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Non-Intrusive Gaze Tracking Using Artificial Neural Networks Shumeet Baluja baluja@cs.cmu.edu School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract Dean Pomerleau pomerleau @cs.cmu.edu School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 We have developed an artificial neural network based gaze tracking system which can be customized to individual users. Unlike other gaze trackers, which normally require the user to wear cumbersome headgear, or to use a chin rest to ensure head immobility, our system is entirely non-intrusive. Currently, the best intrusive gaze tracking systems are accurate to approximately 0.75 degrees. In our experiments, we have been able to achieve an accuracy of 1.5 degrees, while allowing head mobility. In this paper we present an empirical analysis of the performance of a large number of artificial neural network architectures for this task. 1 INTRODUCTION The goal of gaze tracking is to determine where a subject is looking from the appearance of the subject's eye. The interest in gaze tracking exists because of the large number of potential applications. Three of the most common uses of a gaze tracker are as an alternative to the mouse as an input modality [Ware & Mikaelian, 1987], as an analysis tool for human-computer interaction (HCI) studies [Nodine et. aI, 1992], and as an aid for the handicapped [Ware & Mikaelian, 1987]. Viewed in the context of machine vision, successful gaze tracking requires techniques to handle imprecise data, noisy images, and a potentially infinitely large image set. The most accurate gaze tracking has come from intrusive systems. These systems either use devices such as chin rests to restrict head motion, or require the user to wear cumbersome equipment, ranging from special contact lenses to a camera placed on the user's head. The system described here attempts to perform non-intrusive gaze tracking, in which the user is neither required to wear any special equipment, nor required to keep hislher head still. 753 754 Baluja and Pomerleau 2 GAZE TRACKING 2.1 TRADITIONAL GAZE TRACKING In standard gaze trackers, an image of the eye is processed in three basic steps. First, the specular reflection of a stationary light source is found in the eye's image. Second, the pupil's center is found. Finally, the relative position of the light's reflection to the pupil's center is calculated. The gaze direction is determined from information about the relative positions, as shown in Figure 1. In many of the current gaze tracker systems, the user is required to remain motionless, or wear special headgear to maintain a constant offset between the position of the camera and the eye. Specular Reflection ~~~ Looking at Looking Above Looking Below Looking Left of Light Light Light Light Figure 1: Relative position of specular reflection and pupil. This diagram assumes that the light is placed in the same location as the observer (or camera). 2.2 ARTIFICIAL NEURAL NETWORK BASED GAZE TRACKING One of the primary benefits of an artificial neural network based gaze tracker is that it is non-intrusive; the user is allowed to move his head freely. In order to account for the shifts in the relative positions of the camera and the eye, the eye must be located in each image frame. In the current system, the right eye is located by searching for the specular reflection of a stationary light in the image of the user's face. This can usually be distinguished by a small bright region surrounded by a very dark region. The reflection's location is used to limit the search for the eye in the next frame. A window surrounding the reflection is extracted; the image of the eye is located within this window. To determine the coordinates of the point the user is looking at, the pixels of the extracted window are used as the inputs to the artificial neural network. The forward pass is simulated in the ANN, and the coordinates of the gaze are determined by reading the output units. The output units are organized with 50 output units for specifying the X coordinate, and 50 units for the Y coordinate. A gaussian output representation, similar to that used in the ALVINN autonomous road following system [Pomerleau, 1993], is used for the X and Y axis output units. Gaussian encoding represents the network's response by a Gaussian shaped activation peak in a vector of output units. The position of the peak within the vector represents the gaze location along either the X or Y axis. The number of hidden units and the structure of the hidden layer necessary for this task are explored in section 3. The training data is collected by instructing the user to visually track a moving cursor. The cursor moves in a predefined path. The image of the eye is digitized, and paired with the (X,Y) coordinates of the cursor. A total of 2000 image/position pairs are gathered. All of the networks described in this paper are trained with the same parameters for 260 epochs, using standard error back propagation. The training procedure is described in greater Non-Intrusive Gaze Tracking Using Artificial Neural Networks 755 detail in the next section. 3 THE ARTIFICIAL NEURAL NETWORK IMPLEMENTATION In designing a gaze tracker, the most important attributes are accuracy and speed. The need for balancing these attributes arises in deciding the number of connections in the ANN, the number of hidden units needed, and the resolution of the input image. This section describes several architectures tested, and their respective performances. 3.1 EXAMINING ONLY THE PUPIL AND CORNEA Many of the traditional gaze trackers look only at a high resolution picture of the subject's pupil and cornea. Although we use low resolution images, our first attempt also only used an image of the pupil and cornea as the input to the ANN. Some typical input images are shown below, in Figure 2(a). The size of the images is 15x15 pixels. The ANN architecture used is shown in Figure 2(b). This architecture was used with varying numbers of hidden units in the single, divided, hidden layer; experiments with 10, 16 and 20 hidden units were performed. As mentioned before, 2000 image/position pairs were gathered for training. The cursor automatically moved in a zig-zag motion horizontally across the screen, while the user visually tracked the cursor. In addition, 2000 image/position pairs were also gathered for testing. These pairs were gathered while the user tracked the cursor as it followed a vertical zig-zag path across the screen. The results reported in this paper, unless noted otherwise, were all measured on the 2000 testing points. The results for training the ANN on the three architectures mentioned above as a function of epochs is shown in Figure 3. Each line in Figure 3 represents the average of three ANN training trials (with random initial weights) for each of the two users tested. Using this system, we were able to reduce the average error to approximately 2.1 degrees, which corresponds to 0.6 inches at a comfortable sitting distance of approximately 17 inches. In addition to these initial attempts, we have also attempted to use the position of the cornea within the eye socket to aid in making finer discriminations. These experiments are described in the next section. 50 X Output Units 50 Y Output Units 15 x 15 Input Retina Figure 2: (a-left) 15 x 15 Input to the ANN. Target outputs also shown. (b-right) the ANN architecture used. A single divided hidden layer is used. 756 Baluja and Pomerleau Figure 3: Error vs. Epochs for the 15x15 images. Errors shown for the 2000 image test set. Each line represents three ANN trainings per user; two users are tested. JSdSlmages "0 240 1JO '''' 210 3.2 USING THE EYE SOCKET FOR ADDITIONAL INFORMATION to Hidden iii Hidden rO"Hidden In addition to using the information present from the pupil and cornea, it is possible to gain information about the subject's gaze by analyzing the position of the pupil and cornea within the eye socket. Two sets of experiments were performed using the expanded eye image. The first set used the network described in the next section. The second set of experiments used the same architecture shown in Figure 2(b), with a larger input image size. A sample image used for training is shown below, in Figure 4. Figure 4: Image of the pupil and the eye socket, and the corresponding target outputs. 15 x 40 input image shown. 3.2.1. Using a Single Continuous Hidden Layer One of the remaining issues in creating the ANN to be used for analyzing the position of the gaze is the structure of the hidden unit layer. In this study, we have limited our exploration of ANN architectures to simple 3 layer feed-forward networks. In the previous architecture (using 15 x 15 images) the hidden layer was divided into 2 separate parts, one for predicting the x-axis, and the other for the y-axis. Selecting this architecture over a fully connected hidden layer makes the assumption that the features needed for accurate prediction of the x-axis are not related to the features needed for predicting the y-axis. In this section, this assumption is tested. This section explores a network architecture in which the hidden layer is fully connected to the inputs and the outputs. In addition to deciding the architecture of the ANN, it is necessary to decide on the size of the input images. Several input sizes were attempted, 15x30, 15x40 and 20x40. Surprisingly, the 20x40 input image did not provide the most accuracy. Rather, it was the 15x40 image which gave the best results. Figure 5 provides two charts showing the performance of the 15x40 and 20x40 image sizes as a function of the number of hidden units and epochs. The 15x30 graph is not shown due to space restrictions, it can be found in [Baluja & Pomerleau, 1994]. The accuracy achieved by using the eye socket information, for the 15x40 input images, is better than using only the pupil and cornea; in particular, the 15x40 input retina worked better than both the 15x30 and 20x40. Non-Intrusive Gaze Tracking Using Artificial Neural Networks 757 2 &0 260 240 220 I sooo IS x 40 Image 10 Hidden i6"Hfdden iii""fiidden I I I Epochs 10000 I ~OO 20000 2!tO OO 3l1l 320 3 10 300 290 liO I " l70~ I I I sooo 10000 lOx 40 Image ...... ~ . I I 1.5000 20000 10 Hidden i6-Hfdiien 2oHi!iden Figure 5: Performance of 15x40, and 20x40 input image sizes as a function of epochs and number of hidden units. Each line is the average of 3 runs. Data points taken every 20 epochs, between 20 and 260 epochs. 3.2.2. Using a Divided Hidden Layer The final set of experiments which were performed were with 15x40 input images and 3 different hidden unit architectures: 5x2, 8x2 and 10x2. The hidden unit layer was divided in the manner described in the first network, shown in Figure 2(b). Two experiments were performed, with the only difference between experiments being the selection of training and testing images. The first experiment was similar to the experiments described previously. The training and testing images were collected in two different sessions, one in which the user visually tracked the cursor as it moved horizontally across the screen and the other in which the cursor moved vertically across the screen. The training of the ANN was on the "horizontally" collected images, and the testing of the network was on the "vertically" collected images. In the second experiment, a random sample of 1000 images from the horizontally collected images and a random sample of 1000 vertically collected images were used as the training set. The remaining 2000 images from both sets were used as the testing set. The second method yielded reduced tracking errors. If the images from only one session were used, the network was not trained to accurately predict gaze position independently of head position. As the two sets of data were collected in two separate sessions, the head positions from one session to the other would have changed slightly. Therefore, using both sets should have helped the network in two ways. First, the presentation of different head positions and different head movements should have improved the ability of the network to generalize. Secondly, the network was tested on images which were gathered from the same sessions as it was trained. The use of mixed training and testing sets will be explored in more detail in section 3.2.3. The results of the first and second experiments are presented here, see Figure 6. In order to compare this architecture with the previous architectures mentioned, it should be noted that the performance of this architecture, with 10 hidden units, more accurately predicted gaze location than the architecture mentioned in section 3.2.1, in which a single continuous hidden layer was used. In comparing the performance of the architectures with 16 and 20 hidden units, the performances were very similar. Another valuable feature of using the 758 Baluja and Pomerleau divided hidden layer is the reduced number of connections decreases the training and simulation times. This architecture operates at approximately 15hz. with 10 and 16 hidden units, and slightly slower with 20 hidden units. Eno,-o.gr... Separate Hidden Layer & 15x40 Image - Test Set 1 310 300 290 280 240 2 30 2 .0 200 , , """" ............... ... -......... ""~l 10 Hidden i(j·Hraden 2o--fii-dden .. 90c-----ulnn-----,oi=--~.-----.,;I;;OOon-----,,;250i;-,.OOr=" Epochs Erro,-DegI8OS Seperate Hidden Layer & 15x40 Images - Test Set 2 210 240 . 60 .so . 40 '. 10 Hidden i(j-H1aden iifHiCiden Figure 6: (Left) The average of 2 users with the 15x40 images, and a divided hidden layer architecture, using test setup #1. (Right) The average performance tested on 5 users, with test setup #2. Each line represents the average of three ANN trainings per user per hidden unit architecture. 3.2.3. Mixed Training and Testing Sets It was hypothesized, above, that there are two reasons for the improved performance of a mixed training and testing set. First, the network ability to generalize is improved, as it is trained with more than a single head position. Second, the network is tested on images which are similar, with respect to head position, as those on which it was trained. In this section, the first hypothesized benefit is examined in greater detail using the experiments described below. Four sets of 2000 images were collected. In each set, the user had a different head position with respect to the camera. The first two sets were collected as previously described. The first set of 2000 images (horizontal train set 1) was collected by visually tracking the cursor as it made a horizontal path across the screen. The second set (vertical test set 1) was collected by visually tracking the cursor as it moved in a vertical path across the screen. For the third and fourth image sets, the camera was moved, and the user was seated in a different location with respect to the screen than during the collection of the first training and testing sets. The third set (horizontal train set 2) was again gathered from tracking the cursor's horizontal path, while the fourth (vertical test set 2) was from the vertical path of the cursor. Three tests were performed. In the first test, the ANN was trained using only the 2000 images in horizontal training set 1. In the second test, the network was trained using the 2000 images in horizontal training set 2. In the third test, the network was trained with a random selection of 1000 images from horizontal training set 1, and a random selection of 1000 images of horizontal training set 2. The performance of these networks was tested on both of the vertical test sets. The results are reported below, in Figure 7. The last experiment, in which samples were taken from both training sets, provides more accurate results Non-Intrusive Gaze Tracking Using Artificial Neural Networks 759 when testing on vertical test set I, than the network trained alone on horizontal training set 1. When testing on vertical test set 2, the combined network performs almost as well as the network trained only on horizontal training set 2. These three experiments provide evidence for the network's increased ability to generalize if sets of images which contain multiple head positions are used for training. These experiments also show the sensitivity of the gaze tracker to movements in the camera; if the camera is moved between training and testing, the errors in simulation will be large. Error-Degr ... Vertil:al Test Set I combined JOO triiiiisei-j ttaiii""se,"i ' 80 '60 , 40 \. '20 \ \ ' 00 \ "'-" ' 80 '~-Epochs Error-Degr ... 380 : 36:1 )41] ". ,. 300 Z80 \ , 6:1 \ 240 \ ,,. 200 ' 10 Vertical Test Set 1 \ ~ combined iiaiii set -i ifaiii"set2 Figure 7: Comparing the performance between networks trained with only one head position (horizontal train set 1 & 2), and a network trained with both. 4 USING THE GAZE TRACKER The experiments described to this point have used static test sets which are gathered over a period of several minutes, and then stored for repeated use. Using the same test set has been valuable in gauging the performance of different ANN architectures. However, a useful gaze tracker must produce accurate on-line estimates of gaze location. The use of an "offset table" can increase the accuracy of on-line gaze prediction. The offset table is a table of corrections to the output made by a gaze tracker. The network's gaze predictions for each image are hashed into the 2D offset-table, which performs an additive correction to the network's prediction. The offset table is filled after the network is fully trained. The user manually moves and visually tracks the cursor to regions in which the ANN is not performing accurately. The offset table is updated by subtracting the predicted position of the cursor from the actual position_ This procedure can also be automated, with the cursor moving in a similar manner to the procedure used for gathering testing and training images. However, manually moving the cursor can help to concentrate effort on areas where the ANN is not performing well; thereby reducing the time required for offset table creation. With the use of the offset table, the current system works at approximately 15 hz. The best on-line accuracy we have achieved is 1.5 degrees. Although we have not yet matched the best gaze tracking systems, which have achieved approximately 0.75 degree accuracy, our system is non-intrusive, and does not require the expensive hardware which many other systems require. We have used the gaze tracker in several forms; we have used it as an 760 Baluja and Pomerleau input modality to replace the mouse, as a method of selecting windows in an X-Window environment, and as a tool to report gaze direction, for human-computer interaction studies. The gaze tracker is currently trained for 260 epochs, using standard back propagation. Training the 8x2 hidden layer network using the 15x40 input retina, with 2000 images, takes approximately 30-40 minutes on a Sun SPARC 10 machine. 5 CONCLUSIONS We have created a non-intrusive gaze tracking system which is based upon a simple ANN. Unlike other gaze-tracking systems which employ more traditional vision techniques, such as a edge detection and circle fitting, this system develops its own features for successfully completing the task. The system's average on-line accuracy is 1.7 degrees. It has successfully been used in HCI studies and as an input device. Potential extensions to the system, to achieve head-position and user independence, are presented in [Baluja & Pomerleau, 1994]. Acknowledgments The authors would like to gratefully acknowledge the help of Kaari Flagstad, Tammy Carter, Greg Nelson, and Ulrike Harke for letting us scrutinize their eyes, and being "willing" subjects. Profuse thanks are also due to Henry Rowley for aid in revising this paper. Shumeet Baluja is supported by a National Science Foundation Graduate Fellowship. This research was supported by the Department of the Navy, Office of Naval Research under Grant No. NOO014-93-1-0806. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the National Science Foundation, ONR, or the U.S. government. References Baluja, S. Pomerleau, D.A. (1994) "Non-Intrusive Gaze Tracking Using Artificial Neural Networks" CMU-CS-94. Jochem, T.M., D.A. Pomerleau, C.E. Thorpe (1993), "MANIAC: A Next Generation Neurally Based Autonomous Road Follower". In Proceedings of the International Conference on Intelligent Autonomous Systems (IAS-3). Nodine, c.P., H.L. Kundel, L.c. Toto & E.A. Krupinksi (1992) "Recording and analyzing eye-position data using a microcomputer workstation", Behavior Research Methods, Instruments & Computers 24 (3) 475-584. Pomerleau, D.A. (1991) "Efficient Training of Artificial Neural Networks for Autonomous Navigation," Neural Computation 3: I, Terrence Sejnowski (Ed). Pomerleau, D.A. (1993) Neural Network Perception for Mobile Robot Guidance. Kluwer Academic Publishing. Pomerleau, D.A. (1993) "Input Reconstruction Reliability Estimation", Neural Information Processing Systems 5. Hanson, Cowan, Giles (eds.) Morgan Kaufmann, pp. 270-286. Starker, I. & R. Bolt (1990) "A Gaze-Responsive Self Disclosing Display", In CHI-90. Addison Wesley, Seattle, Washington. Waibel, A., Sawai, H. & Shikano, K. (1990) "Consonant Recognition by Modular Construction of Large Phonemic Time-Delay Neural Networks". Readings in Speech Recognition. Waibel and Lee. Ware, C. & Mikaelian, H. (1987) "An Evaluation of an Eye Tracker as a Device for Computer Input", In 1. Carrol and P. Tanner (ed.) Human Factors in Computing Systems -IV. Elsevier.
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Dynamic Modulation of Neurons and Networks Eve Marder Center for Complex Systems Brandeis University Waltham, MA 02254 USA Abstract Biological neurons have a variety of intrinsic properties because of the large number of voltage dependent currents that control their activity. Neuromodulatory substances modify both the balance of conductances that determine intrinsic properties and the strength of synapses. These mechanisms alter circuit dynamics, and suggest that functional circuits exist only in the modulatory environment in which they operate. 1 INTRODUCTION Many studies of artificial neural networks employ model neurons and synapses that are considerably simpler than their biological counterparts. A variety of motivations underly the use of simple models for neurons and synapses in artificial neural networks. Here, I discuss some of the properties of biological neurons and networks that are lost in overly simplified models of neurons and synapses. A fundamental principle in biological nervous systems is that neurons and networks operate over a wide range of time scales, and that these are modified by neuromodulatory substances. The flexible, multiple time scales in the nervous system allow smooth transitions between different modes of circuit operation. 2 NEURONS HA VE DIF'FERENT INTRINSIC PROPERTIES Each neuron has complex dynamical properties that depend on the number and kind of ion channels in its membrane. Ion channels have characteristic kinetics and voltage 511 512 Marder dependencies that depend on the sequence of amino acids of the protein. Ion channels may open and close in several milliseconds; others may stay open for hundreds of milliseconds or several seconds. Some neurons are silent unless they receive synaptic inputs. Silent neurons can be activated by depolarizing synaptic inputs, and many will fire on rebound from a hyperpolarizing input (postinhibitory rebound). Some neurons are tonically active in the absence of synaptic inputs, and synaptic inputs will increase or decrease their firing rate. Some neurons display rhythmic bursts of action potentials. These bursting neurons can display stable patterns of oscillatory activity, that respond to perturbing stimuli with behavior characteristic of oscillators, in that their period can be stably reset and entrained. Bursting neurons display a number of different voltage and time dependent conductances that interact to produce slow membrane potential oscillations with rapid action potentials riding on the depolarized phase. In a neuron such as R15 of Aplysia (Adams and Levitan 1985) or the AB neuron of the stomatogastric ganglion (STG) (Harris-Warrick and Flamm 1987), the time scale of the burst is in the second range, but the individual action potentials are produced in the 5-10msec time scale. Neurons can generate bursts by combining a variety of different conductances. The particular balance of these conductances can have significant impact on the oscillator's behavior (Epstein and Marder 1990; Kepler et aI1990; Skinner et aI1993), and therefore the choice of oscillator model to use must be made with care (Somers and Kopell 1993). Some neurons have a balance of conductances that give them bistable membrane potentials, allowing to produce plateau potentials. Typically, such neurons have two relatively stable states, a hyperpolarized silent state, and a sustained depolarized state in which they fire action potentials. The transition between these two modes of activity can be made with a short depolarizing or hyperpolarizing pulse (Fig. 1). Plateau potentials, like "flip-flops" in electronics, are a "short-term memory" mechanism for neural circuits. The intrinsic properties of neurons can be modified by sustained changes in membrane potential. Because the intrinsic properties of neurons depend on the balance of conductances that activate and inactivate in different membrane potential ranges and over a variety of time scales, hyperpolarization or depolarization can switch a neuron between modes of intrinsic activity (Llinas 1988; McCormick 1991; Leresche et aI1991). An interesting "memory-like" effect is produced by the slow inactivation properties of some K+ currents (McCormick 1991; Storm 1987). In cells with such currents a sustained depolarization can "amplify" a synaptic input from subthreshold to suprathreshold, as the sustained depolarization causes the K+ current to inactivate (Marom and Abbott 1994; Turrigiano, Marder and Abbott in preparation). This is another "short-term memory" mechanism that does not depend on changes in synaptic efficacy. Dynamic Modulation of Neurons and Networks 513 A. CONTROL \20mV I04nA 1..., Figure 1: Intracellular recording from the DG neuron ofthe crab STG. A: control saline, a depolarizing current pulse elicits action potentials for its duration. B: In SDRNFLRFamide, a short depolarization elicits a plateau potential that lasts until a short hyperpolarizing current pulse terminates it. Modified from Weimann et al 1993. 2 INTRINSIC MEMBRANE PROPERTIES ARE MODULATED Biological nervous systems use many substances as neurotransmitters and neuromodulators. The effects of these substances include opening of rapid, relatively nonvoltage dependent ion channels, such as those mediating conventional rapid synaptic potentials. Alternatively, modulatory substances can change the number or type of voltage-dependent conductances displayed by a neuron, and in so doing dramatically modify the intrinsic properties of a neuron. In Fig. 1, a peptide, SDRNFLRFamide transforms the DG neuron of the crab STG from a state in which it fires only during a depolarizing pulse to one in which it displays long-lasting plateau properties (Weimann et al 1993). The salient feature here is that modulatory substances can elicit slow membrane properties not otherwise expressed. 3 SYNAPTIC STRENGTH IS MODULATED In most neural network models synaptic weights are modified by learning rules, but are not dependent on the temporal pattern of presynaptic activity. In contrast, in many biological synapses the amount of transmitter released depends on the frequency of firing of the presynaptic neuron. Facilitation, the increase in the amplitude of the postsynaptic current when' the presynaptic neuron is activated several times in quick succession is quite common. Other synapses show depression. The same neuron may show facilitation at some of its terminals while showing depression at others (Katz et al 1993). The facilitation and depression properties of any given synapse can not be deduced on first principles, but must be determined empirically. Synaptic efficacy is often modified by modulatory substances. A dramatic example is seen in the Aplysia gill withdrawal reflex, where serotonin significantly enhances the amplitude of the monosynaptic connection from the sensory to motor neurons (Clark and Kandel 1993; Emptage and Carew 1993). The effects of modulatory substances can occur on different branches on a neuron independently (Clark and Kandel 1993), and the same modulatory substance may have different actions at different sites of the same neuron. 514 Marder Electrical synapses are also subject to neuromodulation (Dowling, 1989). For example, in the retina dopamine reversibly uncouples horizonal cells. Modulation of synaptic strength can be quite extreme; in some cases synaptic contacts may be virtually invisible in some modulatory environments, while strong in others. The implications of this for circuit ooeration will be discussed below. Hormones .DA LJ5-HT • Oct II CCAP II cCCK IIlomTK • APCH Neuromodulators • ACh • AST .OA • Buc rnm GABA ID cCCK .HA .LK • Oct I'IlomTK • Myomod 1'1 Proc • RPCH • SOAN 1'1 TNRN Sensory Transmitters • ACh } ~~~ . Figure 2: Modulatory substances found in inputs to the STG. See Harris-Warrick et aI., 1992 for details. Figure courtesy of P. Skiebe. 4 TRANS:MITTERS ARE COLOCALIZED IN NEURONS The time course of a synaptic potential evoked by a neurotransmitter or modulator is a characteristic property of the ion channels gated by the transmitter and/or the second messenger system activated by the signalling molecule. Synaptic currents can be relatively fast, such as the rapid action of ACh at the vertebrate skeletal neuromuscular junction where the synaptic currents decay in several milliseconds. Alternatively, second messenger activated synaptic events may have durations lasting hundreds of milliseconds, seconds, or even minutes. Many neurons contain several differen.t neurotransmitters. It is common to find a small molecule such as glutamate or GABA colocalized with an amine such as serotonin or histamine and one or more neuropeptides. To describe the synaptic actions of such neurons, it is necessary to determine for each-signalling molecule how its release depends on the frequency and pattern of activity in the presynaptic Dynamic Modulation of Neurons and Networks SIS terminal, and characterize its postsynaptic actions. This is important, because different mixtures of cotransmitters, and consequently of postsynaptic action may occur with different presynaptic patterns of activity. 5 NEURAL NETWORKS ARE MULTIPLY MODULATED Neural networks are controlled by many modulatory inputs and substances. Figure 2 illustrates the patterns of modulatory control to the crustacean stomatogastric nervous system, where the motor patterns produced by the only 30 neurons of the stomatogastric ganglion are controlled by about 60 input fibers (Coleman et a11992) that contain at least 15 different substances, including a variety of amines, amino acids, and neuropeptides (Marder and Weimann 1992; Harris-Warrick et aI1992). Each of these modulatory substances produces characteristic and different effects on the motor patterns of the STG (Figs. 3,4). This can be understood if one remembers that the intrinsic membrane properties as well as the strengths of the synaptic connections within this group of neurons are all subject to modulation. Because each cell has many conductances, many of which are subject to modulation, and because of the large number of synaptic connections, the modes of circuit operation are theoretically large. 6 CIRCUIT RECONFIGURA TION BY MODULATORY CONTROL Figure 3 illustrates that modulatory substances can tune the operation of a single functional circuit. However, neuromodulatory substances can also produce far more extensive changes in the functional organization of neuronal networks. Recent work on the STG demonstrates that sensory and modulatory neurons and substances can cause neurons to switch between different functional circuits, so that the same neuron is part of several different pattern generating circuits at different times (Hooper and Moulins 1989; Dickinson et al 1990; Weimann et al 1991; Meyrand et al 1991; Heinzel et al 1993). In the example shown in Fig. 4, in control saline the LG neuron is firing in time with the fast pyloric rhythm (the LP neuron is also firing in pyloric time), but there is no ongoing gastric rhythm. When the gastric rhythm was activated by application of the peptide SDRNFLRFNHz, the LG neuron fired in time with the gastric rhythm (Weimann et al 1993). These and other data lead us to conclude that it is the modulatory environment that constructs the functional circuit that produces a given behavior (Meyrand et al 1991). Thus, by tuning intrinsic membrane properties and synaptic strengths, neuromodulatory agents can recombine the same neurons into a variety of circuits, capable of generating remarkably distinct outputs. Acknowledgements I thank Dr. Petra Skiebe for Fig 3 art work. Research was supported by NSI7813. 516 Marder CONTROL PILOCARPINE SEROTONIN Figure 3: Different forms of the pyloric rhythm different modulators. Each panel, the top two traces: simulataneous intracellular recordings from LP and PD neurons of crab STG; bottom trace: extracellular recording, Ivn nerve. Control, rhythmic pyloric activity absent. Substances were bath applied, the pyloric patterns produced were different. Modified from Marder and Weimann 1992. dgn Figure 4: Neurons switch between different pattem-genreating circuits. Left panel, the gastric rhythm not active (monitored by DG neuron), LG neuron in time with the pyloric rhythm (seen as activity in LP neuron). Right panel, gastric rhythm activated by SDRNFLRFamide, monitored by the DG neuron bursts recorded on the dgn. LG now fired in alternation with DG neuron. Pyloric time is seen as the interruptions in the activity of the VD neuron. Modified from Marder and Weimann 1992. Dynamic Modulation of Neurons and Networks 517 References Adams WB and Levitan IB 1985 Voltage and ion dependencies of the slow currents which mediate bursting in Aplysia neurone R,s' J Physiol 360 69-93 Clark GA, Kandel ER 1993 Induction oflong-tenn facilitation in Aplysia sensory neurons by local application of serotonin to remote synapses. Proc Natl Acad Sci USA 90: 1141-11415 Coleman MJ, Nusbaum MP, Cournil I, Claiborne BJ 1992 Distribution of mopdulatory inputs to the stomatogastric ganglion of the crab, Cancer borealis. J Comp Neur 325: 581-594 Dickinson PS, Mecsas C, Marder E 1990 Neuropeptide fusion of two motor-pattern generator circuits. Nature 344: 155-158 Dowling JE 1989 Neuromodulation in the retina: the role of dopamine. Sem Neur 1:3543 Emptage NJ and Carew TJ 1993 Long-term synaptic facilitation in the absence of shortterm facilitation in Aplysia neurons. Science 262 253-256 Epstein IR, Marder E 1990 Multiple modes of a conditional neural oscillator. Bioi Cybern 63: 25-34 Harris-Warrick RM, Flamm RE 1987 Multiple mechanisms of bursting in a conditional bursting neuron. J Neurosci 7: 2113-2128 Harris-Warrick RM, Marder E, Selverston AI, Moulins M eds 1992 Dynamic Biological Networks: The Stomatogastric Nervous System. MIT Press Cambridge Heinzel H-G, Weimann JM, Marder E 1993 The behavioral repertoire of the gastric mill in the crab, Cancer pagurus: An in vivo endoscopic and electrophysiological examination. J Neurosci 13: 1793-1803 Hooper SL, Moulins M 1989 Switching of a neuron from one network to another by sensory-induced changes in membrane properties. Science 244: 1587-1589 Katz PS, Kirk MD, and Govind CK 1993 Facilitation and depression at different branches of the same motor axon: evidence for presynaptic differences in release. J Neurosci 13: 3075-3089 Kepler TB, Marder E, Abbott LF 1990 The effect of electrical coupling on the frequency of model neuronal oscillators. Science 248: 83-85 Leresche N, Lightowler S, Soltesz I, Jassik-Gerschenfeld D, and Crunelli V 1991 Lowfrequency oscillatory activities intrinsic to rat and car thalamocortical cells. J Physiol 441 155-174 Llinas RR 1988 The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function. Science 242 1654-1664 McCormick DA 1991 Functional properties of slowly inactivating potassium current in guinea pig dorsal lateral geniculate relay neurons. J Physiol 66 1176-1189 Marom S and Abbott LF 1994 Modeling state-dependent inactivation of membrane currents. Biophysical J in press Marder E, Weimann JM 1992 Modulatory control of mUltiple task processing in the stomatogastric nervous system. IN: Neurobiology of Motor Programme Selection: new approaches to mechanisms of behavioral choice, Kien J, 518 Marder McCrohan C. Winlow W eds Pergamon Press Oxford Meyrand p. Simmers I. Moulins, M 1991 Construction of a pattern-generating circuit with neurons of different networks. NaJure 351: 60-63 Skinner FK. Turrigiano 00. Marder E 1993 Frequency and burst duration in oscillating neurons and two cell networks. Bioi Cybern 69: 375-383 Somers D. Kopell N 1993 Rapid synchronization through fast threshold modulation. Bioi Cybern 68: 393-407 Storm IF 1987 Temporal integration by a slowly inactivating K+ current in hippocampal neurons. NaJure 336: 379-381 Weimann 1M. Meyrand p. Marder E 1991 Neurons that form multiple pattern generators: Identification and multiple activity patterns of gastric/pyloric neurons in the crab stomatogastric system. J Neurophysiol 65: 111-122 Weimann 1M, Marder E, Evans B. Calabrese RL 1993 The effects of SDRNFLRFNHl and TNRNFLRFNHl on the motor patterns of the stomatogastric ganglion of the crab. Cancer borealis. J Exp Bioi 181: 1-26
