rumelhart, 1990] Michael I. Jordan and David E. Rumelhart. Forward models: Supervised learning with a distal teacher. submitted to Cognitive Science, 1990. [Lin, 19911 Long-Ji Lin. Programming robots using reinforcement learning and teaching. In Proceedings of AAAl-91 , Menlo Park, CA, July 1991. AAAI Press I The MIT Press. [Mitchell et al., 1986] Tom M. Mitchell, Rich Keller, and Smadar Kedar-Cabelli. Explanation-based generalization: A unifying view. Machine Learning, 1(1):47-80, 1986. [Pratt, 1993] Lori Y. Pratt. Discriminability-based transfer between neural networks. Same volume. [Rumelhart et aI., 1986] David E. Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland, editors, Parallel Distributed Processing. Vol. I + II. MIT Press, 1986. [Shavlik and Towell, 1989] Jude W. Shavlik and G.G. Towell. An approach to combining explanation-based and neural learning algorithms. Connection Science, 1(3):231-253, 1989. [Simard et al., 1992] Patrice Simard, Bernard Victorri, Yann LeCun, and John Denker. Tangent prop - a formalism for specifying selected invariances in an adaptive network. In J. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 895-903, San Mateo, CA, 1992. Morgan Kaufmann. [SUlton, 1988] Richard S. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3. 1988. [Sutton, 1990] Richard S. Sutton. Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In Proceedings of the Seventh International Conference on Machine Learning, June 1990, pages 216-224,1990. [Valiant, 1984] Leslie G. Valiant A theory of the learnable. Communications of the ACM, 27:11341142, 1984. [Watkins,1989] Chris J. C. H. Watkins. Learning from Delayed Rewards. PhD thesis, King's College, Cambridge, England,1989.
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Computing with Almost Optimal Size Neural Networks Kai-Yeung Siu Dept. of Electrical & Compo Engineering University of California, Irvine Irvine, CA 92717 V wani Roychowdhury School of Electrical Engineering Purdue University West Lafayette, IN 47907 Thomas Kailath Information Systems Laboratory Stanford University Stanford, CA 94305 Abstract Artificial neural networks are comprised of an interconnected collection of certain nonlinear devices; examples of commonly used devices include linear threshold elements, sigmoidal elements and radial-basis elements. We employ results from harmonic analysis and the theory of rational approximation to obtain almost tight lower bounds on the size (i.e. number of elements) of neural networks. The class of neural networks to which our techniques can be applied is quite general; it includes any feedforward network in which each element can be piecewise approximated by a low degree rational function. For example, we prove that any depth-( d + 1) network of sigmoidal units or linear threshold elements computing the parity function of n variables must have O(dnl/d-£) size, for any fixed i > O. In addition, we prove that this lower bound is almost tight by showing that the parity function can be computed with O(dnl/d) sigmoidal units or linear threshold elements in a depth-(d + 1) network. These almost tight bounds are the first known complexity results on the size of neural networks with depth more than two. Our lower bound techniques yield a unified approach to the complexity analysis of various models of neural networks with feedforward structures. Moreover, our results indicate that in the context of computing highly oscillating symmetric Boolean func19 20 Siu, Roychowdhury, and Kailath tions, networks of continuous-output units such as sigmoidal elements do not offer significant reduction in size compared with networks of linear threshold elements of binary outputs. 1 Introduction Recently, artificial neural networks have found wide applications in many areas that require solutions to nonlinear problems. One reason for such success is the existence of good "learning" or "training" algorithms such as Backpropagation [13] that provide solutions to many problems for which traditional attacks have failed. At a more fundamental level, the computational power of neural networks comes from the fact that each basic processing element computes a nonlinear function of its inputs. Networks of these nonlinear elements can yield solutions to highly complex and nonlinear problems. On the other hand, because of the nonlinear features, it is very difficult to study the fundamental limitations and capabilities of neural networks. Undoubtedly, any significant progress in the applications of neural networks must require a deeper understanding of their computational properties. We employ classical tools such as harmonic analysis and rational approximation to derive new results on the computational complexity of neural networks. The class of neural networks to which our techniques can be applied is quite large; it includes feedforward networks of sigmoidal elements, linear threshold elements, and more generally, elements that can be piecewise approximated by low degree rational functions. 1.1 Background, Related Work and Definitions A widely accepted model of neural networks is the feedforward multilayer network in which the basic processing element is a sigmoidal element. A sigmoidal element computes a function I(X) of its input variables X = (Xl, ... , xn) such that 2 1- e-F(X) I(X) = u(F(X» = 1 + e-F(X) - 1 = 1 + e-F(X) where s F(X) = L: Wi • Xi + WOo i=l The real valued coefficients Wi are commonly referred to as the weights of the sigmoidal function. The case that is of most interest to us is when the inputs are binary, i.e., X E {l, _l}n. We shall refer to this model as sigmoidal network. Another common feed forward multilayer model is one in which each basic processing unit computes a binary linear threshold function sgn(F(X», where F(X) is the same as above, and sgn(F(X» = { _~ if F(X) ~ 0 if F(X) < 0 This model is often called the threshold circuit in the literature and recently has been studied intensively in the field of computer science. Computing with Almost Optimal Size Neural Networks 21 The size of a network/circuit is the number of elements. The depth of a network/circuit is the longest path from any input gate to the output gates. We can arrange the gates in layers so that all gates in the same layer compute concurrently. (A single element can be considered as a one-layer network.) Each layer costs a unit delay in the computation. The depth of the network (which is the number of layers) can therefore be interpreted as the time for (parallel) computation. It has been established that threshold circuit is a very powerful model of computation. Many functions of common interest such as multiplication, division and sorting can be computed in polynomial-size threshold circuits of small constant depth [19, 18, 21]. While many upper bound results for threshold circuits are known in the literature, lower bound results have only been established for restricted cases of threshold circuits. Most of the existing lower bound techniques [10, 17, 16] apply only to depth-2 threshold circuits. In [16], novel techniques which utilized analytical tools from the theory of rational approximation were developed to obtain lower bounds on the size of depth-2 threshold circuits that compute the parity function. In [20], we generalized the methods of rational approximation and our earlier techniques based on harmonic analysis to obtain the first known almost tight lower bounds on the size of threshold circuits with depth more than two. In this paper, the techniques are further generalized to yield almost tight lower bounds on the size of a more general class of neural networks in which each element computes a continuous function. The presentation of this paper will be divided into two parts. In the first part, we shall focus on results concerning threshold circuits. In the second part, the lower bound results presented in the first part are generalized and shown to be valid even when the elements of the networks can assume continuous output values. The class of networks for which such techniques can be applied include networks of sigmoidal elements and radial basis elements. Due to space limitations, we shall only state some of the important results; further results and detailed proofs will appear in an extended paper. Before we present our main results, we shall give formal definitions of the neural network models and introduce some of the Boolean functions, which will be used to explore the computational power of the various networks. To present our results in a coherent fashion, we define throughout this paper a Boolean function as f : {I, _l}n -+ {I, -I}, instead of using the usual {O, I} notation. Definition 1 A threshold circuit is a Boolean circuit in which every gate computes a linear threshold function with an additional property: the weights are integers all bounded by a polynomial in n. 0 Remark 1 The assumption that the weights in the threshold circuits are integers bounded by a polynomial is common in the literature. In fact, the best known lower bound result on depth-2 threshold circuit [10] does not apply to the case where exponentially large weights are allowed. On the other hand, such assumption does not pose any restriction as far as constant-depth and polynomial-size is concerned. In other words, the class of constant-depth polynomial-size threshold circuits (TeO) remains the same when the weights are allowed to be arbitrary. This result was implicit in [4] and was improved in [18] by showing that any depth-d threshold circuit 22 Siu, Roychowdhury, and Kailath with arbitrary weights can be simulated by a depth-(2d + 1) threshold circuit of polynomially bounded weights at the expense of a polynomial increase in size. More recently, it has been shown that any polynomial-size depth-d threshold circuit with arbitrary weights can be simulated by a polynomial-size depth-(2d + 1) threshold circuit. 0 In addition to Boolean circuits, we shall also be interested in the computation of Boolean functions by networks of continuous-valued elements. To formalize this notion, we adopt the following definitions [12]: Definition 2 Let 'Y : R R. A 'Y element with weights WI, ... , Wm E Rand threshold t is defined to be an element that computes the function 'Y(E~1 WiX; -t) where (Xl. ... , xm) is the input. A 'Y-network is a feedforward network of'Y elements with an additional property: the weights Wi are all bounded by a polynomial in n. o For example, when 'Y is the sigmoidal function O'(x), then we have a sigmoidal network, a common model of neural network. In fact, a threshold circuit can also be viewed as a special case of'Y network where 'Y is the sgn function. Definition 3 A 'Y-network C is said to compute a Boolean function f : {I,-l}n {I, -I} with separation (. > 0 if there is some tc E R such that for any input X = (Xl, ... , Xm) to the network C, the output element of C outputs a value C(X) with the following property: If f(X) = 1, then C(X) ~ tc + £. If f(X) = -1, then C(X) ~ tc £. 0 Remark 2 As pointed out in [12], computing with 'Y networks without separation at the output element is less interesting because an infinitesimal change in the output of any 'Y element may change the output bit. In this paper, we shall be mainly interested in computations on 'Y networks Cn with separation at least O(n-k) for some fixed k > o. This together with the assumption of polynomially bounded weights makes the complexity class of constant-depth polynomial-size 'Y networks quite robust and more interesting to study from a theoretical point of view (see [12]). 0 Definition 4 The PARITY function of X = (x}, X2, .. . , xn) E {I, _l}n is defined to be -1 if the number of -1 in the variables x I, ... , Xn is odd and + 1 otherwise. Note that this function can be represented as the product n~=l Xi. 0 Definition 5 The Complete Quadratic (CQ) function [3] is defined to be the following: CQ(X) = (Xl" X2) EEl (Xl" X3) EEl •.• EEl (Xn-l " xn) i.e. CQ(X) is the sum modulo 2 of all AND's between the (~) pairs of distinct variables. Note that it is also a symmetric function. 0 2 Results for Threshold Circuits Fo. the lower bound results on threshold circuits, a central idea of our proof is the use of a result from the theory of rational approximation which states the following Computing with Almost Optimal Size Neural Networks 23 [9]: the function sgn(x) can be approximated with an error of O(e-ck/log(l/€») by a rational function of degree k for 0 < f < Ixl < 1. (In [16], they apply an equivalent result [15] that gives an approximation to the function Ixl instead of sgn(x).) This result allows us to approximate several layers of threshold gates by a rational function oflow (i.e. logarithmic) degree when the size of the circuit is small. Then by upper bounding the degree of the rational function that approximates the PARITY function, we give a lower bound on the size of the circuit. We also give similar lower bound on the Complete Quadratic (CQ) function using the same degree argument. By generalizing the 'telescoping' techniques in [14], we show an almost matching upper bound on the size of the circuits computing the PARITY and the CQ functions. We also examine circuits in which additional gates other than the threshold gates are allowed and generalize the lower bound results in this model. For this purpose, we introduce tools from harmonic analysis of Boolean functions [11, 3, 18, 17]. We define the class of functions called SP such that every function in SP can be closely approximated by a sparse polynomial for all inputs. For example, it can be shown that [18] the class SP contains functions AND, OR, COMPARISON and ADDITION, and more generally, functions that have polynomially bounded spectral norms. The main results on threshold circuits can be summarized by the following theorems. First we present an explicit construction for implementing PARITY. This construction applies to any 'periodic' symmetric function, such as the CQ function. Theorem 1 For every d < logn, there exists a depth-(d + 1) threshold circuit with O(dn1/ d ) gates that computes the PARITY function. 0 We next show that any depth-(d + 1) threshold circuit computing the PARITY function or the CQ function must have size O(dnl/d-£) for any fixed f > o. This result also holds for any function that has strong degree O(n). Theorem 2 Any depth-(d + 1) threshold circuit computing the PARITY (CQ) function must have size O(dnl/d / log:! n). 0 We also consider threshold circuits that approximate the PARITY and the CQ functions when we have random inputs which are uniformly distributed. We derive almost tight upper and lower bounds on the size of the approximating threshold circuits. We next consider threshold circuits with additional gates and prove the following result. Theorem 3 Suppose in addition to threshold gates, we have polynomially many gates E SP in the first layer of a depth-2 threshold circuit that computes the CQ function. Then the number of threshold gates required in the circuit is O(n/ log2 n). o This result can be extended to higher depth circuits when additional gates that have low degree polynomial approximations are allowed. Remark 3 Recently Beigel [2], using techniques similar to ours and the fact 24 Siu, Roychowdhury, and Kailath that the PARITY function cannot be computed in polynomial-size constant-depth circuits of AND, OR gates [7], has shown that any constant-depth threshold circuit • 0(1) With (2n ) AND, OR gates but only o(log n) threshold gates cannot compute the PARITY function of n variables. 0 3 Results for ,-Networks In the second part of the paper, we consider the computational power of networks of continuous-output elements. A celebrated result in this area was obtained by Cybenko [5]. It was shown in [5] that any continuous function over a compact domain can be closely approximated by sigmoidal networks with two layers. More recently, Barron [1] has significantly strengthened this result by showing that a wide class of functions can be approximated with mean squared error of O( n -1 ) by tw<rlayer sigmoidal networks of only n elements. Here we are interested in networks of continuous-output elements computing Boolean functions instead of continuous functions. See Section 1.1 for a precise definition of computation of Boolean functions by a "Y-network. While quite a few techniques have been developed for deriving lower bound results on the complexity of threshold circuits, an understanding of the power and the limitation of networks of continuous elements such as sigmoidal networks, especially as compared to threshold circuits, have not been explored. For example, we would like to answer questions such as: how much added computational power does one gain by using sigmoidal elements or other continuous elements to compute Boolean functions? Can the size of the network be reduced by using sigmoidal elements instead of threshold elements? It was shown in [12] when the depth of the network is restricted to be two, then there is a Boolean function of n variables that can be computed in a depth-2 sigmoidal network with a fixed number of elements, but requires a depth-2 threshold circuit with size that increases at least logarithmic in n. In other words, in the restricted case of depth-2 network, one can reduce the size of the network at least a logarithmic factor by using continuous elements such as the sigmoidal elements instead of threshold elements with binary output values. This result has been recently improved in [6], where it is shown that there exists an explicit function that can be computed using only a constant number of sigmoidal gates, and that any threshold circuit (irrespective of the depth) computing it must have size !l(log n). These results motivate the following question: Can we characterize a class of functions for which the threshold circuits computing the functions have sizes at most a logarithmic factor larger than the sizes of the sigmoidal networks computing them? Because of the monotonicity of the sigmoidal functions, we do not expect that there is substantial gain in the computational power over the threshold elements for computing the class of highly oscillating functions. It is natural to extend our techniques to sigmoidal networks by approximating sigmoidal functions with rational functions. We derive a key lemma that yields a single low degree rational approximation to any function that can be piecewise approximated by low degree rational functions. Computing with Almost Optimal Size Neural Networks 25 Lemma 1 Let f be a continuous function over A = [a, b]. Let Al = [a, c] and A2 = [c,b], a < c < b. Denote II 9 II~,= sUP~e~ Ig(x)l. Suppose there are rational • I functIOns rl and r2 such that II / - rj lI~i ~ { where { > O. Then for each l> 0 and 6 > 0, there is a rational function r such that b - a II / II~ deg r ~ 2 deg rl + 2 deg r2 + Gllog(e + -6-) log(e + l) (1) where w(fj c5)~ is the modulus of continuity of / over A, G1 is a constant. 0 The above lemma is applied to show that both sigmoidal functions and radial basis functions can be closely approximated by low degree rational functions. In fact the above lemma can be generalized to show that if a continuous function can be piecewise approximated by low degree rational functions over k = 10gO(I) n consecutive intervals, then it can be approximated by a single low degree rational function over the union of these intervals. These generalized approximation results enable us to show that many of our lower bound results on threshold circuits can be carried over to sigmoidal networks. Prior to our work, there was no nontrivial lower bound on the size of sigmoidal networks with depth more than two. In fact, we can generalize our results to neural networks whose elements can be piecewise approximated by low degree rational functions. We show in this paper that for symmetric Boolean functions of large strong degree (e.g. the parity function), any depth-d network whose elements can be piecewise approximated by low degree rational functions requires almost the same size as a depth-d threshold circuit computing the function. In particular, if it is the class of polynomially bounded functions that are piecewise continuous and can be piecewise approximated with low degree rational functions, then we prove the following theorem. Theorem 4 Let W be any depth-Cd + 1) neural network in which each element Vj computes a function Ji (Li WiXi) where Ji E it and Li Iwi! ~ nOel) for each element. If the network W computes the PARITY function of n variables with separation 6, where 0 < 6 = n(n- k ) for some k > 0, then for any fixed { > 0, W must have size n(dn 1/ d-(). 0 References [1] A. Barron. Universal Approximation Bounds for Superpositions of a Sigmoidal Function. IEEE Transactions on In/ormation Theory, to appear. [2] R. Beigel. Polylog( n) Majority or O(log log n) Symmetric Gates are Equivalent to One. ACM Symposium on Theory of Computing (STOC), 1992. [3] J. Bruck. Harmonic Analysis of Polynomial Threshold Functions. SIAM Journal on Discrete Mathematics, pages 168-177, May 1990. 26 Siu, Roychowdhury, and Kailath [4] A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. Siam J. Comput., 13:423-439, 1984. [5] G. Cybenko. Approximations by superpositions of a sigmoidal function. Math. Control, Signals, Systems, vol. 2, pages 303-314, 1989. [6] B. Dasgupta and G. Schnitger. Efficient Approximation with Neural Networks: A Comparison of Gate Functions. In 5th Annual Conference on Neural Information Processing Systems - Natural and Synthetic (NIPS'92), 1992. [7] M. Furst, J. B. Saxe, and M. Sipser. Parity, Circuits and the Polynomial-Time Hierarchy. IEEE Symp. Found. Compo Sci., 22:260-270, 1981. [8] M. Goldmann, J. Hastad, and A. Razborov. Majority Gates vs. General Weighted Threshold Gates. Seventh Annual Conference on Structure in Complexity Theory, 1992. [9] A. A. Goncar. On the rapidity of rational approximation of continuous functions with characteristic singularities. Mat. Sbornik, 2(4):561-568, 1967. [10] A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, and G. Turan. Threshold circuits of bounded depth. IEEE Symp. Found. Compo Sci., 28:99-110, 1987. [11] R. J. Lechner. Harmonic analysis of switching functions. In A. Mukhopadhyay, editor, Recent Development in Switching Theory. Academic Press, 1971. [12] W. Maass, G. Schnitger, and E. Sontag. On the computational power of sigmoid versus boolean threshold circuits. IEEE Symp. Found. Compo Sci., October 1991. [13] J. L. McClelland D. E. Rumelhart and the PDP Research Group. Parallel Distributed Processing: Explorations in the Microstructure of Cognition, vol. 1. MIT Press, 1986. [14] R. Minnick. Linear-Input Logic. IEEE Trans. on Electronic Computers, EC 10, 1961. [15] D. J. Newman. Rational Approximation to Ixl. Michigan Math. Journal, 11:11-14, 1964. [16] R. Paturi and M. Saks. On Threshold Circuits for Parity. IEEE Symp. Found. Compo Sci., October 1990. [17] V. P. Roychowdhury, K. Y. Siu, A. Orlitsky, and T. Kailath. A Geometric Approach to Threshold Circuit Complexity. Workshop on Computational Learning Theory (Colt'91), pp. 97-111, 1991. [18] K. Y. Siu and J. Bruck. On the Power of Threshold Circuits with Small Weights. SIAM J. Discrete Math, pp. 423-435, August 1991. [19] K. Y. Siu and J. Bruck. Neural Computation of Arithmetic Functions. Proceedings of the IEEE, Special Issue on Neural Networks, pp. 1669-1675, October 1990. [20] K. Y. Siu, V. P. Roychowdhury, and T. Kailath. Computing with Almost Optimal Size Threshold Circuits. IEEE International Symposium on Information Theory, Budapest, Hungary, June 1991. [21] K.-Y. Siu, J. Bruck, T. Kailath, and T. Hofmeister. Depth-Efficient Neural Networks for Division and Related Problems. to appear in IEEE Trans. Information Theory, 1993.
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Some Solutions to the Missing Feature Problem in Vision Subutai Ahmad Siemens AG, Central Research and Development ZFE ST SN61, Otto-Hahn Ring 6 8000 Miinchen 83, Gennany. ahmad@icsi.berkeley.edu Volker Tresp Siemens AG, Central Research and Development ZFE ST SN41, Otto-Hahn Ring 6 8000 Miinchen 83, Gennany. tresp@inf21.zfe.siemens.de Abstract In visual processing the ability to deal with missing and noisy information is crucial. Occlusions and unreliable feature detectors often lead to situations where little or no direct information about features is available. However the available information is usually sufficient to highly constrain the outputs. We discuss Bayesian techniques for extracting class probabilities given partial data. The optimal solution involves integrating over the missing dimensions weighted by the local probability densities. We show how to obtain closed-form approximations to the Bayesian solution using Gaussian basis function networks. The framework extends naturally to the case of noisy features. Simulations on a complex task (3D hand gesture recognition) validate the theory. When both integration and weighting by input densities are used, performance decreases gracefully with the number of missing or noisy features. Performance is substantially degraded if either step is omitted. 1 INTRODUCTION The ability to deal with missing or noisy features is vital in vision. One is often faced with situations in which the full set of image features is not computable. In fact, in 3D object recognition, it is highly unlikely that all features will be available. This can be due to selfocclusion, occlusion from other objects, shadows, etc. To date the issue of missing features has not been dealt with in neural networks in a systematic way. Instead the usual practice is to substitute a single value for the missing feature (e.g. 0, the mean value of the feature, or a pre-computed value) and use the network's output on that feature vector. 393 394 Ahmad and Tresp y y 5 4 3 Yo ............ Yo x x (a) (b) Figure 1. The images show two possible situations for a 6-class classification problem. (Dark shading denotes high-probability regions.) If the value of feature x is unknown, the correct solution depends both on the classification boundaries along the missing dimension and on the distribution of exemplars. When the features are known to be noisy, the usual practice is to just use the measured noisy features directly. The point of this paper is to show that these approaches are not optimal and that it is possible to do much better. A simple example serves to illustrate why one needs to be careful in dealing with missing features. Consider the situation depicted in Figure 1 (a). It shows a 2 -d feature space with 6 possible classes. Assume a network has already been trained to correctly classify these regions. During classification of a novel exemplar. only feature y has been measured, as Yo; the value of feature x is unknown. For each class Ci , we would like to compute p(Cily). Since nothing is known about x, the classifier should assign equal probability to classes 1, 2, and 3, and zero probability to classes 4,5, and 6. Note that substituting any single value will always produce the wrong result. For example, if the mean value of x is substituted, the classifier would assign a probability near 1 for class 2. To obtain the correct posterior probability, it is necessary to integrate the network output over all values of x. But there is one other fact to consider: the probability distribution over x may be highly constrained by the known value of feature y. With a distribution as in Figure 1 (b) the classifier should assign class 1 the highest probability. Thus it is necessary to integrate over x along the line Y=Yo weighted by the joint distribution p(x,y). 2 MISSING FEATURES We first show how the intituitive arguments outlined above for missing inputs can be formalized using Bayes rule. Let x represent a complete feature vector. We assume the classifier outputs good estimates of p (Cil x) (most reasonable classifiers do - see (Richard & Lippmann, 1991». In a given instance, x can be split up into xc' the vector of known (certain) features, and xu. the unknown features. When features are missing the task is to estimate p (Cil xc) . Computing marginal probabilities we get: Some Solutions to the Missing Feature Problem in Vision 395 Jp (Cil Xc' xu) p (xc' xu) dxu p (xc) (1) Note that p (Cil XC' xu) is approximated by the network output and that in order to use (1) effectively we need estimates of the joint probabilities of the inputs. 3 NOISY FEATURES The missing feature scenario can be extended to deal with noisy inputs. (Missing features are simply noisy features in the limiting case of complete noise.) Let Xc be the vector of features measured with complete certainty, Xu the vector of measured, uncertain features, and xtu the true values of the features in Xu. p (xul XtU) denotes our knowledge of the noise (i.e. the probability of measuring the (uncertain) value Xu given that the true value is xtu). We assume that this is independent of Xc and Ci • i.e. that p (xul xlU, Xc' Ci) = p (xul xlU ) • (Of course the value of xlU is dependent on Xc and Cj .) We want to compute p (Cil Xc' xu) . This can be expressed as: ",,. Jp (xc' xu' xtu, Ci) dxtu p(Cjlxc,xu) = ~ ~ p (xc' xu) (2) Given the independence assumption, this becomes: Jp (Cjl x ,XtU) p (xc' xtu) p (Xul XtU) dxtu p(Cilxc'xu) = ____ c __________ _ Jp (xc' XtU) p (xul XtU) dxtu (3) As before, p (C il Xc> X tu) is given by the classifier. (3) is almost the same as (1) except that the integral is also weighted by the noise model. Note that in the case of complete uncertainty about the features (i.e. the noise is uniform over the entire range of the features), the equations reduce to the miSSing feature case. 4 GAUSSIAN BASIS FUNCTION NETWORKS The above discussion shows how to optimally deal with missing and noisy inputs in a Bayesian sense. We now show how these equations can be approximated using networks of Gaussian basis functions (GBF nets). Let us consider GBF networks where the Gaussians have diagonal covariance matrices (Nowlan, 1990). Such networks have proven to be useful in a number of real-world applications (e.g. Roscheisen et al, 1992). Each hidden unit is characterized by a mean vector ~j and by aj, a vector representing the diagonal of the covariance matrix. The network output is: 4: wijbj (x) Yj (x) = -..::1 ___ _ 396 Ahmad and Tresp with bj (x) = 1tj n (x;a.j , crJ) = d1tj d exp [-r (xi -':;/l I 20'·· 2 II~ JI (21t) O'kj (4) k wji is the weight from the j'th basis unit to the i'th output unit, Ttj is the probability of choosing unit j, and d is the dimensionality of x. 4.1 GBF NETWORKS AND MISSING FEATURES Under certain training regimes sur.h as Gaussian mixture modeling, EM or "soft clustering" (Duda & Hart, 1973; Dempster et ai, 1977; Nowlan, 1990) or an approximation as in (Moody & Darken, 1988) the hidden units adapt to represent local probability densities. In particular Yi (x) "" p (Cil x) and p (x) "" Ijbj (x) . This is a major advantage of this architectur and can be exploited to obtain closed form solutions to (1) and (3). Substituting into (3) we get: J (L, wijbj (xc' XtU» p (xul XtU) dXtu p (C il xc' xu) == ---"j----------J (Lbj (xc' xlU) ) p (xul xtu ) dxlu J (5) For the case of missing features equation (5) can be computed directly. As noted before, equation (1) is simply (3) with p (xui x,u) uniform. Since the infinite integral along each dimension of a multivariate normal density is equal to one we get: '" w .. b· (xc) 4 JI J p(Cilxc)""J", 3. ~bj(xc) j (6) (Here bj (xc) denotes the same function as in except that it is only evaluated over the known dimensions given by xc.) Equation (6) is appealing since it gives us a simple closed form solution. Intuitively, the solution is nothing more than projecting the Gaussians onto the dimensions which are available and evaluating the resulting network. As the number of training patterns increases, (6) will approach the optimal Bayes solution. 4.2 GBF NETWORKS AND NOISY FEATURES With noisy features the situation is a little more complicated and the solution depends on the form of the noise. If the noise is known to be uniform in some region [a, b] then equation (5) becomes: '" w iJb. (xc) II [N (bjal .. , 0'2.) - N (ai;~'" O'~.)] ~ J. IJ IJ IJ IJ p(C'lx,x)== J lEV ICU L 3. II . 2 . 2 bJ.(xc) [N(b"'~ ' :7O' '' ) -N(a,. ,~ .. ,(J .. )] . . IJ IJ IJ IJ J I E V (7) Some Solutions to the Missing Feature Problem in Vision 397 Here ~jj and a~ select the i'th component of the j'th mean and variance vectors. U ranges over the noisy feature indices. Good closed form approximations to the normal distribution function N (x; 1.1., ( 2) are available (Press et al, 1986) so (7) is efficiently computable. With zero-mean Gaussian noise with variance O'~, we can also write down a closed form solution. In this case we have to integrate a product of two Gaussians and end up with: 4, wjjb') (xc' xu) J ~.. ..>. .... 2..>.2 .>. = ~----- with b'j (xc' xu) = n (xu;J..Lju' 0u + 0ju) b/xc)' Lb') (xc' xu) j 5 BACKPROPAGATION NETWORKS With a large training set, the outputs of a sufficiently large network trained with backpropagation converges to the optimal Bayes a posteriori estimates (Richard & Lippmann, 1992). If B j (x) is the output of the i'th output unit when presented with input x, B j (x) "" p (Cj / x) . Unfortunately, access to the input distribution is not available with backpropagation. Without prior knowledge it is reasonable to assume a uniform input distribution, in which case the right hand side of (3) simplifies to: .>. Jp (Cil xc' xtu)p (xul xtu) dxtu p (C -I x ) == -------I C Jp (xul xtu) dxtu (8) The integral can be approximated using standard Monte Carlo techniques. With uniform .>. noise in the interval [a, b] , this becomes (ignoring normalizing constants): " b p(Cjlxc) == JBj(Xc.Xtu)dXtu (9) With missing features the integral in (9) is computed over the entire range of each feature. 6 AN EXAMPLE TASK: 3D HAND GESTURE RECOGNITION A simple realistic example serves to illustrate the utility of the above techniques. We consider the task of recognizing a set of hand gestures from single 2D images independent of 3D orientation (Figure 2). As input, each classifier is given the 2D polar coordinates of the five fingertip positions relative to the 2D center of mass of the hand (so the input space is lO-dimensional). Each classifier is trained on a training set of 4368 examples (624 poses for each gesture) and tested on a similar independent test set. The task forms a good benchmark for testing performance with missing and uncertain inputs. The classification task itself is non-trivial. The classifier must learn to deal with hands (which are complex non-rigid objects) and with perspective projection (which is non-linear and non-invertible). In fact it is impossible to obtain a perfect score since in certain poses some of the gestures are indistinguishable (e.g. when the hand is pointing directly at the screen). Moreover, the task is characteristic of real vision problems. The 398 Ahmad and Tresp "five" "four" "three" "two" "one" "thumbs_up" "pointing" Figure 2. Examples of the 7 gestures used to train the classifier. A 3D computer model of the hand is used to generate images of the hand in various poses. For each training example, we choose a 3D orientation, compute the 3D positions of the fingertips and project them onto 2D. For this task we assume that the correspondence between image and model features are known, and that during training all feature values are always available. position of each finger is highly (but not completely) constrained by the others resulting in a very non-uniform input distribution. Finally it is often easy to see what the classifier should output if features are uncertain. For example suppose the real gesture is "fi ve" but for some reason the features from the thumb are not reliably computed. In this case the gestures "four" and "five" should both get a positive probability whereas the rest should get zero. In many such cases only a single class should get the highest score, e.g. if the features for the little finger are uncertain the correct class is still "five". We tried three classifiers on this task: standard sigmoidal networks trained with backpropagation (BP), and two types of gaussian networks as described in . In the first (GaussRBF), the gaussians were radial and the centers were determined using k-means clustering as in (Moody & Darken, 1988). 0'2 was set to twice the average distance of each point to its nearest gaussian (all gaussians had the same width). After clustering, 1t . was set to J L k [ n (Xk~ ~< ~J~2 ] . The output weights were then determined using LMS gradient L j n(xk,llj,O'J descent. In the second (Gauss-G), each gaussian had a unique diagonal covariance matrix. The centers and variances were determined using gradient descent on all the parameters (Roscheisen et ai, 1992). Note that with this type of training, even though gaussian hidden units are used, there is no guarantee that the distribution information will be preserved. All classifiers were able to achieve a reasonable performance level. BP with 60 hidden units managed to score 95.3% and 93.3% on the training and test sets, respectively. GaussG with 28 hidden units scored 94% and 92%. Gauss-RBF scored 97.7% and 91.4% and required 2000 units to achieve it. (Larger numbers of hidden units led to overfitting.) For comparison, nearest neighbor achieves a score of 82.4% on the test set. 6.1 PERFORMANCE WITH MISSING FEATURES We tested the performance of each network in the presence of missing features. For backpropagation we used a numerical approximation to equation (9). For both gaussian basis function networks we used equation (6). To test the networks we randomly picked samples from the test set and deleted random features. We calculated a performance score as the percentage of samples where the correct class was ranked as one of the top two classes. Figure 3 displays the results. For comparison we also tested each classifier by substituting the mean value of each missing feature and using the normal update equation. As predicted by the theory the performance of Gauss-RBF using (6) was consistently better than the others. The fact that BP and Gauss-G performed poorly indicates that the distribution of the features must be taken into account. The fact that using the mean value is Some Solutions to the Missing Feature Problem in Vision 399 100 90 80 Performanc 70 60 o Performance with milling features ~ · · : ·~ .. "" " 'O '. . '. . Cl&11lIII-RBF -.-.. .. v~o. .... Ga~; ~ ~~"":"'O Gauls-G-MEAN ·0· . '. ." ...• BP-MEAN +. 4: RBF MEAN ·e·1 234 5 No. of milling feat urel Figure 3. The performance of various classifiers when dealing with missing features. Each data point denotes an average over tOOO random samples from an independent test set. For each sample. random features were considered missing. Each graph plots the percentage of samples where the correct class was one of the top two classes. 6 insufficient indicates that the integration step must also be carried out. Perhaps most encouraging is the result that even with 50% of the features missing. Gauss-RBF ranks the correct class among the top two 90% of the time. This clearly shows that a significant amount of information can be extracted even with a large number of missing features. 6.2 PERFORMANCE WITH NOISY FEATURES We also tested the performance of each network in the presence of noisy features. We randomly picked samples from the test set and added uniform noise to random features. The noise interval was calculated as [x . - 2cr., x · + 2cr.J where XI· is the feature value and cr. is I I I I I the standard deviation of that feature over the training set. For BP we used equation (9) and for the GBF networks we used equation (7). Figure 3 displays the results. For comparison we also tested each classifier by substituting the noisy value of each noisy feature and using the normal update equation (RBF-N, BP-N, and Gauss-GN). As with missing features, the performance of Gauss-RBF was significantly better than the others when a large number of features were noisy. 7 DISCUSSION The results demonstrate the advantages of estimating the input distribution and integrating over the missing dimensions, at least on this task. They also show that good classification performance alone does not guarantee good missing feature performance. (Both BP and Gauss-G performed better than Gauss-RBF on the test set.) To get the best of both worlds one could use a hybrid technique utilizing separate density estimators and classifiers although this would probably require equations (1) and (3) to be numerically integrated. One way to improve the performance of BP and Gauss-G might be to use a training set that contained missing features. Given the unusual distributions that arise in vision, in order to guarantee accuracy such a training set should include every possible combination 400 Ahmad and Tresp Performance with noilY featurel Performance: 80 I-Gaull-RBF ___ 70 Io Ga1181-G 0BP -+-Gauu-GN ·0· . BP-N + .. RBF-N .•.. I 1 I I I l l 234 No. of noisy featurel I 1 5 Figure 4. As in Figure 3 except that the performance with noisy features is plotted. 6 of missing features. In addition, for each such combination, enough patterns must be included to accurately estimate the posterior density. In general this type of training is intractable since the number of combinations is exponential in the number of features. Note that if the input distribution is available (as in Gauss-RBF), then such a training scenario is unnecessary. Acknowledgements We thank D. Goryn, C. Maggioni, S. Omohundro, A. Stokke, and R. Schuster for helpful discussions, and especially B. Wirtz for providing the computer hand model. V.T. is supported in part by a grant from the Bundesministerium fUr Forschung und Technologie. References A.P. Dempster, N.M. Laird, and D.H. Rubin. (1977) Maximum-likelihood from incomplete data via the EM algorithm.f. Royal Statistical Soc. Ser. B, 39:1-38. R.O. Duda and P.E. Hart. (1973) Pattern Classification and Scene Analysis. John Wiley & Sons, New York. J. Moody and C. Darken. (1988) Learning with localized receptive fields. In: D. Touretzky, G. Hinton, T. Sejnowski, editors, Proceedings of the 1988 Connectionist Models Summer School, Morgan Kaufmann, CA. S. Nowlan. (1990) Maximum Likelihood Competitive Learning. In: Advances in Neurallnformation Processing Systems 4, pages 574-582. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Veuerling. (1986) Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK. M. D. Richard and R.P. Lippmann. (1991) Neural Network Classifiers Estimate Bayesian a posteriori Probabilities, Neural Computation, 3:461-483. M. Roscheisen, R. Hofman, and V. Tresp. (1992) Neural Control for Rolling Mills: Incorporating Domain Theories to Overcome Data DefiCiency. In: Advances in Neural Information Processing Systems 4, pages 659-666.
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A Note on Learning Vector Quantization Virginia R. de Sa Department of Computer Science University of Rochester Rochester, NY 14627 Dana H. Ballard Department of Computer Science University of Rochester Rochester, NY 14627 Abstract Vector Quantization is useful for data compression. Competitive Learning which minimizes reconstruction error is an appropriate algorithm for vector quantization of unlabelled data. Vector quantization of labelled data for classification has a different objective, to minimize the number of misclassifications, and a different algorithm is appropriate. We show that a variant of Kohonen's LVQ2.1 algorithm can be seen as a multiclass extension of an algorithm which in a restricted 2 class case can be proven to converge to the Bayes optimal classification boundary. We compare the performance of the LVQ2.1 algorithm to that of a modified version having a decreasing window and normalized step size, on a ten class vowel classification problem. 1 Introduction Vector quantization is a form of data compression that represents data vectors by a smaller set of codebook vectors. Each data vector is then represented by its nearest codebook vector. The goal of vector quantization is to represent the data with the fewest code book vectors while losing as little information as possible. Vector quantization of unlabelled data seeks to minimize the reconstruction error. This can be accomplished with Competitive learning[Grossberg, 1976; Kohonen, 1982], an iterative learning algorithm for vector quantization that has been shown to perform gradient descent on the following energy function [Kohonen, 1991] J /Ix - ws·(x) /l2p(x)dx. 220 A Note on Learning Vector Quantization 221 where p(x) is the probability distribution of the input patterns and Ws are the reference or codebook vectors and s*(x) is defined by IIx - WSO(x) I I ~ /Ix - will (for alIt). This minimizes the square reconstruction error of unlabelled data and may work reasonably well for classification tasks if the patterns in the different classes are segregated. In many classification tasks, however, the different member patterns may not be segregated into separate clusters for each class. In these cases it is more important that members ofthe same class be represented by the same codebook vector than that the reconstruction error is minimized. To do this, the quantizer can m&ke use of the labelled data to encourage appropriate quantization. 2 Previous approaches to Supervised Vector Quantization The first use of labelled data (or a teaching signal) with Competitive Learning by Rumelhart and Zipser [Rumelhart and Zipser, 1986] can be thought of as assigning a class to each codebook vector and only allowing patterns from the appropriate class to influence each reference vector. This simple approach is far from optimal though as it fails to take into account interactions between the classes. Kohonen addressed this in his LVQ( 1) algorithm[Kohonen, 1986]. He argues that the reference vectors resulting from LVQ( 1) tend to approximate for a particular class r, P(xICr)P(Cr) ~#rP(xICs)P(Cs). where P( Cj) is the a priori probability of Class i and P(xICj) is the conditional density of Class i. This approach is also not optimal for classification, as it addresses optimal places to put the codebook vectors instead of optimal placement of the borders of the vector quantizer which arise from the Voronoi tessellation induced by the codebook vectors. 1 3 Minimizing Misclassifications In classification tasks the goal is to minimize the numbers of misclassifications of the resultant quantizer. That is we want to minimize: (1) where, P(Classj) is the a priori probability of Classj and P(xIClassj) is the conditional density of Classi and D.Rj is the decision region for class j (which in this case is all x such that I~ - wkll < I~ - wjll (for all i) and Wk is a codebook vector for class j). Consider a One-Dimensional problem of two classes and two codebook vectors wI and w2 defining a class boundary b = (wI + w2)/2 as shown in Figure 1. In this case Equation 1 reduces to: 1 Kohonen [1986] showed this by showing that the use of a "weighted" Voronoi tessellation (where the relative distances of the borders from the reference vectors was changed) worked better. However no principled way to calculate the relative weights was given and the application to real data used the unweighted tessellation. 222 de Sa and Ballard P(CIass i)P(xlClass i) w2 b* b wI % Figure 1: Codebook vectors Wl and'W2 define a border b. The optimal place for the border is at b* where P(Cl)P(xICt} = P(C2)P(xIC2). The extra misclassification errors incurred by placing the border at b is shown by the shaded region. (2) The derivative of Equation 2 with respect to b is That is, the minimum number of misclassifications occurs at b* where P(ClaSS1)P(b*IClasSl) = P(Class2)P(b*IClass2). If f(x) = (Classl)P(xIClassl) - P(Class2)P(xIClass2) was a regression function then we could use stochastic approximation [Robbins and Monro, 1951] to estimate b* iteratively as ben + 1) = ben) + a(n)Z" where Z" is a sample of the random variable Z whose expected value is P(Classl)P(b(n)IClasst) - P(Class2)P(b(n)IClass2» and lim a(n) = 0 ,,-+co l:ia(n) = 00 l:ia2(n) < 00 However, we do not have immediate access to an appropriate random variable Z but can express P( C lassl )P(xIClassl)-P( Class2)P(xIClass2) as the limit of a sequence of regression functions using the Parzen Window technique. In the Parzen window technique, probability density functions are estimated as the sum of appropriately normalized pulses centered at A Note on Learning Vector Quantization 223 the observed values. More formally, we can estimate P(xIClassi) as [Sklansky and Wassel, 1981] Il All 1~ Pi (x) = - L...J'¥II(x-Xj,cll ) n . )=1 where Xj is the sample data point at time j, and 'II II(X- z, c(n)) is a Parzen window function centred at Z with width parameter c(n) that satisfies the following conditions '¥II(X - z, c(n» ~ 0, Vx, Z J~ '¥II(X- Z, c(n»dx = 1 lim '¥;(x- z, c(n))dx = 0 1111-+- n __ lim '¥1I(x-z,c(n» = c5(x-z) II-+We can estimate f(x) = P(Class1)P(xIClasst) - P(Class2)P(xIClass2) as A 1 Il rex) = - LS(Xj)'¥II(x-Xj,c(n» n . 1 J= where S(Xj) is + 1 if Xj is from Class1 and -1 if Xj is from Class2. Then lim j"(X) = P(Class1)P(xIClass1) - P(Class2)P(xIClass2) II-+and lim E[S(X)'¥ix - X, c(n)] = P(Class1)P(xIClassd - P(Class2)P(xIClass2) II-+Wassel and Sklansky [1972] have extended the stochastic approximation method of Robbins and Monro [1951] to find the zero of a function that is the limit of a sequence of regression functions and show rigourously that for the above case (where the distribution of Class1 is to the left of that of Class2 and there is only one crossing point) the stochastic approximation procedure ben + 1) = ben) + a(n)ZII(xlI , Class(n), ben), c(n» (3) using Z _ { 2c(n)'¥(XII - ben), c(n» for XII E Classl II -2c(n)'¥(XII - ben), c(n» for XII E Class2 converges to the Bayes optimal border with probability one where '¥(x - b, c) is a Parzen window function. The following standard conditions for stochastic approximation convergence are needed in their proof a(n), c(n) > 0, lim c(n) = 0 lim a(n) = 0, II-+II-+1:ia(n)c(n) = 00, 224 de Sa and Ballard as well as a condition that for rectangular Parzen functions reduces to a requirement that P( Classl )P(xIClassl) - P( C lass2)P(xlClass2) be strictly positive to the left of b* and strictly negative to the right of b* (for full details of the proof and conditions see [Wassel and Sklansky, 1972]). The above argument has only addressed the motion of the border. But b is defined as b = (wI + w2)/2, thus we can move the codebook vectors according to dE/dwl = dEldw2 = .5dEldb. We could now write Equation 3 as (X" - wj(n - 1» wj(n + 1) = wj(n) + a2(n) IX" _ wj(n _ 1)1 if X" lies in window of width 2c(n) centred at ben), otherwise Wi(n + 1) = wi(n). where we have used rectangular Parzen window functions and X" is from Classj. This holds if Classl is to the right or left of Class2 as long as Wl and W2 are relatively ordered appropriatel y. Expanding the problem to more dimensions, and more classes with more codebook vectors per class, complicates the analysis as a change in two codebook vectors to better adjust their border affects more than just the border between the two codebook vectors. However ignoring these effects for a first order approximation suggests the following update procedure: * * (X" - wren - 1» Wi (n) = Wi (n - 1) + a(n) IIX" _ wren _ 1)11 *. (X" - w;(n - 1» Wj (n) = Wj (n - 1) - a(n) IIX" _ wj(n _ 1)11 where a(n) obeys the constraints above, X" is from Classj, and w;, wj are the two nearest codebook vectors, one each from class i and j U * i) and x" lies within c(n) of the border between them. (No changes are made if all the above conditions are not true). As above this algorithm assumes that the initial positions of the codebook vectors are such that they will not have to cross during the algorithm. The above algorithm is similar to Kohonen's LVQ2.1 algorithm (which is performed after appropriate initialization of the codebook vectors) except for the normalization of the step size, the decreasing size of the window width c(n) and constraints on the learning rate a. A Note on Learning Vector Quantization 225 4 Simulations Motivated by the theory above, we decided to modify Kohonen's LVQ2.1 algorithm to add normalization of the step size and a decreasing window. In order to allow closer comparison with LVQ2.1, all other parts of the algorithm were kept the same. Thus a decreased linearly. We used a linear decrease on the window size and defined it as in LVQ2.1 for easier parameter matching. For a window size of w all input vectors satisfying d;/dj> g:~ where di is the distance to the closest codebook vector and dj is the distance to the next closest codebook vector, fall into the window between those two vectors (Note however, that updates only occur if the two closest codebook vectors belong to different classes). The data used is a version of the Peterson and Barney vowel formant data 2. The dataset consists of the first and second formants for ten vowels in a/hVdj context from 75 speakers (32 males, 28 females, 15 children) who repeated each vowel twice 3. As we were not testing generalization , the training set was used as the test set. ..... u Q) ... ... o U ..... r:: Q) u ... Q) '" 75.------.------.------.-----..-----~ 70 65 ~. ~A ... -:~~.::ra.·-..::: -fA ;,oz:~,,; .. ~ ... ,. ~.~--;:::.-..--....... !\ . .. ; ! \. \ . . ". \ i \ I"" \ .... ' \. .. \ \'" \. \ \ .~ \\ \. \,: \ ~ \ \ ,., " \ \ \, \ •• I ~ \ alpha-0.002 -+alpha-0.030 -t--. alpha-0.080 'B'" alpha-O .150 alpha-0.500 ... \ it: \ \ t \ \ \ \ 60~-----L~----~--~-L--~~~----~ o 0.2 0.4 0.6 0.8 window size Figure 2: The effect of different window sizes on the accuracy for different values of initial a. We ran three sets of experiments varying the number of codebook vectors and the number of pattern presentations. For the first set of experiments there were 20 codebook vectors and the algorithms ran for 40000 steps. Figure 2 shows the effect of varying the window size for different initial learning rates a( 1) in the LVQ2.1 algorithm. The values plotted are averaged over three runs (The order of presentation of patterns is different for the different runs). The sensitivity of the algorithm to the window size as mentioned in [Kohonen, 1990] is evident. In general we found that as the learning rate is increased the peak accuracy is improved at the expense of the accuracy for other window widths. After a certain value 20 btained from Steven Nowlan 33 speakers were missing one vowel and the raw data was linearly transfonned to have zero mean and fall within the range [-3,3] in both components 226 de Sa and Ballard ..... u Qj I-< I-< 0 U ..... r:: Qj u I-< Qj '" 85~----~----~------r------r----~ 80 75 70 orig/20/40000 ~ mod/20/40000 -+orig/20/4000 ·B··· ... _._.-lI'-·-·-···-·- · lIi~T2!t)lrotT(J·"''''':': · ._.;:::~:::::: ... ---·-- ·~~~O..£.4.jlD.oJI._.~=-. ._~~ .. - . mod/100/40000 -•.• .. ; ., ~;.' -,. II'" ----~----+---------------------~ ! ~=~.~'Il ........ " .. ~--- ~ ~-El'.'••••••••••••••••••••••••• 65~----~----~------~----~~--~ o 0.1 0.2 0.3 0.4 0.5 window size Figure 3: The performance of LVQ2.1 with and without the modifications (normalized step size and decreasing window) for 3 different conditions. The legend gives in order [the alg type/ the number of codebook vectors/ the number of pattern presentations] the accuracy declines for further increases in learning rate. Figure 3 shows the improvement achieved with normalization and a linearly decreasing window size for three sets of experiments : (20 code book vectors/40000 pattern presentations), (20 code book vectors/4000 pattern presentations) and (100 code book vectors/40000 pattern presentations). For the decreasing window algorithm, the x-axis represents the window size in the middle of the run. As above, the values plotted were averaged over three runs. The values of a(l) were the same within each algorithm over all three conditions. A graph using the best a found for each condition separately is almost identical. The graph shows that the modifications provide a modest but consistent improvement in accuracy across the conditions. In summary the preliminary experiments indicate that a decreasing window and normalized step size can be worthwhile additions to the LVQ2.1 algorithm and further experiments on the generalization properties of the algorithm and with other data sets may be warranted. For these tests we used a linear decrease of the window size and learning rate to allow for easier comparison with the LVQ2.1 algorithm. Further modifications on the algorithm that experiment with different functions (that obey the theoretical constraints) for the learning rate and window size decrease may result in even better performance. 5 Summary We have shown that Kohonen's LVQ2.1 algorithm can be considered as a variant on a generalization of an algorithm which is optimal for a IDimensional/2 codebook vector problem. We added a decreasing window and normalized step size, suggested from the one dimensional algorithm. to the LVQ2.1 algorithm and found a small but consistent improvement in accuracy. A Note on Learning Vector Quantization 227 Acknowledgements We would like to thank Steven Nowlan for his many helpful suggestions on an earlier draft and for making the vowel formant data available to us. We are also grateful to Leonidas Kontothanassis for his help in coding and discussion. This work was supported by a grant from the Human Frontier Science Program and a Canadian NSERC 1967 Science and Engineering Scholarship to the first author who also received A NIPS travel grant to attend the conference. References [Grossberg, 1976] Stephen Grossberg, "Adaptive Pattern Classification and Universal Recoding: I. Parallel Development and Coding of Neural Feature Detectors," Biological Cybernetics, 23:121-134,1976. [Kohonen,1982] Teuvo Kohonen, "Self-Organized Formation of Topologically Correct Feature Maps," Biological Cybernetics, 43:59--69, 1982. [Kohonen,1986] Teuvo Kohonen, "Learning Vector Quantization for Pattern Recognition," Technical Report TKK-F-A601, Helsinki University of Technology, Department of Technical Physics, Laboratory of Computer and Information Science, November 1986. [Kohonen, 1990] Teuvo Kohonen, "Statistical Pattern Recognition Revisited," In R. Eckmiller, editor, Advanced Neural Computers, pages 137-144. Elsevier Science Publishers, 1990. [Kohonen, 1991] Teuvo Kohonen, "Self-Organizing Maps: Optimization Approaches," In T. Kohonen, K. Makisara, O. Simula, and J. Kangas, editors,Artijicial Neural Networks, pages 981-990. Elsevier Science Publishers, 1991. [Robbins and Monro, 1951J Herbert Robbins and Sutton Monro, "A Stochastic Approximation Method," Annals of Math. Stat., 22:400-407,1951. [Rumelhart and Zipser, 1986] D. E. Rumelhart and D. Zipser, "Feature Discovery by Competitive Learning," In David E. Rumelhart, James L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, volume 2, pages 151-193. MIT Press, 1986. [Sklansky and Wassel, 1981] Jack Sklansky and Gustav N. Wassel, Pattern Classijiers and Trainable Machines, Springer-Verlag, 1981. [Wassel and Sklansky, 1972] Gustav N. Wassel and Jack Sklansky, "Training a OneDimensional Classifier to Minimize the Probability of Error," IEEE Transactions on Systems, Man, and Cybernetics, SMC-2(4):533-541, 1972.
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Diffusion Approximations for the Constant Learning Rate Backpropagation Algorithm and Resistence to Local Minima William Finnoff Siemens AG, Corporate Research and Development Otto-Hahn-Ring 6 8000 Munich 83, Fed. Rep. Germany Abstract In this paper we discuss the asymptotic properties of the most commonly used variant of the backpropagation algorithm in which network weights are trained by means of a local gradient descent on examples drawn randomly from a fixed training set, and the learning rate TJ of the gradient updates is held constant (simple backpropagation). Using stochastic approximation results, we show that for TJ ~ 0 this training process approaches a batch training and provide results on the rate of convergence. Further, we show that for small TJ one can approximate simple back propagation by the sum of a batch training process and a Gaussian diffusion which is the unique solution to a linear stochastic differential equation. Using this approximation we indicate the reasons why simple backpropagation is less likely to get stuck in local minima than the batch training process and demonstrate this empirically on a number of examples. 1 INTRODUCTION The original (simple) backpropagation algorithm, incorporating pattern for pattern learning and a constant learning rate 'T} E (0,00), remains in spite of many real (and 459 460 Finnoff imagined) deficiencies the most widely used network training algorithm, and a vast body of literature documents its general applicability and robustness. In this paper we will draw on the highly developed literature of stochastic approximation theory to demonstrate several asymptotic properties of simple backpropagation. The close relationship between backpropagation and stochastic approximation methods has been long recognized, and various properties of the algorithm for the case of decreasing learning rate 7]n+l < 7]n, n E N were shown for example by White [W,89a], [W,89b] and Darken and Moody [D,M,91]. Hornik and Kuan [H,K,91] used comparable results for the algorithm with constant learning rate to derive weak convergence results. In the first part of this paper we will show that simple backpropagation has the same asymptotic dynamics as batch training in the small learning rate limit. As such, anything that can be expected of batch training can also be expected in simple backpropagation as long as the learning rate of the algorithm is very small. In the special situation considered here (in contrast to that in [H,K,91]) we will also be able to provide a result on the speed of convergence. As such, anything that can be expected of batch training can also be expected in simple backpropagation as long as the learning rate of the algorithm is very small. In the next part of the paper, Gaussian approximations for the difference between the actual training process and the limit are derived. It is shown that this difference, (properly renormalized), converges to the solution of a linear stochastic differential equation. In the final section of the paper, we combine these results to provide an approximation for the simple back propagation training process and use this to show why simple backpropagation will be less inclined to get stuck in local minima than batch training. This ability to avoid local minima is then demonstrated empirically on several examples. 2 NOTATION Define the. parametric version of a single hidden layer network activation function with h inputs, m outputs and q hidden units I: Rd x Rh -+ Rm, (0, x) -+ (lUJ, x), ... , reO, x», by setting for x E Rh, Z = (Xl, ... , Xh, I), 0 = (i:, /3:) and u = 1, ... , m, reo, x) = rA«i:, (3:), x) = t/J (t i'Jt/J({3jzT) + i:+l) , ,=1 where xT denotes the transpose of x and d = m(q + 1) + q(h + 1) denotes the number of weights in the network. Let «Yk, Xk»k=l, ... ,T be a set of training examples, consisting of targets (Yk)k=l, ... ,T and inputs (Xk)k=l, ... ,T. We then define the parametric error function U(y, x, 0) = Ily - 1(0, x)II 2 ,. and for every 0, the cummulative gradient Diffusion Approximations for the Constant Learning Rate Backpropagation Algorithm 461 3 APPROXIMATION WITH THE ODE We will be considering the asymptotic properties of network training processes induced by the starting value 80 , the gradient (or direction) function -W- the learning rate TJ and an infinite training sequence (Yn, xn)neN, where each (Yn, xn) example is drawn at random from the set {(Y1,X1), ... ,(YT,XT)}. One defines the discrete parameter process 8 = 8" = (8~)neZ+ of weight updates by setting (J7J { (Jo for n = 0 n = (J~_l TJ~(Yn' Xn, 8~_d for n E N and the corresponding continuous parameter process «(J" (t»te[o,oo), by setting for t E [en - l)TJ, nTJ), n E N. The first question that we will investigate is that of the 'small learning rate limit' of the continuous parameter process 8", i.e. the limiting properties of the family 8" for TJ -+ O. We show that the family of (stochastic) processes «(J")7J>o converges with probability one to a limit process 8, where 8 denotes the solution to the cummulative gradient equation, 8(t) = (Jo + it h(8(s»ds. Here, for (Jo = a = constant, this solution is deterministic. This result corresponds to a 'law of large numbers' for the weight update process, in which the small learning rate (in the limit) averages out the stochastic fluctuations. Central to any application of the stochastic approximation results is the derivation of local Lipschitz and linear growth bounds for Wand h. That is the subject of the following, Lemma(3.1) i) There exists a constant K > 0 so that sup II ~~ (y, x, (J)I < K(l + 11(11) (1/ ,z) and IIh(8)1I ~ K(l + 11(11). ii) For every G > 0 there exists a constant La so that for any 8, '9 E [-G, Gld, 462 Finnoff Ilau au -II '" sup 8o(Y, x, 0) - ao(Y' x, 0) ~ LGIlO - Oil (y,::) and IIh(8) - h(O)11 ~ LGIIO - 811. Proof: The calculations on which this result are based are tedious but straightforward, making repeated use of the fact that products and sums of locally Lipschitz continuous functions are themselves locally Lipschitz continuous. It is even possible to provide explicit values for the constants given above. • Denoting with P (resp. E) the probability (resp. mathematical expectation) of the processes defined above, we can present the results on the probability of deviations of the process 0 from the limit e. Theorem(3.2) Let r,6 E (0, (0). Then there exists a constant Br (which doesn't depend on 71) so that ii)P (suP,sr IIO(s) - O(s)11 > 6) < bBr71. Proof: The first part of the proof requires that one finds bounds for 0'1 (t) and O(t) for t E [0, r]. This is accomplished using the results of Lemma(3.l) and Gronwall's Lemma. This places 71 independent bounds on Br . The remainder of the proof uses Theorem(9), §1.5, Part II of [Ben,Met,Pri,87]. The required conditions (AI), (A2) follow directly from our hypotheses, and (A3), (A4) from Lemma(3.l). Due to the boundedness of the variables (Yn, xn)neN and 0o, condition (A5) is trivially fulfilled . • It should be noted that the constant Br is usually dependent on r and may indeed increase exponentially (in r) unless it is possible to show that the training process remains in some bounded region for t -- 00. This is not necessarily due exclusively to the difference between the stochastic approximation and the discrete parameter cummulative gradient process, but also to the the error between the discrete (Euler approximation) and continuous parameter versions of (3.3). 4 GAUSSIAN APPROXIMATIONS In this section we will give a Gaussian approximation for the difference between the training process 8" and the limit O. Although in the limit these coincide, for 71 > ° the training process fluctuates away from the limit in a stochastic fashion. The following Gaussian approximation provides an estimate for the size and nature Diffusion Approximations for the Constant Learning Rate Backpropagation Algorithm 463 of these fluctuations depending on the second order statistics (variance/covariance matrix) of the weight update process. Define for any t E [0,00), 8'1(t) = O'1(t) - O(t) . ..ft Further, for i = 1, ... , d we denote with ~~ i (y, x, 0), (resp. hi (6)) the i-th coordinate vector of ~(y,x,O) (resp. h(O)). Then define for i,j = I, ... ,d, 6 E Rd Thus, for any n EN, 6 E R d, R( 0) represents the covariance matrix of the random elements ~(Yn, Xn, 6). We can then define for the symmetric matrix R(6) a further Rdxd valued matrix Ri(6) with the property that R(6) = Ri(6)(R!(6))T. The following result represents a central limit theorem for the training process. This permits a type of second order approximation of the fluctuations of the stochastic training process around its deterministic limit. Theorem( 4.1): Under the assumptions given above, the distributions of the processes 8'1, TJ > 0, converge weakly (in the sense of weak convergence of measures) for TJ ~ 0 to a uniquely defined measure C{O), where '8 denotes the solution to the following stochastic differential equation where W denotes a standard d-dimensional Brownian motion (i.e. with covariance matrix equal to the identity matrix). Proof: The proof here uses Theorem(7), §4.4, Part II of [Ben,Met,Pri,87]. As noted in the proof of Theorem(3.2), under our hypotheses, the conditions (Al)(A5) are fulfilled. Define for i,j = 1, ... ,d, (y,x) E Im+h, 0 E Rd, wij (y,x,6) = pi(y, x, O)pi (y, x, O)-hi(O)hj (0), and 11 = p. Under our hypotheses, h has ~~>ntinuous first and second order derivatives for all 0 E Rd and the function R = (R·'ki=l, ... ,d as well as W = (Wij)i,;=l, ... ,d fulfill the remaining requirements of (AS) as follows: (A8)i) and (A8)ii) are trivial consequence of the definition of Rand W. Finally, setting Pa = P4 = 0 and JJ = 1, (AS)iii) then can be derived directly from the definitions of Wand Rand Lemma(5.1)ii). • 5 RESISTENCE TO LOCAL MINIMA In this section we combine the results of the two preceding sections to provide a Gaussian approximation of simple backpropagation. Recalling the results and 464 Finnoff notation of Theorem(3.2) and Theorem(4.1) we have for any t E [0,(0), 11 (J'1(t) = 8(t) + 7]~(J(t) + 0(7]1). Using this approximation we have: -For 'very small' learning rate 7], simple backpropagation and batch learning will produce essentially the same results since the stochastic portion of the process (controlled by 7]~) will be negligible. -Otherwise, there is a non negligible stochastic eleE1ent in the training process which can be approximated by the Gaussian diffusion (J. -This diffusion term gives simple backpropagation a 'quasi-annealing' character, in which the cummulative gradient is continuously perturbed by the Gaussian term 8, allowing it to escape local minima with small shallow basins of attraction. It should be noted that the rest term will actually have a better convergence rate than the indicated 0(7]~). The calculation of exact rates, though, would require a generalized version of the Berry-Esseen theorem. To our knowledge, no such results are available which would be applicable to the situation described above. 6 EMPIRICAL RESULTS The imperviousness of simple backpropagation to local minima, which is part of neural network 'folklore' is documented here in four examples. A single hidden layer feedforward network with 4J = tanh, ten hidden units n and one output was trained with both simple backpropagation and batch training using data generated by four different models. The data consisted of pairs (Yi, :c,), i = 1, , .. , T, TEN with targets Yi E R and inputs Xi = (:ci, .. " xf) E [-I,l)K, where Yi = g«xl, .. " x1» + Ui, for j, KEN. The first experiment was based on an additive structure 9 having the following form with j = 5 and K = 10, g«xt, .. " x;» = L::~=1 sin(okx:), ok E R, The second model had a product structure 9 with j = 3, K = 10 and g«xt, ... , x:» = n!=1 xf, ok E R , The third structure considered was constructed with j = 5 and K = 10, using sums of Radial Basis " 1 5 8 I (5 ( a •• I-~~ )2) FunctIOns (RBF s) as follows: g«xi' .'" xi» = E,=1 (-1) exp Ek=l 2(12 '• These points were chosen by independent drawings from a uniform distribution on [-1,1)5, The final experiment was conducted using data generated by a feedforward network activation function. For more details concerning the construction of the examples used here consult [F,H,Z,92]. For each model three training runs were made using the same vector of starting weights for both simple backpropagation and batch training. As can be seen, in all but one example the batch training process got stuck in a local minimum producing much worse results than those found using simple backpropagation, Due to the wide array of structures used to generate data and the number of data sets used, it would be hard to dismiss the observed phenomena as being example dependent. Diffusion Approximations for the Constant Learning Rate Backpropagation Algorithm 465 Net Error x 10-3 800.00 .-tl-.-... -.. ....:. .. ..,.~-.... \.\ ~ .OO~~~~~~~~···~· .. ~· ··~ .. ·~·~ .. ·~···=···=··~----------------·····:.:r:::=:::..:::.:-;::=;.~:.:-:.::::::::.::~ .. ::::::::.-:::; 400.00 --+----=~~~.~. ;;:;=~~~; i 0.00 !OO.OO 200.00 300.00 Product Mapping Error x 10-3 : ............. : ...... ~; ..... . ..... ~ ............ ." ............. _ ... .. ..................... . ......... . ................... I . . .. . ... .. 800.00 -·-!I-L--...... :::::::·· :::r+;;::: .. = ... j .. := ..... =.::=::=:::~::\-ij~: :.-::-:.:-::.-.::-::.-.::-::.-.::-::.-. :'.-::'-.::~:::::'.:::'.:::::'.: I 6CXl.OO ---~-- -.+-_.--0.1'1) :oo,C() 200. "() :0000 Sums ~f RBF's Error x : 0'" simple BP ·Sitch·C······· Epochs ~lmpie 3P 3ate·iiI:·· .. ·· SOO.110 . '-'=;::::::::::: ::::~.:.:-:- ' - ----- - - --- -- - --- ---- -_. . -- 51mple :3P .... , ............ , •••••• :;~;;; ~; ::::';':';':':';':';':':';':';';;';':':';':':':':':';';':';';';'!:';';';';';';':':':':';';';';'; ';':';';'; ':';';';';';';';':';';: '3 a tC' n' L::'" .. . SOiJ,GO ~ ... . - --- --- ..J.C{).OO -----.-------=~~;;;::~-~--=0.00 ifX).OO ~OO,OO :00.00 Sums of sin's Error x 10-.) ~lmpie SP ~)().00~~~~r-----------------------------·3ate·i{·::: .. · ... 400.00 -t-------~~~~ ... ~r-:!pocns 0.00 100.00 200.00 466 Finnoff 7 REFERENCES [Ben,Met,Pri,87] Benveniste, A., Metivier, M., Priouret, P., Adaptive Algorithms and Stochastic Approximations, Springer Verlag, 1987. [Bou,85] Bouton C., Approximation Gaussienne d'algorithmes stochastiques a dynamique Markovienne. Thesis, Paris VI, (in French), 1985. [Da,M,91] Darken C. and Moody J., Note on learning rate schedules for stochastic optimization, inAdvances in Neural Information Processing Systems 3, Lippmann, R. Moody, J., and Touretzky, D., ed., Morgan Kaufmann, San Mateo, 1991. [F ,H,Z,92] Improving model selection by nonconvergent methods. To appear in Neural Networks. [H,K,91], Hornik, K. and Kuan, C.M., Convergence of Learning Algorithms with constant learning rates, IEEE Trans. on Neural Networks 2, pp. 484-489, (1991). [Wh,89a] White, H., Some asymptotic results for learning in single hidden-layer feedforward network models, Jour. Amer. Stat. Ass. 84, no. 408, p. 10031013, 1989. [W,89b] White, H., Learning in artificial neural networks: A statistical perspective, Neural Computation 1, p.425-464, 1989.