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Training Neural Networks with Deficient Data Volker Tresp Siemens AG Central Research 81730 Munchen Germany tresp@zfe.siemens.de Subutai Ahmad Interval Research Corporation 1801-C Page Mill Rd. Palo Alto, CA 94304 ahmad@interval.com Ralph N euneier Siemens AG Central Research 81730 Munchen Germany ralph@zfe.siemens.de Abstract We analyze how data with uncertain or missing input features can be incorporated into the training of a neural network. The general solution requires a weighted integration over the unknown or uncertain input although computationally cheaper closed-form solutions can be found for certain Gaussian Basis Function (GBF) networks. We also discuss cases in which heuristical solutions such as substituting the mean of an unknown input can be harmful. 1 INTRODUCTION The ability to learn from data with uncertain and missing information is a fundamental requirement for learning systems. In the "real world" , features are missing due to unrecorded information or due to occlusion in vision, and measurements are affected by noise. In some cases the experimenter might want to assign varying degrees of reliability to the data. In regression, uncertainty is typically attributed to the dependent variable which is assumed to be disturbed by additive noise. But there is no reason to assume that input features might not be uncertain as well or even missing competely. In some cases, we can ignore the problem: instead of trying to model the relationship between the true input and the output we are satisfied with modeling the relationship between the uncertain input and the output. But there are at least two 128 Training Neural Networks with Deficient Data 129 reasons why we might want to explicitly deal with uncertain inputs. First, we might be interested in the underlying relationship between the true input and the output (e.g. the relationship has some physical meaning). Second, the problem might be non-stationary in the sense that for different samples different inputs are uncertain or missing or the levels of uncertainty vary. The naive strategy of training networks for all possible input combinations explodes in complexity and would require sufficient data for all relevant cases. It makes more sense to define one underlying true model and relate all data to this one model. Ahmad and Tresp (1993) have shown how to include uncertainty during recall under the assumption that the network approximates the "true" underlying function. In this paper, we first show how input uncertainty can be taken into account in the training of a feedforward neural network. Then we show that for networks of Gaussian basis functions it is possible to obtain closed-form solutions. We validate the solutions on two applications. 2 THE CONSEQUENCES OF INPUT UNCERTAINTY Consider the task of predicting the dependent variablel y E ~ from the input vector x E ~M consisting of M random variables. We assume that the input data {(xklk = 1,2, ... , K} are selected independently and that P(x) is the joint probability distribution of x. Outputs {(yklk = 1,2, ... , K} are generated following the standard signal-plus-noise model yk = /(xk) + (k where {(klk = 1,2, ... , K} denote zero-mean ran'dom variables with probability density Pc(t:). The best predictor (in the mean-squared sense) of y given the input x is the regressor defined by E(ylx) = J y P(ylx) dx = f(x), where E denotes the expectation. Unbiased neural networks asymptotically (K -+ 00) converge to the regressor. To account for uncertainty in the independent variable we assume that we do not have access to x but can only obtain samples from another random vector z E ~M with zk = xk + Ok where {Ok Ik = 1,2, ... , K} denote independent random vectors containing M random variables with joint density P6(6).2 A neural network trained with data {(zk, yk)lk = 1,2, ... , K} approximates E(ylz) = P~z) J y P(ylx) P(zlx) P(x) dydx = P~z) J /(x) P6(Z - x) P(x) dx. (1) Thus, in general E(ylz) # /(z) and we obtain a biased solution. Consider the case that the noise processes can be described by Gaussians Pc(() = G((j 0, O'Y) and P6(6) = G(Oj 0, 0') where, in our notation, G(Xj m, s) stands for 11M (x· - m·)2 G(x' m s) exp[-- "" J J] , , (211')M/2 n:;l Sj 2 ~ s] lOur notation does not distinguish between a random variable and its realization. 2 At this point, we assume that P6 is independent of x. 130 Tresp, Ahmad, and Neuneier £ f(x) y E(y!x) t E(ylz) I F \j ./ \ ~ Figure 1: The top half of the figure shows the probabilistic model. In an example, the bottom half shows E(Ylx) = f( x) ( continuous), the input noise distribution (dotted) and E(ylz ) (dashed). where m, s are vectors with the same dimensionality as x (here M). Let us take a closer look at four special cases. Certain input. If t7 = 0 (no input noise), the integral collapses and E(ylz) = fez). Uncertain input. If P(x) varies much more slowly than P(zlx), Equation 1 described the convolution of f(x) with the noise process P6(Z - x). Typical noise processes will therefore blur or smooth the original mapping (Figures 1). It is somewhat surprising that the error on the input results in a (linear) convolution integral. In some special cases we might be able to recover f( x) from an network trained on deficient data by deconvolution, although one should use caution since deconvolution is very error sensitive. Unknown input. If t7j 00 then the knowledge of Zj does not give us any information about Xj and we can consider the jth input to be unknown. Our formalism therefore includes the case of missing inputs as special case. Equation 1 becomes an integral over the unknown dimensions weighted by P(x) (Figure 2). Linear approximation. If the approximation (2) is valid, the input noise can be transformed into output noise and E(ylz) = fez). This results can also be derived using Equation 1 if we consider that a convolution of a linear function with a symmetrical kernel does not change the function. This result tells us that if f(x) is approximately linear over the range where P6(6) has significant Training Neural Networks with Deficient Data 131 '."r---~-~--' '.2 ... Figure 2: Left: samples yk = f(xt, x~) are shown (no output noise). Right: with one input missing, P(yIX1) appears noisy. amplitude we can substitute the noisy input and the network will still approximate f(x). Similarly, the mean mean(xi) of an unknown variable can be substituted for an unknown input, if f(x) is linear and xi is independent of the remaining input variables. But in all those cases, one should be aware of the potentially large additional variance (Equation 2). 3 MAXIMUM LIKELIHOOD LEARNING In this section, we demonstrate how deficient data can be incorporated into the training of feedforward networks. In a typical setting, we might have a number of complete data, a number of incomplete data and a number of data with uncertain features. Assuming independent samples and Gaussian noise, the log-likelihood I for a neural network NNw with weight vector W becomes K K 1= 2:logP(zk,yk) = 2: log J G(yk jNNw(x),(1Y) G(zk jX,(1k) P(x) dx. k=1 k=1 Note that now, the input noise variance is allowed to depend on the sample k. The gradient of the log-likelihood with respect to an arbitrary weight Wi becomes3 01 ~ 8IogP(zk, yk) 1 ~ 1 8w. = L...J 8w' = ((1y)2 L...J P(zk yk) X l k=1 l k=1' J(yk - NNw (x)) 8N:~(x) G(yk;NNw(x),(1Y) G(zk;X,(1k) P(x) dx. (3) First, realize that for a certain sample k ((1k --+ 0): 8IogP(zk,yk)/8wi = (yk _ N Nw(zk))/((1Y)2 8N Nw(zk)/8wi which is the gradient used in normal backpropagation. For uncertain data, this gradient is replaced by an averaged gradient. The integral averages the gradient over possible true inputs x weighted by the probability of P(xlzk,yk) = P(zklx) P(yklx) P(x)/p(zk,yk). The term 3This equation can also be obtained via the EM formalism. A similar equation was obtained by Buntine and Weigend (1991) for binary inputs. 132 Tresp, Ahmad, and Neuneier P(yklx) == G(ykjNNw(x),D''') is of special importance since it weights the gradient higher when the network prediction NNw (x) agrees with the target yk. This term is also the main reason why heuristics such as substituting the mean value for a missing variable can be harmful: if, at the substituted input, the difference between network prediction and target is large, the error is also large and the data point contributes significantly to the gradient although it is very unlikely that the substitutes value was the true input. In an implementation, the integral needs to be approximated by a finite sum (i. e. Monte-Carlo integration, finite-difference approximation etc.). In the experiment described in Figure 3, we had a 2-D input vector and the data set consisted of both complete data and data with one missing input. We used the following procedure 1. Train the network using the complete data. Estimate (UIl)2. We used (UII )2 ~ (Ec /(K - H», where Ec is the training error after the network was trained with only the complete data, and H is the number of hidden units in the network. 2. Estimate the input density P(x) using Gaussian mixtures (see next section). 3. Include the incomplete training patterns in the training. 4. For every incomplete training pattern • Let z~ be the certain input and let zt be the missing input, and z1c = (z~, zt) . • Approximate (assuming -1/2 < Xj < 1/2, the hat stands for estimate) J/2 8 log P(z~, y1c) 1 1 1 ~ 8Wi ::::: J (ulI)2 p(z~ y1c) . L..J «y1c - N Nw(z~, j / J» x , J=-J/2 where 4 GAUSSIAN BASIS FUNCTIONS The required integration in Equation 1 is computationally expensive and one would prefer closed form solutions. Closed form solutions can be found for networks which are based on Gaussian mixture densities.4 Let's assume that the joint density is given by N P(x) == L G(x; Ci, Si) P(Wi), i=l where Ci is the location of the center of the ith Gaussian and and Sij corresponds to the width of the ith Gaussian in the jth dimension and P(Wi) is the prior probability of Wi. Based on this model we can calculate the expected value of any unknown 4Gaussian mixture learning with missing inputs is also addressed by Ghahramani and Jordan (1993). See also their contribution in this volume. 0.1 28c 28c, 225m 125c 125c, 128m Training Neural Networks with Deficient Data 133 0.1 28c 225m 28c, mean 225m subst Figure 3: Regression. Left: We trained a feedforward neural network to predict the housing price from two inputs (average number of rooms, percent of lower status population (Tresp, Hollatz and Ahmad (1993». The training data set contained varying numbers of complete data points (c) and data points with one input missing (m). For training, we used the method outlined in Section 3. The test set consisted of 253 complete data. The graph (vertical axis: generalization error) shows that by including the incomplete patterns in the training, the performance is significantly improved. Right: We approximated the joint density by a mixture of Gaussians. The incomplete patterns were included by using the procedure outlined in Section 4. The regression was calculated using Equation 4. As before, including the incomplete patterns in training improved the performance. Substituting the mean for the missing input (column on the right) on the other hand, resulted in worse performance than training of the network with only complete data. 0.86 i 1°·84 § 0.82 I 0.8 _. -.1000 2000 3000 # of data with miss. feat. 0.74 r-----.-----~--__, 1) io.n c: i 0.7 60.68 tf!. 234 # of missing features 5 Figure 4: Left: Classification performance as a function of the number of missing features on the task of 3D hand gesture recognition using a Gaussian mixtures classifier (Equation 5). The network had 10 input units, 20 basis functions and 7 output units. The test set contained 3500 patterns. (For a complete description of the task see (Ahmad and Tresp, 1993).) Class-specific training with only 175 complete patterns is compared to the performance when the network is trained with an additional 350, 1400, and 3325 incomplete patterns. Either 1 input (continuous) or an equal number of 1-3 (dashed) or 1-5 (dotted) inputs where missing. The figure shows clearly that adding incomplete patterns to a data set consisting of only complete patterns improves performance. Right: the plot shows performance when the network is trained only with 175 incomplete patterns. The performance is relatively stable as the number of missing features increases. 134 Tresp, Ahmad, and Neuneier variable XU from any set of known variables xn using (Tresp, Hollatz and Ahmad, 1993) E(xUlxn) = E; ciG(xn; ci,si) P(w.) E.=1 G(xnj cf, sf) P(Wi) (4) Note, that the Gaussians are projected onto the known dimensions. The last equation describes the normalized basis function network introduced by Moody and Darken (1989). Classifiers can be built by approximating the class-specific data distributions P(xlclassi) by mixtures of Gaussians. Using Bayes formula, the posterior class probability then becomes P( 1 I) P(classi)P(xlclassi) c ass, x = " I' L..J; P( class; )P(x class;) (5) We now assume that we do not have access to x but to z where, again, P(zlx) = G(z; x, 0'). The log-likelihood of the data now becomes K N K N 1 = L:log jL:G(X;Ci,Si)P(Wi) G(zk;x,O'k) dx = LlogLG(zk;ci,Sf)P(wi) k=1 i=1 k=1 .=1 where (Sf;)2 = s~; + (O'j)2. We can use the EM approach (Dempster, Laird and Rubin, 1977) to obtain the following update equations. Let Ci;, s.; and P(w.) denote current parameter estimates and let Of; = (Ci;(O'j)2 + z;s~;)/(Sf;? and Df; = «O'j)2 s~;)/(Sf;)2. The new estimates (indicated by a hat) can be obtained using P(wdzk) G(zk; Ci, Sf) P(Wi) (6) Ef=1 G(Zk;C;, Sf) pew;) K P(w.) 1 L: A k (7) K P(w.lz ) k=1 K k A k Ci; Ek=10i; P(wdz ) (8) EK A k k=1 P(Wilz ) A2 Ef=1[Df; + (Of; - Ci;)2] P(wdzk) (9) si; K A Ek=1 P(Wi Izk) These equations can be solved by alternately using Equation 6 to estimate P(wdzk) and Equations 7 to 9 to update the parameter estimates. If uk = 0 for all k (only certain data) we obtain the well known EM equations for Gaussian mixtures (Duda and Hart (1973), page 200). Setting 0': = 00 represents the fact that the jth input is missing in the kth data point and Of; = Cij, Dfj = s~;. Figure 3 and Figure 4 show experimental results for a regression and a classification problem. Training Neural Networks with Deficient Data 135 5 EXTENSIONS AND CONCLUSIONS We can only briefly address two more aspects. In Section 3 we only discussed regression. We can obtain similar results for classification problems if the costfunction is a log-likelihood function (e.g. the cross-entropy, the signal-plus-noise model is not appropriate). Also, so far we considered the true input to be unobserved data. Alternatively the true inputs can be considered unknown parameters. In this case, the goal is to substitute the maximum likely input for the unknown or noisy input. We obtain as log-likelihood function I ~[_~ (y1: - N Nw (x1c)? _ ~ ~ (x;- zt? + I P( 1:)] (X L...J 2 (qy)2 2 ~ (q~)2 og X . 1:=1 1=1 1 The l.earning frocedure consists of finding optimal values for network weights wand true mputs x . 6 CONCLUSIONS Our paper has shown how deficient data can be included in network training. Equation 3 describes the solution for feedforward networks which includes a computationallyexpensive integral. Depending on the application, relatively cheap approximations might be feasible. Our paper hinted at possible pitfalls of simple heuristics. Particularly attractive are our results for Gaussian basis functions which allow closed-form solutions. References Ahmad, S. and Tresp, V. (1993). Some solutions to the missing feature problem in vision. In S. J. Hanson, J. D. Cowan and C. 1. Giles, (Eds.), Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann. Buntine, W. L. and Weigend, A. S. (1991). Bayesian Back-Propagation. Complex systems, Vol. 5, pp. 605-643. Dempster, A. P., La.ird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39, pp. 1-38. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. John Wiley and Sons, New York. Ghahramani, Z. and Jordan, M. I. (1993). Function approximation via density estimation using an EM approach. MIT Computational Cognitive Sciences, TR 9304. Moody, J. E. and Darken, C. (1989). Fast learning in networks oflocally-tuned processing units. Neural Computation, Vol. 1, pp. 281-294. Tresp, V., Hollatz J. and Ahmad, S. (1993). Network structuring and tra.ining using rulebased knowledge. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann. Tresp, V., Ahmad, S. and Neuneier, R. (1993). Uncerta.inty in the Inputs of Neural Networks. Presented at Neural Networks for Computing 1993.
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When Will a Genetic Algorithm Outperform Hill Climbing? Melanie Mitchell Santa Fe Institute 1660 Old Pecos Trail, Suite A Santa Fe, NM 87501 John H. HoUand Dept. of Psychology University of Michigan Ann Arbor, MI 48109 Stephanie Forrest Dept. of Computer Science University of New Mexico Albuquerque, NM 87131 Abstract We analyze a simple hill-climbing algorithm (RMHC) that was previously shown to outperform a genetic algorithm (GA) on a simple "Royal Road" function. We then analyze an "idealized" genetic algorithm (IGA) that is significantly faster than RMHC and that gives a lower bound for GA speed. We identify the features of the IGA that give rise to this speedup, and discuss how these features can be incorporated into a real GA. 1 INTRODUCTION Our goal is to understand the class of problems for which genetic algorithms (GA) are most suited, and in particular, for which they will outperform other search algorithms. Several studies have empirically compared GAs with other search and optimization methods such as simple hill-climbing (e.g., Davis, 1991), simulated annealing (e.g., Ingber & Rosen, 1992), linear, nonlinear, and integer programming techniques, and other traditional optimization techniques (e.g., De Jong, 1975). However, such comparisons typically compare one version of the GA with a second algorithm on a single problem or set of problems, often using performance criteria which may not be appropriate. These comparisons typically do not identify the features that led to better performance by one or the other algorithm, making it hard to distill general principles from these isolated results. In this paper we look in depth at one simple hill-climbing method and an idealized form of the GA, in order to identify some general principles about when and why a GA will outperform hill climbing. 51 52 Mitchell, Holland, and Forrest 81 = 11111111·············································· .......... j C1 =8 82 = ········11111111·········································· ...... j C2 = 8 83 = ················11111111······························ .......... j C3 =8 84 = ························11111111······················ .......... ; C4 =8 85 = ································11111111················ ........ ; Cs = 8 86 = ········································11111111······ .......... ; C6 =8 87 = ················································11111111· ....... ; C7 = 8 8S = ...................................................... ··11111111; Cs = 8 8~t=1111111111111111111111111111111111111111111111111111111111111111 Figure 1: Royal Road function Rl. In previous work we have developed a class of fitness landscapes (the "Royal Road" functions; Mitchell, Forrest, & Holland, 1992; Forrest & Mitchell, 1993) designed to be the simplest class containing the features that are most relevant to the performance of the GA. One of our purposes in developing these landscapes is to carry out systematic comparisons with other search methods. A simple Royal Road function, Rl, is shown in Figure 1. Rl consists of a list of partially specified bit strings (schemas) Si in which '*' denotes a wild card (either o or 1). Each schema 8, is given with a coefficient Ci. The order of a schema is the number of defined (non-'*') bits. A bit string x is said to be an instance of a schema 8, x E 8, if x matches s in the defined positions. The fitness Rl(X) of a bit string x is defined as follows: ~ {I if x E Si Rl(X) = ~ CiOi(X), where o,(x) = 0 otherwise. , For example, if x is an instance of exactly two of the order-8 schemas, Rl (x) = 16. Likewise, Rl (111 ... 1) = 64. The Building Block Hypothesis (Holland, 1975/1992) states that the GA works well w hen instances of low-order, short schemas ("building blocks") that confer high fitness can be recombined to form instances of larger schemas that confer even higher fitness. Given this hypothesis, we initially expected that the building-block structure of Rl would layout a "royal road" for the GA to follow to the optimal string. We also expected that simple hill-climbing schemes would perform poorly since a large number of bit positions must be optimized simultaneously in order to move from an instance of a lower-order schema (e.g., 11111111** ... *) to an instance of a higher-order intermediate schema (e.g., 11111111*****·*·11111111** ... *). However both these expectations were overturned (Forrest & Mitchell, 1993). In our experiments, a simple GA (using fitness-proportionate selection with sigma scaling, single-point crossover, and point mutation) optimized Rl quite slowly, at least in part because of "hitchhiking": once an instance of a higher-order schema is discovered, its high fitness allows the schema to spread quickly in the population, with Os in other positions in the string hitchhiking along with the Is in the schema's defined positions. This slows down the discovery of schemas in the other positions, especially those that are close to the highly fit schema's defined positions. Hitchhiking can in general be a serious bottleneck for the GA, and we observed similar effects When Will a Genetic Algorithm Outperform Hill Climbing? S3 Table 1: Mean and median number of function evaluations to find the optimum string over 200 runs of the GA and of various hill-climbing algorithms on R1. The standard error is given in parentheses. in several variations of our original GA. Our other expectation-that the GA would outperform simple hill-climbing on these functions-was also proved wrong. Forrest and Mitchell (1993) compared the GA's performance on a variation of Rl with three different hill-climbing methods: steepest ascent hill-climbing (SAHC), next-ascent hill-climbing (NAHC), and a zero-temperature Monte Carlo method, which Forrest and Mitchell called ''random mutation hill-climbing" (RMHC). In RMHC, a string is chosen at random and its fitness is evaluated. The string is then mutated at a randomly chosen single locus, and the new fitness is evaluated. If the mutation leads to an equal or higher fitness, the new string replaces the old string. This procedure is iterated until the optimum has been found or a maximum number of function evaluations has been performed. Here we have repeated these experiments for R1. The results (similar to those given for R2 in Forrest & Mitchell, 1993) are given in Table 1. We compare the mean and median number of function evaluations to find the optimum string rather than mean and median absolute run time, because in almost all GA applications (e.g., evolving neural-network architectures), the time to perform a function evaluation vastly dominates the time required to execute other parts of the algorithm. For this reason, we consider all parts of the algorithm excluding the function evaluations to take negligible time. The results on SAHC and NAHC were as expected-while the GA found the optimum on RI in an average of 61,334 function evaluations, neither SAHC nor NAHC ever found the optimum within the maximum of 256,000 function evaluations. However, RMH C found the optimum on Rl in an average of 6179 function evaluationsnearly a factor often faster than the GA. This striking difference on landscapes originally designed to be "royal roads" for the GA underscores the need for a rigorous answer to the question posed earlier: "Under what conditions will a GA outperform other search algorithms, such as hill climbing?" 2 ANALYSIS OF RMHC AND AN IDEALIZED GA To begin to answer this question, we analyzed the RMHC algorithm with respect to R1 • Suppose the fitness function c,onsists of N adjacent blocks of K Is each (in RI, N = 8 and K = 8). What is the expected time (number of function evaluations) E(K, N) to find the optimum string of allIs? We can first ask a simpler question: what is the expected time E(K, 1) to find a single block of K Is? A Markov-chain analysis (not given here) yields E(K, 1) slightly larger than 2K , converging slowly to 2K from above as K -+ 00 (Richard Palmer, personal communication). For 54 Mitchell, Holland, and Forrest example, for K = 8, E(K, 1) = 301.2. Now suppose we want RMHC to discover a string with N blocks of K Is. The time to discover a first block of K Is is E(K, 1), but, once it has been found, the time to discover a second block is longer, since many of the function evaluations are "wasted" on testing mutations inside the first block. The proportion of non-wasted mutations is (K N - K) / K N; this is the proportion of mutations that occur in the KN - K positions outside the first block. The expected time E(K, 2) to find a second block is E(K, 1) + E(K, l)[KN/(KN - K)]. Similarly, the total expected time is: E(K,N) = N N E(K, 1) + E(K, 1) N _ 1 + ... + E(K, 1) N _ (N _ 1) [ 1 1 1] E(K,l)N 1 + "2 + 3 + ... + N . (1) (The actual value may be a bit larger, since E(K,l) is the expected time to the first block, whereas E(K, N) depends on the worst time for the N blocks.) Expression (1) is approximately E(K, l)N(logN + r), where r is Euler's constant. For K = 8, N = 8, the value of expression (1) is 6549. When we ran RMHC on the Rl function 200 times, the average number of function evaluations to the optimum was 6179, which agrees reasonably well with the expected value. Could a GA ever do better than this? There are three reasons why we might expect a GA to perform well on Rl. First, at least theoretically the GA is fast because of implicit parallelism (Holland, 1975/1992): each string in the population is an instance of many different schemas, and if the population is large enough and is initially chosen at random, a large number of different schemas-many more than the number of strings in the population-are being sampled in parallel. This should result in a quick search for short, low-order schemas that confer high fitness. Second, fitness-proportionate reproduction under the GA should conserve instances of such schemas. Third, a high crossover rate should quickly combine instances oflow-order schemas on different strings to create instances of longer schemas that confer even higher fitness. Our previous experiments (Forrest & Mitchell, 1993) showed that the simple GA departed from this "in principle" behavior. One major impediment was hitchhiking, which limited implicit parallelism by fixing certain schema regions sub optimally. But if the GA worked exactly as described above, how quickly could it find the optimal string of Rl? To answer this question we consider an "idealized genetic algorithm" (IGA) that explicitly has the features described above. The IGA knows ahead of time what the desired schemas are, and a "function evaluation" is the determination of whether a given string contains one or more of them. In the IGA, at each time step a single string is chosen at random, with uniform probability for each bit. The string is "evaluated" by determining whether it is an instance of one or more of the desired schemas. The first time such a string is found, it is sequestered. At each subsequent discovery of an instance of one or more not-yet-discovered schemas the new string is instantaneously crossed over with the sequestered string so that the sequestered string contains all the desired schemas that have been discovered so far. This procedure is unusable in practice, since it requires knowing a priori which schemas are relevant, whereas in general an algorithm such as the GA or RMHC When Will a Genetic Algorithm Outperform Hill Climbing? 55 directly measures the fitness of a string, and does not know ahead of time which schemas contribute to high fitness. However, the idea behind the GA is to do implicitly what the IGA is able to do explicitly. This idea will be elaborated below. Suppose again that our desired schemas consist of N blocks of K 1s each. What is the expected time (number of function evaluations) until the saved string contains all the desired schemas? Solutions have been suggested by G. Huber (personal communication), and A. Shevoroskin (personal communication), and a detailed solution is given in (Holland, 1993). The main idea is to note that the probability of finding a single desired block 8 on a random string is p = 1/2K, and the probability of finding s by time t is 1 - (1 - p)t. Then the probability PN(t) that all N blocks have been found by time tis: PN(t) = (1 - (1 - p)t)N, and the probability PN(t) that all N blocks are found at exactly time tis: PN(t) = [1- (1- p)t]N - [1- (1- p)t-l]N. The expected time is then 00 EN = 2:t ([1- (1- p)t]N - [1- (1- p)t-l]N). 1 This sum can be expanded and simplified, and with some work, along with the approximation (1-p)n ~ 1-np for small p, we obtain the following approximation: N 1 EN ~ (lip) I:; ~ 2K(logN + 1)· n=l The major point is that the IGA gives an expected time that is on the order of 2K log N, where RMHC gives an expected time that is on the order of 2K N log N, a factor of N slower. This kind of analysis can help us predict how and when the G A will outperform hill climbing. What makes the IGA faster than RMHC? A primary reason is that the IGA perfectly implements implicit parallelism: each new string is completely independent of the previous one, so new samples are given independently to each schema region. In contrast, RMHC moves in the space of strings by single-bit mutations from an original string, so each new sample has all but one of the same bits as the previous sample. Thus each new string gives a new sample to only one schema region. The IGA spends more time than RMHC constructing new samples, but since we are counting only function evaluations, we ignore the construction time. The IGA "cheats" on each function evaluation, since it knows exactly the desired schemas, but in this way it gives a lower bound on the number of function evaluations that the GA will need on this problem. Independent sampling allows for a speed-up in the IGA in two ways: it allows for the possibility of more than one desirable schema appearing simultaneously on a given sample, and it also means that there are no wasted samples as there are in RMHC. Although the comparison we have made is with RMHC, the IGA will also be significantly faster on Rl (and similar landscapes) than any hill-climbing 56 Mitchell, Holland, and Forrest Levell: 81 82 83 8, 85 8S 81 8a 89 810 811 812 813 8H 815 81S Level 2: (81 82) (83 8,) (85 8S) (81 8a) (89 810) (811 812) (813 81') (815 81S) Level 3: (81 82 83 8,) (85 8S 81 8a) (89 810 811 812) (813 8H 815 81S) Level 4: (81 82 83 8, 85 8S 81 8a) (89 810 811 812 813 8H 815 81S) Figure 2: Royal Road Function R4. method that works by mutating single bits (or a small number of bits) to obtain new samples. The hitchhiking effects described earlier also result in a loss of independent samples for the real GA. The goal is to have the real GA, as much as possible, approximate the IGA. Of course, the IGA works because it explicitly knows what the desired schemas are; the real GA does not have this information and can only estimate what the desired schemas are by an implicit sampling procedure. But it is possible for the real GA to approximate a number of the features of the IGA. Independent samples: The population size has to be large enough, the selection process has to be slow enough, and the mutation rate has to be sufficient to make sure that no single locus is fixed at a single value in every (or even a large majority) of strings in the population. Sequestering desired schemas: Selection has to be strong enough to preserve desired schemas that have been discovered, but it also has to be slow enough (or, equivalently, the relative fitness of the non-overlapping desirable schemas has to be small enough) to prevent significant hitchhiking on some highly fit schemas, which can crowd out desired schemas in other parts of the string. Instantaneous crossover: The crossover rate has to be such that the time for a crossover to occur that combines two desired schemas is small with respect to the discovery time for the desired schemas. Speed-up over RMHC: The string length (a function of N) has to be large enough to make the N speed-up factor significant. These mechanisms are not all mutually compatible (e.g., high mutation works against sequestering schemas), and thus must be carefully balanced against one another. A discussion of how such a balance might be achieved is given in Holland (1993). 3 RESULTS OF EXPERIMENTS As a first step in exploring these balances, we designed R3, a variant of our previous function R2 (Forrest & Mitchell, 1993), based on some of the features described above. In R3 the desired schemas are 81-88 (shown in Fig. 1) and combinations of them, just as in R2. However, in R3 the lowest-level order-8 schemas are each separated by "introns" (bit positions that do not contribute to fitness-see Forrest & Mitchell, 1993; Levenick, 1991) of length 24. In R3, a string that is not an instance of any desired schema receives fitness 1.0. Every time a new level is reached-i.e., a string is found that is an instance of one or more schemas at that level-a small increment u is added to the fitness. Thus strings at level 1 (that are instances of at least one level-l schema) have fitness 1 + u, strings at level 2 have fitness 1 + 2u, etc. For our experiments we set u = 0.2. When Will a Genetic Algorithm Outperfonn Hill Climbing? 57 Table 2: R4: Mean function evaluations (over 37 runs) to attain each level for the GA and for RMHC. In the GA runs, the number of function evaluations is sampled every 500 evaluations, so each value is actually an upper bound for an interval of length 500. The standard errors are in parentheses. The percentage of runs which reached each level is shown next to the heading "% runs." Only runs which successfully reached a given level were included in the function evaluation calculations for that level. The purpose of the introns was to help maintain independent samples in each schema position by preventing linkage between schema positions. The independence of samples was also helped by using a larger population (2000) and the much slower selection scheme given by the function. In preliminary experiments on R3 (not shown) hitchhiking in the GA was reduced significantly, and the population was able to maintain instances of all the lowest-level schemas throughout each run. Next, we studied R4 (illustrated in Figure 2). R4 is identical to R3, except that it does not have introns. Further, R4 is defined over 128-bit strings, thus doubling the size of the problem. In preliminary runs on R4, we used a population size of 500, a mutation rate of 0.005 (mutation always flips a bit), and multipoint crossover, where the number of crossover points for each pair of parents was selected from a Poisson distribution with mean 2.816. Table 2 gives the mean number of evaluations to reach levels 1, 2, and 3 (neither algorithm reached level 4 within the maximum of 106 function evaluations). As can be seen, the time to reach level one is comparable for the two algorithms, but the GA is much faster at reaching levels 2 and 3. Further, the GA discovers level 3 approximately twice as often as RMHC. As was said above, it is necessary to balance the maintenance of independent samples with the sequestering of desired schemas. These preliminary results suggest that R4 does a better job of maintaining this balance than the earlier Royal Road functions. Working out these balances in greater detail is a topic of future work. 4 CONCLUSION We have presented analyses of two algorithms, RMHC and the IGA, and have used the analyses to identify some general principles of when and how a genetic algorithm will outperform hill climbing. We then presented some preliminary experimental results comparing the GA and RMHC on a modified Royal Road landscape. These analyses and results are a further step in achieving our original goals-to design the simplest class of fitness landscapes that will distinguish the GA from other search methods, and to characterize rigorously the general features of a fitness landscape that make it suitable for a GA. S8 Mitchell, Holland, and Forrest Our modified Royal Road landscape R4, like Rl, is not meant to be a realistic example of a problem to which one might apply a GA. Rather, it is meant to be an idealized problem in which certain features most relevant to GAs are explicit, so that the GA's performance can be studied in detail. Our claim is that in order to understand how the GA works in general and where it will be most useful, we must first understand how it works and where it will be most useful on simple yet carefully designed landscapes such as these. The work reported here is a further step in this direction. Acknowledgments We thank R. Palmer for suggesting the RMHC algorithm and for sharing his careful analysis with us, and G. Huber for his assistance on the analysis of the IGA. We also thank E. Baum, L. Booker, T. Jones, and R. Riolo for helpful comments and discussions regarding this work. We gratefully acknowledge the support of the Santa Fe Institute's Adaptive Computation Program, the Alfred P. Sloan Foundation (grant B1992-46), and the National Science Foundation (grants IRI-9157644 and IRI-9224912). References L. D. Davis (1991). Bit-climbing, representational bias, and test suite design. In R. K. Belew and L. B. Booker (eds.), Proceedings of the Fourth International Conference on Genetic Algorithms, 18-23. San Mateo, CA: Morgan Kaufmann. K. A. De Jong (1975). An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Unpublished doctoral dissertation. University of Michigan, Ann Arbor, MI. S. Forrest and M. Mitchell (1993). Relative building-block fitness and the buildingblock hypothesis. In D. Whitley (ed.), Foundations of Genetic Algorithms 2, 109126. San Mateo, CA: Morgan Kaufmann. J. H. Holland (1975/1992). Adaptation in Natural and Artificial Systems. Cambridge, MA: MIT Press. (First edition 1975, Ann Arbor: University of Michigan Press.) J. H. Holland (1993). Innovation in complex adaptive systems: Some mathematical sketches. Working Paper 93-10-062, Santa Fe Institute, Santa Fe, NM. L. Ingber and B. Rosen (1992). Genetic algorithms and very fast simulated reannealing: A comparison. Mathematical Computer Modelling, 16 (11),87-100. J. R. Levenick (1991). Inserting introns improves genetic algorithm success rate: Taking a cue from biology. In R. K. Belew and L. B. Booker (eds.), Proceedings of the Fourth International Conference on Genetic Algorithms, 123-127. San Mateo, CA: Morgan Kaufmann. M. Mitchell, S. Forrest, and J. H. Holland (1992). The royal road for genetic algorithms: Fitness landscapes and GA performance. In F. J. Varela and P. Bourgine (eds.), Proceedings of the First European Conference on Artificial Life, 245-254. Cambridge, MA: MIT Press.