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Analog Cochlear Model for Multiresolution Speech Analysis Weimin Liu~ Andreas G. Andreou and Moise H. Goldstein, Jr. Department of Electrical and Computer Engineering The Johns Hopkins University, Baltimore, Maryland 21218 USA Abstract This paper discusses the parameterization of speech by an analog cochlear model. The tradeoff between time and frequency resolution is viewed as the fundamental difference between conventional spectrographic analysis and cochlear signal processing for broadband, rapid-changing signals. The model's response exhibits a wavelet-like analysis in the scale domain that preserves good temporal resolution; the frequency of each spectral component in a broadband signal can be accurately determined from the interpeak intervals in the instantaneous firing rates of auditory fibers. Such properties of the cochlear model are demonstrated with natural speech and synthetic complex signals. 1 Introduction As a non-parametric tool, spectrogram, or short-term Fourier transform, is widely used in analyzing non-stationary signals, such speech. Usually a window is applied to the running signal and then the Fourier transform is performed. The specific window applied determines the tradeoff between temporal and spectral resolutions of the analysis, as indicated by the uncertainty principle [1]. Since only one window is used, this tradeoff is identical for all spectral components in the signal being analyzed. This implies that conventional spectrographic signal representation and its variations are uniform resolution analysis methods. Such is also the case in parametric analysis methods, such as linear prediction coding (LPC). "Present address: Hughes Network Systems, Inc., 11717 Exploration Lane, Germantown, Maryland 20876 USA 666 Analog Cochlear Model for Multiresolution Speech Analysis 667 In spectrographic analysis of speech, it is frequently necessary to vary the window length, or equivalently the bandwidth in order to obtain appropriate resolution in time or frequency domain. Such a practice has the effect of changing the durationbandwidth tradeoff. Broadband (short window) analysis gives better temporal resolution to the extent that vertical voice pitch stripes can be seen; narrowband (long window) can result in better spectral resolution so that the harmonics of the pitch become apparent. A question arises: if the duty of the biological cochlea were to map a signal onto the time-frequency plane, should it be broadband or narrowband? Neurophysiological data from the study of mammalian auditory periphery suggest that the cochlear filter is effectively broadband with regard to the harmonics in synthetic voiced speech, and a precise frequency estimation of a spectral component, such as a formant, can be determined from the analysis of the temporal patterns in the instantaneous firing rates (IFRs) of auditory nerve fibers (neurograms) [2]. A similar representation was also considered by Shamma [3}. In this paper, we will first have a close look at the spectrogram of speech signals. Then the relevant features of a cochlear model [5, 6} are described and speech processing by the model is presented illustrating good resolution in time and frequency. Careful examination of the model's output reveals that indeed it performs multiresolution analysis. 2 Speech Spectrogram Broadband Narrowband . I o 250 500 0 250 500 time (ms) Figure 1: Broadband (6.4ms Hamming window) and narrowband (25.6ms window) spectrograms for the word "saint" spoken by a male speaker. Figure 1 shows the broadband and narrowband spectrograms of the word "saint" spoken by a male speaker. The broadband spectrogram is usually the choice of speech analysis for several reasons. First, the fundamental frequency is considered of insignificant importance in understanding many spoken languages. Second, broadband reserves good temporal resolution, and meanwhile the representation 668 Liu, Andreou, and Goldstein of formants has been considered adequate. The adequacy of this notion has been seriously challenged, especially for rapidly varying events in real speech [4]. Although the vertical striation in the narrowband spectrogram indicates the pitch period, to accurately estimate the fundamental frequency Po, it is often desirable to look at the narrowband spectrogram in which harmonics of Po are shown. Ideally a speech analysis method should provide multiple resolution so that both formant and harmonic information are represented simultaneously. To further emphasize this, a synthetic signal of a tone/chirp pair was generated. The synthetic signal (Figure 2) consists of tone and chirp pairs that are separated by 100Hz. There are two 1 Oms gaps in both the high and low frequency tones; the chirp pair sweeps from 2900Hz-3000Hz down to 200Hz-300Hz in lOOms. The broadband spectrogram clearly shows the temporal gaps but fails to give a clear representation in frequency; the situation is reversed in the narrowband spectrogram. - ------ ---------- - -Chirp and tone pairs 4K ~""-~'-------- -. o 100 200 300 time (ms) Figure 2: The synthetic tone/chirp pair and its broadband (6.4ms Hamming window) (top) and narrowband (25.6ms window) (bottom) spectrograms. 3 The Analog Cochlear Model Parameterization of speech using software cochlear models has been pursued by several researchers; please refer to [5, 6] for a literature survey. The alternative to software simulations on engineering workstations, is the analog VLSI [7]. Computationally, analog VLSI models can be more effective compared to software simulations. They are also further constrained by fundamental physical limitations and scaling laws; this may direct the development of more realistic models. The constraints imposed by the technology are: power dissipation, physical extent of Analog Cochlear Model for Multiresolution Speech Analysis 669 computing hardware, density of interconnects, precision and noise limitations in the characteristics of the basic elements, signal dynamic range, and robust behavior and stability. Analog VLSI cochlear models have been reported by Lyon and Mead [10] at Caltech with subsequent work by Lazzaro [11] and Watts [12}. Our model [5] consists of the middle ear, the cochlear filter bank, and haircell/synapse modules. All the modules in the model are based on detailed biophysical and physiological studies and it builds on the software simulation and the work in our laboratory b~· Payton [9]. At the present time the model is implemented both as a software simulation package but also as a set of two analog VLSI chips [6} to minimize the simulation times. Even though the silicon implementation of the model is completely functional, adequate interfaces to standard engineering workstations have not been yet fully developed and therefore here we will focus on results obtained through the software simulations. The design of the cochlear filter bank structure is the result of the effective bandwidth concept. The filter structure is flexible enough so that an appropriate set of parameters can be found to fit the neurophysiological data. In particular, the cochlear filter bank is tuned so that the model output closely resemble the auditory fibers' instantaneous firing rates (IFR) in response synthetic speech signals [2}. To do so, a fourth order section is used instead of the second order section of our earlier work [5]. Figure 3 shows the response amplitude and group delay of the filter bank that has been calibrated in this manner. · .. ... . • • •• • I. I • ••• ,. I •• •• • • · . .... . · .. , . . . ! I frl & :t--+--:-+-i+----;---:-~~:.r . . " " : : : : :: . .. . , . , . .. , , , . ' " , 100 1000 Frequency (Hz) ~IJj:t;j:= ; = : B=M=~==:====j·j·j·1·t .. t===t=~LJ~~ HIt 100 1000 FnoquOllC)l (Hz) Figure 3: The amplitude and group delay of the cochlear filter bank. The curves that have higher peak frequencies in the amplitude plot and those having smaller group delays are the filter channels representing locations near the base of the cochlea. The hair cells are the receptor cells for the hearing system. The function of hair cells and synapses in terms of signal processing is more than just rectification; besides the strong compressive nonlinearity in the mechano-electrical transduction, there are also rapid and short-term additive adaptation properties, as seen in the discharge patterns of auditory nerve fibers. Since the auditory fibers have a limited dynamic range of only 2D-30dB, magnitude compression and adaptation become necessary in the transmission of acoustical signals of much wider dynamic range. A neurotransmitter substance reservoir model, proposed by Smith and BrachIOIt 670 Liu, Andreou, and Goldstein man [8], of the hair cell and synapses that characterizes the generation of instantaneous firing rates of nerve fibers has been incorporated in the model. This is computationaly very demanding and the model benefits considerably by the analog VLSI implementation. The circuit output resembles closely the response of mammalian auditory nerve fibers [5]. 4 Multiresolution Analysis The conventional Fourier transform can be considered as a constant- bandwidth analysis scheme, in which the absolute frequency resolution is identical for all frequencies. A wavelet transform, on the other hand, is constant-Q in nature where the relative bandwidth is constant. The cochlear filter that is tuned to fit the experiment data is neither but is more closely related the wavelet transform, even though it required a higher Q at the base than at the apex. The response of the cochlear model is shown in Figure 4, in the form of a neurogram. Each trace shows the IFR of a channel whose characteristic frequency is indicated on the left. The gross temporal aspects of the neurogram are rather obvious. To obtain insight into the fine time structure of IFRs, additional processing is need. One possible feature that can be extracted is the inter-peak intervals (IPI) in the IFR, which is directly related to the main spectral component in the output. The advantage of such a measure over Fourier transform is that it is not affected by the higher harmonics in the IFR. An autocorrelation and peak-picking operation were performed on the IFR output to capture the inter-peak intervals (IPls). The procedure was similar to that by Secker-Walker and Searle [2], except that the window lengths of autocorrelation functions directly depend on the channel peak frequency. That is, for high frequency channels, shorter windows were used. To illustrate the multiresolution nature of cochlear processing, the IPI histogram (Figure 4) across all channels are shown at each response time for the speech input "saint." Both formants and pitch frequencies are clearly shown in the composite IPI histogram. Similarly, the cochlear model's response to the synthetic tone/chirp pairs is shown in Figure 5. The IPI histogram gives high temporal resolution for high frequencies and high spectral resolution for low frequencies such that the 10ms temporal gap in the high frequency tones and the 100Hz spacing between the two low frequency tones are precisely represented. However in the high-frequency regions of the IPI histogram, the fact the each trace consists of a pair of tones or chirps is not clearly depicted. This limitation in the spectral resolution is the result of the relatively broad bandwidths in the highfrequency channels of the basilar membrane filter. U ndoubtly the information about the 100Hz spacing in the tone/chirp pairs is available in the IFRs, which can be estimated from the IFR envelope which exhibits an obvious beat every lOms (Is/100Hz) in the neurogram. Obtaining the beating information calls for a variable resolution IPI analysis scheme. For speech signals, such analysis may be necessary in pitch frequency estimation when only the IFRs of high characteristic frequencies are available. ~ '-" >. u Analog Cochlear Model for Multiresolution Speech Analysis 671 100 Jt .00 ~--------------------~~~N lK f========-===="~======== -g ... r-------~II/AWro~J.JW,kI\II."'/fJ/oWij~......__--'~----____._..~~~ _ ________ _ - ------- " -- - -----8K r-::::===::~~~~~~~~::--=~~:::-::--;:::-:::::-::==:-::=~ --- -------- ----- - - ----- ....... -- --- r---------.-.----T""-- -- .- . -. ~- _.- ' o •• . • .' 1 10k 100 200 . ... 300 Response time (ms) 400 - _ . . - . J .. 500 ~ r f1 {" ',: ' . ... .' " • I .·t It,.. \\ - . ' ') ' . . . . ,., .. 600 ,1 • - -,. ,~ .,. -. • l.t .; ~ .. , .. . J1 • . .. ., . ',.' , ~ : ~ t:V: M,,;Y- -1jj~,.;!1fI:ft ~ Ik1~~~b'~ 7'0 • loo-r-'-'-' --'-'I-r-'- --'~-~~~~~-T--~~~ o 100 200 300 Response time (ms) Figure 4: (Top) Cochlear model output, in response to "saint," in the form of neurogram. Each trace shows the IFR of one channel. Outputs from different channels are arranged according to their characteristic frequency. (Bottom) IPI histograms. 672 Liu, Andreou, and Goldstein 100 200 2K 4K 8K 10k "..... N ::c '-' ~ Ik '" 20 o ~"_"""'_""""IIM"'-""""---~---- ------ -. - - -- ---- ------ --... - ....... - - -_ ... ............. .. .. --~-------- ~-------------------------------------------- ----- -----60 tOO 140 Response time (ms) 100 Response time (ms) IRO '.J" -. 200 220 . ..... '2(,() Figure 5: (Top) Neurogram: cochlear model response to the tone/chirp pairs. (Bottom) IPI histograms. 300 Analog Cochlear Model for Multiresolution Speech Analysis 673 5 Discussion and Conclusions We have presented an analog cochlear model that is tuned to match physiological data of mammalian cochleas in response to complex sounds and uses a small number of realistic model parameters. The response of this model has good resolution in time and in frequency, suitable for speech and broadband signal analysis. Both the analysis performed by the cochlear model and at subsequent stages (IPI) is in the time-domain. Processing information using temporal representations is pervasive in neural information processing systems. From an engineering perspective, it is advantageous because it results in architectures that can be efficiently implemented in analog VLSI. The cochlear model has been implemented as an analog VLSI system [6] operating in real-time. Appropriate interfaces are also being developed that will enable the silicon model to communicate with standard engineering workstations. Furthermore, refinements of the model may find applications as high-performance front-ends for various speech processing tasks. References [1] D. Gabor. (1953) A summary of communication theory. In W. Jackson (ed.), Communication Theory, 1-21. London: Butterworths Scientific Pub. [2] H.E. Secker-Walker and C.L. Searle. (1990) Time-domain analysis of auditorynerve-fiber firing rates. J. Acoust. Soc. Am. 88:1427-1436. [3] S.A. Shamma. (1985) Speech processing in the auditory system. I: representation of speech sounds in the responses of the auditory-nerve. J. Acoust. Soc. Am. 78:1612-1621. [4] H.F. Siverman and Y.-T. Lee. (1987) On the spectrographic representation of rapidly time-varying speech. Computer Speech and Language 2:63-86. [5] W. Liu, A.G. Andreou and M.H. Goldstein. (1992) Voiced-speech representation by an analog silicon model of the auditory periphery. IEEE Trans. Neural Networks, 3(3):477-487. [6] W. Liu. (1992) An analog cochlear model: signal representation and VLSI realization Ph.D. Dissertation, The Johns Hopkins University. [7] C. A. Mead, (1989) Analog VLSI and Neural Systems, Addison-Wesley, Reading MA. [8] R.L. Smith and M.L. Brachman. (1982) Adaptation in auditory-nerve fibers: a revised model. Biological Cybernetics 44:107-120. [9] K.L Payton. (1988) Vowel processing by a model of the auditory periphery: a comparison to eighth-nerve responses. J. Acoust. Soc. Am. 83:155-162. [10] R.F. Lyon and C.A. Mead. (1988) An analog electronic cochlea. IEEE Trans. Acoust. Speech, and Signal Process. 36:1119-1134. [11] J. Lazzaro and C.A. Mead. (1989) A silicon model of auditory localization. Neural Computation 1(1):47-57. [12] L. Watts. (1992) Cochlear mechanics: analysis and analog VLSI. Ph.D. Dissertation, California Institute of Technology.
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A Boundary Hunting Radial Basis Function Classifier Which Allocates Centers Constructively Eric I. Chang and Richard P. Lippmann MIT Lincoln Laboratory Lexington, MA02173-0073, USA Abstract A new boundary hunting radial basis function (BH-RBF) classifier which allocates RBF centers constructively near class boundaries is described. This classifier creates complex decision boundaries only in regions where confusions occur and corresponding RBF outputs are similar. A predicted square error measure is used to determine how many centers to add and to determine when to stop adding centers. Two experiments are presented which demonstrate the advantages of the BHRBF classifier. One uses artificial data with two classes and two input features where each class contains four clusters but only one cluster is near a decision region boundary. The other uses a large seismic database with seven classes and 14 input features. In both experiments the BHRBF classifier provides a lower error rate with fewer centers than are required by more conventional RBF, Gaussian mixture, or MLP classifiers. 1 INTRODUCTION Radial basis function (RBF) classifiers have been successfully applied to many pattern classification problems (Broomhead, 1988, Ng, 1991). These classifiers have the advantages of short training times and high classification accuracy. In addition, RBF outputs estimate minimum-error Bayesian a posteriori probabilities (Richard, 1991). Performing classification with RBF outputs requires selecting the output which is highest for each input. In regions where one class dominates, the Bayesian a posteriori probability for that class will be uniformly "high" and near 1.0. Detailed modeling of the variation of the Bayesian a posteriori probability in these regions is not necessary for classification. Only 139 140 Chang and Lippmann at the boundary between different classes is accurate estimation of the Bayesian a posteriori probability necessary for high classification accuracy. If the boundary between different classes can be located in the input space, RBF centers can be judiciously allocated in those regions without wasting RBF centers in regions where accurate estimation of the Bayesian a posteriori probability does not improve classification perfonnance. In general, having more RBF centers allows better approximation of the desired output. While training a RBF classifier, the number of RBF centers must be selected. The traditional approach has been to randomly choose patterns from the training set as centers, or to perfonn K-means clustering on the data and then to use these centers as the RBF centers. Frequently the correct number of centers to use is not known a priori and the number of centers has to be tuned. Also, with K-means clustering, the centers are distributed without considering their usefulness in classification. In contrast, a constructive approach to adding RBF centers based on modeling Bayesian a posteriori probabilities accurately only near class boundaries provides good perfonnance with fewer centers than are required to separately model class PDF's. Many algorithms have been proposed for constructively building up the structure of a RBF network (Mel, 1991). However. the algorithms proposed have all been designed for training a RBF network to perfonn function mapping. For mapping tasks, accuracy is important throughout the input region and the mean squared error is the criterion that is minimized. In classification tasks, only boundaries between different classes are important and the overall mean squared error is not as important as the error in class boundaries. 2 ALGORITHM DESCRIPTION A block diagram of a new boundary hunting RBF (BH-RBF) classifier that adds centers constructively near class boundaries is presented in Figure 1. A simple unimodal Gaussian classifier is first fonned by clustering the training patterns from a randomly selected class and assigning a center to that class. The confusion matrix generated by using this simple classifier is then examined to determine the pair of classes A and B, which have the most mutual confusion. Training patterns that are close to the boundary between these two classes are detennined by looking at the outputs of the RBF classifier. Boundary patterns ONE RBF CENTER INITIAL RBF NETWORK ADD NEW RBF CENTERS TO CLASS PAIR RESPONSIBLE FOR MOST ERRORS & OVERLAP' INTERMEDIATE RBF NETWORKS CALCULATE PREDICTED SQUARED ERROR SCORE FINAL NETWORK Figure 1: Block Diagram of Training of BH-RBF Network A Boundary Hunting Radial Basis Function Classifier (Allocates Centers Constructively) 141 which produce similar "high" outputs for both classes that are different by less than a "closecall" threshold are used to produce new cluster centers. Figure 2 shows RBF outputs corresponding to class A and B as the input varies over a small range. This figure illustrates how network outputs are used to determine the "closecall" region between classes. Network outputs are high in regions dominated by a particular class and therefore these regions are outside the boundary between different classes. Networlc outputs are close in the region where the absolute difference of the two highest network outputs is less than the closecall threshold. Training patterns which fall into this closecall region plus all the points that are misclassified as the other class in the class pair are considered to be points in the boundary. For example, a pattern in class A which is misclassified as class B would be considered to be in the boundary between class A and B. On the other hand, a pattern in class A which is misclassified as class C would not be placed in the boundary between class A and B. 1 0.9 FCA) F(B) 0.8 CLASS A CLASS B => 0.7 c. => 0.6 CLOSECALL 0 ~ 0.5 THRESHOLD ~ 0.4 Z 0.3 0.2 CLOSECALL 0.1 REGION 0 ~ • -3 -2 -1 0 1 2 3 INPUT Figure 2: Using the Network Output to Determine Closecall Regions After the patterns which belong in the boundary are determined, clustering is performed separately on boundary patterns from different classes using K-means clustering and a number of centers ranging from zero to a preset maximum number of centers. After the centers are found, new RBF classifiers are trained using the new sets of centers plus the original set of centers. The combined set of centers that provides the best performance is saved and the cycle repeats again by fmding the next class pair which accounts for the most remaining confusions. Overfitting by adding too many centers at a time is avoided by using the predicted squared error (PSE) as the criterion for choosing new centers (Barron, 1984): Cxa2 PSE=RMS+-N 142 Chang and Lippmann In this equation, RMS is the root mean squared error on the training set, (12 estimates the variance of the error, C is the total number of centers in the RBF classifier, and N is the total number of patterns in the training set. The error variance (12 is selected empirically using left-out evaluation data. Different values of cr2 are tried and the value which provides the best performance on the evaluation data is chosen. On each cycle, different number of centers are tried for each class of the selected class pair and the PSE is used to select the best subset of centers. The best PSE on each cycle is used to determine when training should be stopped to prevent overfitting. Training stops after the PSE has not decreased for five consecuti ve cycles. 3 EXPERIMENTAL RESULTS Two experiments were performed using the new BH-RBF classifier, a more conventional RBF classifier, a Gaussian mixture classifier (Ng, 1991), and aMLP classifier. Five regular RBF classifiers (RBF) were trained by asSigning 1, 2, 3,4, or 5 centers to each class. Similarly, five Gaussian mixture classifiers (GMIX) were trained with 1,2,3,4, or 5 centers in each class. The means of each center were trained individually using K-means clustering to find the centers for patterns from each class. The diagonal covariance of each center was set using all the patterns that were assigned to a cluster during the last pass of K-means clustering. The structure of the regular RBF classifier and the Gaussian mixture classifier are identical when the number of centers are the same. The only difference between the classifiers is the method used to train parameters. MLP classifiers were trained for 10 independent trials for each data set. The number of hidden nodes was varied from 2 to 30 in increments of 2. The goal of the experiment was to explore the relationship between the complexity of the classifier and the classification accuracy of the classifier. Training was stopped using cross validation to avoid overfitting. 3.1 FOUR-CLUSTER DATABASE The flfst problem is an artificial data set designed to illustrate the difference between BHRBF and other classifiers. There are two classes, each class consist of one large Gaussian cluster with 700 random points and three smaller clusters with 100 points each. Figure 3 shows the distribution of the data and the ideal decision boundary if the actual centers and variances are used to train a Bayesian minimum error classifier. There were 2000 training patterns, 2000 evaluation patterns, and 2000 test patterns. The BH -RBF classifier was trained with the closecall threshold set to 0.75, (12 set to 0.5, and a maximum of two extra centers per class at between each pair of classes. The theoretically optimal Bayesian classifier for this database provides the error rate of 1.95% on the test set. This optimal Bayesian classifier is obtained using the actual centers, variances, and a priori probability used to generate the data in a Gaussian mixture classifier. In a real classification task, these center parameters are not known and have to be estimated from training data. Figure 4 shows the testing error rate of the three different classifiers. The BH-RBF classifier was able to achieve 2.35% error rate with only 5 centers and the error rate gradually decreased to 2.15% with 15 centers. The BH-RBF classifier performed well with few centers because it allocated these centers near the boundary between the two classes. On the other hand, the perfonnance of the RBF classifier and the Gaussian mixture classifier was worse with few centers. These classifiers perfonned worse because they allocated centers A Boundary Hunting Radial Basis Function Classifier (Allocates Centers Constructively) 143 15 10 ex: !! ex: w H 5 o o 60 50 ® ® 40 30 Y 20 o X Figure 3: The Artificially Generated Four-Cluster Problem !~ /\RBF . . ~ , , 0\ J /, l ,- . . \ \ t=:. GMIX . .... /'0. ·t .".' I·! ",. .. ,....... . '--_~=--";"_"'~":':':"':'::'.'::._:,: ,,,,:,:,,,~, .. :=s-::. ••. :.;,:' -=.':.:.!'.~r::.;'· •• ' •• 11:'1 .... .. .... I I • . BH-RBF 5 10 15 20 NUMBER OF CENTERS Figure 4: Testing Error Rate Of The BH-RBF Classifier, The Gaussian Mixture Classifier, And The Regular RBF Classifier On The Four-Cluster Problem. in regions that had many patterns. The training algorithm did not distinguish between patterns that are easily confusable between classes (Le. near the class boundary) and patterns that clearly belong in a given class. Furthermore, adding more centers did not monotoni144 Chang and Lippmann cally decrease the error rate. For example, the RBF classifier had 5% error using two centers, but when the number of centers was increased to four, the error rate jumped to 11 %. Only until the number of centers increased above 14 did the RBF classifier and the Gaussian mixture classifier's error rates converge. The RBF and the Gaussian mixture classifiers performed poorly with few centers because the centers were concentrated away from the decision boundary due to the high concentration of data far away from the boundary. Thus, there weren't enough centers to model the decision boundary accurately. The BHRBF classifier added centers near the boundary and thus was able to define an accurate boundary with fewer centers. Figure 5 presents the results from training MLP classifiers on the same data set using different numbers of hidden nodes. The learning rate was set to 0.001, the momentum term was set to 0.6, and each classifier was trained for 100 epochs. The error rate on a left out evaluation set was checked to assure that the net had not overfitted the training data. As the number of hidden nodes increased, the MLP classifier generally performed better. However, the testing error rate did not decrease monotonically as the number of hidden nodes increased. Furthermore, the random initial condition set by the different random seeds affected the classification error rate of each classifier. In comparison, the training algorithms used for BH -RBF, RBF, and GMIX classifiers do not exhibit such sensitivity to initial conditions. 15 10 a: o a: a: w ~ 5 o 2 4 MAX MIN 6 8 10 12 14 16 18 20 22 24 26 28 NUMBER OF HIDDEN NODES Figure 5: Testing Error Rate Of The MLP Classifiers On The Four-Cluster Problem 3.2 SEISMIC DATABASE The second problem consists of data for classification of seismic events. The input consist of 14 continuous and binary measurements derived from seismic waveform signals. These features are used to classify a wavefonn as belonging to one of 7 classes which represent different seismic phases. There were 3038 training, 3033 evaluation, and 3034 testing patA Boundary Hunting Radial Basis Function Classifier (Allocates Centers Constructively) 145 20 15 GMIX • c: III·I~ • • --.. --~--~~ --.-.. --~10 I ••••• If •••• If ••• "., .................. I ........... I •••• II I • RBF LLJ ~ BH-RBF 5 o ~------~----~~~~--~~~~~~~~~~--~~ o 5 10 15 20 25 30 35 40 NUMBER OF CENTERS Figure 6: Error Rate Comparison Between The BH -RBF Classifier, The Regular RBF Classifier, And The Gaussian Mixture Classifier On The Seismic Problem terns. Once again, the number of centers per class was varied from 1 to 5 for the regular RBF classifier and the Gaussian mixture classifier, while the BH-RBF classifier was started with 1 center in the frrst class and then more centers were automaticallY assigned. The BH-RBF classifier was trained with the closecall threshold set to 0.75, (52 set to 0.5, and a maximum of one extra center per class at each boundary. The parameters were chosen according to the performance of the classifier on the left-out evaluation data. For this problem, the closecall threshold and (52 turned out to be the same as the ones used in the four-cluster problem. Figure 6 shows the error rate on the testing patterns for all three classifiers. The BH-RBF classifier clearly performed better than the regular RBF classifier and the Gaussian mixture classifier. The BH-RBF classifier added centers only at the boundary region where they improved discrimination. Also, the diagonal covariance of the added centers are more local in their influence and can improve discrimination of a particular boundary without affecting other decision region boundaries. MLP classifiers were also trained on this data set with the number of hidden nodes varying from 2 to 32 in increments of 2. The learning rate was set to 0.001, the momentum term was set to 0.6, and each classifier was trained for 100 epochs. The classification error rate on the left-out evaluation set showed that the network had not overfitted on the training data. Once more, the MLP classifiers exhibited great sensitivity to initial conditions, especially when the number of hidden nodes were small. Also, for this high dimensionality classification task, even the best performance of the MLP classifier (15.5%) did not match the best performance of the BH-RBF classifier. This result suggests that for this high 146 Chang and Lippmann dimensionality data, the radially symmetric boundaries fonned with local basis functions such as the RBF classifier are more appropriate than the ridge-like boundaries formed with the MLP classifier. 4 CONCLUSION A new boundary-hunting RBF classifier was developed which adds RBF centers constructively near boundaries of classes which produce classification confusions. Experimental results from two problems differing in input dimension, number of classes, and difficulty show that the BH-RBF classifier performed better than traditional training algorithms used for RBF, Gaussian mixture, and MLP classifiers. Experiments have also been conducted on other problems such as Peterson and Barney's vowel database and the disjoint database used by Ng (Peterson, 1952, Ng, 1990). In all experiments, the BH-RBF constructive algorithm performed at least as well as the traditional RBF training algorithm. These results, and the experiments described above, confirm the hypothesis that better discrimination performance can be achieved by training a classifier to perform discrimination instead of probability density function estimation. Acknowledgments This work was supported by DARPA. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government. Experiments were conducted using LNKnet, a general purpose classifier program developed at Lincoln Laboratory by Richard Lippmann, Dave Nation, and Linda Kukolich. References G. E. Peterson and H. L. Barney. (1952) Control Methods Used in a Study of Vowels. The Journal of the Acoustical Society of America 24:2, 175-84. A. Barron. (1984) Predicted squared error: a criterion for automatic model selection. In S. Farlow, Editor. Self-Organizing Methods in Modeling. New York, Marcel Dekker. D. S. Broomhead and D. Lowe. (1988) Radial Basis Functions, multi-variable functional interpolation and adaptive networks. Technical Report RSRE Memorandum No. 4148, Royal Speech and Radar Establishment, Malvern, Worcester, Great Britain. B. W. Mel and S. M. Omohundro. (1991) How Receptive Field Parameters Affect Neural Learning. In R. Lippmann, J. Moody and D. Touretzky (Eds.), Advances in Neural Information Processing Systems 3, 1991. San Mateo, CA: Morgan Kaufman. K. Ng and R. Lippmann. (1991) A Comparative Study of the Practical Characteristics of Neural Networks and Conventional Pattern Classifiers. In R. Lippmann, 1. Moody and D. Touretzky (Eds.), Advances in Neural Information Processing Systems 3, 1991. San Mateo, CA: Morgan Kaufman. M.D. Richard and R. P. Lippmann. (1991) Neural Network Classifier Estimates Bayesian a posteriori Probabilities. Neural Computation, Volume 3, Number 4.
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Learning to See Where and What: Training a Net to Make Saccades and Recognize Handwritten Characters Gale Martin, Mosfeq Rashid, David Chapman, and James Pittman MCC, 3500 Balcones Center Drive, Austin, Texas 78759 ABSTRACT This paper describes an approach to integrated segmentation and recognition of hand-printed characters. The approach, called Saccade, integrates ballistic and corrective saccades (eye movements) with character recognition. A single backpropagation net is trained to make a classification decision on a character centered in its input window, as well as to estimate the distance of the current and next character from the center of the input window. The net learns to accurately estimate these distances regardless of variations in character width, spacing between characters, writing style and other factors. During testing, the system uses the net~xtracted classification and distance information, along with a set of jumping rules, to jump from character to character. The ability to read rests on multiple foundation skills. In learning how to read, people learn how to recognize individual characters centered in the visual field. They also learn how to move their eyes along a line of text, sequentially centering the visual field on successive characters. We believe that the key to developing optical character recognition (OCR) systems that can mimic human reading capabilities, is to develop systems that can learn these and other skills in an integrated fashion. In this paper, we demonstrate that a backpropagation net can learn to naVigate along a line of handwritten characters, as well as to recognize the characters centered in its visual field. The system, called Saccade, extends the current state of the art in OCR technology by using a single classifier to accurately and efficiently locate and recognize characters, regardless of whether they touch each other or are separate. The Saccade system was described briefly at the last NIPS conference (Martin & Rashid, 1992). In this paper, we describe it mcx-e fully and report on results demonstrating its accuracy and efficiency in recognizing handwritten digits. The Saccade system takes a cue from the ballistic and corrective saccades (eye movements) of natural vision systems. Natural saccades make it possible to efficiently move from one informative area to another by jumping. The eye typically initiates a ballistic saccade to 441 442 Martin, Rashid, Chapman, and Pittman move the center of focus to the general area of interest. followed. if necessary. by one or more ccnective saccades for fme-grained position corrections. Recognition processes are applied only at these multiple fixation points. We have copioo some of these aspects in the artificial Saccade system by training a neural netwcn to know a1>oot the locations of characters in its input window. as well as to know about the identity of the character centered in its input window. During run-time, the Saccade system accesses this information computed by the net for successive input windows, along with a set of simple jumping rules, to yield an OCR system that jumps from character to character, classifying each character in a sequence. 1 TRAINING DETAILS As shown in Figure 1, the Saccade system has a wide input window, large enough to contain No Centered Character Current character ~i~ii3a~i~ } Distance to current character I I I f .. W .. , I I I I Distance to next character 4 Part Output Vector Backprop Net Figure 1. The Saccade system uses an enlarged input window and a 4-part output vector. several characters. Prior to training, each field image of a line of characters is labeled with the hcrizontal center position of each character in the field, as well as with the category of each character. During training, the input window slides hcriwntally across a field of characters, and at each position, the contents of the input window are paired with a four-part target ootput vector, the values of which are computed from the labeled information. The target values answer the following four questions about the contents of the input window: 1. Is a character centered in the input window? 2. What character is closest to the center of the window? 3. How far off~nter (horizontally) is the centennost character? 4. How far is the next character to the right from the center of the window? The first node in the output vector represents the no-centered-character state. It's target value is set high (e.g .• 1.0) when the center of the input window falls between characters, Training a Net to Make Saccades and Recognize Handwritten Characters 443 and set low (e.g .• 0.0) when the center of the input window falls 00 the center of a character. When the net is trained. the value of the rw-cenlered~haracter node indicates whether the input window is centered over a character. oc whether a corrective saccade is needed to better center the character. The second part of the output vector contains a node foc each character category. When the center of the input window falls on a character. the target value for its cocresponding node is set high; otherwise it is set low. When the net is trained. the values in these nodes are used to classify the centered character. The target values for bdh the rw-centeredcharacter and the character-category nodes are defmed continuously across the hcrizontal dimension as trapewidal functions. such that there are plateaus surrounding the off and on positions. with linearly increasing and decreasing values connecting the plateaus. The third and fourth components of the output vector represent distance values. each encoded in a distributed fashion across multiple nodes. using localized receptive fields (Moody & Darken. 1988). The frrst of these two parts represents the distance by which the character closest to the center of the window is off ~nter. The target value can be positive. indicating that the center of the window is to the left of the center of character. or it can be negative. indicating that the center of the window has passed over the character. to the right of it's center. When trained, the value of the current-character-distance set of nodes is accessed to determine the magnitude of a ccxrective saccade. to make a fme-grained position adjustment. The fourth component represems the distance from the center of the window to the center of the next character to the right. The target value can only be positive. When trained, the value of this set of nodes is accessed to detennine the magnitude of a ballistic saccade, to jump to the next character to the right. It is imponant to note that for bdh distance components, the maximum target value can not exceed half the window width. The net is never trained to make a distance judgment that extends beyond its field of view, since it is not given any infonnation about what exists outside of it's input window. For example. when the center of the next character to the right is positioned outside of the current input window. the distance value is set to the maximum value of half the window width. Since the distance values vary with different characters. different writers. and of course, at different positions with respect to a character. the net is forced to learn to use the visual characteristics particular to each window to estimate the distance values. In other words. the net does NOT simply learn average values for each of the two distance metrics. Moreover. as the results will show, the trained net does not seem to use simple density histogram cues to estimate the distance values. It is able to reliably estimate the distance values even when characters overlap. and hence would appear as a single clump in a density histogram. 2 RUN-TIME SACCADE RULES During run-time, the labeled values are. of course, not available. The system uses the computed values in the character classification and distance components of the output vector, and some heuristics, to navigate horizontally along a character field. jumping from one character to the next. and occasionally making a corrective saccade to improve its ability to classify a character. When the net recognizes a character, it executes a ballistic saccade 444 Martin, Rashid, Chapman, and Pittman to the next character. obtaining the distance to jump by reading the next-character-disUlnCe component of the output vector. When this actioo fails to center a character. as indicated by a low value in the no-centered-character output node, the system executes a corrective saccade to better center the character. It obtains the distance and direction to jump by reading the current-character-distance component of the output vector. Multiple corrective saccades can be executed. 3 TFSTING ON NIST HANDWRITTEN DIGIT FIELDS We tested the perfonnance of the system on a set of hand-printed digits collected and distributed by the National Institute of Standards and Technology (NIST). This is a database containing 273,000 samples of handwritten numerals. Each of2100 Census workers filled in a fonn with 33 fields, 28 fields of which only contain handwritten digits. The scanning resolution of the samples was 300 pixels/inch. The neural net was trained on about 80,000 characters from 20,000 fields, written by 800 different individuals. The fields varied in length from 2 characters per field to 6 characters per field. The horiwntal positions of each of the characters in these training-data fields were extracted by a person. The test data contained about 20,000 digits from 5,000 fields, written by a different group of 200 individuals. The test set was chosen to be this large because use of smaller test sets (e.g., 5,000 digits, 1250 fields) yielded significant between-set variations in reported accuracy. Each field image was preprocessed to remove the box around the field of characters, and any surrounding white space. Each field image was size normalized, with respect to the vertical axis. to a height of 20 pixels. Aspect ratio was maintained. An input pattern generator was then passed over the field to create input windows fa training the net. The input window size was 36 pixels wide and 20 pixels high. The input window scanned the field at 2-pixel increments during training. Subsequent experiments have shown that training can be speeded up considerably by training on the character centers and at random points between the character centers. without causing decreased accuracy. The backpropagation netwak architecture is described more fully in Martin & Rashid (1992). It has 2 hidden layers with local, shared connections in the first hidden layer. and local connections in the second hidden layer. Shared weights are not used in the second hidden layer because early experiments showed that this retards learning, presumably because extending the ~ition invariance to second-hidden-Iayer nodes inhibits the net in learning the position specific information regarding what is centered in its input window. The learning rate of the net was initially set at .05. and then successivel y lowered as training reached an asymptcxe. The momentum term was set at .9 throughout training. All nodes in the net used logistic activation functions. Table I reports on the test results in terms of field-based reject rates. for 1 % and .5% percent of the fields rejected. The error rates are field-based in the sense that if the net misclassifies one character in the field. the entire field is considered as mis-classified. Error rates pertain to the fields remaining after rejection. Rejections are based on placing a threshold for the acceptable distance between the highest and next highest running activation total. In this way. by varying the threshold, the error rate can be traded off against the percentage of rejections. In addition. recognized fields were also rejected if the number of recognized digits differed from the expected number of cligits. Training a Net to Make Saccades and Recognize Handwritten Characters 445 Thble 1: Field - Based Error Rates For Saccade System Field Field Field Size Error Rate Reject Rate 2-dIalt. 10K 6.a 0.5111 9.J'11 3-di<. 10K 12.".. 0.5111 19.6'11 4-dJalt. 1.1111 19.5111 0.5111 350K 5-dJalu 1.1'" 23.2111 0.5111 28.J'11 6-di<. 1.1111 26.K 0.5111 35.0111 Figure 2 presents some of the fields of connected characters that the system correctly recognized. Conventional OCR systems typically fail on connected characters because they employ an independent character segmentation stage. in which the character is isolated from its surround using features. such as intervening white spaces. This character segmentation stage typically fails when characters are coonected. The Saccade system goes beyond conventional OCR systems by integrating segmentation and recognition. and thereby is able to recognize touching characters. The Saccade system is also efficient in the sense that it typically jumps from one character to the next without making a corrective saccades. Corrective saccades tend to be mere likely when characters are touching. In addition. there is almost always a corrective saccade for the first character in the field. since the system starts at the beginning of the field. with ·1 41" ,. ~. G9£ ~1 I o~71 ~ ~J ~~~I u.ul ~ ·1~9 ___ q_30 __ 3/,---_r ~l GtiJ· Figure 2. Examples of connected and broken characters that the Saccade system correctly recognizes. 446 Martin, Rashid, Chapman, and Pittman no knowledge of the location of the character. For fields containing two digits, the average number of passes of the net on a small test set was about 2 saccades per character. For field containing five digits, the average number of saccades per character was 1.3. 4 COMPARISONS WITH OTHER SYSTEMS The Saccade system is an extension of a related integrated segmentation and recognition system we reported on at last years NIPS conference (Martin & Rashid, 1992). That system employed an exhaustive scan technique, rather than saccades, to navigate along a line of text. Essentially, the net was convolved hociwntally across a field image at a scan increment of 2 pixels. The net architecture was very similar to that of the Saccade system, except that it did not have the two distance components in the output vector. The accuracy rates of the two systems are essentially equivalent. However, the exhaustive scan version was considerably less efficient, requiring a forward pass of the netwock at every 2-pixel incremental scan position. On average it required about 5.5 forward passes per character, rather than the 1.3 focward passes per character required by the Saccade system. Over the past two years, an approach similar to the exhaustive scan method has been advanced by a number of researchers (Keeler & Rumelhart, 1992; Matan. Burges. Le CuD. & Denker. 1992). This approacb also involves convolving a network across a field image. but uses a time-delay-neural-net (1DNN). or completely local. shared weight. architecture. a smaller input window, and no explicit position labeling of characters. The IDNN approacb has algorithmic advantages over the exhaustive scan version described in the previous paragraph. because the completely shared -weight architecture enables the number of forward passes of the net to be reduced considerably. 5 CONCLUSIONS AND FUTURE WORK As stated at the beginning of this paper, we believe that the key to developing optical character recognition (OCR) systems that can mimic human reading capabilities is to develop systems that can learn the multiple foundation skills underlying human reading. This paper has repocted some progress in this regard. We have demonstrated that a relatively simple back propagation network can integrate its learning of position and category information. thereby enabling efficient navigatioo along a field of text through ballistic and corrective saccades, and accurate recognition of touching or broken characters. There is however, a long way to go before we can claim a system with capabilities similar to human reading. The present StJccade system only moves horizontally, in one dimension. Human reading operates in two-<iimensions. and in a sense. it operates in three-dimensions because it automatically operates across different scales. Human vision also employs auter matic contrast adjusunent; the Saccade system does not. Human vision has a wider field of view and employs a foveal transform. such that objects centered in the field of vision are represented at a higher resolution than objects in the periphery. This effectively expands the field of vision beyond what would be estimated simply by the size of the receptive area on the retina. As a resUlt. saccades enable very effective means of scanning a large visual area. The present artificial Saccade system has only a small field of vision. and no foveal transform. so it's saccades must necessarily be limited in size. The present system is also only oriented toward recognizing a single character centered in its input window at Training a Net to Make Saccades and Recognize Handwritten Characters 447 a time. Human reading typically only makes one or two saccades per woo1. Finally, human reading capabilities clearly integrate recognition processes with higher-level processes, to enable the redundancies of natural language to constrain the recognition decisions. References Keeler, J, & Rumelhart, D. E. (1992) A self-aganizing integrated segmentation and recognition neural network.. In Moody, J.E., Hansoo, S.1., and Lippmann, R.P., (eds.) A.dvances in Neural Information Processing Systems 4. San Mateo. CA: Mocgan Kaufmann Publishers. Malan, 0., Burges, J. C., Le Cun, Y., and Denker. J. S. (1992) Multi-Digit Recognition Using a Space Displacement Neural Netwm. In Moody. J.E .• Hanson. S.1 .• and Lippmann. R.P .• (eds.) Advances in Neural Injormation Processing Systems 4. San Mateo. CA: Mocgan Kaufmann Publishers. 488-495. Martin. O. L. & Rashid. M. (1992) Recognizing overlapping hand-printed characters by centered-OOject integrated segmentation and recognitioo. In Moody, J.E .• Hanson, S.1., and Lippmann, R.P.. (eds.) Advances in Neural Injormation Processing Systems 4. San Mateo. CA: Morgan Kaufmann Publishers. Moody. J. & Darken. C. (1988) Learning with localized receptive fields. Technical Report Yaleu/DCS/RR-649. PART V STOCHASTIC LEARNING AND ANALYSIS
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How Oscillatory Neuronal Responses Reflect Bistability and Switching of the Hidden Assembly Dynamics K. Pawelzik, H.-V. Bauert, J. Deppisch, and T. Geisel Institut fur Theoretische Physik and SFB 185 Nichtlineare Dynamik Universitat Frankfurt, Robert-Mayer-Str. 8-10, D-6000 Frankfurt/M. 11, FRG ttemporary adress:CNS-Program, Caltech 216-76, Pasadena email: klaus@chaos.uni-frankfurt.dbp.de Abstract A switching between apparently coherent (oscillatory) and stochastic episodes of activity has been observed in responses from cat and monkey visual cortex. We describe the dynamics of these phenomena in two parallel approaches, a phenomenological and a rather microscopic one. On the one hand we analyze neuronal responses in terms of a hidden state model (HSM). The parameters of this model are extracted directly from experimental spike trains. They characterize the underlying dynamics as well as the coupling of individual neurons to the network. This phenomenological model thus provides a new framework for the experimental analysis of network dynamics. The application of this method to multi unit activities from the visual cortex of the cat substantiates the existence of oscillatory and stochastic states and quantifies the switching behaviour in the assembly dynamics. On the other hand we start from the single spiking neuron and derive a master equation for the time evolution of the assembly state which we represent by a phase density. This phase density dynamics (PDD) exhibits costability of two attractors, a limit cycle, and a fixed point when synaptic interaction is nonlinear. External fluctuations can switch the bistable system from one state to the other. Finally we show, that the two approaches are mutually consistent and therefore both explain the detailed time structure in the data. 977 978 Pawelzik, Bauer, Deppisch, and Geisel 1 INTRODUCTION A few years ago, oscillatory and synchronous neuronal activity was discovered in cat visual cortex [1-3]. These experiments backed earlier considerations about synchrony in neuronal activity as a mechanism to bind features, e.g., of an object in a visual scene [4]. They triggered broad experimental and theoretical investigations of detailed neuronal dynamics as a means for information processing and, in particular, for feature binding. Many theoretical contributions tried to reproduce and explain aspects of the experimentally observed phenomena [5]. Motivated by the experiments, the models where particularly designed to exhibit spatial synchronization of permanent oscillatory responses upon stimulation by a common, connected stimulus like a bar. Most models consist of elements which exhibit a limit cycle after a simple Hopf bifurcation. The experimental data, however, contain many details which the present models do not yet completely incorporate. One of these details is the coexistence of regular and irregular episodes in the data, which interchange in an apparently stochastic manner. This interchange can be observed in the signals from a single electrode [6] as well as in the time-resolved correlation of the signals from two electrodes [7]. In this contribution we show, that the observed time structure reflects a switching in the dynamics of the underlying neuronal system. This will be demonstrated by two complementary approaches: On the one hand we present a new method for a quantitative analysis of the dynamical system underlying the measured spike trains. Our approach gives a quantitative description of the dynamical phenomena and furthermore explains the relation between the collective excitation in the network which is not accessible experimentally (i.e. hidden) and the contributions of the single observed neurons in terms of transition probability functions. These probabilities are the parameters of our Ansatz and can be estimated directly from multi unit activities (MUA) using the Baum-Welchalgorithm. Especially for the data from cat visual cortex we find that indeed there are two states dominating the dynamics of collective excitation, namely a state of repeated excitation and a state in which the observed neurons fire independently and stochastically. On the other hand using simple statistical considerations we derive a description for a local neuronal subpopulation which exhibits bistability. The dynamics of the subpopulation can either rest on a fixed point - corresponding to the irregular firing pat terns - or can follow a limit cycle - corresponding to the oscillatory firing patterns. The subpopulation can alternate between both states under the influence of noise in the external excitation. It turns out that the dynamics of this formal model reproduces the observed local cortical signals in much detail. 2 Excitability of Neurons and Neuronal Assemblies An abstract model of a neuron under external excitation e is given by its threshold dynamics. The state of the neuron is represented by its phase 4>8, which is the time passed by since the last action potential (4)8 = 0). The threshold 6 is high directly after a spike and falls off in time and the neuron can fire again when e exceeds 6. In case of noise or internal stochasticity, an excitability description of Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics 979 the dynamics of the neuron is more adequate. It gives the probability PI to fire again in dependence of the state l/J6 with PI (l/J6) = 0'( e - B( l/JS)) and 0' some sigmoid function. A monotonously falling threshold B then corresponds to a monotonously increasing excitability PI' Such a description neglects any memory in the neuron going beyond the last spike. In particular this means for an isolated neuron, that PI can be easily calculated from the inter-spike interval histogram (ISIH) Ph using the relation Ph(t) = PI(t) . (1-f: Ph (t')dt'). In that case also the autocorrelation function can be calculated from Ph(t) via G(T) = Ph(T) + f; Ph(T)G(T - t)dt. The excitability formulation sketched above is not valid for a neuron which is embedded in a neuronal assembly. However, we may use this Ansatz of a renewal process to describe the activation dynamics of the whole assembly (see section 5). The phase l/Jb = 0 here corresponds to the state of synchronous activity of many neurons in the assembly, which we call burst for convenience. Since the dynamics of the network can differ from the dynamics of the elements we expect the function pj(l/Jb) which now describes the burst excitability of the whole assembly to be different from·the spike excitability PI(l/Js) of the single neuron. A simple example for this is a system of integrate and fire neurons in which oscillatory and irregular phases emerge under fixed stimulus conditions( [8, 9] and section 5). Contrary to the excitability of the single refractory element the burst excitability pj of the system has a maximum at l/Jb = T which expresses the increased probability to burst again after the typical oscillation period T, i.e. the maximum represents a state 0 of oscillation. The assembly, however, can miss to burst around l/Jb = T with a probability PO-+6 and switch into a second state s in which the probability Ps-+o to burst again is reduced to a constant level. The switching probabilities PO-+6 and Ps-+o can be easily calculated from Pj. In this way the shape of pj distinguishes a system with an oscillatory state from a system which is purely refractory but which nevertheless can still have strong modulations in the autocorrelogram [13]. 3 Hidden states and stochastic observables The single neuron in an assembly, however, need not be strictly coupled to the state of the assembly, i.e. a neuron may also spike for l/Jb > 0 and it may not take part in a burst. This stochastic coupling to an underlying process suffices to destroy the equivalence of Ph and the autocorrelogram C( T) =< s(t)s(t + T) >t of the spike train set) E {O, I} (Fig 1). We therefore include the probability PObs(l/Jb) to observe a spike when the assembly is in the state l/Jb into our description (Fig. 2). The unlikely case where the spike represents the burst corresponds to the choice Pobs = 84>b,O' 4 Application to Experimental Data While our approach is quite general, we here concentrate on the measurements of Gray et al. [2] in cat visual cortex. Because our hidden state model has the structure of a hidden Markov model we can obtain all the parameters POb6(l/J) and 980 Pawelzik, Bauer, Deppisch, and Geisel 0.10 0.08 C(T) 0.0(3 0.04 0.02 20 40 60 80 T[mS] Figure 1: Correlogram of multi unit activities from cat visual cortex (line). Correlaograms predicted from the ISIH (6) and from the hidden state model (+). Measurement Level Network Level 4>: ,. p (n) f Figure 2: The hidden state model. While Pj(q}) governs the dynamics of assembly states q;b, Pobs (q;b) represents the probability to observe a spike of a single neuron. Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics 981 0.25[ 0.20 Pj( 4>b) 0.15 010 005 O.Oob~~~:::t:!~~ ___ ----:,::--_ __ ~. o 020 0.10 o.ooL---_ __ ---=-~:=:!:±.""--~:!T~~~~. o 10 20 Figure 3: Network excitability pJ and single neuron contribution Pobs estimated from experimental spike trains (A17, cat). pJ (¢;) directly from the multi unit activities using the well known Baum-Welch algorithm[lO]. The results can be seen in Fig. 3. The excitability shows a typical peak around the main period at T = 19ms, which indicates a state of oscillation. For larger phases we see a reduced excitability which reveals a state of stochastic activity (Pobs (¢;b > T) > 0). The spike observation probability Pobs (¢;) is peaked near the burst and is about constant elsewhere. This means that we can characterize the data by a stochastic switching between two dynamical states in the underlying system. Because of the stochastic coupling of the single neuron to the assembly state this can only hardly be observed directly. The switching probabilities between either states calculated from pJ coincide with results from other methods [11]. From the excitability pJ and the spike probabilities Pobs we now obtain the autocorrelation function C(r) = f4>f4>' Pobs(¢;')M(¢;',¢;fPobs(¢;)p(¢;)d¢;'d¢;, with M being the transition matrix of the Markov model (see also below). The result is compared to the true autocorrelation C( r) in Fig. 1. The excellent agreement confirms our simple Ansatz of a renewal process for the hidden burst dynamics of the assembly. 5 Bistability and Switching in Networks of Spiking Neurons The above results indicate that the dynamics of a cortical assembly includes bistability rather than a simple Hopf bifurcation. In order to understand how this bistability emerges in a network we go one step back and derive a model for a neuronal subpopulation on the basis of spiking neurons. We assume again that the internal state of the neuron is given by the threshold function e depending on the time since the last spike event and that the excitability of the neuron can be described by a 982 Pawelzik, Bauer, Deppisch, and Geisel c o ':=J Q) c: 4 ~----'l.r---....Il ,..----" 3 2 ti me t Figure 4: Illustration of assembly state representation by a phase density. firing probability Pf. In a network, however, the input to the neuron has external contributions i ext as well as from within iint i.e. Pf(<P,t) = sigm(wextiext(t) + Wintiint(t) - 8(<p)). For a more formal treatment of the dynamics of such a network we characterize the assembly state by a phase density p( <P, t) which gives the relative amount of neurons in the assembly which are in phase <p at time t (Fig 4). Discretizing the internal phases <p, we transform p(<p, t) to p(j), a vector whose components i give the probability to be in phase <Pi E [(i -1)ilt,iilt] at time tj = jilt. The number T of components is chosen large enough to ensure that P f (T, j) = Pf (T ilt, jilt) does not change any more. This vector evolves in time according to p(j + 1) = M(j)p(j) (1) with 0 Pf(I,}) Pf(2,j) Pf(T-l,j) Pf(T,j) 1 0 0 0 I-Pf (1, j) 0 0 M(j) = I-Pi (2,j) 0 0 0 1-Pf(T-1,j) I-Pf(T,j) beeing a matrix that incorporates the effects offiring (reset) via the firing probability Pf(i,j). It remains to define the lateral interaction in the sUbpopulation. Clearly only the fraction of neurons that fire can interact, therefore we have l.int = g(po). Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics 983 ~ 0 .c ... 1.0 0.8 0.6 0.4 l0.2 t-",,"IA~ 0.0 o ltv If'"'" ~~ ~ .AlIJ IA 200 400 600 BOO 1000 Figure 5: Switching in the assembly dynamics under external fluctuations. Note that Po denotes the time dependent firing rate. Numerically iterating the dynamics (1) we find, that the distribution can evolve in two distinct ways, depending on the lateral interaction function 9 and the initialization. First Ii can relax to a fixed point, corresponding to a constant fraction of the neurons firing at a particular time. On the level of an individual neuron, this corresponds to a stochastic firing characteristics, and in the measurements this state corresponds to the irregular periods. Secondly the distribution can evolve according to a limit cycle, with a rather large fraction of the neurons in a narrow band of phases i.e. the neurons are synchronous. We find parameter combinations Wext, Wint where both states coexist, i.e. we find bistability, if the interaction function is nonlinear, like g( x) ex: x2 or g( x) ex: x4. N onlinearities of this kind can be brought about by a different type of preferred synapses with a nonlinear transmission characteristic (like an NMDA-receptor synapse) for the corticocortical projections, as compared to the thalamocortical projections. At this point we only want to make the point that our simple model system can exhibit bistability under reasonable assumptions. In the bistable regime some initializations of Ii lead to the oscillatory state, some to the stochastic state. Adding some noise to the external input the system can also switch between the two states(Fig. 5). In this way our simple local model can capture the switching phenomenon which is inherent in the experimental data [12]. 6 Summary and Synthesis We presented two complementary approaches to the dynamics of neuronal subpopulations, a phenomenological one which captures the time structure in measured spike trains and a neuronal one which provides a formal description of the dynamics of assemblies of spiking neurons. In the phenomenological approach we introduced the hidden state model which revealed that the system underlying the multi unit activities from the cat switches between states of oscillatory and stochastic activity. The analysis of the phase density dynamics showed, that bistability and switching emerges in networks of spiking neurons when the neuronal interaction is nonlinear. It remains to show that these approaches are also quantitatively consistent, i.e. that they are two sides of the same medal. Instead of a formal proof we only remark here that the parameters of the HSM can be extracted directly from the dynamics of the PDD under external noise. For this purpose one only needs to evaluate the interburst interval distribution from which pj can be calculated. Pobs is easily estimated as the average shape of Po(t) between successive bursts. We find, that already this procedure gives HSMs which accurately reproduce the correlation function of the 984 Pawelzik, Bauer, Deppisch, and Geisel full firing dynamics poet). This means that the HSM captures relevant aspects of the assembly dynamics including the relation between the network dynamics and the contributions of single neurons. References [1] Gray C.M., Singer W., Stimulus-Specific Neuronal Oscillations in Cat Visual Cortex: a Cortical Functional Unit, Soc.Neurosci.Abstr. 13.404.3 (1989). [2] Gray C.M., Konig P., Engel A.K., and Singer W., Oscillatory Responses in Cat Visual Cortex Exhibit Inter-Columnar Synchronization which Reflects Global Stimulus Properties Nature 338, pp. 334-337 (1989). [3] Eckhorn R., Bauer R., Jordan W., Brosch M., Kruse W., Munk M., and Reitboeck H.J., Coherent Oscillations: a Mechanism of Feature Linking in the Visual Cortex'l, BioI. Cyb. 60, pp121-130 (1988). [4] C. v.d.Malsburg The Correlation Theory of Brain Function Internal Report 812, Max-Planck-Institute for Biophysical Chemistry, Gottingen, F.R.G. (1981). [5] Schuster H.G. (Ed.), Nonlinear Dynamics and Neuronal Networks, VCH Weinheim, Heidelberg (1991) [6] Pawelzik K., Bauer H.-U., Geisel T. Switching between predictable and unpredictable states in data from cat visual cortex, talk at CNS San Francisco 1992, to appear in the CNS Proceedings. [7] Gray C.M., Engel A.K., Konig P., Singer W., Temporal Properties of Synchronous Oscillatory Interactions in Cat Striate Cortex, in: Nonlinear Dynamics and Neuronal Networks, Ed. H.G. Schuster, VCH Weinheim, pp. 27-55 (1991) [8] Deppisch J., Bauer H.-U., Schillen T., Konig P., Pawelzik K., Geisel T., Stochastic and Oscillatory Burst Activities, accepted for ICANN'92, Brighton, UK. (1992). [9] Deppisch J., Bauer H.-U., Schillen T., Konig P., Pawelzik K., Geisel T., Alternating Oscillatory and Stochastic States in a Network of Spiking Neurons, submitted to Biol.Cyb. (1992). [10] Rabiner, L.R., A Tutorial on Hidden-Markov Models and Selected Applications in Speech Recognition Proc. IEEE 11, 2 pp. 257-286 (1989). [11] Bauer H.TU., Deppisch J., Geisel T., Pawelzik K., in preparation. [12] Bauer H.U., Pawelzik K., Alternating Oscillatory and Stochastic Dynamics in a Model for a Neuronal Assembly, Physica D, submitted. [13] Schuster H.G., Koch C., Burst Synchronization Without Frequency-Locking in a Completely Solvable Network Model, in Moody J.E., Hanson S.J., Lippmann R.P. (Eds.), Neural Information Processing Systems 4, p. 117, Morgan Kauffmann (1992).
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Summed Weight Neuron Perturbation: An O(N) Improvement over Weight Perturbation. Barry Flower and Marwan Jabri SEDAL Department of Electrical Engineering University of Sydney NSW 2006 Australia Abstract The algorithm presented performs gradient descent on the weight space of an Artificial Neural Network (ANN), using a finite difference to approximate the gradient The method is novel in that it achieves a computational complexity similar to that of Node Perturbation, O(N3), but does not require access to the activity of hidden or internal neurons. This is possible due to a stochastic relation between perturbations at the weights and the neurons of an ANN. The algorithm is also similar to Weight Perturbation in that it is optimal in terms of hardware requirements when used for the training ofVLSI implementations of ANN's. 1 INTRODUCTION Optimization of the weights of an ANN may be performed by, the application of a gradient descent teclmique. The gradient may be calculated directly as in Backpropagation, or it may be approximated by a Finite Difference Method which is what we concern ourselves with in this paper. These methods lend themselves to the task of training hardware implementations of ANNs where real estate is at a premium and synaptic density is of great importance. Neuron Perturbation (NP), as described by the Madaline Rule ill (MRllI) (Widrow and Lehr, 1990), is a teclmique that approximates the gradient of the Mean Square Error (MSE) with respect to the change at a given neuron by applying a small perturbation to the input of the neuron and measuring the change in the MSE. The weight dE I1wij = -tl'-:l-'Xr (1) onet. I update is then calculated from the product of this gradient measure and the activation of 212 Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation 213 the neuron from which the weight is fed, as described by (1). Weight Perturbation (WP), as described by Jabri and Flower (Jabri and Flower, 1992) is a neural network training techniques based on gradient descent using a Finite Difference method to approximate the gradient. The gradient of the MSE with respect to a weight is approximated by applying a small pertubation to the weight and measuring the change in the MSE. This gradient is then used to calculated the weight update such that: iJE aWr = -11· :l-.. (2) 'J uw .. I) The advantages of WP over NP are that it performs better when limited precision weights are used, as shown by Xie and Jabri (Xie and Jabri, 1992), and is optimal with respect to hardware requirements when used to train VLSI implementations of ANNs. However, WP has O(~) computational complexity whilst NP has O(N3) computational complexity. Summed Weight Neuron Perturbation (SWNP) is similar to NP in that it has a computational complexity of O(N3) but it has the added advantage that the activation of internal neurons does not need to be known. The cost of this reduced computational complexity is that SWNP needs to save the perturbation vector used. In the following sections a description of the SWNP algorithm is provided and, finally, some experimental results are presented. 2 THE SUMMED WEIGHT NEURON PERTURBATION ALGORITHM A subsection of a feedforward ANN containing N neurons is shown in Figure 1. on which nomenclature the following derivation is based. FIGURE 1: Description Of Indices Used To Describe The Neurons Weights And Perturbations In An ANN. In a feedforward network of size N neurons the activation of a given neuron is determined by: Xi (P) = Ii (neti (p» , and neti (p) = ~WilXl (p) , (3) and Ii (y) is the ith neuron transfer function, Xi (p) is the activation of the ith neuron for the pth pattern, and W iI is the weight connecting the Ith neuron's output to the ith neuron's input. The error function, (MSE), is defmed as in (4), where T is the set of output neurons and dt (p) is the expected value of the output on the kth neuron. The change in E (p) 214 Flower and Jabri with respect to a given weight may then be expressed as (5). 1 E(p) = 2: E (dt(p) -xt (p»2. (4) tE T oE (p) oE (p) dwij = aneti (p) .Xj (p) . (5) The first term of on the right-hand side of (5) can be determined using a Finite Difference, which in this case is a Forward Difference, so that: oE(p) AEr. (p) ~. + 0 (ri), f (6) =-------c- = oneti (p) where, AEr(p) = Er (p) -E(p), (7) , , and r i is the perturbation applied to the ith neuron, E r. (p) is the error for the pth pattern , with a perturbation applied to the ith neuron and E (p) is the error for the pth pattern without a perturbation applied to any neurons. The error introduced by the approximation is represented by the last term on the right-hand side in (6). The perturbation of one or more of the weights that are inputs to the qth neuron can be thought of as being equal to some perturbation applied directly to that neuron. Hence: rq = ~Yqrl(P)' (8) where Yql is the perturbation applied to weight w ql. As will be shown, perturbing the qth neuron by perturbing all the weights feeding into it, enables the sign of the gradient o~~) to be determined without performing the product on the right-hand side of (5). fJ Further more, the activation of hidden neurons, (i.e. Xj (p) in (5» need not be known. The contribution of the perturbation of weight w ij to the perturbation of the ith neuron is Yi/j(p) . (9) Let us take the degenerate case where there is only one weight for the ith neuron. Then the gradient of the MSE with respect to weight wij is: AEr (p) Xj (p) = '() + 0 (r.) y.x). p f fJ ll.Er. (p) = ' +0 (r.) , Yij f (10) Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation 215 noting that Xj (p) has been eliminated. In the general case where the ith neuron has more than one weight the gradient with respect to weight w ij is shown in (11). where, oE(p) Ow .. fJ Mr_ (p) Xj (p) = I r. +O(ri) f Mr. (p) = I + O(r.) ':P ij f (11) r. f ':Pij = x. (p) . (12) J The form of (10) and (11) are the same and it will be shown that y .. can be substituted for fJ ':P .. in (1) due to a stochastic relationship between them. fJ Let us represent the sign of y .. and ':P .. as either + 1 or -1 such that: fJ lJ and I ':P iJ~ v .. = \TI' fJ T .. fJ (13) The set of all possible states for the system represented by the vector (1.1"J v .. ) , assuming fJ f} y .. and ':P .. are never zero, is: lJ fJ {(-1, -1), (-1, 1), (1, -1), (1, 1)} . (14) and it can be seen that when 1.1 .. = v .. then the sign of the gradient of the MSE with fJ fJ respect to weight wij given by (0) is the same as that given by (11). If the sign of Yij is chosen randomly then the probability of 1.1 .. = v .. being true is 0.5, from (4), and so (0) fJ fJ will generate a gradient that is in the correct direction 50% of the time. This in itself is not sufficient to allow the network to be trained as it will take as many steps in the incorrect direction as the correct direction if the steps themselves are of the same size, (i.e. the magnitude of r. is the same for a step in the correct direction as a step in the incorrect direcf tion). Fortunately it can be shown that the size of the steps in the correct direction are greater than those in the incorrect direction. Let us take the case where a particular y .. is chosen f} such that 1.1 .. = v ... fJ IJ (15) Now by substituting (8), (2) and (13) into (5) we get: 216 Flower and Jabri rearranging to give, ~ Yitx t (p) x· J - -----'--Yij ~YitXt (p) x· J Yi/j ~YitXt (p) (16) (17) which implies that the contribution to r. made by the pertUIbation y .. is of the same sign I v as r .. Let us designate this neuron pertUIbation as r. (A) . Now we take the other possible I I case where, JIij * V ij' (18) assuming every other parameter is the same, and only the sign of y .. is changed. The IJ equality in (17) is now untrue and the contribution to r. made by the perturbation y .. is of I IJ the opposite sign as r .. Let us designate this neuron perturbation as r . (B) . From (8) we I I can determine that, (19) Equation (19) shows the relationship between the two possible states of the system where r. (A) represents the summed neuron perturbation for a selected weight perturbation y .. I v that generates a step in the corrected direction and ri (B) is similar but for a step in the incorrect direction. Clearly the correct step is always calculated from an approximated gradient that is larger than that for an incorrect step as the neuron perturbation is larger. The weight update rule then becomes: Mr. (p) ~Wij = -Tl. I Yij (20) The algorithm for SWNP is shown as pseudo code in Figure 2. 2.1 HARDWARE COMPATmILITY OF SWNP This optimisation technique is ideally suited to the training of hardware implementations of ANN's whether they consist of discrete components or are VLSI technology. The speed up over WP of 0 (N) achieved is at the cost of an 0 (N) storage requirement but this sto~ge can be achieved with a single bit per neuron. SWNP is the same order of complexity as NP but does not require access to the activation of internal neurons and therefore can treat a network as a "black box" into which an input vector and weight matrix is fed and an Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation 217 output vector is received. While (total error> error threshold) { For (all patterns in training set) { Select next pattern and training vector, Forward Prop.;Measure, (calculate) and save error; Accumulate total error; For (all non-input neurons) { For (all weights of current neuron) { } Apply & Save perturbation of random polarity; Forward Prop.;Measure, (calculate) and save &!rror; For (all weights of current neuron) { Restore value of weight; Calculate weight delta using saved perturbation value; } If (Online Mode) Update current weight; If (Online Mode) } Forward Prop.; Measure, (calculate) and save new error; If (Batch Mode) { } } } For (all weights) Update current weight; FIGURE 2: Algorithm in Pseudo Code for Summed Weight Neuron Perturbation. 3 TEST RESULTS USING SWNP The results for a series of tests are shown in the next three tables and are summarised in Figure 4. The headings are, N the number of neurons in the network, P the number of patterns in the training set, FF -SWNP the number of feedforward passes for the SWNP Algorithm, FF -WP the number of feedforward passes for the WP Algorithm, and RA no the ratio between the number of feedforward passes for WP against SWNP. The feedforward passes are recorded to 1 significant figure. The results for a series of simulations comparing the performance of SWNP against WP are shown in Table 1. The simulations utilised floating point synaptic and neuron precis1ons. The results for a series of simulations comparing the performance of SWNP against WP are shown in Table 2. The simulations utilised limited synaptic precision, (i.e. 6 bits) and floating point neuron precisions The results for a series of experiments comparing the performance of SWNP against WP are shown in Table 2. Note: the training algorithm are the variations of WP and SWNP that are combined with the Random Search Algorithm (RSA). The results reported are averaged over 10 trials. An example of the training error trajectories of WP and SWNP for the Monk 2 problem are shown in Figure 3. 218 Flower and J abri Table 1: Performance Of SWMP Versus WP, Comparing Feedforward Operations To Convergence. (Simulations With Floating Point Precision) PROBLEM N P FF-SWNP FF-WP ERROR RATIO XOR 3 4 1.6xlQ3 1.9xlcP 0.0125 1. 22 4 Encoder 5 4 0.9xlcP 1.8xlcY 0.0125 1.84 8 Encoder 11 8 1.5xlOS 4.5xlOS 0.0125 2.88 ICEG 15 119 3.7xlOS 7.9xlrf' 0.0125 21.34 Table 2: Performance Of SWNP Versus WP, Comparing Feedforward Operations To Convergence. (Simulations With Limited Precision) PROBLEM N P SWNP WP ERROR RATIO Monk 1 4 129 1.OxlOS 1.9x106 0.001 19.38 MonIa 17 169 3.6xlOS 6.8x106 0.0005 18.71 Monk3 17 122 1.2xl06 7.1xl06 0.022 5.87 IECG 55 5 8 1.6xl04 7.2xl04 0.0001 4.2 Table 3: Performance Of SWNP Versus WP, Comparing Feedforward Operations To Convergence. (Hardware Implementation) PROBLEM N P SWNP WP ERROR RATIO ECG 55 5 8 3.1xlQ3 3.6xlQ3 0.00001 1.13 ECG045 5 8 1.1xl04 2.Oxl04 0.001 1.78 MONX 2 PROBLEM YBKa lIT' ..,CO -lIWRP ",",co _co >:"'co :IOOCO 1I0CO ItDCO ,OOCO 1:"'CO 'OOCO lOCO «lCO ooCO :lOCO 000 OCO 2).(1) «),(1) «).IX) tom 10000 12).00 leO) J!IIIOC2B FIGURE 3: Comparison ofWP and SWNP For Monk 2 Problem Summed Weight Neuron Perturbation: An (O)N Improvement over Weight Perturbation 219 FIGURE 4: Comparison of the number of Feedforward passes performed to achieve convergence on a range of problems using SWNP and WP. 20 18 16 14 12 10 8 6 4 2 XORx~02 4ENCODERxI 8ENCODERxI ICEGxl MONK. 1 xl MONK.2xl MONlOxl mwsm SWNP Feedforward Passes _ WP Feedforward Passes lCEG 55xl 4 CONCLUSION ECG 55xl ECG045xl The algorithm presented, SWNP, performs gradient descent on the weight space of an ANN, using a fInite difference to approximate the gradient. The method is novel in that it achieves 0 (N3) computational complexity similar to that of Node Perturbation but does not require access to the activity of hidden or internal neurons. The algorithm is also similar to Weight Perturbation in that it is optimal in terms of hardware requirements when used for the training of VLSI implementations of ANN's. Results are presented that show the algorithm in operation on floating point simulations, limited precision simulations and an actual hardware implementation of an ANN. References labri, M. and Flower, B. (1992). Weight perturbation: An optimal architecture and learning technique for analog vlsi feedforward and recurrent multilayer networks. IEEE Transactions on Neural Networks, 3(1):154-157. Widrow, B. and Lehr, M. A. (1990). 30 years of adaptive neural networks: Perceptron, madaline, and backpropagation. Proceedings of the IEEE, 78(9):1415-1442. Xie, Y. and lahri, M. (1992). Analysis of the effects of quantization in multilayer neural networks using a statistical model. IEEE Transactions on Neural Networks, 3(2):334-338.
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Extended Regularization Methods for N onconvergent Model Selection W. Finnoff, F. Hergert and H.G. Zimmermann Siemens AG, Corporate Research and Development Otto-Hahn-Ring 6 8000 Munich 83, Fed. Rep. Germany Abstract Many techniques for model selection in the field of neural networks correspond to well established statistical methods. The method of 'stopped training', on the other hand, in which an oversized network is trained until the error on a further validation set of examples deteriorates, then training is stopped, is a true innovation, since model selection doesn't require convergence of the training process. In this paper we show that this performance can be significantly enhanced by extending the 'non convergent model selection method' of stopped training to include dynamic topology modifications (dynamic weight pruning) and modified complexity penalty term methods in which the weighting of the penalty term is adjusted during the training process. 1 INTRODUCTION One of the central topics in the field of neural networks is that of model selection. Both the theoretical and practical side of this have been intensively investigated and a vast array of methods have been suggested to perform this task. A widely used class of techniques starts by choosing an 'oversized' network architecture then either removing redundant elements based on some measure of saliency (pruning), adding a further term to the cost function penalizing complexity (penalty terms), and finally, observing the error on a further validation set of examples, then stopping training as soon as this performance begins to deteriorate (stopped training). The first two methods can be viewed as variations of long established statistical techniques 228 Extended Regularization Methods for N onconvergent Model Selection 229 corresponding in the case of pruning to specification searches, and with respect to penalty terms as regularization or biased regression. The method of stopped training, on the other hand, seems to be one of the true innovations to come out of neural network research. Here, the model chosen doesn't require the training process to converge, rather, the training process is used to perform a directed search of weight space to find a model with superior generalization performance. Recent theoretical ([B,C,91], [F,91], [F,Z,91]) and empirical results ([H,F ,Z,92], [W,R,H,90]) have provided strong evidence for the efficiency of stopped training. In this paper we will show that generalization performance can be further enhanced by expanding the 'nonconvergent method' of stopped training to include dynamic topology modifications (dynamic pruning) and modified complexity penalty term methods in which the weighting of the penalty term is adjusted during the training process. Here, the empirical results are based on an extensive sequence of simulation examples designed to reduce the effects of domain dependence on the performance comparisons. 2 CLASSICAL MODEL SELECTION Classical model selection methods are generally divided into a number of steps that are performed independently. The first step consists of choosing a network architecture, then either an objective function (possibly including a penalty term) is chosen directly, or in a Bayesian setting, prior distributions on the elements of the data generating process (noise, weights in the model, regularizers, etc.) are specified from which an objective function is derived. Next, using the specified objective function, the training process is started and continued until a convergence criterion is fulfilled. The resulting parametrization of the given architecture is then placed in a 'pool' from which a final model will be selected. The next step can consist of a modification of the network architecture (for example by pruning weights/hidden-neurons/input-neurons), or of the penalty term (for example by changing its weighting in' the objective function) or of the Bayesian prior distributions. The last two modifications then result in a modification of the objective function. This establishes a new framework for the training process which is then restarted and continued until convergence, producing another model for the pool. This process is iterated until the model builder is satisfied that the pool contains a reasonable diversity of candidate models, which are then compared with one another using some estimator of generalization ability, (for example, the performance on a validation set). Stopped training, on the other hand, has a fundamentally different character. Although the choice of framework remains the same, the essential innovation consists of considering every parametrization of a given architecture as a potential model. This contrasts with classical methods in which only those parametrizations corresponding to minima of the objective function are taken into consideration for the model pool. Under the weight of accumulated empirical evidence (see [W,R,H,90], [H,F,Z,92]) theorists have begun to investigate the properties of this technique and have been able to show that stopped training has the same sort of regularization effect (Le. reduction of model variance at the cost of bias) that penalty terms provide (see 230 FinnoH, Hergert, and Zimmermann [B,C,91], [F,91]). Since the basic effect of pruning procedures is also to reduce network complexity (and consequent model variance) one sees that there is a close relationship in the instrumental effects of stopped training, pruning and regularization. The question remains whether (or under what circumstances) anyone or combination of these methods produces superior results. 3 THE METHODS TESTED In our expirements a single hidden layer feedforward network with tanh activation functions and ten hidden units was used to fit data sets generated in such a manner that network complexity had to be reduced or constrained to prevent overfitting. A variety of both classical and non convergent methods were tested for this purpose. The first we will discuss used weight pruning. To characterize the relevance of a weight in a given network, three different test variables were used. The first simply measures weight size under the assumption that the training process naturally forces nonrelevant weights into a region around zero. The second test variable is that used in the Optimal Brain Damage (OBD) pruning procedure of Le Cun et al. (see [L,D,S,90]). The final test variables considered are those proposed by Finnoff and Zimmermann in [F,Z,91], based on significance tests for deviations from zero in the weight update process. Two pruning algorithms were used in the experiments, both of which attempt to emulate successful interactive methods. In the first algorithm, one removes a certain fixed percentage of weights in the network after a stopping criterion is reached. The reduced network is then trained further until the stopping criterion is once again fulfilled. This process is then repeated until performance breaks down completely. This method will be referred to in the following as auto-pruning and was implement.ed using all three types of test variables to determine the weights to be removed. The only difference lay in the stopping criterion used. In the case of the OBD test variables, training was stopped after the training process converged. In the case of the statistical and small weight test variables, training was stopped whenever overtraining (defined by a repeated increase in the error on a validation set) was observed. A final (restart) variant of auto-pruning using the statistical test variables was also tested. This version of auto-pruning only differs in that the weights are reinitialized (on the reduced topology) after every pruning step. In the tables of results presented in the appendix, the results for auto-pruning using the statistical (resp. small weight, resp. OBD) test variables will be denoted by p. (resp. G*, resp. 0*). The version of auto-pruning using restarts will be denoted by p •. The second method uses the statistical test variables to both remove and reactivate weights. As in auto-pruning the network is trained until overfitting is observed after a fixed number of epochs, then test values are calculated for all active and inactive weights. Here a fixed number £ > 0 is given, corresponding to some quantile value of a probability distribution. If the test variable for an active weight falls below £ the weight is pruned (deactivated). For weights that have already been set to zero, the value of the test variables are compared with £, and if larger, the weight is reactivated with a small random value. Furthermore, the value of £ is increased by some ~£ > 0 after each pruning step until some value £ma~ is reached. This method is referred to as epsi-pruning. Epsi-pruning was tested in versions both with (e·) Extended Regularization Methods for N onconvergent Model Selection 231 and without restarts (E*). Two complexity penalty terms were considered. These consist of a further term C-\(w) added to the error function which forces the network to achieve a compromise between fit and network complexity during the training process; here, the parameter A E [0,00) controls the strength of the complexity penalty. The first is the quadratic term, the first derivative of which leads to the so-called weight decay term in the weight updates (see [H,P,S9]). The second is the Weigend/Rumelhart penalty term (see [W,R,H,91]). The weight decay penalty term was tested using two techniques. [n the first of these, (D*), A was held constant throughout the training process. In the second, (d*), ..\ was set to zero until overtraining was observed, then turned on and held constant for the remainder of the training process. The Weigend/Rumelhart penalty term was also tested using these two methods (denoted in the following tables by W·, resp. w*). Further, the algorithm suggested by A. Weigend in [W,R,H,91] in which the value of A is varied during training was considered (wF). In addition to the pruning and penalty term methods investigated, two (simple) versions of stopped training were tested, in one case (nN) with a constant learning step throughout, and in the other (nF) with the step size reduced after overtra.ining was observed. Finally three benchmarks were included. All these involved training a network until convergence to emulate the situation when no precautions are taken to prevent overfitting other than varying the number of hidden units. The number of hidden units in these benchmark tests was set at three, six and ten, (#3, #6, ##) this last network having then the same topology as that used in the remaining tests. 4 THE DATA GENERATION PROCESSES To test the methods under consideration, a number of processes were used to generate data sets. By testing on a sufficiently wide range of controlled examples one hopes to reduce the domain dependence that might arise in the performance comparisons. The data used in our experiments was based on pairs (Yi, z~, i = 1, ... , T, TEN with targets 'iii E R and inputs Zi = (xt, ... , zf) E [-1,1] , where Yi = g(zt, ... , xi) + Ui, for j, KEN. Here, 9 represents the structure in the data, xl, ... , z1 the relevant inputs, xJ+1, ... , zK, the irrelevant or decoy inputs and Ui a stochastic disturbance term. The first group of experiments was based on an additive structure 9 having the following form with j = 5 and K = 10, g(xf, ... , z~) = E:=l l(aA: z1), aA: E R and I either the identity on R or sin. The second class of models investigated had a highly nonlinear product structure 9 with j = 3, K = 10 and g(zt, ... , zr) = n:=l I( ale x1), ale E R and I once again either the identity on R or sin. The next structure considered was constructed using sums of Radial Basis Functions (RB ') 11 ( 1 5) - ,,8 (1)' (,,5 (a'""- zon2 ) • h A:,I F s as fo ows, 9 zi' ... , zi L,.../=l exp L,...A:=l 2q2 ,Wlt a E R for k = 1, ... ,5, I = 1, ... ,8. Here, for every 1= 1, ... ,8 the vector parameter (a l ", ... , a lS,,) corresponds to the center of the RBF. The final group of experiments were conducted using data generated by feedforward network activation functions. The network used for this task had fifty input units, two hundred hidden units and 232 Finnoff, Hergert, and Zimmermann one output. In every experiment, the data was divided into three disjoint subsets 1)t,1)",1)g: The first set 1), was used for training, the second 1)" (validation) set to test for overfitting and to steer the pruning algorithms and the third 1), (generalization) set to test the quality of the model selection process. 5 DISCUSSION The results of the experiments are given ~elow. Here we give a short review of the most interesting phanomena observed. Notable in a general sense is a striking domain dependence in the performance, which illustrates the danger of basing a comparison of methods on tests using a single (particularly small) data set. Another valuable observation is that by testing at higher levels of significance, apparent performance differences can dwindle or even disappear. Finally, one sees that even in the examples without noise that overfitting occurs, which contradicts the frequently stated conviction that overfitting is noise fitting. With regard to specific methods, one sees that all the methods tested significantly improved generalization performance when compared to the benchmarks. Further, the results show that the extended non convergent methods are on average superior (sometimes dramatically so) than their classical counterparts. In particular, the performance of penalty terms is greatly enhanced if they are first introduced in the training process after overtraining is observed. Further, dynamic pruning using the statistical or even the small weight test variables produces significantly better results than stopped training alone or using the Optimal Brain Damage (OBD) weight elimination method which requires training to minima of the objective function. A final notable observation is that the pruning methods (especially those using resarts) generally work better in the examples with a great deal of noise, while the penalty term methods are superior when the structure is highly nonlinear. 6 TABLES OF RESULTS The experiments were performed as follows: First, each data generating process was used to produce six independent sets of data and initial weights to increase the statistical significance of observed effects and to help reduce the effects of any data set specific anomalies. In a second step, the parameters of the training processes were optimized for each example by extensive testing, then a fixed value for each parameter was chosen for use across the entire range of experiments. With these parameters, each method was tested on all of the six data sets produced by one data generating process. Both the penalty terms and the pruning methods were tested with different settings of the relevant parameters in each model. The parameter values used in the simulations and an overview of the methods tested are collected in the following two tables. Extended Regularization Methods for Nonconvergent Model Selection 233 6.1 Parameter Settings of the Experiments lze earn tep Vt!V,,/V before/after overfitting exp __ n 4 o 10 . 5 O. 5 exp_3-n 0.3 400/200/1000 0.05/0.005 exp_6-n 0.6 400/200/1000 0.05/0.005 id_7..n 0.7 200/100/1000 0.05/0.005 id_B_n O.B 200/100/1000 0.05/0.005 id_9_n 0.9 200/100/1000 0.05/0.005 n_Ojd 0.0 1400/600/1000 0.05/0.01 n_Lid 0.1 1400/600/1000 0.05/0.01 n_2jd 0.2 1400/600/1000 0.05/0.01 n_O-Bin 0.0 1400/600/1000 0.05/0.01 n_1-Bin 1400/600/1000 0.05/0.01 n_2-Bin 1400/600/1000 0.05/0.01 net_ -n 0 net_3..n 0.3 400/200/1000 0.05/0.005 net_6..n 0.6 400/200/1000 0.05/0.005 sin_O-n 0.0 400/200/1000 0.05/0.005 sin_3_n 0.3 400/200/1000 0.05/0.005 sin_6_n 0.6 400 200/1000 0.05/0.005 6.2 Overview of Methods Tested 234 Finnoff, Hergert, and Zimmermann The following tables give categorical rankings of the results. The rankings were calculated as follows: The method with the best performance was given ranking I, then the performance of each following method was compared with that of the method on the first position using a modified t-test statistic. The first method in the list whose test results deviated from that on the first position to at least the quantile value of the statistic given at the head of the table was then used to start the second category. All those whose test results did not deviate by at least this amount were given the same ranking as the leading method of the category, (in this case 1). Following categories were then formed in an analogous fashion using test results measured against the performance of the leading method at the head of the category. The results are presented in two tables. The first contains the results for the data generating processes without noise and the second for the models with noise. The categorical rankings given were determined using the procedure described above at a 0.9 level of significance. The ordering of the methods given, listed in the first column, is based on the average ranking over all the simulations listed in the table. This average is given in the second column. 6.2.1 Data Generating Processes without Noise Classification by objective function, ta = 0.9 method av exp_O_n n_O.Jd n_O_sm neLO_n slD_O_n d* 1.6 1 3 1 2 1 P'" 1.8 2 2 2 1 2 w'" 2.0 1 2 3 3 1 wf' 2.0 2 1 3 2 2 G* 2.2 2 3 3 1 2 t;'" 2.6 2 5 3 1 2 0'" 2.6 3 4 2 2 2 p* 2.6 4 5 2 1 1 nF 3.0 3 5 3 2 2 e'" 3.8 4 6 4 3 2 nN 3.8 4 7 4 1 3 ## 5.2 5 8 4 3 6 W* 5.6 8 10 7 1 2 D* 5.8 8 11 7 2 1 6# 6.2 6 9 6 5 5 3# 6.4 7 12 5 4 4 Extended Regularization Methods for Nonconvergent Model Selection 235 6.2.2 Data Generating Processes with Noise Classification by objective function, ta = 0.9 method av exp exp id n_l n_l n_2 _3_n _6_n _9_n Jd ..sin Jd P'" 2.1 3 1 4 2 2 1 d'" 2.2 5 5 3 1 1 2 p'" 2.2 2 1 1 3 5 3 wI" 2.2 4 5 3 1 4 1 e'" 2.2 1 2 1 4 5 3 ~'" 2.6 3 3 2 4 4 3 <1'" 2.7 4 4 3 3 3 3 0'" 2.8 5 5 3 3 5 4 w'" 2.8 5 5 3 3 5 3 nF 2.9 5 5 3 3 5 3 nN 3.5 5 5 4 4 5 3 ll'" 3.7 5 4 5 1 5 7 W'" 4.1 5 4 5 5 6 7 ## 5.2 6 6 6 6 5 5 3# 5.3 7 7 7 7 5 'l 6# 5.4 8 8 8 5 4 6 7 REFERENCES n_2 net net sm sm -Bin _3_n _6_n _3_n _6_n 3 3 1 2 2 1 1 1 2 2 2 1 2 1 1 2 2 1 2 1 4 2 2 1 1 3 2 1 2 2 3 2 1 2 2 1 1 1 2 1 3 1 1 2 2 3 1 1 2 2 3 3 1 3 3 5 2 2 1 1 5 2 2 1 1 5 5 4 5 5 6 4 3 4 4 3 4 4 5 5 [B,C,91] Baldi, P. and Chauvin, Y., Temporal evolution of generalization during learning in linear networks, Neural Computation 3, 1991, pp. 589-603. [F,91] Finnoff, W., Complexity measures for classes of neural networks with variable weight bounds, in Proc. Int. J{)int Conf. on Neural Networks, Singapore, 1991. [F,Z,91] Fi'nnofi', W., Zimmermann, H.G., Detecting structure in small datasets by network fitting under complexity constraints, to appear in Proc. of 2nd Ann. Workshop Computational Learning Theory and Natural Learning Systems, Berkeley, 1991. [H,P,89], Hanson, S. J., and Pratt, L. Y., Comparing biases for minimal network construction with back-propagation, in Advances in Neural Information Processing I, D. S. Touretzky, Ed., Morgan Kaufman, 1989. [H,F,Z,92] Hergert, F., Finnofi', W. and H.G. Zimmermann, A comparison of weight elimination methods for reducing complexity in neural networks. To be presented at Int. Joint Con/. on Neural Networks, Baltimore, 1992. [L,D,S,90] Le Cun, Y., Denker J. and Solla, S., Optimal Brain Damage, in Proceedings of Neural Information Processing Systems II, Denver, 1990. [W,R,H,91] Weigend, A., Rumelhart, D., and Huberman, B., Generalization by weight elimination with application to forecasting, Advances in Neural Information Processing III, Ed. R. P. Lippman and J. Moody, Morgan Kaufman, 1991.
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Q-Learning with Hidden-Unit Restarting Charles W. Anderson Department of Computer Science Colorado State University Fort Collins, CO 80523 Abstract Platt's resource-allocation network (RAN) (Platt, 1991a, 1991b) is modified for a reinforcement-learning paradigm and to "restart" existing hidden units rather than adding new units. After restarting, units continue to learn via back-propagation. The resulting restart algorithm is tested in a Q-Iearning network that learns to solve an inverted pendulum problem. Solutions are found faster on average with the restart algorithm than without it. 1 Introduction The goal of supervised learning is the discovery of a compact representation that generalizes well. Such representations are typically found by incremental, gradientbased search, such as error back-propagation. However, in the early stages of learning a control task, we are more concerned with fast learning than a compact representation. This implies a local representation with the extreme being the memorization of each experience. An initially local representation is also advantageous when the learning component is operating in parallel with a conventional, fixed controller. A learning experience should not generalize widely; the conventional controller should be preferred for inputs that have not yet been experienced. Platt's resource-allocation network (RAN) (Platt, 1991a, 1991b) combines gradient search and memorization. RAN uses locally tuned (gaussian) units in the hidden layer. The weight vector of a gaussian unit is equal to the input vector for which the unit produces its maximal response. A new unit is added when the network's error magnitude is large and the new unit's radial domain would not significantly overlap domains of existing units. Platt demonstrated RAN on the supervised learning task 81 82 Anderson of predicting values in the Mackey-Glass time series. We have integrated Platt's ideas with the reinforcement-learning algorithm called Q-Iearning (Watkins, 1989). One major modification is that the network has a fixed number of hidden units, all in a single-layer, all of which are trained on every step. Rather than adding units, the least useful hidden unit is selected and its weights are set to new values, then continue the gradient-based search. Thus, the unit's search is restarted. The temporal-difference errors control restart events in a fashion similar to the way supervised errors control RAN's addition of new units. The motivation for starting with all units present is that in a parallel implementation, the computation time for a layer of one unit is roughly the same as that for a layer with all of the units. All units are trained from the start. Any that fail to learn anything useful are re-allocated when needed. Here the Q-Iearning algorithm with restarts is applied to the problem of learning to balance a simulated inverted pendulum. In the following sections, the inverted pendulum problem and Watkin's Q-Learning algorithm are described. Then the details of the restart algorithm are given and results of applying the algorithm to the inverted pendulum problem are summarized. 2 Inverted Pendulum The inverted pendulum is a classic example of an inherently unstable system. The problem can be used to study the difficult credit assignment problem that arises when performance feedback is provided only by a failure signal. This problem has often used to test new approaches to learning control (from early work by Widrow and Smith, 1964, to recent studies such as Jordan and Jacobs, 1990, and Whitley, Dominic, Das, and Anderson, 1993). It involves a pendulum hinged to the top of a wheeled cart that travels along a track of limited length. The pendulum is constrained to move within the vertical plane. The state is specified by the position and velocity of the cart and the angle between the pendulum and vertical and the angular velocity of the pendulum. The only information regarding the goal of the task is provided by the failure signal, or reinforcement, rt, which signals either the pendulum falling past ±12° or the cart hitting the bounds of the track at ±1 m. The state at time t of the pendulum is presented to the network as a vector, Xt, of the four state variables scaled to be between 0 and 1. For further details of this problem and other reinforcement learning approaches to this problem, see Barto, Sutton, and Ande!'-son (1983) and Anderson (1987). 3 Q-Learning The objective of many control problems is to optimize a performance measure over time. For the inverted pendulum problem, we define a reinforcement signal to be -1 when the pendulum angle or the cart position exceed their bounds, and 0 otherwise. The objective is to maximize the sum of this reinforcement signal over time. Q-Learning with Hidden-Unit Restarting 83 If we had complete knowledge of state transition probabilities we could apply dynamic programming to find the sequence of pushes that maximize the sum of reinforcements. Reinforcement learning algorithms have been devised to learn control strategies when such knowledge is not available. In fact, Watkins has shown that one form of his Q-Iearning algorithm converges to the dynamic programming solution (Watkins, 1989; Watkins and Dayan, 1992). The essence of Q-Iearning is the learning and use of a Q function, Q(x, a), that is a prediction of a weighted sum of future reinforcement given that action a is taken when the controlled system is in a state represented by x. This is analogous to the value function in dynamic programming. Specifically, the objective of Q-Iearning is to form the following approximation: 00 Q(Xt, at) :::::: L .. l7't+k+1 k=O where 0 < 'Y < 1 is a discount rate and 7't is the reinforcement received at time t. Watkins (1989) presents a number of algorithms for adjusting the parameters of Q. Here we focus on using error back-propagation to train a neural network to learn the Q function. For Q-Iearning, the following temporal-difference error (Sutton, 1988) et = 7't+1 + 'Y max [Q(Xt+1, at+t)] - Q(Xt, at). at+l is derived by using max [Q(Xt+l, at+t)] as an approximation to L~=o 'Yk 7't+k+2. See at+l (Barto, Bradtke, and Singh, 1991) for further discussion ofthe relationships between reinforcement learning and dynamic programming. 4 Q-Learning Network For the inverted pendulum experiments reported here, a neural network with a single hidden layer was used to learn the Q( x, a) function. As shown in Figure 1, the network has four inputs for the four state variables of the inverted pendulum, and two outputs corresponding to the two possible actions for this problem, similar to Lin (1992). In addition to the weights shown, wand v, the two units in the output layer each have a single weight with a constant input of 0.5. The activation function of the hidden units is the approximate gaussian function used by Platt. Let dj be the squared distance between the current input vector, x, and the weights in hidden unit j. 4 dj = L(Xi Wj,i)2 i=l Here Xi is the ith component of x at the current time. The output, Yj, of hidden unit j is Yj = { if dj < P; otherwise, 84 Anderson Xl x2 Q(x,-lO) x3 X Q(x,+10) 4 Figure 1: Q-Learning Network where p controls the radius of the region in which the unit's output is nonzero. Unlike Platt, p is constant and equal for all units. The output units calculate weighted sums of the hidden unit outputs and the constant input. The output values are the current estimates of Q(Xt, -10) and Q(Xt, 10), which are predictions of future reinforcement given the current observed state of the inverted pendulum and assuming a particular action will be applied in that state. The action applied at each step is selected as the one corresponding to the larger of Q(Xt, -10) and Q(Xt, 10). To explore the effects of each action, the action with the lower Q value is applied with a probability that decreases with time: _ { 1 - 0.5At, if Q(Xt, 10) > Q(Xt, -10); P 0.5At , otherwise, { 10, at = -10, with probability p; with probability 1 - p. To update all weights, error back-propagation is applied at each step using the following temporal-difference error { ,max[Q(xt+l,at+l)] - Q(Xt, at), if failure does not occur on step t + 1, et = Gt+l rt+l - Q(Xt, at), if failure occurs on step t + l. Note that rt = 0 for all non-failure steps and drops out of the first expression. Weights are updated by the following equations, assuming Unit j is the output unit corresponding to the action taken, and all variables are for the current time t. f3h e yL V· L (x· W· .) '" J,'" I J ,I P ~WL . '" ,I ~V· . J,I f3 e Yi In all experiments, p = 2, A = 0.99999, and, discussed in Section 6. 0.9. Values of f3 and f3h are Q-Learning with Hidden-Unit Restarting 85 5 Restart Algorithm After weights are modified by back-propagation, conditions for a restart are checked. If conditions are met, a unit is restarted, and processing continues with the next time step. Conditions and primary steps of the restart algorithm appear below as the numbered equations. 5.1 When to Restart Several conditions must be met before a restart is performed. First, the magnitude of the error, et, must be larger than usual. To detect this, exponentially-weighted averages of the mean, J1., and variance, u 2, of et are maintained and used to calculate a normalized error, e~ e' t J1.t+l 2 ut+l For our experiments, Ie = 0.99. et (1 _ let)' leJ1.t + (1 - Ie)et, leU; + (1 Ie )e? , Now we can state the first restart condition. A restart is considered on steps for which the magnitude of the error is greater than 0.01 and greater than a constant factor of the error's standard deviation, i.e., whenever le,1 > om and le,1 > aV(1 ~l"n)' (1) Of a small number of tested values, a = 0.2 resulted in the best performance. Before choosing a unit to restart for this step, we determine whether or not the current input vector is already "covered" by a unit. Assuming Yj is the output of Unit j for the current input vector, the restart procedure is continued only if Yj < 0.5, for j = 1, ... ,20 (2) 5.2 Which Ullit to Restart As stated by Mozer and Smolensky (1989), ideally we would choose the least useful unit as the one that results in the largest error when removed from the network. For the Q-network, this requires the removal of one unit at a time, making multiple attempts to balance the pendulum, and determining which unit when removed results in the shortest balancing times. Rather than following this computationally expensive procedure, we simply took the sum of the magnitudes of a hidden unit's output weights as a measure of it's utility. This is one of several utility measures suggested by Mozer and Smolensky and others (e.g., Kloph and Gose, 1969). After a unit is restarted, it may require further learning experience to acquire a useful function in the network. The amount of learning experience is defined as a sum of magnitudes of the error et. The sum of error magnitudes since Unit j was 86 Anderson restarted is given by Cj. Once this sum surpasses a maxImum, Cmax , the unit is again eligible for restarting. Thus, Unit j is restarted when Uj . min (IVI,j 1+ IV2,j I) (3) JE{1 •...• 20} and Cj > Cmax . ( 4 ) Without a detailed search, a value of Cmax = 10 was found to result in good performance. 5.3 New Weights for Restarted Unit Say Unit j is restarted. It's input weights are set equal to the current input vector, x, the one for which the output of the network was in error. One of the two output weights of Unit j is also modified. The output weight through which Unit j modifies the output of the unit corresponding to the action actually taken is set equal to the error, et. The other output weight is not modified. 6 Results W·· }.' where k Xi, for i = 1, ... , 4, { I, 2, if at = -10; if at = 10. (5) (6) The pendulum is said to be balanced when 90,000 steps (1/2 hour of simulated time) have elapsed without failure. After every failure, the pendulum is reset to the center ofthe track with a zero angle (straight up) and zero velocities. Performance is judged by the average number of failures before the pendulum is balanced. Averages were taken over 30 runs. Each run consists of choosing initial values for the hidden units' weights from a uniform distribution from 0 to 1, then training the net until the pendulum is balanced for 90,000 steps or a maximum number of 50,000 failures is reached. To determine the effect of restarting, we ccmpare the performance of the Q-Iearning algorithm with and without restarts. Back-propagation learning rates are given by 13 for the output units and 13h for the hidden units. 13 and 13h were optimized for the algorithm without restarts by testing a large number of values. The best values of those tried are 13 = 0.05 and 13h = 1.0. These values were used for both algorithms. A small number of values for the additional restart parameters were tested, so the restart algorithm is not optimized for this problem. Figure 2 is a graph of the number of steps between failures versus the number of failures. Each algorithm was initialized with the same hidden unit weights. Without restarts the pendulum is balanced for this run after 6,879 failures. With restarts it is balanced after 3,415 failures. The performances of the algorithms were averaged over 30 runs giving the following results. The restart algorithm balanced the pendulum in all 30 runs, within an 100,00010,000Steps Between 1,000 Failures 100 10 - I o With Restarts I 2,000 Q-Learning with Hidden-Unit Restarting 87 Without Restarts ... 1,,\ I " '¥ I I ~ I I J , '.. I, . " _', I', ~ I :~' " \ I "'-------, ... I 4,000 I 6,000 Failures Figure 2: Learning Curves of Balancing Time Versus Failures (averaged over bins of 100 failures) average of 3,303 failures. The algorithm without restarts was unsuccessful within 50,000 failures for two of the 30 runs. Not counting the unsuccessful runs, this algorithm balanced the pendulum within an average of 4,923 failures. Considering the unsuccessful runs, this average is 7 ,928 failures. In studying the timing of restarts, we observe that initially the number of restarts is small, due to the high variance of et in the early stages of learning. During later stages, we see that a single unit might be restarted many times (15 to 20) before it becomes more useful (at least aecording to our measure) than some other unit. 7 Conclusion This first test of an algorithm for restarting hidden units in a reinforcement-learning paradigm led to a decrease in learning time for this task. However, much work remains in studying the effects of each step of the restart procedure. Many alternatives exist, most significantly in the method for determining the utility of hidden units. A significant extension of this algorithm would be to consider units with variable-width domains, as in Platt's RAN algorithm. Acknowledgenlents The work was supported in part by the National Science Foundation through Grant IRI-9212191 and by Colorado State University through Faculty Research Grant 138592. 88 Anderson References C. W. Anderson. (1987). Strategy learning with multilayer connectionist representations. Technical Report TR87-509.3, GTE Laboratories, Waltham, MA, 1987. Corrected version of article that was published in Proceedings of the Fourth International Workshop on Machine Learning, pp. 103-114, June, 1987. A. G. Barto, S. J. Bradtke, and S. P. Singh. (1991). Real-time learning and control using asynchronous dynamic programming. Technical Report 91-57, Department of Computer Science, University of Massachusetts, Amherst, MA, Aug. A. G. Barto, R. S. Sutton, and C. W. Anderson. (1983). Neuronlike elements that can solve difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics, 13:835-846. Reprinted in J. A. Anderson and E. Rosenfeld, Neurocomputing: Foundations of Research, MIT Press, Cambridge, MA, 1988. M. I. Jordan and R. A. Jacobs. (1990). Learning to control an unstable system with forward modeling. In D. S. Touretzky, editor, Advances in Neural Information Processing Systems, volume 2, pages 324-331. Morgan Kaufmann, San Mateo, CA. A. H. Klopf and E. Gose. (1969). An evolutionary pattern recognition network. IEEE Transactions on Systems, Science, and Cybernetics, 15:247-250. L.-J. Lin. (1992). Self-improving reactive agents based on reinforcement learning, planning, and teaching. Machine Learning, 8(3/4):293-32l. M. C. Mozer and P. Smolensky. (1989). Skeltonization: A technique for trimming the fat from a network via relevance assessment. In D. S. Touretzky, editor, Advances in Neural Information Systems, volume 1, pages 107-115. Morgan Kaufmann, San Mateo, CA, 1989. J. C. Platt. (1991a). Learning by combining memorization and gradient descent. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 714-720. Morgan Kaufmann Publishers, San Mateo, CA. J. C. Platt. (1991 b) A resource-allocating network for function interpolation. N eural Computation, 3:213-225. R. S. Sutton. (1988). Learning to predict by the method of temporal differences. Machine Learning, 3:9-44. C. J. C. H. Watkins. (1989). Learning with Delayed Rewards. PhD thesis, Cambridge University Psychology Department. C. J. C. H. Watkins and P. Dayan. (1992). Q-Iearning. Machine Learning, 8(3/4):279-292. D. Whitley, S. Dominic, R. Das, and C. Anderson. (1993). Genetic reinforcement learning for neurocontrol problems. Machine Learning, to appear. B. Widrow and F. W. Smith. (1964). Pattern-recognizing control systems. In Proceedings of the 1963 Computer and Information Sciences (COINS) Symposium, pages 288-317, Washington, DC. Spartan.
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236 Synchronization and Grammatical Inference in an Oscillating Elman Net Bill Baird Dept Mathematics, U .C.Berkeley, Berkeley, Ca. 94720, baird@math.berkeley.edu Todd Troyer Dept Mathematics, U .C.Berkeley, Berkeley, Ca. 94720 Abstract Frank Eeckman Lawrence Livermore National Laboratory, P.O. Box 808 (L-426), Livermore, Ca. 94551 We have designed an architecture to span the gap between biophysics and cognitive science to address and explore issues of how a discrete symbol processing system can arise from the continuum, and how complex dynamics like oscillation and synchronization can then be employed in its operation and affect its learning. We show how a discrete-time recurrent "Elman" network architecture can be constructed from recurrently connected oscillatory associative memory modules described by continuous nonlinear ordinary differential equations. The modules can learn connection weights between themselves which will cause the system to evolve under a clocked "machine cycle" by a sequence of transitions of attractors within the modules, much as a digital computer evolves by transitions of its binary flip-flop attractors. The architecture thus employs the principle of "computing with attractors" used by macroscopic systems for reliable computation in the presence of noise. We have specifically constructed a system which functions as a finite state automaton that recognizes or generates the infinite set of six symbol strings that are defined by a Reber grammar. It is a symbol processing system, but with analog input and oscillatory subsymbolic representations. The time steps (machine cycles) of the system are implemented by rhythmic variation (clocking) of a bifurcation parameter. This holds input and "context" modules clamped at their attractors while 'hidden and output modules change state, then clamps hidden and output states while context modules are released to load those states as the new context for the next cycle of input. Superior noise immunity has been demonstrated for systems with dynamic attractors over systems with static attractors, and synchronization ("binding") between coupled oscillatory attractors in different modules has been shown to be important for effecting reliable transitions. Synchronization and Grammatical Inference in an Oscillating Elman Net 237 1 Introduction Patterns of 40 to 80 Hz oscillation have been observed in the large scale activity (local field potentials) of olfactory cortex [Freeman and Baird, 1987] and visual neocortex [Gray and Singer, 1987], and shown to predict the olfactory [Freeman and Baird, 1987] and visual pattern recognition responses of a trained animal. Similar observations of 40 Hz oscillation in auditory and motor cortex (in primates), and in the retina and EMG have been reported. It thus appears that cortical computation in general may occur by dynamical interaction of resonant modes, as has been thought to be the case in the olfactory system. The oscillation can serve a macroscopic clocking function and entrain or "bind" the relevant microscopic activity of disparate cortical regions into a well defined phase coherent collective state or "gestalt". This can overide irrelevant microscopic activity and produce coordinated motor output. There is further evidence that although the oscillatory activity appears to be roughly periodic, it is actually chaotic when examined in detail. If this view is correct, then oscillatory I chaotic network modules form the actual cortical substrate of the diverse sensory, motor, and cognitive operations now studied in static networks. It must then be shown how those functions can be accomplished with oscillatory and chaotic dynamics, and what advantages are gained thereby. It is our expectation that nature makes ~ood use of this dynamical complexity, and our intent is to search here for novel deSign principles that may underly the superior computational performance of biological systems over man made devices in many task domains. These principles may then be applied in artificial systems to engineering problems to advance the art of computation. We have therefore constructed a parallel distributed processing architecture that is inspired by the structure and dynamics of cerebral cortex, and applied it to the problem of grammatical inference. The construction assumes that cortex is a set of coupled oscillatory associative memories, and is also guided by the principle that at tractors must be used by macroscopic systems for reliable computation in the presence of noise. Present day digital computers are built of flip-flops which, at the level of their transistors, are continuous dissipative dynamical systems with different attractors underlying the symbols we call "0" and "1". 2 Oscillatory Network Modules The network modules of this architecture were developed previously as models of olfactory cortex, or caricatures of "patches"of neocortex [Baird, 1990a]. A particular subnetwork is formed by a set of neural populations whose interconnections also contain higher order synapses. These synapses determine at tractors for that subnetwork independent of other subnetworks. Each subnetwork module assumes only minimal coupling justified by known olfactory anatomy. An N node module can be shown to function as an associative memory for up to N 12 oscillatory and NI3 chaotic memory attractors [Baird, 1990b, Baird and Eeckman, 1992b). Single modules with static, oscillatory, and three types of chaotic attractors - Lorenz, Roessler, Ruelle-Takens - have been sucessfully used for recognition of handwritten characters [Baird and Eeckman, 1992b]. We have shown in these modules a superior stability of oscillatory attractors over static attractors in the presence of additive Gaussian noise perturbations with the 1 If spectral character of the noise found experimentally by Freeman in the brain[Baird and Eeckman, 1992a]. This may be one reason why the brain uses dynamic attractors. An oscillatory attractor acts like a a bandpass filter and is 238 Baird, Troyer, and Eeckman effectively immune to the many slower macroscopic bias perturbations in the thetaalpha-beta range (3 - 25 Hz) below its 40 -80 Hz passband, and the more microscopic perturbations of single neuron spikes in the 100 - 1000 Hz range. The mathematical foundation for the construction of network modules is contained in the normal form projection algorithm[Baird, 1990b]. This is a learning algorithm for recurrent analog neural networks which allows associative memory storage of analog patterns, continuous periodic sequences, and chaotic attractors in the same network. A key feature of a net constructed by this algorithm is that the underlying dynamics is explicitly isomorphic to any of a class of standard, well understood nonlinear dynamical systems - a "normal form" [Guckenheimer and Holmes, 1983]. This system is chosen in advance, independent of both the patterns to be stored and the learning algorithm to be used. This control over the dynamics permits the design of important aspects of the network dynamics independent of the particular patterns to be stored. Stability, basin geometry, and rates of convergence to attractors can be programmed in the standard dynamical system. By analyzing the network in the polar form of these "normal form coordinates", the amplitude and phase dynamics have a particularly simple interaction. When the input to a module is synchronized with its intrinsic oscillation, the amplitudes of the periodic activity may be considered separately from the phase rotation, and the network of the module may be viewed as a static network with these amplitudes as its activity. We can further show analytically that the network modules we have constructed have a strong tendency to synchronize as required. 3 Oscillatory Elman Architecture Because we work with this class of mathematically well-understood associative memory networks, we can take a constructive approach to building a cortical computer architecture, using these networks as modules in the same way that digital computers are designed from well behaved continuous analog flip-flop circuits. The architecture is such that the larger system is itself a special case of the type of network of the submodules, and can be analysed with the same tools used to design the subnetwork modules. Each module is described in normal form or "mode" coordinates as a k-winnertake-all network where the winning set of units may have static, periodic or chaotic dynamics. By choosing modules to have only two attractors, networks can be built which are similar to networks using binary units. There can be fully recurrent connections between modules. The entire super-network of connected modules, however, is itself a polynomial network that can be projected into standard network coordinates. The attractors within the modules may then be distributed patterns like those described for the biological model [Baird, 1990a], and observed experimentally in the olfactory system [Freeman and Baird, 1987]. The system is still equivalent to the architecture of modules in normal form, however, and may easily be designed, simulated, and theoretically evaluated in these coordinates. In this paper all networks are discussed in normal form coordinates. As a benchmark for the capabilities of the system, and to create a point of contact to standard network architectures, we have constructed a discrete-time recurrent "Elman" network [Elman, 1991] from oscillatory modules defined by ordinary differential equations. We have at present a system which functions as a finite state automaton that perfectly recognizes or generates the infinite set of strings defined by the Reber grammar described in Cleeremans et. al. [Cleeremans et al., 1989]. The connections for this network were found by psuedo-inverting to find the connection matrices between a set of pre-chosen automata states for the hidden layer modules Synchronization and Grammatical Inference in an Oscillating Elman Net 239 and the proper possible output symbols of the Reber grammar, and between the proper next hidden state and each legal combination of a new input symbol and the present state contained in the context modules. We use two types of modules in implementing the Elman network architecture. The input and output layer each consist of a single associative memory module with six oscillatory at tractors (six competing oscillatory modes), one for each of the six possible symbols in the grammar. An attractor in these winner-take-all normal form cordinates is one oscillator at its maximum amplitude, with the others near zero amplitude. The hidden and context layers consist of binary "units" composed of a two competing oscillator module. We think of one mode within the unit as representing "I" and the other as representing"O" (see fig. 1). A "weight" for this unit is simply defined to be the weight of a driving unit to the input of the 1 attractor. The weights for the 0 side of the unit are then given as the compliment of these, wO = A - WI. This forces the input to the 0 side of the unit be the complement of the input to the 1 side, If = A - If, where A is a bias constant chosen to divide input equally between the oscillators at the midpoint of activation. .--------------------------Figure 1. I OUTPUT : : \(!)@00<V@)1 ············ .. ···············;······ .... ······) I HIDDEN fr : I 1""'"tJ~~I~~t~_~J~~!~~U ,I .---.:.' --------~-------------~------- ~\:.: .. --I : 1@)01@)01@)01@)01@)01 1~(!)@Cge~1 : I CONTEXT INPUT I L ________________________________________ ~ Information flow in the network is controlled by a "machine cycle" implemented by the sinusoidal clocking of a bifurcation parameter which controls the level of inhibitory inter-mode coupling or "competition" between the individual oscillatory modes within each winner-take-all module. For illustration, we use a binary module represnting either a single hidden or context unit; the behavior of the larger input and output modules is similar. Such a unit is defined in polar normal form coordinates by the following equations: rli Uirli - crli(rii + (d - bsin(wc/od:t»r5i) + L: wijIj cos(Oj - Oli) j rOi UirOi - croi(r5i + (d - bsin(wc/ockt»rii) + L:(A - Wij )Ij cos(Oj - OOi) j Oli Wi + L Wij(Ij !rli) sin(Oj - Oli) j 00i Wi + I)A - Wij )(Ij !rOi) sin(Oj - Ood j 240 Baird, Troyer, and Eeckman The clocked parameter bsin(wclockt) has lower (1/10) frequency than the intrinsic frequency of the unit Wi. Asuming that all inputs to the unit are phase-locked, examination of the phase equations shows that the unit will synchronize with this input. When the oscillators are phase-locked to the input, (J; - (Jli = 0, and the phase terms cos((J; - (Jli) = cos(O) = 1 dissappear. This leaves the amplitude equations rli and rOi with static inputs E; wi;I; and E;(A - wi;)I;. The phase equations show a strong tendency to phase-lock, since there is an attractor at zero phase difference </> = (Jo - (JI = (Jo - wIt = 0, and a repellor at 180 degrees in the phase difference equations ;p for either side of a unit driven by an input of the same frequency, WI - Wo = o. ;p = Wo -WI + (rI/ro)sin(-</», so, ¢ = -sin-1[(ro/rI)(WI - wo)] Thus we have a network module which approximates a static network unit in its amplitude activity when fully phase-locked. Amplitude information is transmitted between modules, with an oscillatory carrier. If the frequencies of attractors in the architecture are randomly dispersed by a significant amount, phase-lags appear, then synchronization is lost and improper transitions begin to occur. For the remainder of the paper we assume the entire system is operating in the synchronized regime and examine the flow of information characterized by the pattern of amplitudes of the oscillatory modes within the network. 4 Machine Cycle by Clocked Bifurcation Given this assumption of a phase-locked system, the amplitude dynamics behave as a gradient dynamical system for an energy function given by where the total input I = E; wii Ii and B = E; Ii. Figures 2a and 2b show the energy landscape with no external input for minimal and maximal levels of competition respectively. External input simply adds a linear "tilt" to the landscape, with large I giving a larger tilt toward the rli axis and small I a larger tilt toward the rOi axis. Note that for low levels of competition, there is a broad circular valley. When tilted by external input, there is a unique equilibrium that is determined by the bias in tilt alon~ one axis over the other. Thinking of rli as the "acitivity" of the unit, this acitlvity becomes an increasing function of I. The module behaves as analog connectionist unit whose transfer function can be approximated by a sigmoid. With high levels of competition, the unit will behave as a binary (bistable) "digital" flip-flop element. There are two deep valleys, one on each axis. Hence the final steady state of the unit is determined by which basin contains the initial state of the system reached during the analog mode of operation before competition is increased by the clock. This state changes little under the influence of external input: a tilt will move the location of the valleys only slightly. Hence the unit performs a winner-take-all choice on the coordinates of its initial state and maintains that choice independent of external input. Figure 2a. Low Competition Synchronization and Grammatical Inference in an Oscillating Elman Net 241 Figure 2b. High Competition We use this bifurcation in the behavior of the modules to control information flow within the network. We think of the input and context modules as "sensory", and the hidden and output modules as "motor" modules. The action of the clock is applied reciprocally to these two sets (grouped by dotted lines in fig.1) so that they alternatively open to receive input from each other and make transitions of attractors. This enables a network completely defined as a set of ordinary differential equations to implement the discrete-time recurrent Elman network. At the beginning of a machine cycle, the input and context layers are at high competition and hence their activity is "clamped" at the bottom of deep attractors. The hidden and output modules are at low competition and therefore behave as traditional feedforward network free to take on analog values. Then the situation reverses. As the competition comes up in the output module, it makes a winnertake-all choice as to the next symbol. Meanwhile high competition has quantized and clamped the activity in the hidden layer to a fixed binary vector. Then competition is lowered in the input and context layers, freeing these modules from their attractors. Identity mappings from hidden to context and from output to input (gray arrows in fig.1) "load" the binarized activity of the hidden layer to the context layer for the next cycle, and "place" the generated output symbol into the input layer. For a Reber grammar there are always two equally possible next symbols being generated in the output layer, and we apply noise to break this symmetry and let the winnertake-all dynamics of the output module chose one. For the recognition mode of operation, these symbols are thought of as "predicted" by the output, and one of them must always match the next actual input of a string to be recognized or the string is instantly rejected. Note that even though the clocking is sinusiodal and these transitions are not sharp, the system is robust and reliable. It is only necessary to set the rates of convergence within modules to be faster than the rate of change of the clocked bifurcation parameter, so that the modules are operating "adiabatically" - i.e. always internally relaxed to an equilibrium that is moved slowly by the clocked parameter. It is the bifurcation in the phase portrait of a module from one to two attractors that contributes the essential "digitization" of the system in time and state. A bifurcation is a discontinuous (topologically sharp) change in the phase portrait of possibilities for the continuous dynamical behavior of a system that occurs as a bifurcation parameter reaches a "critical" value. We can think of the analog mode for a module as allowing input to prepare its initial state for the binary "decision" between attractor basins that occurs as competition rises and the double potential well appears. The feedback between sensory and motor modules is effectively cut when one set is clamped at high competition. The system can thus be viewed as operating in discrete time by alternating transitions between a finite set of attracting states. This kind of clocking and "buffering" (clamping) of some states while other states 242 Baird, Troyer, and Eeckman relax is essential to the reliable operation of digital architectures. The clock input on a flip-flop clamps it's state until its signal inputs have settled and the choice of transition can be made with the proper information available. In our simulations, if we clock all modules to transition at once, the programmed sequences lose stability, and we get transitions to unprogrammed fixed points and simple limit cycles for the whole system. 5 Training When the input and context modules are clamped at their attractors, and the hidden and output modules are in the analog operating mode and synchronized to their inputs, the network approximates the behavior of a standard feedforward network in terms of its amplitude activities. Thus a real valued error can be defined for the hidden and output units and standard learning algorithms like back propagation can be used to train the connections. We can use techniques of Giles et. aI. [Giles et aI., 1992] who have trained simple recurrent networks to become finite state automata that can recognize the regular Tomita languages and others. If the context units are clamped with high competition, they are essentially "quantized" to take on only their 0 or 1 attractor values, and the feedback connections from the hidden units cannot affect them. While Giles, et. aI. often do not quantize their units until the end of training to extract a finite state automaton, they find that quantizing of the context units during training like this increases learning speed in many cases[Giles et aI., 1992]. In preparation for learning in the dynamic architecture, we have sucessfully trained the back propogation network of Cleermans et. aI. with digititized context units and a shifted sigmoid activation function that approximates the one calculated for our oscillatory units. In the dynamic architecture, we have also the option of leaving the competition within the context units at intermediate levels to allow them to take on analog values in a variable sized neighborhood of the 0 or 1 attractors. Since our system is recurrently connected by an identity map from hidden to context units, it will relax to some equilibrium determined by the impact of the context units and the clamped input on the hidden unit states, and the effect of the feedback from those hidden states on the context states. We can thus further explore the impact on learning of this range of operation between discrete time and space automaton and continuous analog recurrent network. 6 Discusion The ability to operate as an finite automaton with oscillatory/chaotic "states" is an important benchmark for this architecture, but only a subset of its capabilities. At low to zero competition, the supra-system reverts to one large continuous dynamical system. We expect that this kind of variation of the operational regime, especially with chaotic attractors inside the modules, though unreliable for habitual behaviors, may nontheless be very useful in other areas such as the search process of reinforcement learning. An important element of intra-cortical communication in the brain, and between modules in this architecture, is the ability of a module to detect and respond to the proper input signal from a particular module, when inputs from other modules which is irrelevant to the present computation are contributing cross-talk and noise. This is smilar to the problem of coding messages in a computer architecture like the Synchronization and Grammatical Inference in an Oscillating Elman Net 243 connection machine so that they can be picked up from the common communication buss line by the proper receiving module. We believe that sychronization is one important aspect of how the brain solves this coding problem. Attractors in modules of the architecture may be frequency coded during learning so that they will sychronize only with the appropriate active attractors in other modules that have a similar resonant frequency. The same hardware (or "wetware") and connection matrix can thus subserve many different networks of interaction between modules at the same time without cross-talk problems. This type of computing architecture and its learning algorithms for computation with oscillatory spatial modes may be ideal for implementation in optical systems, where electromagnetic oscillations, very high dimensional modes, and high processing speeds are available. The mathematical expressions for optical mode competition are nearly identical to our normal forms. Acknowledgements Supported by AFOSR-91-0325, and a grant from LLNL. It is a pleasure to acknowledge the invaluable assistance of Morris Hirsch and Walter Freeman. References [Baird, 1990a] Baird, B. (1990a). Bifurcation and learning in network models of oscillating cortex. In Forest, S., editor, Emergent Computation, pages 365-384. North Holland. also in Physica D, 42. [Baird, 1990b] Baird, B. (1990b). A learning rule for cam storage of continuous periodic sequences. In Proc. Int. Joint Conf. on Neural Networks, San Diego, pages 3: 493-498. [Baird and Eeckman, 1992a] Baird, B. and Eeckman, F. H. (1992a). A hierarchical sensory-motor architecture of oscillating cortical area subnetworks. In Eeckman, F. H., editor, Analysis and Modeling of Neural Systems II, pages 96-204, Norwell, Ma. Kluwer. [Baird and Eeckman, 1992b] Baird, B. and Eeckman, F. H. (1992b). A normal form projection algorithm for associative memory. In Hassoun, M. 11., editor, Associative Neural Memories: Theory and Implementation, New York, NY. Oxford University Press. in press. [Cleeremans et al., 1989] Cleeremans, A., Servan-Schreiber, D., and McClelland, J. (1989). Finite state automata and simple recurrent networks. Neural Computation, 1(3):372-381. [Elman, 1991] Elman, J. (1991). Distributed representations, simple recurrent networks and grammatical structure. Machine Learning, 7(2/3):91. [Freeman and Baird, 19871 Freeman, W. and Baird, B. (1987). Relation of olfactory eeg to behavior: Spatiaf analysis. Behavioral Neuroscience, 101:393-408. [Giles et al., 1992] Giles, C., Miller, C.B.and Chen, D., Chen, H., Sun, G., and Lee, Y. (1992). Learning and extracting finite state automata with second order recurrent neural networks. Neural Computation, pages 393-405. [Gray and Singer, 1987] Gray, C. M. and Singer, W. (1987). Stimulus dependent neuronal oscillations in the cat visual cortex area 17. Neuroscience [Supplj, 22:1301P. [Guckenheimer and Holmes, 1983] Guckenheimer, J. and Holmes, D. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York.