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Unsupervised Parallel Feature Extraction from First Principles .. Mats Osterberg Image Processing Laboratory Dept. EE., Linkoping University S-58183 Linkoping Sweden Reiner Lenz Image Processing Laboratory Dept. EE., Linkoping University S-58183 Linkoping Sweden Abstract We describe a number of learning rules that can be used to train unsupervised parallel feature extraction systems. The learning rules are derived using gradient ascent of a quality function. We consider a number of quality functions that are rational functions of higher order moments of the extracted feature values. We show that one system learns the principle components of the correlation matrix. Principal component analysis systems are usually not optimal feature extractors for classification. Therefore we design quality functions which produce feature vectors that support unsupervised classification. The properties of the different systems are compared with the help of different artificially designed datasets and a database consisting of all Munsell color spectra. 1 Introduction There are a number of unsupervised Hebbian learning algorithms (see Oja, 1992 and references therein) that perform some version of the Karhunen-Loeve expansion. Our approach to unsupervised feature extraction is to identify some desirable properties of the extracted feature vectors and to construct a quality functions that measures these properties. The filter functions are then learned from the input patterns by optimizing this selected quality function. In comparison to conventional unsupervised Hebbian learning this approach reduces the amount of communication between the units needed to learn the weights in parallel since the complexity now lies in the learning rule used. 136 Unsupervised Parallel Feature Extraction from First Principles 137 The optimal (orthogonal) solution to two of the proposed quality functions turn out to be related to the Karhunen-Loeve expansion: the first learns an arbitrary rotation of the eigenvectors whereas the later learns the pure eigenvectors. A common problem with the Karhunen-Loeve expansion is the fact that the first eigenvector is normally the mean vector of the input patterns. In this case one filter function will have a more or less uniform response for a wide range of input patterns which makes it rather useless for classification. We will show that one quality function leads to a system that tend to learn filter functions which have a large magnitude response for just one class of samples (different for each filter function) and low magnitude response for samples from all other classes. Thus, it is possible to classify an incoming pattern by simply observing which filter function has the largest magnitude response. Similar to Intrator's Projection Pursuit related network (see Intrator & Cooper, 1992 and references therein) some quality functions use higher order (> 2) statistics of the input process but in contrast to Intrator's network there is no need to specify the amount of lateral inhibition needed to learn different filter functions. All systems considered in this paper are linear but at the end we will briefly discuss possible non-linear extensions. 2 Quality functions In the following we consider linear filter systems. These can be described by the equation: O(t) W(t)P(t) (1) where P(t) E RM:l is the input pattern at iteration t, W(t) E RN:M is the filter coefficient matrix and O(t) = (01 (t), ... ,0N(t))' E RN:l is the extracted feature vector. Usually M > N, i.e. the feature extraction process defines a reduction of the dimensionality. Furthermore, we assume that both the input patterns and the filter functions are normed; IIP(t)1I = 1 and IIWn(t)1I = 1, "It "In. This implies that 10~(t)1 ~ 1, 'Vi "In. Our first decision is to measure the scatter of the extracted feature vectors around the origin by the determinant of the output correlation matrix: QMS(t) = det EdO(t)O'(t)} (2) QMS(t) is the quality function used in the Maximum Scatter Filter System (MSsystem). The use of the determinant is motivated by the following two observations: 1. The determinant is equal to the product of the eigenvalues and hence the product of the variances in the principal directions and thus a measure of the scattering volume in the feature space. 2. The determinant vanishs if some filter functions are linearly dependent. In (Lenz & Osterberg, 1992) we have shown that the optimal filter functions to QMS(t) are given by an arbitrary rotation of the N eigenvectors corresponding to the N largest eigenvalues of the input correlation matrix: Wopt RUeig (3) where Ueig contains the largest eigenvectors (or principal components) of the input correlation matrix EdP(t)P'(t)}. R is an arbitrary rotation matrix with det( R) = 1. To differentiate between these solutions we need a second criterion. 138 Osterberg and Lenz One attempt to define the best rotation is to require that the mean energy Et { o~ (t)} should be concentrated in as few components on(t) of the extracted feature vector as possible. Thus, the mean energy Ed o~ (t)} of each filter function should be either very high (i.e. near 1) or very low (i.e. near 0). This leads to the following second order concentration measure: N Q2(t) = L Edo~(t)} (1- Edo!(t)}) (4) n=l which has a low non-negative value if the energies are concentrated. Another idea is to find a system that produces feature vectors that have unsupervised discrimination power. In this case each learned filter function should respond selectively, i.e. have a large response for some input samples and low response for others. One formulation of this goal is that each extracted feature vector should be (up to the sign) binary; Oi(t) = ±1 and on(t) = 0, n 1= i, 'Vt. This can be measured by the following fourth order expression: N N Q4(t) = EdL o~(t) (1 o~(t»)} L Edo~(t)} - Edo!(t)} (5) n=l n=l which has a low non-negative value if the features are binary. Note that it is not sufficient to use on(t) instead of o~(t) since Q4(t) will have a low value also for feature vectors with components equal in magnitude but with opposite sign. A third criterion can be found as follows: if the filter functions have selective filter response then the response to different input patterns differ in magnitude and thus the variance of the mean energy Ed o~(t)} is large. The total variance is measured by: N N L Var {o~ (t)} = L Ed ( o~ (t) - Ed o~ (t)} ) 2} n=l n=l N L Edo!(t)} - (Edo!(t)})2 (6) n=l Following (Darlington, 1970) it can be shown that the distribution of o~ should be bimodal (modes below and above Edo~}) to maximize QVar(t). The main difference between QVar(t) and the quality function used by Intrator is the use of a fourth order term Edo!(t)} instead of a third order term Edo~(t)}. With Ed o~(t)} the quality function is a measure of the skewness of the distribution o(t) and it is maximized when one mode is at zero and one (or several) is above Edo~(t)}. In this paper we will examine the following non-parametric combinations of the quality functions above: QMS(t) Q2(t) QMS(t) Q4(t) QVar(t)QM set) (7) (8) (9) Unsupervised Parallel Feature Extraction from First Principles 139 We refer to the corresponding filter systems as: the Karhunen-Loeve Filter System (KL-system), the Fourth Order Filter System (FO-system) and the Maximum Variance Filter System (MV-system). Since each quality function is a combination of two different functions it is hard to find the global optimal solution. Instead we use the following strategy to determine a local optimal solution. Definition 1 The optimal orthogonal solution to each quality function is of the form: W opt (10) where Ropt is the rotation of the largest eigenvectors which minimize Q2(t), Q4(t) or maximize QYar(t). In (Lenz & Osterberg, 1992 and Osterberg, 1993) we have shown that the optimal orthogonal solution to the KL-system are the N pure eigenvectors if the N largest eigenvalues are all distinct (i.e. Ropt = I). If some eigenvalues are equal then the solution is only determined up to an arbitrary rotation of the eigenvectors with equal eigenvalues. The fourth order term Edo~(t)} in Q4(t) and QYar(t) makes it difficult to derive a closed form solution. The best we can achieve is a numerical method (in the case of Q4(t) see Osterberg, 1993) for the computation of the optimal orthogonal filter functions. 3 Maximization of the quality function The partial derivatives of QMS(t), Q2(t), Q4(t) and QYar(t) with respect to w~(t) (the mth weight in the nth filter function at iteration t) are only functions of the input pattern pet), the output values OCt) = (OI(t), ... , ON(t» and the previous values of the weight coefficients (w~ (t - 1), ... , w~ (t - 1» within the filter function (see Osterberg, 1993). Especially, they are not functions of the internal weights ((wlCt - 1), ... , wf1(t -1», i;/; n) of the other filter functions in the system. This implies that the filter coefficients can be learned in parallel using a system of the structure shown in Figure 1. In (Osterberg, 1993) we used on-line optimization techniques based on gradient ascent. We tried two different methods to select the step length parameter. One rather heuristical depending on the output On (t) of the filter function and one inverse proportional to the second partial derivative of the quality function with respect to w~ (t). In each iteration the length of each filter function was explicitly normalized to one. Currently, we investigate standard unconstrained optimization methods (Dennis & Schnabel, 1983) based on batch learning. Now the step length parameter is selected by line search in the search direction Set): mrc Q(W(t) + AS(t» (11) Typical choices of Set) include Set) = I and Set) = H-l. With the identity matrix we get Steepest Ascent and with the inverse Hessian the quasi-Newton algorithm. U sing sufficient synchronism the line search can be incorporated in the parallel structure (Figure 1). To incorporate the quasi-Newton algorithm we have to assume 140 Osterberg and Lenz Inpul pall.ra P(I) --...---.1 OuIPl'I ~----.,.--+ 0.(1) - .... '(1)1'(1) Oulpul 0,(1) - ... ,'(1)1'(1) QuIP'" ~--+-++--+ o . .{t) - ...... (1)1'(1) Figure 1: The architecture of the filter system that the Hessian matrix is block diagonal, i.e. the second partial derivatives with respect to wr(t)w,(t), k f. I, "1m are assumed to be zero. In general this is not the case and it is not clear if a block diagonal approximation is valid or not. The second partial derivatives can be approximated by secant methods (normally the BFGS method). Furthermore the condition of normalized filter functions can be achieved by optin4izing in hyperspherical polar coordinates. Preliminary experiments (mustly with Steepest Ascent) show that more advanced optimization techniques lead to a more robust convergence of the filter functions. 4 Experiments In (Osterberg, 1993) we describe a series of experiments in which we investigate systematically the following properties of the MS-system, the KL-system and the FO-system: convergence speed, dependence on initial solution W(O) , distance between learned solution and optimal (orthogonal) solution, supervised classification of the extracted feature vectors using linear regression and the degree of selective response of the learned filter functions. We use training sets with controlled scalar products between the cluster centers of three classes of input patterns embedded in a 32-D space. The results of the experiments can be summarized as follows. In contrast to the MS-system, we noticed that the KL- and FO-system had problems to converge to the optimal orthogonal solutions for some initial solutions. All systems learned orthogonal solutions regardless of W(O). The supervised classification power was independent of the filter system used. Only the FO-system produced Unsupervised Parallel Feature Extraction from First Principles 141 Table 1: Typical filter response to patterns from (a)-(c) Tsetl and (d) Tset2 using the filter functions learned with (a) the KL-system, (b) the FO-system and (c)-(d) the MV-system. (e)-(f) Output covariance matrix using the filter functions learned with (e) the KL-system and (f) the MV-system. [( -0.12) (-0.46) (0.73)] 0.92 , 0.83 , 0.66 -0.38 0.32 0.14 (a.) [( 0.28) ( 0.10) ( 0.98)] -0.91 , -0.39 , -0.23 0.44 0.95 0.11 ( 0.0340 0.0001 0.0005 (c) 0.0001 0.0005) 0.9300 0.0000 0.0000 0.0353 (e) [ ( -0.71) (-0.99) (-0.22)] 0.59 , -0.08 , -0.04 0.28 0.01 0.97 (b) [ ( -0.50) (-0.49) (-0.81)] -0.80 , -0.50 , -0.49 0.50 0.81 0.50 ( 0.3788 0.3463 -0.3473 (d) 0.3463 0.3760 -0.3467 (f) -0.3473 ) -0.3467 0.3814 filter functions which mainly react for patterns from just one class and only if the similarity (measured by the scalar product) between the classes in the training set was smaller than approximately 0.5. Thus, the FO-system extracts feature vectors which have unsupervised discrimination power. Furthermore, we showed that the FO-system can distinguish between data sets having identical correlation matrices (second order statistics) but different fourth order statistics. Recent experiments with more advanced optimization techniques (Steepest Ascent) show better convergence properties for the KL- and FO-system. Especially the distance between the learned filter functions and the optimal orthogonal ones becomes smaller. We will describe some experiments which show that the MV-system is more suitable for tasks requiring unsupervised classification. We use two training sets Tsetl and Tset2. In the first set the mean scalar product between class one and two is 0.7, between class one and three 0.5 and between class two and three 0.3. In the second set the mean scalar products between all classes are 0.9, i.e. the angle between all cluster centers is arccos(0.9) = 26 0 • In Table 4(a)-( c) we show the filter response of the learned filter functions with the KL-, FO- and MV-system to typical examples of the input patterns in the training set Tsetl. For the KL-system we see that the second filter function gives the largest magnitude response for both, patterns from class one and two. For the FO-system the feature vectors are more binary. Still the first filter function has the largest magnitude response for patterns from class one and two. For the MV -system we see that each filter function has largest magnitude response for only one class of input patterns and thus the extracted feature vectors support unsupervised discrimination. In Table 4( d) (computed from Tset2) we see that this is the case even then the scalar products between the cluster centers are as high as 0.9. The filter functions learned by the MV -system are approximately orthogonal. The system learns thus the rotation of the largest eigenvectors which maximizes QVa.r(t). Therefore it will not extract uncorrelated features (see Ta142 Osterberg and Lenz 02 015 0.1 o 02 ........ '(\\ I • , -' ,", , , , , , , , , , , , (a) , '. '. " ',- - ... , , ISO , \ ..... . , ............... ,/ .•..•• 1.,' ,'., I I , , , , , I ..••.• " ,-. I I , , , ", " " ......... . , . . .),' ,', ./ " " ''II. .. (c) , . • 'lUG 025 02 0.5 o. OOS 0 -0 os -0' -0 '5 -02 -o~ (b) 031~--~--~--~--~--~--~ .. ...... .. 02 0.6 O' 006 ,', I , I , I ' , : ' , , ... , \ , t ". " \ I : , : , , , , , , , ,~. \ , .. \ ,,' \. " .................... . ~~~~~~~~~~~~_~-4I~m--~~ (d) Figure 2: (a) Examples of normalized reflectance spectra of typical reddish (solid curve), greenish (dotted curve) and bluish (dashed curve) Munsell color chips. (b) The three largest eigenvectors belonging to the correlation matrix of the 1253 different reflectance spectra. (c) The learned filter functions with the MV-system. (d) The learned non-negative filter functions with the MV-system. In all figures the x-axes show the wave length (nm) ble 4(f» but the variances (e.g. the diagonal elements of the covariance matrix) of the features are more or less equal. In Table 4( e) we see that the KL-system extracts uncorrelate features with largely different variance. This demonstrates that the KL-system tries to learn the pure eigenvectors. Recently, we have applied the MV-system to real world data. The training set consists of normalized reflectance spectra of the 1253 different color chips in the Munsell color atlas. Figure 2(a) shows one typical example of a red, a green and a blue color chip and Figure 2(b) the three largest eigenvectors belonging to the correlation matrix of the training set. We see that the first eigenvector (the solid curve) has a more or less uniform response for all different colors. On the other hand, the MV -system (Figure 2 (c» learns one bluish, one greenish and one reddish filter function. Thus, the filter functions divide the color space according to the primary colors red, green and blue. We notice that the learned filter functions are orthogonal and tend to span the same space as the eigenvectors since IIW.ol - RoptUeigliF = 0.0199 (the Frobenius norm) where Ropt maximizes QVa.r(t). Figure 2(d) show one preliminary attempt to include the condition of non-negative filter functions in the Unsupervised Parallel Feature Extraction from First Principles 143 optimization process (Steepest Ascent). We see that the learned filter functions are non-negative and divide the color space according to the primary colors. One possible real word application is optical color analysis where non-negative filter functions are much easier to realize using optical components. Smoother filter functions can be optained by incorporating additional constraints into the quality function. 5 Non-linear extensions The proposed strategy to extract feature vectors apply to nonlinear filter systems as well. In this case the input output relation OCt) = W(t)P(t) is replaced by OCt) = I(W(t)P(t» where I describes the desired non-linearity. The corresponding learning rule can be derived using gradient based techniques as long as the non-linearity 1(·) is differentiable. The exact form of 1(,) will usually be application oriented. Node nonlinearities of sigmoid type are one type of nonlinearities which has received a lot of attention (see for example Oja & Karhunen, 1993). Typical applications include: robust Principal Component Analysis PCA (outlier protection, noise suppression and symmetry breaking), sinusoidal signal detection in colored noise and robust curve fitting. Acknowledgements This work was done under TFR-contract TFR-93-00192. The visit of M. Osterberg at the Dept. of Info. Tech., Lappeenranta University of Technology was supported by a grant from the Nordic Research Network in Computer Vision. The Munsell color experiments were performed during this visit. References R. B. Darlington. (1970) Is Kurtosis really peakedness? American Statistics 24(2):19-20. J. E. Dennis & Robert B. Schnabel. (1983) Numerical Methods lor Unconstrained Optimization and Nonlinear Equations. Prentice-Hall. N. Intrator & L.N. Cooper. (1992) Objective Function Formulation of the BCM Theory of Visual Cortical Plasticity: Statistical Connections, Stability Conditions. Neural Networks 5:3-17. R. Lenz & M. Osterberg. (1992) Computing the Karhunen-Loeve expansion with a parallel, unsupervised filter system. Neural Computations 4(3):382-392. E. Oja. (1992) Principal Components, Minor Components, and Linear Neural Networks. Neural Networks 5:927-935. E. Oja & J. Karhunen. (1993) Nonlinear PCA: algorithms and Applications Technical Report AlB, Helsinki University 01 Technology, Laboratory of Computer and Information Sciences, SF -02150 Espoo, Finland. M. Osterberg. (1993) Unsupervised Feature Extraction using Parallel Linear Filters. Linkoping Studies in Science and Technology. Thesis No. 372.
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Classification of Electroencephalogram using Artificial Neural Networks A C Tsoi*, D S C So*, A Sergejew** *Department of Electrical Engineering **Department of Psychiatry University of Queensland St Lucia, Queensland 4072 Australia Abstract In this paper, we will consider the problem of classifying electroencephalogram (EEG) signals of normal subjects, and subjects suffering from psychiatric disorder, e.g., obsessive compulsive disorder, schizophrenia, using a class of artificial neural networks, viz., multi-layer perceptron. It is shown that the multilayer perceptron is capable of classifying unseen test EEG signals to a high degree of accuracy. 1 Introduction The spontaneous electrical activity of the brain was first observed by Caton in 1875. Although considerable investigations on the electrical activity of the non-human brain have been undertaken, it was not until 1929 that a German neurologist Hans Berger first published studies on the electroencephalogram (EEG) recorded on the scalp of human. He lay the foundation of clinical and experimental applications of EEG between 1929 and 1938. Since then EEG signals have been used in both clinical and experimental work to discover the state which the brain is in (see e.g., Herrmann, 1982, Kolb and Whishaw, 1990, Lindsay and Holmes, 1984). It has served as a direct indication of any brain activities. It is routinely being used in clinical diagnosis of epilepsy (see e.g., Basar, 1980; Cooper, 1980). Despite advances in technology, the classification of EEG signals at present requires a trained personnel who either "eyeballs" the direct EEG recordings over time, 1151 1152 Tsoi, So, and Sergejew or studies the contour maps representing the potentials generated from the "raw" electrical signal (see e.g., Cooper, 1980). This is both a highly skillful job, as well as a laborious task for a neurologist. With the current advances in computers, a logical question to ask: can we use the computer to perform an automa'(.ic classification of EEG signals into different classes denoting the psychiatric states of the subjects? This type of classification studies is not new. In fact, in the late 1960's there were a number of attempts in performing the automatic classification using discriminant analysis techniques. However, this work was largely abandoned as most researchers concluded that classification based on discriminant techniques does not generalise well, i.e., while it has very good classification accuracies in classifying the data which is used to train the automatic classification system, it may not have high accuracy in classifying the unseen data which are not used to train the system in the first instance. Recently, a class of classification techniques, called artificial neural network (ANN), based on nonlinear models, has become very popular (see e.g., Touretzky, 1989, 1990, Lippmann et aI, 1991). This type of networks claims to be inspired by biological neurons, and their many inter-connections. This type of artificial neural networks has limited pattern recognition capabilities. Among the many applications which have been applied so far are sonar signal classification (see e.g., Touretzky, 1989), handwritten character recognition (see .e.g., Touretzky, 1990), facial expression recognition (see e.g., Lippmann et a1. 1991). In this paper, we will investigate the possibility of using an ANN for EEG classifications. While it is possible to extract features from the time series using either time domain or frequency domain techniques, from some preliminary work, it is found that the time domain techniques give much better results. The structure of this paper is as follows: In section 2, we will give a brief discussion on a popular class of ANNs, viz., multi-layer perceptrons (MLP). In section 3, we will discuss various feature extractions using time domain techniques. In section 4, we will present results in classifying a set of unseen EEG signals. 2 Multi-layer Perceptrons Artificial neural network (ANN) consists of a number of artificial neurons interconnected together by synaptic weights to form a network (see e.g, Lippmann, 1987). Each neuron is modeled by the following mechanical model: n y = f(L WiXi + 0) (1) i=l where y is the output of the neuron, Wi, i = 1,2, ... , n are the synaptic weights, Xi, i = 1,2 ... , n are the inputs, and 0 is a threshold function. The nonlinear function f(.) can be a sigmoid function, or a hyperbolic tangent function. An ANN is a network of inter-connected neurons by synapses (Hertz, Krogh and Palmer, 1991). There are many possible ANN architectures (Hertz, Krogh, Palmer, 1991). A popClassification of Electroencephalogram Using Artificial Neural Networks 1153 ular architecture is the multi-layer perceptron (MLP) (see e.g., Lippmann, 1987). In this class of ANN, signal travels only in a forward direction. Hence it is also known as a feedforward network. Mathematically, it can be described as follows: Y = !(Az + 0ll) z=!(Bu+Oz ) (2) (3) where y is a m x 1 vector, representing the output of the output layer neurons; z is a p x 1 vector, representing the outputs of the hidden layer neurons; u is a n x 1 vector, representing the input feature vector; OJ! is a m x 1 vector, known as the threshold vector for the output layer neurons; Oz is a p x 1 vector, representing the threshold vector for the hidden layer neurons; A and B are matrices of m x p and p x n respectively. The matrices A, and B are the synaptic weights connecting the hidden layer neuron to the output layer neuron; and the input layer neurons, and the hidden layer neurons respectively. For simplicity sake, we will assume the nonlinearity function to be a sigmoid function, i.e., 1 f(a)=I+e- a (4) The unknown parameters A, B, OJ!, Oz can be obtained by minimizing an error criterion: p J = L(di - Yi)2 (5) .=1 where P is the total number of examplars, di , i = 1,2, ... , P are the desired outputs which we wish the MLP to learn. By differentiating the error criterion J with respect to the unknown parameters, learning algorithms can be obtained. The learning rules are as follows: (6) where Anew is the next estimate of the matrix A, T denotes the transpose of a vector or a matrix. TJ is a learning constant. A(y) is a m x m diagonal matrix, whose dia~onal elements are / (Y')' i = 1,2, ... , m. The vector e is m x 1, and it is given by e = [(d1 - yd, (d2 - Y2), ... , (dm - Ym)]T. The updating equation for the B matrix is given by the following (7) where 6 is a p x 1 vector, given by 1154 Tsoi, So, and Sergejew fJ = AT A(y)e and the other parameters are as defined above. The threshold vectors can be obtained as follows: and (8) (9) (10) Thus it is observed that once a set of initial conditions for the unknown parameters are given, this algorithm will find a set of parameters which will converge to a value, representing possibly a local minimum of the error criterion. 3 Pre-processing of the EEG signal A cursory glance at a typical EEG signal of a normal subject, or a psychiatrically ill subject would convince anyone that one cannot hope to distinguish the signal just from the raw data alone. Consequently, one would need to perform considerable feature extraction (data pre-processing) before classification can be made. There are two types of simple feature extraction techniques, viz., frequency domain and time domain (see e.g., Kay, 1988, Marple, 1987). In the frequency domain, one performs a fast Fourier transform (FFT) on the data. Often it is advantageous to modify the signal by a window function. This will reduce the sidelobe leakage (Kay, and Marple, 1981, Harris, 1978). it is possible to use the average spectrum, obtained by averaging the spectrum over a number of frames, as the input feature vector to the MLP. In the time domain, one way to pre-process the data is to fit a parametric model to the underlying data. There are a number of parametric models, e.g., autoregressive (AR) model, an autoregressive moving average (ARMA) model (see e.g., Kay, 1988, Marple, 1987). The autoregressive model can be described as follows: N Se = L OjSe_j + fe j=1 (11) where Se is the signal at time t; ft is assumed to be a zero mean Gaussian variable with variance (T2. The unknown parameters OJ, j = 1,2, ... , N describe the spectrum of the signal. They can be obtained by using standard methods, e.g., Yule-Walker equations, or Levinson algorithm (Kay, 1988, Marple, 1987). The autoregressive moving average (ARMA) model can be seen as a parsimonious model for an AR model with a large N. Hence, as long as we are not concerned Classification of Electroencephalogram Using Artificial Neural Networks 1155 about the interpretation of the AR model obtained, there is little advantage to use the more complicated ARMA model. Subsequently, in this paper, we will only consider the AR models. Once the AR parameters are determined, then they can be used as the input features to the MLP. It is known that the AR parametric model basically produces a smoothed spectral envelope (Kay, 1988, Marple, 1987). Thus, the model parameters of AR is another way to convey the spectral information to the MLP. This information is different in quality to that given by the FFT technique in that the FFT transforms both signal and noise alike, while the parametric models tend to favor the signal more and is more effective in suppressing the noise effect. In some preliminary work, we find that the frequency domain extracted features do not give rise to good classification results using MLP. Henceforth we will consider only the AR parameters as input feature vectors. 4 Classification Results In this section, we will summarise the results of the experiments in using the AR parametric method of feature extraction as input parameters to the MLP. We obtained EEG data pertaining to normal subjects, subjects who have been diagnosed as suffering from severe obsessive compulsive disorder (OCD), and subjects who have been diagnosed as suffering from severe schizophrenia. Both the OCD and the schizophrenic subjects are under medication. The subjects are chosen so that their medication as well as their medical conditions are at a steady state, i.e., they have not changed over a long period of time. The diagnosis is made by a number of trained neurologists. The data files are chosen only if the diagnosis from the experts concur. We use the standard 10-20 recording system (Cooper, 1980), i.e., there are 19 channels of EEG recording, each sampled at 128 Hz. The recording were obtained while the subject is at rest. Some data screening has been performed to screen out the segment of data which contains any artifact. In addition, the data is anti-aliased first by a low pass filter before being sampled. The sampled data is then low pass filtered at 30 Hz to get rid of any higher frequency components. We have chosen one channel, viz., the Cz channel (the channel which is the recording of the signal at the azimuth of the scalp). This channel can be assumed to be representative of the brain state from the overall EEG recording of the scalp. 1 This time series is employed for feature extraction purposes. For time domain feature extraction, we first convert the time series into a zero mean one. Then a data frame of one second duration is chosen 2 as the basic time segmentation of the series. An AR model is fitted to this one second time frame to 1 From some preliminary work, it can be shown that this channel can be considered as a linear combination of the other channels, in the sense that the prediction error variance is small. 2It has been found that the EEG signal is approximately stationary for signal length of one second. Hence employing a data frame width of one second ensures that the underlying assumptions in the AR modelling technique are valid (Marple, 1988) 1156 Tsoi, So, and Sergejew extract a feature vector formed by the resulting AR coefficients. An average feature vector is acquired from the first 250 seconds, as in practice, the first 250 seconds usually represent a state of calm in the patient, and therefore the EEG is less noisy. After the first 250 seconds, the patient may enter an unstable condition, such as breathing faster and muscle contraction which can introduce artifacts. We use an AR model of length between 8 to 15. We have chosen 15 such data file to form our training data set. This consists of 5 data files from normal subjects, 5 from OeD subjects, and 5 from subjects suffering from schizophrenia. In the time domain extracted feature vectors, we use a MLP with 8 input neurons, 15 hidden layer neurons, and 3 output neurons. The MLP's are trained accordingly. We use a learning gain of 0.01. Once trained, the network is used to classify unseen data files. These unseen data files were pre-classified by human experts. Thus the desired classification of the unseen data files are known. This can then be used to check the usefulness of the MLP in generalising to unseen data files. The results 3 are shown in table 1. The unseen data set consists of 6 normal subjects, 8 schizophrenic subjects, and 10 obsessive compulsive disorder subjects. It can be observed that the network correctly classifies all the normal cases, makes one mistake in classifying the schizophrena cases, and one mistake in classifying the OeD cases. Also we have experimented on varying the number of hidden neurons. It is found that the classification accuracy does not vary much with the variation of hidden layer neurons from 15 to 50. We have also applied the MLP on the frame by frame data, i.e., before they are being averaged over the 250 second interval. However, it is found that the classification results are not as good as the ones presented. We were puzzled by this result as intuitively, we would expect the frame by frame results to be better than the ones presented. A plausible explanation for this puzzle is given as follows: the EEG data is in general quite noisy. In the frame by frame analysis, the features extracted may vary considerably over a short time interval, while in the approach taken here, the noise effect is smoothed out by the averaging process. One may ask: why would the methods presented work at all? In traditional EEG analysis (Lindsay & Holmes, 1984), FFT technique is used to extract the frame by frame frequency responses. The averaged frequency response is then obtained over this interval. Traditionally only four dominant frequencies are observed, viz., the "alpha", "beta", "delta", and "theta" frequencies. It is a basic result in EEG research that these frequencies describe the underlying state of the subject. For example, it is known that the "alpha" wave indicates that the subject is at rest. An EEG technologist uses data in this form to assist in the diagnosis of the subject. On the other hand, it is relatively well known in signal processing literature (Kay, 3The results shown are typical results. We have used different data files for training and testing. In most cases, the classification errors on the unseen data files are small, similar to those presented here. Classification of Electroencephalogram Using Artificial Neural Networks 1157 original activation of activation of activation of predicted classes normal schiz ocd classes normall 0.905 0.008 0.201 normal normal2 0.963 0.006 0.103 normal normal3 0.896 0.021 0.086 normal normal4 0.870 0.057 0.020 normal normal5 0.760 0.237 0.000 normal norma16 0.752 0.177 0.065 normal schiz1 0.000 0.981 0.042 schiz schiz2 0.000 0.941 0.163 schiz schiz3 0.002 0.845 0.050 schiz schiz4 0.015 0.989 0.004 schiz schiz5 0.000 0.932 0.061 schiz schiz6 0.377 0.695 0.014 schiz schiz7 0.062 0.898 0.000 schiz schiz8 0.006 0.086 0.921 ocd ocdl 0.017 0.134 0.922 ocd ocd2 0.027 0.007 0.940 ocd ocd3 0.000 0.033 0.993 ocd ocd4 0.000 0.014 0.997 ocd ocd5 0.015 0.138 0.889 ocd ocd6 0.000 0.150 0.946 ocd ocd7 0.002 0.034 0.985 ocd ocd8 0.006 0.960 0.003 schiz ocd9 0.045 0.005 0.940 ocd ocdlO 0.085 0.046 0.585 ocd Table 1: Classification of unseen EEG data files 1988, Marple, 1987) to view the AR model as indicative of the underlying frequency content of the signal. In fact, an 8th order AR model indicates that the signal can be considered to consist of 4 underlying frequencies. Thus, intuitively, the 8th order AR model averaged over the first 250 seconds represents the underlying dominant frequencies in the signal. Given this interpretation, it is not surprising that the results are so good. The features extracted are similar to those used in the diagnosis of the subjects. The classification technique, which in this case, the MLP, is known to have good generalisation capabilities (Hertz, Krogh, Palmer, 1991). This contrasts the techniques used in previous attempts in the 1960's, e.g., the discriminant analysis, which is known to have poor generalisation capabilities. Thus, one of the reasons why this approach works may be attributed to the generalisation capabilities of the MLP. 5 Conclusions In this paper, a method for classifying EEG data obtained from subjects who are normal, OCD or schizophrenia has been obtained by using the AR parameters as 1158 Tsoi, So, and Sergejew input feature vectors. It is found that such a network has good generalisation capabili ties. 6 Acknowledgments The first and third author wish to acknowledge partial financial support from the Australian National Health and Medical Research Council. In addition, the first author wishes to acknowledge partial financial support from the Australian Research Council. 7 References Basar, E. (1980). EEG-Brain Dynamics - Relation between EEG and Brain Evoked Potentials. Elsevier/North Holland Biomedical Press. Cooper, R. (1980). EEG Technology. Butterworths. Third Editions. Harris, F.J. (1978). "On the Use of windows for Harmonic Analysis with the Discrete Fourier Transform". Proceedings IEEE. Vol. 66, pp 51-83. Herrmann, W.M. (1982). Electroencephalography in Drug Research. Butterworths. Hertz, J. Krogh, A, Palmer, R. (1991) Introduction to The Theory of Neural Computation. Addison Wesley, Redwood City, Calif. Kay, S.M., Marple, S.L., Jr. (1981). "Spectrum Analysis - A Modern Perspective". Proceeding IEEE. Vol. 69, No. 11, Nov. pp 1380 - 1417. Kay, S.M. (1988) Modern Spectral Estimation - Theory and Applications Prentice hall. Kolb, B., Whishaw, I.Q. (1990). Fundamentals of Human Neuropsychology. Freeman, New York. Lindsay, D.F., Holmes, J.E. (1984). Basic Human Neurophysiology. Elsevier. Lippmann, R.P. (1987) " An introduction to computing with neural nets" IEEE Acoustics Speech and Signal Processing Magazine. Vol. 4, No.2, pp 4-22. Lippmann, R.P., Moody, J., Touretzky, D.S. (Ed.) (1991). Advances in Neural Information Processing Systems 9. Morgan Kaufmann, San Mateo, Calif. Marple, S.L., Jr. (1987). Digital Spectral Analysis with Applications. Prentice Hall. Touretzky, D.S. (Ed.) (1989). Advances in Neural Information Processing Systems 1. Morgan Kaufmann, San Mateo, Calif. Touretzky, D.S. (Ed.) (1990). Advances in Neural Information Processing Systems 2. Morgan Kaufmann, San Mateo, Calif. PART XII WORKSHOPS
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Comparisoll Training for a Resclleduling Problem ill Neural Networks Didier Keymeulen Artificial Intelligence Laboratory Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels Belgium Abstract Martine de Gerlache Prog Laboratory Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels Belgium Airline companies usually schedule their flights and crews well in advance to optimize their crew pools activities. Many events such as flight delays or the absence of a member require the crew pool rescheduling team to change the initial schedule (rescheduling). In this paper, we show that the neural network comparison paradigm applied to the backgammon game by Tesauro (Tesauro and Sejnowski, 1989) can also be applied to the rescheduling problem of an aircrew pool. Indeed both problems correspond to choosing the best solut.ion from a set of possible ones without ranking them (called here best choice problem). The paper explains from a mathematical point of view the architecture and the learning strategy of the backpropagation neural network used for the best choice problem. We also show how the learning phase of the network can be accelerated. Finally we apply the neural network model to some real rescheduling problems for the Belgian Airline (Sabena). 1 Introduction Due to merges, reorganizations and the need for cost reduction, airline companies need to improve the efficiency of their manpower by optimizing the activities of their crew pools as much as possible. A st.andard scheduling of flights and crews is usually made well in advance but many events, such as flight delays or the absence of a crew member make many schedule cha.nges (rescheduling) necessary. 801 802 Keymeulen and de Gerlache Each day, the CPR 1 team of an airline company has to deal with these perturbations. The problem is to provide the best answer to these regularly occurring perturbations and to limit their impact on the general schedule. Its solution is hard to find and usually the CPR team calls on full reserve crews. An efficient rescheduling tool taking into account the experiences of the CPR team could substantially reduce the costs involved in rescheduling notably by limit.ing the use of a reserve crew. The paper is organized as follow. In the second section we describe the rescheduling task. In the third section we argue for the use of a neural network for the rescheduling task and we apply an adequate architecture for such a network. Finally in the last section, we present results of experiments with schedules based on actual schedules used by Sabena. 2 Rescheduling for an Airline Crew Pool When a pilot is unavailable for a flight it becomes necessary to replace him, e.g. to reschedule the crew. The rescheduling starts from a list of potential substitute pilots (PSP) given by a scheduling program based generally on operation research or expert syst.em technology (Steels, 1990). The PSP list obtained respects legislation and security rules fixing for example t.he number of flying hours per month, the maximum number of consecutive working hour and the number of training hours per year and t.heir schedule. From the PSP list, the CPR team selects the best candidat.es taking into account t.he schedule stability and equity. The schedule stability requires that possible perturbations of the schedule can be dealt with with only a minimal rescheduling effort. This criterion ensures work stability t.o the crew members and has an important influence on their social behavior. The schedule equity ensures the equal dist.ribution of the work and payment among the crew members during the schedule period. One may think to solve this rescheduling problem in t.he same way as the scheduling problem itself using software t.ools based on operational research or expert system approach. But t.his is inefficient. for t.wo reasons, first., the scheduling issued from a scheduling system and its adapt.at.ion t.o obt.ain an acceptable schedule takes days. Second this system does not t.ake into account the previous schedule. It follows that the updat.ed one may differ significantly from the previous one after each perturbation. This is unaccept.able from a pilot's point of view. Hence a specific procedure for rescheduling is necessary. 3 Neural Network Approach The problem of reassigning a new crew member to replace a missing member can be seen as the problem of finding the best pilot in a pool of potential substitute pilots (PSP), called the best choice problem. To solve the best choice problem, we choose the neural network approach for two reasons. First the rules llsed by the expert. are not well defined: to find the best PSP, lCrew Pool Rescheduler Comparison Training for a Rescheduling Problem in Neural Networks 803 the expert associates implicit.ly a score value to each profile. The learning approach is precisely well suited to integrate, in a short period of time, t.he expert knowledge given in an implicit form. Second, t.he neural network approach was applied with success to board-games e.g. the Backgammon game described by Tesauro (Tesauro and Sejnowski, 1989) and the Nine l\llen's Morris game described by Braun (Braun and al., 1991). These two games are also exa.mples of best choice problem where the player chooses the best move from a set of possible ones. 3.1 Profile of a Potential Substitute Pilot To be able to use the neural network approach we have to identify the main features of the potential substitute pilot and to codify them in terms of rating values (de Gerlache and Keymeulen, 1993). We based our coding scheme on the way the expert solves a rescheduling problem. He ident.ifies the relevant parameters associated with the PSP and the perturbed schedule. These parameters give three types of information. A first type describes the previous, present and future occupation of the PSP. The second t.ype represents information not in the schedule such as the human relationship fadars. The assocjat.ed values of t.hese two t.ypes of parameters differ for f'ach PSP. The last t.ype of paramet.ers describes the context of the rescheduling, namely t.he characteristics of t.he schedule. This last type of parameters are the same for all the PSP. All t.hese paramet.ers form the profile of a PSP associated to a perturbed schedule. At each rescheduling problem corresponds one perturbed schedule j and a group of 11 PSpi to which we associate a Projile~ = (PSpi, PertU1·berLSchedulej) . Implicitly, the expert associates a rating value between 0 and 1 to each parameter of the P1'ojile; based on respectively its little or important impact on the result.ing schedule if the P S pi was chosen. The rating value reflects the relative importance of the parameters on the stability and the equity of the resulting schf'dnle obt.ained after the pilots substitution. 3.2 Dual Neural Network It would have been possible to get more information from the expert than only the best profile. One of the possibilities is to ask him to score every profile associated with a perturbed planning. From this associat.ion we could immediately construct a scoring function which couples each profile with a specific value, namely its score. Another possibility is to ask the expert to rank all profiles associated with a perturbed schedule. The corresponding ranking function couples each profile with a value such that the values associat.ed with the profiles of the same perturbed schedule order the profiles according t.o t.heir rank. The decision making process used by the rescheduler team for the aircrew rescheduling problem does not consist in the evaluation of a scoring or ranking function . Indeed only the knowledge of the best profile is useful for the rescheduling process. From a neura.l net.work architectural point of view, because the ranking problem is a generalization of the best choice problem, a same neural net.work architecture can be used. But the difference between the best choice problem and t.he scoring problem is such that two different neural network architectures are associated to them. As we show in this section, although a backpropagatian network is sufficient to learn a scoring function, its architecture, its learning and its retrieval procedures must be 804 Keymeulen and de Gerlache adapted to learn the best profile. Through a mathematical formulation of the best choice problem, we show that the comparison paradigm of Tesauro (Tesauro, 1989) is suited to the best choice problem and we suggest how to improve the learning convergence. 3.2.1 Comparing Function For the best choice problem the expert gives the best profile Projilefest associated with the perturbed schedule j and that for m pert.urbed schedules. The problem consists then to learn the mapping of the m * n profiles associated with the m perturbed schedules into the m best profiles, one for each pert.urbed schedule. One way to represent this association is through a comparing function. This function has as input a profile, represented by a vector xj, and returns a single value. When a set of profiles associated with a perturbed schedule are evaluated by the function, it returns the lowest value for the best profile. This comparing function integrates the information given by the expert and is sufficient to reschedule any perturbed schedule solved in the past by the expert. Formally it is defined by: Comp(J.1>e) = C(Projile)) (1) C Best C ,. {V)' with)' = 1, ... ,111. omparej < ompcl1>Cj Vi=fBest with i=l, ... ,n The value of Comp(J.1>e) are not known a priori and have only a meaning when they are compared to the value Comp(J.1>efest of the comparing function for the best profile. 3.2.2 Geometrical Interpretation To illustrate the difference between the neural network learning of a scoring function and a comparing function, we propose a geometrical interpretation in the case of a linear network having as input vect.ors (profiles) XJ, ... ,XJ, ... ,Xp associated with a perturbed schedule j. The learning of a scoring function which associat.es a score Score; with each input vector xj consists in finding a hyperplane in the input vector space which is tangent to the circles of cent.er xf and radius SC01>e{ (Fig. 1). On the contrary the learning of a comparing function consists t.o obt.ain t.he equation of an hyperplane such that the end-point of the vector Xfest is nearer the hyperplane than the end-points of the other input vectors XJ associated with the same perturbed schedule j (Fig. 1). 3.2.3 Learning We use a neural network approach to build the comparing function and the mean squared error as a measure of the quality of t.he approximation. The comparing function is approximated by a non-linear function: C(P1>ojile;) = N£(W,Xj) where W is the weight. vector of the neural network (e.g backpropagat.ion network). The problem of finding C which has the property of (1) is equivalent to finding the function C that minimizes the following error function (Braun and al., 1991) where <I> is the sigmoid function : Comparison Training for a Rescheduling Problem in Neural Networks 805 I ... --_ .. _,-"""", , , x,, -.,' ,.,1' , \ , , , , I I , , , , , , , " x..,,, ', ....... --,' W(Wl ,w2'''')' .. ,WL) wlh .w. - 1 Figure 1: Geometrical Interpretation of the learning of a Scoring Function (Rigth) and a Comparing Function (Left) n I: (2) i = 1 i -::f Best To obtain t.he weight vector which mll1UTIlzes the error funct.ion (2), we use the property that t.he -gr~ld£~(W) point.s in the direct.ion in which the error function will decrease at the fastest possible rate. To update t.he weight we have thus to calculate the partial derivative of (2) with each components of the weight vector ltV: it is made of a product of three factors. The evaluation of the first two factors (the sigmoid and the derivative of the sigmoid) is immediate. The third factor is the partial derivative of the non-linear function N £, which is generally calculated by using the generalized delta rule learning law (Rumelhart. and McClelland, 1986), Unlike the linear associator network, for the backpropagation network, the error function (2) is not equivalent to the error function where the difference Xl e3t - X; is associated with the input vector of the backpropagation network because: (3) By consequence to calculate t.he three factors of the partial derivative of (2), we have to introduce separately at the bottom of the network t.he input vector of the best profile X !e3t and the input vector of a less good profile XJ. Then we have to memorize theIl' partial contribution at each node of the network and multiply their contributions before updating the weight. Using this way to evaluate the derivative of (2) and to update t.he weight, the simplicity of the generalized delta rule learning law has disappeared. 806 Keymeulen and de Gerlache 3.2.4 Architecture Tesauro (Tesauro and Sejnowski, 1989) proposes an architecture, t.hat we call dual neural network, and a learning procedure such that the simplicity of the generalized delta rule learning law can still be used (Fig. 2). The same kind of architecture, called siamese network, was recently used by Bromley for the signature verification (Bromley and al., 1994). The dual neural network architecture and the learning strategy are justified mathematically at one hand by the decomposition of the partial derivative of the error function (2) in a sum of two terms and at the other hand by the asymmetry property of the sigmoid and its derivative. The architecture of the dual neural network consists to duplicate the multi-layer network approximating the comparing function (1) and to connect the output of both to a unique output node through a positive unit weight for the left network and negative unit weight. for the right network. During the learning a couple of profiles is presented to the dual neural network: a best profile X f e3t and a less good profile X!. The desired value at the output node of the dual neural network is 0 when the left network has for input the best profile and the right network has for input a less good profile and 1 when these profiles are permuted. During the recall we work only with one of the two multi-layer networks, suppose the left one (the choice is of no importance because they are exactly the same). The profiles JY~ associated with a perturbed schedule j are presented at the input of the left. multi-layer network. The best profile is the one having the lowest value at the output of the left multi-layer network. Through this mathematical formulation we can use the suggestion of Braun to improve the learning convergence (Braun and al., 1991). They propose to replace the positive and negative unit weight het.ween the output node of the multi-layer networks and the output. node of the dual neural network by respect.ively a weight value equal to V for the left net.work and - V for the right. network. They modify the value of V by applying the generalized delt.a rule which has no significant impact on the learning convergence. By manually increasing the factor V during the learning procedure, we improve considerably the learning convergence due to its asymmetric impact on the derivative of £<I>(W) with W: the modification of the weight vector is greater for couples not yet learned than for couples already learned. 4 Results The experiments show the abilit.y of our model to help the CPR team of the Sabena Belgian Airline company to choose the best profile in a group of PSPs based on the learned expertise of the team. To codify the profile we identify 15 relevant parameters. They constitute the input of our neural network. The training data set was obtained by analyzing the CPR team at work during 15 days from which we retain our training and test perturbed schedules. We consider that the network has learned when the comparing value of the best profile is less than the comparing value of t.he other profiles and that for all training perturbed schedules. At that time £cJ>(W) is less t.han .5 for every couple of profiles. The left graph of Figure 3 shows t.he evolution of t.he mean error over t.he couples Comparison Training for a Rescheduling Problem in Neural Networks 807 C_ .. n Belt. Dual Neural Network ·1 C_,.n 'J .1Ir.(~.l ) BeltoJ ZMulI.La,'"" -'-od ..... .11 r.(~. t .) ,~ -+ -+ (X .X ) But~ toJ Nouol NoI",ort.o 10.6lo.~ 10,910", 10.110.710.11 XB .... I I63IO.2!O-.1 b.61 0.21 0.91 0.21 X2•1 10.310.210.11 0.61 0.21 0.91 0.21 X 2•1 10,616316.91 631 0.11 6.71 0.21 X B .... I • wdb XIJ = X But.l it.t ) 10.610.710.1/ 0.11 0.711.01 0.21 XB ..... lo. 1Io.9Io.7! 0.910.110.11 0.&1 X I .. ,~ B.... 10.11 0,91 0.71 0.91 0.11 0.11 0,61 X I.. 10.610.710.1( 0.11 0.711.0 I 0.11 XB ..... Wllb 'S .. = XBe ... Figure 2: The training of a dual neural network. during the training. The right graph shows the improvement of the convergence when the weight V is increased regularly during the training process. OJ OA 0.1 M~--~~~---------------------0.1 liIl) liIO Nwnbcr of'lhilitg lnclasilll V or the Dull NaIl'll Ndworll o.s 0 .• OJ 02 ~--~~-----------------------01 Nunt.crci ~~~~~~~~~~~~~~~~ nW~ 111(1) IlO) Figure 3: Convergence of the dual neural network architecture. The network does not converge when we introduce contradictory decisions in our training set. It is possible to resolve them by adding new context parameters in the coding scheme of the profile. After learning, our network shows generalization capacity by retrieving the best profile for a new perturbed schedule that is similar to one which has already been learned. The degree of similarity required for the generalization remains a topic for further study. 808 Keymeulen and de Gerlache 5 Conclusion In conclusion, we have shown that the rescheduling problem of an airline crew pool can be stated as a decision making problem, namely the identification of the best potential substitute pilot. We have stressed the importance of the codification of the information used by the expert to evaluate the best candidate. We have applied the neural network learning approach to help the rescheduler team in the rescheduling process by using the experience of already solved rescheduling problems. By a mathematical analysis we have proven the efficiency of the dual neural network architecture. The mathematical analysis permits also to improve the convergence of the network. Finally we have illustrated the method on rescheduling problems for the Sabena Belgian Airline company. Acknowledgments We thank the Scheduling and Rescheduling team of Mr. Verworst at Sabena for their valuable information given all along this study; Professors Steels and D'Hondt from the VUB and Professors Pastijn, Leysen and Declerck from the Military Royal Academy who supported this research; Mr. Horner and Mr. Pau from the Digital Europe organization for their funding. We specially thank Mr. Decuyper and Mr. de Gerlache for their advices and attentive reading. References H. Braun, J. Faulner & V. Uilrich. (1991) Learning strategies for solving the problem of planning using backpropagation. In Proceedings of Fourth International Conference on Neural Networks and their Applications, 671-685. Nimes, France. J. Bromley, I. Guyon, Y. Lecun, E. Sackinger, R. Shah . (1994). Signature verification using a siamese delay neural network. In J. Cowan, G. Tesauro & J. Alspector (eds.), Advances in Neural Information Processing Systems 1. San Mateo, CA: Morgan Kaufmann. M. de Gerlache & D. Keymeulen. (1993) A neural network learning strategy adapted for a rescheduling problem. In Proceedings of Fourth International Conference on Neural Networks and their Applications, 33-42. Nimes, France. D. Rumelhart & J. McClelland. (1986) Parallel Distributed Processing: Explorations in the Microstructure of Cognition I [1 II. Cambridge, MA: MIT Press. L. Steels. (1990) Components of expertise. AI Maga.zine, 11(2):29-49. G. Tesauro. (1989) Connectionist learning of expert preferences by comparison training. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 1, 99-106. San Mateo, CA: Morgan Kaufmann. G. Tesauro & T.J. Sejnowski. (1989) A parallel network that learns to play backgammon. Artificial Intelligence, 39:357-390.