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Attractor Neural Networks with Local Inhibition: from Statistical Physics to a Digital Programmable Integrated Circuit E. Pasero Dipartimento di Elettronica Politecnico di Torino 1-10129 Torino, Italy R. Zecchina Dipartimento di Fisica Teorica e INFN U niversita. di Torino 1-10125 Torino, Italy Abstract Networks with local inhibition are shown to have enhanced computational performance with respect to the classical Hopfield-like networks. In particular the critical capacity of the network is increased as well as its capability to store correlated patterns. Chaotic dynamic behaviour (exponentially long transients) of the devices indicates the overloading of the associative memory. An implementation based on a programmable logic device is here presented. A 16 neurons circuit is implemented whit a XILINK 4020 device. The peculiarity of this solution is the possibility to change parts of the project (weights, transfer function or the whole architecture) with a simple software download of the configuration into the XILINK chip. 1 INTRODUCTION Attractor Neural Networks endowed with local inhibitory feedbacks, have been shown to have interesting computational performances[I]. Past effort was concentrated in studying a variety of synaptic structures or learning algorithms, while less attention was devoted to study the possible role played by different dynamical schemes. The definition of relaxation dynamics is the central problem for the study of the associative and computational capabilities in models of attractor neural networks and might be of interest also for hardware implementation in view of the 805 806 Pasero and Zecchina constraints on the precision of the synaptic weights. In this paper, we give a brief discussion concerning the computational and physical role played by local inhibitory interactions which lead to an effective non-monotonic transfer function for the neurons. In the last few years others models characterized by non-monotonic neurons have been proposed[2,3]. For Hebbian learning we show, numerically, that the critical capacity increases with respect to the Hopfield case and that such result can be interpreted in terms of a twofold task realized by the dynamical process. By means of local inhibition, the system dynamically selects a subspace (or subnetwork) of minimal static noise with respect to the recalled pattern; at the same time, and in the selected subspace, the retrieval of the memorized pattern is performed. The dynamic behaviour of the network, for deterministic sequential updating, range from fixed points to chaotic evolution, with the storage ratio as control parameter, the transition appearing in correspondence to the collapse of the associative performance. Resorting to two simplified versions of the model, we study the problem of their optimal performance by the replica method; in particular the role of non-monotonic functions and of subspaces dynamical selection are discussed. In a second part of the work, the implementation of the discussed model by means ofaXILINK programmable gate array is discussed. The circuit implements a 16-32 neurons network in which the analogical characteristics (such as a capacitive decay) are emulated by digital solutions. As expected, the limited resolution of the weghts does not represent a limit for the performance of the network. 2 THE MODEL: theory and performance We study an attractor neural network composed of N three state ±1,O formal neurons. The ± 1 values code for the patterns (the patterns are indeed binary) and are thus used during the learning phase, while the O-state is a don't care state, not belonging to the patterns code, which has only a dynamical role. The system is assumed to be fully connected and its evolution is governed by sequential or parallel updating of the following equations (1) N hi(t + 1) = >'hi(t) + L JijSj(t) i = l, ... ,N (2) j=l where,), is a dynamic threshold of the local inhibitory feedback (typically we take ')'(t) = fr 2:i Ihi(t - 1)1), the {Jij} are the synaptic conductances and>' is a capacitive decay factor of the input potential (>. = e~, where T = RC). The performance of the network are described in terms of two parameters which have both a dynamical and a computational simple interpretation. In particular we define the retrieval activity as the fraction of neurons which are not not in the zero state Attractor Neural Networks with Local Inhibition 807 (3) while the parameter that defines the retrieval quality is the scaled overlap p. _ 1 ""' tP.S m N ~,"i i· a . , (4) where the {er = ±1, i = 1, N; J.L = 1, P} are the memorized binary patterns. The scaled overlap can be thought simply as the overlap computed in the subspace M of the active neurons, M = {i / Si # 0, i = 1, N}. Given a set of P random independent binary patterns {er}, the Hebb-Hopfield learning rule corresponds to fix the synaptic matrix Jij by the additive relation p Jij = ~ L ere: (with Jii = 0). The effect of the dynamical process defined by p.=1 (1) and (2) is the selection of subspaces M of active neurons in which the static noise is minimized (such subspaces will be hereafter referred to as orthogonal subspaces). Before entering in the description of the results, it is worthwhile to remember that, in Hopfield-like attractor neural networks, the mean of cross correlation Huctuations produce in the local fields of the neurons a static noise, referred to as cross-talk of the memories. Together with temporal correlations, the static noise is responsible of the phase transition of the neural networks from associative memory to spin-glass. More pTecisely, when the Hopfield model is in a fixed point elF which belongs to the set of memories, the local fields are given by hier = 1 + Rf where Rf = ~ L L er e: er t; is the static noise (gaussian distribution with 0 mean and P.¢lF j¢i variance va). The preliminary performance study of the model under discussion have revealed several new basic features, in particular: (i) the critical capacity, for the Hebb learning rule, results increased up to Q c ::::; 0.33 (instead of 0.14[4]); (ii) the mean cross correlation Huctuations computed in the selected subspaces is minimized by the dynamical process in the region Q < Q c; (iii) in correspondence to the associative transition the system goes through a dynamic transition from fixed points to chaotic trajectories. The quantitative results concerning associative performance, are obtained by means of extended simulations. A typical simulation takes the memorized patterns as initial configurations and lets the system relax until it reaches a stationary point. The quantity describing the performance of the network as an associative memory is the mean scaled overlap m between the final stationary states and the memorized patterns, used as initial states. As the number of memorized configurations grows, one observes a threshold at Q = Q c ::::; 0.33 beyond which the stored states become unstable. (numerical results were performed for networks of size up to N = 1000). We observe that since the recall of the patterns is performed with no errors (up to 808 Pasero and Zecchina a ~ 0.31), also the number of stored bits in the synaptic matrix results increased with respect to the Hopfield case. The typical size of the sub-networks DM, like the network capacity, depends on the threshold parameter l' and on the kind of updating: for 1'(t) = fv I:i Ihi(t -1)1 and parallel updating we find DM ::: N/2 (a c = 0.33). The static noise reduction corresponds to the minimization of the mean fluctuation of the cross correlations (cross talk) in the subspaces, defined by where ei = 1 if i E M in pattern a and zero otherwise, as a function of a. Under the dynamical process (1) and (2), C does not follow a statistical law but undergoes a minimization that qualitatively explains the increase in the storage capacity. For a < a c , once the system has relaxed in a stationary subspace, the model becomes equivalent (in the subspace) to a Hopfield network with a static noise term which is no longer random. The statistical mechanics of the combinatorial task of minimizing the noise-energy term (5) can be studied analytically by the replica method; the results are of general interest in that give an upper bound to the performance of networks endowed with Hebb-like synaptic matrices and with the possibility selecting optimal subnetworks for retrieval dynamics of the patterns[8]. As already stated, the behaviour of the neural network as a dynamical system is directly related to its performance as an associative memory. The system shows an abrupt transition in the dynamics, from fixed points to chaotic exponentially long transients, in correspondence to the value of the storage ratio at which the memorized configurations become unstable. The only (external) control parameter of the model as a dynamical system is the storage ratio a = P / N. Dynamic complex behaviour appears as a clear signal of saturation of the attractor neural network and does not depend on the symmetry of the couplings. As a concluding remark concerning this short description of the network performance, we observe that the dynamic selection of subspaces seems to take advantage of finite size effects allowing the storage of correlated patterns also with the simple Hebb rule. Analytical and numerical work is in progress on this point, devoted to clarify the performance with spatially correlated patterns[5]. Finally, we end this theoretical section by addressing the problem of optimal performance for a different choice of the synaptic weights. In this direction, it is of basic interest to understand whether a dynamical scheme which allows for dynamic selection of subnetworks provides a neural network model with enhanced optimal capacity with respect to the classical spin models. Assuming that nothing is known about the couplings, one can consider the Jij as dynamical variables and study the fractional volume in the space of interactions that makes the patterns fixed points of the dynamics. Following Gardner and Derrida[6] , we describe the problem in terms of a cost-energy function and study its statistical mechanics: for a generic choice of the {Jij }, the cost function Ei is defined to be the number of patterns such that a given site i is wrong (with respect to (1)) Attractor Neural Networks with Local Inhibition 809 p Ei( {Jij }, {€f}) = L [€f (e(hfer +,,) - e(hfer)) + (1 - €f)eb 2 - hf2)] (6) 1-'=1 where e is the step function, the hf = ~ L Jijej €} are the local fields, " is the vN . J threshold of the inhibitory feedback and with €f = {O, I} being the variables that identify the subspace M (€f = 1 if i E M and zero otherwise). In order to estimate the optimal capacity, one should perform the replica theory on the following partition function (7) Since the latter task seems unmanageable, as a first step we resort to two simplified version of the model which, separately, retain its main characteristics (subspaces and non-monotonicity); in particular: (i) we assume that the {€f} are quenched random variables, distributed according to P(€f) = (1 - A)6(€f) + A6(€f - 1), A E [0,1]; (ii) we consider the case of a two-state (±1) non-monotonic transfer function. For lack of space, here we list only the final results. The expressions of the R.S. critical capacity for the models are, respectively: { (' A 1 00 }-l Q~.s · b; A) = 2(1 - A) Jo D(b - ()2 + "2 + A"y D(b - ()2 (8) (9) 1 ~ where D( = J7Le 2 d( (for (9) see also Ref.[4]). V 271" The values of critical capacity one finds are much higher than the monotonic perceptron capacity (Qc = 2). Unfortunately, the latter results are not reliable in that the stability analysis shows that the RS solution are unstable. Replica symmetry breaking is thus required. All the details concerning the computation with one step in replica symmetry breaking of the critical capacity and stabilities distribution can be found in Ref.[9]. Here we just quote the final quantitative result concerning optimal capacity for the non-monotonic two-state model: numerical evaluation of the saddle-point equations (for unbiased patterns) ~ives Qcbopt) ~ 4.8 with "opt ~ 0.8, the corresponding R.S. value from (9) being Q c .5. ~ 10.5. 810 Pasero and Zecchina 3 HARDWARE IMPLEMENTATION: a digital programmable integrated circuit The performance of the network discussed in the above section points out the good behavior of the dynamical approach. Our goal is now to investigate the performance of this system with special hardware. Commercial neural chips[9] and[10] are not feasible: the featu res of our net require non monotonic transfer characteristic due to local inhibitions. These aspects are not allowed in traditional networks. The implementation of a full custom chip is, on the other side, an hasty choice. The model is still being studied: new developments must be expected in the next future. Therefore we decided to build a prototype based on programmable logic circuits. This solution allows us to implement the circuit in a short time not to the detriment of the performance. Moreover the same circuit will be easily updated to the next evolutions. After an analysis of the existing logic circuits we decided to use the FPGAs devices[ll]. The reasons are that we need a large quantity of internal registers, to represent both synapses and capacitors, and the fastest interconnections. Xilinx 4000 family[12] offers us the most interesting approach: up tp 20000 gates are programmable and up to 28 K bit of Ram are available. Moreover a 3 ns propagation delay between internal blocks allow to implement very fast systems. We decided to use a XC4020 circuit with 20000 equivalent gates. Main problems related to the implementation of our model are the following: (a) number of neurons, (b) number of connections and (c) computation time parameters. (a) and (b) are obviously related to the logic device we have at our disposal. The number of gates we can use to implement the transfer function of our non monotonic neurons are mutually exclusive with the number of bits we decide to assign to the weights. The 20000 gates must be divided between logic gates and Ram cells. The parameter (c) depends on our choices in implementing the neural network. We can decide to connect the logic blocks in a sequential or in a parallel way. The global propagation time is the sum of the propagation delays of each logic block, from the the input to the output. Therefore if we put more blocks in parallel we don't increase the propagation delay and the time performance is better. Unfortunately the parallel solution clashes with the limitations of available logic blocks of our device. Therefore we decided to design two chips: the first circuit can implement 16 neurons in a faster parallell implementation and the second circuit allow us to use 32 (or more) neurons in a slower serial approach. Here we'll describe the fastest implementation. Figure 1 shows the 16 neurons of the neural chip. Each neuron, described in figure 2, performs a sequential sum and multiplication of the outputs of the other 15 neurons by the synaptic values stored inside the internal Ram. A special circuit implements the activation function described in the previuos section. All the neurons perform these operations in a parallel way: 15 clock pulses are sufficient to perform the complete operation for the system. Figure 2 shows the circuit of the neuron. Tl is a Ram where the synapses Tij are stored after the training phase. Ml and Al perform sums and multiplications according to our model. Dl simulates the A decay factor: every 15 clock cycles, which correspond to a complete cycle of sum and multiplication for all the 16 neurons, this circuit decreases the input of the neuron of a factor A. The activation function (1) is realized by Fl. Such circuit emulates a three levels logic, based on -1, 0 and +1 values by using two full adder blocks. Limitation due to the electrical characteristic of the circuit, impose a maximum Attractor Neural Networks with Local Inhibition 811 clock cycle of 20 MHz . The 16 neurons verSlOn of the chip takes from 4 to 6 complete computations to gain stability and every computation is 16 clock cycles long. Therefor e the network gives a stable state after 3 J.1.S at maximum. The second version of this circuit allows to use more neurons at a lower speed. We used the Xilinx device to implement one neuron while the synapses and the capacitors are stored in an external fast memory. The single neuron is time multiplexed in order to emulate a large number identical devices. At each step, both synapses and state variables are downloaded and uploaded from an external memory. This solution is obviously slower than the tirst one but a larger number of nerons can be implemented. A 32 neurons version takes about 6 J.1.S to reach a stable configuration. po t (I: odCI~:O) C0t"'l"'O L3 : lood D) DJ~ _oD (K u.~ n(1:0) /I I out(LO) n{ I :0) /I 2 out(l :0) n{I,:O) /I 3 outO ·O) L--.-",,"'{I:O) /I /I out(l :O) CO/olTROtlIR Figure 1 - Neural Chip : Al : I T- 8111.81 . .44.vU2 ~lUl.I!IJ L.. 1(11.81 If Ul,ll ( r= ( Q (J I. 11 U.QJ ~ (0""012 II r!.Iloon, •• , ~(1I'811 4 ••• ",'_' MJ ~c TJ & DJ ..... e.l .4 N,." •• 1I1.1Ir- I ... (l,e: .~d.u =r ... ll.1 • ..,l' 1, OJ Figure 2 - Neuron I OUTl'UT(2n: 1) c::: W 17: SOl (\ 013 Cl :::: oulf Olle, ~~ "o12 CIlI.al outr I 131 1 :el 1 :81 III : 131 1\ ;: (3) f""" • ..,t (I, el ....... (], eJ Dou\!I:OI ' __ at 812 Pasero and Zecchina 4 CONCLUSION A modified approach to attractor neural networks and its implementation on a XILINK XC4020 FPGA was discussed. The chip is now under test. Six /-L8 are sufficient for the relaxation of the system in a stable state, and the recognition of an input pattern is thus quite fast. A next step will be the definition of a multiple chip system endowed with more than 32 neurons, with the weights stored in an external fast memory. Acknowledgements This work was partially supported by the Annethe-INFN Italian project and by Progetto /inalizzato sistemi informatici e calcolo parallelo of CNR under grant N. 91.00884.PF69. References [1] R. Zecchina, "Computational and Physical Role of Local Inhibition in Attractor Neural Networks: a Simple Model," Parallel Architectures and Neural Networks, edt. E.R. Caianiello, World Scientific (1992). [2] M. Morita, S. Yoshizawa, H. Nakano, "Analysis and Improvement of the Dynamics of Autocorrelated Associative Memory," IEleE Trans. J73-D-ll, 232 (1990). [3] K. Kobayashi, "On the Capacity of a Neuron with a Non-Monotone Output Function," Network, 2, 237 (1991). [4] D.J. Amit, M. Gutfreund, H. Sompolinsky, "Storing Infinite Numbers of Patterns in a Spin-Glass Model of Neural Networks," Phy&. Rev. Lett., 55, 1530 (1985). [5] G. Boifetta, N. BruneI, R. Monasson, R. Zecchina, in preparation (1993). [6] E. Gardner, B. Deridda, "Optimal Storage Properties of Neural Network Models," J. Phys., A21, 271 (1988). [7] G. Boifetta, R. Monasson, R. Zecchina, "Symmetry Breaking in NonMonotonic Neural Networks," in preparation (1992). [8] N. BruneI, R. Zecchina, "Statistical Mechanics of Optimal Memory Retrieval in the Space of Dynamic Neuronal Activities," pre print (1993). [9] "An electrical Trainable Artificial Neural Network", proceedings of IJCNN, 1989, S. Diego. [10] M. Dzwonczyk, M.Leblanc, "INCA: An Integrated Neurocomputing Architecture", proceedings of AlA A Computing in Aerospace, October 1991 [11] W.R. Moore, W. Luk, "FPGAs", Abingdon EE-CS Books, 1991 [12] "The XC4000 Data Book", Xilinx, 1991
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Metamorphosis Networks: An Alternative to Constructive Methods Brian v. Bonnlander Michael C. Mozer Department of Computer Science & Institute of Cognitive Science University of Colorado Boulder, CO 80309-0430 Abstract Given a set oft raining examples, determining the appropriate number of free parameters is a challenging problem. Constructive learning algorithms attempt to solve this problem automatically by adding hidden units, and therefore free parameters, during learning. We explore an alternative class of algorithms-called metamorphosis algorithms-in which the number of units is fixed, but the number of free parameters gradually increases during learning. The architecture we investigate is composed of RBF units on a lattice, which imposes flexible constraints on the parameters of the network. Virtues of this approach include variable subset selection, robust parameter selection, multiresolution processing, and interpolation of sparse training data. 1 INTRODUCTION Generalization performance on a fixed-size training set is closely related to the number of free parameters in a network. Selecting either too many or too few parameters can lead to poor generalization. Geman et al. (1991) refer to this problem as the bias/variance dilemma: introducing too many free parameters incurs high variance in the set of possible solutions, and restricting the network to too few free parameters incurs high bias in the set of possible solutions. Constructive learning algorithms (e.g., Fahlman & Lebiere, 1990; Platt, 1991) have 131 132 Bonnlander and Mozer output layer RBFunit layer iDpIlt layer Figure 1: Architecture of an RBF network. been proposed as a way of automatically selecting the number of free parameters in the network during learning. In these approaches, the learning algorithm gradually increases the number of free parameters by adding hidden units to the network. The algorithm stops adding hidden units when some validation criterion indicates that network performance is good enough. We explore an alternative class of algorithms-called metamorphosis algorithmsfor which the number of units is fixed, but heavy initial constraints are placed on the unit response properties. During learning, the constraints are gradually relaxed, increasing the flexibility of the network. Within this general framework, we develop a learning algorithm that builds the virtues of recursive partitioning strategies (Breiman et aI., 1984j Friedman, 1991) into a Radial Basis Function (RBF) network architecture. We argue that this framework offers two primary advantages over constructive RBF networks: for problems with low input variable interaction, it can find solutions with far fewer free parameters, and it is less susceptible to noise in the training data. Other virtues include multiresolution processing and built-in interpolation of sparse training data. Section 2 introduces notation for RBF networks and reviews the advantages of using these networks in constructive learning. Section 3 describes the idea behind metamorphosis algorithms and how they can be combined with RBF networks. Section 4 describes the advantages of this class of algorithm. The final section suggests directions for further research. 2 RBF NETWORKS RBF networks have been used successfully for learning difficult input-output mappings such as phoneme recognition (Wettschereck & Dietterich, 1991), digit classification (Nowlan, 1990), and time series prediction (Moody & Darken, 1989j Platt, 1991). The basic architecture is shown in Figure 1. The response properties of each RBF unit are determined by a set of parameter values, which we'll call a pset. The pset for unit i, denoted ri, includes: the center location of the RBF unit in the input space, pij the width of the unit, Uij and the strength of the connection(s) from the RBF unit to the output unit(s), hi. One reason why RBF networks work well with constructive algorithms is because Metamorphosis Networks: An Alternative to Constructive Methods 133 the hidden units have the property of noninterference: the nature of their activation functions, typically Gaussian, allows new RBF units to be added without changing the global input-output mapping already learned by the network. However, the advantages of constructive learning with RBF networks diminish for problems with high-dimensional input spaces (Hartman & Keeler, 1991). For these problems, a large number of RBF units are needed to cover the input space, even when the number of input dimensions relevant for the problem is small. The relevant input dimensions can be different for different parts of the input space, which limits the usefulness of a global estimation of input dimension relevance, as in Poggio and Girosi (1990). Metamorphosis algorithms, on the other hand, allow RBF networks to solve problems such as these without introducing a large number of free parameters. 3 METAMORPHOSIS ALGORITHMS Metamorphosis networks contrast with constructive learning algorithms in that the number of units in the network remains fixed, but degrees of freedom are gradually added during learning. While metamorphosis networks have not been explored in the context of supervised learning, there is at least one instance of a metamorphosis network in unsupervised learning: a Kohonen net. Units in a Kohonen net are arranged on a lattice; updating the weights of a unit causes weight updates of the unit's neighbors. Units nearby on the lattice are thereby forced to have similar responses, reducing the effective number of free parameters in the network. In one variant of Kohonen net learning, the neighborhood of each unit gradually shrinks, increasing the degrees of freedom in the network. 3.1 MRBF NETWORKS We have applied the concept of metamorphosis algorithms to ordinary RBF networks in supervised learning, yielding MRBF networks. Units are arranged on an n-dimensional lattice, where n is picked ahead of time and is unrelated to the dimensionality of the input space. The response of RBF unit i is constrained by deriving its pset, ri, from a collection of underlying psets, each denoted Uj, that also reside on the lattice. The elements of Uj correspond to those of ri: Uj = (I-'i, uj, hj). Due to the orderly arrangement of the Uj, the lattice is divided into nonoverlapping hyperrectangular regions that are bounded by 2n Uj. Consequently, each ri is enclosed by 2n Uj. The pset ri can then be derived by linear interpolation of the enclosing underlying psets Uj, as shown in Figure 2 for a one-dimensional lattice. Learning in MRBF networks proceeds by minimizing an error function E in the Uj components via gradient descent: where NEIGHj is the set of RBF units whose values are affected by underlying pset i, and k indexes the input units of the network. The update expression is similar for uj and hiTo better condition the search space, instead of optimizing the 134 Bonnlander and Mozer (a) (b) Figure 2: Constrained RBF units. (a) Four RBF units with psets rl-r4 are arranged on a one-dimensional lattice, enclosed by underlying psets Ul and U2. (b) An input space representation of the constrained RBF units. RBF center locations, widths, and heights are linearly interpolated. 0'[ directly, we follow Nowlan and Hinton's (1991) suggestion of computing each RBF unit width according to the transformation O'i = ezp{"Yi!2) and searching for the optimum value of "Yi. This forces RBF widths to remain positive and makes it difficult for a width to approach zero. When a local optimum is reached, either learning is stopped or additional underlying psets are placed on the lattice in a process called metamorphosis. 3.2 METAMORPHOSIS Metamorphosis is the process that gradually adds new degrees of freedom to the network during learning. For the MRBF network explored in this paper, introducing new free parameters corresponds to placing additional underlying psets on the lattice. The new psets split one hyperrectangular region-an n-dimensional sublattice bounded by 2n underlying psets-into two nonoverlapping hyperrectangular regions. To achieve this, 2n - 1 additional underlying psets, which we call the split group, are required (Figure 3). The splitting process implements a recursive partitioning strategy similar to the strategies employed in the CART (Breiman et aI., 1984) and MARS (Friedman, 1991) statistical learning algorithms. Many possible rules for region splitting exist. In the simulations presented later, we consider every possible region and every possible split of the region into two subregions. For each split group k, we compute the tension of the split, defined as jE'Pl~QUP .11 :! II' We then select the split group that has the greatest tension. This heuristic is based on the assumption that the error gradient at the point in weight space where a split would take place reflects the long-term benefit of that split. It may appear that this splitting process is computationally expensive, but it can be implemented quite efficiently; the cost of computing all possible splits and choosing Metamorphosis Networks: An Alternative to Constructive Methods 135 Key • RBF llllit 1*[ "deriv.live of RBF 1*[ 0 split &JOIIP pse[ ., derivalive of "Pli[ sroup poe[ 0 UDdel'1ying pse[ ~::: ::::: Latria: repOll boundary Figure 3: Computing the tension of a split group. Arrows are meant to represent derivatives of corresponding pset components. the best one is linear in the number of RBF units on the lattice. 4 VIRTUES OF METAMORPHOSIS NETS 4.1 VARIABLE SUBSET SELECTION One advantage ofMRBF networks is that they can perform variable subset selection; that is, they can select a subset of input dimensions more relevant to the problem and ignore the other input dimensions. This is also a property of other recursive partitioning algorithms such as CART and MARS. In MRBF networks, however, region splitting occurs on a lattice structure, rather than in the input space. Consequently, the learning algorithm can orient a small number of regions to fit data that is not aligned with the lattice to begin with. CART and MARS have to create many regions to fit this kind of data (Friedman, 1991). To see if this style oflearning algorithm could learn to solve a difficult problem, we trained an MRBF network on the Mackey-Glass chaotic time series. Figure 4(a) compares normalized RMS error on the test set with Platt's (1991) RAN algorithm as the number of parameters increases during learning. Although RAN eventually finds a superior solution, the MRBF network requires a much smaller number of free parameters to find a reasonably accurate solution. This result agrees with the idea that ordinary RBF networks must use many free parameters to cover an input space with RBF units, whereas MRBF networks may use far fewer by concentrating resources on only the most relevant input dimensions. 4.2 ROBUST PARAMETER SELECTION In RBF networks, the local response of a hidden unit makes it difficult for back propagation to move RBF centers far from where they are originally placed. Consequently, the choice of initial RBF center locations is critical for constructive al136 Bonnlander and Mozer (a) 1S RANII) MRBF ..... ! § " t.t-1 -'. en ::;g ~ 0.1 0 100 200 300 400 500 600700 800 900 Degrees of Freedom (b) 8 ~ S ~ 0« 70 60 50 40 30 20 RAN "'*MRBF -tIo 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Level of Noise Figure 4: (a) Comparison on the Mackey-Glass chaotic time series. The curves for RAN and MRBF represent an average over ten and three simulation runs, respectively. The simulations used 300 training patterns and 500 test patterns as described in (Platt 1991). Simulation parameters for RAN match those reported in (Platt 1991) with € = 0.02. (b) Gaussian noise was added to the function y = sin81[':l, 0 < :l < 1, where the task was to predict y given x. The horizontal axis represents the standard deviation of the Gaussian distribution. For both algorithms, 20 simulations were run at each noise level. The number of degrees of freedom (DOF) needed to achieve a fixed error level was averaged. gorithms. Poor choices could result in the allocation of more RBF units than are necessary. One apparent weakness of the RAN algorithm is that it chooses RBF center locations based on individual examples, which makes it susceptible to noise. Metamorphosis in MRBF networks, on the other hand, is based on the more global measure of tension. Figure 4(b) shows the average number of degrees of freedom allocated by RAN and an MRBF network on a simple, one-dimensional function approximation task, Gaussian noise was added to the target output values in the training and test sets. As the amount of noise increases, the average number of free parameters allocated by RAN also increases, whereas for the MRBF network, the average remains low. One interesting property of RAN is that allocating many extra RBF units does not necessarily hurt generalization performance. This is true when RAN starts with wide RBF units and decreases the widths of candidate RBF units slowly. The main disadvantage to this approach is wasted computational resources. 4.3 MULTIRESOLUTION PROCESSING Our approach has the property of initially finding solutions sensitive to coarse problem features and using these solutions to find refinements more sensitive to finer features (Figure 5). This idea of multiresolution processing has been studied in the context of computer vision relaxation algorithms and is a property of algorithms proposed by other authors (e.g. Moody, 1989, Platt, 1991). Metamorphosis Networks: An Alternative to Constructive Methods 137 (a) (b) (c) two underlying psets three Wlderlying psets five Wlderlying psets Figure 5: Example of multiresolution processing. The figure shows performance on a two-dimensional classification task, where the goal is to classify all inputs inside the U-shape as belonging to the same category. An MRBF network is constrained using a one-dimensional lattice. Circles represent RBF widths, and squares represent the height of each RBF. 4.4 INTERPOLATION OF SPARSE TRAINING DATA For a problem with sparse training data, it is often necessary to make assumptions about the appropriate response at points in the input space far away from the training data. Like nearest-neighbor algorithms, MRBF networks have such an assumption built in. The constrained RBF units in the network serve to interpolate the values of underlying psets (Figure 6). Although ordinary RBF networks can, in principle, interpolate between sparse data points, the local response of an RBF unit makes it difficult to find this sort of solution by back propagation. :t .,. ............... \ ""-1 \/~da' .. MRBF Network Output I-Nearest Neighbor Asswnption :t Plain RBFNetwork Output Figure 6: Assumptions made for sparse training data on a task with a onedimensional input space and one-dimensional output space. Target output values are marked with an 'x'. Like nearest-neighbor algorithms, the assumption made by MRBF networks causes network response to interpolate between sparse data points. This assumption is not built into ordinary RBF networks. 138 Bonnlander and Mozer 5 DIRECTIONS FOR FURTHER RESEARCH In our simulations to date, we have not observed astonishingly better generalization performance with metamorphosis nets than with alternative approaches, such as Platt's RAN algorithm. Nonetheless, we believe the approach worthy of further exploration. We've examined but one type of metamorphosis net and in only a few domains. The sorts of investigations we are considering next include: substituting finite-element basis functions for RBFs, implementing a "soft" version of the RBF pset constraint using regularization techniques, and using a supervised learning algorithm similar to Kohonen networks, where updating the weights of a unit causes weight updates of the unit's neighbors. A cknowledgeIllent s This research was supported by NSF PYI award IRI-9058450 and grant 90-21 from the James S. McDonnell Foundation. We thank John Platt for providing the Mackey-Glass time series data, and Chris Williams, Paul Smolensky, and the members of the Boulder Connectionist Research Group for helpful discussions. References L. Breiman, J. Friedman, R. A. Olsen & C. J. Stone. (1984) Clauification and Regreuion Trees. Belmont, CA: Wadsworth. S. E. Fahlman & C. Lebiere. (1990) The cascade-correlation learning architecture. In D. S. Touretzky (ed.), Advance! in Neural Information Proceuing Sy!tem! ~, 524-532. San Mateo, CA: Morgan Kaufmann. J. Friedman. (1991) Multivariate Adaptive Regression Splines. Annab of Stati&tic! 19:1141. S. Geman, E. Bienenstock & R. Doursat. (1992) Neural networks and the bias/variance dilemma. Neural Computation 4(1):1-58. E. Hartman & J. D. Keeler. (1991) Predicting the future: advantages of semilocal units. Neural Computation 3( 4):566-578. T. Kohonen. (1982) Self-organized formation of topologically correct feature maps. Biological Cybernetic! 43:59-69. J. Moody & C. Darken. (1989) Fast learning in networks oflocally-tuned processing units. Neural Computation 1(2):281-294. J. Moody. (1989) Fast learning in multi-resolution hierarchies. In D. S. Touretzky (ed.), Advance! in Neural Information Proceuing 1, 29-39. San Mateo, CA: Morgan Kaufmann. S. J. Nowlan. (1990) Maximum likelihood competition in RBF networks. Tech. Rep. CRG-TR-90-2, Department of Computer Science, University of Toronto, Toronto, Canada. S. J. Nowlan & G. Hinton. (1991) Adaptive soft weight-tying using Gaussian Mixtures. In Moody, Hanson, & Lippmann (eds.), Advance! in Neural Information Proce!!ing 4, 993-1000. San Mateo, CA: Morgan-Kaufmann. J. Platt. (1991) A resource-allocating network for function interpolation. Neural Computation 3(2):213-225. T. Poggio & F. Girosi. (1990) Regularization algorithms for learning that are equivalent to multilayer networks. Science 247:978-982. D. Wettschereck & T. Dietterich. (1991) Improving the performance of radial basis function networks by learning center locations. In Moody, Hanson, & Lippmann (eds.), Advances in Neural Info. Proceuing 4, 1133-1140. San Mateo, CA: Morgan Kaufmann.
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A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems Wei-Tsih Lee John Pearson David Sarnoff Research Center CN5300 Princeton, NJ 08543 Abstract Channel equalization problem is an important problem in high-speed communications. The sequences of symbols transmitted are distorted by neighboring symbols. Traditionally, the channel equalization problem is considered as a channel-inversion operation. One problem of this approach is that there is no direct correspondence between error probability and residual error produced by the channel inversion operation. In this paper, the optimal equalizer design is formulated as a classification problem. The optimal classifier can be constructed by Bayes decision rule. In general it is nonlinear. An efficient hybrid linear/nonlinear equalizer approach has been proposed to train the equalizer. The error probability of new linear/nonlinear equalizer has been shown to be better than a linear equalizer in an experimental channel. 1 INTRODUCTION In a typical communication system, a sequence of symbols {ld are transmitted though a linear time-dispersive channel h(t). Let x(t) be the received signal, it can be written as x (t) = L/jh (t-n1) + W (I) j (1) where h(t) denotes the elementary pulse waveform, and wet) represents the random noise with iid Gaussian distribution. In a Quadrature Amplitude Modulation (QAM), symbols (Id are represented by complex numbers. During the transmission, interferences from neighboring symbols may distort the received signals. It is called Intersymbol Interference (lSI). It mainly because following reasons: nonideal channel which introduces phase or amplitude distortions, phase jitter, and impulse noise. Thus, equalization techniques are used to reduce the lSI. 674 A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems 675 2 ADAPTIVE LINEAR/RADIAL BASIS FUNCTION APPROACH TO EQUALIZER DESIGN Traditionally, the channel equalization problem is considered as a channel-inversion operation. The idea is that an equalizer is constructed as to undo the interference from neighboring symbols as they passing through a linear dispersive channel. It can be used to explain different equalizer structures (Zero-forcing, Least mean square, and decision feedback) and their performance [Proakis, 1989]. One problem of this approach is that in general there is no direct correspondence between error probability and residual error produced by the channel inversion operation. In [Gibson, etal, 1991], authors proposed a classification viewpoint for the equalizer design. They suggested that the optimal equalizer should be a classifier whose decision boundary is constructed according to Bayes decision rule. Compared with the channel inversion approach, the outputs of receiver are used as features for a classifier. The decision is made solely based on the classifier output, hence, on feature distribution. As it is well-known in [Fukunaga, 1978], the optimal decision boundaries can rapidly be computed if the features are Gaussian distributed. However, there is no idea about the structure of the optimal equalizer (classifier)for timedispersive channel outputs. In next section, we prove that for a linear channel, the optimal equalizer is nonlinear. 2.1 THE OPTIMAL EQUALIZER OF A LINEAR TIME-DISPERSIVE CHANNEL Let us first consider a two-value equalization problem. Symbols with two possible values ( -1, I) are transmitted. Let the channel be represented in a discrete form as a FIR of (hi), i=O,N-l. The output Xi can be written as N-l X· = ~ I . . h .+w. , L '-I I , j=o (2) The optimal equalizer design is equivalent to the following Bayes decision problem. Given (xi}, decide Ii by 1 /. = { , -1 if P(lj=-II Xj,xj+l'·····,xi+N-l) >P(lj= lI X j,xj+l'·····,xj+N-l) (3) where P (lj = 11 xi,xi + 1 , ••.•. ,xi+N-l) is the posterior probability of the transmitted symbol Ii being 1 given channel output (xd. By Bayes theorem, expression(3) can be expanded to the following form: (4) i+N-l IT L P(xjl Ii = 1,···Jj _ N+ 1 = ki - N+ 1) P(lj = 1, ... J j - N+ 1 = kj _ N+ 1) j = j kl' k;z •.... ,kj _ N + I e {I. -1 } (5) Since conditional probability P (xii Ij = 1, ... J j -N + 1 = kj _ N + 1) in (5) is a Gaussian distribution, the numerator in (5) is a mixture of Gaussian distribution. Plugging (5) into (3), Bayes decision rule determines the optimal decision boundary as the solution of eqUality. Since denominator is the same on both sides, it can be ignored. Rearranging the equation, 676 Lee and Pearson it can be written as summation of exponential functions. The solution of this equation is nonlinear function of {xil. In general, no analytical form can be found. However, it can be solved by numerical methods. Thus, the optimal decision boundary can be determined. The result can be extended to multi-class problems. Based on the result established above, we provide a theoretical justification of a nonlinear equalizer approach to linear time-dispersive channel. The theoretical comparison of performances of linear and optimal equalizers can be found in [Gibson, et.al, 1991]. They concluded that performance of linear equalizers can not be improved by increasing tap length. This also suggests that a nonlinear equalizer approach is necessary. Another reason for nonlinear equalization approach is due to channels with spectrum hulls [Proakis, 1989]. In this case, the linear equalizer can not achieve the desired performance due to "noise enhancement". 2.2 NONLINEAR EQUALIZER DESIGN PROBLEM There are several approaches to nonlinear equalizer design. To reduce the Least Mean Square (LMS) error, Voterra-series approach uses high-order product terms of input as new features. The tree-structured linear equalizer method [Gelfand, et.al., 1991] partitions the feature-space, and makes a piecewise linear approximation to the optimal nonlinear equalizer. As reported in [Gelfand, et.al., 1991], the tree-structured linear equalizer approach provides reasonable fast convergence and lower error probability as compared with linear and Voterra series approaches. The problem of this approach is that a lot of training samples are needed to achieve good performance. A neural network approach, MultiLayer Perceptron(MLP) [Gibson, et.al, 1991], trains 3 or 4 layers interconnected Perceptrons to form the nonlinear decision boundary. It is observed in [Gibson, et.al, 1991] that the performance of a MLP equalizer is close to optimal Bayes classifier. However, the training time is long and a fine-turing procedure is used. A nonlinear equalizer approach using radial basis functions is also reponed in [Chen, et.al., 1991]. To put equalizers into use, the long training time is unpractical, and a fine-adjusting procedure is not allowed. Hence, it is desired to have an efficient, automatic procedure for nonlinear equalize design. To achieve this goal, we propose a hybrid linear and radial basis functions approach for automatic nonlinear equalizer design. Although the optimal equalizer should be nonlinear, all these nonlinear design methods require long training time or large amount of training samples. Linear equalizers are not optimal, but with following advantages: easy training, fast convergence. It is also reported that the linear equalizer is relatively robust [Fukunaga, 1978]. Hence, it is desirable to combine the advantages of both linear and nonlinear equalizers. However, the hybrid structure should provide desired properties: fast convergence, automatic training procedure, and low error rate. To satisfy these constraints, we propose a feature-space partitioning approach to hybrid equalizer design. 2.3 FEATURE·SPACE PARTITIONING APPROACH TO HYBRID EQUALIZER DESIGN To design a hybrid linear/nonlinear equalizer, we adopt the feature-space partitioning concept. The idea is similar to the one developed in [Gelfand, et.al, 1991]. Here, we consider a partitioning method based on geometrical reasoning for eqUalization problems. The idea is based on the fact that linear equalizers can recover distorted signals, except the cases when strong noise push samples into boundaries where two classes overlaid with each A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems 677 other. We consider the "confused" samples as these samples near decision boundaries. The separation of "confused" samples can be accomplished based on the output values of linear equalizers. If the distance between output value and the closest point in signal constellation [Proakis, 1989] is greater than a threshold, then we consider current sample is "confused". This means that the sample is the one close to decision boundary. To achieve an accurate classification, we classify it by a nonlinear equalizer, which is constructed for separating the samples near Bayes decision boundary. The hybrid structure consisted of a linear equalizer, followed by a radial basis function (RBS) network, as shown in Fig. 1. A RBS network (Fig.2) is a two-layered network with radial_basis_function nodes in first layer, and a weighted linear combination of outputs of these nodes. Each feature vector consisted of a collection of consecutive data from the channel. It is assumed that these data are properly time and carrier synchronized [Proakis, 1989]. For the QAM, a complex-valued linear equalizer is adopted. The distance between output value of linear equalizer and the closest point is then computed and compared with threshold as described before. The "confused" samples are classified by a nonlinear RBS equalizer. The output of a RBS network can be written as weighted summation of outputs of nodes as follows: (6) where f(x) is the output of network. The oUfut value of each node is computed according to the bell-shaped function centered at ci' (J is the width of a node. Wi is the weight associated with ith node. In our experiments, the width of all nodes are fixed. The first N training samples are assigned to the centers of N -nodes network. The weights are adjusted according to stochastic gradient decent rule: (7) where 11 is the learning rate. dk is the desired output of network To train a hybrid LE/RBS equalizer, a collection of training samples is used to adjust the parameters of linear equalizer. The training samples for RBS network are collected according to the distance rule described above. They are used to adjust weights of RBS network only. The classification of a unknown sample follows a similar rule. The output value of the linear equalizer is computed. If the distance between output value and the closest point in signal constellation is smaller than the threshold, then the closest point is considered as the recovered signal. If not, the output of the RBS network is used to classify the sample. The closest point in signal constellation from output of RBS network is then used for sample class. Note, however, that there is a different interpretation for the output of linear and RBS equalizer. The function of linear equalizers can be considered as an approximation of channel inversion. Hence, it is similar to a deconvolution computation [Proakis, 1989]. However, for a RBS network, the output is the summation of weighted local Gaussian functions. For closely located points, the network is asked to give same output by then training procedure. Thus, it is more a classification approach than a deconvolution method. The approach provides a design method for hybrid LE/RBS equalizer. The linear equaliz678 Lee and Pearson {Xn } line:lI' Equalizer yes Radial_basis _ function network Fig. 1 System Diagram of Hybrid Linear/Nonlinear Equalizer I(x) Fig. 2 A RadiaCBasis_Function Network A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems 679 ers perfonn the channel inversion or partitioning of the feature space, depending on the output value. More complicated tree-structured equalizer [Gelfand, et.al., 1991] can be adopted for this proposes. The nonlinear RBS networks are used for classifying "confused" samples. They can be replaced by MLPs. Hence, the approach provides a general method for designing hybrid structure eqalizers. However, the trade-off between complexity and efficiency of these combinations has to to be considered. For example, a multilayer tree-structured equalizer can divide the space into smaller regions for finer classification. However, the small amount of training samples in practice can be a problem for this method. A MLP network can be used for nonlinear classifier. Nevertheless, convergence time will be a major concern. 3 EXPERIMENT We have applied our hybrid design method to a 4-QAM system. The channel is modeled by X j _ 1 = 0.406/j +0.814/j _ 1 +0.407/j _ 2 +wj (8) where Ij = {- 1 - j,- 1 + j,l- j,1 + j} . A 7-tapped complex linear equalizer is used for classifying the input. Threshold for nonlinear equalizer is 0.1. We use 4,000training samples and 5,000 testing samples. A 400 nodes RBS network is used for nonlinear equalizer. The first 400 "confused" training samples are used for the center of network. The network is trained according to (6). Learning coefficient T) is chosen to be 0.01. The width of a RBS node is 1.0. Fig. 3 shows the symbol error probability vs. SNR. The error probability is evaluated over 5,000 testing samples. The hybrid LE/RBS network produces nearly 10% reduction of error rate compared with linear equalizer. This shows that a hybrid linear/RBS network equalizer can reduce the error rate by classifying "confused" samples near decision boundaries. No comparison with Bayes classifier has been made. In our experiments, it is observed that the error rate can be reduced further by increasing the number of RBS nodes. This seems imply that a large-size RBS network will in general produce better classification result. However, since there is always a limitation of the computation resources: computation time and memory storage, the perfonnance of the hybrid linear/RBS network is limited, especially in high signal constellation case discussed below. Equalization in high signal constellation, 16 and 64-QAM, have been tried. The result shows no significant improvement This can be explained by the increasing of complexity. Recall that the RBS network is to separate the samples near the boundary. To deal with the increasing of number of classes due to high signal constellation, the number of nodes of network must increase proportionally. Since the increasing rate is exponential in terms of number of classes, it implies a straight-forward implementation of RBS network method can not be used for high signal constellation. In [Chen, et.al., 1991], authors suggest a dynamical RBS network with adjustable center location and width. The algorithm runs in batch mode. It is reported that the size of network can be reduced dramatically by the dynamical RBS network method. However, for equalizer application, on-line version of the algorithm is needed. 680 Lee and Pearson · · · : :< _CUlt · 1 .. om SNR Fig. 3 Error Probability of Hybrid Linear/Radial_Basis_Function Network Equalizer for a Linear Channel xi- 1 = 0.406Ii + 0.814/i _ 1 + 0.407li _ 2 +wi with 4-QAM. 4 CONCLUSION AND DISCUSSION FOR HYBRID LE/RBS EQUALIZER DESIGN By combining feature-space partitioning and nonlinear equalizers, we have developed a hybrid linear/nonlinear equalization approach. The major contribution of this research is to provide a theoretical justification of nonlinear equalization approach for linear time-dispersive channels. A feature-space partitioning method by linear equalizer is proposed. RBS networks for nonlinear equalizers are integrated into the design to separate the samples near decision boundary. The experiments for 4-QAM equalization have demonstrated the feasibility of the approach. For high signal constellation modulation, a dynamical RBS network method [Chen, etal., 1991] has been suggested to overcome the problem of increasing complexity. The hybrid Linear/nonlinear equalization approach combines the strength of linear and nonlinear equalizers. It offers a framework to integrate the deconvolution and classification methods. The approach can be generalized to include complicated partitioning A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems 681 scheme and other nonlinear networks, such as :MLP, as well. More researches need to be conducted to make this approach practical for general use. The relationship between the performance of hybrid equalizer and taps length of linear equalizers, the width and the number ofRBS nodes need to be investigated. The on-line version of dynamical RBS network [Chen, et.al., 1991] need to be developed. Reference: Proakis, J. G., Digital Communications, McGrwa-Hill company, New York, 1989. Gibson, GJ., Siu, S., Cowan, C.F.N., "The Application of Nonlinear Structures to Reconstruction of Binary Signals," IEEE. Trans. on Signal Processing, vol. 39, No.8, Aug.,pp. 1877-1884, 1991. Fukunaga, K., Introduction to Statistical Pattern Recognition, Academic Press, New York, 1978. Gelfand, S.B., Ravishankar, C.S., and Delp, EJ., "Tree-structured Piecewise Linear Adaptive Equalization," ICC91, 001383, 1386. Chen, S., Gilbson, GJ., Cowan, C.F.N., and Grant, P.M., "Reconstruction of binary signals using an adaptive radial-bas is-function equalizer," Signal Processing, 22, pp. 77-93, 1991. Chen, S., Cowan, C.F.N., and Grant, P.M., "Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks," IEEE. Trans. on Neural Networks, vol. 2, no., 2, March, pp.302-309, 1991.
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Physiologically Based Speech Synthesis ~akoto Hirayanaa t ATR Human Information Processing Research Laboratories 2-2, Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan Eric Vatikiotis-Bateson tATR Auditory and Visual Perception Research Laboratories Kiyoshi Hondat Yasuharu Koiket ~itsuo Kawatot* Abstract This study demonstrates a paradigm for modeling speech production based on neural networks. Using physiological data from speech utterances, a neural network learns the forward dynamics relating motor commands to muscles and the ensuing articulator behavior that allows articulator trajectories to be generated from motor commands constrained by phoneme input strings and global performance parameters. From these movement trajectories, a second neural network generates PARCOR parameters that are then used to synthesize the speech acoustics. 1 INTRODUCTION Our group has attempted to model speech production computationally as a process in which linguistic intentions are realized as speech through a causal succession of patterned behavior. Our aim is to gain insight into the cognitive and neurophysiological mechanisms governing this complex skilled behavior as well as to provide plausible models of speech synthesis and possibly recognition based on the physiology of speech production. It is the use of physiological data (EMG) representing * Also, Laboratory of Parallel Distributed Processing, Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 060, Japan 658 Physiologically Based Speech Synthesis 659 motor commands to muscles that distinguishes our modeling effort from those of others who use neural networks for articulation-based synthesis and/or inference of the dynamical constraints on speech motor control (Jordan, 1986, Jordan, 1990, Bailly, Laboissiere, and Schwalz, 1992, Saltzman, 1986, Bengio, Houde, and Jordan, 1992). This paper reports two areas in which implementation of the speech production scheme shown in Figure 1 has progressed. Initially, we concentrated on modeling the dynamics underlying articulation so that phoneme strings can specify motor commands to muscles, which then specify phoneme-specific articulator behavior (Hirayama, Vatikiotis-Bateson, Kawato, and Jordan, 1992). A neural network learned the forward dynamics relating motor commands to muscles and the ensuing articulator behavior associated with prosodic ally intact, but phonemic ally simplified, reiterant speech utterances. Then, a cascade neural network (Kawato, Maeda, Uno, and Suzuki, 1990) containing the forward dynamics model along with a suitable smoothness criterion (Uno, Kawato, and Suzuki, 1989) was used to produce continuous motor commands from a sequence of discrete articulatory targets corresponding to the phoneme input string. From this sequence of motor commands, appropriate articulator trajectories were then generated. Intention to Speak Intended Phoneme Global Performance Sequence Parameters Articulator Movement Figure 1: Conceptual scheme of speech production Although the results of this early work were encouraging, there were two technical limitations obstructing our effort to model real speech. First, using optoelectronic transduction techniques, only simple speech samples whose primary articulators were the lips and jaw could be recorded, hence the use of reiterant ba. Without dynamic tongue data, real speech could not be modeled. Also, the reiterant paradigm introduced a degree of rhythmical movement behavior not observed in real speech. The second limitation was that activity of only four muscles and generally only one dimension of articulator motion could be recorded simultaneously. Thus, agonistantagonist muscle activity was not represented even for this limited set of articulators. Technical improvements in data acquisition and their consequences for the subsequent dynamical modeling of real speech are presented in the next two sections. The second area of progress has been to implement the transform from model- generated articulator trajectories to acoustic output. A neural network is 660 Hirayama, Vatikiotis-Bateson, Honda, Koike, and Kawato used to acquire the mapping between articulation and acoustics in terms of PARCOR parameters (Itakura and Saito, 1969), which are correlated with vocal tract area functions. Speech signals are then generated using a PARCOR synthesizer from articulator input and appropriate glottal sources (currently, the residual of the PARCOR analysis). The results of this modeling for real and reiterant speech are reported in the final section of the paper. 2 EMPIRICAL DEVELOPMENTS In order to acquire data more suitable for real speech modeling, two additional experiments were run in which articulator position, EMG and acoustic data were recorded while the same subject produced real and reiterant speech utterances 5-8 seconds long at different speaking rates and styles (e.g., casual vs. precise). In the first of these, a sophisticated optoelectronic device, OPTOTRAK (Northern Digital, Inc.), was used because it permitted simultaneous recording of numerous 3D articulator positions for the lips, jaw and head, ten EMG channels, the speech acoustics, and even dynamic tongue-palate contact patterns. These data were used for modeling of the forward dynamics (see Figure 2) and the forward acoustics. Real speech utterances collected with this system were heavily loaded with labial stops, /p,b,m/, and labiodental fricatives, /f,v /, as well as many low vowels /a, ae/. Since surface EMG was used, it was difficult to obtain reliable recordings of jaw opening (anterior belly of the digastric), and closing (medial pterygoid) muscles. More recently, an electromagnetic position traking system, EMMA (Perkell, Cohen, Svirsky, Matthies, Garabieta, and Jackson, 1992), was used to transduce midsagittal motions of the tongue tip and tongue blade as well as the lips, jaw, and head. Data were collected for the same speech utterances used in the OPTOTRAK and original experiments as well as more natural utterances. Reiterant speech was also recorded for tao For this experiment, surface and hooked-wire EMG techniques were combined, which enabled nine orofacial and extrinsic tongue muscles to be recorded for jaw opening and closing, lip opening and closing, and tongue raising and lowering. The most important aspects of the signal processing for modeling the forward dynamics concern the numerical differentiation of articulator position to obtain velocity and acceleration, and the severe low-pass filtering (including rectification and integration) of the EMG to from 2000 Hz to 20-40 Hz. Both of these introduce spatiotemporal distortions, whose effects on the forward dynamics model are currently being examined. 3 MODELING THE FORWARD DYNAMICS The forward dynamics model was obtained using a 3-layer perceptron with back propagation (Rumelhart, Hinton, and Williams, 1986). Inputs to the network were instantaneous position and velocity for each dimension of articulator motion, and the EMG signals of 9-10 related muscles, which serve as the record of motor commands to muscles; outputs were accelerations for each dimension of motion. Figure 2 shows an example of predicting lip and jaw accelerations from 10 orofacial muscles for the 'natural' test utterance, "Pam put the bobbin in the frying pan and added more puppy parts to the boiling potato soup." As shown by the generalization results in Figure 2, the acquired model produced appropriate acceleration trajectories Physiologically Based Speech Synthesis 661 for real speech utterances, suggesting that utterance complexity is not a limiting factor in this approach. -Network Output ....... ·Experimental Data Upper Lip I---"-'~ Lower Lip Jaw o 200 400 600 800 1000 Figure 2: Estimated acceleration over time (5 ms samples) for vertical motion of the three articulators is compared to that of the test sentence: "Pam put the bobbin in the frying pan and added more puppy parts to the boiling potato soup". r Motor Commands .Jln~ra.!!d EMG.L Accelerlation l---j .... -V-E~L~ Predictor P~S ( Forwa~d ) ACC .~....-JI~ DynamiCs Movement Trajectory , Figure 3: The musculo-skeletal forward dynamics model for producing articulator movement trajectories is implemented as a recurrent network. Continuous motor command (EMG) input drives the network, which uses estimated acceleration at time tn, to predict new velocity (integration) and position (double integration) values at the next time step tn+l. D is a one-sample delay unit. The network is initialized with position and velocity values taken from the test utterance at to. Network training resulted in a one-step look-ahead predictor of the articulator dynamics, and was connected recurrently as shown in Figure 3. Using only initial values of articulator position and velocity for the first sample and continuous EMG input, estimated acceleration is looped back and summed with the velocities and positions of the input layer to predict their values for each time step. This is perhaps an overly stringent test of the acquired model because errors are cumulative over the entire utterance 5-8 second utterance. Yet the network outputs appropriate articulator trajectories for the entire utterance. Figure 4 shows the generated trajectory for vertical motion of the jaw during reiterant production of ba (recorded with the electromagnetometer). While the trajectory generated by the network tends to underestimate movement amplitude and introduce a small DC offset, it preserves the temporal properties of the test utterance very well everywhere except before a phrasal pause. Although good results have been obtained for the analysis 662 Hirayama, Vatikiotis-Bateson, Honda, Koike, and Kawato JAW (Vertical) o Network Output ........ Experimental Data 2 4 Time (5) 6 8 Figure 4: Jaw trajectories, generated by the forward dynamics network are compared with experimental data. of real speech using the larger sets of articulator and muscle inputs, network complexity has greatly increased. Performance of the full network has been poorer than before in modeling simple reiterant speech, which suggests some form of modularity should be introduced. Also, the addition of tongue data has increased the number of apparent many-to-one mappings between muscle activity and articulator motion. We are now incorporating as a boundary constraint the midsagittal profile of the hard palate and alveolar ridge, against which tongue-tip articulations are made. 4 MODELING THE FORWARD ACOUSTICS Articulator Positions Glottal Source t----------......;-.:;~.)) ) "----------' Acoustic wa;e Figure 5: Forward acoustics network. The final stage of our speech production model entails using a neural network to acquire a model of the relation between articulator motion and the ensuing acoustics. As shown in Figure 5, a 3-layer perceptron, using articulator position as input, was used to learn PARCOR analysis and generate appropriate 16-order PARCOR parameters for subsequent speech synthesis (Itakura and Saito, 1969). We chose PARCOR parameters, rather than more commonly used formant values, because the parameters have some relation to specific cross-sections of the vocal tract e.g., the first PARCOR corresponds to the cross-sectional area closest to the lips Physiologically Based Speech Synthesis 663 Network Output ........ Experimental Data ............. • ••••• • •••• k1 k2 .k3 . ~ ~ . k4 k5 k6 o • 0, " -,." .: -: 2 , .1f":'_ ..... 4 Time (s) 6 8 Figure 6: PARCOR parameter values (IS-order, 30ms Hanning window at 200Hz) for reiterant ba are predicted by the network. Only the first six PARCOR parameters are shown. The range of each parameter is -1 to 1 (small tick beside each wave label indicates 0). The value of kl is about 1 during vowels, and network output generally matches the desire wave almost perfectly. (Wakita, 1973). Also, PARCOR estimation errors do not have the radical consequences that formant estimation errors show. Finally, there is a unique mapping from PARCOR to formant values, but not the reverse (Itakura and Saito, 1969). Figure 6 shows the performance of the PARCOR estimation network for the first 6 parameters out of 16 parameters. Using the learned PARCOR coefficients and a sound source, acoustic signals can be synthesized. Currently, we are investigating various models for controlling sound source as well as prosodic characteristics. However, for this preliminary test of the network's ability to learn PARCOR parameters, the residual signal of PARCOR analysis served as the source waveform. Figure 7 shows an example of the network-learned PARCOR synthesis for reiterant ba. In this case, the training result is good as can be seen in the waveform (and frequency spectrum), or by listening the synthesized sound. However, the results have not been as good, so far, for real speech utterances containing a lot of abrupt changes and variability in vocal tract shape. One reason for this may be that learning has not yet converged, because the number articulator input channels is still too limited. So far, we have only two markers on tongue, which is not enough to recover the full vocal tract shape. This situation, hopefully, will improve as data for more tongue positions, or perhaps more functionally motivated placements, are collected. Another reason may be the inherent weakness of PARCOR analysis for modeling dynamic changes in vocal tract shape. 5 SUMMARY This paper outlines two areas of progress in our effort to develop a computational model of speech production. First, we extended our data acquisition to include more 664 Hirayama, Vatikiotis-Bateson, Honda, Koike, and Kawato Experimental Source (Residual) Synthesized Experimental Source (Residual) Synthesized 0 / b a/ •• , •• 1_'. II' ............ .t .......... I 2 b a , . ,_.". ....... -_.tl. I 4 Time (s) ", .. . ..... tt ••• I I 6 8 3.00 3.02 3.04 3.06 3.08 3.10 Time (s) Figure 7: Speech acoustics are synthesized by driving network-learned PARCOR parameters with a glottal source (the residual). The test sentence is reiterant speech using 00. Top and bottom graphs differ only in time scale. muscles and dimensions of motion for more articulators, especially the tongue, so that we could begin modeling the articulatory dynamics of real speech. As hoped, increasing the scope of the data demonstrated the applicability of our network approach to real speech. However, this also increases the size of the network, which has introduced some interesting problems for modeling simple speech samples. We are now considering modifications to the network architecture that will enable adaptive modeling of speech samples, whose complexity (e.g., number of physiological/ articulatory components) may vary. Second, we have employed a simple neural network for modeling the articulatory-to-acoustic transform based on PARCOR analysis, whose parameters are correlated with vocal tract shape. Although PARCOR can be used to synthesize speech, its main use for us is as a tool for assessing empirical issues associated with articulatory-acoustic interface. Acknowledgments We thank Haskins Laboratories for use of their facilities (NIH grant DC-00121), Vincent Gracco and Kiyoshi Ohsima for muscle insertions, M. I. Jordan for insightful Physiologically Based Speech Synthesis 665 discussion, and Yoh'ichi Toh'kura for continuous encouragement. Further support was provided by HFSP grants to M. Kawato. References [1] Bailly, G., Laboissiere, R. and Schwarz, J. L. (1992) Formant trajectories as audible gestures: an alternative for speech synthesis. Journal of Phonetics, 19,9-23. [2] Bengio, Y., Houde, J., and Jordan, M. I. (1992) Representations based on articulatory dynamics for speech recogmtion. Presented at Neural Networks for Computing, Snowbird, Utah. [3] Hirayama, M., Vatikiotis-Bateson, E., Kawato, M., and Jordan, M. I. (1992) Forward dynamics modeling of speech motor control using physiological data. In Moody, J. E., Hanson, S. J., and Lippmann, R. P. (eds.) Advances in neural information processing systems 4. San Mateo, CA: Morgan Kaufmann Publishers. [4] Itakura, F., and Saito, S. (1969) Speech analysis and synthesis by partial correlation parameters. Proceeding of Japan Acoust. Soc., 2-2-6. [5] Jordan, M. I. (1986) Serial order: a parallel distributed processing approach. ICS Report, 8604. [6] Jordan, M. I. (1990) Motor learning and the degrees of freedom problem. In M. Jeannerod (ed.) Attention and performance XIII, 796-836, Hillsdale, NJ: Erlbaum. [7] Kawato, M., Maeda, M., Uno, Y., and Suzuki, R. (1990). Trajectory formation of arm movement by cascade neural-network model based on minimum torquechange criterion. Bioi. Cybern., 62, 275-288. [8] Perkell, J., Cohen, M., Svirsky, M., Matthies, M., Garabieta, I., and Jackson, M., Electromagnetic midsagittal articulometer systems for transducing speech articulatory movements. J. Acoust. Soc. Am., 92, 3078-3096. [9] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986) Learning representations by back-propagating errors. Nature, 323, 533-536. [10] Saltzman, E. L. (1986) Task dynamic coordination of the speech articulators: A preliminary model. In H. Heuer and C. Fromm (eds.) Generation and modulation of action patterns, Berlin: Springer-Verlag. [11] Uno, Y., Kawato, M., and Suzuki, R. (1989) Formation and control of optimal trajectory in human multijoint arm movement - minimum torque-change model. Bioi. Cybern., 61, 89-101. [12] Wakita, H. (1973) Direct estimation of the vocal tract shape by inverse filtering of acoustic speech waveforms. IEEE Trans. Audio Electroacoust., AV-21 417-427.
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Weight Space Probability Densities in Stochastic Learning: II. Transients and Basin Hopping Times Genevieve B. Orr and Todd K. Leen Department of Computer Science and Engineering Oregon Graduate Institute of Science & Technology 19600 N.W. von Neumann Drive Beaverton, OR 97006-1999 Abstract In stochastic learning, weights are random variables whose time evolution is governed by a Markov process. At each time-step, n, the weights can be described by a probability density function pew, n). We summarize the theory of the time evolution of P, and give graphical examples of the time evolution that contrast the behavior of stochastic learning with true gradient descent (batch learning). Finally, we use the formalism to obtain predictions of the time required for noise-induced hopping between basins of different optima. We compare the theoretical predictions with simulations of large ensembles of networks for simple problems in supervised and unsupervised learning. 1 Weight-Space Probability Densities Despite the recent application of convergence theorems from stochastic approximation theory to neural network learning (Oja 1982, White 1989) there remain outstanding questions about the search dynamics in stochastic learning. For example, the convergence theorems do not tell us to which of several optima the algorithm 507 508 Orr and Leen is likely to converge 1 • Also, while it is widely recognized that the intrinsic noise in the weight update can move the system out of sUb-optimal local minima (for a graphical example, see Darken and Moody 1991), there have been no theoretical predictions of the time required to escape from local optima, or of its dependence on learning rates. In order to more fully understand the dynamics of stochastic search, we study the weight-space probability density and its time evolution. In this paper we summarize a theoretical framework that describes this time evolution. We graphically portray the motion of the density for examples that contrast stochastic and batch learning. Finally we use the theory to predict the statistical distribution of times required for escape from local optima. We compare the theoretical results with simulations for simple examples in supervised and unsupervised learning. 2 Stochastic Learning and Noisy Maps 2.1 Motion of the Probability Density We consider stochastic learning algorithms of the form w(n+1} = w(n) + JJH[w(n},x(n)] (1) where w(n} E 1R,m is the weight, x(n} is the data exemplar input to the algorithm at time-step n, JJ is the learning rate, and H[ ... ] E 1R, m is the weight update function. The exemplars x(n} can be either inputs or, in the case of supervised learning, input/target pairs. We assume that the x(n) are i.i.d. with density p(x). Angled brackets ("'):t denote averaging over this density. In what follows, the learning rate will be held constant. The learning algorithm (1) is a noisy map on w. The weights are thus random variables described by the probability density function P(w, n). The time evolution of this density is given by the Kolmogorov equation P(w, n + 1) = f dw' P(w', n) W(w' -+ w) (2) where the single time-step transition probability is given by (Leen and Orr 1992, Leen and Moody 1993) W(w' -+ w) = (8{w-w'-JJH[w',xJ) ):t and 8 ( ... ) is the Dirac delta function. (3) The Kolmogorov equation can be recast as a differential-difference equation by expanding the transition probability (3) as a power series in JJ. This gives a KramersMoya! expansion (Leen and Orr 1992, Leen and Moody 1993) IHowever Kushner (1987) has proved convergence to global optima for stochastic approximation algorithms with added Gaussian noise subject to logarithmic annealing schedules. Weight Space Probability Densities in Stochastic Learning 509 P(w, n + 1) P(w, n) 00 . m L (_~)t L I. (4) i=l ;l •... ;j=l where Wj" and Hj" are the j~h component of weight, and weight update, respectively. Truncating (4) to second order in J1, leaves a Fokker-Planck equation2 that is valid for small 1 J1,H I. The drift coefficient (H}z is simply the average update. It is important to note that the diffusion coefficients, (Hj"HjIJ ) , can be strongly dependent on location in the weight-space. This spatial depe~dence influences both equilibria and transient phenomena. In section 3.1 we will use both the Kolmogorov equation (2), and the Fokker-Planck equation to track the time evolution of network ensemble densities. 2.2 First Passage Times Our discussion of basin hopping will use the notion of the first passage time (Gardiner, 1990); the time required for a network initialized at Wo to first pass into an €-neighborhood D of a global or local optimum w. (see Figure 1). The first passage time is a random variable. Its distribution function P( n; wo) is the probability that a network initialized at Wo makes its first passage into D at the nth iteration of the learning rule. Figure 1: Sample search path. To arrive at an expression for P( n; wo), we first examine the probability of passing from the initial weight Wo to the weight w after n iterations. This probability can be expressed as pew, n I Wo, 0) = J dw' pew, n I w', 1) W( Wo -7 w'). (5) Substituting the single time-step transition probability (3) into the above expression, integrating over w', and making use of the time-shift invariance of the system3 we find pew, n 1 wo, 0) = ( pew, n - 1 1 Wo + J1,H(wo, x), O)}z . (6) Next, let G(n; wo) denote the probability that a network initialized at Wo has not passed into the region D by the nth iteration. We obtain G(n; wo) by integrating pew, n 1 Wo, 0) over weights w not in D; G(n;wo) = f dwP(w,nlwo,O) (7) JDe 2See (Ritter and Schulten 1988) and (Radons et al. 1990) for independent derivations. 3With our assumptions of a constant learning rate J.l and stationary sample density p{x), the system is time-shift invariant. Mathematically stated, P{w,n I w',m) = P{w,n -1/ w',m -1) 510 Orr and Leen where DC is the complement of D. Substituting equation (6) into (7) and integrating over w we obtain the recursion G(n;wo) = (G(n - l;wo + JJH[wo,x]) L: . (8) Before any learning takes place, none of the networks in the ensemble have entered D. Thus the initial condition for G is G(O;wo) = 1, woEDc. (9) Networks that have entered D are removed from the ensemble (i.e. aD is an absorbing boundary). Thus G satisfies the boundary condition G(n; wo) = 0, Wo ED. (10) Finally, the probability that the network has not passed into the region D on or before iteration n - 1 minus the probability the network has not passed into D on or before iteration n is simply the probability that the network has passed into D exactly at iteration n. This is just the probability for first passage into D at time-step n. Thus P(n;wo) = G(n -1;wo) G(n;wo) . (11) Finally the recursion (8) for G can be expanded in a power series in JJ to obtain the backward Kramers-Moyal equation G(n;w) G(n -1;w) = ex> . m J.lt L"1 L I. (12) ;=1 ;1, ... ;;=1 Truncation to second order in JJ results in the backward Fokker-Planck equation. In section 3.2 we will use both the full recursion (8) and the Fokker-Planck approximation to (12) to predict basin hopping times in stochastic learning. 3 Backpropagation and Competitive Nets We apply the above formalism to study the time evolution of the probability density for simple backpropagation and competitive learning problems. We give graphical examples of the time evolution of the weight space density, and calculate times for passage from local to global optima. 3.1 Densities for the XOR Problem Feed-forward networks trained to solve the XOR problem provide an example of supervised learning with well-characterized local optima (Lisboa and Perantonis, 1991). We use a 2-input, 2-hidden, I-output network (9 weights) trained by stochastic gradient descent on the cross-entropy error function in Lisboa and Perantonis (1991). For computational tractability, we reduce the state space dimension by Weight Space Probability Densities in Stochastic Learning 511 constraining the search to one- or two-dimensional subs paces of the weight space. To provide global optima at finite weight values, the output targets are set to 8 and 1 - 8, with 8 < < 1. Figure 2a shows the cost function evaluated along a line in the weight space. This line, parameterized by v, is chosen to pass through a global optimum at v = 0, and a local optimum at v = 1.0. In this one-dimensional slice, another local optimum occurs at v = 1.24 . Figure 2b shows the evolution of P( v, n) obtained by numerical integration of the Fokker-Planck equation. Figure 2c shows the evolution of P( v, n) estimated by simuhtion of 10,000 networks, each receiving a different random sequence of the four input/target patterns. Initially the density is peaked up about the local optimum at v = 1.24. At intermediate times, there is a spike of density at the local optimum at v = 1.0. This spike is narrow since the diffusion coefficient is small there. At late times the density collects at the global optimum. We note that for the learning rate used here, the local optimum at v = 1.24 is asymptotically stable under true gradient descent, and no escape would occur. CD It) ii 8· .., (II a) ..().5 0.0 0.5 1.0 1.5 2.0 v c) Figure 2: a) XOR cost function. b) Predicted density. c) Simulated density. Figure 3 shows a series of snapshots of the density superimposed on the cost function for a 2-D slice through the XOR weight space. The first frame shows the weight evolution under true gradient descent. The weights are initialized at the upper right-hand corner of the frame, travel down the gradient and settle into a local optimum. The remaining frames show the evolution of the density calculated by direct integration of the Kolmogorov equation (2). Here one sees an early spreading of the initial density and the ultimate concentration at the global optimum. 3.2 Basin Hopping Times The above examples graphically illustrate the intuitive notion that the noise inherent in stochastic learning can move the system out of local optima4 In this section we calculate the statistical distribution of times required to pass between basins. 4The reader should not infer from these examples that stochastic update necessarily converges to global optima. It is straightforward to construct examples for which stochastic learning convergences to local optima with probability one. 512 Orr and Leen True Gradient Descent Time =0 Time = 10 Time-28 Time =34 Time = 100 Figure 3: Weight evolution for 2-D XOR. The density is superimposed on top of the cost function. The first frame shows density using true gradient descent for all 100 timesteps. The remaining frames show the density for selected timesteps using stochastic descent. 3.2.1 Basin Hopping in Back-propagation For the search direction used in the example of Figure 2, we calculated the distribution of times required for networks initialized at v = 1.2 to first pass within € = 0.1 of the global optimum at v = 0.0. For this example we numerically integrated the backward Fokker-Planck equation. We verified the theoretical predictions by obtaining first passage times from an ensemble of 10,000 networks initialized at v = 1.2. See Figure 4. For this example the agreement is good at the small learning rate (JJ = 0.025) used, but degrades for larger JJ as higher order terms in the expansion (12) become significant. o o o 200 400 600 800 1000 First Passage Time Figure 4: XOR problem. Simulated (histogram) and theoretical (solid line) distributions of first passage times for the cost function of Figure la. Weight Space Probability Densities in Stochastic Learning 513 When the Fokker-Planck approximation fails, results obtained from the exact expression (8) are in excellent agreement with experimental results. One such example is shown in Figure 5. Similar to Figure 2a, we have chosen a one-dimensional subspace of the XOR weight space (but in a different direction). Here, the FokkerPlanck solution is quite poor because the steepness of the cost function results in large contributions from higher order terms in (12). As one would expect, the exact solution obtained using (8) agrees well with the simulations. It) -0.5 0.0 0.5 a) v 1.0 1.5 b) o Exact Fokker-Planck 200 400 600 800 First Passage Time Figure 5: Second I-D XOR example. a) Cost function. b) Simulated (histogram) and theoretical (lines) distributions of first passage times. 3.2.2 Basin Hopping in Competitive Learning As a final example, we consider competitive learning with two 2-D weight vectors symmetrically placed about the center of a rectangle. Inputs are uniformly distributed in a rectangle of width 1.1 and height 1. This configuration has both global and local optima. Figure 6a shows a sample path with weights started near the local optimum (crosses) and switching to hover around the global optimum. The measured and predicted (from numerical integration of (8)) distribution of times required to first pass within a distance € = 0.1 of the global optimum are shown in Figure 6b. w2 ~:~. : .. :...: .<~~:\.:··i a) 4--~:-O---=,,"-:----'=''''':::-''''''' -:-:---r-- w1 0.2 0.4 0.6 O.S 1 § 0 ~ ~8 -80 Q: q 0 b) 0 100 200 300 400 First Passage Time 500 Figure 6: Competitive Learning a) Data (small dots) and sample weight path (large dots). b) First passage times. 4 Discussion The dynamics of the time evolution of the weight space probability density provides a direct handle on the performance of learning algorithms. This paper has focused 514 Orr and Leen on transient phenomena in stochastic learning with constant learning rate. The same theoretical framework can be used to analyze the asymptotic properties of stochastic search with decreasing learning rates, and to analyze equilibrium densities. For a discussion of the latter, see the companion paper in this volume (Leen and Moody 1993). Acknowledgements This work was supported under grants N00014-90-J-1349 and NOOO-HI-J-1482 from the Office of Naval Research. References E. Oja (1982), A simplified neuron model as a principal component analyzer. J. Math. Biology, 15:267-273. Halbert White (1989), Learning in artificial neural networks: A statistical perspective. Neural Computation, 1:425-464. J.J. Kushner (1987), Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: Global minimization via monte carlo. SIAM J. Appl. Math., 47:169-185. Christian Darken and John Moody (1991), Note on learning rate schedules for stochastic optimization. In Advances in Neural Information Processing Systems 3, San Mateo, CA, Morgan Kaufmann. Todd K. Leen and Genevieve B. Orr. (1992), Weight-space probability densities and convergence times for stochastic learning. In International Joint Conference on Neural Networks, pages IV 158-164. IEEE. Todd K. Leen and John Moody (1993), Probability Densities in Stochastic Learning: Dynamics and Equilibria. In Giles, C.L., Hanson, S.J., and Cowan, J.D. (eds.), Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann Publishers. H. Ritter and K. Schulten (1988), Convergence properties of Kohonen's topology conserving maps: Fluctuations, stability and dimension selection, Bioi. Cybern., 60, 59-71. G. Radons, H.G. Schuster and D. Werner (1990), Fokker-Planck description oflearning in backpropagation networks, International Neural Network Conference, Paris, II 993-996, Kluwer Academic Publishers. C.W. Gardiner (1990), Handbook of Stochastic Methods, 2nd Ed. Springer-Verlag, Berlin. P. Lisboa and S. Perantonis (1991), Complete solution of the local minima in the XOR problem. Network: Computation in Neural Systems, 2:119.
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Performance Through Consistency: MS-TDNN's for Large Vocabulary Continuous Speech Recognition Joe Tebelskis and Alex Waibel School of Computf'f Science Carnegie MeHon University Pittsburgh, PA 15213 Abstract Connectionist Rpeech recognition systems are often handicapped by an inconsistency between training and testing criteria. This problem is addressed by the Multi-State Time Delay Neural Network (MS-TDNN), a hierarchical phonf'mp and word classifier which uses DTW to modulate its connectivit.y pattern, and which is directly trained on word-level targets. The consistent use of word accuracy as a criterion during bot.h t.raining and testing leads to very high system performance, even wif II limited training dat.a. Until now, the MS-TDN N has been appli('d primarily to small vocabulary recognition and word spotting tasks. In this papf'f we apply the architecture to large vocabulary continuous speech recognition, and demonstrate that our MS-TDNN outperforms all ot,hf'r systems that have been tested on tht' eMU Conference Registration database. 1 INTRODUCTION Neural networks hold great promise in the area of speech recognition. But in order to fulfill their promise, they must be used "properly". One ohviolls conditioll of "proper" use is that both training and testing should use a consistent error criterion. Unfortunately, in speech recognition, this ohvious condition is often violafeo: networks are frequently trained using phoneme-level criteria (phoneme classificaf iun 696 MS-TDNN's for Large Vocabulary Continuous Speech Recognition 697 or acoustic prediction), while the testing criterion is word recognit.ion accuracy. If phoneme recognition were perfect, then word recognition would also be perfect; but of course this is not. the case, and the errors which are inevitably made are opt.imized for the wrong criterion, resulting in subopt.imal word recognition accuracy. The Multi-State Time Delay Neural Network (MS-TDNN) has recently been proposed as a solution to this problem [1]. The MS-TDNN is a hierarchically struct.ured classifier which consistently uses word accuracy as a criterion for bot.h training and testing. It has so far yielded excellent results on small vocabulary recognition [1, 2, 3] and a word spotting task [4]. In the present paper, we review the MS-TDNN architecture, discuss its application to large vocabulary continuous speech recognition, and present some favorable experimental results on this task. 2 RESOLVING INCONSISTENCIES: MS-TDNN ARCHITECTURE In this section we motivate the design of our MS-TDNN by showing how a series of intermediate designs resolve successive inconsistencies. The preliminary system architecture, shown in Figure l(a), simply consists of a phoneme classifier, in this case a TDNN, whose outputs are copied into a DTW matrix, in which continuous speech recognition is performed. Many existing systems are based on this type of approach. \rVhile this is fine for bootstrapping purposes, the design is ultimately suboptimal because the training criterion is inconsistent with the testing criterion: phoneme classification is not word classification. To address this inconsistency we must train t.he network explicitly to perform word classification. To this end, we define a word layer with one unit for each word in the vocabulary; the idea is illustrated in Figure 1 (b) for the word "cat". We correlate the activation of the word unit with the associat.ed DTW score by estahlishing connections from the DTW alignment path to the word unit. Also, we give the phonemes within a word independently trainable weights, to enhance word discrimination (for example, to discriminate "cat" from "mat" it may be useful to give special emphasis to the first phoneme); these weights are tif'd over all frames in which the phoneme occurs. Thus a word unit is an ordinary unit, except that its connectivity to the preceding layer is determined dynamically, and its net input should be normalized by the total duration of the word. The word unit is trained on a target of 1 or 0, depending if the word is correct or incorrp.ct for the current segment of speech, and the resulting error is backpropagat.ed t.hrough the entire net.work. Thus, word discrimination is treated very much like phoneme discrimination. Although network (b) resolves the original inconsistency, it now suffers from a secondary one - namely, that the weight.s leading to a word unit are used during training but ignored during testing, since nTW is still performed entirely in the DTW layer. We resolve this inconsistency hy "pushing down j , tht'se weights one level, as shown in Figure l(c). Now the phoneme activations are no longer directly copied into the DTW layer, but instead are modulated by a weight and bias before being stored there (DTW units are linear); and t.he word unit has constant weights, and no bias. During word-level training, error is still backpropagat.ed from targets at the word level, but bia.ses and weight.s are modified only at t.he DTW lev(') and 698 Tebelskis and Waibel Word [ "CAT" r~n DTW[ T7eBt ... ... oC oC U U copy copy N N ... ... ! ; u u • • oC • • oC •• TDNN hidden hiddell Speech: "CAT" Speech: "CAT" (a) (b) Word [ "CAT" "CAT" r"f-in r .. f-;n DTW[ ... ... oC oC U u w2 w3 N N ; ; ... ... ! ! U U • • oC oC • • TDNN hidtUn Speech: "CAT" Speech: "CAT' (e) (d) Figure 1: Resolving inconsistencies: (a) TDNN+DTW. (b) Adding word layer. (c) Pushing down weights. (d) Linear word units, for continuous speech recognition. MS-TDNN's for Large Vocabulary Continuous Speech Recognition 699 below. Note that this transformed network is not exactly equivalent to the previous one, but it preserves the properties that there are separate learned weights associated with each phoneme, and there is an effective bias for each word. Network (c) is still flawed by a minor inconsistency, arising from its sigmoidal word unit. The problem does not exist for isolat.ed word recognition, since any monotonic function (sigmoidal or otherwise) will correlate the highest word activation with the highest DTW score. However, for continuous speech recognition, which concatenates words into a sentence, the optimal sum of sigmoids may not correspond to the optimal sigmoid of a Slim, leading to an inconsistency between word and sentence recognition. Linear word units, as shown in (d), resolve this problem; in practice we have found that linear word units perform slightly better than sigmoidal word units. The resulting architecture is called a "Multi-State TDNN" because it integrates the DTW alignment of multiple states into a TDNN to perform word classification. While an MS-TDNN for small vocabulary recognition can be based on word models with non-shared states (4], a large vocabulary MS-TDNN must be based on shared units of speech, such as phonemes. In our system, the TDNN (first three layers) is shared by all words in the vocabulary, while each word requires only oue nOI1shared weight and bias for each of its phonemes. Thus the number of parameters in the MS-TDNN remains moderate even for a large vocabulary, and it can make the most of limited training data. Moreover, new words can be added to the vocabulary without retraining, by simply defining a new DTW layer for each new word, with incoming weights and biases initialized to 1.0 and 0.0, respectively. Given constant weights under the word layer, word level training is really just another way of viewing DTW level training; but the former is conceptually simpler because there is a single binary target for each word, which makes word level discrimination very straightforward. For a large vocabulary, discriminating against all incorrect words would be very expensive, so we discriminate against only a small number of close matches (typically 1). Word level training yields better word classification than phoneme level traiuing. In one experiment, for example, we bootstrapped our system with phoneme level training (as shown in Figure la), and found that word recognition accuracy asymptoted at 71% on a test set. We then continued training from the word level (as shown in Figure 1d), and found that word accuracy improved to 81% on the test set. It is worth noting that in an intermediate experiment, even when we held the DTW layer's incoming weights and biases constant (at 1.0 and 0.0 respectively), thus adding no new trainable parameters to the system, we found that word level training still improved the word accuracy from 71% to 75% on the test set, as a consequence of word-level discrimination. 3 BALANCING THE TRAINING SET The MS-TDNN must be first bootstrapped wit.h phoneme level training. In our early experiments we had difficulty bootst.rapping the TDNN, not only because our training set was unbalanced, but also because the vast majority of phonemes were being trained on a target of 0, so that t.he negative training was overwhelming and 700 Tebelskis and Waibel A E o N T Y Z A E o N T Y Z NON A T YET 0 0 0 • 0 0 0 0 0 0 0 0 0 0 • 0 0 • 0 0 0 0 0 0 • 0 • 0 0 0 0 0 0 0 0 0 • 0 0 • 0 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 Scale backprop error by: NON A T YET -117 -117 -117 1.0 -117 -117 -117 -117 -117 -117 -117 -117 -117 -117 1.0 -117 . . .. 112 -1/6 112 -1/6 -116 -116 -1/6 -116 -1/6 -116 -116 -1/6 112 -1/6 -1/6 112 . . .. -118 -1/8 -1/8 -1/8 -1/8 -118 -118 -118 Figure 2: Balancing the training set: "No not yet" . defeating the positive training. In order to address these problpms, we normalized the amount of error back propagated from each phoneme unit so that the l'elative influence of positive and negative training was balanced out over the entire training set. This apparently novel technique is illustrated in Figure 2. Given the utterance "No not yet" , for example, we observe that thpre are t.wo frames each of "N" and "T", one frame of several other phonemes, and zero of others. Based on these counts, we compute a backpropagation scaling fact.or for each phoneme in each frame, as shown in the bottom half of the figure. We found that this technique was indisppnsible when bootst.rapping with the squared error criterion, E = 2:(Tj Y;)2. In subsequent experiments, we found that it was still somewhat helpful but no longer necessary when training with the McClelland error function, E = - 2: 10g(1- (T; - y;)2), or with the Cross Entropy error function, E = - 2:(1;;logYi) + (1- Td/og(1- Yi). We attribute this difference to the fact that the sum squared error function is merely a quadratic error function, whereas the latter two functions tend towards infinite error as the difference between the target and actual activation approaches its maximum value, compensating more forcefully for the flat behavior encouraged hy all the negative training. 4 EXPERIMENTAL RESULTS Wp have trained and tested our MS-TDNN on two recordings of 200 English sentences from the CMU Conference Registrat.ion database, recorded by one male speaker using a close-speaking microphone. Our speaker-dependpnt t.esting results MS-TDNN's for Large Vocabulary Continuous Speech Recognition 701 Table 1: Comparison of speech recognition systems applied to the CMU Conference Registration Database. HMM-n = HMM with n mixture densities [5]. LPNN = Linked Predictive Neural Network [6]. HCNN = Hidden Control Neural Network [7]. LVQ = Learned Vector Quantization [8]. TDNN corresponds to MS-TDNN without word-level training. (Perplexity 402( a) used only 41 test sentences; 402(b) used 204 test sentences.) " System 7 perplexity 111 402(a) 402(b) II HMM-I 55% HMM-5 96% 71% 58% HMM-10 97% 75% 66% LPNN 97% 60% 41% HCNN 75% LVg 98% 84% 74% 61% TDNN 98% 78% 72% 64% MS-TDNN 98% 82% 81 '70 70% are given in Table 1, along with comparative results from several other systems. It can be seen that the MS-TDNN has outperformed all other systems that have been compared on this database. This particular MS-TDNN used 16 melscale spectral input units, 20 hidden units, 120 phoneme units (40 phonemes with a 3-state model), 5,487 DTW units, and 402 word units, with 3 and 5 delays respectively in the first two layers of weights, giving a total of 24,074 weights; it used symmetric (-1..1) unit activations and inputs, and linear DT'\! units and word units. Word level training was performed using the Classification Figure of Merit (CFM) error function, E = (1 + (Ye - Yc))2, in which the correct word (with activation Yc ) is explicitly discriminated from the best incorrect word (with activation Yc); CFM was somewhat better than MSE for word level training, although the opposite was true for phoneme level training. Negative word level training was performed only if the two words were sufficiently confusable, in order to avoid disrupting the network on behalf of words that had already been well-learned. More recently we have begun experiments on the speaker independent Resource Management database, containing nearly 4000 training sentences. To date we have primarily focused on bootstrapping the phoneme level TDNN on this database, without doing much word level training; but. early experiments suggest we may reasonably expect another 4% improvement from word level training. Our preliminary results are shown in Table 2, compared against. two other systems: an early version of Sphinx [9], and a simple but large MLP which has been trained as a phoneme classifier [10]. Each of these systems uses cont.ext independent phoneme models, with multiple states per phoneme, and includes differenced coefficients in its input representation. It can be seen that our TDNN outperforms this version of Sphinx while using a comparable number of parameters, but is outperformed by the MLP which has an order of magnitude more paramet.ers. (We note that t.he MLP also uses phonological models to enhance its performance, and uses online training with ran702 Tebelskis and Waibel Table 2: Context-independent systems applied to the Resource Management database. " System I parameters perplexity II 60 1000 Early Sphinx 35,000 76% 36% MLP iICSI-SRIl 300,000 94% 75% TDNN 42,000 79% 43% MS-TDNN 75,000 (+4%) dom sampling rather than updating the weights after each sentence.) In any case, we suggest that the MLP might further improve its performance by incorporating word level training. 5 REMAINING INCONSISTENCIES While the MS-TDNN was designed for consistency, it is not yet entirely consistent. For example, the MS-TDNN's training algorithm assumes that the network connectivity is fixed; but in fact the connect.ivit.y at the word level varies, depending on the DTW alignment path during the current iteration. We presume that this is a negligible factor, however, by the time the training has asymptoted and the segmentation has stabilized. A more serious inconsistency arises during discriminative training. In our MSTDNN, negative training is performed at known word boundaries; this is inconsistent because word boundaries are in faet. unknown during testing. It would be better to discriminate against words found hy a free alignment, as suggested by Hild [3]. Unfortunately this is an expensive operation, and it proved impractical for our system. 6 CONCLUSION We have shown that the perfOl'mance of a connectionist speech recognition system can be improved by resolving inconsistf'ncie:s in its design. Specifically, by introducing word level training into a TDNN phoneme classifier (thus defining an MS-TDNN), the training and testing crit,e:ria become consistent, enhancing the system's word recognition accuracy. We applied our MS-TDNN archit.ecture to the task of large vocabulary continuous speech recognition, and found that it outperforms all other systems that have been evaluated on the CMU Conference Registration database. In addition, preliminary results suggest that t.he MS-TDNN may perform well on the large vocabulary Rf'source Management database, using a relatively small number of free paramet.ers. Our future work will focus on t.his investigation. MS· TDNN's for Large Vocabulary Continuous Speech Recognition 703 Acknowledgements The authors gratefully acknowledge the support of DARPA and the National Science Foundation. References [1] P. Haffner, M. Franzini, and A. Waibel. Integrating Time Alignment and Connectionist Networks for High Performance Continuous Speech Recognition. In Proc. International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 1991. [2] P. Haffner. Connectionist Word-Level Classification in Speech Recognition. In Proc. ICASSP, 1992. [3] H. Hild and A. Waibel. Connected Letter Recognition with a Multi-State Time Delay Neural Network. In Advances in Neural Information Processing Systems 5, Morgan Kaufmann Publishers, 1993. [4] T. Zeppenfeld and A. ,-vaibel. A Hybrid Neural Network, Dynamic Programming Word Spotter. In Proc. ICASSP, 1992. [5] O. Schmidbauer. An LVQ Based Reference Model for Speaker-Independent and -Adaptive Speech Recognition. Technical Report, Carnegie Mellon University, 1991. [6] J. Tebelskis, A. Waibel, B. Petek, and O. Schmidbauer. Continuous Speech Recognition using Linked Predictive Neural Networks. In Proc. ICASSP, 1991. [7] B. Petek and J. Tebelskis. Context-Dependent Hidden Control Neural Network Architecture for Continuous Speech Recognition. In Proc. ICASSP, 1992. [8] O. Schmidbauer and J. Tebelskis. An LVQ Based Reference Model for Speaker Adaptive Speech Recognition. In Proc. ICASSP, 1992. [9] K. F. Lee. Large Vocabulary Speaker-Independent Continuous Speech Recognition: The SPHINX System. PhD Thesis, Carnegie Mellon University, 1988. [10] M. Cohen, H. Franco, N. Morgan, D. Rumelhart, and V. Abrash. Cont.extDependent Multiple Distribution Phonetic Modeling with MLP's. In Advances in Neural Information Processing Systems 5, Morgan Kaufmann Publishers, 1993.
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Generic Analog Neural Computation The EPSILON Chip Stepben Cburcber Dept. of Elee. Engineering University of Edinburgh King's Buildings Edinburgh. EH9 3JL Donald J. Baxter Dept of Elec. Engineering University of Edinburgh King's Buildings Edinburgh. EH9 3JL Alister Hamilton DeptofE~.En~ring University of Edinburgh King's Buildings Edinburgh. EH9 3JL Alan F. Murray as above H. Martin Reekie as above Abstract An analog CMOS VLSI neural processing chip has been designed and fabricated. The device employs "pulse-stream" neural state signalljng, and is capable of computing some 360 million synaptic connections per secood. In addition to basic characterisation results. the performance of the chip in solving "real-world" problems is also demonstrated. 1 INTRODUCTION Inspired by biology. and borne out of a desire to perform analogue computation with fundamentally digital fabrication processes. the so-called "pulse-stream" arithmetic system has been steadily evolved and improved since its inception in 1986 (Murrayl990a. Murray 1989a). In addition to this continuous development at Edinburgh. many other research groups around the world (mast notably Meador et al (Meadorl990a» have experimented with their own pulse-firing neural circuits. In pulsed implementations. each neural state is represented by some variable attribute (e.g. the width of fixed frequency pulses. or the rate of fixed width pulses) of a train (or "stream") of pulses. The neuron design therefore reduces to a form. of oscillatcr. Each neuron is fed by a column of synapses. which multiply incoming neural states by the synaptic weights. In contrast with the original circuits of Murray and Smith (MurrayI987a). the synapse design which will be discussed herein utilises analog circuit techniques to perform. the multiplication of neural state by synaptic weight This paper describes the Edinburgh Pulse-Stream Implementation of a Learning Oriented Network (EPSn.ON) chip. EPSILON was developed as a ftexible neural processor. capable of addressing a variety of applications. The main design criteria were as follows: • That it be large enough to be of use in practical problems. • It should be capable of implementing networks of arbitrary size and architecture. • It must be able to act as both a "slave" acceleratcr to a conventional computer. and as an "autonomous" processor. As will be seen. these constraints resulted in a chip which could realise only a single layer of synaptic weights. but which could be cascaded to form. large. useful networks for solving real-wcrld problems. 773 774 Churcher, Baxter, Hamilton, Murray, and Reekie The remaining sections of this paper describe the attributes of pulse-coded neural systems in general. befoce detailing the circuits which were employed on EPSILON. Finally. results frOOl a vowel recognition application are presented. in order to illustrate the performance of EPSILON when applied to real tasks. 2 PULSE CODED NEURAL SYSTEMS As already mentioned. EPSILON is a pulse coded analog neural processing chip. In such implementations. neural states are encoded as digital pulses. The states themselves may then be represented either by varying the width of the pulses (pulse width modulation PWM). oc by varying the rate of the pulses (pulse frequency modulation PFM). The arguments for using pulses in this way are strong. Firstly. they provide a very effective and robust method for communicating states both on- and between-chip. since pulses are extremely resistant to noise. Secondly. the use of pulses to represent states renders interfacing to digital circuits and computer peripherals straightforward. Finally. pulsed signalling leads to simplification of artihmetic circuits (i.e. synapses). resulting in much higher inter-connection densities. Unfortunately. pulse-based systems do have drawbacks. In common with all analog circuits. the synaptic computing elements have limited precision (usually equivalent to about 7 bits). and their performance is subject to the vagaries of fabrication process variations. This results in a situation whereby supposedly "matched" circuits vary marlredly in their characteristics. Furthermore. the switching which is inherent in any pulsed circuit results in increased levels of system noise. most usually in the form of power supply transients. An additional problem with pulse frequency modulation (PPM) systems is that computation rates are dependent on the data: this is an impatant consideration in speedcritical applicatioos. 3 THE EPSILON DESIGN This section describes the circuits which were used in the EPSILON design. The operating principles of each circui~ are discussed. and characterisation results presented. In accordance with the demerits mentioned in the previous section. all circuits were designed to be tolerant to noise and process variations. to cause as little noise as possible themselves. and to be easy to "set up" in practice. Finally. the specification of the EPSn..ON chip is presented. 3.1 SYNAPSE The synapse design was based on the standard transconductance multiplier circuit. which had previously been the basis for monolithic analogue transversal filters for use in signal processing applications (DenyerI981a). Such multipliers use MOS transistors in their linear region of operation to generate output currents proportional to a product of two input voltages. This concept was adapted for use in pulsed neural networks by fixing one of the input voltages. and using a neural state to gate the output current. In this manner. the synaptic weight controls the magnitude of the output current. which is multiplied by the incoming neural pulses. The resultant charge packets are subsequently integrated to yield the total post-synaptic activity voltage. Figure 1 shows the basic pulsed multiplier cell. where MI and M2 form the transconductance multiplier. and M3 is the output pulse transistor. By ensuring that the drain-source voltages for MI and M2 are the same and constant (the differential amplifier and transistors M4 and M5 are used to satisfy this constraint). non-linearities in the transistor responses can be cancelled out. such that I OUT is linearly dependent on the ctifference of V GS I and V GS2 (Murray 1992a). Multiplication is achieved by pulsing this current by the neural state, V)' An "instantaneous" representation of the aggregated post-synaptic activity is given by the output voltage. Vour: this must subsequently be integrated in order to provide an activity input to a neuron. Generic Analog Neural Computation-The EPSILON Chip 775 --~----------------.--~ v. 1 ~ ~ •......... -... -.---------------------, ---------..---- ... --_. --- _.~ ~- . -._VIWV .. Figure 1: Transconductance Multiplier Synapse 14.0 5.0 -. 13.0 II! 4.i 1 12.0 4.4 ~ 11.0 4.1 .!t .s 10.0 £ 3.75 !I ~ 3.4 '; 9.0 Q. 3.1 '; 8.0 0 2.8 7.0 0.0 10.0 20.0 30.0 40.0 50.0 ro.O 70.0 80.0 90.0 Input Stale (microseconds) Figure 2: Synapse Characterisation Results Results from characterisation tests of the synapse are presented in Figure 2. which shows output state against input state. for different synaptic weight voltages. As seen from the Figure. the linearity of the synapses. with respect to input state. is very high. The variation of synapse respoose with synaptic weight voltage is also fairly uniform. The graphs depict mean performance over all the synaptic columns in all the chips tested. The associated standard deviations were more or less constant. representing a variation of approximately ± 300 ns in the values of the output pulse widths. The effects of intra- and interchip process mismatches would therefore seem to be well contained by the circuit design. The "zero point" in the synaptic weight range was set at 3.75 V and. as can be seen from the Figure. each graph shows an offset problem when the input neural state is zero. This was attributable to an imbalance in the operating conditions of the transist(Xs in the synapse. induced by the non-ideal nature of the power supplies (i.e. the non-zero sheet resistance of the power supply tracks). resulting in an offset in the input voltage to the post-synaptic integrator. This problem is easily obviated in practice. by employing three synapses per column to cancel the offset. 776 Churcher, Baxter, Hamilton, Murray, and Reekie 3.2 NEURONS In order to reflect the diversity of neural network forms. and possible applications. two different neuron designs were included on the EPSn.,oN chip. The first, a synchronous pulse width modulation neuron was designed with vision applicatioos in mind. This circuit could guarantee netwcrk computation times. thereby eliminating the data dependency inherent in pulse frequency systems. The second neuron design used asynchronous pulse frequency modulation; the asynchronous nature of these circuits is advantageous in feedback and recurrent neural architectures. where temporal characteristics are important. As with the synapse. both circuits were designed to minimise transient noise injection. and to be tolerant of process variations. 3.2.1 Pulse Width Modulation As already stated. this system retains all the advantages of using pulses for communication/calculation. whilst being able to guarantee a maximum network evaluation time. In the first instance. the main disadvantage with this technique appeared to be its synchronous nature - neurons would all be switching together causing larger power supply transients than in an asynchronous system. This problem has. however. been circumvented via a "double-sided" pulse modulation scheme. which will be more fully explained later. V~ ~~... .......... ~ Xc;~ ~.~' ~.:.~~.:.':.:.:' .~.::.~.:. : ....... :.:.".".'.:: ..... :.:.:':.'.:.::.:.:.' VOUT Figure 3: Pulse-Width Modulation Neuron The operation of the pulse-width modulation neuron is illustrated in Figure 3. The neuron itself is nothing more elaborate than a 2-stage comparator. with an inverter output driver stage. The inputs to the circuit are the integrated post-synaptic activity voltage. V A.CT. and a reference voltage. V RAMP. which is generated off-chip and is globally distributed to all neurons in parallel. As seen from the waveforms in Figure 3. the output of the neuron changes state whenever the reference signal crosses the activity voltage level. An output pulse. which is some function of the input activation. is thus generated. The transfer function is entirely dependent on the shape of the reference signal - when this is generated by a RAM look-up table. the function can become completely arbitrary and hence user programmable. Figure 3 shows the signal which should be applied if a sigmoidal transfer characteristic is desired. Note that the sigmoid signals are "on their sides" - this is because the input (or independent variable) is on the vertical axis rather than the horizontal axis. as would normally be expected. The use of a "double-sided" ramp for the reference signal was alluded to earlier - this mechanism generates a pulse which is symmetrical about the mid-point of the ramp. thereby greatly reducing the likelihood of coincident Generic Analog Neural Computation-The EPSILON Chip 777 edges. This edge-asynchronicity obviates the problem of larger switching transients on the power supplies. Furthermore. because the analogue element (i.e. the ramp voltage) is effectively removed from the chip. and the circuit itself merely functions as a digital block. the system is immune to process variations. 100 80 ~ B 60 S tfl Ol 40 .... '" 0 z 20 / / -.- -. " i I .. . ,,' .... ., .. , ," . ' I .. .. " .' I . ' ,,' . I / • , 0' f ,' , " .: • •• I · , .' J.;:.:" Temperatures : l.0 0.8 ···· 0.6 --0.4 . .. 0.2 - 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 Activity (V) Figure 4: PWM Neuron Performance Figure 4 shows plots of output state (measured as a percentage of a maximum possible 20 fJS pulse) versus input activity voltage. fer five different sigmoid "temperatures". averaged over all the neurons on one chip. As can be seen. the fidelity of the sigmoids is extremely high. and it should be noted that all the curves are symmetrical about their midpoints sOOlething which is difficult to achieve using standard analog circuits. 3.2.2 Pulse Frequency Modulation The second neuron design which was included in EPSn..ON used pulse frequency encoding of the neural state. Although hampered by data dependent calculation times. its wholly asynchronous nature makes it ideal for neural network architectures which embody temporal characteristics Le. feedback networks. and recurrent networks. VIH 01---------< c T Figure 5: Pulse Frequency Modulation Neuron 778 Churcher, Baxter, Hamilton, Murray, and Reekie The neuron design is illustrated in Figure 5. and is basically a Voltage Controlled Oscillatoc (VCO) with a variable gain sigmoidal transfer characteristic. Oscillation is achieved via the hysteretic charge and discharge of capacitor C. by the currents fH and fL respectively. The output pulse width is constant. and is set by fH. whilst the inter-pulse spacing (and hence output frequency) is controlled by fL. fL itself is determined by the activity voltage. VXf. via the differential stage constituted by transistors M3 to M6. It is this latter which gives the veo its sigmoidal characteristic. and gain variations may be achieved by injecting and removing additional current at appropriate points in this stage (note that the circuitry for this has been omitted from Figure 5. for the sake of clarity). 60 I 50 40 ~ ! JO ! 20 10 0 ·1.5 ·1 ~.5 0 0.5 1.5 vm - VXl . VMJJ) (V) Figure 6: PFM Neuron Performance The characterisation results for the veo are presented in Figure 6. The Figure shows plots of output percentage duty cycle versus input differential voltage. for different values of sigmoid gain. Note that the curves are fair approximations to sigmoids. although. in contrast with the pulse width modulation neuron. they are not symmetrical about their mid-points. It can also be seen that the range of possible sigmoid gains is smaller than the range available with the PWM system. although this is not a crucial factor in many applications. 3.3 EPSILON SPECIFICATIOJlli The circuits described in the previous section were combined to form the EPSILON chip. This was subsequently fabricated by European Silicoo Structures (ES2) using their ECPD15 (i.e. 1.5 pm. double metal. single poly CMOS). As already stated. each chip was capable of implementing a single layer of synaptic connections. and could accept inputs as either analog voltages (for direct interface to sensors) oc as pulses (for communication with other chips. and with digital systems). The full specification is given in Table 1. 4 APPLICATION VOWEL RECOGNITION Mter the device characterisation experiments had been completed. EPSILON was used to implement a multi-layer perceptron (MLP) for speech data classification. The MLP had 54 inputs. 27 hidden units. and 11 outputs. and the task was to classify 11 different vowel sounds spoken by each of 33 speakers. The input vectors were formed by the analog outputs of 54 band-pass filters. The MLP was initially trained on a SPARC station. using a subset of 22 patterns. Learning (using the VIrtual Targets algorithm. with 0 % noise (Murray 1991a» proceeded until the maximum bit erroc in the output vector was S; O. 3. at which point the weight set was Generic Analog Neural Computation-The EPSILON Chip 779 Table 1: EPSn.,ON Specifications EPSn.,ON Specification No. of State Input Pins 30 No. of Actual State Inputs 120. Muxed in Banks of 30 Input Modes Analog. PW. or PF No. of State Outputs 30. Directly Pinned Out Output Modes PW orPF No. of Synapses 3600 No. of Weight Load Channels 2 Weight Load TilDe 3.6 ms Weight Storage Dynamic Maximum Speed (cps) 360 Mcps Technology 1.5 pm. Double Metal CMOS Die Size 9.5 mm x 10. 1 mm Maximum Power Dissipation 350mW 6 0.3 tE 0.25 (1) ~ 0.2 ::s cr' CZ) 0.15 c:: c:u (1) ::E 0.1 0.05 200 400 600 800 1000 1200 1400 Epoch Number Figure 7: EPSn..ON Under Training downloaded to EPSn.,ON. Training was then restarted under the same regime as before (using the same "stop" criterion). although this time EPSn..ON was used to evaluate the "forward pass" phases of the network. Figure 7 shows the evolution of mean square error with number of epochs during this period; at the end of training. EPSn..ON could correctly identify all 22 training patterns. Subsequent to this. 176 "unseen" test patterns were presented to the EPSILON network. with the result that 65.34 % of these vectors were correctly classified. This compared very favourably with similar generalisation experiments which were carried out on a SPARC : in this case. the best result obtained was 67.61 %. 5 CONCLUSIONS In conclusion. a large analog VLSI neural chip. composed of process tolerant circuits. with useful characteristics has been fabricated. Although not a self-learning device. it has been proved that EPSn.,ON will support learning. and can be applied successfully to real780 Churcher, Baxter, Hamilton, Murray, and Reekie world problems. Indeed. when correctly trained. the performance of the EPSn..ON chip has been shown to be comparable with that of software simulations on a SPARC station. Work is currently under way to apply EPSll..ON to computer vision tasks: more specifically. it it will be used to implement an MLP which is capable of recognising regions in segmented images ci natural scenes. Furthermcn. the success of the learning experiments has given us sufficient coofidence to undertake the development ci a self-learning analog neural chip. It is envisaged that this will employ EPSILON-type circuits. and will implement the VutuaI Targets (Murrayl991a) training algorithm: the design of a small prototype is currently nearing completion. Acknowledgements The authors would like to thank the Science and Engineering Research Council for their continued funding of this work. In addition. Stephen Churcber and Donald Baxter are grateful to British Aerospace PLC and Thorn-EMI CRL respectively for sponsorship and technical suppat during the course of their PhD's. Lionel Tarassenko (Dept. of Engineering Science. University of Oxford) must also be thanked for his invaluable comments. and for supplying the vowel database. References Murray 1990a. A. F. Murray. D. Baxter. Z. Butler. S. Churcber. A. Hamilton. H. M. Reekie. and L. Tarassenko. "Innovations in Pulse Stream Neural VLSI : Arithmetic and COOlJIlUnieations". IEEE Workshop on Microelectronics for Neural Networks, Dortmund 1990. pp. 8-15. 1990. Murray 1989a. A. F. Murray. "Pulse Arithmetic in VLSI Neural Networks". IEEE MICRO. vol. 9. no.6.pp.64-74.1989. Me adorl99Oa. J. Meador. A. Wu. C. Cole. N. Nintunze. and P. Chintrakulchai. "Programmable Impulse Neural Circuits". IEEE Transactions on Neural Networks. vol. 2. no. 1. pp. 101-109. 1990. Murray 1987a. A. F. Murray and A. V. W. Smith. "Asynchronous Arithmetic fer VLSI Neural Systems". Electronics Letters. vol. 23. no. 12. pp. 642-643. June. 1987. Denyerl981a. P. B. Denyer and 1. Maver. "MOST Transconductance Multipliers for Array Applications" . lEE Proc. Pt. 1. vol. 128. no. 3. pp. 81-86. June 1981. Murray 1992a. A.F. Murray. A. Hamilton. DJ. Baxter. S. Churcber. H.M. Reekie. and L. Tarassenko. "Integrated Pulse-Stream Neural Networks - Results. Issues and Pointers" . IEEE Trans. Neural Networks. pp. 385-393. 1992. Murray 1991a. A. F. Murray. "Analog VLSI and Multi-Layer Perceptrons - Accuracy. Noise and On-Chip Learning". Proc. Second International Conference on Microelectronics for Neural NeMorks, Munich (Germany). pp. 27-34. 1991.
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Analog VLSI Implementation of Multi-dimensional Gradient Descent David B. Kirk, Douglas Kerns, Kurt Fleischer, Alan H. Barr California Institute of Technology Beckman Institute 350-74 Pasadena, CA 91125 E-mail: dkIDegg.gg . cal tech. edu Abstract We describe an analog VLSI implementation of a multi-dimensional gradient estimation and descent technique for minimizing an onchip scalar function fO. The implementation uses noise injection and multiplicative correlation to estimate derivatives, as in [Anderson, Kerns 92]. One intended application of this technique is setting circuit parameters on-chip automatically, rather than manually [Kirk 91]. Gradient descent optimization may be used to adjust synapse weights for a backpropagation or other on-chip learning implementation. The approach combines the features of continuous multi-dimensional gradient descent and the potential for an annealing style of optimization. We present data measured from our analog VLSI implementation. 1 Introduction This work is similar to [Anderson, Kerns 92], but represents two advances. First, we describe the extension of the technique to multiple dimensions. Second, we demonstrate an implementation of the multi-dimensional technique in analog VLSI, and provide results measured from the chip. Unlike previous work using noise sources in adaptive systems, we use the noise as a means of estimating the gradient of a function f(y), rather than performing an annealing process [Alspector 88]. We also estimate gr-;:dients continuously in position and time, in contrast to [Umminger 89] and [J abri 91], which utilize discrete position gradient estimates. 789 790 Kirk, Kerns, Fleischer, and Barr It is interesting to note the existence of related algorithms, also presented in this volume [Cauwenberghs 93] [Alspector 93] [Flower 93]. The main difference is that our implementation operates in continuous time, with continuous differentiation and integration operators. The other approaches realize the integration and differentiation processes as discrete addition and subtraction operations, and use unit perturbations. [Cauwenberghs 93] provides a detailed derivation of the convergence and scaling properties of the discrete approach, and a simulation. [Alspector 93] provides a description of the use of the technique as part of a neural network hardware architecture, and provides a simulation. [Flower 93] derived a similar discrete algorithm from a node perturbation perspective in the context of multi-layered feedforward networks. Our work is similar in spirit to [Dembo 90] in that we don't make any explicit assumptions about the "model" that is embodied in the function fO. The function may be implemented as a neural network. In that case, the gradient descent is on-chip learning of the parameters of the network. We have fabricated a working chip containing the continuous-time multidimensional gradient descent circuits. This paper includes chip data for individual circuit components, as well as the entire circuit performing multi-dimensional gradient descent and annealing. 2 The Gradient Estimation Technique d/dt d/dt Figure 1: Gradient estimation technique from [Anderson, Kerns 92] Anderson and Kerns [Anderson, Kerns 92] describe techniques for one-dimensional gradient estimation in analog hardware. The gradient is estimated by correlating (using a multiplier) the output of a scalar function f( v(t)) with a noise source n(t), as shown in Fig. 1. The function input y(t) is additively "contaminated" by the noise n(t) to produce v(t) = y(t) + n(t). A scale factor B is used to set the scale of the noise to match the function output, which improves the signal-to-noise ratio. The signals are "high-pass" filtered to approximate differentiation (shown as d/ dt operators in Fig. 1) directly before the multiplication. The results of the multiplication are "low-pass" filtered to approximate integration. The gradient estimate is integrated over time, to smooth out some of the noise and to damp the response. This smoothed estimate is compared with a "zero" reference, using an amplifier A, and the result is fed back to the input, as shown in Fig. 2. Th~ contents of Fig. 1 are represented by the "Gradient Estimation" box in Fig. 2. We have chosen to implement the multi-dimensional technique in analog VLSI. We Analog VLSI Implementation of Multi-dimensional Gradient Descent 791 Gradient [ Estimation J dt "zero" Figure 2: Closing the loop: performing gradient descent using the gradient estimate. will not reproduce here the one-dimensional analysis from [Anderson, Kerns 92]' but summarize some ofthe more important results, and provide a multi-dimensional derivation. [Anderson 92] provides a more detailed theoretical discussion. 3 Multi-dimensional Derivation The multi-dimensional gradient descent operation that we are approximating can be written as follows: (1) where y and y' are vectors, and the solution is obtained continuously in time t, rather than at discrete ti. The circuit described in the block diagram in Fig. 1 computes an approximation to the gradient: (2) We approximate the operations of differentiation and integration in time by realizable high-pass and low-pass filters, respectively. To see that Eq. 2 is valid, and that this result is useful for approximating Eq. 1, we sketch an N-dimensional extension of [Anderson 92]. Using the chain rule, d ~ of dtf 0l.(t) + !let)) = L.-J (yj(t) + nj(t)) ~ . Y3 3 Assuming nj(t) ~ yj (t), the rhs is approximated to produce dd f0l.(t)+n(t)) = Lnj(t)oo~ t . Y3 3 (3) (4) Multiplying both sides by ni(t), and taking the expectation integral operator E[ ] of each side, E [nitt) ~ f 0t(t) + !!(t») 1 = E [n;(t) ~>;(t) :~ ] (5) If the noise sources ni(t) and nj (t) are unc.orrelated, nat) is independent of nj (t) when i =P j, and the sum on the right has a contribution only when i = j, E [ni(t) :t f 0l.(t) + net)) 1 = E [n~(t)n~(t) :~ 1 (6) 792 Kirk, Kerns, Fleischer, and Barr [ d 1 of E ni(t)- f ~(t) + n(t)) ~ an- = a\l f dt UYi (7) The expectation operator E[] can be used to smooth random variations of the noise nj(t). So, we have Since the descent rate k is arbitrary, we can absorb a into k. can approximate the gradient descent technique as follows: y~(t) ~ -k E [ni(t) ~ f (1t(t) + n(t)) 1 (8) U sing equation 8, we (9) 4 Elements of the Multi-dimensional Implementation We have designed, fabricated, and tested a chip which allows us to test these ideas. The chip implementation can be decomposed into six distinct parts: noise source(s): an analog VLSI circuit which produces a noise function. An independent, correlation-free noise source is needed for each input dimension, designated ni(t). The noise circuit is described in [Alspector 91]. target function: a scalar function f(Yl , Y2, ... , YN) of N input variables, bounded below, which is to be minimized [Kirk 91]. The circuit in this case is a 4dimensional variant of the bump circuit described in [Delbriick 91]. In the general case, this fO can be any scalar function or error metric, computed by some circuit. Specifically, the function may be a neural network. input signal(s): the inputs Yi(t) to the function fO. These will typically be onchip values, or real-world inputs. multiplier circuit(s): the multiplier computes the correlation between the noise values and the function output. Offsets in the multiplication appear as systematic errors in the gradient estimate, so it is important to compensate for the offsets. Linearity is not especially important, although monotonicity is critical. Ideally, the multiplication will also have a "tanh-like" character, limiting the output range for extreme inputs. integrator: an integration over time is approximated by a low-pass filter differentiator: the time derivatives of the noise signals and the function are approximated by a high-pass filter. The N inputs, Yi(t), are additively "contaminated" with the noise signals, ni(t), by capacitive coupling, producing Vi(t) = Yi(t) + ni(t), the inputs to the function fO. The function output is differentiated, as are the noise functions. Each differentiated noise signal is correlated with the differentiated function output, using the multipliers. The results are low-pass filtered, providing N partial derivative estimates, for the N input dimensions, shown for 4 dimensions in Fig. 3. The function fO is implemented as an 4-dimensional extension of Delbriick's [D~lbriick 91] bump circuit. Details ofthe N-dimensional bump circuit can be found in [Kirk 93]. For learning and other applications, the function fO can implement some other error metric to be minimized. Analog VLSI Implementation of Multi-dimensional Gradient Descent 793 , ............................................................................................ : : : : ~' ........................................................................................... : n1(t) ...... ------------n: : , ......................................................................................... ;. .. n2(t) • ! ! r .. ······················· .. ·························· .................................. .L.! ~ro : : n4(t) : d/dt ! y1(t) y2(t) y3(t) y4(t) v1(t) v2(t) ----4 I ~-+---=-'::....j f(.) J dt d/dt . . . . \0 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• , .......... ... Figure 3: Block diagram for a 4-dimensional gradient estimation circuit. 5 Chi p Results We have tested chips implementing the gradient estimation and gradient descent techniques described in this paper. Figure 4 shows the gradient estimate, without the closed loop descent process. Figure 5 shows the trajectories of two state variables during the 2D gradient descent process. Figure 6 shows the gradient descent process in operation on a 2D bump surface, and Fig. 7 shows how, using appropriate choice of noise scale, we can perform annealing using the gradient estimation hardware. , .. " 06 " DO DO i : " i : '1 1 1 0 0 ~ ~ g ! .. .. L-__ ~-----___ ~ ..... ~~ ·11 11 " DO ·06 ." ·18 08 11 '" 018 '" '" lOS 'I '" 011 007 I" '" TUD! (RCIKlda) nm.(",oodI} Figure 4: Measured Chip Data: 1D Gradient Estimate. Upper curves are 1D bump output as the input yet) is a slow triangle wave. Lower curves are gradient estimates. (left) raw data, and (right) average of 1024 runs. 794 Kirk, Kerns, Fleischer, and Barr 11 ·1' ·lJ ·1' 1. ·17 i ·11 ;> :> ., ·'1 .,. i .. ~ ~ :> ·1' \V.~ ., ·31 ~~i ·' 1 ·16 " lit III '" I.~ III 011 "' I~ 1.01 1ID 1"'-} 1ID1"'_} Figure 5: Measured Chip Data: 2D Gradient Descent. The curves above show the function optimization by gradient descent for 2 variables. Each curve represents the path of one of the state variables ¥..(t) from some initial values to the values for which the function 10 is minimized. (left) raw data, and (right) average of S runs. 6 Conclusions We have implemented an analog VLSI structure for performing continuous multidimensional gradient descent, and the gradient estimation uses only local information. The circuitry is compact and easily extensible to higher dimensions. This implementation leads to on-chip multi-dimensional optimization, such as is needed to perform on-chip learning for a hardware neural network. We can also perform a kind of annealing by adding a schedule to the scale of the noise input. Our approach also has some drawbacks, however. The gradient estimation is sensitive to the input offsets in the multipliers and integrators, since those offsets result in systematic errors. Also, the gradient estimation technique adds noise to the input signals. We hope that with only small additional circuit complexity, the performance of analog VLSI circuits can be greatly increased by permitting them to be intrinsically adaptive. On-chip implementation of an approximate gradient descent technique is an important step in this direction. Acknowledgements This work was supported in part oy an AT&T Bell Laboratories Ph.D. Fellowship, and by grants from Apple, DEC, Hewlett Packard, and IBM. Additional support was provided by NSF (ASC-S9-20219), as part of the NSF/DARPA STC for Computer Graphics and Scientific Visualization. All opinions, findings, conclusions, or recommendations expressed in this document are those of the author and do not necessarily reflect the views of the sponsoring agencies. References [Alspector 93] Alspector, J., R. Meir, B. Yuhas, and A. Jayakumar, "A Parallel Gradient Analog VLSI Implementation of Multi-dimensional Gradient Descent 795 Inltlol Stote Figure 6: Measured Chip Data: 2D Gradient Descent. Here we see the results for 2D gradient descent on a 2D bump surface. Both the bump surface and t.he descent path are actual data measured from our chips. Figure 7: Measured Chip Data: 2D Gradient Descent and Annealing. Here we see the effects of varying the amplitude of the noise. The dots represent points along the optimization pat.h. At left, with small magnitude noise, the process descends to a local minimum. At right, with larger magnitude, the descent process escapes to the global minimum. A schedule of gradually decreasing noise amplitude could reduce the probability of getting caught in undesirable local minima, and increase the probability of converging to a small region near a more desirable minimum, or even the global minimum. 796 Kirk, Kerns, Fleischer, and Barr Descent Method for Learning in Analog VLSI Neural Networks," in Advances in Neural Information Processing Systems, Vol. 5, Morgan Kaufman, San Mateo, CA, 1993. [Alspector 91] Alspector, J., J. W. Gannett, S. Haber, M. B. Parker, and R. Chu, "A VLSI-Efficient Technique for Generating Multiple Uncorrelated Noise Sources and Its Application to Stochastic Neural Networks," IEEE Transactions on Circuits and Systems, Vol.38, no.l, pp.l09-123, January, 1991. [Alspector 88] Alspector, J., B. Gupta, and R. B. Allen, "Performance of a stochastic learning microchip," in Advances in Neural Information Processing Systems, vol. I, Denver Colorado, Nov. 1988. D. S. Touretzky, ed., Morgan Kauffman Publishers, 1989, pp. 748-760. [Anderson, Kerns 92] Anderson, Brooke P., and Douglas Kerns, "Using Noise Injection and Correlation in Analog Hardware to Estimate Gradients," submitted to IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. [Anderson 92] Anderson, Brooke P., "Low-pass Filters as Expectation Operators for Multiplicative Noise," submitted to IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. [Cauwenberghs 93] Cauwenberghs, Gert, "A Fast Stochastic Error-Descent Algorithm for Supervised Learning and Optimization," in Advances in Neural Information Processing Systems, Vol. 5, Morgan Kaufman, San Mateo, CA, 1993. [Delbriick 91] Delbriick, Tobias, "'Bump' Circuits for Computing Similarity and Dissimilarity of Analog Voltages," Proceedings of International Joint Conference on Neural Networks, July 8-12, 1991, Seattle Washington, pp 1-475-479. (Extended version as Caltech Computation and Neural Systems Memo Number 10.) [Dembo 90] Dembo, A., and T. Kailath, "Model-Free Distributed Learning," IEEE Transactions on Neural Networks, Vol. 1, No.1, pp. 58-70, 1990. [Flower 93] Flower, B., and M. Jabri, "Summed Weight Neuron Perturbation: An O(n) Improvement over Weight Perturbation," in Advances in Neural Information Processing Systems, Vol. 5, Morgan Kaufman, San Mateo, CA, 1993. [Jabri 91] Jabri, M., S. Pickard, P. Leong, Z. Chi, and B. Flower, "Architectures and Implementations of Right Ventricular Apex Signal Classifiers for Pacemakers," IEEE Neural Information Processing Systems 1991 (NIPS 91), Morgan Kaufman, San Diego, 1991. [Kerns 92] Kerns, Douglas, "A Compact Noise Source for VLSI Applications," submitted to IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. [Kirk 91] Kirk, David, Kurt Fleischer, and Alan Barr, "Constrained Optimization Applied to the Parameter Setting Problem for Analog Circuits," IEEE Neural Information Processing Systems 1991 (NIPS 91), Morgan Kaufman, San Diego, 1991. [Kirk 93] Kirk, David, "Accurate and Precise Computation using Analog VLSI, with Applications to Computer Graphics and Neural Networks," Ph.D. Thesis, California Institute of Technology, Caltech-CS-TR-93-??, June, 1993. [Mead 89] Mead, Carver, "Analog VLSI and Neural Systems," Addison-Wesley, 1989. [Platt 89] Platt, John, "Constrained Optimization for Neural Networks and Computer Graphics," Ph.D. Thesis, California Institute of Technology, Caltech-CS-TR-89-07, June, 1989. [Umminger 89] Umminger, Christopher B., and Steven P. DeWeerth, "Implementing Gradient Following in Analog VLSI," Advanced Research in VLSI, MIT Press, Boston, 1989, pp. 195-208.
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Improving Performance in Neural Networks Using a Boosting Algorithm Harris Drucker AT&T Bell Laboratories Holmdel, NJ 07733 Robert Schapire AT&T Bell Laboratories Murray Hill, NJ 07974 Abstract Patrice Simard AT &T Bell Laboratories Holmdel, NJ 07733 A boosting algorithm converts a learning machine with error rate less than 50% to one with an arbitrarily low error rate. However, the algorithm discussed here depends on having a large supply of independent training samples. We show how to circumvent this problem and generate an ensemble of learning machines whose performance in optical character recognition problems is dramatically improved over that of a single network. We report the effect of boosting on four databases (all handwritten) consisting of 12,000 digits from segmented ZIP codes from the United State Postal Service (USPS) and the following from the National Institute of Standards and Testing (NIST): 220,000 digits, 45,000 upper case alphas, and 45,000 lower case alphas. We use two performance measures: the raw error rate (no rejects) and the reject rate required to achieve a 1% error rate on the patterns not rejected. Boosting improved performance in some cases by a factor of three. 1 INTRODUCTION In this article we summarize a study on the effects of a boosting algorithm on the performance of an ensemble of neural networks used in optical character recognition problems. Full details can be obtained elsewhere (Drucker, Schapire, and Simard, 1993). The "boosting by filtering" algorithm is based on Schapire's original work (1990) which showed that it is theoretically possible to convert a learning machine with error rate less than 50% into an ensemble of learning machines whose error rate is arbitrarily low. The work detailed here is the first practical implementation of this boosting algorithm. As applied to an ensemble of neural networks using supervised learning, the algorithm proceeds as follows: Assume an oracle that generates a large number of independent 42 Improving Performance in Neural Networks Using a Boosting Algorithm 43 training examples. First, generate a set of training examples and train a first network. After the first network is trained it may be used in combination with the oracle to produce a second training set in the following manner: Flip a fair coin. If the coin is heads, pass outputs from the oracle through the first learning machine until the first network misclassifies a pattern and add this pattern to a second training set. Otherwise, if the coin is tails pass outputs from the oracle through the first learning machine until the first network finds a pattern that it classifies correctly and add to the training set. This process is repeated until enough patterns have been collected. These patterns, half of which the first machine classifies correctly and half incorrectly, constitute the training set for the second network. The second network may then be trained. The first two networks may then be used to produce a third training set in the following manner: Pass the outputs from the oracle through the first two networks. If the networks disagree on the classification, add this pattern to the training set. Otherwise, toss out the pattern. Continue this until enough patterns are generated to form the third training set. This third network is then trained. In the final testing phase (of Schapire's original scheme), the test patterns (never previously used for training or validation) are passed through the three networks and labels assigned using the following voting scheme: If the first two networks agree, that is the label. Otherwise, assign the label as classified by the third network. However, we have found that if we add together the three sets of outputs from each of the three networks to obtain one set of ten outputs (for the digits) or one set of twenty-size outputs (for the alphas) we obtain better results. Typically, the error rate is reduced by .5% over straight voting. The rationale for the better performance using addition is as follows: A voting criterion is a hard-decision rule. Each voter in the ensemble has an equal vote whether in fact the voter has high confidence (large difference between the two largest outputs in a particular network) or low confidence (small difference between the two largest outputs). By summing the outputs (a soft-decision rule) we incorporate the confidence of the networks into the total output. As will be seen later, this also allows us to build an ensemble with only two voters rather than three as called for in the original algorithm. Conceptually, this process could be iterated in a recursive manner to produce an ensemble of nine networks, twenty-seven networks, etc. However, we have found significant improvement in going from one network to only three. The penalty paid is potentially an increase by a factor of three in evaluating the performance (we attribute no penalty to the increased training time). However it can show how to reduce this to a factor of 1.75 using sieving procedures. 2 A DEFORMATION MODEL The proof that boosting works depends on the assumption of three independent training sets. Without a very large training set, this is not possible unless that error rates are large. After training the first network, unless the network has very poor performance, there are not enough remaining samples to generate the second training set. For example, suppose we had 9000 total examples and used the first 3000 to train the first network and that network achieves a 5% error rate. We would like the next training set to consist of 1500 patterns that the first network classifies incorrectly and 1500 that the first network 44 Drucker, Schapire, and Simard classifies incorrectly. At a 5% error rate, we need approximately 30,000 new images to pass through the first network to find 1500 patterns that the first network classifies incorrectly. These many patterns are not available. Instead we will generate additional patterns by using small deformations around the finite training set based on the techniques of Simard (Simard, et al., 1992). The image consists of a square pixel array (we use both 16x16 and 20x20). Let the intensity of the image at coordinate location (ij) be Fjj(x,y) where the (x,y) denotes that F is a differentiable and hence continuous function of x and y. i and j take on the discrete values 0,1, ... ,15 for a 16x16 pixel array. The change in F at location (ij) due to small x-translation, y-translation, rotation, diagonal deformation, axis deformation, scaling and thickness deformation is given by the following respective matrix inner products: aFjj(x,y) ax aFjj(x,y) ay where the k's are small values and x and yare referenced to the center of the image. This construction depends on obtaining the two partial derivatives. ap.·(x y) For example, if all the k' s except k 1 are zero, then M'jj(X,y) = k 1 '~X ' is the amount by which Fij(x,y) at coordinate location (ij) changes due to an x-translation of value k 1. The diagonal deformation can be conceived of as pulling on two opposite comers of the image thereby stretching the image along the 45 degree axis (away from the center) while simultaneously shrinking the image towards the center along a - 45 degree axis. If k4 changes sign, we push towards the center along the 45 degree axis and pull away along the - 45 degree axis. Axis deformation can be conceived as pulling (or pushing) away from the center along the x-axis while pushing (or pulling) towards the center along the y-axis. If all the k's except k7 are zero, then M'jj(x,y) = k711 VFjj(x,y) I j2 is the norm squared of the gradient of the intensity. It can be shown that this corresponds to varying the "thickness" of the image. Typically the original image is very coarsely quantized and not differentiable. Smoothing of the original image is done by numerically convolving the original image with a 5x5 _ (x2 + y2) square kernel whose elements are values from the Gaussian: exp cr to give us Improving Performance in Neural Networks Using a Boosting Algorithm 45 . a 16x 16 or: 20x20 square matrix of smoothed values. A matrix of partial derivatives (with respect to x) for each pixel --Iocation is obtained by convolving the original image with a kernel whose elements are the derivatives with respect to x of the Gaussian function. ' We can similarly form a matrix of parti~ derivatives with respect to y. A new image can then be constructed by adding together the smoothed image and a differential matrix whose elements are given by the above equation. Using the above equation, we may simulate an oracle by cycling through a finite sized training set, picking random values (uniformly distributed in some small range) of the constants k for each new image. The choice of the range of k is somewhat critical: too small and the new image is too close to the old image for the neural network to consider it a "new" pattern. Too large and the image is distorted and nonrepresentative of "real" data. We will discuss the proper choice of k later. 3 NETWORK ARCHITECTURES We use as the basic learning machine a neural network with extensive use of shared weights (LeCun, et. al., 1989, 1990). Typically the number of weights is much less than the number of connections. We believe this leads to a better ability to reject images (i.e., no decision made) and thereby minimizes the number of rejects needed to obtain a given error rate on images not rejected. However, there is conflicting evidence (Martin & Pitman, 1991) that given enough training patterns, fully connected networks give similar performance to networks using weight sharing. For the digits there is a 16 by 16 input surrounded by a six pixel border to give a 28 by 28 input layer. The network has 4645 neurons, 2578 different weights, and 98442 connections. The networks used for the alpha characters use a 20 by 20 input surrounded by a six pixel border to give a 32 by 32 input layer. There are larger feature maps and more layers, but essentially the same construction as for the digits. 4 TRAINING ALGORITHM The training algorithm is described in general terms: Ideally, the data set should be broken up into a training set, a validation set and a test set. The training set and validation set are smoothed (no deformations) and the first network trained using a quasi-Newton procedure. We alternately train on the training data and test on the validation data until the error rate on the validation data reaches a minimum. Typically, there is some overtraining in that the error rate on the training data continues to decrease after the error rate on the validation set reaches a minimum. Once the first network is trained, the second set of training data is generated by cycling deformed training data through the first network. After the pseudo-random tossing of a fair coin, if the coin is heads, deformed images are passed though the first network until the network makes a mistake. If tails, deformed images are passed through the network until the network makes a correct labeling. Each deformed image is generated from the original image by randomly selecting values of the constants k. It may require multiple passes through the training data to generate enough deformed images to form the second training set 46 Drucker, Schapire, and Simard Recall that the second training set will consist equally of images that the first network misclassifies and images that the the first network classifies correctly. The total size of the training set is that of the first training set. Correctly classified images are not hard to find if the error rate of the first network is low. However, we only accept these images with probability 50%. The choice of the range of the random variables k should be such that the deformed images do not look distorted. The choice of the range of the k' s is good if the error rate using the first network on the deformed patterns is approximately the same as the error rate of the first network on the validation set (NOT the first training set). A second network is now trained on this new training set in the alternate train/test procedure using the original validation set (not deformed) as the test set. Since this training data is much more difficult to learn than the first training data, typically the error rate on the second training set using the second trained network will be higher (sometimes much higher) than the error rates of the first network on either the first training set or the validation set. Also, the error rate on the validation set using the second network will be higher than that of the first network because the network is trying to generalize from difficult training data, 50% of which the first network could not recognize. The third training set is formed by once again generating deformed images and presenting the images to both the first and second networks. If the networks disagree (whether both are wrong or just one is), then that image is added to the third training set. The network is trained using this new training data and tested on the original validation set. Typically, the error rate on the validation set using the third network will be much higher than either of the first two networks on the same validation set. The three networks are then tested on the third set of data, which is the smoothed test data. According to the original algorithm we should observe the outputs of the first two networks. If the networks agree, accept that labeling, otherwise use the labeling assigned by the third network. However, we are interested in more than a low error rate. We have a second criterion, namely the percent of the patterns we have to reject (i.e. no classification decision) in order to achieve a 1 % error rate. The rationale for this is that if an image recognizer is used to sort ZIP codes (or financial statements) it is much less expensive to hand sort some numbers than to accept all and send mail to the wrong address or credit the wrong account. From now on we shall call this latter criterion the reject rate (without appending each time the statement "for a 1 % error rate on the patterns not rejected"). For a single neural network, a reject criterion is to compare the two (of the ten or twentysix) largest outputs of the network. If the difference is great, there is high confidence that the maximum output is the correct classification. Therefore, a critical threshold is set such that if the difference is smaller then that threshold, the image is rejected. The threshold is set so that the error rate on the patterns not rejected is 1 %. 5 RESULTS The boosting algorithm was first used on a database consisting of segmented ZIP codes from the United States Postal Service (USPS) divided into 9709 training examples and 2007 validation samples. Improving Performance in Neural Networks Using a Boosting Algorithm 47 The samples supplied to us from the USPS were machine segmented from zip codes and labeled but not size normalized. The validation set consists of approximately 2% badly segmented characters (incomplete segmentations. decapitated fives, etc.) The training set was cleaned thus the validation set is significantly more difficult than the training set. The data was size normalized to fit inside a 16x16 array. centered, and deslanted. There is no third group of data called the "test set" in the sense described previously even though the validation error rate has been commonly called the test error rate in prior work (LeCun. et. al., 1989, 1990). Within the 9709 training digits are some machine printed digits which have been found to improve performance on the validation set. This data set has an interesting history having been around for three years with an approximate 5% error rate and 10% reject rate using our best neural network. There has been a slight improvement using double backpropagation (Drucker & LeCun. 1991) bringing down the error rate to 4.7% and the reject rate to 8.9% but nothing dramatic. This network. which has a 4.7% error rate was retrained on smoothed data by starting from the best set of weights. The second and third networks were trained as described previously with the following key numbers: The retrained first network has a training error rate of less than 1%, a test error rate of 4.9% and a test reject rate of 11.5% We had to pass 153,000 deformed images (recycling the 9709 training set) through the trained first network to obtain another 9709 training images. Of these 9709 images. approximately one-half are patterns that the first network misclassifies. This means that the first network has a 3.2% error rate on the deformed images, far above the error rate on the original training images. A second network is trained and gives a 5.8% test error rate. To generate the last training set we passed 195,000 patterns (again recycling the 9709) to give another set of 9709 training patterns. Therefore, the first two nets disagreed on 5% of the deformed patterns. The third network is trained and gives a test error rate of 16.9% Using the original voting scheme for these three networks, we obtained a 4.0% error rate. a significant improvement over the 4.9% using one network. As suggested before. adding together the three outputs gives a method of rejecting images with low confidence scores (when the two highest outputs are too close). For curiosity, we also determined what would happen if we just added together the first two networks: Original network: 4.9% test error rate and 11.5% reject rate. Two networks added: 3.9% test error rate and 7.9% reject rate. Three networks added: 3.6% test error rate and 6.6% reject rate. The ensemble of three networks gives a significant improvement, especially in the reject rate. In April of 1992, the National Institute of Standards and Technology (NIST) provided a labeled database of 220.000 digits. 45.000 lower case alphas and 45.000 upper case 48 Drucker, Schapire, and Simard alphas. We divided these into training set, validation set, and test set. All data were resampled and size-normalized to fit into a 16x16 or 20x20 pixel array. For the digits, we deslanted and smoothed the data before retraining the first 16x16 input neural network used for the USPS data. After the second training set was generated and the second network trained the results from adding the two networks together were so good (Table 1) that we decided not to generate the third training set For the NIST data, the error rates reported are those of the test data. TABLE 1. Test error rate and reject rate in percent DATABASE USPS NIST NIST NIST digits digits upper lower alphas alpha ERROR RATE 5.0 1.4 4.0 9.8 SINGLE NET ERROR RATE 3.6 .8 2.4 8.1 USING BOOSTING REJECT RATE 9.6 1.0 9.2 29. SINGLE NET REJECT RATE 6.6 * 3.1 21. USING BOOSTING * Reject rate is not reported if the error rate is below 1 %. 6 CONCLUSIONS In all cases we have been able to boost performance above that of single net. Although others have used ensembles to improve performance (Srihari, 1990; Benediktsson and Swain, 1992; Xu, et. al., 1992) the technique used here is particularly straightforward since the usual multi-classifier system requires a laborious development of each classifier. There is also a difference in emphasis. In the usual multi-classifier design, each classifier is trained independently and the problem is how to best combine the classifiers. In boosting, each network (after the first) has parameters that depend on the prior networks and we know how to combine the networks (by voting or adding). 7 ACKNOWLEDGEMENTS We hereby acknowledge the United State Postal Service and the National Institute of Standards and Technology in supplying the databases. Improving Performance in Neural Networks Using a Boosting Algorithm 49 References J.A. Benediktsson and P.H. Swain, "Consensus Theoretic Classification Methods", IEEE trans. on Systems, Man, and Cybernetics, Vol. 22, No.4, July/August 1992, pp. 688-704. H. Drucker. R. Schapire, and P. Simard "Boosting Perfonnance in Neural Networks", International Journal of Pattern Recognition and Artificial Intelligence, (to be published, 1993)d H. Drucker and Y. LeCun, "Improving Generalization Perfonnance in Character Recognition", Proceedings of the 1991 IEEE Workshop on Neural Networks for Signal Processing, IEEE Press,pp. 198 - 207. Y. LeCun, et. aI., "Backpropagation Applied to Handwritten Zip Code Recognition", Neural Computation 1,1989, pp. 541-551 Y. LeCun, et. aI., Handwritten Digit Recognition with a Back-Propagation Network", In D.S. Touretsky (ed), Advances in Neural Information Processing Systems 2, (1990) pp. 396-404, San Mateo, CA: Morgan Kaufmann Publishers G. L. Martin and J. A. Pitman, "Recognizing Handed-Printed Letters and Digits Using Backpropagation Learning", Neural Computation, Vol. 3, 1991, pp. 258-267. R. Schapire, "The Strength of Weak Learnability", Machine Learning, Vol. 5, #2, 1990, pp. 197-227. P. Simard. "Tangent Prop - A fonnalism for specifying selected invariances in an adaptive network", In J.E. Moody, SJ. Hanson, and R.P. Lippmann (eds.) Advances in Neural Information Processing Systems 4, (1992) p. 895-903, San Mateo, CA: Morgan Kaufmann Publishers Sargur Srihari, "High-Perfonnance Reading Machines", Proceeding of the IEEE, Vol 80, No.7, July 1992, pp. 1120-1132. C.Y. Suen, et. aI., "Computer Recognition of Unconstrained Handwritten Numerals", Proceeding of the IEEE, Vol 80, No.7, July 1992, pp. 1162-1180. L. Xu, et. al.. "Methods of Combining Multiple Classifiers", IEEE Trans. on Systems Man, and Cybernetics, Vol. 22. No.3, May/June 1992, pp. 418-435.