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Bounds on the complexity of recurrent neural network implementations of finite state machines Bill G. Horne NEC Research Institute 4 Independence Way Princeton, NJ 08540 Don R. Hush EECE Department University of New Mexico Albuquerque, NM 87131 Abstract In this paper the efficiency of recurrent neural network implementations of m-state finite state machines will be explored. Specifically, it will be shown that the node complexity for the unrestricted case can be bounded above by 0 ( fo) . It will also be shown that the node complexity is 0 (y'm log m) when the weights and thresholds are restricted to the set {-I, I}, and 0 (m) when the fan-in is restricted to two. Matching lower bounds will be provided for each of these upper bounds assuming that the state of the FSM can be encoded in a subset of the nodes of size rlog m 1. 1 Introduction The topic of this paper is understanding how efficiently neural networks scale to large problems. Although there are many ways to measure efficiency, we shall be concerned with node complexity, which as its name implies, is a calculation of the required number of nodes. Node complexity is a useful measure of efficiency since the amount of resources required to implement or even simulate a recurrent neural network is typically related to the number of nodes. Node complexity can also be related to the efficiency of learning algorithms for these networks and perhaps to their generalization ability as well. We shall focus on the node complexity of recurrent neural network implementations of finite state machines (FSMs) when the nodes of the network are restricted to threshold logic units. 359 360 Home and Hush In the 1960s it was shown that recurrent neural networks are capable of implementing arbitrary FSMs. The first result in this area was due to Minsky [7], who showed that m-state FSMs can be implemented in a fully connected recurrent neural network. Although circuit complexity was not the focus of his investigation it turns out that his construction, yields 0 (m) nodes. This construction was also guaranteed to use weight values limited to the set {I, 2}. Since a recurrent neural network with k hard-limiting nodes is capable of representing as many as 2k states, one might wonder if an m-state FSM could be implemented by a network with log m nodes. However, it was shown in [1] that the node complexity for a standard fully connected network is n ((m log m)1/3). They were also able to improve upon Minsky's result by providing a construction which is guaranteed to yield no more than 0 (m3/ 4 ) nodes. In the same paper lower bounds on node complexity were investigated as the network was subject to restrictions on the possible range of weight values and the fan-in and fan-out of the nodes in the network. Their investigation was limited to fully connected recurrent neural networks and they discovered that the node complexity for the case where the weights are restricted to a finite size set is n (y'm log m) . Alternatively, if the nodes in the network were restricted to have a constant fan-in then the node complexity becomes n (m) . However, they left open the question of how tight these bounds are and if they apply to variations on the basic architecture. Other recent work includes investigation of the node complexity for networks with continuous valued nonlinearities [14]. However, it can also be shown that when continuous nonlinearities are used, recurrent neural networks are far more powerful than FSMs; in fact, they are Turing equivalent [13]. In this paper we improve the upper bound on the node complexity for the unrestricted case to 0 (yIm). We also provide upper bounds that match the lower bounds above for various restrictions. Specifically, we show that a node complexity of 0 ( y'm log m) can be achieved if the weights are restricted to the set {-I, I} , and that the node complexity is 0 (m) for the case when the fan-in of each node in the network is restricted to two. Finally, we explore the possibility that implementing finite state machines in more complex models might yield a lower node complexity. Specifically, we explore the node complexity of a general recurrent neural network topology, that is capable of simulating a variety of popular recurrent neural network architectures. Except for the unrestricted case, we will show that the node complexity is no different for this architecture than for the fully connected case if the number of feedback variables is limited to rlog m 1, i.e. if the state of the FSM is encoded optimally in a subset of the nodes. We leave it as an open question if a sparser encoding can lead to a more efficient implementation. 2 Background 2.1 Finite State Machines FSMs may be defined in several ways. In this paper we shall be concerned with Mealy machines, although our approach can easily be extended to other formulations to yield equivalent results. Bounds on the Complexity of Recurrent Neural Network Implementations 361 Definition 1 A Mealy machine is a quintuple M = (Q, qo, E, d, <1», where Q is a finite set of states; qo is the initial state; E is the input alphabet; d is the output alphabet; and <I> : Q x E Q x d is the combined transition and output function. o Throughout this paper both the input and output alphabets will be binary (i.e. E = d = {a, I}). In general, the number of states, m = IQI, may be arbitrary. Since any element of Q can be encoded as a binary vector whose minimum length is pog m 1 , the function <I> can be implemented as a boolean logic function of the form <I> : {a, l}pogm1+l _ {a, l}pogm1+l . (1) The number, N M , of different minimal FSMs with m states will be used to determine lower bounds on the number of gates required to implement an arbitrary FSM in a recurrent neural network. It can easily be shown that (2m)m :S NM [5]. However, it will be convenient to reexpress N M in terms of n = flog m 1 + 1 as follows (2) 2.2 Recurrent Neural Networks The fundamental processing unit in the models we wish to consider is the perceptron, which is a biased, linearly weighted sum of its inputs followed by a hard-limiting nonlinearity whose output is zero if its input is negative and one otherwise. The fan-in of the perceptron is defined to be the number of non-zero weights. When the values of Xi are binary (as they are in this paper) , the perceptron is often referred to as a threshold logic unit (TL U). A count of the number of different partially specified threshold logic functions, which are threshold logic functions whose values are only defined over v vertices of the unit hypercube, will be needed to develop lower bounds on the node complexity required to implement an arbitrary logic function. It has been shown that this number, denoted L~, is [15] 2vn L~:S -,-. n. (3) As pointed out in [10], many of the most popular discrete-time recurrent neural network models can be implemented as a feedforward network whose outputs are fed back recurrently through a set of unit time delays. In the most generic version of this architecture, the feed forward section is lower triangular, meaning the [th node is the only node in layer I and receives input from all nodes in previous layers (including the input layer). A lower triangular network of k threshold logic elements is the most general topology possible for a feedforward network since all other feedforward networks can be viewed as a special case of this network with the appropriate weights set equal to zero. The most direct implementation of this model is the architecture proposed in [11]. However, many recurrent neural network architectures can be cast into this framework. For example, fully connected networks [3] fit this model when the the feedforward network is simply a single layer of nodes. Even models which appear very different [2, 9] can be cast into this framework. 362 Home and Hush 3 The unrestricted case The unrestricted case is the most general, and thus explores the inherent power of recurrent neural networks. The unrestricted case is also important because it serves as a baseline from which one can evaluate the effect of various restrictions on the node complexity. In order to derive an upper bound on the node complexity of recurrent neural network implementations of FSMs we shall utilize the following lemma, due to Lupanov [6]. The proof of this lemma involves a construction that is extremely complex and beyond the scope of this paper. Lemma 1 (Lupanov, 1973) Arbitrary boolean logic functions with x inputs and y outputs can be implemented in a network of perceptrons with a node complexity of o ( J x ~~:g y) . o Theorem 1 Multilayer recurrent neural networks can implement FSMs having m states with a node complexity of 0 (.Jffi) . 0 Proof: Since an m-state FSM can be implemented in a recurrent neural network in which the multilayer network performs a mapping of the form in equation (1), then using n = m = flog m 1 + 1, and applying Lemma 1 gives an upper bound of O(.Jffi). Q.E.D. Theorem 2 Multilayer recurrent neural networks can implement FSMs having m states with a node complexity of n (fo) if the number of unit time delays is flog m 1. o Proof: In order to prove the theorem we derive an expression for the maximum number of functions that a k-node recurrent neural network can compute and compare that against the minimum number of finite state machines. Then we solve for k in terms of the number of states of the FSM. Specifically, we wish to manipulate the inequality 2(n-l)2 n - 2 < n! ( k - 1 ) krr-l 2n(n+i~+1 n - 1 . (n + z)! ,=0 (a) (b) where the left hand side is given in equation (2), (a) represents the total number of ways to choose the outputs and feedback variables of the network, and (b) represents the total number of logic functions computable by the feed forward section of the network, which is lower triangular. Part (a) is found by simple combinatorial arguments and noting that the last node in the network must be used as either an output or feedback node. Part (b) is obtained by the following argument: If the state is optimally encoded in flog m 1 nodes, then only flog m 1 variables need Bounds on the Complexity of Recurrent Neural Network Implementations 363 to be fed back. Together with the external input this gives n = rlog m 1 + 1 local inputs to the feedforward network. Repeated application of (3) with v = 2n yields expression (b). Following a series of algebraic manipulations it can easily be shown that there exists a constant c such that n2n < ck2n. Since n = flog ml + 1 it follows that k = f2 (fo). Q.E.D. 4 Restriction on weights and thresholds All threshold logic functions can be implemented with perceptrons whose weight and threshold values are integers. It is well known that there are threshold logic functions of n variables that require a perceptron with weights whose maximum magnitude is f2(2n) and O( nn/2) [8]. This implies that if a perceptron is to be implemented digitally, the number of bits required to represent each weight and threshold in the worst case will be a super linear function of the fan-in. This is generally undesirable; it would be far better to require only a logarithmic number of bits per weight, or even better, a constant number of bits per weight. We will be primarily be interested in the most extreme case where the weights are limited to values from the set {-I , I}. In order to derive the node complexity for networks with weight restrictions, we shall utilize the following lemma, proved in [4]. Lemma 2 Arbitrary boolean logic functions with x inputs and y outputs can be implemented in a network ofperceptrons whose weights and thresholds are restricted to the set {-I, I} with a node complexity of e (Jy2 x ) . 0 This lemma is not difficult to prove, however it is beyond the scope of this paper. The basic idea involves using a decomposition of logic functions proposed in [12]. Specifically, a boolean function f may always be decomposed into a disjunction of 2r terms of the form XIX2. ' . Xr fi(X r +1 , .. . , x n ) , one for each conjunction of the first r variables, where Xj represents either a complemented or uncomplemented version of the input variable Xj and each Ii is a logic function of the last n r variables. This expression can be implemented directly in a neural network. With negligible number of additional nodes, the construction can be implemented in such a way that all weights are either -lor 1. Finally, the variable r is optimized to yield the minimum number of nodes in the network. Theorem 3 Multilayer recurrent neural networks that have nodes whose weights and thresholds are restricted to the set {-I , I} can implement FSMs having m states with a node complexity of 0 (Jm log m) . 0 Proof: Since an m-state FSM can be implemented in a recurrent neural network in which the multilayer network performs a mapping of the form in equation (1), then using n = m = flog m 1 + 1, and applying Lemma 2 gives an upper bound of o (Jmlogm) . Q.E.D. 364 Home and Hush Theorem 4 Multilayer recurrent neural networks that have nodes whose weights and thresholds are restricted to a set of size IWI can implement FSMs having m states with a node complexity of n ( if the number of unit time delays is flogml. 0 Proof: The proof is similar to the proof of Theorem 2 which gave a lower bound for the node complexity required in an arbitrary network of threshold logic units. Here, the inequality we wish to manipulate is given by ) k-l k - 1 II IWln+i+ 1. n-I i=O (a) (b) where the left hand side and (a) are computed as before and (b) represents the maximum number of ways to configure the nodes in the network when there are only IWI choices for each weight and threshold. Following a series of algebraic manipulations it can be shown that there exists a constant c such that n2n ::; ck 2 log IWI. Since n = pog m 1 + 1 it follows that k = n ( mlogm) loglWI . Q.E.D. Clearly, for W = {-I, I} this lower bound matches the upper bound in Theorem 3. 5 Restriction on fan-in A limit on the fan-in of a perceptron is another important practical restriction. In the networks discussed so far each node has an unlimited fan-in. In fact, in the constructions described above, many nodes receive inputs from a polynomial number of nodes (in terms of m) in a previous layer. In practice it is not possible to build devices that have such a large connectivity. Restricting the fan-in to 2, is the most severe restriction, and will be of primary interest in this paper. Once again, in order to derive the node complexity for restricted fan-in, we shall utilize the following lemma, proved in [4]. Lemma 3 Arbitrary boolean logic functions with x inputs and y outputs can be implemented in a network of perceptrons restricted to fan-in 2 with a node complexityof ( y2X ) e x + logy . o This proof of this lemma is very similar to the proof of Lemma 2. Here Shannon's decomposition is used with r = 2 to recursively decompose the logic function into a set of trees, until each tree has depth d. Then, all possible functions of the last n - d variables are implemented in an inverted tree-like structure, which feeds into the bottom of the trees. Finally, d is optimized to yield the minimum number of nodes. Bounds on the Complexity of Recurrent Neural Network Implementations 365 Theorem 5 Multilayer recurrent neural networks that have nodes whose fan-in is restricted to two can implement FSMs having m states with a node complexity of Oem) 0 Proof: Since an m-state FSM can be implemented in a recurrent neural network in which the multilayer network performs a mapping of the form in equation (1), then using n = m = rlog m 1 + 1, and applying Lemma 3 gives an upper bound of o (m). Q.E.D. Theorem 6 Multilayer recurrent neural networks that have nodes whose fan-in is restricted to two can implement FSMs having m states with a node complexity of n (m) if the number of unit time delays is rlog m 1. 0 Proof: Once again the proof is similar to Theorem 2, which gave a lower bound for the node complexity required in an arbitrary network of threshold logic units. Here, the inequality we need to solve for is given by 2(n-1)2'-' :s n! ( ~:= ~ ) D. 14 ( n t i ) ,----_V~----A~----_V~----~ (a) (b) where the left hand side and (a) are computed as before and (b) represents the maximum number of ways to configure the nodes in the network. The term ( n t i ) is used since a node in the ith layer has n + i possible inputs from which two are chosen. The constant 14 represents the fourteen possible threshold logic functions of two variables. Following a series of algebraic manipulations it can be shown that there exists a constant c such that n2n ~ ck logk Since n = rlog m 1 + 1 it follows that k = n (m) . 6 Summary Q.E.D. In summary, we provide new bounds on the node complexity of implementing FSMs with recurrent neural networks. These upper bounds match lower bounds developed in [1] for fully connected recurrent networks when the size of the weight set or the fan-in of each node is finite. Although one might speculate that more complex networks might yield more efficient constructions, we showed that these lower bounds do not change for restrictions on weights or fan-in, at least when the state of the FSM is encoded optimally in a subset of flog m 1 nodes. When the network is unrestricted, this lower bound matches our upper bound. We leave it as an open question if a sparser encoding of the state variables can lead to a more efficient implementation. One interesting aspect of this study is that there is really not much difference in efficiency when the network is totally unrestricted and when there are severe restrictions placed on the weights. Assuming that our bounds are tight, then there 366 Home and Hush is only a y'log m penalty for restricting the weights to either -1 or 1. To get some idea for how marginal this difference is consider that for a finite state machine with m = 18 x 1018 states, y'log m is only eight! A more detailed version of this paper can be found in [5]. References [1] N. Alon, A.K. Dewdney, and T.J. Ott. Efficient simulation of finite automata by neural nets. JACM, 38(2):495-514, 1991. [2] A.D. Back and A.C. Tsoi. FIR and I1R synapses, a new neural network architecture for time series modeling. Neural Computation, 3(3):375-385, 1991. [3] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci., 79:2554-2558, 1982. [4] B.G. Horne and D.R. Hush. On the node complexity of neural networks. Technical Report EECE 93-003, Dept. EECE, U. New Mexico, 1993. [5] B.G. Horne and D.R. Hush. Bounds on the complexity of recurrent neural network implementations of finite state machines. Technical Report EECE 94-001, Dept. EECE, U. New Mexico, 1994. [6] O.B. Lupanov. The synthesis of circuits from threshold elements. Problemy Kibernetiki, 26:109-140, 1973. [7] M. Minsky. Computation: Finite and infinite machines. Prentice-Hall, 1967. [8] S. Muroga. Threshold Logic and Its Applications. Wiley, 1971. [9] K.S. Narendra and K. Parthasarathy. Identification and control of dynamical systems using neural networks. IEEE Trans. on Neural Networks, 1:4-27, 1990. [10] O. Nerrand et al. Neural networks and nonlinear adaptive filtering: Unifying concepts and new algorithms. Neural Computation, 5(2):165-199, 1993. [11] A.J. Robinson and F. Fallside. Static and dynamic error propagation networks with application to speech coding. In D.Z. Anderson, editor, Neural Information Processing Systems, pages 632-641, 1988. [12] C. Shannon. The synthesis of two-terminal switching circuits. Bell Sys. Tech. 1., 28:59-98, 1949. [13] H. Siegelmann and E.D. Sontag. Neural networks are universal computing devices. Technical Report SYCON-91-08, Rutgers Ctr. for Sys. and Cont., 1991. [14] H.T. Siegelmann, E.D. Sontag, and C.L. Giles. The complexity of language recognition by neural networks. In Proc. IFIP 12th World Compo Cong., pages 329-335, 1992. [15] R.O. Winder. Bounds on threshold gate realizability. IEEE Trans. on Elect. Comp., EC-12:561-564, 1963.
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Classification of Multi-Spectral Pixels by the Binary Diamond Neural Network Yehuda Salu Department of Physics and CSTEA, Howard University, Washington, DC 20059 Abstract A new neural network, the Binary Diamond, is presented and its use as a classifier is demonstrated and evaluated. The network is of the feed-forward type. It learns from examples in the 'one shot' mode, and recruits new neurons as needed. It was tested on the problem of pixel classification, and performed well. Possible applications of the network in associative memories are outlined. 1 INTRODUCTION: CLASSIFICATION BY CLUES Classification is a process by which an item is assigned to a class. Classification is widely used in the animal kingdom. Identifying an item as food is classification. Assigning words to objects, actions, feelings, and situations is classification. The purpose of this work is to introduce a new neural network, the Binary Diamond, which can be used as a general purpose classification tool. The design and operational mode of the Binary Diamond are influenced by observations of the underlying mechanisms that take place in human classification processes. An item to be classified consists of basic features. Any arbitrary combination of basic features will be called a clue. Generally, an item will consist of many clues. Clues are related not only to the items which contain them, but also to the classes. Each class, that resides in the memory, has a list of clues which are associated with it. These clues 1143 1144 Salu are the basic building blocks of the classification rules. A classification rule for a class X would have the following general form: Classification rule: If an item contains clue Xl, or clue X2, ... , or clue Xn• and if it does not contain clue Xl. nor clue X2, ...• nor clue Xm• it is classified as belonging to class X. Clues Xl •... ,Xn are the excitatory clues of class X, and clues Xl, ... ,xmare the inhibitory clues of class X. When classifying an item, we frrst identify the clues that it contains. We then match these clues with the classification rules, and fmd the class of the item. It may happen that a certain item satisfies classification rules of different classes. Some of the clues match one class, while others match another. In such cases, a second set of rules, disambiguation rules, are employed. These rules select one class out of those tagged by the classification rules. The disambiguation rules rely on a hierarchy that exists among the clues. a hierarchy that may vary from one classification scheme to another. For example, in a certain hierarchy clue A is considered more reliable than clue B, if it contains more features. In a different hierarchy scheme, the most frequent clue is considered the most reliable. In the disambiguation process, the most reliable clue, out of those that has actively contributed to the classification. is identified and serves as the pointer to the selected class. This classification approach will be called classification by clues (CRC). The classification rules may be 'loaded' into our memory in two ways. FIrst, the precise rules may be spelled out and recorded (e.g. 'A red light means stop'). Second, we may learn the classification rules from examples presented to us, utilizing innate common sense learning mechanism. These mechanisms enable us to deduce from the examples presented to us, what clues should serve in the classification rules of the adequate classes, and what clues have no specificity, and should be ignored. For example, by pointing to a red balloon and saying red, an infant may associate each of the stimuli red and balloon as pointers to the word red. After presenting a red car, and saying red, and presenting a green balloon and saying green. the infant has enough information to deduce that the stimulus red is associated with the word red, and the stimulus balloon should not be classified as red. 2 THE BINARY DIAMOND 2.1 STRUCTURE In order to perform a CRe in a systematic way, all the clues that are present in the item to be classified have to be identified frrst, and then compared against the classification rules. The Binary Diamond enables carrying these tasks fast and Classification of Multi-Spectral Pixels by the Binary Diamond Neural Network 1145 effectively. Assume that there are N different basic features in the environment. Each feature can be assigned to a certain bit in an N dimensional binary vector. An item will be represented by turning-on (from the default value of ° to the value of 1) all the bits that correspond to basic features, that are present in the item. The total number of possible clues in this environment is at most 2N. One way to represent these possible clues is by a lattice, in which each possible clue is represented by one node. The Binary Diamond is a lattice whose nodes represent clues. It is arranged in layers. The frrst (bottom) layer has N nodes that represent the basic features in the environment. The second layer has N'(N-l)/2 nodes that represent clues consisting of 2 basic features. The K'th layer has nodes that represent clues, which consist of K basic features. Nodes from neighboring layers which represent clues that differ by exactly one basic feature are connected by a line. Figure 1 is a diagram of the Binary Diamond for N = 4. Figure 1: The Binary Diamond of order 4. The numbers inside the nodes are the binary codes for the feature combination that the node represents, e.g 1 < = > (0,0,0,1),5< = >(0,1,0,1),14 < = > (1,1,1,0), 15 < = > (1,1,1,1). 2.2 THE BINARY DIAMOND NEURAL NElWORK The Binary Diamond can be turned into a feed-forward neural network by treating each node as a neuron, and each line as a synapse leading from a neuron in a lower layer (k) to a neuron in the higher layer (k + 1). All synaptic weights are set to 0.6, and 1146 Salu all thresholds are set to 1, in a standard Pitts McCulloch neuron. The output of a flring neuron is 1. An item is entered into the network by turning-on the neurons in the flrst layer, that represent the basic features constituting this item. Signals propagate forward one layer at a time tick, and neurons stay active for one time tick. It is easy to verify that all the clues that are part of the input item, and only such clues, will be turned on as the signals propagate in the network. In other words, the network identifies all the clues in the item to be classified. An item consisting of M basic features will activate neurons in the fIrst M layers. The activated neuron in the M'th layer is the representation of the entire item. As an example, consider the input item with feature vector (0,1,1,1), using the notations of figure 1. It is entered by activating neurons 1, 2, and 4 in the first layer. The signals will propagate to neurons 3, 5, 6, and 7, which represent all the clues that the input item contains. 2.3 INCORPORATING CLASS INFORMATION Each neuron in the Binary Diamond represent a possible clue in the environment spun by N basic features. When an item is entered in the frrst layer, all the clues that it contains activate their representing neurons in the upper layers. This is the first step in the classification process. Next, these clues have to point to the appropriate class, based upon the classification rule. The possible classes are represented by neurons outside of the Binary Diamond. Let x denote the neuron, outside the Binary Diamond, that represents class X. An excitatory clue Xi (from the Binary Diamond) will synapse onto x with a synaptic weight of 1. An inhibitory clue Xl (in the Binary Diamond) will synapse onto x with an inhibitory weight of -z, where z is a very large number (larger than the maximum number of clues that may point to a class). This arrangement ensures that the classification rule formulated above is carried out. In cases of ambiguity, where a number of classes have been activated in the process, the class that was activated by the clue in the highest layer will prevail. This clue has the largest number of features, as compared with the other clues that actively participated in the classification. 2.4 GROWING A BINARY DIAMOND A possible limitation on the processes described in the two previous sections is that, if there are many basic features in the environment, the 2N nodes of the Binary Diamond may be too much to handle. However, in practical situations, not all the clues really occur, and there is no need to actually represent all of them by nodes. One way of taking advantage of this simplifying situation is to grow the network one event (a training item and its classification) at a time. At the beginning, there is just the frrst layer with N neurons, that represent the N basic features. Each event adds its neurons to the network, in the exact positions that they would occupy in the regular Classification of Multi-Spectral Pixels by the Binary Diamond Neural Network 1147 Binary Diamond. A clue that has already been represented in previous events, is not duplicated. After the new clues of the event have been added to the network, the information about the relationships between clues and classes is updated. This is done for all the clues that are contained in the new event. The new neurons send synapses to the neuron that represent the class of the current event. Neurons of the current event, that took part in previous events, are checked for consistency. If they point to other classes, their synapses are cut-off. They have just lost their specificity. It should be noted that there is no need to present an event more than one time for it to be correctly recorded (' one shot learning'). A new event will never adversely interfere with previously recorded information. Neither the order of presenting the events, nor repetitions in presenting them will affect the final structure of the network. Figure 2 illustrates how a Binary Diamond is grown. It encodes the information contained in two events, each having three basic features, in an environment that has four basic features. The first event belongs to class A, and the second to class B. (0,1,1,1) -> A @ Figure 2. Growing a Binary Diamond. Left: All the feature combinations of the threefeature item (0,1,1,1) are represented by a 3'rd order Binary Diamond, which is grown from the basic features represented by neurons 1, 2, and 4. All these combinations, marked by a wavy background, are, for the time being, specific clues to class A. Right: The three-feature item, (1,1,1,0) is added, as another 3'rd order Binary Diamond. At this point, only neurons l,3~,and 7 represent specific clues to class A. Neurons 8,10,12, and 14 represent specific clues to class B, and neurons 2,4, and 6 represent non-specific clues. 3 CLASSIFICATION OF MULTI-SPECfRAL PIXELS 3.1 THE PROBLEM Spectral information of land pixels, which is collected by satellites, is used in preparation of land cover maps and similar applications. Depending on the satellite and its instrumentation, the spectral information consists of the intensities of several 1148 Salu light bands, usually in the visible and infra-red ranges, which have been reflected from the land pixels. One method of classification of such pixels relies on independent knowledge of the land cover of some pixels in the scene. These classified pixels serve as the training set for a classification algorithm. Once the algorithm is trained, it classifies the rest of the pixels. The actual problem described here involves testing the Binary Diamond in a pixel classification problem. The tests were done on four scenes from the vicinity of Washington DC, each consisting of approximately 22,000 pixels. The spectral information of each pixel consisted of intensities of four spectral bands, as collected by the Thematic Mapper of the Landsat 4 satellite. Ground covers of these scenes were determined independently by ground and aerial surveys. There were 17 classes of ground covers. The following list gives the number of pixels per class in one of the scenes. The distributions in the other scenes were similar. 1) water (28). 2) miscellaneous crops (299). 3) corn-standing (0). 4) com-stubble (349). 5) shrub-land (515). 6) grass/ pasture (3,184). 7) soybeans (125). 8) baresoil, clear land (535). 9) hardwood, canopy> 50% (10,169). 10) hardwood, canopy < 50% (945). 11) conifer forest (2,051). 12) mixed wood forest (616). 13) asphalt (390). 14) single family housing (2,220). 15) multiple family housing (26). 16) industrial/ commercial (118). 17) bare soil-plowed field (382). Total 21,952. 3.2 METHODS Approximately 10% of the pixels in each of the four scenes were randomly selected to become a training set. Four Binary Diamond networks were grown, based on these four training sets. In the evaluation phase, each network classified each scene. The intensity of the light in each band was discretized into 64 intervals. Each interval was considered as a basic feature. So, each pixel was characterized by four basic features (one for each band), out of 4x64=256 possible basic features. The fust layer of the Binary Diamond consisted of 256 neurons, representing these basic features. Pixels of the training set were treated like events. They were presented sequentially, one at a time, for one time, and the neurons that represent their clues were added to the network, as explained in section 2.4. After the training phase, the rest of the pixels were presented, and the network classified them. The results of this classification were kept for comparisons with the observed ground cover values. The same training sets were used to train two other classification algorithms; a backpropagation neural network, and a nearest neighbor classifier. The backpropagation network had four neurons in the input layer, each representing a spectral band. It had seventeen neurons in the output layer, each representing a class, and a hidden layer of ten neurons. The nearest neighbor classifier used the pixels of the training set as models. The Euclidean distance between the feature vector of a pixel to Classification of Multi-Spectral Pixels by the Binary Diamond Neural Network 1149 be classified and each model pixel was computed. The pixel was classified according to the class of its closest model. 3.3 RESULTS In auto-classification, the pixels of a scene are classified by an algorithm that was trained using pixels from the same scene. In cross-classification, the classification of a scene is done by an algorithm that was trained by pixels of another scene. It was found that in both auto-classification and cross-classification, the results depend on the consistency of the training set. Boundary pixels, which form the boundary (on the ground) between two classes, may contain a combination of two ground cover classes. If boundary pixels were excluded from the scene, the results of all the classification methods improved significantly. Table 1 compares the overall performance of the three classification methods in auto-classification and cross-classification, when only boundary pixels were considered. Similar ordering of the classification methods was obtained when all the pixels were considered. 1 2 3 4 1 2 3 4 1 2 3 4 1 83 58 71 74 1 83 41 46 61 1 73 60 33 64 2 41 78 50 44 2 27 73 28 17 2 25 55 38 26 3 49 48 75 52 3 43 38 62 35 3 33 38 52 37 4 54 44 57 76 4 52 36 39 70 4 48 43 42 60 Binary Diamond Nearest Neighbor Back-Propagation Table 1: The percent of correctly classified pixels for the implementations of the three methods, for non-boundary pixels only, as tested on the four maps. Column's index is the training map, rows index is the testing map. Table 2 compares the performances of the three methods class by class, as obtained in the classification of the flfst scene. Similar results were obtained for the other scenes. 1= 1 2 3 4 5 6 7 8 9 10 11 U 13 14 15 16 17 BD 48 10 0 33 7 44 10 53 88 37 34 5 32 69 33 34 41 BDp 57 8 0 48 10 14 10 58 87 37 35 6 30 69 42 25 27 bNN 63 54 0 72 47 19 52 77 60 63 48 60 70 43 64 62 72 BP 68 5 0 68 0 1 11 66 80 74 11 1 26 45 54 52 27 Table 2: The percent of pixels from category I that have been classified as category I. Auto-classification of scene 1. All the pixels are included. BD; results of Binary Diamond where the feature vectors are in the standard Cartesian representation. BDp = results of Binary Diamond where the feature vectors are in four dimensional polar coordinates. bNN results of nearest neighbor, and BP of back-propagation. 1150 Salu The overall performance of the Binary Diamond was better than those of the nearest neighbor and the back-propagation classifiers. This was the case in auto-classification and in cross-classification, in scenes that included all the pixels, and in scenes that consisted only of non-boundary pixels. However, when comparing individual classes, it was found that different classes may have different best classifiers. In practical applications, the prices of correct or the wrong classifications of each class, as well as the frequency of the classes in the environment will determine the optimal classifier. All the networks recruited their neurons as needed, during the training phase. They all started with 256 neurons in the first layer, and with seventeen neuron in the class layer, outside the Binary Diamond. At the end of the training phase of the first scene, The Binary Diamond consisted of 5,622 neurons, in four layers. This is a manageable number, and it is much smaller than the maximum number of possible clues, 644=224. 4 OrnER APPLICATIONS OF mE BINARY DIAMOND The Binary Diamond, as presented here, was the core of a network that was used as a classifier. Because of its special structure, the Binary Diamond can be used in other related problems, such as in associative memories. In associative memory, a presented clue has to retrieve all the basic features of an associated item. If we start from any node in the Binary Diamond, and cascade down in the existing lines, we reach all the basic features of this clue in the frrst layer. So, to retrieve an associated item, the signals of the input clue have frrst to climb up the binary diamond till they reach a node, which is the best generalization of this clue, and then to cascade down and to activate the basic features of this generalization. The synaptic weights in the upward direction can encode information about causality relationships and the frequency of co-activations of the pre and post-synaptic neurons. This information can be used in the retrieval of the most appropriate generalization to the given clue. An associative memory of this kind retrieves information in ways similar to human associative retrieval (paper submitted). REFERENCES A reference list, as well as more details about pixel classification can be found in: Classification of Multi-Spectral Image Data by the Binary Diamond Neural Network and by Non-Parametric Pixel-by-Pixel Methods, by Yehuda Salu and James Tilton. IEEE Transactions On Geoscience And Remote Sensing, 1993 (in press).