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Synaptic Weight Noise During MLP Learning Enhances Fault-Tolerance, Generalisation and Learning Trajectory Alan F. Murray Dept. of Electrical Engineering Edinburgh University Scotland Peter J. Edwards Dept. of Electrical Engjneering Edinburgh University Scotland Abstract We analyse the effects of analog noise on the synaptic arithmetic during MultiLayer Perceptron training, by expanding the cost function to include noise-mediated penalty terms. Predictions are made in the light of these calculations which suggest that fault tolerance, generalisation ability and learning trajectory should be improved by such noise-injection. Extensive simulation experiments on two distinct classification problems substantiate the claims. The results appear to be perfectly general for all training schemes where weights are adjusted incrementally, and have wide-ranging implications for all applications, particularly those involving "inaccurate" analog neural VLSI. 1 Introduction This paper demonstrates both by consjderatioll of the cost function and the learning equations, and by simulation experiments, that injection of random noise on to MLP weights during learning enhances fault-tolerance without additional supervision. We also show that the nature of the hidden node states and the learning trajectory is altered fundamentally, in a manner that improves training times and learning quality. The enhancement uses the mediating influence of noise to distribute information optimally across the existing weights. 491 492 Murray and Edwards Taylor [Taylor , 72] has studied noisy synapses, largely in a biological context, and infers that the noise might assist learning. vVe have already demonstrated that noise injection both reduces the learning time and improves the network's generalisation ability [Murray, 91],[Murray, 92]. It is established[Matsuoka, 92],[Bishop, 90] that adding noise to the training data in neural (MLP) learning improves the "quality" of learning, as measured by the trained network's ability to generalise. Here we infer (synaptic) noise-mediated terms that sculpt the error function to favour faster learning, and that generate more robust internal representations, giving rise to better generalisation and immunity to smaIl variations in the characteristics of the test data. Much closer to the spirit of this paper is the work of Hanson[Hanson, 90]. His stochastic version of the delta rule effectively adapts weight means and standard deviations. Also Sequin and Clay [Sequin , 91] use stuck-at faults during training which imbues the trained network with an ability to withstand such faults. They also note, but do not pursue, an increased generalisation ability. This paper presents an outline of the mathematical predictions and verification simulations. A full description of the work is given in [Murray, 93]. 2 Mathematics Let us analyse an MLP with I input, J hidden and ]{ output nodes, with a set of P training input vectors Qp = {Oip}, looking at the effect of noise injection into the error function itself. We are thus able to infer, from the additional terms introduced by noise, the characteristics of solutions that tend to reduce the error, and those which tend to increase it. The former will clearly be favoured, or at least stabilised, by the additional terms. while the latter will be de-stabilised. Let each weight Tab be augmented by a random noise source, such that Tab -+Tab + ~abTab, for all weights {Tab}. Neuron thresholds are treated in precisely the same way. Note in passing, but importantly, that this synaptic noise is not the same as noise on the input data. Input noise is correlated across the synapses leaving an input node, while the synaptic noise that forms the basis of this study is not. The effect is thus quite distinct. Considering, therefore, an error function of the form ;1 K-l 1 K-l ftot,p ="2 L fk/ ="2 L(okp({Tab}) -Okp)2 (1) k=O k=O Where Okp is the target output. We can now perform a Taylor expansion of the output Okp to second order, around the noise-free weight set, {TN}, and thus augment the error function ;""' (aOkP ) 1 ""' ( a20kp ) Okp -+ Okp + L..J Tab~ab aT. +"2 L..J Tab~abTcd~cd aT. aT. +0(> 3) (2) b ab b d ab cd a a ,c If we ignore terms of order ~ 3 and above, and taking the time average over the learning phase, we can infer that two terms are added to the error function ;< ftot >=< (tot( {TN}) > + 2~ t"I:l ~2 LTab 2 [(~;kP) 2 + (kp (:~k~)l (3) p=l k=O ab ab ab Synaptic Weight Noise During MLP Learning 493 Consider also the perceptron rule update on the hidden-output layer along with the expanded error function :"'" I ~ 2 "'" I "'" 2 {)2 ° k P < 6Tkj >= -T L..J < fkpOjpOkp > -T2 L..J < OjpOkp > X L..J Tab --2 p P ab aTab (4) averaged over several training epochs (which is acceptable for small values of T the adaption rate parameter). 3 Simulations The simulations detailed below are based on the virtual targets algorithm [Murray, 92], a variant on backpropagation, with broadly similar performance. The "targets" algorithm was chosen for its faster convergence properties. Two contrasting classification tasks were selected to verify the predictions made in the following section by simulation. The first, a feature location task, uses real world normalised greyscale image data. The task was to locate eyes in facial images - to classify sections of these as either "eye" or "not-eye". The network was trained on 16 x 16 preclassified sections of the images, classified as eyes and not-eyes. The not-eyes were random sections of facial images, avoiding the eyes (see Fig. 1). The second, 16x 16 section .:J "eye" ------=>~ c="not-eye" Figure 1: The eye/not-eye classifier. a more artificial task, was the ubiquitous character encoder (Fig. 2) where a 251B-~111111111111111111 Figure 2: The character encoder task. :> 26 I I I I dimensional binary input vector describing the 26 alphabetic characters (each 5 x 5 pixels) was used to train the network with a one-out-of-26 output code. During the simulations noise was added to the weights at a level proportional to the weight size and at a probability distribution of uniform density (i.e. -~max < ~ < ~max). Levels of up to 40% were probed in detail - although it is clear that the expansion above is not quantitatively valid at this level. Above these percentages further improvements were seen in the network performance, although the dynamics of the training algorithm became chaotic. The injected noise level was reduced 494 Murray and Edwards smoothly to a minimum value of 1% as the network approached convergence (as evidenced by the highest output bit error). As ill all neural network simulations, the results depended upon the training parameters, network sizes and the random start position of the network. To overcome these factors and to achieve a meaningful result 35 weight sets were produced for each noise level. All other characteristics of the training process were held constant. The results are therefore not simply pathological freaks. 4 Prediction/Verification 4.1 Fault Tolerance Consider the first derivative penalty term in the expanded cost function (3), averaged over all patterns, output nodes and weights :[{ X A' [Ta" ( ~~: ) '] (5) The implications of this term are straightforward. For large values of the (weighted) average magnitude of the derivative, the overall error is increased. This term therefore causes solutions to be favoured where the dependence of outputs on individual weights is evenly distributed across the entire weight set. Furthermore, weight saliency should not only have a lower average value, but a smaller scatter across the weight set as the training process attempts to reconcile the competing pressures to reduce both (1) and (5) . This more distributed representation should be manifest in an improved tolerance to faulty weights. ~ U 80 OJ ... ... 0 U 60 -0 OJ :-S II! ~ 40 0 II! 0:: ~ 20 '" Q. 0 0 5 '. '. ~ . 10 15 nolSe=O% noise::;: 1 fJro noise = 20% ----noise =30% noise=40% --20 Synapses Removed ('¥o) 25 Figure 3: Fault tolerance in the character encoder problem. Simulations were carried out on 35 weight sets produced for each ofthe two problems at each of 5 levels of noise injected during training. Weights were then randomly removed and the networks tested on the training data. The resulting graphs (Fig. 3, 4) show graceful degradation with an increased tolerance to faults with injected noise during training. The networks were highly constrained for these simulations to remove some of the natural redundancy of the MLP structure. Although the eye/not-eye problem contains a high proportion of redundant information, the Synaptic Weight Noise During MLP Learning 495 -- -I nDise~O% noise = 10% noise = ~ -._. __ noise~ 30% noise = 40% - - 1 ~L-____ ~ ____ -L ____ ~ ______ ~ ____ ~ o 5 15 20 25 Synapses Removed (%) Figure 4: Fault tolerance enhancement in the eye/not-eye classifier. improvement in the networks ability to withstand damage, with injected noise, is clear. 4.2 Generalisation Ability Considering the derivative in equation 5, and looking at the input-hidden weights. The term that is added to the error function, again averaged over all patterns, output nodes and weights is :(6) If an output neuron has a non-zero connection from a particular hidden node (Tkj "I 0), and provided the input Oip is non-zero and is connected to the hidden node (Tji "I 0), there is also a term oJp that will tend to favour solutions with the hidden nodes also turned firmly ON or OFF (i.e. Ojp = 0 or 1). Remembering, of course, that all these terms are noise-mediated, and that during the early stages of training, the "actual" error fkp, in (1), will dominate, this term will de-stabilise final solutions that balance the hidden nodes on the slope of the sigmoid. Naturally, hidden nodes OJ that are firmly ON or OFF are less likely to change state as a result of small variations in the input data {Oi}. This should become evident in an increased tolerance to input perturbations and therefore an increased generalisation ability. Simulations were again carried out on the two problems using 35 weight sets for each level of injected synaptic noise during training. For the character encoder problem generalisation is not really an issue, but it is possible to verify the above prediction by introducing random gaussian noise into the input data and noting the degradation in performance. The results of these simulations are shown in Fig. 5, and clearly show an increased ability to withstand input perturbation, with injected noise into the synapses during training. Generalisation ability for the eye/not-eye problem is a real issue. This problem therefore gives a valid test of whether the synaptic noise technique actually improves generalisation performance. The networks were therefore tested on previously unseen facial images and the results are shown in Table 1. These results show 496 Murray and Edwards 100 t : ~" ~ I:: 0 u 11 70 -= 'iii 60 " '" ./. 0 50 '" E .!:! 40 .,. llI 30 0 -.~ . , 10 .. -.20 -_.------------.. perturbation = 0,(5 perturbation = 0.10 . perturbation = 0.15 ..... perturbation = 0.20 .. . 30 40 Noise Level (%) Figure 5: Generalisation enhancement shown through increased tolerance to input perturbation, in the character encoder problem. Correctly Classified (%) Noise Levels 0% I 10% I 20% I 30% J 40% Test Patterns 67.875 I 70.406 I 70.416 I 72.454 I 75.446 Table 1: Generalisation enhancement shown through increased ability to classifier previously unseen data, in the eye/not-eye task. dramatically improved generalisation ability with increased levels of injected synaptic noise during training. An improvement of approximately 8% is seen - consistent with earlier results on a different "real" problem [Murray, 91]. 4.3 Learning Trajectory Considering now the second derivative penalty term in the expanded cost function (2). This term is complex as it involves second order derivatives, and also depends upon the sign a.nd magnitude of the errors themselves {flep}. The simplest way of looking at its effect is to look at a single exemplar term :K t:,. 2 f T. 2 ({)2 Olep ) lep ab {)Tab 2 (7) This term implies that when the combination of flep ~~::~ is negative then the overall cost function error is reduced and vice versa. The term (7) is therefore constructive as it can actually lower the error locally via noise injection, whereas (6) always increases it. (7) can therefore be viewed as a sculpting of the error surface during the early phases of training (i.e. when flep is sUbstantial). In particular, a weight set with a higher "raw" error value, calculated from (1), may be favoured over one with a lower value if noise-injected terms indicate that the "poorer" solution is located in a promising area of weight space. This "look-ahead" property should lead to an enhanced learning trajectory, perhaps finding a solution more rapidly. In the augmented weight update equation (4), the noise is acting as a medium projecting statistical information about the character of the entire weight set on to Synaptic Weight Noise During MLP Learning 497 the update equation for each particular weight. So, the effect of the noise term is to account not only for the weight currently being updated, but to add in a term that estimates what the other weight changes are likely to do to the output, and adjust the size of the weight increment/decrement as appropriate. To verify this by simulation is not as straightforward as the other predictions. It is however possible to show the mean training time for each level of injected noise. For each noise level, 1000 random start points were used to allow the underlying properties of the training process to emerge. The results are shown in Fig. 6 and 600 ~ u S50 0 Q., ~ .. SOO 6 l= 00 450 .5 c: .. '" 400 .. ...J c: i! 350 ::E 300 0 10 20 30 40 50 Synaptic Noise Level Used During Training Figure 6: Training time as a function of injected synaptic noise during training. clearly show that at low noise levels (::; 30% for the case of the character encoder) a definite reduction in training times are seen. At higher levels the chaotic nature of the "noisy learning" takes over. It is also possible to plot the combination of fkp ~~::~. This is shown in Fig. 7, again for the character encoder problem. The term (7) is reduced more quickly 1.0 ~ O.S ~ .. 0.0 . ~ ~ noise =7% i;j :> -0.5 ·c .. Q -1.0 'tl .G -I.S " .. ~ -2.0 .. -2.5 g I.<i -3.0 0 SOO 1000 ISOO 2000 Learning Epochs Figure 7: The second derivative x error term trajectory for injected synaptic noise levels 0% and 7%. with injected noise, thus effecting better weight changes via (4). At levels of noise > 7% the effect is exaggerated, and the noise mediated improvements take place 498 Murray and Edwards during the first 100-200 epochs of training. The level of 7% is displayed simply because it is visually clear what is happening, and is also typical. 5 Conclusion We have shown both by mathematical expansion and by simulation that injecting random noise on to the synaptic weights of a MultiLayer Perceptron during the training phase enhances fault-tolerance, generalisation ability and learning trajectory. It has long been held that any inaccuracy during training is detrimental to MLP learning. This paper proves that analog inaccuracy is not. The mathematical predictions are perfectly general and the simulations relate to a non-trivial classification task and a "real" world problem. The results are therefore important for the designers of analog hardware and also as a non-invasive technique for producing learning enhancements in the software domain. Acknowledgements We are grateful to the Science and Engineering Research Council for financial support, and to Lionel Tarassenko and Chris Bishop for encouragement and advice. References [Taylor, 72] [Murray, 91] [Murray, 92] J. G. Taylor, "Spontaneous Behaviour in Neural Networks" , J. Theor. Bioi., vol. 36, pp. 513-528, 1972. A. F. Murray, "Analog Noise-Enhanced Learning in Neural Network Circuits," Electronics Letters, vol. 2, no. 17, pp. 1546-1548, 1991. A. F. Murray, "Multi-Layer Perceptron Learning Optimised for OnChip Implementation - a Noise Robust System," Neural Computation, vol. 4, no. 3, pp. 366-381, 1992. [Matsuoka, 92] K. Matsuoka, "Noise Injection into Inputs in Back-Propagation Learning", IEEE Trans. Systems, Man and Cybernetics, vol. 22, no. 3, pp. 436-440, 1992. [Bishop, 90] [Hanson, 90] [Sequin, 91] [Murray, 93] C. Bishop, "Curvature-Driven Smoothing in Backpropagation Neural Networks," IJCNN, vol. 2, pp. 749-752, 1990. S. J. Hanson, "A Stochastic Version of the Delta Rule", Physica D, vol. 42, pp. 265-272, 1990. C. H. Sequin, R. D. Clay, "Fault Tolerance in Feed-Forward Artificial Neural Networks" , Neural Networks: Concepts, Applications and Implementations, vol. 4, pp. 111-141, 1991. A. F. Murray, P. J. Edwards, "Enhanced MLP Performance and Fault Tolerance Resulting from Synaptic Weight Noise During Training", IEEE Trans. Neural Networks, 1993, In Press.
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An Object-Oriented Framework for the Simulation of Neural Nets A. Linden Th. Sudbrak Ch. Tietz F. Weber German National Research Center for Computer Science D-5205 Sankt Augustin 1, Germany Abstract The field of software simulators for neural networks has been expanding very rapidly in the last years but their importance is still being underestimated. They must provide increasing levels of assistance for the design, simulation and analysis of neural networks. With our object-oriented framework (SESAME) we intend to show that very high degrees of transparency, manageability and flexibility for complex experiments can be obtained. SESAME's basic design philosophy is inspired by the natural way in which researchers explain their computational models. Experiments are performed with networks of building blocks, which can be extended very easily. Mechanisms have been integrated to facilitate the construction and analysis of very complex architectures. Among these mechanisms are t.he automatic configuration of building blocks for an experiment and multiple inheritance at run-time. 1 Introduction In recent years a lot of work has been put into the development of simulation systems for neural networks [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12]. Unfortunately their importance has been largely underestimated. In future, software environments will provide increasing It-vels of assistance for the design, simulation and analysis of neural networks as well as for other pattern and signal processing architectures. Yet large improvements are still necessary in order to fulfill the growing demands of the research community. Despite the existence of at least 100 software simulators, only very few of them can deal with, e. g. multiple learning paradigms and applications, 797 798 Linden, Sudbrak, Tietz, and Weber very large experiments. In this paper we describe an object oriented framework for the simulation of neural networks and try to illustrate its flexibility, transparency and extendability. The prototype called SESAME has been implemented using C++ (on UNIX workstations running X-Windows) and currently consists of about 39.000 lines of code, implementing over 80 classes for neural network algorithms, pattern handling, graphical output and other utilities. 2 Philosophy of Design The main objective of SESAME is to allow for arbitrary combinations of different learning and pattern processing paradigms (e. g. supervised, unsupervised, selfsupervised or reinforcement learning) and different application domains (e. g. pattern recognition, vision, speech or control). To some degree the design of SESAME has been based on the observation that many researchers explain their neural information processing systems (NIPS) with block-diagrams. Such a block diagram consists of a group of primitive elements (building blocks). Each building block has inputs and outputs and a functional relationship between them. Connections describe the flow of data between the building blocks. Scripts related to the building blocks specify the flow of control. Complex NIPS are constructed from a library of building blocks (possibly themselves whole NIPS), which are interconnected via uniform communication links. 3 SESAME Design and Features All building blocks share a list of common components. They all have insites and outsites that build the endpoints of communication links. Datafields contain the data (e. g. weight matrices or activation vectors) which is sent over the links. Action functions process input from the insites, update the internal state and compute appropriate outputs, e. g. performing weight updates and propagating activation or error vectors. Command functions provide a uniform user interface for all building blocks. Scripts control the execution of action or command functions or other script.s. They may contain conditional statements and loops as control structures. Furthermore a symbol table allows run-time access to parameters of the building hlock as learning rat.E's, sizes, data ranges etc. Many other internal data structures and routines are provided for the administration and maintainance of building blocks. The description of an experiment may be divided into the functional description of the building blocks (which can be done either in C++ or in the high-level description language, see below), the connection topology of the building blocks used, the cont.rol flow defined by scripts and a set of parameters for each of the building blocks. An Object-Oriented Framework for the Simulation of Neural Nets 799 Design Highlights 3.1 User Interface The user interface is text oriented and may be used interactively as well as script driven. This implies that any command that the user may choose interactively can also be used in a command file that is called non-interactively. This allows the easy adaption of arbitrary user interface structures from a primitive batch interface for large offline simulatiors to a fancy graphical user interface for online experiments. Another consequence is that experiments are specified in the same command language that is used for the user interface. The user may thus easily switch from description files from previously saved experiments to the interactive manipulation of already loaded ones. Since the complete structure of an experiment is accessible at runtime, this not only means manipulation of parameters but also includes any imaginable modification of the experiment topology. The experienced user can, for example, include new building blocks for experiment observation or statistical evaluation and connect them to any point of the communication structure. Deletion of building blocks is possible, as well as modifying control scripts. The complete state of the experiment (i. e. the current values of all relevant data) can be saved for later experiments. 3.2 Hierarchies In SESAME we distinguish two kinds of building blocks: terminal and non-terminal blocks. Non-terminal building blocks are used to structure a complex experiment into hierarchies of abstract building blocks containing substructures of an experiment that may themselves contain hierarchies of substructures. Terminal building blocks provide the data structures and primitive functions that are used in scripts (of non-terminal blocks) to compose the network algorithms. A non-terminal building block hides its internal structure and provides abstract sites and scripts as an interface to its internals. Therefore it appears as a terminal building block to the outside and may be used as such for the construction of an experiment. This construction is equivalent. to the building of a single non-terminal building block (the Experiment) that encloses the complete experiment structure. 3.3 Construction of New Building Blocks The functionality of SESAME can be extended using two different approaches. New terminal building blocks can be programmed deriving from existing C++ classes or new non-terminal building blocks may be assembled by using previously defined building blocks: 3.3.1 Programming New Terminal Building Blocks Terminal building blocks can be designed by derivation from already existing C++ classes. The complete administration structure and possible predefined properties are inherited from the parent classes. In order to add new properties e. g. new action functions, symbols, datafields, insites or outsites a set of basic operations is being provided by the framework. One should note that new algorithms 800 Linden, Sudbrak, Tietz, and Weber and structures can be added to a class without any changes to the framework of SESAME. 3.3.2 Composing New Non-Terminal Building Blocks Non-terminal building blocks can be combined from libraries of already designed terminal or non-terminal blocks. See for an example fig. ??, where a set of building blocks build a multilayer net which can be collapsed into one building block and reused in other contexts. Here insites and outsites define an interface between building blocks on adjacent levels of the experiment hierarchy. The flow of data inside the new building block is controlled by scripts that call action functions or scripts of its components. Such an abstract building block may be saved in a library for reuse. Even whole experiments can be collapsed to one building block leaving a lot of possibilities for the experimenter to cope with very large and complicated experiments. 3.3.3 Deriving New Non-Terminal Building Blocks A powerful mechanism for organizing very complex experiments and allowing high degrees of flexibility and reuse is offered by the concept of inheritance. The basic mechanism executes the description of the parent building block and thereafter the description of the refinements for the derived block. All this may be done interactively, thus additional refinements can be added at runtime. Even the set of formal parameters of a block may be inherited and/or refined. Multiple inheritance is also possible. For an example consider a general function approximator which may be used at many points in a more complex architecture. It can be implemented as an abstract base building block, only supplying basic structure as input and output and basic operations as ((propagate input" and ((train" . Derivations of it then implement the algorithm and structure actually used. Statistical routines, visualization facilities, pattern handling and other utilities can be added as further specializations to a basic function approximator. 3.3.4 Parameters and Generic Building Blocks A building block may also define formal parameters that allow the user to configure it at the time of its instantiation or inclusion into some other non-terminal building block. Thus non-terminal building blocks can be generic. They may be parameterized with types for interior building blocks, names of scripts etc. With this mechanism a multilayer net can be created with an arbitrary type of node or weight layers. 3.4 Autoconfiguration When a user defines an experiment, only parameters that are really important must be specified. Redundant parameters, that depend on other paremeters of other building blocks, can often be determined automatically. In SESAME this is done via a constraint satisfaction process. Not only does this mechanism avoid specification of redundant information and check experiment parameters for consistency, but it An Object-Oriented Framework for the Simulation of Neural Nets 801 also enables the construction of generic structures. Communication links between outsites and insites of building blocks check data for matching types. Building blocks impose additional constraints on the data formats of their own sites. Constraints are formed upon information about the base types, dimensions, sizes and ranges of the data sent between the sites. The primary source of information are the parameters given to the building blocks at the time of their instantiation. After building the whole experiment, a propagation mechanism iteratively tries to complete missing information in order to satisfy all constraints. Thus information which is determined in one building block of the experiment may spread all over the experiment topology. As an example one can think of a building block which loads patterns from a file. The dimensionality of these patterns may be used automatically to configure building blocks holding weight layers for a multilayer network. This autoconfiguration can be considered as finding the unique solution of set of equations where three cases may occur: inconsistency (contradiction between two information sources at one site), deadlock (insufficient information for a site) or success (unique solution). Inconsistencies are a proof of an erroneous design. Deadlocks indicate that the user has missed something. 3.5 Experiment Observation Graphical output, file I/O or statistical analysis are usually not performed within the normal building blocks which comprise the network algorithms. These features are built into specialized utility building blocks that can be integrated at any point of the experiment topology, even during experiment runs. 4 Classes of Building Blocks SESAME supports a rich taxonomy of building blocks for experiment construction: For neural networks one can use building blocks for complete node and weight layers to construct multilayer networks. This granulation was chosen to allow for a more efficient way of computation than with building blocks that contain single neurons only. This level of abstraction still captures enough flexibility for many paradigms ,of NIPS. However, terminal building blocks for complete classes of neural nets are also provided if efficiency is first demand. Mathematical building blocks perform arithmetic, trigonometric or more general mathematical transformations, as scaling and normalization. Building blocks for coding provide functionality to encode or decode patterns. Utility building blocks provide access to the filesystem, where not only input or output files can be dealt with but also other UNIX processes by means of pipes. Others simply store structured or unstructured patterns to make them randomly accessible. Graphical building blocks can be used to display any kind of data no matter if weight matrices, activation or error vectors are involved. This is a consequence of the abstract view of combining building blocks with different functionality but a uniform data interface. There are special building blocks for analysis which allow for clustering, averaging, error analysis, plotting and other statistical evaluations. 802 Linden, Sudbrak, Tietz, and Weber Finally simulations (cart pole, robot-arms etc.) can also be incorporated into building blocks. Real-world applications or other software packages can be accessed via specialized interface blocks. 5 Examples Some illustrative examples for experiments can be found in [?] and many additional and more complex examples in the SESAME documentation. The full documentation as well as the software are available via ftp (see below). Here we sketch only briefly, how paradigms and applications from different domains can be easily glued together as a natural consequence of the design of SESAME. Figure?? shows part of an experiment in which a robot arm is controlled via a modified Kohonen feature map and a potential field path planner. The three building blocks, workspace, map and planner form the main part of the experiment. Workspace contains the simulation for the controlled robot arm and its graphical display and map contains the feature map that is used to transform the map coordinates proposed by planner to robot arm configurations. The map has been trained in another experiment to map the configuration space of the robot arm and the planner may have stored the positions of obstacles with respect to the map coordinates in still another experiment. The configuration and obstacle map have been saved as the results of the earlier experiments and are reused here. The map was taken from a library that contains different flavors of feature maps in form of nonterminal building blocks and hides the details of its complicated inner structure. The Views help to visualize the experiment and the Buffers are used to provide start values for the experiment runs. A Subtractor is shown that generates control inputs for the workspace by simply performing vector subtraction on subsequently proposed state vectors for the robot arm simulation. 6 Epilogue We designed an object-oriented neural network simulator to cope with the increasing demands imposed by the current lines of research. Our implementation offers a high degree of flexibility for the experimental setup. Building blocks may be combined to build complex experiments in short development cycles. The simulator framework provides mechanisms to detect errors in the experiment setup and to provide parameters for generic subexperiments. A prototype was built, that is in use as our main research tool for neural network experiments and is constantly refined. Future developments are still necessary, e. g. to provide a graphical interface and more elegant mechanisms for the reuse of predefined building blocks. Further research issues are the parallelization of SESAME and the compilation of experiment parts to optimize their performance. The software and its preliminary documentation can be obtained via ftp at :ftp.gmd.de in the directory gmd/as/sesame. Unfortunately we cannot provide professional support at this moment. Acknowledgments go to the numerous programmers and users of SESAME for all t.he work, valuable discussions and hints. An Object-Oriented Framework for the Simulation of Neural Nets 803 References [1] B. Angeniol and P. '"rreleaven. The PYGMALION neural network programming environment. In R. Eckmiller, editor, Advanced Neural Computers, pages 167 - 175, Amsterdam, 1990. Elsevier Science Publishers B. V. (North-Holland). [2] N. Goddard, K. Lynne, T. Mintz, and 1. Bukys. Rochester connectionist simulator. Technical Report TR-233 (revised), Computer Science Dept, University of Rochester, 1989. [3] G. 1. Heileman, H. K. Brown, and Georgiopoulos. Simulation of artificial neural network models using an object-oriented software paradigm. In Proceedings of the International Joint Conference on Neural Networks, pages 11-133 - 11-136, Washington, DC, 1990. [4] NeuralWare Inc. Neuralworks professional ii user manual. 1989. [5] T. T. Kraft. AN Spec tutorial workbook. San Diego, CA, 1990. [6] T. Lange, J .B. Hodges, M. Fuenmayor, and L. Belyaev. Descartes: Development environment for simulating hybrid connectionist architectures. In Proceedings of the Eleventh A nnual Conference of the Cognitive Science Society, Ann Arbor, MI, August 1989, 1989. [7] A. Linden and C. Tietz. Combining mUltiple neural network paradigms and applications using SESAME. In Proceedings of the Internation Joint Conference on Neural Networks IJCNN - Baltimore. IEEE, 1992. [8] Y. Miyata. A user's guide to Sun Net version 5.6 - a tool for constructing, running, and looking into a PDP network. 1990. [9] J. M. J. Murre and S. E. Kleynenberg. The MetaNet network environment for the development of modular neural networks. In Proceedings of the International Neural Network Conference, Paris, 1990, pages 717 - 720. IEEE, 1990. [10] M.A. Wilson, S.B. Upinder, J.D. Uhley, and J.M. Bower. GENESIS: A system for simulating neural networks. In David S. Touretzky, editor, Advances in Neural Information Processing Systems I, pages 485-492. Morgan Kaufmann, 1988. Collected papers of the IEEE Conference on Neural Information Processing Systems - Natural and Synthetic, Denver, CO, November 1988. [11] A. ZeIl, N. Mache, T. Sommer, and T. Korb. Recent developments of the snns neural network simulator. In SPIE Conference on Applications of Artificial Neural Networks. Universit"at Stuttgart, April 1991. 804 Linden, Sudbrak, Tietz, and Weber 1 ............ tet : : .. : . uu.a f: ' 0 ...... :: ' .PN .... 1IIcI •• Bl'Lleu IIWape. B ..... all ... :: .:;.' : : . . Nt ....... Nt . ~; ':;.;. ... ---1 • Nt "' Sea_Is d.,.1s I .. .. I Figure 1: Integration of several terminal building blocks into a non-terminal building block with the Backpropagation example. L~ I"""" .... loaIID I t*!MapO" SUb.,...,...~b ,leldPlaa .... inJliD2 ,..a ... T ~ ."IIB1a I .weOu! .atPo.Out j ' I ctP .. Ja I ooatrolIa I-L TwoA .... VIcw: : wert...•• rr., ow ,., .alrcr ow .lIIr1aNlr I 10lIIla t ~ I8fennal ia2 rcul&!u_vj4,9, oIIltm.p p; ... l~rOu! DotIbieMap •• p W.I wID .... I iDdea Plal.Vlew Figure 2: Robot arm control with a hybrid controller
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A Practice Strategy for Robot Learning Control Terence D. Sanger Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology, room E25-534 Cambridge, MA 02139 tds@ai.mit.edu Abstract "Trajectory Extension Learning" is a new technique for Learning Control in Robots which assumes that there exists some parameter of the desired trajectory that can be smoothly varied from a region of easy solvability of the dynamics to a region of desired behavior which may have more difficult dynamics. By gradually varying the parameter, practice movements remain near the desired path while a Neural Network learns to approximate the inverse dynamics. For example, the average speed of motion might be varied, and the inverse dynamics can be "bootstrapped" from slow movements with simpler dynamics to fast movements. This provides an example of the more general concept of a "Practice Strategy" in which a sequence of intermediate tasks is used to simplify learning a complex task. I show an example of the application of this idea to a real 2-joint direct drive robot arm. 1 INTRODUCTION The most general definition of Adaptive Control is one which includes any controller whose behavior changes in response to the controlled system's behavior. In practice, this definition is usually restricted to modifying a small number of controller parameters in order to maintain system stability or global asymptotic stability of the errors during execution of a single trajectory (Sastry and Bodson 1989, for review). Learning Control represents a second level of operation, since it uses Adaptive Con335 336 Sanger trol to modify parameters during repeated performance trials of a desired trajectory so that future trials result in greater accuracy (Arimoto et al. 1984). In this paper I present a third level called a "Practice Strategy", in which Learning Control is applied to a sequence of intermediate trajectories leading ultimately to the true desired trajectory. I claim that this can significantly increase learning speed and make learning possible for systems which would otherwise become unstable. 1.1 LEARNING CONTROL During repeated practice of a single desired trajectory, the actual trajectory followed by the robot may be significantly different. Many Learning Control algorithms modify the commands stored in a sequence memory to minimize this difference (Atkeson 1989, for review). However, the performance errors are usually measured in a sensory coordinate system, while command corrections must be made in the motor coordinate system. If the relationship between these two coordinate systems is not known, then command corrections might be in the wrong direction and inadvertently worsen performance. However, if the practice trajectory is close to the desired trajectory, then the errors will be small and the relationship between command and sensory errors can be approximated by the system Jacobian. An alternative to a stored command sequence is to use a Neural Network to learn an approximation to the inverse dynamics in the region of interest (Sanner and Slotine 1992, Yabuta and Yamada 1991, Atkeson 1989). In this case, the commands and results from the actual movement are used as training data for the network, and smoothness properties are assumed such that the error on the desired trajectory will decrease. However, a significant problem with this method is that if the actual practice trajectory is far from the desired trajectory, then its inverse dynamics information will be of little use in training the inverse dynamics for the desired trajectory. In fact, the network may achieve perfect approximation on the actual trajectory while still making significant errors on the desired trajectory. In this case, learning will stop (since the training error is zero) leading to the phenomenon of "learning lock-up" (An et al. 1988). So whether Learning Control uses a sequence memory or a Neural Network, learning may proceed poorly if large errors are made during the initial practice movements. 1.2 PRACTICE STRATEGIES I define a "practice strategy" as a sequence of trajectories such that the first element in the sequence is any previously learned trajectory, and the last element in the sequence is the ultimate desired trajectory. A well designed practice strategy will result in a seqence for which learning control of the trajectory for any particular step is simplified if prior steps have already been learned. This will occur if learning of prior trajectories reduces the initial performance error for subsequent trajectories, so that a network will be less likely to experience learning lock-up. One example of a practice strategy is a three-step sequence in which the intermediate step is a set of independently executable subtasks which partition the desired trajectory into discrete pieces. Another example is a multi-step sequence in which intermediate steps are a set of trajectories which are somehow related to the desired trajectory. In this paper I present a multi-step sequence which gradually A Practice Strategy for Robot Learning Control 337 ---~-------, I " I A A " N u P y a. Figure 1: Training signals for network learning. transforms some known trajectory into the desired trajectory by varying a single parameter. This method has the advantage of not requiring detailed knowledge of the task structure in order to break it up into meaningful subtasks, and conditions for convergence can be stated explicitly. It has a close relationship to Continuation Methods for solving differential equations, and can be considered to be a particular application of the Banach Extension Theorem. 2 METHODS As in (Sanger 1992), we need to specify 4 aspects of the use of a neural network within a control system: 1. the networks' function in the control system, 2. the network learning algorithm which modifies the connection weights, 3. the training signals used for network learning, and 4. the practice strategy used to generate sample movements. The network's function is to learn the inverse dynamics of an equilibrium-point controlled plant (Shadmehr 1990). The LMS-tree learning algorithm trains the network (Sanger 1991b, Sanger 1991a). The training signals are determined from the actual practice data using either "Actual Trajectory Training" or "Desired Trajectory Training", as defined below. And the practice strategy is "Trajectory Extension Learning", in which a parameter of the movement is gradually modified during training. 338 Sanger 2.1 TRAINING SIGNALS Figure 1 shows the general structure of the network and training signals. A desired trajectory y is fed into the network N to yield an estimated command U. This command is then applied to the plant Pcx where the subscript indicates that the plant is parameterized by the variable a. Although the true command u which achieves y is unknown, we do know that the estimated command u produces y, so these signals are used for training by comparing the network response to y given by ~ = Ny to the known value u and subtracting these to yield the training error 6,. Normally, network training would use this error signal to modify the network output for inputs near y, and I refer to this as "Actual Trajectory Training". However, if y is far from y then no change in response may occur at y and this may lead even more quickly to learning lock-up. Therefore an alternative is to use the error 6fJ to train the network output for inputs near y. I refer to this as "Desired Trajectory Training", and in the figure it is represented by the dotted arrow. The following discussion will summarize the convergence conditions and theorems presented in (Sanger 1992). Define Ru . (1 - N P(x))u = u - U to be an operator which maps commands into command errors for states x on the desired trajectory. Similarly, let Ru = (1 - N P( x))u = u ~ map commands into command errors for states x on the actual trajectory. Convergence depends upon the following assumptions: A1: The plant P is smooth and invertible with respect to both the state x and the input u with Lipschitz constants k'z; and ku, and it has stable zero-dynamics. A2: The network N is smooth with Lipschitz constant kN. A3: Network learning reduces the error in response to a pair (y, 6y ). A4: The change in network output in response to training is smooth with Lipschitz constant kL. A5: There exists a smoothly controllable parameter a such that an inverse dynamics solution is available at a = ao, and the desired performance occurs when a = ad. A6: The change in command required to produce a desired output after any change in a is bounded by the change in a multiplied by a constant kcx • A 7: The change in plant response for any fixed input is bounded by the change in a multiplied by a constant kp • Under assumptions A1-A3 we can prove convergence of Desired Trajectory Training: Theorem 1: If there exists a k Rn such that IIRnu - Rnull < kRn lI u - ull A Practice Strategy for Robot Learning Control 339 then if the learning rate 0 < 'Y :::; 1, If k Rn < 1 and 'Y :::; 1, then the network output u approaches the correct command u. Under assumptions A1-A4, we can prove convergence of Actual Trajectory Training: Theorem 2: If there exists a kRn such that IIRn u - Rnull < kRn lIu - illl then if the learning rate 0 < 'Y :::; 1, 2.2 TRAJECTORY EXTENSION LEARNING Let a be some modifiable parameter of the plant such that for a = ao there exists a simple inverse dynamics solution, and we seek a solution when a = ad. For example, if the plant uses Equilibrium Point Control (Shadmehr 1990), then at low speeds the inverse dynamics behave like a perfect servo controller yielding desired trajectories without the need to solve the dynamics. We can continue to train a learning controller as the average speed of movement (a) is gradually increased. The inverse dynamics learned at one speed provide an approximation to the inverse dynamics for a slightly faster speed, and thus the performance errors remain small during practice. This leads to significantly faster learning rates and greater likelihood that the conditions for convergence at any given speed will be satisfied. Note that unlike traditional learning schemes, the error does not decrease monotonically with practice, but instead maintains a steady magnitude as the speed increases, until the network is no longer able to approximate the inverse dynamics. The following is a summary of a result from (Sanger 1992). Let a change from al to a2, and let P = Pal and P' = Pa2 . Then under assumptions AI-A7 we can prove convergence of Trajectory Extension Learning: Theorem 3: If there exists a kR such that for a = al then for a = a2 IIR'u' - R'illl < kRllu' - ull + (2ka + kNkp)la2 - all This shows that given the smoothness assumptions and a small enough change in a, the error will continue to decrease. 340 Sanger 3 EXAMPLE Figure 2 shows the result of 15 learning trials performed by a real direct-drive twojoint robot arm on a sampled desired trajectory. The initial trial required 11.5 seconds to execute, and the speed was gradually increased until the final trial required only 4.5 seconds. Simulated equilibrium point control was used (Bizzi et al. 1984) with stiffness and damping coefficients of 15 nm/rad and 1.5 nm/rad/sec, respectively. The grey line in figure 2 shows the equilibrium point control signal which generated the actual movement represented by the solid line. The difference between these two indicates the nontrivial nature of the dynamics calculations required to derive the control signal from the desired trajectory. Note that without Trajectory Extension Learning, the network does not converge and the arm becomes unstable. The neural network was an LMS tree (Sanger 1991b, Sanger 1991a) with 10 Gaussian basis functions for each of the 6 input dimensions, and a total of 15 subtrees were grown per joint (see (Sanger 1992) for further explanation). 4 CONCLUSION Trajectory Extension Learning is one example of the way in which a practice strategy can be used to improve convergence for Learning Control. This or other types of practice strategies might be able to increase the performance of many different types of learning algorithms both within and outside the Control domain. Such strategies may also provide a theoretical model for the practice strategies used by humans to learn complex tasks, and the theoretical analysis and convergence conditions could potentially lead to a deeper understanding of human motor learning and successful techniques for optimizing performance. Acknowledgements Thanks are due to Simon Giszter, Reza Shadmehr, Sandro Mussa-Ivaldi, Emilio Bizzi, and many people at the NIPS conference for their comments and criticisms. This report describes research done within the laboratory of Dr. Emilio Bizzi in the department of Brain and Cognitive Sciences at MIT. The author was supported during this work by a National Defense Science and Engineering Graduate Fellowship, and by NIH grants 5R37 AR26710 and 5ROINS09343 to Dr. Bizzi. References An C. H., Atkeson C. G., Hollerbach J. M., 1988, Model-Based Control of a Robot Manipulator, MIT Press, Cambridge, MA. Arimoto S., Kawamura S., Miyazaki F., 1984, Bettering operation of robots by learning, Journal of Robotic Systems, 1(2):123-140. Atkeson C. G., 1989, Learning arm kinematics and dynamics, Ann. Rev. Neurosci., 12:157-183. Bizzi E., Accornero N., Chapple W., Hogan N., 1984, Posture control and trajectory formation during arm movement, J. Neurosci, 4:2738-2744. Sanger T. D., 1991a, A tree-structured adaptive network for function approximation in high dimensional spaces, IEEE Trans. Neural Networks, 2(2):285-293. A Practice Strategy for Robot Learning Control 341 Sanger T. D., 1991b, A tree-structured algorithm for reducing computation in networks with separable basis functions, Neural Computation, 3(1):67-78. Sanger T. D., 1992, Neural network learning control of robot manipulators using gradually increasing task difficulty, submitted to IEEE Trans. Robotics and Automation. Sanner R. M., Slotine J.-J. E., 1992, Gaussian networks for direct adaptive control, IEEE Trans. Neural Networks, in press. Also MIT NSL Report 910303, 910503, March 1991 and Proc. American Control Conference, Boston pages 2153-2159, June 1991. Sastry S., Bodson M., 1989, Adaptive Control: Stability, Convergence, and Robustness, Prentice Hall, New Jersey. Shadmehr R., 1990, Learning virtual equilibrium trajectories for control of a robot arm, Neural Computation, 2:436-446. Yabuta T., Yamada T., 1991, Learning control using neural networks, Proc. IEEE Int'l ConJ. on Robotics and Automation, Sacramento, pages 740-745. Figure 2: Dotted line is the desired trajectory, solid line is the actual trajectory, and the grey line is the equilibrium point control trajectory.
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Modeling Consistency in a Speaker Independent Continuous Speech Recognition System Yochai Konig, Nelson Morgan, Chuck Wooters International Computer Science Institute 1947 Center Street, Suite 600 Berkeley, CA 94704, USA. Victor Abrash, Michael Cohen, Horacio Franco SRI International 333 Ravenswood Ave. Menlo Park, CA 94025, USA Abstract We would like to incorporate speaker-dependent consistencies, such as gender, in an otherwise speaker-independent speech recognition system. In this paper we discuss a Gender Dependent Neural Network (GDNN) which can be tuned for each gender, while sharing most of the speaker independent parameters. We use a classification network to help generate gender-dependent phonetic probabilities for a statistical (HMM) recognition system. The gender classification net predicts the gender with high accuracy, 98.3% on a Resource Management test set. However, the integration of the GDNN into our hybrid HMM-neural network recognizer provided an improvement in the recognition score that is not statistically significant on a Resource Management test set. 1 INTRODUCTION Earlier work [Bourlard and Morgan, 1991l has shown the ability of Multilayer Perceptrons (MLPs) to estimate emission probabilities for Hidden Markov Models (HMM). As shown in their report, with a few assumptions, an MLP may be viewed as estimating the probability P ( q I x ) where q is a sub word model (or a state of a subword model) and x is the input acoustic 682 Modeling Consistency in a Speaker Independent Continuous Speech Recognition System 683 speech data. In this hybrid HMMIMLP recognizer, it was shown that these estimates led to improved performance over standard estimation techniques when a fairly simple HMM was used. More recent results have shown improvements using hybrid HMMIMLP probability estimation over a state-of-the-art pure HMM-based system[Cohen et ai., 1993; Renals et ai., 19921. Some speaker dependencies exist in common parametric representations of speech, and it is possible that making the dependencies explicit may improve performance for a given speaker (essentially enabling the recognizer to soften the influence of the speaker dependency). The basic problem with modeling and estimating explicitly speaker dependent p~ rameters is the lack of training data. In the limit, the only available information about a new speaker is the utterance to be recognized. This limit is our starting point for this study. Even with this limited information, we can incorporate constraints on analysis that rely on the same speaker producing all the frames in an utterance, thus ensuring consistency. As has been observed for some mainstream Hidden Markov Models (HMM) systems [Murveit et ai., 19901, given enough training data, separate phonetic models for male and female speakers can be used to improve performance. Our first attack on consistency, then, is to incorporate gender consistency in the recognition process. In contrast to non-connectionist HMM systems, our proposed architecture attempts to share the gender-independent parameters. Our study had two steps: first we trained an MLP to estimate the probability of gender. Then, we investigated ways to integrate the gender consistency constraint into our existing MLP-HMM hybrid recognizer, resulting in our GDNN architecture. We will give a short description of some related work, followed by an explanation of the two steps described above. We conclude with some discussion and thoughts about future work. 2 RELATED AND PREVIOUS WORK Our previous experiments with the Gender-Dependent Neural Network (GDNN) are described in [Abrash et ai., 1992; Konig and Morgan, 1992]. Other researchers have worked on related problems. For example Hampshire and Waibel presented the "Meta-Pi" architecture [Hampshire and Waibel, 19901. The building blocks for the "Meta-Pi" architecture are multiple TDNN's that are trained to recognize the speech of an individual speaker. These building blocks are integrated by another multiple TDNN trained in a Bayesian MAP scheme to maximize the phoneme recognition rate of the overall architecture. The performance of the "Meta-Pi" architecture on a six speaker /b,d,g/ task was comparable to a speaker dependent system on the same task. Another example of related work is speaker normalization, which attempts to minimize between-speaker variations by transforming the data of a new speaker to that of a reference speaker, and then applying the speaker dependent system for the reference speaker [Huang et ai., 1991]. 3 THE CLASSIFICATION NET In order to classify the gender of a new speaker we need features that distinguish between speakers, in contrast to the features that are used for phoneme recognition that are chosen to suppress speaker variations. Given our constraint that the only available information 684 Konig, Morgan, Wooters, Abrash, Cohen, and Franco about the new speaker is the sentence to be recognized, we chose features that are a rough estimate of the vocal tract properties and the fundamental frequency of the new speaker. Furthermore, we tried to suppress the linguistic information in our estimate. More specifically, the goal was to build a net that estimates the probability P(GenderIData). After some experimentation, the first twelve LPC cepstral coefficients were calculated over a 20 msec window every 10 msec (50% overlap) and averaged along each sentence. The sampling rate was 16khz. These features were augmented by an estimate of the fundamental frequency for a total of 13 features per sentence. The MLP had one hidden layer with 24 hidden units. There were two output units, one for each gender. The training set was the 109-speaker DARPA Resource Management corpus. 3500 sentences were used for the training set and 490 in the cross validation set. The size of the test set was 600 sentences, and it was a combination of the DARPA Resource Management speaker-independent Feb89 and Oct89 test sets. The trained MLP predicts the gender for the test set with less than 1.7% error on the sentence level. 4 INCORPORATING GENDER CONSISTENCY INTO OUR HYBRID HMMIMLP RECOGNIZER 4.1 DISCUSSION Our goal is to find an architecture that shares the gender independent parameters and models the gender dependent parameters. Given our gender consistency constraint we estimate a probability that is explicitly conditioned on gender, as if the phonetic models were simply doubled to permit male and female forms of each phoneme. We can express P( male, phoneldata) (which is then divided by priors to get the corresponding data likelihood) by expansion to P(phonelmale, data) x P(maleldata). This factorization is realized by two separate MLP's: P(maleldata) is estimated by the classification net described above, and P(phonelmale, data) is realized by our GDNN described below. For further description on how to factorize probabilities by neural networks see [Morgan and Bourlard, 19921. The final likelihood for the male case can be expressed as: P(d I h I ) _ P(phonelmale, data) x P(maleldata) x P(data) ata pone, ma e P( h I I) P( I) pone ma e x ma e (1) Note that during recognition, P(data) can be ignored. Similarly, a female-assumed probability can be computed for each hypothesized phone. These male and female-assumed probabilities can then be used in separate Viterbi calculations (since we do not permit any hypothesis to switch gender in the midst of an utterance). In other words, dynamic programming is used with the framewise network outputs to evaluate the best hypothesized utterance assuming male gender, and then the same is done for the female case. The case with the lowest cost (highest probability) is then chosen. Note that the output of the classification net only helps in choosing between the sentence recognized according to female gender or male gender. The critical question is how to estimate P(phonelgender, data). A possible answer is to have two separate nets, one trained only on males, and the other trained only on females. This approach has the potential disadvantages of doubling the number of parameters in the system, and of not sharing the gender independent parameters. We have experimented with a such a net [Konig and Morgan, 1992] and it improved our result over the baseline system. Modeling Consistency in a Speaker Independent Continuous Speech Recognition System 685 l J ~ P(phonelgender,data) t t t t 69 units 1000 units t t t f f 9 x (12 mel cepstral + log energy + first derivatives) = 234 if(gender == Male) {M=l,F=O} otherwise {M=O,F=l} Figure 1: A Gender Dependent Neural Network(GDNN) We present here a hybrid GDNN architecture that has the flexibility to tune itself to each gender. The idea is to have extra binary inputs that specify the gender of the speaker and then the probabilities that the network estimates will be conditioned on the gender. The architecture is shown in figure 1. 4.2 EXPERIMENTS AND RESULTS We have compared four different architectures. The first architecture is our baseline system, namely, one large net that was trained on all the sentences in the training set. The second uses two separate nets, for males and females. The third is the hybrid GDNN architecture described in figure 1. The fourth architecture is a variant of the third architecture, the difference being that the binary units are connected to the output units instead of to the hidden units. All the nets have 1000 hidden units and 69 output units, including the totally separate male and female nets. While one might think that the consequent doubling of 686 Konig, Morgan, Wooters, Abrash, Cohen, and Franco Table 1: Result Summary Architecture Test Set Word Error Baseline 10.6% Two Separate Nets 10.9% Hybrid Architecture - Variant(Binary to Output) 10.9% GDNN Binary to hidden 10.2% the number of parameters in the system might explain the observed degradation for the perfonnance for the second architecture, we have also experimented with several sizes of male and female separate nets, by changing the number of the hidden units and the number of input frames. None of these experiments resulted in a significant improvement. We used 12 mel-cepstral features and the log-energy along with their first derivatives, so the number of input features per frame was 26. The features were calculated from 20ms of speech, computed every 10 msec (as before). The length of the temporal window (the number of input frames) was 9, so the total number of input features was 234. The training set was the 109-speaker DARPA Resource Management corpus. 3500 sentences were used for the training set and 490 in the cross validation set. The size of the test was the 600 sentences making up the DARPA Feb89 and Oct89 test sets. The results are summarized in table I, and are achieved using the standard Resource Management wordpair grammar(perplexity = 60) with a simple context-independent HMM recognizer. We should note here that these results are all somewhat worse than our other results published in [Renals et al., 1992; Cohen et al., 1993], as the latter were achieved using SRI's phonological models, and these were done with a single-pronunciation single-state HMM (with each state repeated for a rough duration model). 5 DISCUSSION AND FUTURE WORK The best results were achieved by the GDNN hybrid architecture that shares the genderindependent parameters while modeling the gender dependent parameters . . However the improvement over our baseline is not statistically significant for this test set, although it is consistent with our experiments with other test sets, not reported here. A possible source for further improvement is using a training set with a more balanced representation of gender. In the DARPA Resource Management speaker independent training set there are 2830 sentences uttered by males versus only 1160 sentences uttered by females. Thus, perfonnance may have suffered from an insufficient number of female training sentences. A reasonable extension to this work would be the modeling of additional speaker dependent parameters such as speech rate, accent, etc. Another direction that might be more fruitful is to combine the gender-dependent models in the local estimation of phonemes, and not to do separate Viterbi recognitions for each gender. We are currently examining this latter alternative. Modeling Consistency in a Speaker Independent Continuous Speech Recognition System 687 Acknowledgements Thanks to Steve Renals for his comments along the way. Computations were done on the RAP machine, with support from software guru Phil Kohn, and hardware wiz Jim Beck. Thanks to Hynek Hermansky for advising us about the features for the gender classification net. Thanks to the other members of the speech group at ICSI for their helpful comments. This work was partially funded by DARPA contract MDA904-90-C-5253. References [Abrash et al., 1992] V. Abrash, H. Franco, M. Cohen, N. Morgan, and Y. Konig. Connectionist gender adaptation in a hybrid neural network / hidden markov model speech recognition system. In Proc. Int'l Conf. on Spoken Lang. Processing, Banff, Canada, October 1992. [Bourlard and Morgan, 19911 H. Bourlard and N. Morgan. Merging multilayer perceptrons & hidden markov models: Some experiments in continuous speech recognition. In E. Gelenbe, editor, Artificial Neural Networks: Advances and Applications. North Holland Press, 1991. [Cohen et al., 1993] M. Cohen, H. Franco, N. Morgan, D. Rumelhart, and V. Abrash. Context-dependent multiple distribution phonetic modeling. In C.L. Giles, Hanson SJ, and J.D. Cowan, editors, Advances in Neural Information Processing Systems, volume 5. Morgan Kaufmann, San Mateo, 1993. [Hampshire and Waibel, 1990] J.B. Hampshire and A. Waibel. Connectionist architectures for multi-speaker phoneme recognition. In D.S. Touretzky, editor, Advances in Neural Information Processing Systems 2, San mateo, CA, 1990. Morgan Kaufman. [Huang et al., 19911 X.D. Huang, K.F. Lee, and A. Waibel. Connectionist speaker normalization and its application to speech recognition. In Neural Networks for Siganl Processing, proc. of 1991 IEEE Workshop" Princeton, New Jersey, October 1991. [Konig and Morgan, 1992] Y. Konig and N. Morgan. Gdnn: A gender -dependent neural network for continuous speech recognition. In Proc. international loint Conference on Neural Networks, Baltimore, Maryland, June 1992. [Morgan and Bourlard, 1992] N. Morgan and H. Bourlard. Factoring neural networks by a statistical method. Neural Computation, (4):835-838,1992. [Murveit et al., 1990] H. Murveit, M. Weintraub, and M. Cohen. Training set issues in sri's decipher speech recognition system. In Proc. speech and Natural Language Workshop. pages 337-340,June 1990. [Renals et al., 1992] S. Renals, N. Morgan, M. Cohen, H. Franco, and H. Bourlard. Connectionist probability estimation in the decipher speech recognition system. In Proceedings IEEE Inti. Conf. on Acoustics, Speech, and Signal Processing, San Francisco, California, March 1992. IEEE.
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Learning to categorize objects using temporal coherence Suzanna Becker· The Rotman Research Institute Baycrest Center 3560 Bathurst St. Toronto, Ontario, M6A 2E1 Abstract The invariance of an objects' identity as it transformed over time provides a powerful cue for perceptual learning. We present an unsupervised learning procedure which maximizes the mutual information between the representations adopted by a feed-forward network at consecutive time steps. We demonstrate that the network can learn, entirely unsupervised, to classify an ensemble of several patterns by observing pattern trajectories, even though there are abrupt transitions from one object to another between trajectories. The same learning procedure should be widely applicable to a variety of perceptual learning tasks. 1 INTRODUCTION A promising approach to understanding human perception is to try to model its developmental stages. There is ample evidence that much of perception is learned. Even some very low level perceptual abilities such as stereopsis (Held, Birch and Gwiazda, 1980; Birch, Gwiazda and Held, 1982) are not present at birth, and appear to be learned. Once rudimentary feature detection abilities have been established, the infant can learn to segment the sensory input, and eventually classify it into familiar patterns. These earliest stages of learning seem to be inherently unsuper• Address as of July 1993: Department of Psychology, McMaster University, 1280 Main Street West, Hamilton Ontario, Canada, L8S 4K1 361 362 Becker vised (or "self-supervised"). Gradually, the infant learns to detect regularities in the world. One kind of structure that is ubiquitous in sensory information is spatiotemporal coherence. For example, in speech signals, speaker characteristics such as the fundamental frequency are relatively constant over time. At shorter time scales, individual words are typically composed of long intervals having relatively constant spectral characteristics, corresponding to vowels, with short intervening bursts and rapid transitions corresponding to consonants. The consonants also change across time in very regular ways. This temporal coherence at various scales makes speech predictable, to a certain degree. As one moves about in the world, the visual field flows by in characteri3tic patterns of expansion, dilation and translation. Since most objects in the visual world move slowly, if at all, the visual scene changes slowly over time, exhibiting the same temporal coherence as other sensory sources. Independently moving rigid objects are invariant with respect to shape, texture and many other features, up to very high level properties such as the object's identity. Even under nonlinear shape distortions, images like clouds drifting across the sky are perceived to have coherent features, in spite of undergoing highly non-rigid transformations. Thus, temporal coherence of the sensory input may provide important cues for segmenting signals in space and time, and for object localization and identification. 2 PREVIOUS WORK A common approach to training neural networks to perform transformationinvariant object recognition is to build in hard constraints which enforce invariance with respect to the transformations of interest. For example, equality constraints among feature-detecting kernels have been used to enforce translation-invariance (Fukushima, 1988; Le Cun et al., 1990). Various other higher-order constraints have been used to enforce viewpoint-invariance (Hinton and Lang, 1985; Zemel, Hinton and Mozer, 1990) and invariance with respect to arbitrary group transformations (Giles and Maxwell, 1987). While in the case of translation-invariance it is straightforward to hard-wire the appropriate constraints, more general linear transformation-in variance requires rather cumbersome machinery, and for arbitrary non-linear transformations the approach is difficult if not impossible. In contrast to the above approaches, Foldiak's model of complex cell development results in translation-invariant orientation detectors without the imposition of any hard constraints (Foldiak, 1991). Further, his method is unsupervised. He proposed a modified Hebbian learning rule, in which each weight change depends on the unit's output history: ~Wij(t) = a Yi(t) (Xj(t) - Wij(t)) where Xj(t) is the activity of the jth presynaptic unit at the tth time step, and Yi(t) is a temporally low-pass filtered trace of the postsynaptic activity of the ith unit. Whereas a standard Hebb-rule encourages a unit to detect correlations between its inputs, this rule encourages a unit to produce outputs which are correlated over time. A single unit can therefore learn to group patterns which have zero overlap. Foldiak demonstrated this by presenting trajectories of moving lines, with line orientation held constant within each trajectory, to a network whose input features were local orientation detectors. Units became tuned to particular orientations, Learning to categorize objects using temporal coherence 363 independent of location. While Foldiak's work is of interest as a model of cell development in early visual cortex, there are several reasons why it cannot be applied directly to the more general problem of transformation-invariant object recognition. One reason that Foldiak's learning rule worked well on the line trajectory problem is that the input representation (oriented line features) made the problem linearly separable: there was no overlap between input features present in successive trajectories, hence it was easy to categorize lines of the same orientation. Generally, in more difficult pattern classification problem:l (such as digit or speech recognition) the optimal input features cannot be preselected but must be learned, and there is considerable overlap between the component features of different pattern classes. Hence, a multilayer network is required, and it must be able to optimally select features so as to improve its classification performance. The question of interest here is whether it is possible to train such a network entirely unsupervised? As mentioned above, the temporal coherence of the sensory input may be an important cue for solving this problem in biological systems. 3 TEMPORAL-COHERENCE BASED LEARNING One way to capture the constraint of temporal coherence in a learning procedure is to build it into the objective function. For example, we could try to build representations that are relatively predictable, at least over short time scales. We also need a constraint which captures the notion of high information content; for example, we could require that the network be unpredictable over long time scales. A measure which satisfies both criteria is the mutual information between the classifications produced by the network at successive time steps. If the network produces classification C(t) at time t and classification C(t+ 1) at time t+ 1, the mutual information between the two successive classifications, averaged over the entire sequence of patterns, is given by H(Ct) + H(Ct+t) - H(Ct, Ct+t) -L (p/)t log (p/)t - L (p/+!)t log (p/+1)t ; + L {p/p/+1)t log {p/p/+l)t i; where the angle brackets denote time-averaged quantities. A set of n output units can be forced to represent a probability distribution over n classes, C E {Cl ... cn }, by adopting states whose probabilities sum to one. This can be done, for example, by using the "soft max" activation function suggested by Bridle (1990): eXi(t) t Pi = ",n x ·(t) ~j=l e 1 = P(C(t) = Ci) where Xi is the total weighted summed input to the ith unit, and Pit, the output of the ith unit, stands for the probability of the ith class, P( C(t) = CIi). 364 Becker Once we know the probability of the network assigning each pattern to each class, we can compute the mutual information between the classifications produced by the network at neighboring time steps, e(t) and e(t + 1). This requires sampling, over the entire training set, the average probability of each class, as well as the joint probabilities of each possible pair of classifications being produced as successive time steps. The learning involves adjusting the weights in the network so as to maximize the mutual information between the representations produced by the network at adjacent time steps. In the experiments reported here, a gradient ascent procedure was used with the method of conjugate gradients. One problem with maximizing the information measure described above is that for a fixed amount of entropy in the classifications, H(et ), the network can always improve the mutual information by decreasing the joint entropy, H(et, etJ. In order to achieve low joint entropy, the network must try to assign class probabilities with high certainty, i.e., produce output values near zero or one. Thus the network can always improve its current solution by simply make the weights very large. Unfortunately, this often occurs during learning. To discourage the network from getting stuck in such locally optimal (but very poor) solutions, we introduce a constant A to weight the importance of the joint entropy term in the objective function, so as to maximize the following: In the simulations reported here, we used a value of 0.5 for A. This effectively prevents the network from concentrating all its effort on reducing the joint entropy, and forces it to learn more gradually, resulting in more globally optimal solutions. We have tested this learning procedure on a simple signal classification problem. The pattern set consisted of trajectories of random intensity patterns, drawn from six classes, shown in figure 1. Members of the same class consisted of translated versions of the same pattern, shifted one to five pixels with wrap-around. A trajectory consisted of a block of ten randomly selected patterns from the same class. Between trajectories, the pattern class changed randomly. The network had six input units, twenty hidden units, and six output units. The hidden units used the logistic nonlinearity, 1+~-'" and the output units used the softmax activation function. The hidden units had biases but the outputs did not. 1 After training the network on 1200 patterns (20 trajectories of 10 examples of each of the six patterns) for 300 conjugate gradient iterations, the output units always became reasonably specific to particular pattern classes, as shown for a typical run in Figure 2a). The general pattern is that each output unit responds maximally to one or two pattern classes, although some of the units have mixed responses. This classification problem is extremely difficult for an unsupervised learning procedure, as there is considerable overlap between patterns in different classes, and essentially no overlap between patterns in the same class. It is therefore easy to see why a single unit might end up capturing a few patterns from one class and a few from another. We can create an easier subproblem by only training the network on half the patterns in each class. In this case, the network always learns to separate the six pattern classes either perfectly, or nearly so, as shown in figure 2b). 1 Removing biases from the outputs helps prevent the network from getting trapped in local maxima during learning. Learning to categorize objects using temporal coherence 365 :····::I:··--~.::.···:::···:::.: • .1, I' •• ' ," • •• " ---"....... ,'-- ,' •• _-_ ••• • __ I Figure 1: The set of 6 random patterns used to create pattern trajectories. Each pattern was created by randomly setting the intensities of the 6 pixels, and normalizing the intensity profile to have zero mean. 4 DISCUSSION Becker and Hinton (1992) showed that a network could learn to extract a continuous parameter of visual scenes which is coherent across space, by maximizing the mutual information between the outputs of two network modules that receive input from spatially adjacent parts of the input. Here, we have shown how the same idea can be applied to the temporal domain, to perform a discrete classification of the input assuming temporal coherence. We could also apply the same algorithm to the problem of unsupervised multi-sensory integration, by forming classifications which are coherent across different sensory modalities, as well as across time. One advantage of the approach presented here over unsupervised learning procedures such as competitive learning is that units must co-operate to try to find a globally optimal solution. There is therefore incentive for each unit to try to improve the temporal predictability of all of the output units' classifications over time, including its own; this discourages anyone unit from trying to model all of 366 Becker a) Mean: Unit 1 Unit 2 Unt 3 Unit 4 Unit 5 Unit 6 o.~---+---------+--------~--------;---------+---------+---------; 0 .. 7~---+--------~---------~--------+---------~ o. 0 O. 0 O. o. b) Mean: Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unlt6 1 ,,,, ,. !0.90 -• O.Se) !;: I ~ -~ ~ • ... " ~ • .. -~ ~ • O . ~:: ~ == • -~ ;; ;; ~ n c" ~ ~ • I-~ ! ~ ~ o oil" f-~ • =: ~ • !; • O.3~ f-P~ III t; ;; • t; ~ : 0'''' f-""" == ,,~v ;: ~ : 0.10--- tit :! e • :::: mm • ;: t: ~ .vv Figure 2: The probability of each output unit responding for each of the six classes of patterns, averaged over 1200 cases. In a) the pattern trajectories contained six shifted examples of each class, while in b) there were three examples of each class. Learning to categorize objects using temporal coherence 367 the patterns. Additionally, because we have a well-defined objective function for the learning, the procedure can be applied to multi-layer networks which discover features specifically tuned to the classification problem. However, there are a few drawbacks to using this learning procedure. One is that if any lower-order temporally coherent structure exists, the network will invariably discover it. So, for example, if the pattern classes differ in their average intensity, the network can easily learn to separate them simply by detecting the average intensity of the inputs and ignoring all other information. Similarly, if the spatial location of pattern features varies slowly .md predictably over time, the network tends to learn a spatial map rather than solving the higher-order problem of pattern classification. On the other hand, this suggests that a sequential approach to modelling temporally coherent structure may be possible: an initial processing stage could try to model low-order temporal structure such as local spatial correlations, a second processing stage could model the remaining structure in the output of the first over a larger spatio-temporal extent, and so on. A second drawback is the space complexity of the algorithm: for a network with n output units, each must store n2 joint probability statistics and n individual probabilities.2 The storage complexity can be reduced from n 2 + n to just two statistics per output unit by optimizing a more constrained objective function in which each output unit assumes a maximum entropy distribution for the other n - 1 units. It then need only consider the average probability of its own output, and the joint probability of its output at successive time steps. In this case, the mutual information can be approximated by a sum of n terms: L H(Ci,t) + H(Ci,t+d - H(Ci,t, Ci,t+d i where H(Ci,t) = - (Pit)t log (Pit)t n~l (1 - Pit)t log n~l (1 - Pit)t is the entropy of the ith output unit under the maximum entropy assumption for the other output units, and the other constrained entropies are computed similarly. A final drawback of the learning procedure presented here, as discussed earlier, is its tendency to become trapped in local optima with very large weights. We dealt with this by introducing a constant parameter, A, to dampen the importance of the joint entropy term. A more principled way to deal with the problem of local optima is to use stochastic rather than deterministic output units, resulting in a stochastic gradient descent learning procedure (although this would increase the simulation time considerably). Another way of obtaining more globally optimal solutions might be to consider the predictability of classifications over longer time scales rather than just at pairwise time steps, as was done in Foldicik's model (1991). The network could thus maximize the mutual information between its current response and a weighted average of its responses over the last few time steps. 2Note, however, that the complexity (both in time and space) of the computation of these statistics is negligible relative to that of the gradient calculations, assuming there are many more weights than the squared number of output units in the network. 368 Becker 5 CONCLUSIONS The invariance of an objects' identity over time, with respect to transformations it may undergo as it and/or the observer move, provides a powerful cue for perceptual learning. We have demonstrated that a network can learn, entirely unsupervised, to build translation-invariant object detectors based on the assumption of temporal coherence about the input. This procedure should be widely applicable to a variety of perceptual learning tasks, such as identifying phonemes in speech, segmenting objects in images of trajectories, and classifying textures in tactile input. Acknowledgments I thank Geoff Hinton for many fruitful discussions that led to the ideas presented in this paper. References Becker, S. and Hinton, G. E. (1992). A self-organizing neural network that discovers surfaces in random-dot stereograms. Nature, 355:161-163. Birch, E. E., Gwiazda, J., and Held, R. (1982). Stereoacuity development for crossed and uncrossed disparities in human infants. Vison research, 22:507-513. Bridle, J. S. (1990). Training stochastic model recognition algorithms as networks can lead to maximum mutual information estimation of parameters. In Touretzky, D. S., editor, Neural Information Processing Systems, Vol. 2, pages 111217, San Mateo, CA. Morgan Kaufmann. Foldiak, P. (1991). Learning invariance from transformation sequences. Neural Computation, 3(2):194-200. Fukushima, K. (1988). Neocognition: A hierarchical neural network capable of visual pattern pattern recognition. Neural networks, 1:119-130. Giles, C. 1. and Maxwell, T. (1987). Learning, invariance, and generalization in high-order neural networks. Applied Optics, 26(23):4972-4978. Held, R., Birch, E. E., and Gwiazda, J. (1980). Stereoacuity of human infants. Proceedings of the national academy of sciences USA, 77(9):5572-5574. Hinton, G. E. and Lang, K. (1985). Shape recognition and illusory conjunctions. In IlCAI 9, Los Angeles. Le Cun, Y., Boser, B., Denker, J., Henderson, D., Howard, R., Hubbard, W., and Jackel, 1. (1990). Handwritten digit recognition with a back-propagation network. In Touretzky, D., editor, Advances in Neural Information Processing Systems, pages 396-404, Denver 1989. Morgan Kaufmann, San Mateo. Zemel, R. S., Hinton, G. E., and Mozer, M. C. (1990). TRAFFIC: object recognition using hierarchical reference frame transformations. In Advances in Neural Information Processing Systems 2, pages 266-273. Morgan Kaufmann Publishers.
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Learning Curves, Model Selection and Complexity of Neural Networks Noboru Murata Department of IVIathematical Engineering and Information Physics University of Tokyo, Tokyo 113, JAPAN E-mail: mura~sat.t.u-tokyo.ac.jp Shuji Yoshizawa Dept. Mech. Info. University of Tokyo ShUll-ichi Amari Dept. Math. Eng. and Info. Phys. University of Tokyo Abstract Learning curves show how a neural network is improved as the number of t.raiuing examples increases and how it is related to the network complexity. The present paper clarifies asymptotic properties and their relation of t.wo learning curves, one concerning the predictive loss or generalization loss and the other the training loss. The result gives a natural definition of the complexity of a neural network. Moreover, it provides a new criterion of model selection. 1 INTRODUCTION The leal'lI ing Cl1l've shows how well t hE' behavior of a neural network is improved as t.he nurnber of training examples increast"'s and how it is I'elated with the complexity of neural net.works. This provides liS with a criterion for choosing an adequate network ill relat.ion t.o the number of training examples. Some researchers have attacked this problem by using statistical mechanical met.hods (see Levin et al. [1990], Seung et al. [1991]' etc.) and some by informat.ion theory and algorithmic methods (see Baum and Haussler 607 608 Murata, Yoshizawa, and Amari [1989], et.c.). The present. paper elucidates asympt.otic properties of the learning CUl"ve from the statistical point of view, giving a new criterion for model selection. 2 STATEMENT OF THE PROBLEM Let us consider a stochastic neural network, which is parameterized by a set of m weights 0 = (0 1 , ..• ,om) and whose input-output relation is specified by a conditional probability p(ylx, 0). In other words, for an input signal is x E R"·n, the probability distribution of output y E R"oU! is given by p(ylx, 0). A typical form of the stochastic neural network is as follows: let us consider a multi-layered network !(x, 0) where 0 is a set of m parameters 0 = (0 1 , ••• , om) and its components correspond to weights and thresholds of the network. When some input x is given, the network produce an output y = /(x,() + TJ(X), (1 ) where TJ(x) is noise whose conditional distribut.ion is given by a(TJlx). Then the condit.ional dist.ribution of the net.work. which specifies the input-output relation, is given by p(yl1.·,O) = a(y - /(x, ()Ix). (2) \Ve define a t.raining sample e = {(Xl, Yd, .. " (Xt, Yt)} as a set of t examples generated from the true conditional distribution q(ylx), where Xi is generated from a probability distribution 1'(X) independently. We should note that both r(x) and q(ylx) are unknown and we need not assume the faithfulness of the model, that is, we do not a'3sume that there exists a parameter 0* which realize the true distribution q(ylx) such that p(Ylx, 0·) = q(ylx). Our purpose is t.o find an appropriate parameter () which realizes a good approximation IJ(ylx, 0) t.o q(yl:r). For this purpose, we use a loss function L(O) = D(1'; qlp(O)) + 8(0) (3) as a Cl'it.erioll t.o be minimized, where D( 1'; qlp( 0) represent.s a general divergence measure between t.wo conditional probabilit.ies q(ylx) and p(ylx, 0) in the expectat.ion form under t.he true input-output probability D(1'; qlp(O») = J 1'(x)q(Ylx)k(x, y, O)dxdy (4) and S(O) is a regulal'ization t.erm to fit. the smoothness condition of outputs (Moody [1992]), So t.he loss functioll is rewritten as a expectation form L(O)= j1'(J;)Q( Y1 X)d(x,y'(l)dxdy, d(x,y,()=k(x,y,O)+S(O), (5) and d(:t,!I, 0) is raIled t.he pointwise loss funct.ioll. A typical rase of the divergence D of t.he multi-layered network f( X, 0) with noise is the squared error D( 1'; qllJ( 0») = j 1'( X )q( ylx )lly - /( x, 0)11 2dxdy, (6) Learning Curves, Model Selection and Complexity of Neural Networks 609 The error function of an ordinary multi-layered network is in this form, and the conventional Back-Pr'opagation met.hod is derived from this type of loss function. Anot.her t.ypical case is the Kullhaek-Leibler divergence J q(ylx) D(I';qlp(O)) = r(.r)lJ(ylx)log dxdy. p(ylx,B) (7) The integration J 1'(x)q(ylx) logq(ylx)dxdy is a constant called a conditional entropy, and we usually use the following abbreviated form instead of the previous divergence: D(7'; qlp((})) = - J 1'(x)q(ylx) logp(y/x, B)dxdy. (8) Next, we define an optimum of the parameter in the sense of the loss function that we introduced. We denote by B* the optimal parameter that minimizes the loss function L( 0), that is, L(O*) = min L(O), (J and we regard p(ylx, 0*) as the best realization of the model. (9) \t\'hen a trailling sample e is given, we can also define an empirical loss function: 1.(0) = D(1'; qlp(B)) + S((n, (10) where i', If are the empirical distribut.ions given by the sample e, that is, 1 t D(l·;tj/p(O)) = t Lk(Xi'Yi,(}), (xi,yd E e. (11) i=l In practical case, we consider t.he empirical loss function and search for the quasioptimal paramet.er 0 defined hy L(O) = min L(O), (J (12) because the trw·' distribut.ions 1'{x) and q(ylx) are unkllown and we can only use examplps (XidJd observed from t.he tl'lle distribution ,,(x)IJ(ylx), We should note that. the quasi-optilllal paramet.er 0 is a rallc\OI1l variable depending on the sample e, each element of which is chosen randOlnly. The following lemma guarantees that we can use the empirical loss function instead of the actual loss funct.ion when t.he number of examples t is large. Lenllna 1 If fhe 11'11111ber of examples t is large e1lough, it is shown that the quasioptimal pam7llcier 0 -is lIormally dist7'ib'utcd al'ound the optimal parameter B*, that lS, where (-. •. 1 Q / r(.t)I/(yl;L')\'c!(.l', y. 0* )'Vd(;L', V, 0* )Td.tdy, J l'(x)IJ(ylx)'V'Vd(x,y,O*)dxdy, and 'V denote~ fhe di.fJer·en/utl oper'ator with respect to B, (13) ( 14) (15) 610 Murata, Yoshizawa, and Amari This lemma is proved hy using t.he uSllal statistical methods. 3 LEARNING PROCEDURE In many cases, however, it is difficult to obtain the quasi-optimal parameter 9 by minimizing the equation (10) direct.ly. VVe therefore often use a stochastic descent method to get an approximation to the quasi-optimal parameter 9. Definition 1 (Stochastic Descent Method) In each learning step, an example is re-sampled from the given sample e randomly, and the following modification is applied to the parameter On at step 71, (16) where c is a positit,e value called a learni7lg coefficient and (Xi(n), Yi(n)) 2S the resampled example at step 71. This is a sequent.ial learning method and the operations of random sampling frol11 e in eacll lcarning step is called the re-sampling plan. The parameter 011 at. st.ep 11 is a random variable as a function of the re-sampled sequence ...; = {( J'i( 1) • .lJi( 1) ) •.•. , (J: i( ,t!, lji( Il d }. However, if the initial value of 0 is appropriate (this assumpt.ion prevent.s being stuck in local minima) and if the learning step n is large enough, it. is shown that the learned parameter On is normally distributed around the qnasi-opt.imal parameter. Lenuua 2 If the learning step n is large enough and the learning coefficient c is small enough, the parameter 0" is normally distributed asymptotically, that is, Oil '" N(O,EV), (17) where' \I satisfies the followi7lg T"Clatio71 G = QF + VQ, (18) t (,' = f L \1 d ( J ' / , Yi , 0 rv d ( .l: i , !Ii , 0) T , , It Q = t L V' V' d ( Xi, Yi , 0) . i= I i==l In the following discussion, we assume that. 11 is large enough and c is small enough, and we denot.e the learned parameter by (19) The dist.ribut.ion of t.he randolll variable 0, therefore, can be regarded a<; the normal distribllt.ioll N(O.EV). 4 LEARNING CURVES It. is import.allt. to evalll<:l\.t> the difl'crellce bet.ween two quantities L(O) and 1.(0). The quantit.y 1.(0) is calkd the predict.ive loss or the generalization error, which shows Learning Curves, Model Selection and Complexity of Neural Networks 611 t.he average loss of t.he tl"ained network when a novel example is given. On the other hand, the quant.ity L(O) is called the training loss or the training error, which shows the average loss evaluated by the examples used in tl·aining. Since these quantities depend all t.he sample e and the I'e-sampled sequence w, we take the expectation E and the variance Val' with respect to the sample e and the re-sampling sequence w. First., let. us consider the predictive loss which is t.he average loss of the trained network when a new example (which does not belong to the sample e) is given. This averaging operation is replaced by averaging all over the input-output pairs, because the measure of the sample e is z€'ro. Then the predictive loss is written as L(O) = J 1·(x)q(Ylx)d(x,y,O)dxdy. (20) From the properties of ° and B, we can prove the following important relations. Theorem 1 Th.e predictive loss asymptotically satisfies 1 E E[L(O)] L(()*) + 2t trCQ-1 + '2trQv, (21) I ['2 \lar[L(O)] 21.'.! t)'{,'Q-I(,'Q-1 + 2"trQ VQV + 7t.rG\I. (22) Roughly speaking, thel'!' exist t.wo raudOll1 values Y1 and }"~. and the predictive loss can he writ t.en as t.he following forl1l: 1 E L(O) L(O·) + 2tt.rCQ-l + 2t.rQll +fYl + EY2 + Op(~) + Op(E), (23) where Y1 aud Y2 satisfy E[Yd = 0, E[Y'.!] = o. Cov[Y) }''.!] Var[Yd = ~t.rCQ-1CQ-l, . I Vad}":!] = 1t.rQV QV, '.rGV, E, Val' and Cov dellol.e t.he I'xpect.al.ioll, t.he variance and the covariance respectively. Next, we consider t.he> t.railling loss, i.e., t.lw average loss evaluated by the examples used ill t.l'<lining . .Just. as we did in t.he previolls theorem, we can get the following re la.tions. ThCOl'Clll 2 The training loss aSY1l71ltotically satisfy 1 E L(O·) 2t t.I'GQ-I + 2 t.l'Q V, (24) 1(/' " f . 1'(J:)q(YI·t)d(J:,y,O*)-dxdy - (./ /'(.1: )q( Y IJ: )d( x, y, o· )d.tdy) :!) . (25) 612 Murata, Yoshizawa, and Amari Intuitively speaking like the predictive loss, the training loss can be expanded as (26) where Y3 satisfies 0, / r(x)q(ylx)d(x,y,O*)2dxdy- (/ r(x)q(Ylx)d(x,y,O*)dxdy)2. When we look at two curves E[L(O)] and E[L(o)] as functions of t, they are called learning curves which represent the characteristics of learning. The expectations of the predictive loss and the training loss look quite similar. They are different in the sign of the term lit. As the learning coefficient £ increases, the expectations E[L(lJ)] and E[L(o)] increase, but as the number of examples t increases, the average predictive loss E[L(O')] dec.reases and t.he average training loss E[.L(8)] conversely increases. Moreover, their variances are different in the Ol·der of t. The coefficients trGQ-l, trQV, etc. are calculated from the matrices G, Q and V, which reflect the architecture of the network and the loss criterion t.o be minimized. We can consider t.hese mat.rices as representing the complexity of the network. In earlier work, Amari and tvillrata [1991] introduced an effective complexity of the network, trCQ-l, by analogy to Akaike's Information Criterion (AIC) (see Akaike [1974]). 5 AN APPLICATION FOR MODEL SELECTION These results nat.urally leads us to a model selection criterion, which is like the AlC criterion of statistical model selection and which is related those proposed by some researchers (see Murata et a1. [1991], Moody [1992]). From the previous relations, we can easily show the following relation (27) where c is a quant.it.y of order 1/ Jl and common to all the net.works of the same archit.ecture. We compare the abilities of two different networks, which have the same al'chitecture and are tl'ained by the same sample, but differ in the number of weights or nemons (see Fig.l). We can use a quantity, NIC (Network Information Criterion), (28) where (29) fO!" selecting an opt.imal net.work model. Note that. this quantity NIC is directly calculable, since all elements of it. L(O). G, Q, are given by summing over the Learning Curves, Model Selection and Complexity of Neural Networks 613 sample e. When we have two models 1111 and M2, and the NIC of A11 is smaller than that of .Hz, the predictive loss of Afl is expected smaller than that of M 2 , so All can be l'egal'ded as a better model in the sense of the loss function. This criterion cannot he used when we compare two networks of different architectures, for example a multi-layered network and a radial basis expansion network. This is because the value c of the order 1/ Vi term is common only to two networks in which one is included in the other as a submodel. The criterion is in general valid only for such a family of networks (see Fig.2). 6 CONCLUSIONS In this paper, we show that. there is nice relation between the expectation of the predictive loss and that of the training loss. This result naturally leads us to a new model selection criterion. We will consider the application of this result as an algorithm for automatically changing the number of hidden units in the learning as future work. References H. Aka.ike. (1974) A new look at the statistical model identification. IEEE Trans. AC,19(6):716-723. S. Amari. (1967) Theol'Y of adaptive pattern classifiers. IEEE Trans. EC, 16(3}:29U-:307. S. Amari and N. l'vI urata.. (1991) Stat.ist.ical theory of learning curves under entropic loss criterioll. Technical Report METR 91-12, University of Tokyo, Tokyo, Japan. E. B. Baum and D. Haussler. (1989) What size net gives valid generalization? Neural Computation, 1:151-160. E. Levin, N. Tishby, and S. A. Solla. (1990) A statistical approach to learning and generalization in layered neural networks. Proc. of IEEE, 78(1O}:1568-1574. J . E. l\'Ioody. (1992) The effective number of parameters: An analysis of generalization alld regularization in nonlinear learning systems. In J . E. Moody, S. J. Hanson, and R. P. Lippmann, (eds.), Advances ill Neural JlIfonnation Processing Systems 4San Mateo, CA: Morgan Ka.ufmanll . N. Murata. (1992) Statistical aSY17l1liotic study on learning (In Japanese). PhD thesis, University of Tokyo, Tokyo, Japan . N. Murata, S. Yoshizawa, and S. Amari. {1991} A criterion for determining the number of paramet.ers in an artificial neural network model. In T. Kohonen et al., (eds.), Artificial Ne'ural Networks, 9-14. Holland: Elsevier Science Publishers. H. S. Seung, H. Sompolinsky, and N. Tishby. (1991) Statistical mechanics of learning from examples II. quenched theory a.nd unrealizable rules. Submitted to Physical Review A. 614 Murata, Yoshizawa, and Amari origin of the large q(y Ix) variance ~)1[\ ~ ", q(y Ix) /!\ \ I I \ \ I • • \ I I \ \ \ Figure 1: Geomet.rical represent.ation of hierarchical models: the solid lines between q(vlx) and Oi show predictive losses, anel the dashed lines between q(Ylx) and OJ show t.raining losses. The large variance of t.he trailling loss originated in the discrepancy of q(YI.r) alld q(yl.l'). Whell we est.illiatt' I,he I)l"('dinion loss from t.he t.ra.ining loss, the large variallef' st.ill ,'('maills. Bllt. ill t.he case t.hat t.he model M 1 includes the model IIf:!, t.his variance is common to two models, so we do not have to take care of it. variance Figure 2: Geomet.rical representat.ion of non-hierarchical models: the solid lines bet.ween q(Ylx) and Oi show predictive losses, and the dashed lines between q(ylx) and OJ show t.raining losses. The discrepancy of (J(ylx) and q(Ylx) works differently on two models M I alld 11-12 ill est.imating predict.ivf' losses.
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Optimal Depth Neural Networks for Multiplication and Related Problems Kai-Yeung Siu Dept. of Electrical & Compo Engineering University of California, Irvine Irvine, CA 92717 Abstract Vwani Roychowdhury School of Electrical Engineering Purdue University West Lafayette, IN 47907 An artificial neural network (ANN) is commonly modeled by a threshold circuit, a network of interconnected processing units called linear threshold gates. The depth of a network represents the number of unit delays or the time for parallel computation. The SIze of a circuit is the number of gates and measures the amount of hardware. It was known that traditional logic circuits consisting of only unbounded fan-in AND, OR, NOT gates would require at least O(log n/log log n) depth to compute common arithmetic functions such as the product or the quotient of two n-bit numbers, unless we allow the size (and fan-in) to increase exponentially (in n). We show in this paper that ANNs can be much more powerful than traditional logic circuits. In particular, we prove that that iterated addition can be computed by depth-2 ANN, and multiplication and division can be computed by depth-3 ANNs with polynomial size and polynomially bounded integer weights, respectively. Moreover, it follows from known lower bound results that these ANNs are optimal in depth. We also indicate that these techniques can be applied to construct polynomial-size depth-3 ANN for powering, and depth-4 ANN for mUltiple product. 1 Introduction Recent interest in the application of artificial neural networks [10, 11] has spurred research interest in the theoretical study of such networks. In most models of neural networks, the basic processing unit is a Boolean gate that computes a linear 59 60 Siu and Roychowdhury threshold function, or an analog element that computes a sigmoidal function. Artificial neural networks can be viewed as circuits of these processing units which are massively interconnected together. While neural networks have found wide application in many areas, the behavior and the limitation of these networks are far from being understood. One common model of a neural network is a threshold circuit. Incidentally, the study of threshold circuits, motivated by some other complexity theoretic issues, has also gained much interest in the area of computer science. Threshold circuits are Boolean circuits in which each gate computes a linear threshold function, whereas in the classical model of unbounded fan-in Boolean circuits only AND, OR, NOT gates are allowed. A Boolean circuit is usually arranged in layers such that all gates in the same layer are computed concurrently and the circuit is computed layer by layer in some increasing depth order. We define the depth as the number of layers in the circuit. Thus each layer represents a unit delay and the depth represents the overall delay in the computation of the circuit. 2 Related Work Theoretical computer scientists have used unbounded fan-in Boolean circuits as a model to understand fundamental issues of parallel computation. To be more specific, this computational model should be referred to as unbounded fan-in parallelism, since the number of inputs to each gate in the Boolean circuit is not bounded by a constant. The theoretical study of unbounded fan-in parallelism may give us insights into devising faster algorithms for various computational problems than would be possible with bounded fan-in parallelism. In fact, any nondegenerate Boolean function of n variables requires at least O(log n) depth to compute in a bounded fan-in circuit. On the other hand, in some practical situations, (for example large fan-in circuits such as programmable logic arrays (PLAs) or multiple processors simultaneously accessing a shared bus), unbounded fan-in parallelism seems to be a natural model. For example, a PLA can be considered as a depth-2 AND/OR circuit. In the Boolean circuit model, the amount of resources is usually measured by the number of gates, and is considered to be 'reasonable' as long as it is bounded by a polynomial (as opposed to exponential) in the number of the inputs. For example, a Boolean circuit for computing the sum of two n-bit numbers with O(n3 ) gates is 'reasonable', though circuit designers might consider the size of the circuit impractical for moderately large n. One of the most important theoretical issues in parallel computation is the following: Given that the number of gates in the Boolean circuit is bounded by a polynomial in the size of inputs, what is the minimum depth (i.e. number of layers) that is needed to compute certain functions? A first step toward answering this important question was taken by Furst et al. [4] and independently by Ajtai [2]. It follows from their results that for many basic functions, such as the parity and the majority of n Boolean variables, or the multiplication of two n-bit numbers, any constant depth (i. e. independent of n) classical Boolean circuit of unbounded fan-in AND/OR gates computing these functions must have more than a polynomial (in n) number of gates. This lower bound on the size was subsequently improved by Yao [18] and Hastad [7]; it was proved that Optimal Depth Neural Networks for Multiplication and Related Problems 61 indeed an exponential number of AND/OR gates are needed. So functions such as parity and majority are computationally 'hard' with respect to constant depth and polynomial size classical Boolean circuits. Another way of interpreting these results is that circuits of AND/OR gates computing these 'hard' functions which use polynomial amount of chip area must have unbounded delay (i. e. delay that increases with n). In fact, the lower bound results imply that the minimum possible delay for multipliers (with polynomial number of AND/OR gates) is O(logn/loglogn). These results also give theoretical justification why it is impossible for circuit designers to implement fast parity circuit or multiplier in small chip area using AND, OR gates as the basic building blocks. One of the 'hard' functions mentioned above is the majority function, a special case of a threshold function in which the weights or parameters are restricted. A natural extension is to study Boolean circuits that contain majority gates. This type of Boolean circuit is called a threshold circuit and is believed to capture some aspects of the computation in our brain [12]. In the rest of the paper, the term 'neural networks' refers to the threshold circuits model. With the addition of majority gates, the resulting Boolean circuit model seems much more powerful than the classical one. Indeed, it was first shown by Muroga [13] three decades ago that any symmetric Boolean function (e.g. parity) can be computed by a two-layer neural network with (n + 1) gates. Recently, Chandra et al. [3] showed that multiplication of two n-bit numbers and sorting of n n-bit numbers can be computed by neural networks with 'constant' depth and polynomial size. These 'constants' have been significantly reduced by Siu and Bruck [14, 15] to 4 in both cases, whereas a lower bound of depth-3 was proved by Hajnal et al. [6] in the case of multiplication. It is now known [8] that the size of the depth-4 neural networks for multiplication can be reduced to O(n2 ). However, the existence of depth-3 and polynomial-size neural networks for multiplication was left as an open problem [6, 5, 15] since the lower bound result in [6]. In [16], some depth-efficient neural networks were constructed for division and related arithmetic problems; the networks in [16] do not have optimal depth. Our main contribution in this paper is to show that small constant depth neural networks for multiplication, division and related problems can be constructed. For the problems such as iterated addition, multiplication, and division, the neural networks constructed can be shown to have optimal depth. These results have the following implication on their practical significance: Suppose we can use analog devices to build threshold gates with a cost (in terms of delay and chip area) that is comparable to that of AND, OR, logic gates, then we can compute many basic functions much faster than using traditional circuits. Clearly, the particular weighting of depth, fan-in, and size that gives a realistic measure of a network's cost and speed depends on the technology used to build it. One case where circuit depth would seem to be the most important parameter is when the circuit is implemented using optical devices. We refer those who are interested in the optical implementation of neural networks to [1]. Due to space limitations, we shall only state some of the important results; further results and detailed proofs will appear in the journal version of this paper [17]. 62 Siu and Roychowdhury 3 Main Results Definition 1 Given n n-bit integers, Zi = Lj~; zi,i2i, i = 1, ... , n, zi,i E {O, I}, We define iterated addition to be the problem of computing the (n + log n )-bit sum L~=l Zi of the n integers. Definition 2 Given 2 n-bit integers, x = Lj==-~ xi2i and Y = Lj==-~ Yi2i. We define multiplication to be the problem of computing the (2n)-bit product of x and y. Using the notations of [15], let us denote the class of depth-d polynomial-size neural networks where the (integer) weights are polynomially bounded by & d and the corresponding class where the weights are unrestricted by LTd. It is easy to see that if it~ated addition can be computed in &2, then multiplication can be computed in LT3 . We first prove the result on iterated addition. Our result hinges on a recent striking result of Goldmann, Hcistad and Razborov [5]. The key observation is that iterated addition can be computed as a sum of polynomially many linear threshold (LTd functions (with exponential weights). Let us first state the result of Goldmann, Hastad and Razborov [5]. Lemma 1 [5] Let LTd denote the class of depth-d polynomial-size neural networks where the weights at the output gate are polynomially bounded integers (with no restriction on the weights of the other gates). Then LTd = & d for any fixed integer d ~ 1. The following lemma is a generalization of the result in [13]. Informally, the result says that if a function is 1 when a weighted sum (possibly exponential) of its inputs lies in one of polynomially many intervals, and is 0 otherwise, then the function can be computed as a sum of polynomially many LTI functions. Lemma 2 Let S = L7=1 WiXi and f(X) be a function such that f = 1 if S E [Ii, ud for i = 1, ... , Nand f = 0 otherwise, where N is polynomially bounded. The~ can be computed as a sum of polynomially many LTI functions and thus f E LT2 · Combining the above two lemmas yields a depth-2 neural network for iterated addition. .Theorem 1 Iterated addition is in LT2 • It is also easy to see that iterated addition cannot be computed in LTI . Simply observe that the first bit of the sum is the parity function, which does not belong to LT1 . Thus the above neural network for iterated addition has minimum possible depth. Theorem 2 Multiplication of 2 n-bit integers can be computed in LT3. It follows from the results in [6] that the depth-3 neural network for multiplication stated in the above theorem has optimal depth. Optimal Depth Neural Networks for Multiplication and Related Problems 63 We can further apply the results in [5] to construct small depth neural networks for division, powering and multiple product. Let us give a formal definition of these problems. Definition 3 Let X be an input n-bit integer ~ O. We define powering to be the n2-bit representation of xn. Definition 4 Given n n-bit integers Zi, i = 1, ... , n, We define multiple product to be the n2-bit representation of n~=l Zi. Suppose we want to compute the quotient of two integers. Some quotient in binary representation might require infinitely many bits, however, a circuit can only compute the most significant bits of the quotient. If a number has both finite and infinite binary representation (for example 0.1 = 0.0111 ... ), we shall always express the number in its finite binary representation. We are interested in computing the truncated quotient, defined below: Definition 5 Let X and Y ~ 1 be two input n bit integers. Let X /Y = L~;~oo zi 2i be the quotient of X divided by Y. We define DIVk(X/Y) to be X/Y truncated to the (n + k)-bit number, i.e. o In particular, DIVo(X /Y) is l X /Y J, the greatest integer ~ X /Y. Theorem 3 -. 1. Powering can be computed in LT3 . 2. DIVk(x/y) can be computed in Lr3 . 3. Multiple Product can be computed in LT4 . It can be shown from the lower-bound results in [9] that the neural networks for division are optimal in depth. References [1] Y. S. Abu-Mostafa and D. Psaltis. Optical Neural Computers. Scientific American , 256(3):88-95, 1987. [2] M. Ajtai. L~ -formulae on finite structures. Annals of Pure and Applied Logic, 24:1-48, 1983. [3] A. K. Chandra, 1. Stockmeyer, and U. Vishkin. Constant depth reducibility. Siam J. Comput., 13:423-439, 1984. [4] M. Furst, J. B. Saxe, and M. Sipser. Parity, Circuits and the Polynomial-Time Hierarchy. IEEE Symp. Found. Compo Sci., 22:260-270, 1981. [5] M. Goldmann, J. Hastad, and A. Razborov. Majority Gates vs. General Weighted Threshold Gates. preprint, 1991. 64 Siu and Roychowdhury [6] A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, and G. Turan. Threshold circuits of bounded depth. IEEE Symp. Found. Compo Sci., 28:99-110, 1987. [7] J. H1stad and M. Goldmann. On the power of small-depth threshold circuits. InProceedings of the 31st IEEE FOCS, pp. 610-618, 1990. [8] T. Hofmeister, W. Hohberg and S. Kohling . Some notes on threshold circuits and multiplication in depth 4. Information Processing Letters, 39:219-225, 1991. [9] T. Hofmeister and P. PudIa.k, A proof that division is not in TC~. Forschungsbericht Nr. 447, 1992, Uni Dortmund. [10] J. J. Hopfield. Neural Networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, 79:2554-2558, 1982. [11] J. L. McClelland D. E. Rumelhart and the PDP Research Group. Parallel Distributed Processing: Explorations in the Microstructure of Cognition, vol. 1. MIT Press, 1986. [12] W. S. McCulloch and W. Pitts. A Logical Calculus of Ideas Immanent in Nervous Activity. Bulletin of Mathematical Biophysics, 5:115-133, 1943. [13] S. Muroga. The principle of majority decision logic elements and the complexity of their circuits. Inti. Con/. on Information Processing, Paris, France, June 1959. [14] K. Y. Siu and J. Bruck. Neural Computation of Arithmetic Functions. Proc. IEEE, 78, No. 10:1669-1675, October 1990. Special Issue on Neural Networks. [15] K.-Y. Siu and J. Bruck. On the Power of Threshold Circuits with Small Weights. SIAM J. Discrete Math., 4(3):423-435, August 1991. [16] K.-Y. Siu, J. Bruck, T. Kailath, and T. Hofmeister. Depth-Efficient Neural Networks for Division and Related Problems . to appear in IEEE Trans. Information Theory, 1993. [17] K.-Y. Siu and V. Roychowdhury. On Optimal Depth Threshold Circuits for Mulitplication and Related Problems. to appear in SIAM J. Discrete Math. [18] A. Yao. Separating the polynomial-time hierarchy by oracles. IEEE Symp. Found. Compo Sci., pages 1-10, 1985.
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Time Warping Invariant Neural Networks Guo-Zheng Sun, Hsing-Hen Chen and Yee-Chun Lee Institute for Advanced Computer Studies and Laboratory for Plasma Research, University of Maryland College Park, MD 20742 Abstract We proposed a model of Time Warping Invariant Neural Networks (TWINN) to handle the time warped continuous signals. Although TWINN is a simple modification of well known recurrent neural network, analysis has shown that TWINN completely removes time warping and is able to handle difficult classification problem. It is also shown that TWINN has certain advantages over the current available sequential processing schemes: Dynamic Programming(DP)[I], Hidden Markov Model(HMM)[2], Time Delayed Neural Networks(TDNN) [3] and Neural Network Finite Automata(NNFA)[4]. We also analyzed the time continuity employed in TWINN and pointed out that this kind of structure can memorize longer input history compared with Neural Network Finite Automata (NNFA). This may help to understand the well accepted fact that for learning grammatical reference with NNF A one had to start with very short strings in training set. The numerical example we used is a trajectory classification problem. This problem, making a feature of variable sampling rates, having internal states, continuous dynamics, heavily time-warped data and deformed phase space trajectories, is shown to be difficult to other schemes. With TWINN this problem has been learned in 100 iterations. For benchmark we also trained the exact same problem with TDNN and completely failed as expected. I. INTRODUCTION In dealing with the temporal pattern classification or recognition, time warping of input signals is one of the difficult problems we often encounter. Although there are a number of schemes available to handle time warping, e.g. Dynamic Programming (DP) and Hidden Markov Model(HMM), these schemes also have their own shortcomings in certain aspects. More depressing is that, as far as we know, there are no efficient neural network schemes to handle time warping. In this paper we proposed a model of Time Warping Invariant Neural Networks (TWINN) as a solution. Although TWINN is only a simple modification to the well known neural net structure, analysis shows that TWINN has the built-in ability to remove time warping completely. The basic idea ofTWINN is straightforward. If one plots the state trajectories of a continuous 180 Time Warping Invariant Neural Networks 181 dynamical system in its phase space, these trajectory curves are independent of time warping because time warping can only change the time duration when traveling along these trajectories and does not affect their shapes and structures. Therefore, if we normalize the time dependence of the state variables with respect to any phase space variable, say the length of trajectory, the neural network dynamics becomes time warping invariant. To illustrate the power of the TWINN we tested it with a numerical example of trajectory classification. This problem, chosen as a typical problem that the TWINN could handle, has the following properties: (1). The input signals obey a continuous time dynamics and are sampled with various sampling rates. (2). The dynamics of the de-warped signals has internal states. (3). The temporal patterns consist of severely time warped signals. To our knowledge there have not been any neural network schemes which can deal with this case effectively. We tested it with TDNN and failed to learn. In the next section we will introduce the TWINN and prove its time warping invariance. In Section III we analyze its features and identify the advantages over other schemes. The numerical example of the trajectory classification with TWINN is presented in Section IV. II. TIME WARPING INVARIANT NEURAL NETWORKS (TWINN) To process temporal signals, we consider a fully recurrent network, which consists of two groups of neurons: the state neurons (or recurrent units) represented by vector S(t) and the input neurons that are clamped to the external input signals {I(t), t = 0, I, 2, ...... , T-l). The Time Warping Invariant Neural Networks (TWINN) is simply defined as: S(t+ 1) = S(t) +1(t)F(S(t), W,/(t» (1) where W is the weight matrix, [(t) is the distance between two consecutive input vectors defined by the norm l(t) = 11/(t+ 1) -/(t) II (2) and the mapping function F is a nonlinear function usually referred as neural activity function. For example of first order networks, it could take the form: Fj(S(t), W,/(t» = Tanh(~Wij(S(t) EfH(t») J (3) where Tanh(x) is Hyperbolic Tangent function and symbol Ef> stands for the vector concatenation. For the purpose of classification (or recognition), we assign the target final state Sk> (k= 1,2,3, ... K), for each category of patterns. After we feed into the TWINN the whole sequence {J(O), 1(1), 1(2), ...... ,/(T-l)}, the state vector S(t) will reach the final state SeT). We then need to compare S(n with the target final state Sk for each category k, (k=I,2,3, ... K), and calculate the error: (4) The one with minimal error will be classified as such. The ideal error is zero. For the purpose of training, we are given a set of training examples for each category. We then minimize the error functions given by Eq. (4) using either back-propagation[7] or forward propagation algorithm[8]. The training process can be terminated when the total error reach its minimum. The formula of TWINN as shown in Eq. (1) does not look like new. The subtle difference from wildly used models is the introduction of normalization factor let) as in Eq. (1). The main advantage by doing this lies in its built-in time warping ability. This can be directly seen from its continuous version. As Eq. (1) is the discrete implementation of continuous dynamics, we can easily convert it into a continuous version by replacing "t +1" by "t+~t" and let ~t --? O. By doing so, we get 182 Sun, Chen, and Lee . S(t+~t) -Set) dS 11m ------61-.01l/(t+M) -/(t) II - dL (5) where L is the input trajectory length, which can be expressed as an integral I L (t) = III ~~II dt o (6) or summation (as in discrete version) I L(t) = L II/(t+ 1) - ·/(t) II 1:=0 (7) For deterministic dynamics, the distance L(t) is a single-valued function. Therefore, we can make a unique mapping from t to L, TI: t --7 L, and any function of t can be transformed into a function of L in terms of this mapping. For instance, the input trajectory I(t) and the state trajectory Set) can be transformed into I(L) and S(L). By doing so, discrete dynamics of Eq. (1) becomes, in the continuous limit, ~~ = F (S (L), W, I (L) ) (8) It is obvious that there is no explicit time dependence in Eq. (8) and therefore the dynamics represented by Eq. (8) is time warping independent. To be more specific, if we draw the trajectory curves of l(t) and S(t) in their phase spaces respectively, these two curves would not be deformed if we only change the time duration when traveling along the curves. Therefore, if we generate several input sequences {J(t)} using different time warping functions and feed them into TWINN, represented by Eq. (8) or Eq. (1), the induced state dynamics of S(L) would be the same. Meanwhile, the final state is the solo criterion for classification. Therefore, any time warped signals would be classified by the TWINN as the same. This is the so called "time warping invariant". III. ANALYSIS OF TWINN VS. OTHER SCHEMES We emphasize two points in this section. First, we would analyze the advantages of the TWINN over the other neural network structures, like TDNN, and other mature and well known algorithms for time warping, such as HMM and Dynamics Programming. Second, we would analyze the memory capacity of input history for both the continuous dynamical networks as illustrated in Eq. (1) and its discrete companion, Neural Network Finite Automata used in grammatical inference by Liu [3], Sun [4] and Giles [5]. And, we will show by mathematical estimation that the continuity employed in TWINN increases the power of memorizing history compared with NNFA The Time Delayed Neural Networks (TDNN)[3] has been a useful neural network structure in processing temporal signals and achieves successes in several applications, e.g. speech recognition. The traditional neural network structures are either feedforward or recurrent. The TDNN is something in between. The power of TDNN is in its dynamic combination of the spatial processing (as in a feedforward net) and sequential processing (as in a recurrent net with short time memory). Therefore, the TDNN could detect the local features within each windowed frame and store their voting scores into the short time memory neurons, and then make a final decision at the end of input sequence. This technique is suitable for processing the temporal patterns where the classification is decided by the integration of local features. But, it could not handle the long time correlation across time frames like a state machine. It also does not tolerate time warping effectively. Each of time warped patterns will be treated as a new feature. ThereforG, TDNN would not be able to handle the numerical example given in this paper which has both the severe time warping and the internal states (long time correlation). The benchmark test has been performed and it proved our prediction. Actually, it can be seen later that in our examTime Warping Invariant Neural Networks 183 pIes, no matter which category they belong to, all windowed frames would contain similar local features, the simple integration of local features do not contribute directly to the final classification, rather the whole sinal history will decide the classification. As for the Dynamic Programming, it is to date the most efficient way to cope with time warping problem. The most impressing feature of dynamic programming is that it accomplishes a global search among all NN possible paths using only -0(N2) operations, where N is the length of the input time series and, of course, one operation here represents all calculations involved in evaluating the 'score" of one path. But, on the other hand this is not ideal. If we can do the time warping using recurrent network, the number of uperations will be reduced to -O(N). This is a dramatic saving. Another undesirable feature of current dynamic warping scheme is that the recognition or classification result heavily depends on the pre-selected template and therefore one may need a large number of templates for a better classification rate. By adding one or two template we actually double or triple the number of operations. Therefore, search for a neural network time warping scheme is a pressing task. Another available technique for time warping is Hidden Markov Model (HMM), which has been successfully applied in speech recognition. The way for HMM to deal with time warping is in terms of statistical behavior of its hidden state transition. Starting from one state qj, HMM allows a certain probability ~j to forward to another state qj. Therefore, for any given HMM one could generate various state sequences, say, qlq2q2q3q4q4qS' QlQ2Q2Q2q3Q3q4q4qS' etc., each with a certain occurrence probability. But, these state sequences are "hidden", the observed part is a set of speech data or symbol represented by {Sk} for example. HMM also includes a set of observation probability B=={bjk}, so that when it is in a certain state, say Qj' HMM allows each symbol from the set {sk} to occur with the probability bjk. Therefore, for any state sequence one can generate various series of symbols. As an example, let us consider one simple way to generate symbols: in state Qj we generate symbol Sj (with probability bjj). By doing so, the two state sequences mentioned above would correspond to two possible symbol sequences: sl s2s2s3s4s4sS and sl s2s2s2s3s3s4S4sS' Examining the two strings closely, we find that the second one may be considered as the time warped version of the first one, or vice versa. If we present these two strings to the HMM for testing, it will accept them with similar probabilities. This is the way that HMM tolerates time warping. And, these state transition probabilities of HMM are learned from the statistics of training set by using re-estimation formula. In this sense, HMM does not deal with time warping directly, instead, it learns statistical distribution of training set which contains time warped patterns. Consequently, if one presents a test pattern with time warped signals which is far away from the statistical distribution of training set, it is very unlikely for a HMM to recognize this pattern. On the contrary, the model of TWINN we proposed here has intrinsic built-in time warping nature. Although the TWINN itself has internal states, these internal states are not used for tolerating time warping. Instead, they are used to learn more complex behavior of the "de-warped" trajectories. In this sense, TWINN could be more powerful than HMM. Another feature ofTWINN needs be mention is its explicit expression of continuous mapping from S(t) to S(t+1) as shown in Eq. (1). In our early work of [4,5,6], to train a NNFA (Neural Network Finite Automaton), we used a discrete mapping S(t+ 1) = F(S(t), W,/(t» (9) where F is a nonlinear function, say Sigmoid function g(x) == 1 l(l+e' X). This model has been successfully applied into the grammatical inference. The reason we call Eq. (1) a continuous mapping but Eq. (9) a discrete one, even though both of them are implemented in discrete time steps, is because there is an explicit infinitesimal factor let) used in Eq. (1). Due to this factor the continuous state dynamics is guaranteed, by which we mean that the state variation S(t+ I) - S(t+1) approaches zero if the input variation 1(t+l) -1{t+I) does so. But, In general, the state 184 Sun, Chen, and Lee variation S(t+ 1) - S(t+ 1) generated by Eq. (9) is of order of one, regardless of what input variations are. If one starts from random initial weights, Eq. (9) provides a discrete jump between different, randomly distributed states, which is far away from any continuous dynamics. We did numerical test using NNFA of Eq. (9) to learn the classification problem of continuous trajectories as shown in Section V. For simplicity we did not include time warping, but the NNFA still failed to learn. The reason is that when we tried to train a NNF A to learning the continuous dynamics, we were actually forcing the weights to generate an almost identical mapping F from Set) to S(t+ 1). This is a very strong constrain on the weight parameters, such that it drives the diagonal terms to positive infinity and off-diagonal terms to negative infinity (Sigmoid function is used). When this happens, the learning is stuck due to the saturation effect. The failure of NNF A may also comes from the short history memory capacity compared to the continuous mapping ofEq. (1). It has been shown by many numerical experiments on grammatical inference [3,4,5] that to train an NNFA as in Eq. (9) effectively, one has to start with short training patterns (usually, the sentence length ~ 4). Otherwise, learning will fail or be very slow. This is exactly what happened to learning the trajectory classification using NNFA, where the lengths of our training patterns are in general considerably long (normally,- 60). But, TWINN learned it easily. To understand the NNFA's failure and TWINN's success, in the following, we will analyze how the history information enters the learning process. Consider the example of learning grammatical inference. Before training since we have no (I priori knowledge about the target values of weights, we normally start with random initial values. On the other hand, during training the credit assignment (or the weight correction ~ W) can only be done at the end of each input sequence. Consequently, each ~W should explicitly contain the information about all symbols contained in that string, otherwise the learning is meaningless. But, in numerical implementation, every variable, including both ~W and W, has a finite precision and any information beyond the precision range will be lost. Therefore, to compare which model has the longer history memory we need to examine how the history information relates to the finite precisions of ~ Wand W. Let us illustrate this point with a simple second-order connected fully recurrent network and write both Eq. (1) and Eq. (9) in a unified form S(t+l) =G,+l (10) such that Eq. (1) is represented by G' + I = S (1) + I (1) g (K (1) ) (11 ) and Eq. (9) is just G,+l = g(K(t)) (12) where K(t) is the weighted sum of concatenation of vectors Set) and /(t) Kj(t) = LWjj(S(t) EfH(t»j (13) j For a grammatical inference problem the error is calculated from the final state S(I) as E= (S(T)-Starget)2 (14) Learning is to minimize this error function. According to the standard error back-propagation scheme, the recurrent net can be viewed as a multi-layered net with identical weights between neurons at adjacent time step: w(t) = W, where w(t) is the "till layer" weights connecting input S(t-I) to output S(t). The total weight correction is the summation of all weight corrections at each layer. By using the gradient descent scheme one immediately has T T a E T aE aG I ~W= LOW(t) =-llLaW(t) =-llL aS(t) · aW(t) (15) 1=1 1=1 1=1 If we define new symbols: vector u(t), second-order tensor A(t) and third-order tensor B(t) as Time Warping Invariant Neural Networks 185 the weight correction can be simply written as T a G~+ I A .. (t) == a I IJ S.(t) J (16) ~W=-1\~U(t).B(t) (17) and the "error rate" u(t) can be back-propagated using the Derivative Chain Rule u (t) = u (t + 1) . A (t) t = 1, 2, ... , T - 1 ; (18) so that it is easy to have u(t) = u(n ·A(T-I) · A(T-2) · ... ·A(t) ==u(n'tJ~t~(t) t = 1,2, ... ,T-I; (19) First, let us examine the model ofNNFA in Eq. (9). Using Eqs. (12), (13) and (16), Ai/t) and Bijk(t) can be written as A .. (t) = g' (K. (t) ) W .. lj I ~ B··k(f) = a .. (S(t-I) El)/(t-I»k IJ IJ (20) where g'(x) == dg/dx = gO-g) is the derivative of Sigmoid function and 8ij is Kronecker delta function. If we substitute Bijk(t) into Eq. (17), ~Wbecomes a weighted sum of all input symbols {/(O), 1(1), 1(2), ...... J(T-I)}, each with different weighting factor u(t). Therefore, to guarantee that ~ W contain the information of all input symbols {/(O), 1(1), 1(2), ...... J(T-I)}, the ratio of lu(t)lmaxllu(t)lmin should be within the range of precision of ~W. This is the main point. The exact mathematical analysis has not been done, but from a rough estimate we can gain some good understanding. From Eq. (9), u(t) is a matrices product of Aij(t), and u(1) the coefficient of 1(0) contains the highest order product of Ai/t). The key point is that the coefficient ratio between the adjacent symbols: lu(t)"lu(t+l) is of the order of lAi/t)I, which is a small value, therefore the earlier symbol information could be lost from ~ W due to its finite precision. It can be shown that xg'(x) =x g(x)( l-g(x)< 0.25 for any real value of x. Then, we roughly have lAij(t)1 = Ig' Wijl = Ig(l-g)Wij 1< 0.25, if we assume the values of weights Wij to be order 1. Thus, the ratio R=lu(t)lmax"u(t)lmin is estimated as 1 R- IU(1)l/lu(nl-"p_ 1IA(f')1 <2-2. (T-l) (21) From Eq. (21) we see that if the input pattern length is T= lOwe need at least 2(T -1) == 18 bits computer memory to store weight variables (including u, W and ~ W). If T= 60, as in the trajectory classification problem, it requires at least 128 bit weight variables. This is why the NNFA Eq. (9) could not work. Similarly, for the dynamics of Eq. 0), we use Eqs. (11), (13) and (16), and obtain Aij(l) = 1+1(t) (g'(Kj(t»Wi} Bijk(l) = 1(1) (aij(S(t-l) GH(t-l»k) (22) From Eq. (22) we see that no matter how small the factor let) will be, lAi/!)1 remains a value of order of one, therefore the ratio R=lu(t)lmax"u(t)lmin which is estimated as a product of lAij(1)1 would be of order of one compared with result of discrete case as in Eq. (21).Therefore, the contributions from all {I(O), 1(1), 1(2), ...... J(T-I)} to the weight correction ~ Ware of the same order. This prevents the information loss during learning. IV NUMERICAL SIMULATION We demonstrate the power of TWINN with a trajectory classification problem. The three 2186 Sun, Chen, and Lee D trajectory equations are artificially given by (xCt) =sin(t+~)lsinCt)1 (xCt) =sin(O.5t+~)sin(1.5t) (xU) =sinU+~)sin(2t) (23) \y (t) = cos (t+~) I sin (t) I \y (t) = cos (O.5t +~) sin (1.5t) ~ (1) = cos (t +~) sin (2t) where ~ is a unifonnly distributed random parameter. When ~ is changed, these trajectories are distorted accordingly. Some examples (three for each class) are shown in Fig. I. Class 1 Class 2 Class 3 -G .7 !. -0.5 Fig.l PHASE SPACE TRAJECTORIES Three different shapes of 2-D trajecwry, each is shown in one column with three examples. Recurrent neural networks are trained to recognize the different shapes of trajectory. The trajectory data are the time series of two dimensional coordinate pairs {x(t), y(t)} sampled along three different types of curves in the phase space. The neural net dynamics of TWINN is (24) where we used 6 input neurons I = {I, x(t), y(t), .?(t), y2(t), x(t)y(t)} (normalized to norm = 1.0) and 4 (N=4) state neurons S ={ Sl' S2' S3' S4}. The neural network structure is shown in Fig. 2. Fig.2 Time Warping Invariant Neural Network for Trajectory Classification Fig.3 Time Delayed Neural Network for Trajectory ClassificatiolJ For training, we assign the desired final output for the three trajectory classes to be (1,0,0), Time Warping Invariant Neural Networks 187 (0,1,0) and (0,0,1) respectively. For recognition, each trajectory data sequence needs to be fed to the input neurons and the state neurons evolve according to the dynamics in Eq. (24). At the end of input series we check the last three state neurons and classify the input trajectory according to the "winner-take-all" rule. In each iteration of training we randomly picked up 150 deformed trajectories, 50 for each of the three categories, by choosing different values of ~ within O$~ $27t. To simulate time warping we randomly sampled the data by choosing the random time step ~t = 27trff along each trajectory, where r is a random number between 0 and 2 and the sampling rate T=60 for training patterns, and T=20 to 200 for testing patterns. Therefore, each training pattern is a time warped trajectory data with averaged length = 60. Using RTRL algorithm[8] to minimize the error function, after 100 iterations of training it converged to Mean Square Error of == 0.03. We tested the trained network with hundreds of randomly picked input sequences with different sampling rate (from 20/27t to 200127t) and different wrapping functions (non-uniform step length). All input trajectories are classified correctly. If the sampling rates are too large (>200) or too small( <20), some classification errors will occur. We test the same example with TDNN. See Fig.3 for its parameters. The top layer contains three output neurons for the three classes of trajectories. The classification rules, error function and training patterns are the same as those ofTWINN. After three days of training with DEC3100 Workstation the training error (MSE) approaches 0.5 and in testing the error rate is 70%. V. CONCLUSION We have proposed a model of Time Warping Invariant Neural Network to handle temporal pattern classification where the severely time warped and deformed data may occur. This model is shown to have built-in time warping ability. We have analyzed the properties ofTWINN and shown that for trajectory classification it has several advantages over other schemes: HMM, DP, TDNN and NNFA. We also numerically implemented the TWINN and trained a trajectory classification easily. This problem is shown by analysis to be difficult to other schemes. It has been trained with TDNN but failed. References [1] H.Sakoe and S. Chiba, "Dynamic Programming Algorithm Optimization for Spoken Word Recognition", IEEE Transactions on Acoustics Speech and Signal Processing, Vol. ASSP-26, pp.43-49, Feb. 1978. [2] L.R.Rabiner and B.H.Juang, "An Introduction to Hidden Markov Models", IEEE, ASSP Mag., Vol.3, No.1, pp. 4-16, 1986. [3]A. Weibel, T. Hanazawa, G. Hinton, K.shikano and K. Lang, "Phoneme Recognition Using Time-Delay Neural Networks", IEEE Transactions on Acoustics Speech and Signal Processing, March, 1989. [4]. Y.D. Liu, G.Z. Sun, H.H. Chen, c.L. Giles and Y.c. Lee, "Grammatic Inference and Neural Network State Machine", Proceedings of the International Joint Conference on Neural Networks, pp. 1-285, Washington D.C. (1990). [5]. G.Z. Sun, H.H. Chen, c.L. Giles, Y.c. Lee and D. Chen, "Connectionist Pushdown Automata that Learn Context-Free Grammars", Proceedings of the International Joint Conference on Neural networks, pp. 1-577, Washington D.C. (1990). [6]Giles, C.L., Sun, G.Z., Chen, H.H., Lee,Y.C., and Chen, D. (1990). "Higher Order Recurrent Networks & Grammatical Inference". Advances in Neurallnformation Processing Systems 2, D.S. Touretzky (editor), 380-386, Morgan Kaufmann, San Mateo, c.A. (7) [7] D.Rumelhart, G. Hinton, and R. Williams. "Learning internal representations by error propagation", In PDP: VoU MIT press 1986. P. Werbos, "Beyond Regression: New tools for prediction and analysis in the behavior sciences", Ph.D. thesis, Harvard university, 1974. [8] R. Williams and D. Zipser, "A learning algorithm for continually running fully recurrent neural networks", Neural Computation 1 (1989), pp.270-280.