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Optimal Brain Surgeon: Extensions and performance comparisons Babak Hassibi* David G. Stork Takahiro Watanabe Ricoh California Research Center 2882 Sand Hill Road Suite 115 Menlo Park, CA 94025-7022 and Gregory Wolff * Department of Electrical Engineering 105B Durand Hall Stanford University Stanford, CA 94305-4055 Abstract We extend Optimal Brain Surgeon (OBS) a second-order method for pruning networks to allow for general error measures, and explore a reduced computational and storage implementation via a dominant eigenspace decomposition. Simulations on nonlinear, noisy pattern classification problems reveal that OBS does lead to improved generalization, and performs favorably in comparison with Optimal Brain Damage (OBD). We find that the required retraining steps in OBD may lead to inferior generalization, a result that can be interpreted as due to injecting noise back into the system. A common technique is to stop training of a large network at the minimum validation error. We found that the test error could be reduced even further by means of OBS (but not OBD) pruning. Our results justify the t ~ 0 approximation used in OBS and indicate why retraining in a highly pruned network may lead to inferior performance. 263 264 Hassibi, Stork, Wolff, and Watanabe 1 INTRODUCTION The fundamental theory of generalization favors simplicity. For a given level of performance on observed data, models with fewer parameters can be expected to perform better on test data. In practice, we find that neural networks with fewer weights typically generalize better than large networks with the same training error. To this end, LeCun, Denker and Solla's (1990) Optimal Brain Damage method (OED) sought to delete weights by keeping the training error as small as possible. Hassibi and Stork (1993) extended OED to include the off-diagonal terms in the network's Hessian, which were shown to be significant and important for pruning in classical and benchmark problems. OED and Optimal Brain Surgeon (OES) share the same basic approach of training a network to (local) minimum in error at weight w*, and then pruning a weight that leads to the smallest increase in the training error. The predicted functional increase in the error for a change in full weight vector 8w is: ( BE )T 1 T B2E 8E= ·8w+-8w·-Bw 2 Bw2 'V' ~ ~O =H ·8w + O(1I8wI1 3 ) , "-v-" ~O (1 ) where H is the Hessian matrix. The first term vanishes because we are at a local minimum in error; we ignore third- and higher-order terms (Gorodkin et al., 1993). Hassibi and Stork (1993) first showed that the general solution for minimizing this function given the constraint of deleting one weight was: (2) Here, eq is the unit vector along the qth direction in weight space and Lq is the saliency of weight q an estimate of the increase in training error if weight q is pruned and the other weights updated by the left equation in Eq. 2. 2 GENERAL ERROR MEASURES AND FISHER'S METHOD OF SCORING In this section we show that the recursive procedure for computing the inverse Hessian for sum squared errors presented in Hassibi and Stork (1993) generalizes to any twice differentiable distance norm and that the key approximation based on Fisher's method of scoring is still valid. Consider an arbitrary twice differentiable distance norm d(t, 0) where t is the desired output (teaching vector) and 0 = F(w, in) the actual output. Given a weight vector w, F maps the input vector in to the output; the total error over P patterns is E = J; 2:f=l d(tlkJ, olkJ). It is straightforward to show that for a single output unit network the Hessian is: Optimal Brain Surgeon: Extensions and Performance Comparisons 265 _ 2. ~ 8P(w, in[kJ) . 82d(t[kJ,0[kJ) . 8pT(w, in[kJ) H- P L 8 8 2 8w + k=1 W 0 ~ ~ 8d(t[kJ, O[kJ) . 8 2 P(w, in[kJ) L (3) P k=1 80 8w2 · The second term is of order O(lIt - 011); using Fisher's method of scoring (Sever & Wild, 1989), we set this term to zero. Thus our Hessian reduces to: H = 2. ~ 8P(w, in[kJ) . 82d(t[kJ, o[kJ) . 8pT(w, in[kJ) P L 8w 802 8w· (4) k=1 8F(W in[k1) 8 2d(t[k1 o[kl) . . .. We define Xk 8W and ak 802 , and followmg the lOgIC of HasSlbl and Stork (1993) we can easily show that the recursion for computing the inverse Hessian becomes: H- 1 .X .XT .H- 1 H- 1 = H- 1 _ k k+l k+l k H- 1 = a-II and Hp-l = H- 1 , k+l k .E. + XT . H- 1 . X '0 , ak k+l k k+l (5) where a is a small parameter effectively a weight decay constant. Note how different error measures d(t,o) scale the gradient vectors X k forming the Hessian (Eq. 4). For the squared error d(t,o) = (t - 0)2, we have ak = 1, and all gradient vectors are weighted equally. The cross entropy or Kullback-Leibler distance, d(t, 0) = o log ~ + (1- 0) log i~ = ~? ' 0::; 0, t::; 1 (6) yields ak otkl(I~O[l'l). Hence if o[kJ is close to zero or one, Xk is given a large weight in the Hessian; conversely, the smallest value of ak occurs when o[kJ = 1/2. This is desirable and makes great intuitive sense, since in the cross entropy norm the value of o[kJ is interpreted as the probability that the kth input pattern belongs to a particular class, and therefore we give large weight to Xk whose class we are most certain and small weight to those which we are least certain. 3 EIGENSPACE DECOMPOSITION Although DES has been shown to be a powerful method for small and intermediate sized networks Hassibi, Stork and Wolff (1993) applied OES successfully to NETtaik its use in larger problems is difficult because of large storage and computation requirements. For a network of n weights, simply storing the Hessian requires 0(n2/2) elements and 0(Pn2 ) computations are needed for each pruning step. Reducing this computational burden requires some type of approximation. Since OES uses the inverse of the Hessian, any approximation to DES will at some level reduce to an approximation of H. For instance OED uses a diagonal approximation; magnitude-based methods use an isotropic approximation; and dividing the network into subsets (e.g., hidden-to-output and input-to-hidden) corresponds to the less-restrictive block diagonal approximation. In what follows we explore the dominant eigenspace decomposition of the inverse Hessian as our approximation. It should be remembered that all these are subsets of the full DBS approach. 266 Hassibi, Stork, Wolff, and Watanabe 3.1 Theory The dominant eigendecomposition is the best low-rank approximation of a matrix (in an induced 2-norm sense). Since the largest eigenvalues of H- 1 are the smallest eigenvalues of H, this method will, roughly speaking, be pruning weights in the approximate nullspace of H. Dealing with a low rank approximation of H-l will drastically reduce the storage and computational requirements. Consider the eigendecomposition of H: (7) where ~s contains the largest eigenvalues of H and ~N the smallest ones. (We use the subscripts Sand N to loosely connote signal and noise.) The dimension of the noise subspace is typically m« n. Us and UN are n x (n - m) and n x m matrices that span the dominant eigenspace of Hand H-l, and * denotes matrix transpose and complex conjugation. If, as suggested above, we restrict the weight prunings to lie in UN, we obtain the following saliency and full weight change when removing the qth weight: 1 w~ Lq = -----...:.~--2 ef . UN . ~N 1 . UN . e q (8) Wq 1 8w = 1 ~N Uiveq , e T . UN . ~. U N* . e q N q (9) where we have used 'bars' to indicate that these are approximations to Eq. 2. Note now that we need only to store ~N and UN, which have roughly nm elements. Likewise the computation required to estimate ~N and UN is O(Pnm). The bound on Lq is: LqLq 1 Lq < Lq < Lq + 2 w~ . a(s) , (10) where a(8) is the smallest eigenvalue of ~s. Moreover if Q:.(8) is large enough so that Q:.( 8) > [H! l)qq we have the following simpler form: (11) In either case Eqs. 10 and 11 indicate that the larger a(8) is, the tighter the bounds are. Thus if the subspace dimension m is such that the eigenvalues in Us are large, then we will have a good approximation. LeCun, Simard and Pearlmutter (1993) have suggested a method that can be used to estimate the smallest eigenvectors of the Hessian. However, for 0 BS (as we shall see) it is best to use the Hessian with the t ~ 0 approximation, and their method is not appropriate. Optimal Brain Surgeon: Extensions and Performance Comparisons 267 3.2 Simulations We pruned networks trained on the three Monk's problems (Thrun et al., 1991) using the full OBS and a 5-dimensional eigenspace version of OBS, using the validation error rate for stopping criterion. (We chose a 5-dimensional subspace, because this reduced the computational complexity by an order of magnitude.) The Table shows the number of weights obtained. It is clear that this eigenspace decomposition was not particularly successful. It appears as though the the off-diagonal terms in H beyond those in the eigenspace are important, and their omission leads to bad pruning. However, this warrants further study. unpruned OBS 5-d eigenspace Monk1 58 14 28 Monk2 39 16 27 Monk3 39 4 11 4 OBS/OBD COMPARISON General criteria for comparing pruning methods do not exist. Since such methods amount to assuming a particular prior distribution over the parameters, the empirical results usually tell us more about the problem space, than about the methods themselves. However, for two methods, such as OBS and OBD, which utilize the same cost function, and differ only in their approximations, empirical comparisons can be informative. Hence, we have applied both OBS and OBD to several problems, including an artificially generated statistical classification task, and a real-world copier voltage control problem. As we show below, the OBS algorithm usually results in better generalization performance. 4.1 MULTIPLE GAUSSIAN PRIORS We created a two-catagory classification problem with a five-dimensional input space. Category A consisted of two Gaussian distributions with mean vectors /-LA! = (1,1,0,1, .5) and /-LA2 = (0,0,1,0, .5) and covariances ~A! = Diag[0.99, 1.0, 0.88, 0.70, 0.95] and ~A2 = Diag[1.28, 0.60, 0.52, 0.93, 0.93] while category B had means /-LB! = (0,1,0,0, .5) and /-LB2 = (1,0,1,1, .5) and covariances ~Bl = Diag[0.84, 0.68, 1.28, 1.02,0.89] and ~B2 = Diag[0.52, 1.25, 1.09,0.64,1.13]. The networks were feedforward with 5 input units, 9 hidden units, and a single output unit (64 weights total). The training and the test sets consisted of 1000 patterns each, randomly chosen from the equi-probable categories. The problem was a difficult one: even with the somewhat large number of weights it was not possible to obtain less than 0.15 squared error per training pattern. We trained the networks to a local error minimum and then applied OBD (with retraining after each pruning step using backpropagation) as well as 0 BS. Figure 1 (left) shows the training errors for the network as a function of the number of remaining weights during pruning by OBS and by OBD. As more weights are pruned the training errors for both OBS and OBD typically increase. Comparing the two graphs for the first pruned weights, the training error for OBD and OBS are roughly equal, after which the training error of OBS is less until the 24th weight 268 Hassibi, Stork, Wolff, and Watanabe E Train Test E . 17 r . 22 r .'~ I , " . ~.,OBD .215 .. ' . 165 ',' tI _. ••••• • ._. . . .... . 21 .16 .205 .155 .2 aBS .15 .195 ' 30 35 40 45 50 55 60 65 30 35 40 45 50 55 60 65 number of weights number of weights Figure 1: DES and OED training error on a sum of Gaussians prior pattern classification task as a function of the number of weights in the network. (Pruning proceeds right to left.) DES pruning employed 0: = 10-6 (cf., Eq. 5); OED employed 60 retraining epochs after each pruning. is removed. The reason OED training is initially slightly better is that the network was not at an exact local minimum; indeed in the first few stages the training error for OED actually becomes less than its original value. (Training exhaustively to the true local minimum took prohibitively long.) In contrast, due to the t ---+ 0 approximation DES tries to keep the network response close to where it was, even if that isn't the minimum w*. We think it plausible that if the network were at an exact local minimum DES would have had virtually identical performance. Since OED is using retraining the only reason why OES can outperform after the first steps is that OED has removed an incorrect weight, due to its diagonal approximation. (The reason DES behaves poorly after removing 24 weights a radically pruned net may be that the second-order approximation breaks down at this point.) We can see that the minimum on test error occurs before this breakdown, meaning that the failed approximation (Fig. 2) does not affect our choice of the optimal network, at least for this problem. The most important and interesting result is the test error for these pruned networks (Figure 1, right). The test error for OED does not show any consistent behaviour, other than the fact that on the average it generally goes up. This is contrary to what one would expect of a pruning algorithm. It seems that the retraining phase works against the pruning process, by tending to reinforce overfitting, and to reinject the training set noise. For DES, however, the test error consistently decreases until after removing 22 weights a minimum is reached, because the t ---+ 0 approximation avoids reinjecting the training set noise. 4.2 OBS/OBD PRUNING AND "STOPPED" NETWORKS A popular method of avoiding overfitting is to stop training a large net when the validation error reaches a minimum. In order to explore whether pruning could improve the performance on such a "stopped" network (Le., not at w*), we monitored the test error for the above problem and recorded the weights for which a minimum on the test set occured. We then applied OES and OED to this network. E ·204 .202 .200 . 198 .196 .194 Optimal Brain Surgeon: Extensions and Performance Comparisons 269 . , , , , , , '- " aBO , ,\ "'--, , -' .... -, ,'" , .. " 35 40 45 50 55 60 number of weights Figure 2: A 64-weight network was trained to minimum validation error on the Gaussian problem not w* and then pruned by OBD and by OBS. The test error on the resulting network is shown. (Pruning proceeds from right to left.) Note es,pecially that even though the network is far from w*, OBS leads lower test error over a wide range of prunings, even through OBD employs retraining. The results shown in Figure 2 indicate that with OBS we were able to reduce the test error, and this reached a minimum after removing 17 weights. OBD was not able to consistently reduce the test error. This last result and those from Fig. 2 have important consequences. There are no universal stopping criteria based on theory (for the reasons mentioned above), but it is a typical practice to use validation error as such a criterion. As can be seen in Figure 2, the test error (which we here consider a validation error) consistantly decreases to a unique miniumum for pruning by OBS. For the network pruned (and continuously retrained) by OBD, there is no such structure in the validation curves. There seems to be no reliable clue that would permit the user to know when to stop pruning. 4.3 COPIER CONTROL APPLICATION The quality of an image produced by a copier is dependent upon a wide variety of factors: time since last copy, time since last toner cartridge installed, temperature, humidity, overall graylevel of the source document, etc. These factors interact in a highly non-linear fashion, so that mathematical modelling of their interrelationships is difficult. Morita et al. (1992) used backpropagation to train an 8-4-8 network (65 weights) on real-world data, and managed to achieve an RMS voltage error of 0.0124 on a critical control plate. We pruned his network with both OBD with retraining as well as with OBS. When the network was pruned by OBD with retraining, the test error continually increased (erratically) such that at 34 remaining weights, the RMS error was 0.023. When also we pruned the original net by OBS, and the test error gradually decreased such that at the same number of weights the test error was 0.012 significantly lower than that of the net pruned by OBD. 270 Hassibi, Stork, Wolff, and Watanabe 5 CONCLUSIONS We compared pruning by OES and by OED with retraining on a difficult non-linear statistical pattern recognition problem and found that OES led to lower generalization error. We also considered the widely used technique of training large nets to minimum validation error. To our surprise, we found that subsequent pruning by OES lowered generalization error, thereby demonstrating that such networks still have over fitting problems. We have found that the dominant eigenspace approach to OES leads to poor performance. Our simulations support the claim that the t ---+ 0 approximation used in OBS avoids reinjecting training set noise into the network. In contrast, including such t - 0 terms in OES reinjects training set noise and degrades generalization performance, as does retraining in OBD. Acknowledgements Thanks to T. Kailath for support of B.H. through grants AFOSR 91-0060 and DAAL03-91-C-0010. Address reprint requests to Dr. Stork: stork@crc.ricoh.com. References J. Gorodkin, L. K. Hansen, A. Krogh, C. Svarer and O. Winther. (1993) A quantitative study of pruning by Optimal Brain Damage. International Journal of Neural Systems 4(2) 159-169. B. Hassibi & D. G. Stork. (1993) Second order derivatives for network pruning: Optimal Brain Surgeon. In S. J. Hanson, J. D. Cowan and C. L. Giles (eds.), Advances in Neural Information Processing Systems 5, 164-171. San Mateo, CA: Morgan Kaufmann. B. Hassibi, D. G. Stork & G. Wolff. (1993) Optimal Brain Surgeon and general network pruning. Proceedings of ICNN 93, San Francisco 1 IEEE Press. 293-299. Y. LeCun, J. Denker & S. Solla. (1990) Optimal Brain Damage. In D. Touretzky (ed.), Advances in Neural Information Processing Systems 2, 598-605. San Mateo, CA: Morgan Kaufmann. Y. LeCun, P. Simard & B. Pearlmutter. (1993) Automatic learning rate maximization by on-line estimation of the Hessian's eigenvectors. In S. J. Hanson, J. D. Cowan & C. L. Giles (eds.), Advances in Neural Information Processing Systems 5, 156-163. San Mateo, CA: Morgan Kaufmann. T. Morita, M. Kanaya, T. Inagaki, H. Murayama & S. Kato. (1992) Photo-copier image density control using neural network and fuzzy theory. Second International Workshop on Industrial Fuzzy Control €3 Intelligent Systems December 2-4, College Station, TX, 10. S. Thrun and 23 co-authors. (1991) The Monk's Problems A performance comparison of different learning algorithms. CMU-CS-91-197 Carnegie-Mellon University Dept. of Computer Science Technical Report.
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Agnostic PAC-Learning of Functions on Analog Neural Nets (Extended Abstract) Wolfgang Maass Institute for Theoretical Computer Science Technische Universitaet Graz Klosterwiesgasse 32/2 A-BOlO Graz, Austria e-mail: maass@igi.tu-graz.ac.at Abstract: There exist a number of negative results ([J), [BR), [KV]) about learning on neural nets in Valiant's model [V) for probably approximately correct learning ("PAC-learning"). These negative results are based on an asymptotic analysis where one lets the number of nodes in the neural net go to infinit.y. Hence this analysis is less adequate for the investigation of learning on a small fixed neural net. with relatively few analog inputs (e.g. the principal components of some sensory data). The latter type of learning problem gives rise to a different kind of asymptotic question: Can the true error of the neural net be brought arbitrarily close to that of a neural net with "optimal" weights through sufficiently long training? In this paper we employ some new arguments ill order to give a positive answer to this question in Haussler's rather realistic refinement of Valiant's model for PAC-learning ([H), [KSS)). In this more realistic model no a-priori assumptions are required about the "learning target" , noise is permitted in the training data, and the inputs and outputs are not restricted to boolean values. As a special case our result implies one of the first positive results about learning on multi-layer neural net.s in Valiant's original PAC-learning model. At the end of this paper we will describe an efficient parallel implementation of this new learning algorit.hm. 311 312 Maass We consider multi-layer high order feedforward neural nets N with arbitrary piecewise polynomial activation functions. Each node g of fan-in m > 0 in N is called a computation node. It is labelled by some polynomial Q9(Yl, ... , Ym) and some piecewise polynomial activation funetion ,9 : R --+ R. We assume that ,9 consists of finitely many polynomial pieces and that its definition involves only rational parameters. The computation node g computes the function (Yl, ... ,Ym) t-+ ,9 (Q9 (Yl, ... , Ym)) from R minto R. The nodes of fan-in 0 in N ("input nodes") are labelled by variables Xl, ... , Xk. The nodes g of fan-out 0 in N ("output nodes") are labelled by 1, ... , I. We assume that the range B of their activation functions ,9 is bounded. Any parameters that occur in the definitions of the ,9 are referred to as architectural parameters of N. The coefficient.s of all the polynomials Q9 are called the programmable parameters (or weights) of N. Let w be the number of programmable parameters of N. For any assignment a E R W to the programmable parameters of N the network computes a function from Rk into RI which we will denote by N!!... We write Q n for the set of rational numbers that can be written as quotients of I integers with bit-length::; n. For;,. = (Zl, .. . ,ZI) E RI we write 11;,.lh for E Iz;l. ;=1 Let F : Rk --+ RI be some arbitrary function, which we will view as a "prediction rule". For any given instance (~, 1/) E R k X Rl we measure the error of F by "F(~) - 111 II· For any distribution A over some subset of R k x Rl we measure the true error of F with regard to A by E(£,Y)EA [IIF(~) -lllll]' i.e. the expected value of the error of F with respect to distribution A. Theorelll 1: Let N be some arbitrary high order feedforward neural net with piecewise polynomial activation functions. Let tv be the number of programmable parameters of N (we assume that w = 0(1)). Then one can construct from N some first order feedforward neural net jj with piecewise linear activation functions and the quadratic activation function ,(x) = x2, which has the following property: There exists a polynomial m(:, i) and a learning algorithm LEARN such that for any given €, 6, E (0,1) and s, n E N and any distribution A over Q~ x (Qn n B)l the following holds: For any sample ( = ({Xi, Yi) )i=l, ... ,m of m ~ m(:, i) points that are independently drawn according to A the algorithm LEARN computes in polynomially in m, s, n computation steps an assignment ii of rational numb~rs to the programmable parameters of jj such that with probability ~ 1 - 6: or in other words: The true error of jjli with regard to A is within € of the least possible true error that can be achieved by any N!!.. with a E Q:. Remarks a) One can easily see (see [M 93b] for details) that Theorem 1 provides a positive learning result in Haussler's extension of Valiant's model for PAClearning ([H], [KSS]). The "touchstone class" (see [KSS)) is defined as the Agnostic PAC-Learning of Functions on Analog Neural Nets 313 class of function f : Rk -+ Rl that are computable on N with programmable parameters from Q. This fact is of some general interest, since so far only very few positive results are known for any learning problem in this rather realistic (but quite demanding) learning model. b) Consider the special case where the distribution A over Q~ x (Qn n B)l is of the form { D(~) ADIO'T(~' y) = 0 otherwise for some arbitrary distribution D over the domain Q~ and some arbitrary Q:T E Q~. Then the term inf EC~IY}EA[IINQ.(~) -lllhl a EQw 3 is equal to O. Hence the preceding theorem states that with learning algorithm LEARN the "learning network" jj can "learn" with arbitrarily small true error any target function NQT that is computable on N with rational "weights" aT' Thus by choosing N sufficiently large, one can guarantee that the associated "learning network" jj can learn any target-function that might arise in the context of a specific learning problem. In addition the theorem also applies to the more realistic situation where the learner receives examples (~, y) of the form (~, NQT (~)+ noise), or even if there exists no "target function" NQT that would "explain" the actual distribution A of examples (~, ll) ("agnostic learning"). The proof of Theorem 1 is mathematically quite involved, and we can give here only an outline. It consists of three steps: (1) Construction of the auxiliary neural net fl . (2) Reducing the optimization of weights in jj for a given distribution A to a finite nonlinear optimization problem. (3) Reducing the resulting finite nonlinear optimization problem to a family of finite linear optimization problems. Details to step (1): If the activation functions ,9 in N are piecewise linear and all computation nodes in N have fan-out::; 1 (this occurs for example if N has just one hidden layer and only one output) then one can set fI := N. If the ,9 are piecewise linear but not all computation nodes in N have fan-out::; lone defines jj as the tree of the same depth as N, where sub circuits of computation nodes with fan-out m > 1 are duplicated 111 times. The activation functions remain unchanged in this case. If the activation functions ,9 are piecewise polynomial but not piecewise linear, one has to apply a rather complex construction which is described in detail in the Journal version of [M 93a]. In any case if has the property that all functions that 314 Maass are computable on N can also be computed on N, the depth of N is bounded by a constant, and the size of N is bounded by a polynomial in the size of N (provided that the depth and order of N, as well as the number and degrees of the polynomial pieces of the "'(9 are bounded by a constant). Details to step (2): Since the VC-dimension of a neural net is only defined for neural nets with boolean output, one has to consider here instead the pseudodimension of the function class F that is defined by N. Definition: (see Haussler (H]). Let X be some arbitrary domain, and let F be an arbitrary class of functions from X into R. Then the pseudo-dimension of F is defined by dimp(F) := max{ISI: S ~ X and 3h : S --+ R such that Vb E {O, l}s 31 E F Vx E S (I(x) ~ hex) ~ b(x) = I)}. Note that in the special case where F is a concept class (i.e. all 1 E Fare ° - 1 valued) the pseudo-dimension dimp(F) coincides with the VC-dimension of F. The pseudo-dimension of the function class associated with network architectures N with piecewise polynomial activation functions can be bounded with the help of Milnor's Theorem [Mi] in the same way as the VC-dimension for the case of boolean network output (see [GJ)): Theorenl 2: Consider arbitrary network architectures N of order v with k input nodes, I output nodes, and w programmable parameters. Assume that each gate in N employs as activation function some piecewise polynomial (or piecewise rational) function of degree ~ d with at most q pieces. For some arbitrary p E {I, 2, ... } we define F { 1 : R k+1 --+ R : 30: E R W Vx E Rk V1!. E Rl(l(~,1!.) IINQ'.(.~) -1!.lIp)}· Then one has dimp(F) = 0(w2 10gq) if v, d, 1= 0(1). • With the help of the pseudo-dimension one can carry out the desired reduction of the optimization of weights in N (with regard to an arbitrary given distribution A of examples (~, 11.) to a finite optimization problem. Fix some interval [b1 , b2] ~ R such that B ~ [b1 , b2], b1 < b2, and such that the ranges of the activation functions of the output gates of N are contained in [b1 , b2]. We define b := I· (b2 - bt) , and F:= {f :RkX[b1,b2]I--+[0,b]: 30:ERwV~ERkV1!.E[bl,b2F(f(~,1!.)= IINQ'.(~) - YIII)}· Assume now that parameters c, 6 E (0,1) with c ~ band s, n E N have been -fixed. For convenience we assume that s is sufficiently large so that all architectural parameters in N are from Qs (we assume that all architectural parameters in Ai are rational). We define (11) 257·b2 (. 33eb 8) 771 €'"8 := c2 2· dllnp(F) .In-c- + In"8 . By Corollary 2 of Theorem 7 in Haussler [H) one has for 771 ~ 771(:, i), I< := y~57 E (2,3), and any distribution A over Q~ x (Qn n [b 1,b2))1 (1) 1 ~ c P7'(EAm[{31 E F: 1(771 L...J /(!1.,1!.») E(~,.~)EA[f(!1.'1!.)]I > I<}] < 6, (~,~)E( Agnostic PAC-Learning of Functions on Analog Neural Nets 315 where E(~.!!)EA [f(~, u)] is the expectation of f(~, u) with regard to distribution A. We design an algorithm LEARN that computes for any mEN, any sample (= ((Xi,yi))iE{l •..• m} E (Q~ x (Qn n [b1,b2])I)m, and any given sEN in polynomially in m, s, n computation steps an assignment a of rational numbers to the parameters in j\( such that the function it that is computed by j\(!i. satisfies 1 m _ 2 (2) Tn L Ilh(xd - ydh ~ (1 - ]{)e + i=l m inf ~ ~ IIN£(xd - ydh· w m~ -a E Q" i=l This suffices for the proof of Theorem 1, since (1) and (2) together imply that, for any distribution A over Q~ x (Qn n [b1 , b2])1 and any m ~ m( 1, i), with probability ~ 1 - 6 (with respect to the random drawing of ( E Am) the algorithm LEARN outputs for inputs ( and s an assignment a of rational numbers to the parameters in j\( such that E(~'1!:)EA[IIN!i.(~) -ulld ~ c + inf E(!:.Y)EA[IIN£(~) -ulh]· a E Q~ Details to step (3): The computation of weights a that satisfy (2) is nontrivial, since this amounts t.o solving a nonlinear optimization problem. This holds even if each activation function in N is piecewise linear, because weights from successive layers are multiplied with each other. We employ a method from [M 93a] that allows us to replace the nonlinear conditions on the programmable parameters a of N by linear conditions for a transformed set .£, f3 of parameters. We simulate j\(£ by another network architecture N[£]~ (which one may view as a "normal form" for j\(£) that uses the same graph (V, E) as N, but different activation functions and different values f3 for its programmable parameters. The activation functions of N[.£] depend on IVI new architectural parameters .£ E RI vI, which we call scaling parameters in the following. Whereas the architectural parameters of a network architecture are usually kept fixed, we will be forced to change the scaling parameters of N along with its programmable parameters f3. Although this new network architecture has the disadvantage that it requires IVI additional parameters .£, it has the advantage that we can choose in N[£] all weights on edges between computation nodes to be from {-I,O, I}. Hence we can treat them as constants with at most 3 possible values in the system of inequalities that describes computations of N[£]. Thereby we can achieve that all variables that appear in the inqualities that describe computations of N[£J for fixed network inputs (the variables for weights of gates on levell, the variables for the biases of gates on all levels, and the new variables for the scaling parameters .£) appear only linearly in those inqualities. We briefly indicate the construction of N in the case where each activation function "I in N is piecewise linear. For any c > ° we consider the associated piecewise linear activation function "Ic with T;f x E R( "Ic (c . x) = c . "I ( x ) ). 316 Maass Assume that fr is some arbitrary given assignment to the programmable parameters in jj. We transform jjsr through a recursive process into a "normal form" N(£]t in which all weights on edges between computation nodes are from {-I, 0, I}, such that \:f ll. E R k (jjsr(ll.) = N(£]t(ll.») . q Assume that an output gate gout of jjsr receives as input L: aiYi + ao, where i=l al, ... , a q , ao are the weights and the bias of gout (under the assignment a) and Yl, ... ,Yq are the (real valued) outputs of the immediate predecessors g1, ... ,gq of g. For each i E {I, ... , q} with 0i =/:- 0 such that gi is not an input node we replace the activation function "fi of gi by "f!a,l, and we multiply the weights and the bias of gate gi with lail. Finally we replace the weight ai of gate gout by sgn( ad, where sgn(ad := 1 ifai > 0 and sgn(ai) := -1 ifai < o. This operation has the effect that the multiplication with IOj I is carried out before the gate gi (rather than after gj, as done in jjsr), but that the considered output gate gout still receives the same input as before. If aj = 0 we want to "freeze" that weight at O. This can be done by deleting gi and all gates below gi from N. The analogous operations are recursively carried out for the predecessors gi of gout (note however that the weights of gj are no longer the original ones from jjsr, since they have been changed in the preceding step). We exploit here the assumption that each gate in jj has fan-out::; 1. Let f3 consist of the new weights on edges adjacent to input nodes and of the resulting biases of all gates in N. Let f consist of the resulting scaling parameters at the gates of N. Then we have \:f~ E Rk (jjsr(~) = N[.£]t(~»). Furthermore c > 0 for all scaling parameters c in f. At the end of this proof we will also need the fact that the previously described parameter transformation can be inverted, i.e. one can compute from Q, f3 an equivalent weight assignment a for jj (with the original activation functions "f). We now describe how the algorithm LEARN computes for any given sample (= ({Xi,Yi)i=l ..... m E (Q~ x (Q" n[bl,b2W)m and any given sEN with the help of linear programming a new assignment .£, ~ to the parameters in N such that the function It that is computed by N@]i satisfies (2). For that purpose we describe the computations of N for the fixed inputs Xi from the sample ( = ((Xi, Yi) )i=l .... ,m by polynomially in m many systems L l , . .. , Lp(m) that each consist of Oem) linear inequalities with the transformed parameters Q, f3 as variables. Each system Lj reflects one possibility for employing specific linear pieces of the activation functions in N for specific network inputs Xl, ... , X m , and for employing different combinations of weights from {-I, 0, I} for edges between computation nodes. One can show that it suffices to consider only polynomially in Tn many systems of inequalities Lj by exploiting that all inequalities are linear, and that the input space for N has bounded dimension k. Agnostic PAC-Learning of Functions on Analog Neural Nets 317 We now expand each of the systems Lj (which has only 0(1) variables) into a linear programming problem LPj with Oem) variables. We add to Lj for each of the I output nodes IJ of N 2m new variables ur, vr for i = 1, ... , m, and the 4m inequalities tj(xd :S (Y;)II + ui - vi, tj(xd ~ (Ydll + ui - vi, ui ~ 0, vi ~ 0, where ((Xi, Yi) )i=l , .. . ,m is the fixed sample ( and (Yi)1I is that coordinate of Yj which corresponds to the output node IJ of N. In these inequalities the symbol tj(xd denotes the term (which is by construction linear in the variables f, (3) that represents the output of gate IJ for network input Xi in this system Lj. One-should note that these terms tj( Xi) will in general be different for different j, since different linear pieces of the activation functions at preceding gates may be used in the computation of N for the same network input Xi. We expand the system Lj of linear inequalities to a linear programming problem LPj in canonical form by adding the optimization requirement m mmlmlze i=l IJ output node The algorithm LEARN employs an efficient algorithm for linear programming (e.g. the ellipsoid algorithm, see [PS]) in order to compute in altogether polynomially in m, sand n many steps an optimal solution for each of the linear programming problems LP1 , ... , LPp(m). We write hj for the function from Rk into Rl that is computed by N[f]~ for the optimal solution £, (3 of LPj. The algorithm LEARN m computes ~ '" Ilhj(xj) - Yilll for j = 1, . .. ,p(m). Let] be that index for which mL...J i=l this expression has a minimal value. Let f, ~ be the associated optimal solution of LPl (i.e. N@)l computes hl). LEARN employs the previously mentioned backwards transformation from f, j3 into values Ii for the programmable parameters of jj such that 'V~ E Rk (jjQ.(~) = N[f.]l(~)). These values a are given as output of the algorithm LEARN. We refer to [M 93b] for the verification that this weight assignment a has the property that is claimed in Theorem 1. We also refer to [M 93b] for the proof in the more general case where the activation functions of N are piecewise polynomial .• Reillark: The algorithm LEARN can be speeded up substantially on a parallel machine. Furthermore if the individual processors of the parallel machine are allowed to use random bits, hardly any global control is required for this parallel computation. We use polynomially in m many processors. Each processor picks at random one of the systems Lj of linear inequalit.ies and solves the corresponding linear programming problem LPj . Then the parallel machine compares in a "competitive m phase" the costs L: Ilhj(Xi) - ydh of the solutions hj that have been computed by i=l the individual processors. It outputs the weights a for jj that correspond to the 318 Maass best ones of these solutions hj . If one views the number w of weights in N no longer as a c.onstant, one sees that the number of processores that are needed is simply exponential in w, but that the parallel computation time is polynomial in m and w. Acknowledgements I would like to thank Peter Auer, Phil Long and Hal White for their helpful comments. References [BR] [GJ] [H] [J] [KV] [KSS] [M 93a] [M 93b] [Mi] [PS] [V] A. Blum, R. L. Rivest, "Training a 3-node neural network is NPcomplete", Proc. of the 1988 Workshop on Computational Learning Theory, Morgan Kaufmann (San Mateo, 1988), 9 - 18 P. Goldberg, M. Jerrum, "Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers", Proc. of the 6th Annual A CM Conference on Computational Learning Theory, 361 - 369. D. Haussler, "Decision theoretic generalizations of the PAC model for neural nets and other learning applications", Information and Computation, vol. 100, 1992, 78 - 150 J. S. Judd, "Neural Network Design and the Complexity of Learning" , MIT-Press (Cambridge, 1990) M. Kearns, L. Valiant, "Cryptographic limitations on learning boolean formulae and finite automata", Proc. of the 21st ACM Symposium on Theory of Computing, 1989,433 - 444 M. J. Kearns, R. E. Schapire, L. M. Sellie, "Toward efficient agnostic learning", Proc. of the 5th A CM Workshop on Computational Learning Theory, 1992, 341 - 352 W. Maass, "Bounds for t.he c.omputational power and learning c.omplexity of analog neural nets" (extended abstract), Proc. of the 25th ACM Symposium on Theory of Computing, 1993,335 - 344. Journal version submitted for publication W. Maass, "Agnostic PAC-learning of functions on analog neural nets" (journal version), to appear in Neural Computation. .J. Milnor, "On the Betti numbers ofreal varieties", Proc. of the American Math. Soc., vol. 15, 1964, 275 - 280 C. H. Papadimitrioll, K. Steiglitz, "Combinatorial Optimization: Algorithms and Complexity" , Prent.ice Hall (Englewood Cliffs, 1982) L. G. Valiant, "A theory of the learnable", Comm. of the ACM, vol. 27, 1984, 1134 - 1142