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Deriving Receptive Fields Using An Optimal Encoding Criterion Ralph Linsker IBM T. J. Watson Research Center P. O. Box 218, Yorktown Heights, NY 10598 Abstract An information-theoretic optimization principle ('infomax') has previously been used for unsupervised learning of statistical regularities in an input ensemble. The principle states that the inputoutput mapping implemented by a processing stage should be chosen so as to maximize the average mutual information between input and output patterns, subject to constraints and in the presence of processing noise. In the present work I show how infomax, when applied to a class of nonlinear input-output mappings, can under certain conditions generate optimal filters that have additional useful properties: (1) Output activity (for each input pattern) tends to be concentrated among a relatively small number of nodes. (2) The filters are sensitive to higher-order statistical structure (beyond pairwise correlations). If the input features are localized, the filters' receptive fields tend to be localized as well. (3) Multiresolution sets of filters with subsampling at low spatial frequencies - related to pyramid coding and wavelet representations - emerge as favored solutions for certain types of input ensembles. 1 INTRODUCTION In unsupervised network learning, the development of the connection weights is influenced by statistical properties of the ensemble of input vectors, rather than by the degree of mismatch between the network's output and some 'desired' output. An implicit goal of such learning is that the network should transform the input so that salient features present in the input are represented at the output in a 953 954 Linsker more useful form. This is often done by reducing the input dimensionality in a way that preserves the high-variance components of the input (e.g., principal component analysis, Kohonen feature maps). The principle of maximum information preservation ('infomax') is an unsupervised learning strategy that states (Linsker 1988): From a set of allowed input-output mappings (e.g., parametrized by the connection weights), choose a mapping that maximizes the (ensemble-averaged) Shannon information that the output vector conveys about the input vector, in the presence of noise. Such a mapping maximizes the ensemble-averaged mutual information (MI) between input and output. This paper (a) summarize earlier results on info max solutions for linear networks, (b) identifies some limitations of these solutions (ways in which very different filter sets are equally optimal from the infomax standpoint), and (c) shows how, by adding a small nonlinearity to the network, one can remove these limitations and at the same time improve the utility of the output representations. We show that infomax, acting on the modified network, tends to favor sparsely coded representations and (depending on the input ensemble) sets of filters that span multiple resolution scales (related to wavelets and 'pyramid coding'). 2 INFOMAX IN LINEAR NETWORKS For definiteness and brevity, we consider a linear network having a particular type of noise model and input statistical properties. For a more detailed discussion of related models see (Linsker 1989). Since the computation of the MI (which involves the output entropy) is in general intractable for continuous-valued output vectors, previous work (and the present paper) makes use of a surrogate MI, which we will call the 'as-if-Gaussian' MI. This quantity is, by definition, computed as though the output vectors comprised a multivariate Gaussian distribution having the same mean and covariance as the actual distribution of output vectors. Although expedient, this substitution has lacked a principled justification. The Appendix shows that, under certain conditions, using this 'surrogate MI' (and not the full MI) is indeed appropriate and justified. Denote the input vector by S = {Si} (Si is the activity at input node i), the output vector by Z = {Zn}, the matrix of connection weights by C = {Cni}, noise at the input nodes by N = {Nd, and noise at the output nodes by v = {vn }. Then our processing model is, in matrix form, Z = C(S + N) + v. Assume that N and v are Gaussian random variables, (S) = (N) = (v) = 0, \fNT) = (SvT) = (NvT) = 0, and, for the covariance matrices, (SsT) = Q, (N N ) = TJI, (vvT) = {3I'. (Angle brackets denote an ensemble average, superscript T denotes transpose, and I and I' denote unit matrices on the input and output spaces, respectively.) In general, MI = Hz - (Hzls) where Hz is the output entropy and HZls is the entropy of the output for given S. Replacing MI by the 'as-if-Gaussian' MI means replacing Hz by the expression for the entropy of a multivariate Gaussian distribution, which is (apart from an irrelevant constant term) H~ = (1/2) lndet Q', where Q' = (ZZT) = CQCT + TJCCT + (3I' is the output covariance. Note that, when S is fixed, Z = CS+(CN +v) is a Gaussian distribution centered on CS, so that we have (Hzls) = (1/2)lndetQII where Q" = ((CN + v)(CN + v)T) = TJCCT + {3I'. Therefore the Deriving Receptive Fields Using An Optimal Encoding Criterion 955 'as-if-Gaussian' MI is MI' = (1/2)[lndetQ' -lndetQ"]. (1) The variance of the output at node n (prior to adding noise lin) is Vn = ([C(S + N)]~) = (CQCT + TJCCT)nn. We will constrain the dynamic range of each output node (limiting the number of output values that can be discriminated from one another in the presence of output noise) by requiring that Vn = 1 for each n. Subject to this constra.int, we are to find a matrix C that maximizes MI'. For a local Hebbian algorithm that accomplishes this maximization, see (Linsker 1992). Here, in order to proceed analytically, we consider a special case of interest. Suppose that the input statistics are shift-invariant, so that the covariance (SiSj) is a function of (j - i). We then use a shift-invariant filter Ansatz, Cni = C(i - n). Infomax then determines the optimal filter gain as a function of spatial frequency; i.e., the magnitude of the Fourier components c(k) of C(i - n). The derivation is summarized below. Denote by q(k), q'(k), and q"(k) the Fourier transforms of QU - i), Q'(m - n), and Q"(m - n) respectively. Since Q' = CQCT + TJccf1' + (3I', therefore q'(k) = [q(k) + TJ) I c(k) 12 +(3. Similarly, q"(k) = TJ I c(k) 12 +(3. We obtain MI' = (1/2)~dlnq'(k) - Inq"(k)]. Each node's output variance Vn is equal to V = (I/K)~dq(k)+TJ] I c(k) 12 where K is the number of terms in the sum over k. To maximize MI' subject to the constraint on V we use the Lagrange multiplier method; that is, we maximize MI" = MI' + J.L(V - 1) with respect to each I c(k) 12. !his yields an equation for each k that is quadratic in I c(k) 12. The unique solution IS 2_ q(k) 2TJK 1/2 (TJ/{3) I c(k) I - -1 + 2[q(k) + TJ]{1 + [1- J.L{3q(k)] } (2) if the RHS is positive, and zero otherwise. The Lagrange multiplier J.L( < 0) is chosen so that the {I c( k) I} satisfy V = 1. Starting from a differently-stated goal (that of reducing redundancy subject to a limit on information loss), which turns out to be closely related to infomax, (Atick & Redlich 1990a) found an expression for the optimal filter gain that is the same as that of Eq. 2 except for the choice of constraint. Filter properties found using this approach are related to those found in early stages of biological sensory processing. Smoothing and bandpass (contrast-enhancing) filters emerge as infomax solutions (Linsker 1989, Atick & Redlich 1990a) in certain cases, and good agreement with retinal contrast sensitivity measurements has been found (Atick & Redlich 1990b). Nonetheless, the value of the infomax solution Eq. 2 is limited in two important ways. First, the phases of the {c(k)} are left undetermined. Any choice of phases is equally good at maximizing MI' in a linear network. Thus the real-space response function C(i - n), which determines the receptive field properties of the output nodes, is nonunique (and indeed may be highly nonlocalized in space). Second, it is useful to extend the solution Ansatz to allow a number of different filter types a = 1, ... , A at each output site, while continuing to require that each type 956 Linsker satisfy the shift-invariance condition Cni(a) == C(i - n;a). For example, one may want to model a topographic 'retinocortical' mapping in which each patch of cortex (each 'site') contains multiple filter types, yet each patch carries out the same set of processing functions on its input. For this Ansatz, one again obtains Eq. 2 (derivation omitted here), but with 1 c(k) 12 on the LHS replaced by ~ap(a)1 c(k; a) 12, where c(k; a) is the F.T. of C(i - n; a), and p(a) is the fraction of the total number of filters (at each site) that are of type a. The partitioning of the overall (sumsquared) gain among the multiple filter types is thus left undetermined. The higher-order statistical structure of the input (beyond covariance) is not being exploited by infomaxin the above analysis, because (1) the network is linear and (2) only pairwise correlations among the output activities enter into MI'. We shall show that if we make the network even mildly nonlinear, MI' is no longer independent of the choice of phases or of the partitioning of gain among multiple filter types. 3 NETWORK WITH WEAK NONLINEARITY We consider the weakly nonlinear input-output relation Zn = Un + fU~ + ~iCniNi + lin, where Un = ~iCniSi, for small f. This differs from the linear network analyzed above by the term in U~. (For simplicity, terms nonlinear in the noise are not included.) The cubic term increases the signal-to-noise ratio selectively when Un is large in absolute value. We maximize MI' as defined in Eq. l. Heuristically, the new term will cause infomax to favor solutions in which some output nodes have large (absolute) activity values, over solutions in which all output nodes have moderate activities. The output layer can thus encode information about the input vector (e.g., signal the presence of a feature) via the high activity of a small number of nodes, rather than via the particular activity values of many nodes. This has several (interrelated) potential advantages. (1) The concentration of activity among fewer nodes is a type of sparse coding. (2) The resulting output representation may be more resistant to noise. (3) The presence of a feature can be signaled to a later processing stage using fewer connections. (4) Since the particular nodes that have high activity depend upon the input vector, this type of mapping transforms a set of continuous-valued inputs at each site into a partially place-coded representation. A model of this sort may thus be useful for understanding better the formation of place-coded representations in biological systems. 3.1 MATHEMATICAL DETAILS This section may be skipped without loss of continuity. In matrix form, U CS, Wn U~ for each n, and Z = U + f W + C N + II. Keeping terms through first order in f, the output covariance is Q' = {ZZT} = CQCT + 1]CCT + (3I' + fF, where F = {WUT)+{UWT} . [As an aside, Fnm = (UnUm(U~+U~)} resembles the covariance (UnUm), except that presentations having large U~ +U~ are given greater weight in the ensemble average.] For shift-invariant input statistics and one filter type Cni = C( i-n), taking the Fourier transform yields q'(k) = [q(k)+1]]1 c(k; a) 12+ (3 + ff(k) where f(k) is the F.T. of F(m - n) = Fnm. So In det Q' = ~k lnq'(k) = ~ln{[q(k)+1]] 1 c(k) 12 +(3}+f~g(k) where g(k) _ [f(k)/{[q(k) +1]]1 c(k;a) 12+{3}]. Using a Lagrange multiplier as before, the quantity to be maximized is MI" = Deriving Receptive Fields Using An Optimal Encoding Criterion 957 ~~J[~ ~~ -+-11~ Figure 1: Breaking of phase degeneracy. See text for discussion. MI"(e = 0) + (e/2)~g(k). Now suppose there are multiple filter types a = 1, ... , A at each output site. For each k define d(k) to be the A x A matrix whose elements are: d(k)a.b = [q(k) + TJ]c(k; a)c*(k; b) + [f3/ p(a)]c5a.b where c5a.b is the Kronecker delta. Also define f(k) to be the A x A matrix each of whose elements f(k)ab is the F.T. of F(m - n;a,b) where F(m - n; a, b) = (Un(a)Wm(b)) + (Wn(a)Um(b)). Then the O{e) part of MI" is: (e/2)~kTr{[d(k)]-lf(k)}. Note that [d(k)]-l is the inverse of the matrix d(k), and that 'Tr' denotes the trace. [Outline of derivation: In the basis defined by the Fourier harmonics, Q' is block diagonal (one A x A block for each k). So In det Q' = ~k Indetq'(k) where each q'(k) is an A x A matrix of the form q~(k) + eq~(k). Expanding In det q' (k) through O( e) yields the stated result.] The infomax calculation to lowest order in e [i.e., O( eO)] is the same as for the linear network. Here, for simplicity, we determine the sum-squared gain, ~ap(a)1 c(k; a) 12, as in the linear case; then seek to maximize the new term, of O(e), subject to this constraint on the value of the sum-squared gain. How the nonlinear term breaks phase and gain-apportionment degeneracies is of interest here; a small O( e) correction to the sum-squared gain is not. 4 ILLUSTRATIVE RESULTS Two examples will show how adding the nonlinear perturbative term to the network's output breaks a degeneracy among different filter solutions. In each case the input space is a one-dimensional 'retina' with wraparound. 4.1 BREAKING THE PHASE DEGENERACY In this example (see Figure 1) there is one filter type at each output site. We consider two types of input ensembles: (1) Each input vector (Fig. la shows one example) is drawn from a multivariate Gaussian distribution (so there is no higher-order statistical structure beyond pairwise correlations). The input covariance matrix Q(j - i) is a Gaussian function of the distance between the sites. (2) Each input 958 Linsker vector is a random sum of Gaussian 'bumps': Si = Ej aj [s( i - j) - so] where s( i - j) is a Gaussian (shown in Fig. 1b for j=20j there are 64 nodes in all); So is the mean value of s( i - j); and each aj is independently and randomly chosen (with constant probability) to be 1 or O. This ensemble does have higher-order structure, with each input presentation being characterized by the presence of localized features (the bumps) at particular locations. The infomax solution for I c(k) 12 is plotted versus spatial frequency k in Fig. 1c for a particular choice of noise parameters (1], (3). As stated earlier, MI' for a linear network is indifferent to the phases of the Fourier components {c(k)}. A particular random choice of phases produces the real-space filter C(i - n) shown in Fig. Id, which spans the entire 'retina.' Setting all phases to zero produces the localized filter shown in Fig. 1£. If the Gaussian 'bump' of Fig. 1b is presented as input to a network of filters each of which is a shifted version of Fig. 1d, the linear response of the network (i.e., the convolution of the 'bump' with the filter) is shown in Fig. Ie. Replacing the filter of Fig. 1d by that of Fig. 1£, but keeping the input the same, produces the output response shown in Fig. 19. The cubic nonlinearity causes MI' to be larger for the filter of Fig. 1£ than for that of Fig. 1d. Heuristically, if we focus on the diagonal elements of the output covariance Q', the nonlinear term is 2€(U~). Maximizing MI' favors increasing this term (subject to a constraint on output variance) hence favors filter solutions for which the Un distribution is non-Gaussian with a preponderance of large values. Projection pursuit methods also use a measure of the non-Gaussianity of the output distribution to construct filters that extract 'interesting' features from high-dimensional data (cf. Intrator 1992). 4.2 BREAKING THE PARTITIONING DEGENERACY FOR MULTIPLE FILTER TYPES In this example (see Fig. 2), the input ensemble comprises a set of self-similar patterns (each is a sine-Gabor 'ripple' as in Fig. 2a) that are related by translation and dilation (scale change over a factor of 80). Figure 2b shows the input power spectrum vs. k; the scaling region goes as 11k. Figure 2c shows the infomax solution for the gain I c(k;a) I vs. k when there is just one filter type. When the input SNR is large (as in the scaling region) the infomax filters 'whiten' the output; note the flat portion of the output power spectrum (Fig. 2d). [We modify the infomax solution by extending the power-law form of I c(k) I to low k (dotted line in Figs. 2c,d). This avoids artifacts resulting from the rapid increase in I c(k) I, which is in turn caused by our having omitted low-k patterns from the input ensemble for reasons of numerical efficiency.] The dotted envelope curve in Figure 2e shows the sum-squared gain Eap(a) I c(k) 12 when multiple filter types a are allowed. The quantity plotted is just the square of that shown in Fig. 2c, but on a linear rather than log-log plot (note values greater than 5 are cut off to save space). The network nonlinearity has the following effect. We first allow two filter types to share the overall gain. Optimizing MI' over various partitionings, we find that info max favors a crossover between filter types at k ~ 400. Allowing three, then four, filter types produces additional crossovers at lower k. For an Ansatz in which each filter's share of the sum-squared gain is tapered linearly near its cutoff frequencies, Deriving Receptive Fields Using An Optimal Encoding Criterion 959 the best solution found for each p( a) 1 c( k) 12 is shown in Fig. 2e (semilog plot vs. k ). Figure 2f plots the corresponding 1 c( k; a) 1 vs. k on a linear scale. Note that the three lower-k filters appear roughly self-similar. (The peak in the highest-k filter is an artifact due to the cutoff of the input ensemble at high k.) The four real-space filters C(i-n; a) are plotted vs. (i-n) in Fig. 2g [phases chosen to make C(i-n; a) antisymmetric] . The resulting filters span multiple resolution scales. The density p( a) is less for the lower-frequency filters (spatial subsampling). When more filter types are allowed, the increase in MI' becomes progressively less. Although in our model the filters are present with density p at each output site, a similar MIl is obtained if one spaces adjacent filters of type a by a distance <X 1/ p( a). The resulting arrangement of filters resembles the 'tiling' of the joint space and spatial-frequency domain that is used in wavelet and 'pyramid coding' approaches to image processing. [The infomax filters overlap, rather than disjointly tiling the (:I;, k) domain.] Using the infomax method, the region of (:1;, k) space spanned by an optimal filter has an aspect ratio that depends upon the relative distances - along the :I; and k axes - over which the input feature is 'coherent' (possesses higher-order correlations). One may thus be able to use infomax to predict relationships between statistical measures of coherence in natural scenes and observed (:1;, k) aspect ratios for, e.g., orientation-selective cells. See (Field 1989) for a discussion of this issue that is not based on infomax. 5 APPENDIX: HEURISTIC JUSTIFICATION FOR USING A SURROGATE, 'AS-IF-GAUSSIAN,' MUTUAL INFORMATION The mutual information between input S and output Z is MI !dSdZPszln(Psz/PsPz) !dSPsKD(Pzls;Pz) where KD(Pzls;PZ) ! dZPzlsln(Pzls/PZ) is a Kullback divergence. So, maximizing MI means maximizing the average (over S) of KD(Pzls; Pz). What does the KD represent? Suppose that the network has somehow learned the distribution Pz. Before being presented with a particular input S, the network 'expects' an output vector drawn from Pz. The actual output response to S, however, is a vector drawn from PZls. The KD measures the 'surprise' (i.e., the amount of information gained) upon seeing the actual distribution PZls when one expected Pz. Infomax maximizes this average 'surprise.' However, the network cannot in general have access to the full distribution Pz, which contains far too much information (including all higher-order statistics) to be stored in the connections and nodes of the network. Let us suppose for definiteness that the system remembers only the mean and the covariance matrix of Z. Define pff to be the multivariate Gaussian distribution that has the same mean and covariance as Pz. Then we may think of the system as a priori 'expecting' the output vector to be drawn from the distribution pff. We accordingly modify the principle so that we maximize the average (over S) of KD(PzIS; pff) (note the superscript G). This equals 960 Linsker 10 (d) 0.1 I 10 100 1000 1000 100 10 10 100 1000 o __~ ,...=;; ' 0.1 10 100 1000 (9) = 80 Figure 2: Partitioning among multiple filter types. See text. J dSPs J dZ PZls In( PZ1s / pi) = (-H zls) s - J dZ Pz In pi (where H denotes entropy). Using a property of the Gaussian distribution, we have - J dZPz In pi = - J dZpi InPff = He;. We conclude that the average of KD equals He; - (HZls)s, which is exactly equal to the surrogate 'as-if-Gaussian' MI defined preceding Eq. 1. This argument provides a principled justification for using the surrogate MI, when the system has stored information about the output vectors' mean and covariance, but not about higher-order statistics. References J. J. Atick & A. N. Redlich. (1990a) Towards a theory of early visual processing. Neural Computation 2:308-320. J. J. Atick & A. N. Redlich. (1990b) Quantitative tests of a theory of retinal processing: contrast sensitivity curves. Inst. Adv. Study IASSNS-HEP-90/51. D. J. Field. (1989) What the statistics of natural images tell us about visual coding. In Proc. SPIE 1077:269-276. N.lntrator. (1992) Feature extraction using an unsupervised neural network. Neural Computation 4:98-107. R. Linsker. (1988) Self-organization in a perceptual network. Computer 21(3):105117. R. Linsker. (1989) An application of the principle of maximum information preservation to linear systems. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 1, 186-194. San Mateo, CA: Morgan Kaufmann. R. Linsker. (1992) Local synaptic learning rules suffice to maximize mutual information in a linear network. Neural Computation 4(5 ):691-702.
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Connected Letter Recognition with a Multi-State Time Delay Neural Network Hermann Hild and Alex Waibel School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891, USA Abstract The Multi-State Time Delay Neural Network (MS-TDNN) integrates a nonlinear time alignment procedure (DTW) and the highaccuracy phoneme spotting capabilities of a TDNN into a connectionist speech recognition system with word-level classification and error backpropagation. We present an MS-TDNN for recognizing continuously spelled letters, a task characterized by a small but highly confusable vocabulary. Our MS-TDNN achieves 98.5/92.0% word accuracy on speaker dependent/independent tasks, outperforming previously reported results on the same databases. We propose training techniques aimed at improving sentence level performance, including free alignment across word boundaries, word duration modeling and error backpropagation on the sentence rather than the word level. Architectures integrating submodules specialized on a subset of speakers achieved further improvements. 1 INTRODUCTION The recognition of spelled strings of letters is essential for all applications involving proper names, addresses or other large sets of special words which due to their sheer size can not be in the basic vocabulary of a recognizer. The high confusability of the English letters makes the seemingly easy task a very challenging one, currently only addressed by a few systems, e.g. those of R. Cole et. al. [JFC90, FC90, CFGJ91] for isolated spoken letter recognition. Their connectionist systems first find a broad phonetic segmentation, from which a letter segmentation is derived, which is then 712 unity Connected Letter Recognition with a Multi-State Time Delay Neural Network 713 • A Sil ----..... Word Layer 27 word units (only 4 shown) DTW Layer 27 word templates (only 4 shown) Phoneme Layer 59 phoneme units (only 9 shown) Hidden Layer 15 hidden units Input Layer 16 melscale FFT coefficients Figure 1: The MS-TDNN recognizing the excerpted word 'B'. Only the activations for the words 'SIL', 'A', 'B', and 'c' are shown. classified by another network. In this paper, we present the MS-TDNN as a connectionist speech recognition system for connected letter recognition. After describing the baseline architecture, training techniques aimed at improving sentence level performance and architectures with gender-specific sub nets are introduced. Baseline Architecture. Time Delay Neural Networks (TDNNs) can combine the robustness and discriminative power of Neural Nets with a time-shift invariant architecture to form high accuracy phoneme classifiers [WHH+S9]. The Multi-State TDNN (MS-TDNN) [HFW91, Haf92, HW92], an extension of the TDNN, is capable of classifying words (represen ted as sequences of phonemes) by integrating a nonlinear time alignment procedure (DTW) into the TDNN architecture. Figure 1 shows an MS-TDNN in the process of recognizing the excerpted word 'B', represented by 16 melscale FFT coefficients at a 10-msec frame rate. The first three layers constitute a standard TDNN, which uses sliding windows with time delayed connections to compute a score for each phoneme (state) for every frame, these are the activations in the "Phoneme Layer". In the "DTW Layer" , each word to be recognized is modeled by a sequence of phonemes. The corresponding activations are simply 714 Hild and Waibel copied from the Phoneme Layer into the word models of the DTW Layer, where an optimal alignment path is found for each word. The activations along these paths are then collected in the word output units. All units in the DTW and Word Layer are linear and have no biases. 15 (25 to 100) hidden units per frame were used for speaker-dependent (-independent) experiments, the entire 26 letter network has approximately 5200 (8600 to 34500) parameters. Training starts with "bootstrapping", during which only the front-end TDNN is used with fixed phoneme boundaries as targets. In a second phase, training is performed with word level targets. Phoneme boundaries are freely aligned within given word boundaries in the DTW layer. The error derivatives are backpropagated from the word units through the alignment path and the front-end TDNN. The choice of sensible objective functions is of great importance. Let Y = (Yl, ... ,Yn) the output and T = (tl, ... , tn ) the target vector. For training on the phoneme level (bootstrapping), there is a target vector T for each frame in time, representing the correct phoneme j in a "1-out-of-n" coding, i.e. ti = Dij. To see why the standard Mean Square Error (MSE = l:?:l (Yi - ti)2) is problematic for "1-out-of-n" codings for large n (n = 59 in our case), consider for example that for a target (1.0,0.0, ... ,0.0) the output vector (0.0, ... ,0.0) has only half the error than the more desirable output (1.0,0.2, ... ,0.2). To avoid this problem, we are usmg n i=1 which (like cross entropy) punishes "outliers" with an error approaching infinity for Iti - yd approaching 1.0. For the word level training, we have achieved best results with an objective function similar to the "Classification Figure of Merit (CFM)" [HW90], which tries to maximize the distance d = Yc - Yhi between the correct score Yc and the highest incorrect score Yhi instead of using absolute target values of 1.0 and 0.0 for correct and incorrect word units: ECFM(T, Y) = f(yc - Yhd = f(d) = (1 - d)2 The philosophy here is not to "touch" any output unit not directly related to correct classification. We found it even useful to apply error backpropagation only in the case of a wrong or too narrow classification, i.e. if Yc Yhi < DllaJety...margin. 2 IMPROVING CONTINUOUS RECOGNITION 2.1 TRAINING ACROSS WORD BOUNDARIES A proper treatment of word1 boundaries is especially important for a short word vocabulary, since most phones are at word boundaries. While the phoneme boundaries within a word are freely aligned by the DTW during "word level training" , the word boundaries are fixed and might be error prone or suboptimal. By extending the alignment one phoneme to the left (last phoneme of previous word) and the right (first phoneme of next word), the word boundaries can be optimally adjusted 1 In our context, a "word" consists of one spelled letter, and a "sentence" is a continuously spelled string of letters. Connected Letter Recognition with a Multi-State Time Delay Neural Network 715 B ~ c b 410 1 .. ~ :-1 .. : b .u p b u:~ ... 2 ~ • u 4 ~ .. . . . . : . . . . . . . word boundaries ~ word boundary found by free alignment (a) alignment across word boundaries B ~~~+---~----~~--~ probed) d (durat1.on) (b) duration dependent word penalties b b h" P B b ':~ all P b X>. ' A::!,~ .:. aU'~ . I-- word bound_ri •• ~ word boundary found by fr_ allgnlngnt (C) training on the sentence level Figure 2: Various techniques to improve sentence level recognition performance 716 Hild and Waibel in the same way as the phoneme boundaries within a word. Figure 2(a) shows an example in which the word to recognize is surrounded by a silence and a 'B', thus the left and right context (for all words to be recognized) is the phoneme 'sil' and 'b', respectively. The gray shaded area indicates the extension necessary to the DTW alignment. The diagram shows how a new boundary for the beginning of the word 'A' is found. As indicated in figure 3, this techniques improves continuous recognition significantly, but it doesn't help for excerpted words. 2.2 WORD DURATION DEPENDENT PENALIZING OF INSERTION AND DELETION ERRORS In "continuous testing mode" , instead of looking at word units the well-known "One Stage DTW" algorithm [Ney84) is used to find an optimal path through an unspecified sequence of words. The short and confusable English letters cause many word insertion and deletion errors, such as "T E" vs. ''T'' or "0" vs. "0 0", therefore proper duration modeling is essential. As suggested in [HW92], minimum phoneme duration can be enforced by "state duplication". In addition, we are modeling a duration and word dependent penalty Penw(d) = log(k + probw(d», where the pdf probw(d) is approximated from the training data and k is a small constant to avoid zero probabilities. Penw (d) is added to the accumulated score AS of the search path, AS = AS + Aw * Penw (d), whenever it crosses the boundary of a word w in Ney's "One Stage DTW" algorithm, as indicated in figure 2(b). The ratio Aw , which determines the degree of influence of the duration penalty, is another important degree of freedom. There is no straightforward mathematically exact way to compute the effect of a change of the "weight" Aw to the insertion and deletion rate. Our approach is a (pseudo) gradient descent, which changes Aw proportional to E(w) = (#insw - #delw)/#w, i.e. we are trying to maximize the relative balance of insertion and deletion errors. 2.3 ERROR BACKPOPAGATION AT THE SENTENCE LEVEL Usually the MS-TDNN is trained to classify excerpted words, but evaluated on continuously spoken sentences. We propose a simple but effective method to extend training on the sentence level. Figure 2( c) shows the alignment path of the sentence "e A B", in which a typical error, the insertion of an 'A', occurred. In a forced alignment mode (i.e. the correct sequence of words is enforced), positive training is applied along the correct path, while the units along the incorrect path receive negative training. Note that the effect of positive and negative training is neutralized if the paths are the same, only differing parts receive non-zero error backpropagation. 2.4 LEARNING CURVES Figure 3 demonstrates the effect of the various training phases. The system is bootstrapped (a) during iteration 1 to 130. Word level training starts (b) at iteration 110. Word level training with additional "training across word boundaries" (c) is started at iteration 260. Excerpted word performance is not improved after (c), but continuous recognition becomes significantly better, compare (d) and (e). In (d), sentence level training is started directly after iteration 260, while in (e) sentence level training is started after additional "across boundaries (word level) training". Connected Letter Recognition with a Multi-State Time Delay Neural Network 717 (~) ~( c) 95.0 90 . 0 ---~---~---~~-.-- - - ----.----- --~ -- --- - -- -· . . · . . . . 85.0 ~ . ~ - - - - ~ - - - -:- - - - i - - - -:- - - - -! - - - - - - -:- - - - ~ - - - - :- - - - -:- - - · . . . . . . . · . . . . . . . • , I I I . I • . . . .. . o 20 40 60 80 ~00120140160~80200220 2 402602803003 2 03403603804004 2 0 Figure 3: Learning curves (a = bootstrapping, b,c = word level (excerpted words), d,e = sentence level training (continuous speech)) on the training (0), crossvalidation (-) and test set (x) for the speaker-independent RM Spell-Mode data. 3 GENDER SPECIFIC SUBNETS A straightforward approach to building a more specialized system is simply to train two entirely individual networks for male and female speakers only. During training, the gender of a speaker is known, during testing it is determined by an additional "gender identification network" , which is simply another MS-TDNN with two output units representing male and female speakers. Given a sentence as input, this network classifies the speaker's gender with approx. 99% correct. The overall modularized network improved the word accuracy from 90.8% (for the "pooled" net, see table 1) to 91.3%. However, a hybrid approach with specialized gender-specific connections at the lower, input level and shared connections for the remaining net worked even better. As depicted in figure 4, in this architecture the gender identification network selects one of the two gender-specific bundles of connections between the input and hidden layer. This technique improved the word accuracy to 92.0%. More experiments with speaker-specific subnetworks are reported in [HW93] . 4 EXPERIMENTAL RESULTS Our MS-TDNN achieved excellent performance on both speaker dependent and independent tasks. For speaker dependent testing, we used the "eMU AlphData", with 1000 sentences (Le. a continuously spelled string of letters) from each of 3 male and 3 female speakers. 500, 100, and 400 sentences were used as train718 Hild and Waibel Phoneme Layer Hidden Layer Input Layer F F T Figure 4: A network architecture with gender-sp ecific and shared connections. Only the front-end TDNN is shown. ing, cross-validation and test set, respectively. The DARPA Resource Management Spell-Mode Data were used for speaker independent testing. This data base contains about 1700 sentences, spelled by 85 male and 35 female speakers. The speech of 7 male and 4 female speakers was set aside for the test set, one sentence from all 109 and all sentences from 6 training speakers were used for crossvalidation. Table 1 summarizes our results. With the help of the training techniques described above we were able to outperform previously reported [HFW91] speaker dependent results as well as the HMM-based SPHINX System. 5 SUMMARY AND FUTURE WORK We have presented a connectionist speech recognition system for high accuracy connected letter recognition. New training techniques aimed at improving sentence level recognition enabled our MS-TDNN to outperform previous systems of its own kind as well as a state-of-the art HMM-based system (SPHINX). Beyond the gender specific subnets, we are experimenting with an MS-TDNN which maintains several "internal speaker models" for a more sophisticated speaker-independent system. In the future we will also experiment with context dependent phoneme models. Acknowledgements The authors gratefully acknowledge support by the National Science Foundation and DARPA. We wish to thank Joe Tebelskis for insightful discussions, Arthur McNair for keeping our machines running, and especially Patrick Haffner. Many of the ideas presented have been developed in collaboration with him. References [CFGJ91] R. A. Cole, M. Fanty, Gopalakrishnan, and R. D.T. Janssen. SpeakerIndependent Name Retrival from Spellings Using a Database of 50,000 Names. Connected Letter Recognition with a Multi-State Time Delay Neural Network 719 Speaker Dependent (eMU Alph Data) 500/2500 train, 100/500 crossvalidation, 400/2000 test sentences/words speaker SPHINX[HFW91] MS-TDNN[HFW91] our MC:_'T'nNN rnjrnt 96.0 97.5 98.5 rndbs 83.9 89.7 91.1 rnaern 94.6 fcaw 98.8 flgt 86.9 fee 91.0 Speaker Independent (Resource Management Spell-Mode) 109 (ca. 11000) train, 11 (ca. 900) test speaker (words). SPHINX[HH92] our MS-TDNN + Sen one gender specific 88.7 90.4 90.8 92.0 Table 1: Word accuracy (in % on the test sets) on speaker dependent and speaker independent connected letter tasks. In Proceedings of the International Conference on Acoustics, Speech and Signal Processing, Toronto, Ontario, Canada, May 1991. IEEE. [FC90] M. Fanty and R. Cole. Spoken letter recognition. In Proceedings of the Neural Information Processing Systems Conference NIPS, Denver, November 1990. [Haf92] P. Haffner. Connectionist Word-Level Classification in Speech Recognition. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing. IEEE, 1992. [HFW91] P. Haffner, M. Franzini, and A. Waibel. Integrating Time Alignment and Neural Networks for High Performance Continuous Speech Recognition. In Proc. Int. Conf. on Acoustics, Speech, and Signal Processing. IEEE, 1991. [HH92] M.-Y. Hwang and X. Huang. Subphonetic Modeling with Markov States Senone. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, pages 133 - 137. IEEE, 1992. [HW90] J. Hampshire and A. Waibel. A Novel Objective Function for Improved Phoneme Recognition Using Time Delay Neural Networks. IEEE Transactions on Neural Networks, June 1990. [HW92] P. Haffner and A. Waibel. Multi-state Time Delay Neural Networks for Continuous Speech Recognition. In NIPS(4). Morgan Kaufman, 1992. [HW93] H. Hild and A. Waibel. Multi-SpeakerjSpeaker-Independent Architectures for the Multi-State Time Delay Neural Network. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing. IEEE, 1993. [JFC90] R.D.T. Jansen, M. Fanty, and R. A. Cole. Speaker-independent Phonetic Classification in Continuous English Letters. In Proceedings of the JJCNN 90, Washington D.C., July 1990. [Ney84] H. Ney. The Use of a One-Stage Dynamic Programming Algorithm for Connected Word Recognition. In IEEE Transactions on Acoustics, Speech, and Signal Processing, pages 263-271. IEEE, April 1984. [WHH+89] A. Waibel, T. Hanazawa, G. Hinton, K. Shikano, and K. Lang. Phoneme Recognition Using Time-Delay Neural Networks. IEEE, Transactions on Acoustics, Speech and Signal Processing, March 1989. PART IX ApPLICATIONS
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Input Reconstruction Reliability Estimation Dean A. Pomerleau School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract This paper describes a technique called Input Reconstruction Reliability Estimation (IRRE) for determining the response reliability of a restricted class of multi-layer perceptrons (MLPs). The technique uses a network's ability to accurately encode the input pattern in its internal representation as a measure of its reliability. The more accurately a network is able to reconstruct the input pattern from its internal representation, the more reliable the network is considered to be. IRRE is provides a good estimate of the reliability of MLPs trained for autonomous driving. Results are presented in which the reliability estimates provided by IRRE are used to select between networks trained for different driving situations. 1 Introduction In many real world domains it is important to know the reliability of a network's response since a single network cannot be expected to accurately handle all the possible inputs. Ideally, a network should not only provide a response to a given input pattern, but also an indication of the likelihood that its response is "correct". This reliability measure could be used to weight the outputs from multiple networks and to determine when a new network needs to be trained. This paper describes a technique for estimating a network's reliability called Input Reconstruction Reliability Estimation (IRRE). IRRE relies on the fact that the hidden representation developed by an artificial neural network can be considered to be a compressed representation of important input features. For example, when the network shown in Figure 1 is trained to produce the correct steering direction from images of the road ahead, the hidden units learn to encode the position and orientation of important features like the road edges and lane markers (See [pomerleau, 1991] for more details). Because there are many fewer hidden units than input units in the network, the hidden units cannot accurately represent all the details of an 279 280 Pomerleau SUp Ri&hI 30 Output Units 3Ox32 SelUlor Input Retina Figure 1: Original driving network architecture. arbitrary input pattern. Instead, the hidden units learn to devote their limited representational capabilities to encoding the position and orientation of consistent, frequently-occurring features from the training set. When presented with an atypical input, such as a road with a different number of lanes, the feature detectors developed by the hidden units will not be capable of accurately encode all the actual input features. Input Reconstruction Reliability Estimation exploits this limitation in representational capacity to estimate a network's reliability. In IRRE, the network's internal representation is used to reconstruct in the input pattern being presented. The more closely the reconstructed input matches the actual input, the more familiar the input and hence the more reliable the network's response. 2 Reconstructing the Input IRRE utilized an additional set of output units to perform input reconstruction called the encoder output array, as depicted in Figure 2. This second set of output units has the same dimensionality as the input retina. In the experiments described in this paper, the input layer and encoder output array have 30 rows and 32 columns. The desired activation for each of these additional output units is identical to the activation of the corresponding input unit. In essence, these additional output units turn the network into an autoencoder. The network is trained using backpropagation both to produce the correct steering response on the steering output units, and to reconstruct the input image as accurately as possible on the encoder output array. During the training process, the network is presented with several hundred images taken with a camera onboard our test vehicle as a person drives (See [pomerleau, 1991] for more details). Training typically requires approximately 3 minutes during which the person drives over a 1/4 to 1/2 mile stretch of road. Input Reconstruction Reliability Estimation 281 Figure 2: Network architecture augmented to include an encoder output array. During testing on a new stretch of road, images are presented to the network and activation is propagated forward through the network to produce a steering response and a reconstructed input image. The reliability of the steering response is estimated by computing the correlation coefficient p(l , R) between the activation levels of units in the actual input image I and the reconstructed input image R using the following formula: p(I ,R) = iR - I . II 01 OR where I and R are the mean activation value of the actual and the reconstructed images, IR is the mean of the set formed by the unit-wise product of the two images, and u/ and UR represent the standard deviations of the activation values of each image. The higher the correlation between the two images, the more reliable the network's response is estimated to be. The reason correlation is used to measure the degrees of match between the two images is that, unlike Euclidean distance, the correlation measure is invariant to differences in the mean and variance between the two images. This is important since the mean and variance of the input and the reconstructed images can sometimes vary, even when the input image depicts a familiar situation. 3 Results and Applications The degree of correlation between the actual and the reconstructed input images is an extremely good indicator of network response accuracy in the domain of autonomous driving, as shown in Figure 3. It shows a trained network's steering error and reconstruction error as the vehicle drives down a quarter mile stretch of road that starts out as a single lane path and eventually becomes a two-lane street. The solid line indicates the network's steering error, as measured by the difference in turn curvature between the network's steering response and a person's 282 Pomerleau 0.04 ~ g -;;- 0.03 ~~ ..... b.OV 'S e 0.02 v ........ v..... en 0.01 Steering Error CoJTelation Coeff"u:ienl = 0.92 ii.PU;Reoo~n&ror Intersection ® ~l. 1.25 ~ 1.00 ~ c::~ o~ 'a .. 0.75 ~b: jg~ 0.50 8-~ 000 O~ • L-__________________________________ ~ I ..... t__---- One Lane Road Im.gel----~ .. ~nll"' .. ---.r--t .. ~1 Two Lane Road Images Figure 3: Reconstruction error obtained using autoencoder reconstruction versus network steering error over a stretch of one-lane and two-lane road. steering response at that point along the road. The dashed line represents the network's "reconstruction error", which is defined to be the degree of statical independence between the actual and reconstructed images, or 1 - p(I , R}. The two curves are nearly identical, having a correlation coefficient of 0.92. This close match between the curves demonstrates that when the network is unable to accurately reconstruct the input image, it is also probably suggesting an incorrect steering direction. Visual inspection of the actual and reconstructed input images demonstrates that the degree of resemblance between them is a good indication of the actual input's familiarity, as shown in Figure 4. It depicts the input image, network response, and reconstructed input at the three points along the road,labeled A, Band C in Figure 3. When presented with the image at point A, which closely resembles patterns from training set, the network's reconstructed image closely resembles the actual input, as shown by the close correspondence between the images labeled "Input Acts" and "Reconstructed Input" in the left column of Figure 4. This close correspondence between the input and reconstructed images suggests that the network can reliably steer in this situation. It in fact it can steer accurately on this image, as demonstrated by the close match between the network's steering response labeled "Output Acts" and the desired steering response labeled ''Target Acts" in the upper left corner of Figure 4. When presented with a situation the network did not encounter during training, such as the fork image and the two-lane road image shown in the other two columns of Figure 4, the reconstructed image bears much less resemblance to the original input. This suggests that the network is confused. This confusion results in an incorrect steering response, illustrated in the discrepancy between the network's steering response and the target steering response for the two atypical images. The reliability prediction provided by IRRE has been used to improve the performance of the neural network based autonomous driving system in a number of ways. The simplest is Input Reconstruction Reliability Estimation 283 Figure 4: The actual input. the reconstructed input and the point-wise absolute difference between them on a road image similar to those in the training set (labeled A), and on two atypical images (labeled B and C). 284 Pomerleau One Lane Net Rcconstuction Error TW~ line-Net Rct;;n;tiiCtion Eir~; 1.00 0.80 0.60 0.40 1 .' ~ L i\ I !'i, I !iJ •• a I • t-' a :!I\ II I .. : v·, it ,,':: \.:1 i != \ f \ : f ~ "\ ." • II • II \ i . / \ ii I !i II I \.. • .: 1 II ';. 1 'V: \\! y ~ OJ \.J , (\ : \ . : \.-: : 0.20 ,. , , . ., : . = , : ~. J ....: '\i \_ •• .: O.OOL.-__________________ _ I.. One Lane Images ---11-- Two Lane Images -..j Figure 5: Reconstruction error of networks trained for one-lane road driving (solid line) and two-lane road driving (dashed line), to use IRRE to control vehicle speed. The more accurate the input reconstruction, the more confident the network, and hence the faster the system drives the vehicle. A second use the system makes of the reliability estimate provided by IRRE is to update the vehicle's position on a rough map of the area being driven over. When the map indicates there should be an intersection or other confusing situation up ahead, a subsequent sharp rise in reconstruction error is a good indication that the vehicle has actually reached that location, allowing the system to pinpoint the vehicle's position. Knowing the vehicle's location on a map is useful for integrating symbolic processing, such as planning, into the neural network driving system (for more details, see [pomerleau et al., 1991]). Figure 5 illustrates how IRRE can be used to integrate the outputs from multiple expert networks. The two lines in this graph illustrate the reconstruction error of two networks, one trained to steer on one-lane roads (solid line), and the other trained to steer on two-lane roads (dashed line). The reconstruction error for the one-lane network is low on the one-lane road images, and high on the two-lane road images. The opposite is true for the network trained for two-lane road driving. The reliability estimate provided by IRRE allows the system to determine which network is most reliable for driving in the current situation. By simulating multiple networks in parallel, and then selecting the one with the highest reliability, the system have been able to drive the Navlab test vehicle on one, two and four lane roads automatically at speeds of up to 30 miles per hour. 4 Discussion The effectiveness of input reconstruction reliable estimation stems from the fact that the network has a small number of hidden units and is only trained in a narrow range of situations. These constraints prevent the network from faithfully encoding arbitrary input patterns. Instead, the hidden units learn to encode features in the training images that are most important for the task. Baldi and Hornik [Baldi & Hornik, 1989] have shown that if an autoencoder Input Reconstruction Reliability Estimation 285 network with a single layer of N linear hidden units is trained with back-propagation, the activation levels of the hidden units will represent the first N principal components of the training set. Since the units in the driving network are non-linear, this assertion does not strictly hold in this case. However, Cottrell and Munro [Cottrell & Munro, 1988] have found empirically that autoencoder networks with a sigmoidal activation function develop hidden units that span the principal subspace of the training images, with some noise on the first principal component due to network non-linearity. Because the principal components represent the dimensions along which the training examples varies most, it can be shown that using linear combinations of the principal components to represent the individual training patterns optimally preserves the information contained in the training set [Linsker, 1989]. However the compressed representation developed by a linear autoencoder network is only optimal for encoding images from the same distribution as the training set. When presented with images very different from those in the training set, the image reconstructed from the internal representation is not as accurate. The results presented in this paper demonstrate that this reconstruction error can be employed to estimate the likelihood and magnitude of error in MLPs trained for autonomous driving. However the input reconstruction technique presented here has a serious potential shortcoming, namely that it forces the network's hidden units to encode all input features, including potentially irrelevant ones. While this increased representation load on the hidden units has the potential to degrade network performance, this effect has not been observed in the tests conducted so far. In support of this finding, Gluck [Gluck, personal communications] has found that forcing a network to autoencode its input frequently improves its generalization. In [Gluck & Myers, 1992], Gluck and Myers use the representation developed by an autoencoder network as a model for simple types of learning in biological systems. The model suggests that the hippocampus acts as an autoencoder, developing internal representations that are then used to perform other tasks. But if interference from the autoencoder task proves to be a problem, one way to eliminate it would be to have separate groups of hidden units connected exclusively to one group of outputs or the other. Having a separate set of hidden units for the autoencoder task would ensure that the representation developed for the input reconstruction does not interfere with representation developed for the "normal" task. It remains to be seen if this decoupling of internal representations will adversely affect IRRE's ability to predict network errors. As a technique for multi-network integration, IRRE has several advantages over existing connectionist arbitration methods, such as Hampshire and Waibel's Meta-Pi architecture [Hampshire & Waibel, 1992] and the Adaptive Mixture of Experts Model of Jacobs et aI. [Jacobs et al., 1991]. It is a more modular approach, since each expert can be trained entirely in isolation and then later combined with other experts without any additional training by simply selecting the most reliable network for the current input. Since IRRE provides an absolute measure of a single network's reliability, and not just a measure of how appropriate a network is relative to others, IRRE can be also used to determine when none of the experts is capable of coping with the current situation. A potentially interesting extension to IRRE is the development of techniques for reasoning about the difference between the actual input and the reconstructed input. For instance, it should be possible to recognize when the vehicle has reached a fork in the road by the characteristic mistakes the network makes in reconstructing the input image. Another important component of future work in is to test the ability of IRRE to estimate network reliability in domains other than autonomous driving. 286 Pomerleau Acknowledgements I thank Dave Touretzky, Chuck Thorpe and the entire Unmanned Ground Vehicle Group at CMU for their support and suggestions. Principle support for this research has come from DARPA, under contracts "Perception for Outdoor Navigation" (contract number DACA7689-C-OOI4, monitored by the US Army Topographic Engineering Center) and ''Unmanned Ground Vehicle System" (contract number DAAE07-90-C-R059, monitored by TACOM). References [Baldi & Hornik,1989] Baldi, P. and Hornik, K. (1989) Neural networks and principal component analysis: Learning from examples without local minima. Neural Networks, Vol. 2 pp. 53-58. [Cottrell & Munro, 1988] Cottrell, G.W. and Munro, P. (1988) Principal components analysis ofimages via back-propagation. Proc. Soc. of Photo-Opticallnstr. Eng., Cambridge MA. [Gluck, personal communications] Gluck, M.A. (1992) Personal Communications. Rutgers Univ., Newark NJ. [Gluck & Myers, 1992] Gluck, M.A. and Myers, C.E. (1992) Hippocampal function in representation and generalization: a computational theory. Proc.1992 Cogn. Sci. Soc. Con/. Hillsdale, NJ: Erlbaum Assoc. [Hampshire & Waibel, 1992] Hampshire J.B. and Waibel, A.H. (1989) The Meta-Pi network: building distributed knowledge representations for robust pattern recognition. IEEE Trans. on Pattern Analysis and Machine Intelligence. [Jacobs et al., 1991] Jacobs, R.A., Jordan, M.I. Nowlan, SJ. and Hinton, G.E. (1991) Adaptive mixtures of local experts. Neural Computation, 3 :1, Terrence Sejnowski (ed). [Linsker, 1989] Linsker, R. (1989) Designing a sensory processing system: What can be learned from principal component analysis? IBM Technical Report RC 14983 (#66896). [pomerleau, 1991] Pomerleau, D.A. (1991) Efficient Training of Artificial Neural Networks for Autonomous Navigation. Neural Computation 3:1, Terrence Sejnowski (ed). [pomerleau et al., 1991] Pomerleau, D.A., Gowdy, J., Thorpe, C.E. (1991) Combining artificial neural networks and symbolic processing for autonomous robot guidance. Engineering Applications of Artificial Intelligence, 4:4 pp. 279-285.
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Backpropagation without Multiplication Patrice Y. Simard AT &T Bell Laboratories Holmdel, NJ 07733 Abstract Hans Peter Graf AT&T Bell Laboratories Holmdel, NJ 07733 The back propagation algorithm has been modified to work without any multiplications and to tolerate comput.ations with a low resolution, which makes it. more attractive for a hardware implementatioll. Numbers are represented in float.ing point format with 1 bit mantissa and 3 bits in the exponent for the states, and 1 bit mantissa and 5 bit exponent. for the gradients, while the weights are 16 bit fixed-point numbers. In this way, all the computations can be executed with shift and add operations. Large nehvorks with over 100,000 weights were t.rained and demonstrat.ed the same performance as networks comput.ed with full precision. An estimate of a circuit implementatioll shows that a large network can be placed on a single chip, reaching more t.han 1 billion weight updat.es pel' second. A speedup is also obtained on any machine where a multiplication is slower than a shift operat.ioJl. 1 INTRODUCTION One of the main problems for implement.ing the backpropagation algorithm in hardware is the large number of multiplications t.hat. have to be executed. Fast multipliers for operands wit.h a high resolution l'eqllire a large area. Hence the multipliers are the elements dominating t.he area of a circuit. i\'Iany researchers have tried to reduce the size of a circuit by limit.ing the resolution of the computation. Typically, this is done by simply reducing the number of bits utilized for the computation. For a forward pass a reduction tOjllst a few , 4 to 6. bits, often degl'ades the performance very little, but. learning requires considerably more resolution. Requirements ranging anywhere from 8 bits to more than 16 bits were report.ed to be necessary to make learning converge relia.bly (Sakaue et al., 1993; Asanovic, I\'Iorgan and \Va.wrzYllek, 1993; Reyneri and Filippi, 1991). But t.here is no general theory, how much resolution is enough, and it depends on several factors, such as the size and architecture of the network as \-vell as on the t.ype of problem to be solved . 232 Backpropagation without Multiplication 233 Several researchers have tried to tl'ain networks where the weights are limited to powers of two (Kwan and Tang, 1993; White and Elmasry, 1992; l'vlarchesi et. al., 1993). In this way all the multiplications can be reduced to shift operations, an operation that can be implemented with much less hardware than a multiplication. But restricting the weight values severely impacts the performance of a network, and it is tricky to make t.he learning procedure converge. III fact , some researchers keep weights with a full resolution off-line and update t.hese weights in the backward pass, while the weights with reduced resolution are used in the forward pass (Marchesi et al., 1993). Similar tricks are usually used when networks implemented in analog hardware are trained. Weight.s with a high resolution are stored in an external, digital memory while the analog net.work with its limited resolution is used in the forward pass. If a high resolution copy is not stored, the weight update process needs to be modified. This is typically done by using a stochastic update technique, such as weight dithering (Vincent and l\lyers, 1 9~)2), or weight perturbation (.J abri and Flower, 1992). We present here an algorithm that instead of reducing the resolut.ion of the weights, reduces the resolution of all t.he other values, namely those of the states, gradients and learning rates, to powers of two. This eliminates multiplications without affecting the learning capabilities of t.he network. Therefore we ohtain the benefit of a much compacter circuit without any compromises on the learning performance. Simulations of large net.works with over 100,000 weights show that this algorithm perf?r.ms as well as standal'd backpwpagation computed with 32 bit floating point preCIsIon. 2 THE ALGORITHM The forward propagat.ion for each unit i. is given by the pquation: Xj = j~(L wji.t'il ( 1 ) where f is the unit functjoll, Wji is the weight from unit i to unit j, and Xi is the activation of unit i. The backpwpagation algorithm is wbust with regard to the unit function as long as the function is nonlinear, monotonically increasing, and a derivative exists (the most commonly used function is depicted in Figure 1, left. A saturated ramp function (see Figure 1, middle), for instance, performs as well as the differentiable sigmoid. The binary threshold function, however, is too much of a simplification and results in poor performance. The choice of OUl' function is dictated by the fact that we would like t.o have only powers of two for the unit values. This function is depicted ill Figure 1, right. It gives performances comparable to the sigmoid or the saturated ramp. Its values can be represented by a 1 bit mantissa (the sign) with a. 2 or 3 bit exponent. (negative powers of t.wo). The derivative of this funct.ion is a. SlIm of Dirac delta functions, but we take instead the derivative of a piecewise linear ramp funct.ion (see Figure 1) . 0\1(" could actually consider this a low pass filtered version of the real derivat.ive. After the gradients of all the units have been computed using the equation. [h = If ( s 11 In i ) L U'j i [lj j (2) we will discretize the values to be a power of two (wit h sign) . This introduces noise into the gradient and its effect, on the learning has to be considered carefully. This 234 Simard and Graf Sigmoid F\Jflctl.On 1.. ... -, -l.$_I:-.--:_,:--: .• ~_":", ~_.~ .• ~. -=.--=-.• ~, ""7,--=-.,~, Functlon derl.Vatlve 1.. -o.S -, -1 • '_':-, ~_,~ •• :-':"_,-_-:•• :-. "'=.~ •. :-, "':,--;" .. :-, ~, Piecewise linear Functlon 1., -, -l.~_':-, -:_,:'": .• ~_":", -:_,~ . ,~, -:,--=-. ,~, ""7,"": .• -:, FunctIon derlvatl.Ve 1.. -o.S -, -1. S_I:-, --:_,:--:. ,-_":", ~_.~. ,~. -=,'"'". ,~, ""7,'"'". ,--:, Power of two F'Unction .. , .., - o. S -'1----1 - ) . s _':-, -:_,-=. ,~_.,.., -:_.-=. ,--::-. -,.'"'"., -.-, -:",'"'".,-!, FunctIon derl.vatl.ve (apprOXlmationJ .., -O.! -, -1. '-'="2 --:""1. ':-':"-1--":"" •. :-, "'=o~ •. :-, -:"'--:"'1.:-, -:, Figure 1: Left: sigmoid function with its derivative. ]\>'1iddle: piecewise linear function with its derivative. Right.: Sat.urated power of two function with a power of two approximation of its derivative (ident.ical to t.he piecewise linear derivative). problem will be discussed in section 4. The backpropagat.ion algorithm can now be implemented with addit.ions (or subtract.iolls) and shifting only. The weight update is given by the equa.tion: D.1.Vji = -119jXi (3) Since both 9j and Xi are powers of two, the weight update also reduces to additions and shifts. 3 RESULTS A large structured network wit.h five layers and overall 11l00'e t.han 100,000 weights was used to test this algorithm. The applicat.ion analyzed is recognizing handwrit.ten character images. A database of 800 digits was used for training and 2000 handwritten digits were used for test.ing. A description of this network can be found in (Le Cun et aI., 1990). Figure 2 shows the learning curves on t.he test set for various unit functions and discretization processes. First, it should be noted that t.he results given by the sigmoid function and the saturated ramp with full precision on unit values, gradients, and weights are very similar. This is actually a well known behavior. The surprising result comes from the fact that reducing the precision of the unit values and the gradients to a 1 bit mantissa does not reduce the classification accuracy and does not even slow down the learning process. During these tests the learning process was interrupted at various stages to check that both the unit values (including the input layer, but excluding the output layer) and t.he gradient.s (all gradients) were restricted to powers of two. It was further confirmed that. ollly 2 bits wet'e sufficient. for the exponent of the unit Backpropagation without Multiplication 235 Training Testing 100 error 100 error go • sigmoid 90 • sigmoid o piecewise lin D piecewise lin eo o power of 2 80 o power of 2 70 70 60 60 50 50 40 40 30 30 20 20 10 10 IlohU:e::a.n:u ~ 0 0 0 2 4 6 8 10 12 14 18 18 20 22 24 0 2 4 6 8 10 12 14 16 16 20 22 24 age (in 1000) age (in 1000) Figure 2: Training and testing error during leaming. The filled squares (resp. empty squared) represent the points obtained with the vanilla backpropagation and a sigmoid function (resp. piecewise linear function) used a<; an activation function. The circles represent the same experiment done wit.h a power of t.wo function used as the activation function, and wit.h all lInit. gradients discretized to the nearest power of two. values (from 2° to 2-3 ) and 4 bit.s were sufficient for the exponent. of the gradients (from 2° to 2- 15). To test whether there was any asymptot.ic limit on performance, we ran a long term experiment (several days) with our largest network (17,000 free parameters) for handwritten character recognition. The training set (60,000 patterns) was made out 30,000 patterns of the original NIST t.raining set (easy) and 30,000 patterns out of the original NIST testing set (hard). Using the most basic backpropagation algorithm (with a guessed constant learning rate) we got the training raw error rate down to under 1 % in 20 epochs which is comparable to our standard learning time. Performance on the test set was not as good with the discrete network (it took twice as long to reach equal performance with the discrete network). This was attributed to the unnecessary discretization of the output units 1. These results show that gradients and unit activations can be discretized to powers of two with negligible loss in pel"formance and convergence speed! The next section will present theoretical explanations for why this is at. all possible and why it is generally the case. lSince the output units are not multiplied by anything, t.here is no need to use a discrete activation funct.ion. As a matter of fact the continuous sigmoid function can be implemented by just changing the target. values (using the inverse sigmoid function) and by using no activation function for the output units. This modificat.ion was not introduced but we believe it would improves the performance on t.he t.est. set. especially when fancy decision rules (with confidencE' evaluatioll) are used, since t.hey require high precision on the output units. 236 Simard and Graf ~:s:.ogram 2000 lROO 1600 1"00 1200 1000 aDD 600 '00 200 histogram Best case: Noise is uncorrelated and a" weights are equal Worse case: NOise is correlated or the weights are unequal Figure 3: Top left: histogram of t.he gradients of one output unit after more than 20 epochs of learning over a training set of GO,OOO pallel'lIs. Bottom left: same histogram assuming that the distt'ibutioll is constant between powers of two. Right: simplified network architectlll'es fOl' noise effect. analysis. 4 DISCUSSION Discretizing the gradient is potentially very dangerous. Convergence may no longer be guaranteed, learning may hecome prohibitively slow, and final performance after learning may be be too poor to be interesting, "Ve will now explain why these problems do not arise for our choice of discret.ization. Let gi(p) be the error gradient at weight i and pattern p. Let 1'.,: and Ui be the mean and standard deviation of gi(p) over the set of patterns. The mean Pi is what is driving the weights to their final values, the standard deviation Ui represents the amplitudes of the variations of the gradients from pattern to pattern. In batch learning, only Pi is used for the weight upda.te, while in stochastic gradient descent, each gi(p) is used for the weight update. If the learning rate is small enough the effects of the noise (measured by u;) of the stochastic variable Ui (p) are negligible, but the frequent weight updates in stochastic gradient descent result. in important speedups, To explain why the discretization of the gradient to a power of two has negligible effect on the pel'formance, consider that in stochastic gradient descent, the noise on the gradient is already so large that it is minimally affected by a rounding (of the gradient) to the nearest power of two. Indeed asymptotically, t.he gradient a.verage (Pi) tends to be negligible compared to it.s standard deviation (ui), meaning that from pattern to pattern the gradient can undergo sign reversals, Rounding to the nearest power of two in comparison is a change of at. most 33%, but never a change in sign. This additional noise can therefore be compensated by a slight. decrease in the learning rate which will hardly affect the leal'l1ing process. Backpropagation without Multiplication 237 The histogram of gi(p) after learning in the last experiment described in the result section, is shown in Figure 3 (over the training set of 60,000 patterns). It is easy to see in the figure that J.li is small wit.h respect to (7i (in this experiment J.li was one to two orders of magnitude smaller than (7i depending on the layer). vVe can also see that rounding each gradient to the nearest power of two will not affect significantly the variance (7i and therefore the learning rate will not need to be decreased to achieve the same performance. We will now try to evaluate the rounding to the nearest power of two effect more quantitatively. The standard deviation of the gradient for any weight can be written as 'I 1~ ') l~ ') ') l~ 2 (7; = N ~(gi(P) - pi)- = N ~ gi(p)- - J.l- ~ N ~ gi(p) (4) p p p This approximation is very good asymptotically (after a few epochs of learning). For instance if lJ.li I < (7;/ 10, the above formula gives the standard deviation to 1 %. Rounding the gradient gi to the nearest power of two (while keeping the sign) can be viewed as the effect of a multiplicative noise 11i in the equation g/ = 2k = ad 1 + nd for some k (5) where g/ is the nearest power of two from gj. It can be easily verified that this implies that 11.i ranges from -1/3 and 1/3. From now on , we will view Hi as a random variable which models as noise the effect of discretization . To simplifY the computation we will assume that 11j has uniform distribution. The effect of this assumption is depicted in figure :3, where the bottom histogram has been assumed constant between any two powers of t.wo. To evaluate the effect of the noise ni in a multilayer network , let 7Ili be the multiplicative noise introduced at layer I (l = 1 for the output, and I = L for the first layer above the input) for weight i. Let's further assume that there is only one unit per layer (a simplified diagra.m of the network architecture is shown on figure 3. This is the worst case analysis. If there are several units per layer, the gradients will be summed to units in a lower layer. The gradients within a layer are correlated from unit to unit (they all originate from the same desired values), but the noise introduced by the discretization can only be less correlated, not more. The summation of the gradient in a lower layer can therefore only decrease the effect of the discretization. The worst case analysis is t.herefore when there is only one unit. per layer as depicted in figure :3, extreme right. \Ve will further assume that the noise introduced by the discretizat.ion ill one layer is independent from the Iloise introduced in the next layer . This is not ~'eally true but it greatly simplifies the derivation. Let J.l~ and (7i be the mean and standard deviation of Oi (p)'. Since nli has a zero mean, J.l~ = J.li and J1~ is negligible with respect to gd}J)· In the worst case, when the ~radient has to be backpropagated all the way to t.he input, the standard deviation IS: L 1 (3 j1/3 ) (1 ) L N L gi(p)2 II -2 (1 + 11 Ii )2d7l/i /1 2 ~ (7; 1 + -. p 1 -1/3 27 (6) 238 Simard and Graf As learning progresses, the minimum average distance of each weight to the weight corresponding to a local minimum becomes proportional to the variance of the noise on that weight, divided by the learning rate. Therefore, asymptotically (which is where most of the time is spent), for a given convergence speed, the learning rate should be inversely proportional to the variance of the noise in the gradient. This means that to compensClte the effect of the discretization. the learning rate should be divided by (1" I L (11+ ;7) '" 1.02£ (7) Even for a 10 layer network this value is only 1.2, (u~ is 20 % larger than ud. The assumption that the noise is independent from layer to layer tends to slightly underestimate this number while the assumption that the noise from unit to unit in the same layer is completely correlated tends to overestimate it. All things considered, we do not expect that the learning rate should be decrea'Sed by more than 10 to 20% for any practical application. In all our simulations it was actually left unchanged! 5 HARDWARE This algorithm is well suited for integrating a large network on a single chip. The weights are implemented with a resolution of 16 bits, while the states need only 1 bit in the mantissa and 3 bits in the exponent, the gradient 1 bit in the mantissa and 5 bits in the exponent, and for the learning rate 1 bits mantissa and 4 bits exponent suffice. In this way, all the multiplications of weights with states, and of gradients with learning rates and st.at.t's. reduce to add operations of the exponents. For the forward pass the weights are multiplied with the states and then summed. The mUltiplication is executed as a shift operation of the weight values. For summing two products their mantissae have to be aligned, again a shift operation, and then they can be added. The partial sums are kept at full resolution until the end of the summing process. This is necessary to avoid losing the influence of many small products. Once the sum is computed, it is then quantized simply by checking the most significant bit in the mantissa. For the backward propagation the computation runs in the same way, except t.hat now the gradient is propagated through the net, and the learning rate has to be taken into account.. The only operations required for this algorithm are 'shift' and 'add'. An ALU implementing these operations with the resolution ment.ioned can be built with less than 1,000 transistors. In order to execut.e a network fast, its weights have to be stored on-chip. Ot.herwise, t.he time required to t.ransfer weight!; from external memory onto the chip boundary makes the high compute power all but useless. If storage is provided for 10,000 weights plus 2,000 states, this requires less than 256 kbit of memory. Together with 256 ALUs and circuitry for routing the data, this leads to a circuit with about 1.7 million transistors, where over 80% of them are contained in the memory. This assumes that the memory is implemented with static cells, if dynamic memory is used instead the transistor count drops considerably .. An operating speed of 40 MHz resnlts in a compute rate of 10 billion operat.ions per second. \-\lith such a chip a network may be trained at a speed of more than 1 billion weight updates per second. Backpropagation without Multiplication 239 This algorithm has been optimized for an implementation on a chip, but it can also provide a considerable speed up when executed on a standard computer. Due to the small resolution of the numbers, several states can be packed into a 32 bit number and hence many more fit int.o a chache. Moreover on a machine without a hardware multiplier, where the multiplication is executed with microcode, shift operations may be much faster than multiplications. Hence a suhstancial speedup may be observed. References Asanovic, K., Morgan, N., and \Vawrzynek, J. (1993). Using Simulations of Reduced Precision Arithmetic t.o Design a Neura- Microprocessor. 1. FLSI Signal Processing, 6(1):33-44. Jabri, M. and Flower, B. (1992). 'Veight Perturbation: An optimal architecture and learning technique for analog VLSI feedforward and l'ecmrent multilayer networks. IEEE Trans. Neural Networks, 3(3):154-157. Kwan, H. and Tang, C. (1993). Multipyerless Multilayer Feedforward Neural Network Desi~n Suitable for Continuous Input-Output Mapping. Elecironic Leiters,29(14):1259-1260. Le Cun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W., and Jackel, L. D. (1990). Handwritten Digit Recognition with a BackPropagation Network . In Touretzky, D., editor, Neural Injo1'lnaiio71 Processing Systems, volume 2, (Denver, 1989). l'vIorgan Kaufman. Marchesi, M., Orlando, G., Piazza, F., and Uncini, A. (1993). Fast Neural Networks without Multipliers. IEEE Trall.5. Ne11ral Networks, 4(1):53-62. Reyneri, L. and Filippi, E. (1991). An analysis on the Performance of Silicon Implementations of Backpropagation Algorithms for Artificial Nemal Networks. IEEE Trans. Computer's, 40( 12): 1380-1389. Sakaue, S., Kohda, T., Yamamoto, II., l\'laruno, S., and Shimeki, Y. (1993). Reduction of Required Precision Bits for Back-Propagation Applied to Pattern Recognition. IEEE Tralls. Neural Neiworks, 4(2):270-275. Vincent, J. and Myers, D. (1992). Weight dithering and \Vordlength Selection for Digital Backpropagation Networks. BT Tech7lology J., 10(3):124-133. White, B. and Elmasry, M. (1992). The Digi-Neocognitron: A Digit.al Neocognit.ron Neural Network Model for VLSI. IEEE Trans. Nel/r'al Networks, 3( 1 ):73-85.
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