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Learning Curves: Asymptotic Values and Rate of Convergence Corinna Cortes, L. D. Jackel, Sara A. Solla, Vladimir Vapnik, and John S. Denker AT&T Bell Laboratories Holmdel, NJ 07733 Abstract Training classifiers on large databases is computationally demanding. It is desirable to develop efficient procedures for a reliable prediction of a classifier's suitability for implementing a given task, so that resources can be assigned to the most promising candidates or freed for exploring new classifier candidates. We propose such a practical and principled predictive method. Practical because it avoids the costly procedure of training poor classifiers on the whole training set, and principled because of its theoretical foundation. The effectiveness of the proposed procedure is demonstrated for both single- and multi-layer networks. 1 Introd uction Training classifiers on large data.bases is computationally demanding. It is desirable to develop efficient procedures for a reliable prediction of a classifier's suitability for implementing a given task. Here we describe such a practical and principled predictive method. The procedure applies to real-life situations with huge databases and limited resources. Classifier selection poses a problem because training requires resources especially CPU-cycles, and because there is a combinatorical explosion of classifier candidates. Training just a few of the many possible classifiers on the full database might take up all the available resources, and finding a classifier particular suitable for the task requires a search strategy. 327 328 Cortes, Jackel, SoBa, Vapnik, and Denker test error ,~ ----------------....... -~~~~,~----------........... 10,000 30,000 50,000 training set size Figure 1: Test errors as a function of the size of the training set for three different classifiers. A classifier choice based on best test error at training set size 10 = 10,000 will result in an inferior classifier choice if the full database contains more than 15,000 patterns. The naive solution to the resource dilemma is to reduce the size of the database to 1 = 10 , so that it is feasible to train all classifier candidates. The performance of the classifiers is estimated from an independently chosen test set after training. This makes up one point for each classifier in a plot of the test error as a function of the size 1 of the training set. The naive search strategy is to keep the best classifier at 10 , under the assumption that the relative ordering of the classifiers is unchanged when the test error is extrapolated from the reduced size 10 to the full database size. Such an assumption is questionable and could easily result in an inferior classifier choice as illustrated in Fig. 1. Our predictive method also utilizes extrapolation from medium sizes to large sizes of the training set, but it is based on several data points obtained at various sizes of the training set in the intermediate size regime where the computational cost of training is low. A change in the representation of the measured data points is used to gain confidence in t.he extrapolation. 2 A Predictive Method Our predictive method is based on a simple modeling of the learning curves of a classifier. By learning curves we mean the expectation value of the test and training errors as a function of the training set size I. The expectat.ion value is taken over all the possible ways of choosing a training set of a given size. A typical example of learning curves is shown in Fig. 2. The test error is always larger than the training error, but asymptotically t.hey reach a common value, a. We model the errors for large siz€'s of the training s€'t as power-law decays to the Learning Curves: Asymptotic Values and Rate of Convergence 329 error a·· ..•. ···· •.• ·~.~·~.~.~.~.~.~·~Trr.P.~ _------------training error ~~training set size, I Figure 2: Learning curves for a typical classifier. For all finite values of the training set size I the test error is larger t han the training error. Asymptotically they converge to the same value a. asymptotic error value, a: b ['test = a + ler and c ['train = a 1i3 where I is the size of the training set, and a and f3 are positive exponents. From these two expressions the sum and difference is formed: ['test + ['train b c 2a + ler 1i3 b c ler + 1i3 ['test - ['train If we make the assumption 0'= f3 and b = c the equation (1) and (2) reduce to ['test + [train 2a [test - [train 2b ler (1) (2) (3) These expressions suggest a log-log representation of the sum and difference of the test and training errors as a function of the the training set size I, resulting in two straight lines for large sizes of the training set: a constant "-' log(2a) for the sum, and a straight line with slope -a and intersection log(b + c) "-' log(2b) for the difference, as shown in Fig. 3. The assumption of equal amplitudes b = c of the two convergent terms is a convenient but not crucial simplification of the model. \Ve find experimentally that for classifiers where this approximation does not hold, the difference ['test - ['train still forms a straight line in a log-log-plot. From this line the sum s = b + c can be extracted as the intersection, as indicated on Fig. 3. The weighted sum 330 Cortes, Jackel, Solla, Vapnik, and Denker log(error) log(b+c) log(2b) .. .. .. .. .. log(£tesl +£train) -log(2a) log(trainlng set size, l) Figure 3: Within the validity of the power-law modeling of the test and training errors, the sum and difference between the two errors as a function of training set size give two straight lines in a log-log-plot: a constant"" log(2a) for the sum, and a straight line with slope -0' and intersection log(b + c) ,..., log(2b) for the difference. c . Etest + b . Etrain will give a constant for an appropriate choice of band c, with b + c = s. The validity of the above model was tested on numerous boolean classifiers with linear decision surfaces. In all experiments we found good agreement with the model and we were able to extract reliable estimates of the three parameters needed to model the learning curves: the asymptotic value a, and the power 0', and amplitude b of the power-law decay. An example is shown in Fig. 4, (left). The considered task is separation of handwritten digits 0-4 from the digits 5-9. This problem is unrealizable with the given database and classifier. The simple modeling of the test and training errors of equation (3) is only assumed to hold for large sizes of the training set, but it appears to be valid already at intermediate sizes, as seen in Fig. 4, (left). The predictive model suggested here is based on this observation, and it can be illustrated from Fig. 4, (left): with test and training errors measured for I ~ 2560 it is possible to estimate the two straight lines, extract approximate values for the three parameters which characterize the learning curves, and use the resulting power-laws to extrapolate the learning curves to the full size of the database. The algorithm for the predictive method is therefore as follows: 1. Measure Etest and Etrain for intermediate sizes of the training set. 2. Plot 10g(Etest + Etrain) and 10g(Etest - Etrain) versus log I. 3. Estimate the two straight lines and extract the asymptotic value a the amplitude b, and the exponent 0'. 4. Extrapolate the learning curves to the full size of the database. Learning Curves: Asymptotic Values and Rate of Convergence 331 log (error) -1 -2 -3r---------~~----o 1 2 log (1/ 256) I • 256 2560 25600 I Figure 4: error 0.25 + points used for prediction •••• predicted learning curves 0.2 0.15 0.1 ...... ~. -4-··-... m A.~-·-A· -~ .•. _ •• -.a 0.05 I training error 0+------------2560 7680 15360 training set size, I Left: Test of the model for a 256 dimensional boolean classifier trained by minimizing a mean squared error. The sum and difference of the test and training errors are shown as a function of the normalized training set size in a log-log-plot (base 10). Each point is the mean with standard deviation for ten different choices of a training set of the given size. The straight line with a = 1, corresponding to a 1/1 decay, is shown as a reference. Right: Prediction of learning curves for a 256 dimensional boolean classifier trained by minimizing a mean squared error. Measured errors for training set size of I ~ 2560 are used to fit the two proposed straight lines in a log-log plot. The three parameters which characterize the learning curves are extracted and used for extrapolation. A prediction for a boolean classifier with linear decision surface is illustrated in Fig. 4, (right). The prediction is excellent for this type of classifiers because the sum and difference of the test and training errors converge quickly to two straight lines in a log-log-plot. Unfortunately, linear decision surfaces are in general not adequate for many real-life applications. The usefulness of the predictive method proposed here can be judged from its performance on real-life sophisticated multi-layer networks. Fig. 5 demonstrates the validity of the model even for a fully-connected multi-layer network operating in its non-linear regime to implement an unrealizable digit recognition task. Already for intermediate sizes of the training set the sum and difference between the test and training errors are again observed to follow straight lines. The predictive method was finally tested on sparsely connected multi-layer networks. Fig. 6, (left), shows the test and training errors for two networks trained for the recognition of handwritten digits. The network termed "old" is commonly referred to as LeNet [LCBD+90]. The network termed "new" is a modification of LeN et with additional feature maps. The full size of the database is 60,000 patterns, 332 Cortes, Jackel, SoHa, Vapnik, and Denker log (error) -1 -2 -3 E lest + E train log ( 11100) , . 1000 10000 100000 I Figure 5: Test of the model for a fully-connected 100-10-10 network. The sum and the difference of the test and training error are shown as a function of the normalized training set size in a log-log-plot. Each point is the mean with standard deviation for 20 different choices of a training set of the given size. a 50-50 % mixture of the NIST1 training and test sets. After training on 12,000 patterns it becomes obvious that the new network will outperform the old network when trained on the full database, but we wish to quantify the expected improvement. If our predictive method gives a good quantitative estimate of the new network's test error at 60,000 patterns, we can decide whether three weeks of training should be devoted to the new architecture. A log-log-plot based on the three datapoints from the new network result in values for the three parameters that determine the power-laws used to extrapolate the learning curves of the new network to the full size of the database, as illustrated in Fig. 6, (right). The predicted test error at the full size of the database I = 60,000 is less than half of the test error for the old architecture, which strongly suggest performing the training on the full database. The result of the full training is also indicated in Fig. 6, (right). The good agreement between predicted and measured values illustrates the power and applicability of the predictive method proposed here to real-life applications. 3 Theoretical Foundation The proposed predictive method based on power-law modeling of the learning curves is not just heuristic. A fair amount of theoretical work has been done within the framework of statistical mechanics [SST92] to compute learning curves for simple classifiers implementing unrealizable rules with non-zero asymptotic error value. A key assumption of this theoretical approach is that the number of weights in the network is large. 1 National Institute for Standards and Technology, Special Database 3. error 0.03 • 0.02 Learning Curves: Asymptotic Values and Rate of Convergence 333 . . • : old network -: new network error 0.02 : new network - - - : new network predicted .................. a C; 0.01 . o 20 Figure 6: 0.01 ...... --.-..... ---------- - - 't:) , ---..------------n ... -30 40 50 60 training set size, 111000 o ~~---------------------. 20 30 40 50 60 training set size, 111000 Left: Test (circles) and training ( triangles) errors for two networks. The "old" network is what commonly is referred to as LeNet. The network termed "new" is a modification of LeNet with additional feature maps. The full size of the database is 60,000 patterns, and it is a 50-50 % mixture of the NIST training and test set. Right: Test (circles) and training (triangles) errors for the new network. The figure shows the predicted values of the learning curves in the range 20,000 - 60,000 training patterns for the "new" network, and the actually measured values at 60,000 patterns. The statistical mechanical calculations support a symmetric power-law decay of the expected test and training errors to their common asymptotic value. The powerlaws describe the behavior in the large I regime, with an exponent a which falls in the interval 1/2 ~ a ~ 1. Our numerical observations and modeling of the test and training errors are in agreement with these theoretical predictions. We have, moreover, observed a correlation between the exponent a and the asymptotic error value a not accounted for by any of the theoretical models considered so far. Fig. 7 shows a plot of the exponent a versus the asymptotic error a evaluated for three different tasks. It appears from this data that the more difficult the target rule, the smaller the exponent, or the slower the learning. A larger generalization error for intermediate training set sizes is in such cases due to the combined effect of a larger asymptotic error and a slower convergence. Numerical results for classifiers of both smaller and larger input dimension support the explanation that this correlation might be due to the finite size of the input dimension of the classifier (here 256). 4 Summary In this paper we propose a practical and principled method for predicting the suitability of classifiers trained on large databases. Such a procedure may eliminate 334 Cortes, Jackel, Solla, Vapnik, and Denker exponent, (X. 0.9 • 0.8 ~I 0.7 0. 0.6 0 0.1 0.2 asymptotic error, a Figure 1: Exponent of extracted power-law decay as a function of asymptotic error for three different tasks. The un-realizability of the tasks, as characterized by the asymptotic error a, can be changed by tuning the strength of a weight-decay constraint on the norm of the weights of the classifier. poor classifiers at an early stage of the training procedure and allow for a more intelligent use of computational resources. The method is based on a simple modeling of the expected training and test errors, expected to be valid for large sizes of the training set. In this model both error measures are assumed to follow power-law decays to their common asymptotic error value, with the same exponent and amplitude characterizing the power-law convergence. The validity of the model has been tested on classifiers with linear as well as nonlinear decision surfaces. The free parameters of the model are extracted from data points obtained at medium sizes of the training set, and an extrapolation gives good estimates of the test error at large size of the training set. Our numerical studies of learning curves have revealed a correlation between the exponent of the power-law decay and the asymptotic error rate. This correlation is not accounted for by any existing theoretical models, and is the subject of continuing research. References [LCBD+90] Y. Le Cun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel. Handwritten digit recognition with a back-propagation network. In Advances in Neural Information Processing Systems, volume 2, pages 396-404. Morgan Kaufman, 1990. [SST92] H. S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples. Physical Review A, 45:6056-6091, 1992.
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Mixtures of Controllers for Jump Linear and Non-linear Plants Timothy W. Cacciatore Department of Neurosciences University of California at San Diego La Jolla, CA 92093 Abstract Steven J. Nowlan Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 We describe an extension to the Mixture of Experts architecture for modelling and controlling dynamical systems which exhibit multiple modes of behavior. This extension is based on a Markov process model, and suggests a recurrent network for gating a set of linear or non-linear controllers. The new architecture is demonstrated to be capable of learning effective control strategies for jump linear and non-linear plants with multiple modes of behavior. 1 Introduction Many stationary dynamic systems exhibit significantly different behaviors under different operating conditions. To control such complex systems it is computationally more efficient to decompose the problem into smaller subtasks, with different control strategies for different operating points. When detailed information about the plant is available, gain scheduling has proven a successful method for designing a global control (Shamma and Athans, 1992). The system is partitioned by choosing several operating points and a linear model for each operating point. A controller is designed for each linear model and a method for interpolating or 'scheduling' the gains of the controllers is chosen. The control problem becomes even more challenging when the system to be controlled is non-stationary, and the mode of the system is not explicitly observable. One important, and well studied, class of non-stationary systems are jump linear systems of the form: ~~ = A(i)x + B(i)u. where x represents the system state, 719 720 Cacciatore and Nowlan u the input, and i, the stochastic parameter that determines the mode of the system, is not explicitly observable. To control such a system, one must estimate the mode of the system from the input-output behavior of the plant and then choose an appropriate control strategy. For many complex plants, an appropriate decomposition is not known a priori. One approach is to learn the decomposition and the piecewise solutions in parallel. The Mixture of Experts architecture (Nowlan 1990, Jacobs et a11991) was proposed as one approach to simultaneously learning a task decomposition and the piecewise solutions in a neural network context. This architecture has been applied to control simple stationary plants, when the operating mode of the plant was explicitly available as an input to the gating network (Jacobs and Jordan 1991). There is a problem with extending this architecture to deal with non-stationary systems such as jump linear systems. The original formulation of this architecture was based on an assumption of statistical independence oftraining pairs appropriate for classification tasks. However, this assumption is inappropriate for modelling the causal dependencies in control tasks. We derive an extension to the original Mixture of Experts architecture which we call the Mixture of Controllers. This extension is based on an nth order Markov model and can be implemented to control nonstationary plants. The new derivation suggests the importance of using recurrence in the gating network, which then learns t.o estimate the conditional state occupancy for sequences of outputs. The power of the architecture is illustrated by learning control and switching strategies for simple jump linear and non-stationary nonlinear plants. The modified recurrent architecture is capable of learning both the control and switching for these plants. while a non-recurrent architecture fails to learn an adequate control. 2 Mixtures of Controllers The architecture of the system is shown in figure 1. Xt denotes the vector of inputs to the controller at time t and Yt is the corresponding overall control output. The architecture is identical to the Mixture of Experts architecture, except that the gating network has become recurrent, receiving its outputs from the previous time step as part of its input. The underlying statistical model, and corresponding training procedure for the Mixture of Controllers, is quite different from that originally proposed for the Mixture of Experts. We assume that the system we are interested in controlling has N different modes or statesl and we will have a distinct control l\·h for each mode. In general we are interested in the likelihood of producing a sequence of control outputs Yl, ... , YT given a sequence of inputs Xl, ... , XT. This likelihood can be computed as: I1L.:P(YtI St = k,Xt)P(St = kIYl .. ·Yt-I,Xl·· .xd k (1) IThis is an idealization and if N is unknown it is safest to overestimate it. Mixtures of Controllers for Jump Linear and Non-Linear Plants 721 1 Yt 2 1---i-iY t 3 L-J-H--"1Yt Yt Figure 1: The Mixture of Controllers architecture. MI, M2 and M3 are feedforward networks implementing controls appropriate for different modes of the system to be controlled. The gating network (Sel.) is recurrent and uses a softmax non-linearity to compute the weight to be assigned to each of the control out.puts. The weighted sum of the controls is then used as the overall control for the plant. where bf represents the probability of producing the desired control Yt given the input Xt and that the system is in state k. If represents the conditional probability of being in state k given the sequence of inputs and outputs seen so far. In order to make the problem tractable, we assume that this conditional probability is completely determined by the current input to the system and the previous state of the system: I: = fW'Y(Xt, {it-I})' Thus we are assuming that our control can be approximated by a Markov process, and since we are assuming that the mode of the system is not explicitly available, this becomes a hidden Markov model. This Markov assumption leads to the particular recurrent gating architecture used in the Mixture of Controllers. If we make the same gaussian assumptions used in the original Mixture of Experts model, we can define a gradient descent procedure for maximizing the log of the likelihood given in Equation 1. Assume b~ = 1 e-(Yt-y~)2/2(72 y'2iu and define f3f = P(YT,"" Yt\Sk, XT,···, Xt), Lt = Lk f3f,f and j3k k R:=~. Lt Then the derivative of the likelihood with respect to the output of one of the controllers becomes: ologL r k( k) a k = l\ Rt Yt - Yt . Yt (2) 722 Cacciatore and Nowlan The derivative of the likelihood with respect to a weight in one of the control networks is computed by accumulating partial derivatives over the sequence of control outputs: For the gating network, we once again use a softmax non-linearity so: k k exp gt It = .. Lj eXP9~ Then a log L _ '""'(Rk _ k) k a k ~ t I't It-I' 9t t (3) (4) The derivatives for the weights in the gating network are again computed by accumulating partial derivatives over output sequences: (5) Equations (2) and (4) turn out to be quite similar to those derived for the original Mixture of Experts architecture. The primary difference is the appearance of (3; rather than bf in the expression for R:. The appearance of /3 is a direct. result of the recurrence introduced into the gating network. {3 can be computed as part of a modified back propagation through time algorithm for the gating network using the recurrence: where /3: = b: + L W kjf3f+l j O;f+l Wkj = olf (6) Equation (6) is the analog of the backward pass in the forward-backward algorithm for standard hidden Markov models. In the simulations reported in the next section, we used an online gradient descent procedure which employs an approximation for (3f which uses only one step of back propagation through time. This approximation did not appear to significantly affect the final performance of the recurrent architecture. 3 Results The performances of the recurrent Mixture of Controllers and non-recurrent Mixture of Experts were compared on three control tasks: a first order jump linear system, a second order jump linear system, and a tracking task that required two nonlinear controllers. The object of the first two jump-linear tasks was to control a plant which switched randomly between two linear systems. The resulting overa.ll systems were highly non-linear. In both the first. and second order cases it was Mixtures of Controllers for Jump Linear and Non-Linear Plants 723 Arst Order Model Traming Error Arst Order Model Trajectory 20c00 200 N~I'OJmnl 150 '5000 b .OCOO j t W <'3 , 00 5000 ·SO RlClJ1Tlnl 0000 .0000 20c00 3OJOO 40000 S<X110 -100 00 SOD 1000 .SOO E_ To"" Figure 2: Left: Training convergence of Mixtures of Experts and Mixtures of Controllers on first order jump linear system. The vertical axis is average squared error over training sequences and horizontal axis is the number of training sequences seen. Right: Sample test trajectory of first order jump linear system under control of Mixture of Controllers. The system switches states at times 50 and 100. desired to drive all plant outputs to zero (zero-forcing control). Neither the first or second order systems could be successfully controlled by a single linear controller. For both jump-linear tasks, the architecture of the MixtUre of Controllers and Mixture of Experts consisted of two linear experts, and a one layer gating network. The input to the experts was the plant output at the previous time step, while the input to the gating network was the ratio of the plant outputs at the two preceding time steps. An ideal linear controller was designed for each mode of the system. Training targets were derived from outputs of the appropriate ideal controller, using the known mode of the system for the training trajectories. The parameters of the gating and control networks were updated after each pass through sample trajectories which contained several state transitions. The recurrent Mixture of Controllers could be trained to successfully control the first order jump linear system (figure 2), and once trained generalized successfully to novel test trajectories. The non-recurrent Mixture of Experts failed to learn even the training data for the first order jump linear system (note the high asymptote for the training error without recurrence in figure 2). The recurrent Mixture of Controllers was also able to learn to control the second order jump linear system (figure 3), however, it was necessary to teacher force the system during the first 5000 epochs of training by providing the true mode of the system as an extra input to the gating network. This extra input was removed at epoch 5000 and the error initially increases dramatically but the system is able to eventually learn to control the second order jump linear system autonomously. Note that the Mixture of Experts system is actually able to learn a successful control even more rapidly than the Mixture of Controllers when the additional teacher input is provided, however learning again completely fails once this input is removed at epoch 5000 (figure 3). 724 Cacciatore and Nowlan Second Order Model Training Error Second Order Model Trajectory Aecurren I Ccnlroler eoo .----~--~--_, '000 3000 -oulpUtl Ideal t ---- oulpU12 ldul2 TIme Figure 3: Left: Training convergence of Mixt.ures of Experts and Mixtures of Controllers on second order jump linear system. Right: Sample test trajectory of second order jump linear system under control of MixtUre of Controllers. The system again switches states at times 50 and 100. In both first and second order cases, the trained Mixture of Controllers is able to control the system in both modes of system behavior, and to deted mode changes automatically. The difficulty in designing a control for a jump linear system usually lies in identifying the state of the system. No explicit law describing how to identify and switch between control modes is necessary to train the Mixture of Controllers, as this is learned automatically as a byproduct of learning to successfully control the system. Performance of the Mixture of Controllers and the Mixture of Experts was also compared on a more complex task requiring a non-linear control law in each mode. The task wa:s to control the trajectory of a ship to track an object traveling in a straight line, or flee from an object having a random walk trajectory (figure 4). There is a high degree of task interference between the controls appropriate during the two modes of object behaviors. The ship dynamics were t.aken from Miller and Sutton (1990). For both the Mixture of Controllers and the Mixture of Experts two experts were used. The experts received past and present measurements of the object bearing, distance, velocity, and the ship heading and turn rate. The controllers specified the desired turn rate of the ship. A one layer gating network was used which received the velocity of the object as input. Training targets were produced from ideal controllers designed for each object behavior. The ideal controller for the random walk behavior produced a turn rate that headed directly away from the object. The ideal controller for intercepting the object used future information about object position to determine the turn rate which would lead to the closest possible intercept point. Both ideal controllers made use of information not available to the Mixture of Experts or Mixture of Controllers. The Mixture of Controllers and the Mixture of Experts were trained on sequences of a) 200 8 Ilil actual ~ ~ IDD Mixtures of Controllers for Jump Linear and Non-Linear Plants 725 ~/1 target! i I ;!l '0 60 80 100 111) X pOliition b) :: 'f\·/fJ''''r·'(r~!-{,·~~N\''1·'I·hf('~/~"r! ·''.l,;''·..t,:·+·i'i'II'·~·y. correct o. 0' incorrect time Figure 4: (a) Actual and desired trajectories of ship under control of Mixture of Experts while attempting to intercept target. (b) Gating unit activities as a function of time for trajectory in (a). trajectories where the object changed behaviors multiple times. The weights of the networks were updated after each pass through the trajectories. The input to the gating net in this ta.sk provided more instantaneous information about the mode of object behavior than was provided in the jump linear tasks. As a result, the nonrecurrent Mixture of Experts was able to achieve a minimum level of performance on the overall task. The recurrent Mixture of Controllers performed much better. The differences between two architectures are revealed by examining the gating network outputs. Without recurrence, the Mixture of Experts gating network could not determine the state of the object with certainty, and compromised by selecting a combination of the correct and incorrect control (figure 4b). Since the two controls are incompatible, this uncertainty degrades the performance of the overall controller. With recurrence in the ga.ting network, the Mixture of Controllers is able to determine the target state with greater certainty by integrating information from many observations of object behavior. The sharper decisions about which control to use greatly improve tracking performance (figure 5). We explored the ability of the Mixture of Controllers to learn the dynamics of switching by training on trajectories where the object switched behavior with varying frequency. The gating network trained on an object that switched behaviors infrequently was sluggish to respond to transitions, but more noise tolerant than the gating network trained on a frequently switching object. Thus, the gating network is able to incorporate the frequency of transition into its state model. 4 Discussion \Ve have described an extension to the Mixture of Experts architecture for modelling and controlling dynamical systems which exhibit multiple modes of behavior. The algorithm we have presented for updating the parameters of the model is a simple gradient descent procedure. Application of the technique to large scale problems 726 Cacciatore and Nowlan a) 250 alii ] uo 1'00 :>. ~ ~ .2t) actual . . desired i i target! i 2D ~ ~ ~ ~ rn x position b) 09 :' , 08 • I 0' , 03 ." ... -.... _.02f\ O\r ~ correct incarect °0 51! .00 1 !I) 200 2Slt :a 3SO tOO time Figure 5: (a) Actual and desired trajectories of ship under control of Mixture of Controllers while attempting to intercept target. (b) Gating unit activities as a function of time for trajectory in (a). Note that these are much less noisy than the activities seen in figure 4(b). may require the development of faster converging update algorithms, perhaps based on the generalized EM (GEM) family of algorithms, or a variant of the iterative reweighted least squares procedure proposed by Jordan and Jacobs (1993) for hierarchies of expert networks. Additional work is also required to establish the stability and convergence rate of the algorithm for use in adaptive control applications. References Jacobs, R.A. and Jordan, M.I. A competitive modular connectionist architecture. Neural Information Processing Systems 3 (1991). Jacobs, R.A., Jordan, M.I., Nowlan, S.J. and Hinton, G.E. Adaptive Mixtures of Local Experts. Neural Computation, 3, 79-87, (1991). Jordan, M.I. and Jacobs, R.A. Hierarchical Mixtures of Experts and the EM algorithm. Neural Computation, (1994). Miller, W.T., Sutton, R.S. and Werbos, P.J. Neural Networks for Control, MIT Press (1993). Nowlan, S.J. Competing Experts: An Experimental Investigation of Associative Mixture Models. Technical Report CRG- TR-90-5, Department of Computer Science, University of Toronto (1990). Shamma, J.S., and Athans, M. Gain scheduling: potential hazards and possible remedies. IEEE Control Systems Magazine, 12:(3), 101-107 (1992).
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A Hybrid Radial Basis Function Neurocomputer and Its Applications Steven S. Watkins ECE Department UCSD La Jolla. CA. 92093 Raoul Tawel JPL Bjorn Lambrigtsen JPL Paul M. Chau ECE Department UCSD La Jolla, CA. 92093 Caltech Pasadena. CA. 91109 Caltech Pasadena. CA. 91109 Mark Plutowski CSE Department UCSD La Jolla. CA. 92093 Abstract A neurocomputer was implemented using radial basis functions and a combination of analog and digital VLSI circuits. The hybrid system uses custom analog circuits for the input layer and a digital signal processing board for the hidden and output layers. The system combines the advantages of both analog and digital circuits. featuring low power consumption while minimizing overall system error. The analog circuits have been fabricated and tested, the system has been built, and several applications have been executed on the system. One application provides significantly better results for a remote sensing problem than have been previously obtained using conventional methods. 1.0 Introduction This paper describes a neurocomputer development system that uses a radial basis function as the transfer function of a neuron rather than the traditional sigmoid function. This neurocOOlputer is a hybrid system which has been implemented with a combination of analog and digital VLSI technologies. It offers the low-power advantage of analog circuits operating in the subthreshold region and the high-precision advantage of digital circuits. The system is targeted for applications that require low-power operation and use input data in analog form, particularly remote sensing and portable computing applications. It has already provided significantly better results for a remote sensing 850 A Hybrid Radial Basis Function Neurocomputer and Its Applications 851 NEURON ,-------- - ----- - Yo '0 2 (c I k - 'k) :E EXPONENTlAL '-'NPUTS MUL TlPL Y AND ACCUMULATE Figure I: Radial Basis Function Network NEURoN [~ Figure 2: Mapping of RBF Network to Hardware Analog Board = = PC Figure 3: The RBF Neurocomputer Development System 852 Watkins, Chau, Tawel, Lambrigsten, and Plutowski climate problem than have been previously obtained using conventional methods. Figure 1 illustrates a radial basis functioo (RBF) network. Radial basis functions have been used to solve mapping and function estimation problems with positive results (Moody and Darken. 1989; Lippman, 1991). When coupled with a dynamic neuron allocation algorithm such as Platt's RANN (platt. 1991). RBF networks can usually be trained much more quickly than a traditional sigmoidal. back-propagation network. RBF networlcs have been implemented with completely-analog (platt, Anderson and Kirk. 1993), c<mpletely-digital (Watkins. Chau and Tawel, Nov .• 1992). and with hybrid analogi digital approaches (Watkins. Chau and Tawel, Oct., 1992). The hybrid approach is optimal for applications which require low power consumption and use input data that is naturally in the analog domain while also requiring the high precision of the digital domain. 2.0 System Architecture and Benefits Figure 2 shows the mapping of the RBF network to hardware. Figure 3 shows the neurocomputer development system. The system consists of a PC controller, a DSP board with a Motorola 56000 DSP chip and a board with analog multipliers. The benefits of the hybrid approach are lower-cost parallelism than is possible with a completely-digital system, and more precise computation than is possible with a completely-analog system. The parallelism is available for low cost in terms of area and power, when the inputs are in the analog domain. When comparing a single analog multiplier to a 100bit fixed point digital multiplier, the analog cell uses less than one-quarter the area and approximately five orders of magnitude less power. When comparing an array of analog multipliers to a Motorola 56000 DSP chip, 1000 Gilbert multipliers can fit in an area about half the size of the DSP chip, while consuming .003% of the power. The restriction of requiring analog inputs is placed on the system. because if the inputs were digital, the high cost of D to A conversion would remove the low cost benefit of the system. lbis restriction causes the neurocomputer to be taIgeted for applications using inputs that are in the analog domain, such as remote sensing applications that use microwave or infrared sensors and speech recognitioo applications that use analog filters. The hybrid system reduces the overall system error when compared with a completelyanalog solution. The digital circuits compute the hidden and output layers with 24 bits of precision while analog circuits are limited to about 8 bits of precision. Also the RANN algorithm requires a large range of width variatioo for the Gaussian function and this is more easily achieved with digital computation. Completely analog solutions to this problem are severely limited by the voltage rails of the chip. 3.0 Circuits Several different analog circuit approaches were explored as possible implementations of the network. Mter the dust settled, we chose to implement only the input layer with analog circuits because it offers the greatest opportunity for parallelism, providing parallel performance benefits at a low cost in terms of area and power. The input layer requires more than 0 UP) computations (where N is the number of neurons). while the hidden and output layers require only 0 (N) computations (because there is one hidden layer computatioo per neuron and the number of outputs is either one or very small). A Hybrid Radial Basis Function Neurocomputer and Its Applications 853 The analog circuits used in the input layer are Gilbert multipliers (Mead. 1989). 'The circuits were fabricated with 2.0 micron. double-poly, P-well. CMOS technology. The Gilbert cell performs the operation of multiplying two voltage differences: (Vi-V2)x(V3V4). In this system. Vi =V3 and V2=V4. which causes the circuit to compute the square of the difference between a stored weight and the input. The current outputs of the Gilbert cells in a row are wired together to sum their currents. giving a sum of squared errors. This current is converted to a voltage. fed to an A to D converter and then passed to the DSP board where the hidden and output layers are computed. The radial basis function (Gaussian) of the hidden layer is computed by using a lookup table. The system uses the fast multiply/accumulate operation of the DSP chip to compute the output layer. 4.0 Applications The low-power feature of the hybrid system makes it attractive for applications where power consumption is a prime consideration, such as satellite-based applications and portable computing (using battery power). The neurocomputer has been applied to three problems: a remote sensing climate problem. the Mackey-Glass chaotic time series estimation and speech phoneme recognitim. The remote sensing application falls into the satellite category. The Mackey-Glass and speech recognition applications are potentially portable. Systems fa these applications are likely to have inputs in the analog domain (eliminating the need for D to A conversion. as already discussed) making it feasible to execute them on the hybrid neurocomputer. 4.1 The Remote Sensing Application The remote sensing problem is an inverse mapping problem that uses microwave energy in different bands as input to predict the water vapor content of the atmosphere at different altitudes. Water vapor content is a key parameter for predicting weather in the tropics and mid-latitudes (Kakar and Lambrigtsen. 1984). The application uses 12 inputs and 1 output. The system input is naturally in analog form. the result of amplified microwave signals, so no D to A conversion of input data is required. Others have used neural networks with success to perform a similar inverse mapping to predict the temperature gradient of the atmosphere CMotteler et al .. 1993). Section 5 details the improved results of the RBF network over conventional methods. Since water vapor content is a very important compment of climate models. improved results in predicted water vapor content means improved climate models. Remote sensing problems require satellite hardware where power consumptim is always a major constraint.The low-power nature of the hybrid network would allow the network to be placed on board a satellite. With future EOS missions requiring several thousand sensors. the on-board network would reduce the bandwidth requirements of the data being sent back to earth. allowing the reduced water vapor content data to be transmitted rather than the raw sensor data. This data bandwidth reduction could be used either to send back more meaningful data to further improve climate models. or to reduce the amount of data transmitted. saving energy. 4.2 The Mackey-Glass Application The Mackey-Glass chaotic time series application uses several previous time sample values to predict the current value of a time series which was generated by the MackeyGlass delay-difference equation. It was used because it has proved to be difficult for 854 Watkins, Chau, Tawel, Lambrigsten, and Plutowski sigmoidal neural networks (platt. 1991). The applicatioo uses 4 inputs and 1 output. The Mackey-Glass time series is representative of time series found in medical applications such as detecting arrhythmias in heartbeats. It could be advantageous to implement this application with portable hardware. 4.3 The Speech Phoneme Recognition Application The speech phoneme recognition problem used the same data as Waibel (Waibel et ai .• 1989) to learn to recognize the acoustically similar phonemes of b. d and g. The application uses 240 inputs and 3 outputs. The speech phoneme recognition problem represents a sub problem of the more difficult continuous speech recognition problem. Speech recognition applications also represent opportunities for portable computing. 5.0 Results 5.1 The Remote Sensing Application Using the RBF neural network 00 the remote sensing climate problem produced significantly better results than had been previously obtained using conventional statistical methods (Kakar and Lambrigtsen. 1984). The input layer of the RBF network was implemented in two different ways: 1) it was simulated with 32-bit floating point precision to represent a digital input layer. and 2) it was implemented with the analog Gilbert multipliers as the input layer. Both implementations produced similar results. At an altitude corresponding to 570 mb pressure, the RBF neural network with a digital input layer produced results with .33 absolute rms error vs. .42 rms error for the best results using conventional methods. This is an improvement of 21 %. Figure 4 shows the plot of retrieved vs. actual water vapor content for both the RBF network and the conventional method. Using the hybrid neurocomputer with the analog input layer for the data at 570 mb pressure produced results with .338 rms error. This is an improvement of 19.5% over the conventional method. Using the analog input layer produced nearly as much improvement as a completely-digital system. demonstrating the feasibility of placing the network on board a satellite. Similar results were obtained for other altitudes. The RBF network also was compared to a sigmoidal network using back propagation learning enhanced with line-search capability (to automatically set step-size). Both networks used eight neurons in the hidden layer. As Figure 5 shows. the RBF network learned much faster than the sigmoidal network. 6 , --~-.~ --/ Key ' o - neural network /' 0 + f .. ::c J!I ! ] ] 2 .;! + _ startatical method o fit+ 0 + 0.- 0 0 Act •• 1 Specific l/...,td,1y + 6 Figure 4: Comparison of Retrieved vs. Actual Water Vapor Content for 570 mb Pressure for RBF Network and Conventional Statistical Method 1 1 09 ~08 ~0 .7 06 05 '_ A Hybrid Radial Basis Function Neurocomputer and Its Applications 855 ------~solid _ r bl software daahed - rbl analog hardware dotted - sigmoid b.IiCkprop o 4 -----=-.::..::.~-:.~ ~-::...:..~_- ------____ . __ _ 03 ---;-·-2--3"4 5 e -7 8 ..L...9 number 01 passes through training patter!'!s Figure 5: Comparison of Learning Curves for RBF and Sigmoidal Networks for Water Vaptt Application 03 -- -,--,-----,.--,---,-, --,--025 . 02 Key' solid _ rbl software dashed - rbl analog hardware dotted - Sigmoid b.Iickprop ~O 15 .... .. .. ..... . ............... ..... .. . , .. .. .......... .... ..... . § 01 \ ... .... -..... ~ ........ --- .... .,-- " .. .... ,. ........ %~-~ 0 '5~~; ·--~1~5--~2~~2 .5~-73--~3~.5--~4 number 01 paaaes through training patterns x 10. Figure 6: Comparison of Learning Curves for RBF and Sigmoidal Networks for MackeyGlass Application 5.2 The Mackey-Glass Application The RBF network was not compared to any non-neural network method for the MackeyGlass time series estimation. It was only compared to a traditional sigmoidal networlc using back propagation learning enhanced with line search. Both networks used four neurons. As Figure 6 shows. applying the RBF neural network to the Mackey-Glass chaotic time series estimation produced much faster learning than the sigmoidal network. The RBF network with a digital input layer and the RBF hybrid network with an analog input layer both produced similar results in dropping to an rms error of about .025 after only 5 minutes of training on a PC using a 486 CPU. Using the digital input layer. the RBF network reached a minimum absolute rms error of .017. while the sigmoidal network reached a minimum absolute rms error of .025. This is an improvement of 32% over the sigmoidal network. Using the hybrid neurocomputer with the analog input layer produced a minimum absolute rms error of .022. This is an improvement of 12% over the sigmoidal network 856 Watkins, Chau, Tawel, Lambrigsten, and Plutowski 5.3 The Speech Phoneme Recognition Application The RBF network was not compared to any non-neural network method for the speech phooeme recognition problem. It was only compared to Waibel's Tme Delay Neural Network (IDNN) (Waibel et al .. 1989). The IDNN uses a topology matched to the timevarying nature of speech with two hidden layers of eight and three neurons respectively. The RBF network used a single hidden layer with the number of neurons varying between eight and one hundred. The IDNN achieved a 98% accuracy on the test set discriminating between the phooemes b. d and g. The RBF network achieved over 99% accuracy in training. but was only able to achieve an 86% accuracy on the test set. To obtain better results. it is clear that the topology of the RBF network needs to be altered to more closely match Waibel's IDNN. However. this topology will complicate the VLSI implementation. 5.4 The Feasibility of Using the Analog Input Layer One potential problem with using an analog input layer is that every individual hybrid RBF neurocomputer might need to be trained on a problem. rather than being able to use a common set of weights obtained from another RBF neurocomputer (which had been previously trained). This potential problem exists because every analog circuit is unique due to variation in the fabrication process. A set of experiments was designed to test this possibility. The remote sensing application and the Mackey-Glass application were trained and tested two different ways: 1) hardware-trainedlhardware-tested. that is. the analog input layer was used for both training and testing; 2) software-trainedlhardware-tested. that is the analog input layer was simulated with 32-bit floating point precision for training and then the analog hardware was used for testing . . The hardwarelhardware results provided a benchmark. The softwarelhardware results demonstrated the feasibility of having a standard set of weights that are not particular to a given set of analog hardware. For both the remote sensing and the Mackey-Glass applications. the rms error performance was only slightly degraded by using weights learned during software simulation. The remote sensing results degraded by only .Oll in terms of absolute rms error. and the MackeyGlass results degraded by only .002 in terms of absolute rms error. The results of the experiment indicate that each individual hybrid RBF neurocomputer only needs to be calibrated. not trained. 6.0 Conclusions A low-power. hybrid analog/digital neurocomputer development system was constructed using custom hardware. The system implements a radial basis function (RBF) network and is targeted for applications that require low power consumption and use analog data as their input. particularly remote sensing and portable applications. Several applications were executed and results were obtained for a remote sensing application that are superior to any previous results. Comparison of the results of a completely-digital simulation of the RBF network and the hybrid analog/digital RBF network demonstrated the feasibility of the hybrid approach. A Hybrid Radial Basis Function Neurocomputer and Its Applications 857 Acknowledgments The research described in this paper was performed at the Center for Space Microelectronics Technology. Jet Propulsion Laboratory. California Institute of Technology, and was sponsored by the National Aerooautics and Space Admjnjstration. One of the authors. Steven S. Watkins. acknowledges the receipt of a Graduate Student Researcher's Center Fellowship from the Natiooal Aeronautics and Space Administration. Useful discussions with Silvio Eberhardt, Roo Fellman. Eric Fossum. Doug Kerns. Fernando Pineda, John Platt, and Anil Thakoor are also gratefully acknowledged. References Ramesh Kakar and Bjorn Lambrigtsen. "A Statistical Correlation Method for the Retrieval of Atmospheric Moisture Profiles by Microwave Radiometry," Journal of Climate and Applied Meteorology. vol. 23, no. 7. July 1984, pp. 1110-1114. R. P. Lippman. "A Critical Overview of Neural Network Pattern Oassifiers." Proceedings of the IEEE Neural Networks for Signal Processing Workshop, 1991, Princeton. NJ .• pp. 266-275. Carver Mead, Analog VLSI and Neural Systems. Addison-Wesley. 1989, pp. 90-94. J. Moody and C. Darken. "Fast Learning in Networks of Locally-Tuned Processing Units," Neural Computation, vol. 1. no. 2, Summer 1989. pp. 281-294. Howard Motteler, lA. Gualtieri. LL. Strow and Larry McMillin. "Neural Networks for Atmospheric Retrievals," NASA Goddard Conference on Space Applications of Artificial Intelligence. 1993, pp. 155-167. John Platt, "A Resource-Allocating Neural Network for Function Interpolation," Neural Computation, vol. 3. no. 2, Summer 1991, pp. 213-225. John Platt. Janeen Anderson and David B. Kirk. "An Analog VLSI Qrip for Radial Basis Functions," NIPS 5. 1993, pp. 765-772. Alexander Waibel. T. Hanazawa. G. Hinton. K. Shikano and K. Lang. "Phoneme Recognition Using Tune-Delay Neural Networks." IEEE International Conference on Acoustics, Speech and Signal Processing, May 1989, pp. 393-404. Steve Watkins, Paul Chau and Raoul Tawel. "A Radial Basis Functioo Neurocomputer with an Analog Input Layer." Proceedings of the IJCNN, Beijing. China. November 1992. pp. ill 225-230. Steve Watkins. Paul Chau and Raoul Tawel, "Different Approaches to Implementing A Radial Basis Function Neurocomputer." RNNSIlEEE Symposium on Neuroinformatics and Neurocomputing. Rostov-on-Don. Russia. October 1992, pp. 1149-1155.
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Resolving motion ambiguities K. I. Diamantaras Siemens Corporate Research 755 College Rd. East Princeton, NJ 08540 Abstract D. Geiger* Courant Institute, NYU Mercer Street New York, NY 10012 We address the problem of optical flow reconstruction and in particular the problem of resolving ambiguities near edges. They occur due to (i) the aperture problem and (ii) the occlusion problem, where pixels on both sides of an intensity edge are assigned the same velocity estimates (and confidence). However, these measurements are correct for just one side of the edge (the non occluded one). Our approach is to introduce an uncertamty field with respect to the estimates and confidence measures. We note that the confidence measures are large at intensity edges and larger at the convex sides of the edges, i.e. inside corners, than at the concave side. We resolve the ambiguities through local interactions via coupled Markov random fields (MRF) . The result is the detection of motion for regions of images with large global convexity. 1 Introduction In this paper we discuss the problem of figure ground separation, via optical flow, for homogeneous images (textured images just provide more information for the disambiguation of figure-ground). We address the problem of optical flow reconstruction and in particular the problem of resolving ambiguities near intensity edges. We concentrate on a two frames problem, where all the motion ambiguities we discuss can be disambiguiated by the human visual system. *work done when the author was at the Isaac Newton Institute and at Siemens Corporate Research 977 978 Diamantaras and Geiger Optical flow is a 2D (two dimensional) field defined as to capture the projection of the 3D (three dimensional) motion field into the view plane (retina). The Horn and Schunk[8] formulation of the problem is to impose (i) the brightness constraint dE(;~y,t) = 0, where E is the intensity image, and (ii) the smoothness of the velocity field. The smoothness can be thought of coming from a rigidity or quasi-rigidity assumption (see Ullman [12]). We utilize two improvements which are important for the optical flow computation, (i) the introduction of the confidence measure (Nagel and Enkelman [10], Anandan [1]) and (ii) the application of smoothness while preserving discontinuities (Geman and Geman [6], Blake and Zisserman [2], Mumford and Shah [9]). It is clear that as an object moves with respect to a background not only optical flow discontinuities occur, but also occlusions occur (and revelations). In stereo, occlusions are related to discontinuities (e.g. Geiger et. al 1992 [5]) , and for motion a similar relation must exist. We study ambiguities ocuring at motion discontinuities and occlusions in images. The paper is organized as follows: Section 2 describes the problem with examples and a brief discussion on possible approaches, section 3 presents our approach, with the formulation of the model and a method to solve it, section 4 gives the results. 2 Motion ambiguities Figure 1 shows two synthetic problems involving a translation and a rotation of simple objects in front of stationary backgrounds. Consider the case of the square translation (see figure Ia.). Humans perceive the square translating, although block-matching (and any other matching technique) gives translation on both sides of the square edges. Moreover, there are other interpretations of the scene, such as the square belonging to the stationary background and the outside being a translating foreground with a square hole. The examples are synthetic, but emphasize the ambiguities. Real images may have more texture, thus many times helping resolve these ambiguities, but not everywhere. (a) (b) Figure 1: Two image sequences of 128 x 128. (a) Square translation of 3 pixels; (b) "Eight" rotation of 10 0 • Note that the "eight" has concave and convex regions. Resolving Motion Ambiguities 979 3 A Markov random field model We describe a model capable of solving these ambiguities. It is based on coupled Markov random fields and thus, based on local processes. Our main contribution is to introduce the idea of uncertainty on the estimates and confidence measures. We propose a Markov field that allows the estimates of each pixel to be chosen among a large neighborhood, thus each pixel estimate can be neglected. We show that convex regions of the image do bias the confidence measures such that the final motion solutions are expected to be the ones with global larger convexity Note that locally, one can have concave regions of a shape that give "wrong" bias (see figure 1 b). 3.1 Block Matching Block matching is the process of correlating a block region of one image, say of size (2w M + 1) x (2w M + 1), with a block regIOn of the other image. Block-matching yields a set of matching errors dir, where (i, j) is a pixel in the image and v = [m, n] is a displacement vector in a search window of size (2ws + 1) x (2ws + 1) around the pixel. We define the velocity measurements gij and the covariance matrix Cij as the mean and variance of the-vector v = [m, n] averaged according to the distribution _kdmn e ']: gij = ~ _kdmn ~ e .] m,n Figure 2 shows the block matching data gij for the two problems discussed above and figure 3 shows the correspondent confidence measurse (inverse of the covariance matrix as defined below). 3.2 The aperture problem and confidence The aperture problem [7] occurs where there is a low confidence on the measurements (data) in the direction along an edge; In particular we follow the approach by [1]. The eigenvalues AI, A2, of C ij correspond to the variance of distribution of v along the directions of the corresponding eigenvectors VI, V2. The confidence of the estimate should be inversely proportional to the variance of the distribution, i.e. the confidence along direction VI (V2) is ex 1/ Al (ex 1/ A2)' All this confidence information can be packaged inside the confidence matrzx defined as follows: Rij=f(Cij+f)-1 (1) where € is a very small constant that guarantees invertibility. Thus the eigenvalues of Rij are values between 0 and 1 corresponding to the confidence along the directions VI and V2, whereas VI and V2 are still eigenvectors of Rij . The confidence measures at straight edges is high perpendincular to the edges and low (zero) along the edges. However, at corners, the confidence is high on both 980 Diamantaras and Geiger directions thus through smoothness this result can be propagated through the other parts of the image, then resolving the aperture problem. 3.3 The localization problem and a binary decision field The localization problem arises due to the local symmetry at intensity edges, where both sides of an edge give the same correspondences. These cases occur when occluded regions are homogeneous and so, block matching, pixel matching or any matching technique can not distinguish which side of the edge is being occluded or is occluding. Even if one considers edge based methods, the same problem arises in the reconstruction stage, where the edge velocities have to be propagated to the rest of the image. In this cases a localization uncertainty is introduced. More precisely, pixels whose matching block contains a strong feature (e.g. a corner) will obtain a high-confidence motion estimate along the direction in which this feature moved. Pixels on both sides of this feature , and at distances less than half the matching window size, W M , will receive roughly the same motion estimates associated with high confidences. However, it could have been just one of the two sides that have moved in this direction. In that case this estimate should not be taken into account on the other side. We note however a bias towards inside of corner regions from the confidence measures. Note that in a corner, despite both sides getting roughly the same velocity estimate and high confidence measures, the inside pixel always get a larger confidence. This bias is due to having more pixels outside the edge of a closed contour than outside, and occurs at the convex regions (e.g. a corner). Thus, in general, the convex regions will have a stronger confidence measure than outside them. Note that at concavities in the "eight" rotation image, the confidence will be higher outside the "eight" and correct at convex regions. Thus, a global optimization will be required to decide which confidences to "pick up" . Our approach to resolve this ambiguity is to allow for the motion estimate at pixel (i, j) to select data from a neighborhood Nij , and its goal is to maximize the total estimates (taking into account the confidence measures). More precisely, let iij be the vector motion field at pixel (i, j) . We introduce a binary field ai'r that indicates which data gi+m,j+n in a neighborhood Nij of (i,j) should correspond to a motion estimate iij . The size of Nij is given by W M + 1 to overcome the localization uncertainty. For a given lattice point (i , j) the boolean parameters ai'r should be mutually exclusive, i.e. only one of them, a~·n· , should be equal to 1 indicating that iij should correspond to gi+m. ,j+n. , While the rest a'?;n , m =f:. m* , n =f:. n*, should be zero (or 2:m.n.EN'J a7r n• = 1). The conditional probability reflects both an uncertainty due to noise and an uncertainty due to spatial localization of the data Resolving Motion Ambiguities 981 3.4 The piecewise smooth prior The prior probability of the motion field fij is a piecewise smoothness condition, as in [6]. P(f. a. h, v) = ~1 exp{ -(I: J.L( i-hi) )llitJ - il-1.) 112+J-L( I-Vi] )lIiij - ii ,J _1112+~'i) (hlJ +Vi) )) } . IJ I (3) where hij = 0 (Vij = 0) if there is no motion discontinuity separating pixels (i ,j) , (i - l,j) ((i,j), (i,j - 1) , otherwise hij = 1 (vii = 1). The parameter J.L has to be estimated. We have considered that the cost to create motion discontinuities should be lowered at intensity edges (see Poggio et al. [11]) , i.e Iii = 1(1 - 6eij), where eij is the intensity edge and 0 ~ 6 ~ 1 and I have to be estimated. 3.5 The posterior distribution The posterior distribution is given by Bayes' law P(f a h vlg R) = 1 P(g Rlf a)P(f a h v) = !e-V(f,o,h,v;g) (4) , ' " P(g, R) " ' " Z where V(j, a , h, v) L { L a~TIIRi+m,j+n(jij - gi+m,j+n)11 2 ij mneN'J + J.L(I- hij)llfij - fi_i,jI12 + J.L(I- Vij)llfij - !i,j_1112 + lij(hij+Vij)} (5) is the energy of the system. Ideally, we would like to mInImIZe V under all possible configurations of the fields f , h, v and a , while obeying the constraint EmneN'J ai]n = 1. 3.6 Mean field techniques Introducing the inverse temperature parameter (3(= liT) we can obtain the transformed probability distribution 1 P{3(j, alg, R) = _e-{3V(f,o) Z{3 (6) where Z{3 L exp{ -(3 L J.L?j Ilfij - Ii-i,j W + J.L~j Ilfij - Ii ,i _1112} U} ij x (LexP{-(3L L a~TIIRi+m ,j+n(jij - gi+m,j+n)11 2 }) (7) {o} ij mneN'J 982 Diamantaras and Geiger where J-lij = J-l(1 - Vij) and J-l?j = J-l(1 - hij ). We have to obey the constraint LmnEN'l o:ir = 1. For the sake of simplicity we have assumed that the neighborhood Nij around site (i, j) is Nij = {( i + m, j + n): -1 ~ m ~ 1, -1 ~ n ~ I}. The second factor in (7) can be explicitly computed. Employing the mean field techniques proposed in [3] and extended in [4] we can average out the variables h, v and 0: (including the constraint) and yield 1 :LII( L (exp{-j1"~+m,j+n(fij - 9i+m,j+n)11 2}) {J} ij m,n=-1 (elll +el-'lIf'l-f,-1,lI12)(el'l +el-'llf.l-f.o1-dI 2)) (8) which yields the following effective energy Veff(f) since Z{3 = L{f} e-{3Veff(f) . Using the saddle point approximation, i.e. considering Z{3 ~ e -(3Veff Cf) with! minimizing Veff (/; g). the mean field equations become 0= L cii.t ~+m,j+n(hj - 9i+m,J+n) + J-lij!1v lij + J-l7j !1 h iij mn with J-lij = J-l( 1 - Vij) and J-lt (hj - h,j -1), and (9) The normalization constant Z{3 called the partition function, has the important property that lim - ~ In Z{3 = min {V(f, 0:, h, v)} {3-00 fJ {f,cr,h,v} (10) Then using an annealing method we let j1 00 and the minimum of V{3 = - ~ In Z{3 approaches asymptotically the desired minimum. Resolving Motion Ambiguities 983 4 Results We have applied an iterative method along with an annealing schedule to solvE' the above mean field equations for f3 ~ 00. The method was run on the two examples already described. Figure 4 depicts the results of the experiments. The system chooses a natural interpretation (in agreement with human perception), namely it interprets the object (e.g. the square in the first example or the eight-shaped region in the second example) moving and the background being stationary. In the beginning of the annealing process the localization field a may produce "erroneous" results, however the neighbor information eventually forces the pixels outside the moving object to coincide with the rest of the background which has zero motion. For the pixels inside the object, on the contrary, the neighbor information eventually reinforces the adoption of the motion of the edges. References [1] P. Anandan, "Measuring Visual Motion from Image Sequences", PhD thesis. COINS Dept., Univ. Massachusetts, Amherst, 1987. [2] A. Blake and A. Zisserman, "Visual Reconstruction", Cambridge, Mass, MIT press, 1987. [3] D. Geiger and F. Girosi, "Parallel and Deterministic Algorithms for MRFs: Surface Reconstruction and Integration", IEEE PAMI: 13(5), May 1991. [4] D. Geiger and A. Yuille, "A Common Framework for Image Segmentation" , Int. J. Comput. Vision, 6(3) , pp. 227-243, 1991. [5] D. Geiger and B. Ladendorf and A. Yuille, "Binocular stereo with occlusion" , Computer Vision- ECCV92, ed. G. Sandini, Springer-Verlag, 588, pp 423-433, May 1992. [6] S. Geman and D. Geman, "Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images", IEEE PAMI 6, pp. 721-741 , 1984. [7] E. C. Hildreth, "The measurement of visual motion" , MIT press, 1983. [8] B.K.P. Horn and B.G. Schunk, "Determining optical flow", Artificial Intelligence, vol 17, pp. 185-203, August 1981. [9] D. Mumford and J. Shah, "Boundary detection by minimizing functionals, 1" , Proc. IEEE Conf. on Computer Vision & Pattern Recognition, San Francisco, CA,1985. [10] H.-H. Nagel and W. Enkelmann, "An Investigation of Smoothness Constraints for the Estimation of Displacement Vector Fields from Image Sequences" , IEEE PAMI: 8, 1986. [11] T. Poggio and E. B. Gamble and J. J. Little, "Parallel Integration of Vision Module" , Science, vol 242, pp. 436-440, 1988. [12] S. Ullman, "The Interpretation of Visual Motion" , Cambridge, Mass, MIT press, 1979. 984 Diamantaras and Geiger (a) (b) (c) Figure 2: Block matching data giJ' Both sides of the edges have the same data (and same confidence). White represents motion to the right (x-direction) or up (y-direction). Black is the complement. (a) The x-component of the data for the square translation. (b) The x-component of the data for the rotation and (c) the y-component of the data. (a) (b) Figure 3: The confidence R extracted from the block matching data gij. The display is the sum of both eigenvalues, i.e. the trace of R. Both sides of the edges have the same confidence. White represents high confidence. (a) For the square translation. (b) For the rotation. (a) (b) (c) Figure 4: The final motion estimation, after 20000 iterations, resolved the ambiguities with a natural interpretation of the scene. J.l = 10, 6 = 1" = 100. (a) square translation (b) x component of the motion rotation (c) y component of the motion rotation
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Developing Population Codes By Minimizing Description Length Richard S. Zemel CNL, The Salk Institute 10010 North Torrey Pines Rd. La J oUa, CA 92037 Geoffrey E. Hinton Department of Computer Science University of Toronto Toronto M5S 1A4 Canada Abstract The Minimum Description Length principle (MDL) can be used to train the hidden units of a neural network to extract a representation that is cheap to describe but nonetheless allows the input to be reconstructed accurately. We show how MDL can be used to develop highly redundant population codes. Each hidden unit has a location in a low-dimensional implicit space. If the hidden unit activities form a bump of a standard shape in this space, they can be cheaply encoded by the center ofthis bump. So the weights from the input units to the hidden units in an autoencoder are trained to make the activities form a standard bump. The coordinates of the hidden units in the implicit space are also learned, thus allowing flexibility, as the network develops a discontinuous topography when presented with different input classes. Population-coding in a space other than the input enables a network to extract nonlinear higher-order properties of the inputs. Most existing unsupervised learning algorithms can be understood using the Minimum Description Length (MDL) principle (Rissanen, 1989). Given an ensemble of input vectors, the aim of the learning algorithm is to find a method of coding each input vector that minimizes the total cost, in bits, of communicating the input vectors to a receiver. There are three terms in the total description length: • The code-cost is the number of bits required to communicate the code that the algorithm assigns to each input vector. 11 12 Zemel and Hinton • The model-cost is the number of bits required to specify how to reconstruct input vectors from codes (e.g., the hidden-to-output weights) . • The reconstruction-error is the number of bits required to fix up any errors that occur when the input vector is reconstructed from its code. Formulating the problem in terms of a communication model allows us to derive an objective function for a network (note that we are not actually sending the bits). For example, in competitive learning (vector quantization), the code is the identity of the winning hidden unit, so by limiting the system to 1i units we limit the average code-cost to at most log21i bits. The reconstruction-error is proportional to the squared difference between the input vector and the weight-vector of the winner, and this is what competitive learning algorithms minimize. The model-cost is usually ignored. The representations produced by vector quantization contain very little information about the in put (at most log21i bits). To get richer representations we must allow many hidden units to be active at once and to have varying activity levels. Principal components analysis (PCA) achieves this for linear mappings from inputs to codes. It can be viewed as a version of MDL in which we limit the code-cost by only having a few hidden units, and ignoring the model-cost and the accuracy with which the hidden activities must be coded. An autoencoder (see Figure 2) that tries to reconstruct the input vector on its output units will perform a version of PCA if the output units are linear. We can obtain novel and interesting unsupervised learning algorithms using this MDL approach by considering various alternative methods of communicating the hidden activities. The algorithms can all be implemented by backpropagating the derivative of the code-cost for the hidden units in addition to the derivative of the reconstruction-error backpropagated from the output units. Any method that communicates each hidden activity separately and independently will tend to lead to factorial codes because any mutual information between hidden units will cause redundancy in the communicated message, so the pressure to keep the message short will squeeze out the redundancy. In (Zemel, 1993) and (Hinton and Zemel, 1994), we present algorithms derived from this MDL approach aimed at developing factorial codes. Although factorial codes are interesting, they are not robust against hardware failure nor do they resemble the population codes found in some parts of the brain. Our aim in this paper is to show how the MDL approach can be used to develop population codes in which the activities of hidden units are highly correlated. For a more complete discussion of the details of this algorithm, see (Zemel, 1993). Unsupervised algorithms contain an implicit assumption about the nature of the structure or constraints underlying the input set. For example, competitive learning algorithms are suited to datasets in which each input can be attributed to one of a set of possible causes. In the algorithm we present here, we assume that each input can be described as a point in a low-dimensional continuous constraint space. For instance, a complex shape may require a detailed representation, but a set of images of that shape from multiple viewpoints can be concisely represented by first describing the shape, and then encoding each instance as a point in the constraint space spanned by the viewing parameters. Our goal is to find and represent the constraint space underlying high-dimensional data samples. Developing Population Codes by Minimizing Description Length 13 size • • • • • • • • • • • • ·x. • • • • • • • • • orientation Figure 1: The population code for an instance in a two-dimensional implicit space. The position of each blob corresponds to the position of a unit within the population, and the blob size corresponds to the unit's activity. Here one dimension describes the size and the other the orientation of a shape. We can determine the instantiation parameters of this particular shape by computing the center of gravity of the blob activities, marked here by an "X". 1 POPULATION CODES In order to represent inputs as points drawn from a constraint space, we choose a population code style of representation. In a population code, each code unit is associated with a position in what we call the implicit space, and the code units' pattern of activity conveys a single point in this space. This implicit space should correspond to the constraint space. For example, suppose that each code unit is assigned a position in a two-dimensional implicit space, where one dimension corresponds to the size of the shape and the second to its orientation in the image (see Figure 1). A population of code units broadly-tuned to different positions can represent any particular instance of the shape by their relative activity levels. This example illustrates that population codes involve three quite different spaces: the input-vector space (the pixel intensities in the example); the hidden-vector space (where each hidden, or code unit entails an additional dimension); and this third, low-dimensional space which we term the implicit space. In a learning algorithm for population codes, this implicit space is intended to come to smoothly represent the underlying dimensions of variability in the inputs, i.e., the constraint space. For instance, the Kohonen (1982) algorithm defines the implicit space topology through fixed neighborhood relations, and the algorithm then manipulates hiddenvector space so that neighbors in implicit space respond to similar inputs. This form of coding has several computational advantages, in addition to its significance due to its prevalence in biological systems. Population codes contain some redundancy and hence have some degree of fault-tolerance, and they reflect underlying structure of the input, in that similar inputs are mapped to nearby implicit positions. They also possess a hyperacuity property, as the number of implicit positions that can be represented far exceeds the number of code units. 14 Zemel and Hinton 2 LEARNING POPULATION CODES WITH MDL Autoencoders are a general way of addressing issues of coding, in which the hidden unit activities for an input are the codes for that input which are produced by the input-hidden weights, and in which reconstruction from the code is done by the hidden-output mapping. In order to allow an autoencoder to develop population codes for an input set, we need some additional structure in the hidden layer that will allow a code vector to be interpreted as a point in implicit space. While most topographic-map formation algorithms (e.g., the Kohonen and elastic net (Durbin and Willshaw, 1987) algorithms) define the topology of this implicit space by fixed neighborhood relations, in our algorithm we use a more explicit representation. Each hidden unit has weights coming from the input units that determine its activity level. But in addition to these weights, it has another set of adjustable parameters that represent its coordinates in the implicit space. To determine what implicit position is represented by a vector of hidden activities, we can average together the implicit coordinates of the hidden units, weighting each coordinate vector by the activity level of the unit. Suppose, for example, that each hidden unit is connected to an 8x8 retina and has 2 implicit coordinates that represent the size and orientation of a particular kind of shape on the retina, as in our earlier example. If we plot the hidden activity levels in the implicit space (not the input space), we would like to see a bump of activity of a standard shape (e.g., a Gaussian) whose center represents the instantiation parameters of the shape (Figure 2 depicts this for a 1D implicit space). If the activities form a perfect Gaussian bump of fixed variance we can communicate them by simply communicating the coordinates of the mean of the Gaussian; this is very economical if there are many less implicit coordinates than hidden units. It is important to realize that the activity of a hidden unit is actually caused by the input-to-hidden weights, but by setting these weights appropriately we can make the activity match the height under the Gaussian in implicit space. If the activity bump is not quite perfect, we must also encode the bump-error-the misfit between the actual activity levels and the levels predicted by the Gaussian bump. The cost of encoding this misfit is what forces the activity bump in implicit space to approximate a Gaussian. The reconstruction-error is then the deviation of the output from the input. This reconstruction ignores implicit space; the output activities only depend on the vector of hidden activities and weights. 2.1 The objective function Currently, we ignore the model-cost, so the description length to be minimized is: Et Bt + Rt 1£ N I)bj - bj)2 /2VB + L(a~ - c~)2 /2VR (1) j=l k=l where a, b, c are the activities of units in the input, hidden, and output layers, respectively, VB and VR are the fixed variances of the Gaussians used for coding the Developing Population Codes by Minimizing Description Length 15 Output (1...N) IIidden (l...H) NETWOHI< ... ~ - ------~ Inpllt. 0 () ... () 0 (l...N) Activity (b) .l JJ Xl x\ IMPLICIT SPACE (1/ = 1) I; I: I: I: Ii I: I' X(j X2 i7 Xi X8 X7 Posit ion (x) II} ,;/ J beRt-fit Gaussian .' X~ Figure 2: Each of the 1t hidden units in the autoencoder has an associated position in implicit space. Here we show a ID implicit space. The activity h; of each hidden unit j on case t is shown by a solid line. The network fits the best Gaussian to this pattern of activity in implicit space. The predicted activity, h;, of unit j under this Gaussian is based on the distance from Xj to the mean j..lt; it serves as a target for hj. bump-errors and the reconstruction-errors, and the other symbols are explained in the caption of Figure 2. We compute the actual activity of a hidden unit, h;, as a normalized exponential of its total input. 1 Note that a unit's actual activity is independent of its position in implicit space. Its expected activity is its normalized value under the predicted Gaussian bump: 1{. hj = exp( -(Xj - j..lt)2 /2(7'2)/ L exp( -(xi - j..lt)2/2(7'2) (2) i=l where (7' is the width of the bump, which we assume for now is fixed throughout training. We have explored several methods for computing the mean of this bump. Simply computing the center of gravity of the representation units' positions, weighted by their activity, produces a bias towards points in the center of implicit space. Instead, on each case, a separate minimization determines j..lt; it is the position in implicit space that minimizes Bt given {Xj' hj} . The network has full inter-layer connectivity, and linear output units. Both the network weights and the implicit coordinates of the hidden units are adapted to minimize E. 1b~ = exp(net~)/ 2::::1 exp(net~), where net~ is the net input into unit j on case t. 16 Zemel and Hinton Unit 18 - Epoch 0 Unit 18 - Epoch 23 0.08 0.2 0.06 0.15 ActivityO.04 Activity 0.1 0.05 x posi tion Y position X position y position 10 10 Figure 3: This figure shows the receptive field in implicit space for a hidden unit. The left panel shows that before learning, the unit responds randomly to 100 different test patterns, generated by positioning a shape in the image at each point in a 10xlO grid. Here the 2 dimensions in implicit space correspond to x and y positions. The right panel shows that after learning, the hidden unit responds to objects in a particular position, and its activity level falls off smoothly as the object position moves away from the center of the learned receptive field. 3 EXPERIMENTAL RESULTS In the first experiment, each 8x8 real-valued input image contained an instance of a simple shape in a random (x, y)-position. The network began with random weights, and each of 100 hidden units in a random 2D implicit position; we trained it using conjugate gradient on 400 examples. The network converged after 25 epochs. Each hidden unit developed a receptive field so that it responded to inputs in a limited neighborhood that corresponded to its learned position in implicit space (see Figure 3). The set of hidden units covered the range of possible positions. In a second experiment, we also varied the orientation of the shape and we gave each hidden unit three implicit coordinates. The network converged after 60 epochs of training on 1000 images. The hidden unit activities formed a population code that allowed the input to be accurately reconstructed. A third experiment employed a training set where each image contained either a horizontal or vertical bar, in some random position. The hidden units formed an interesting 2D implicit space in this case: one set of hidden units moved to one corner of the space, and represented instances of one shape, while the other group moved to an opposite corner and represented the other (Figure 4). The network was thus able to squeeze a third dimension (i.e., which shape) into the 2D implicit space. This type of representation would be difficult to learn in a Kohonen network; the fact that the hidden units learn their implicit coordinates allows more flexibility than a system in which these coordinates are fixed in advance. Developing Population Codes by Minimizing Description Length 17 Implicit Spare (Epoch 0) Implicit Spare (Epoch 120) Y 6.s0 --- -----,--, --- - - ,- - -T - ----,-- - 1 - 6~ - --,--r--r-----.-~-·--, Xposn x x x °mean.V 6.00 )( x 0>< x "x )( * 0 0 6.00 x D~ )( x °mean.H 0 )( 5.50 x x o x Da° o 0 to · ~}o xcIX 5.50 XX l,~oWt ° !i.00 .,P • cP r 0'01 rmx x • ~ " 5.00 x 'bl:r$J 0,(' x x ,(' x qfJb °X0'b ° "x 4.50 x x xX x o 0 o~ x 4.50 ~c xBij /1x x * ~. nil X x n 4.00 • '6 x o )(o~ '" '''xc 4.00 . )( 0 )( • x x x 3.50 -IV x 0 c. x x x III X I ,,>!)1(0 ° ., 3.50 x " x y )( )( x )( 3.00 [] . f,~ x x " x f no • x x ~ 3.00 • ••• x ~ x 2.50 • II. J< x x If " • 2.50 ~~ )( ·oX • .J,' • • X ° 'ic x x 2.00 •••• lff~~ , 0 r8 x x )( .~x I x ~ t u 2.00 "x 'b x • x ~~~ UiO x • CI X • X x' '\, ~ ~ o • • ~ I~O r;/~o<J1' )( ~.~n jroX"x • ,8 Ii' x x • ~.'oxx x 1.00 xn n • c x 1.00 0 c • -t .~,,,P 0050 0 x fiX Xx "1, X x" x n n x • • x ~ IJ 0.50 )(Jf. x x x )f,.o I) x x 0.00 x x x x _L - 1L _ L . L L _LL_ .. L _L _ L _ _ __ _ L. _L _J X 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Figure 4: This figure shows the positions of the hidden units and the means in the 2D implicit space before and after training on the horizontal/vertical task. The means in the top right of the second plot all correspond to images containing vertical bars, while the other set correspond to horizontal bar images. Note that some hidden units are far from all the means; these units do not playa role in the coding of the input, and are free to be recruited for other types of input cases. 4 RELATED WORK This new algorithm bears some similarities to several earlier algorithms. In the experiments presented above, each hidden unit learns to act as a Radial Basis Function (RBF) unit. Unlike standard RBFs, however, here the RBF activity serves as a target for the activity levels, and is determined by distance in a space other than the input space. Our algorithm is more similar to topographic map formation algorithms, such as the Kohonen and elastic-net algorithms. In these methods, however, the populationcode is in effect formed in input space. Population coding in a space other than the input enables our networks to extract nonlinear higher-order properties of the inputs. In (Saund, 1989), hidden unit patterns of activity in an autoencoder are trained to form Gaussian bumps, where the center of the bump is intended to correspond to the position in an underlying dimension of the inputs. In addition to the objective functions being quite different in the two algorithms, another crucial difference exists: in his algorithm, as well as the other earlier algorithms, the implicit space topology is statically determined by the ordering of the hidden units, while units in our model learn their implicit coordinates. 18 Zemel and Hinton 5 CONCLUSIONS AND CURRENT DIRECTIONS We have shown how MDL can be used to develop non-factorial, redundant representations. The objective function is derived from a communication model where rather than communicating each hidden unit activity independently, we instead communicate the location of a Gaussian bump in a low-dimensional implicit space. If hidden units are appropriately tuned in this space their activities can then be inferred from the bump location. Our method can easily be applied to networks with multiple hidden layers, where the implicit space is constructed at the last hidden layer before the output and derivatives are then backpropagated; this allows the implicit space to correspond to arbitrarily high-order input properties. Alternatively, instead of using multiple hidden layers to extract a single code for the input, one could use a hierarchical system in which the code-cost is computed at every layer. A limitation of this approach (as well as the aforementioned approaches) is the need to predefine the dimensionality of implicit space. We are currently working on an extension that will allow the learning algorithm to determine for itself the appropriate number of dimensions in implicit space. We start with many dimensions but include the cost of specifying j-tt in the description length. This obviously depends on how many implicit coordinates are used. If all of the hidden units have the same value for one of the implicit coordinates, it costs nothing to communicate that value for each bump. In general, the cost of an implicit coordinate depends on the ratio between its variance (over all the different bumps) and the accuracy with which it must be communicated. So the network can save bits by reducing the variance for unneeded coordinates. This creates a smooth search space for determining how many implicit coordinates are needed. Acknowledgements This research was supported by grants from NSERC, the Ontario Information Technology Research Center, and the Institute for Robotics and Intelligent Systems. Geoffrey Hinton is the Noranda Fellow of the Canadian Institute for Advanced Research. We thank Peter Dayan for helpful discussions. References Durbin, R. and Willshaw, D. (1987). An analogue approach to the travelling salesman problem. Nature, 326:689-691. Hinton, G. and Zemel, R. (1994). Autoencoders, minimum description length, and Helmholtz free energy. To appear in Cowan, J.D., Tesauro, G., and Alspector, J. (eds.), Advances in Neural Information Processing Systems 6. San Francisco, CA: Morgan Kaufmann. Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43:59-69. Rissanen, J. (1989). Stochastic Complexity in StatisticalInquiry. World Scientific Publishing Co., Singapore. Saund, E. (1989). Dimensionality-reduction using connectionist networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(3):304-314. Zemel, R. (1993). A Minimum Description Length Framework for Unsupervised Learning. Ph.D. Thesis, Department of Computer Science, University of Toronto.
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Learning in Computer Vision and Image Understanding Hayit Greenspan Department of Electrical Engineering California Institute of Technology, 116-81 Pasadena, CA 91125 There is an increasing interest in the area of Learning in Computer Vision and Image Understanding, both from researchers in the learning community and from researchers involved with the computer vision world. The field is characterized by a shift away from the classical, purely model-based, computer vision techniques, towards data-driven learning paradigms for solving real-world vision problems. Using learning in segmentation or recognition tasks has several advantages over classical model-based techniques. These include adaptivity to noise and changing environments, as well as in many cases, a simplified system generation procedure. Yet, learning from examples introduces a new challenge - getting a representative data set of examples from which to learn. Applications of learning systems to practical problems have shown that the performance of the system is often critically dependent on both the size and quality of the training set. Federico Girosi of MIT suggested the use of prior information as a general method for synthesizing many training examples from few exemplars. Prototypical transformations are used for general 3D object recognition. Face-recognition was presented as a particular example. Dean Pomerleau of Carnegie Mellon addressed the training data problem as well, within the context of ALVINN, a neural network vision system which drives an autonomous van without human intervention. Some general problems emerge, such as getting sufficient training data for the more unexpected scenes including passing cars and intersections. Several techniques for exploiting prior geometric knowledge during training and testing of the neural-network, were presented. A somewhat different perspective was presented by Bartlett Mel of Caltech. Bartlett introduced a 3D object recognition approach based on concepts from the human visual system. Here the assumption is that a large database of examples exists, with varying viewing angles and distances, as is available to human observers as they manipulate and inspect common objects. A different issue of interest was using learning schemes in general recognition frameworks which can handle several different vision problems. Hayit Greenspan of Caltech suggested combining unsupervised and supervised learning approaches within a multiresolution image representation space, for texture and shape recognition. It was suggested that shifting the input pixel representation to a more robust representation (using a pyramid filtering approach) in combination with learning 1182 Learning in Computer Vision and Image Understanding 1183 schemes can combine the advantages of both approaches. Jonathan Marshall of Univ. of North Carolina concentrated on unsupervised learning and proposed that a common set of unsupervised learning rules might provide a basis for communication between different visual modules (such as stereopsis, motion perception, depth and so forth). The role of unsupervised learning in vision tasks, and its combination with supervised learning, was an issue of discussion. The question arose on how much unsupervised learning is actually unsupervised. Some a-priori knowledge, or bias, is always present (e.g., the metric chosen for the task). Eric Saund of Xerox introduced the window registration problem in unsupervised learning of visual features. He argued that there is a strong dependence on the window placement as slight shifts in the window placement can represent confounding assignments of image data to the input units of the classifying network. Chris Williams of Toronto introduced the use of unsupervised learning for classifying objects. Given a set of images, each of which contains one instance of a small but unknown set of objects imaged from a random viewpoint, unsupervised learning is used to discover the object classes. Data is grouped into objects via a mixture model which is trained with the EM algorithm. Real-world computer vision applications in which learning can playa major role, and the challenges involved, was an additional theme in the workshop. Yann Le Cun of AT&T described a handwritten word recognizer system of multiple modules, as an example of a large scale vision system. Yann suggested that increasing the role of learning in all modules allows one to minimize the amount of hand-built heuristics and improves the robustness and generality of the system. Challenges include training large learning machines which are composed of multiple, heterogeneous modules, and what the modules should contain. Padhraic Smyth of JPL introduced the challenges for vision and learning in the context of large scientific image databases. In this domain there is often a large amount of data which typically has no ground truth labeling. In addition, natural objects can be much more difficult to deal with than man made objects. Learning can be valuable here, as a low-cost solution and sometimes the only solution (with model-based schemes being impractical). The task of face recognition was addressed by Joachim Buhmann of Bonn. Elastic matching was introduced for translation, rotation and scale invariant recognition. Methods to combine unsupervised and supervised data clustering with elastic matching to learn a discriminant metric and enhance saliency of prototypes were discussed. Related issues from a recent AAAI forum on Machine Learning in Computer Vision, were presented by Rich Zemel of the Salk Institute. In Conclusion The vision world is very diverse with each different task introducing a whole spectrum of challenges and open issues. Currently, many of the approaches are very application dependent. It is clear that much effort still needs to be put in the definition of the underlying themes of the field as combined across the different application domains. There was general agreement at the workshop that the issues brought up should be pursued further and discussed at future follow-up workshops. Special thanks to Padhraic Smyth, Tommy Poggio, and Rama Chellappa for their contribution to the organization of the workshop.
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Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes Alan Lapedes Complex Systems Group (TI3) LANL, MS B213 Los Alamos N .M. 87545 and The Santa Fe Institute, Santa Fe, New Mexico Evan Steeg Department of Computer Science University of Toronto, Toronto, Canada Robert Farber Complex Systems Group (TI3) LANL, MS B213 Los Alamos N .M. 87545 Abstract We use two co-evolving neural networks to determine new classes of protein secondary structure which are significantly more predictable from local amino sequence than the conventional secondary structure classification. Accurate prediction of the conventional secondary structure classes: alpha helix, beta strand, and coil, from primary sequence has long been an important problem in computational molecular biology. Neural networks have been a popular method to attempt to predict these conventional secondary structure classes. Accuracy has been disappointingly low. The algorithm presented here uses neural networks to similtaneously examine both sequence and structure data, and to evolve new classes of secondary structure that can be predicted from sequence with significantly higher accuracy than the conventional classes. These new classes have both similarities to, and differences with the conventional alpha helix, beta strand and coil. 809 810 Lapedes, Steeg, and Farber The conventional classes of protein secondary structure, alpha helix and beta sheet, were first introduced in 1951 by Linus Pauling and Robert Corey [Pauling, 1951] on the basis of molecular modeling. Prediction of secondary structure from the amino acid sequence has long been an important problem in computational molecular biology. There have been numerous attempts to predict locally defined secondary structure classes using only a local window of sequence information. The prediction methodology ranges from a combination of statistical and rule-based methods [Chou, 1978] to neural net methods [Qian, 1988], [Maclin, 1992], [Kneller, 1990], [Stolorz, 1992]. Despite a variety of intense efforts, the accuracy of prediction of conventional secondary structure is still distressingly low. In this paper we will use neural networks to generalize the notion of protein secondary structure and to find new classes of structure that are significantly more predictable. We define protein "secondary structure" to be any classification of protein structure that can be defined using only local "windows" of structural information about the protein. Such structural information could be, e.g., the classic cI>'lf angles [Schulz, 1979] that describe the relative orientation of peptide units along the protein backbone, or any other representation of local backbone structure. A classification of local structure into "secondary structure classes", is defined to be the result of any algorithm that uses a representation of local structure as Input, and which produces discrete classification labels as Output. This is a very general definition of local secondary structure that subsumes all previous definitions. We develop classifications that are more predictable than the standard classifications [Pauling, 1951] [Kabsch, 1983] which were used in previous machine learning projects, as well as in other analyses of protein shape. We show that these new, predictable classes of secondary structure bear some relation to the conventional category of "helix", but also display significant differences. We consider the definition, and prediction from sequence, of just two classes of structure. The extension to multiple classes is not difficult, but will not be made explicit here for reasons of clarity. We won't discuss details concerning construction of a representative training set, or details of conventional neural network training algorithms, such as backpropagation. These are well studied subjects that are addressed in e.g., [Stolorz, 1992] in the context of protein secondary structure prediction. We note in passing that one can employ complicated network architectures containing many output neurons (e.g. three output neurons for predicting alpha helix, beta chain, random coil), or many hidden units etc. (c.f. [Stolorz, 1992], [Qian, 1988], [Kneller, 1990]). However, explanatory figures presented in the next section employ only one output unit per net, and no hidden units, for clarity. Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes 811 A widely adopted definition of protein secondary structure classes is due to Kabsch and Sander [Kabsch, 1983]. It has become conventional to use the Kabsch and Sander definition to define, via local structural information, three classes of secondary structure: alpha helix, beta strand, and a default class called random coil. The Kabsch and Sander alpha helix and beta strand classification captures in large part the classification first introduced by Pauling and Corey [Pauling, 1951]. Software implementing the Kabsch and Sander definitions, which take a local window of structural information as Input, and produce the Kabsch and Sander secondary structure classification of the window as Output, is widely available. The key ideas of this paper are contained in Fig. (1). [ F = I,-C-o-r:e- Ia-h-' o-n-( (-)-l-P)- O-(-P-)----. _ J _P_~ __ L_' R_~ Left Net Maps AA sequence to "secondary structure" . Right Net Maps <l>.\f' to "secondary structure". In this figure the Kabsch and Sander rules are represented by a second neural network. The Kabsch and Sander rules are just an Input/Output mapping (from a local window of structure to a classification of that structure) and may in principle be replaced with an equivalent neural net representing the same Input/Output mapping. We explicitly demonstrated that a simple neural net is capable of representing rules of the complexity of the Kabsch and Sander rules by training a network to perform the same structure classification as the Kabsch and Sander rules, and obtained high accuracy. The representation of the structure data in the right-hand network uses cI>\i' angles. The right-hand net sees a window of cI>\i' angles corresponding to the window of amino acids in the left-hand network. Problems due to the angular periodicity of the cI>\i' angles (i.e., 360 degrees and 0 degrees are different numbers, but represent the same angle) are eliminated by utilizing both the sin and cos of each angle. 812 Lapedes, Steeg, and Farber The representation of the amino acids in the left-hand network is the usual unary representation employing twenty bits per amino acid. Results quoted in this paper do not use a special twenty-first bit to represent positions in a window extending past the ends of a protein. Note that the right-hand neural network could implement extremely general definitions of secondary structure by changing the weights. We next show how to change the weights in a fashion so that new classifications of secondary structure are derived under the important restriction that they be predictable from amino acid sequence. In other words, we require that the synaptic weights be chosen so that the output of the left-hand network and the output of the right-hand network agree for each sequence-structure pair that is input to the two networks. To achieve this, both networks are trained simultaneously, starting from random initial weights in each net, under the sole constraint that the outputs of the two networks agree for each pattern in the training set. The mathematical implementation of this constraint is described in various versions below. This procedure is a general, effective method of evolving predictable secondary structure classifications of experimental data. The goal of this research is to use two mutually self-supervised networks to define new classes of protein secondary structure which are more predictable from sequence than the standard classes of alpha helix, beta sheet or coil. 3 CONSTRAINING THE TWO NETS TO AGREE One way to impose agreement between the outputs of the two networks is to require that they covary when viewed as a stream of real numbers. Note that it is not sufficient to merely require that the outputs of the left-hand and right-hand nets agree by, e.g., minimizing the following objective function (1) P Here, LeftO(p) and RightO(p) represent the outputs of the left-hand and righthand networks, respectively, for the pth pair of input windows: (sequence window -left net) and (structure window -right net). It is necessary to avoid the trivial minimum of E obtained where the weights and thresholds are set so that each net presents a constant Output regardless of the input data. This is easily accomplished in Eqn (1) by merely setting all the weights and thresholds to 0.0. Demanding that the outputs vary, or more explicitly co-vary, is a viable solution to avoiding trivial local minima. Therefore, one can maximize the correlation, P, between the left-hand and right-hand network outputs. The standard correlation measure between two objects, LeftO(p) and RightO(p) is: p = '2:)LeftO(p) - LeftO)(RightO(p) - RightO) (3) P where LeftO denotes the mean of the left net's outputs over the training set, and respectively for the right net. p is zero if there is no variation, and is maximized Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes 813 if there is simultaneously both individual variation and joint agreement. In our situation it is equally desirable to have the networks maximally anti-correlated as it is for them to be correlated. (Whether the networks choose correlation, or anti-correlation, is evident from the behavior on the training set). Hence the minimization of E = _p2 would ensure that the outputs are maximally correlated (or anti-correlated). While this work was in progress we received a preprint by Schmidhuber [Schmidhuber, 1992] who essentially implemented Eqn. (1) with an additional variance term (in a totally different context). Our results using this measure seem quite susceptible to local minima and we prefer alternative measures to enforce agreement. One alternative to enforce agreement, since one ultimately measures predictive performance on the basis of the Mathews correlation coefficient (see, e.g., [Stolorz, 1992]), is to simultaneously train the two networks to maximize this measure. The Mathews coefficient, Gi, for the ith state is defined as: c. _ Pini - UiOi I [(ni + ui)(ni + Oi)(Pi + Ud(pi + Oi»)1/2 where Pi is the number of examples where the left-hand net and right-hand net both predict class i, ni is the number of examples where neither net predicts ;, Ui counts the examples where the left net predicts i and the right net does not, and 0i counts the reverse. Minimizing E = -Gi 2 optimizes Gi. Other training measures forcing agreement of the left and right networks may be used. Particularly suitable for the situation of many outputs (i.e., more than twoclass discrimination) is "mutual information". Use of mutual information in this context is related to the IMAX algorithm for unsupervised detection of regularities across spatial or temporal data [Becker, 1992]. The mutual information is defined as '" p " M = LJ Pi; log ....!2L. . . Pi.P.; I,J (4) where Pij is the joint probability of occurrence of the states of the left and right networks. (In previous work [Stolorz, 1992] we showed how Pij may be defined in terms of neural networks). Minimizing E = -M maximizes M. While M has many desirable properties as a measure of agreement between two or more variables [Stolorz, 1992] [Farber, 1992] [Lapedes, 1989] [Korber, 1993], our preliminary simulations show that maximizing M is often prone to poor local maxima. Finally, an alternative to using mutual information for multi-class, as opposed to dichotomous classification, is the Pearson correlation coefficient, X 2 • This is defined in terms of Pi; as (5) Our simulations indicate that X 2 , Gi and p are all less susceptible to local minima 814 Lapedes, Steeg, and Farber than M. However, these other objective functions suffer the defect that predictability is emphasized at the expense of utility. In other words, they can be maximal for the peculiar situation where a structural class is defined that occurs very rarely in the data, but when it occurs, it is predicted perfectly by the other network. The utility of this classification is therefore degraded by the fact that the predictable class only occurs rarely. Fortunately, this effect did not cause difficulties in the simulations we performed. Our best results to date have been obtained using the Mathews objective function (see Results). 4 RESULTS The database we used consisted of 105 proteins and is identical to that used in previous investigations [Kneller, 1990] [Stolorz, 1992]. The proteins were divided into two groups: a set of 91 "training" proteins, and a distinct "prediction" set of 14 proteins. The resulting database is similar to the database used by Qian & Sejnowski [Qian, 1988] in their neural network studies of conventional secondary structure prediction. When comparison to predictability of conventional secondary structure classes was needed, we defined the conventional alpha, beta and coil states using the Kabsch and Sander definitions and therefore these states are identical to those used in previous work [Kneller, 1990] [Stolorz, 1992]. A window size of 13 residues resulted in 16028 train set examples and 3005 predict set examples. Effects of other windows sizes have not yet been extensively tested. All results, including conventional backpropagation training of Kabsch and Sander classifications, as well as two-net training of our new secondary structure classifications, did not employ an extra symbol denoting positions in a window that extended past the ends of a protein. Use of such a symbol could further increase accuracy. We found that random initial conditions are necessary for the development of interesting new classes. However, random initial conditions also suffer to a certain extent from local minima. The mutual information function, in particular, often gets trapped quickly in uninteresting local minima when evolved from random initial conditions. More success was obtained with the other objective functions discussed above. We have not exhaustively investigated strategies to avoid local minima, and usually just chose new initial conditions if an uninteresting local minimum was encountered. Results were best for two class discrimination using the Mathews objective function and a layer of five hidden units in each net. If one assigns the name "Xclass" to the newly defined structural class, then the Mathews coefficient on the prediction set for the Xclass dichotomy is -0.425. The Mathews coefficient on the train set for the Xclass dichotomy is -0.508. For comparison, the Mathews coefficient on the same predict set data for dichotomization (using standard backpropagation training with no hidden units), into the standard secondary structure classes Alpha/NotAlpha, Beta/NotBeta, and CoilJNotCoil is 0.33, 0.26, and 0.39, respectively. Adding hidden units gives negligible accuracy increase in predicting the conventional classes, but is important for improved prediction of the new classes. The negative sign of the two-net result indicates anti-correlation - a feature allowed by our objective function. The sign of the correlation is easily assessed on the train set and then can be trivially compensated for in prediction. Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes 815 A natural question to ask is whether the new classes are simply related to the more conventional classes of alpha helix, beta, coil. A simple answer is to compute the Mathews correlation coefficient of the new secondary structure classes with each of the three Kabsch and Sander classes, for those examples in which the sequence network agreed with the structure network's classification. The correlation with Kabsch and Sander's alpha helix is highest: a Mathews coefficient of 0.248 was obtained on the train set, while a Mathews coefficient of 0.247 was obtained on the predict set. There is therefore a significant degree of correlation with the conventional classification of alpha helix, but significant differences exist as well. The new classes are a mixture of the conventional classes, and are not solely dominated by either alpha, beta or coil. Conventional alpha-helices comprise roughly 25% of the data (for both train and predict sets), while the new Xclass comprises 10%. It is quite interesting that an evolution of secondary structure classifications starting from random initial conditions, and hence completely unbiased towards the conventional classifications, results in a classification that has significant relationship to conventional helices but is more predictable from amino acid sequence than conventional helices. Graphical analysis (not shown here) of the new Xclass shows that the Xclass that is most closely related to helix typically extends the definition of helix past the standard boundaries of an alpha-helix. 5 CONCLUSIONS A primary goal of this investigation is to evolve highly predictable secondary structure classes. Ultimately, such classes could be used, e.g., to provide constraints on tertiary structure calculations. Further work remains to derive even more predictable classes and to analyze their physical meaning. However, it is now clear that the use of two, co-evolving, adaptive networks defines a novel and useful machine learning paradigm that allows the evolution of new definitions of secondary structure that are significantly more predictable from primary amino acid sequence than the conventional definitions. Related work is that of [Hunter, 1992], [Hunter, 1992], [Zhang, 1992], [Zhang, 1993] in which clustering either only in sequence space, or only in structure space, is attempted. However, no condition on the compatibility of the clustering is required, so new classes of structure are not guaranteed to be predictable from sequence. Finally, we note that the methods described here might be usefully applied to other cognitive/perceptual or engineering tasks in which correlation of two or more different representations of the same data is required. In this regard the relation of our work to that of independent work of Becker [Becker, 1992], and of Schmidhuber [Schmidhuber, 1992], should be noted. Acknowledgements We are grateful for useful discussions with Geoff Hinton, Sue Becker, and Joe Bryngelson. Sue Becker's contribution of software that was used in the early stages of this project is much appreciated. The research of Alan Lapedes and Robert Farber was supported in part by the U.S. Department of Energy. The authors would like to acknowledge the hospitality of the Santa Fe Institute, where much of this work 816 Lapedes, Steeg, and Farber was performed. References [Becker, 1992] [Becker, 1992] [Chou, 1978] [Farber, 1992] [Hunter, 1992] [Hunter, 1992] [Kabsch, 1983] [Kneller, 1990] [Korber, 1993] [Lapedes, 1989] [Maclin, 1992] [Pauling, 1951] [Qian, 1988] S. Becker. An Information-theoretic Unsupervised Learning Algorithm for Neural Networks. PhD thesis, University of Toronto (1992) S. Becker, G. Hinton, Nature 355, 161-163 (1992) P. Chou, G. Fasman Adv. Enzymol. 47, 45 (1978) R. Farber, A. Lapedes J. Mol. Bioi. 226 , 471, (1992) 1. Hunter, N. Harris, D. States Proceedings of the Ninth International Conference on Machine Learning, San Mateo, California, Morgan Kaufmann Associates (1992) L. Hunter, D. States, IEEE Expert, 7(4) 67-75 (1992) W. Kabsch, C. Sander Biopolymers 22, 2577 (1983) D. Kneller, F. Cohen, R. Langridge J. Mol. Bioi. 214, 171 (1990) B. Korber, R. Farber, D. Wolpert, A. Lapedes P.N.A.S. - in press (1993) A. Lapedes, C.Barnes, C. Burks, R.Farber, K. Sirotkin in Computers and DNA editors: G.Bell, T. Marr, (1989) R. Maclin, J. W. Shavlik Proceedings of the Tenth National Conference on Artificial Intelligence, San Jose, California, Morgan Kauffman Associates (1992) L. Pauling, R. Corey Proc. Nat. Acad. Sci. 37,205 (1951) N. Qian, T. Sejnowski J. Mol. Bioi. 202, 865 (1988) [Schmidhuber, 1992] J. Schmidhuber Discovering Predictable Classifications, Tech[Schulz, 1979] [Stolorz, 1992] [Zhang, 1992] [Zhang, 1993] nical report CU-CS-626-92, Department of Computer Science, University of Colorado (1992) G. Schulz, R. Schirmer Principles of Protein Structure Springer Verlag, New York, (1979) P. Stolorz, A. Lapedes, X. Yuan J. Mol. Bioi. 225, 363 (1992) X. Zhang, D. Waltz in Artificial Intelligence and Molecular Biology, editor: L. Hunter, AAAI Press (MIT Press) (1992) X. Zhang, J. Fetrow, W. Rennie, D. Waltz, G. Berg, in Proceedings: First International Conference on Intelligent Systems For Molecular Biology, p. 438, editors: L. Hunter, D. Searls, J. Shavlik, AAAI Press, Menlo Park, CA. (1993)
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Coupled Dynamics of Fast Neurons and Slow Interactions A.C.C. Coolen R.W. Penney D. Sherrington Dept. of Physics - Theoretical Physics University of Oxford 1 Keble Road, Oxford OXI 3NP, U.K. Abstract A simple model of coupled dynamics of fast neurons and slow interactions, modelling self-organization in recurrent neural networks, leads naturally to an effective statistical mechanics characterized by a partition function which is an average over a replicated system. This is reminiscent of the replica trick used to study spin-glasses, but with the difference that the number of replicas has a physical meaning as the ratio of two temperatures and can be varied throughout the whole range of real values. The model has interesting phase consequences as a function of varying this ratio and external stimuli, and can be extended to a range of other models. 1 A SIMPLE MODEL WITH FAST DYNAMIC NEURONS AND SLOW DYNAMIC INTERACTIONS As the basic archetypal model we consider a system of Ising spin neurons (J'i E {-I, I}, i E {I, ... , N}, interacting via continuous-valued symmetric interactions, Iij, which themselves evolve in response to the states of the neurons. The neurons are taken to have a stochastic field-alignment dynamics which is fast compared with the evolution rate of the interactions hj, such that on the time-scale of Iii-dynamics the neurons are effectively in equilibrium according to a Boltzmann distribution, (1) 447 448 Cooien, Penney, and Sherrington where HVoj} ({O"d) = - L JijO"iO"j (2) i<j and the subscript {Jij} indicates that the {Jij} are to be considered as quenched variables. In practice, several specific types of dynamics which obey detailed balance lead to the equilibrium distribution (1), such as a Markov process with single-spin flip Glauber dynamics [1]. The quantity /3 is an inverse temperature characterizing the stochastic gain. For the hj dynamics we choose the form d 1 1 T' dthj = N(O"iO"j)V,i} - jjJij + viirJij(t) (i < j) (3) where ( .. ')ViJ} refers to a thermodynamic average over the distribution (1) with the effectively instantaneous {Jij}, and TJij (t) is a stochastic Gaussian white noise of zero mean and correlation (TJij(t)TJkl(t')) = 2T'ffi- 1o(ij),(kl)O(t - t') The first term on the right-hand side of (3) is inspired by the Hebbian process in neural tissue in which synaptic efficacies are believed to grow locally in response to the simultaneous activity of pre- an~ post-synaptic neurons [2]. The second term acts to limit the magnitude of hj; f3 is the characteristic inverse temperature of the interaction system. (A related interaction dynamics without the noise term, equivalent to ffi = 00, was introduced by Shinomoto [3]; the anti-Hebbian version of the above coupled dynamics was studied in layered systems by Jonker et al. [4, 5].) Substituting for (O"iO"j) in terms of the distribution (1) enables us to re-write (3) as d a NT' dthj = - aJij 11. ({Jij}) + VNTJij(t) (4) where the effective Hamiltonian 11. ({ hj}) is given by 1 1 ~ 2 11. ( { Jij }) = - /3 In Z {3 ( { Jij } ) + 2 jjN ~<. Jij l J (5) where Z{3 ({ hj}) is the partition function associated with (2): 2 COUPLED SYSTEM IN THERMAL EQUILIBRIUM We now recognise (4) as having the form of a Langevin equation, so that the equilibrium distribution of the interaction system is given by a Boltzmann form. Henceforth, we concentrate on this equilibrium state which we can characterize by a partition function Z t3 an d an associated 'free energy' F t3: Z {3 = J P dJij [Z{3 ({ Jij}) r exp [- ~ ffijjN ~ Ji~] S<J S<J --1F{3 = -f3 In Z{3 (6) Coupled Dynamics of Fast Neurons and Slow Interactions 449 where n _ ~/j3. We may use Z~ as a generating functional to produce thermodynamic averages of state variables <I> ( {O"d; {Jij}) in the combined system by adding suitable infinitesimal source terms to the neuron Hamiltonian (2): HP.j}({O"d) -+ Hp.j} ({O"d) + A<p({ud;{Jij}) oplim £:J: = (<p({O"d;{Jij})){J } A-+O UA 'J _ IfL<j dhj (<p({O"d; {Jij}))plj}e-~1l(Plj}) IfL<jdhj e-~1l({J·j}) where the bar refers to an average over the asymptotic {hj} dynamics. (7) The form (6) with n -+ 0 is immediately reminiscent of the effective partition function which results from the application of the replica trick to replace In Z by limn-+o ~(zn - 1) in dealing with a quenched average for the infinite-ranged spinglass [6], while n = 1 relates to the corresponding annealed average, although we note that in the present model the time-scales for neuron and interaction dynamics remain completely disparate. These observations correlate with the identification of n with fi / j3, which implies that n -+ 0 corresponds to a situation in which the interaction dynamics is dominated by the stochastic term T)ij (t), rather than by the behaviour of the neurons, while for n = 1 the two characteristic temperatures are the same. For n -+ 00 the influence of the neurons on the interaction dynamics dominates. In fact, any real n is possible by tuning the ratio between the two {3's. In the formulation presented in this paper n is always non-negative, but negative values are possible if the Hebbian rule of (3) is replaced by an anti-Hebbian form with (UiO"j) replaced by - (O"iO"j) (the case of negative n is being studied by Mezard and co-workers [7]). The model discussed above is range-free/infinite-ranged and can therefore be analyzed in the thermodynamic limit N -+ 00 by the replica mean-field theory as devised for the Sherrington-Kirkpatrick spin-glass [6, 8, 9]. This can be developed precisely for integer n [6, 8, 9, 10] and analytically continued. In the usual manner there enters a spin-glass order parameter (, f- b) where the superscripts are replica labels. q"{6 is given by the extremum of F({q1'6})=_LL:[q1'O]2+ ln Tr exp [ ~ L:O"1'q1'OO"O] 2J-ln2 1'<6 {O"1'} J-ln2 1'<6 while Z~ is proportional to exp [NextrF ({q1'6})]. In the replica-symmetric region (or ansatz) one assumes q1'O = q. We will first choose as the independent variables nand j3 and briefly discuss the phase picture of our model (full details can be found in [11]). The system exhibits a transition from a paramagnetic state (q = 0) to an ordered state (q > 0) at a critical j3c(n). For n ::; 2 this transition is second order at j3c = 1, down to the SK 450 Coolen, Penney, and Sherrington spin-glass limit, n 0, but for n > 2 the coupled dynamics leads to a qualitative, as well as quantitative, change to first order. Replica symmetry is stable above a critical value nc(!3), at which there is a de Almeida-Thouless (AT) transition (c.f. Kondor [12]). As expected from spin-glass studies, n c(f3) goes to zero as {3 ! 1 but rises for larger /3, having a maximum of order 0.3 at {3 of order 2. Thus, for n > nc(max) ::::: 0.3 there is no instability against small replica-symmetry breaking fluctuations, while for smaller n there is re-entrance in this stability. The transition from a paramagnetic to an ordered state and the onset of local RS instability for various temperatures is shown in Figure 1. 3 EXTERNAL FIELDS Several simple modifications of the above model are possible. One consists of adding external fields to the spin dynamics and/or to the interaction dynamics, by making the substitutions HV,j} ({O"d) ~ HV'J} ({O"d) - LOiO"i i 1£ ( {Jii }) ~ 1£ ( {Jij }) - L hi Kij i<i in (2) and (5) respectively. These external fields may be viewed as generating fields in the sense of (7); for example For neural network models a natural first choice for the external fields would be Oi = hei and Kij = Keiej, ei E {-I, I}, where the ei are quenched random variables corresponding to an imposed pattern. Without loss of generality all the ei can be taken as +1, via the gauge transformation O"i ~ O"iei, Jii ~ Jiieiei. Henceforth we shall make this choice. The neuron perturbation field h induces a finite 'magnetization' characterized by a new order parameter m a = (O"f) which is independent of Q: in the replica-symmetric assumption (which turns out to be stable with respect to variation in this parameter). As in the case of the spin-glass, there is now a critical surface in (h, n, {3) space characterizing the onset of replica symmetry breaking. In introducing the interaction perturbation field K we find that K/ J-l is the analogue of the mean exchange Jo in the SK spin-glass model, ]2 = ({3nJ-l )-1 being the analogue of the variance. If large enough, this field leads to a spontaneous 'ferromagnetic' order. Again we find further examples of both second and first order transitions (details can be found in [11]). For the paramagnetic (P; m = 0, q = 0) to ferromagnetic (F; m I=- 0, q I=- 0) case, the transition is second order at the SK value f3Ja = 1 so long as ({3])-2 ~ 3n - 2. Only when ({3])-2 < 3n - 2 do the interaction dynamics Coupled Dynamics of Fast Neurons and Slow Interactions 451 1.2 ll--_P_A_RAM __ A_GN_ET _____ -.-.-------.. ~ .• -·,...· 0.8 T 0.6 0.2 1 WA'M'IS GLASS SPIN GLASS 2 3 n Figure 1: Phasediagrarn for j = 1. Dotted line: first order transition, solid line: second order transition. The separation between Mattis-glass and spin-glass phase is defined by the de Almeida-Thouless instability 452 Coolen, Penney, and Sherrington influence the transition, changing it to first order at a lower temperature. Regarding the ferromagnetic to spin-glass (SGj m = 0, q "# 0) transition, this exhibits both second order (lower .70) and first order (higher Jo) sections separated by a tricritical point for n less than a critical value of the order of 3.3. This tricritical point exhibits re-entrance as a function of n. 4 COMPARISON BETWEEN COUPLED DYNAMICS AND SK MODEL In order to clarify the differences, we will briefly summarize the two routes that lead to an SK-type replica theory: Coupled Dynamics: Fast Ising spin neurons + slow dynamic interactions, Free energy: Define: Thermodynamics: d 1 K -J .. = -((J'(J'){J .} + - - IIJ .. + GWN dt lJ N I J 'J N r lJ 1 f - --_-logZ, f3N z = fIT dhj e-1ht({J,j}) i<j io = K/ /-t, N-+oo: 1 D f = - f3n extr G ({q'Y }; {m'Y}) + const. SK spin-glass: Ising spins + fixed random interactions, P(Jij) = [27rJ2]-~e-~[J;j-Jo]2/J2 Free energy: f 1 1. l[n ] =--logZ=--hm- Z -1 f3N f3N n-O n Selt-averaging: Physical scaling: Jo = Jo/N, J = J/Vii Thermodynamics: f = - lim 131 extr G ({q'YD}j {m'Y}) + const. n_O n Coupled Dynamics of Fast Neurons and Slow Interactions 453 5 DISCUSSION \Ve have obtained a solvable model with which a coupled dynamics of fast stochastic neurons and slow dynamic interactions can be studied analytically. Furthermore it presents the replica method from a novel perspective, provides a direct interpretation of the replica dimension n in terms of parameters controlling dynamical processes and leads to new phase transition characters. As a model for neural learning the specific example analyzed here is however only a first step, with hand K as introduced corresponding to only a single pattern. Its adaptation to treat many patterns is the next challenge. One type of generalization is to consider the whole system as of lower connectivity with only pairs of connected sites being available for interaction upgrade. For example, the system could be on a lattice, in which case the corresponding coupled partition function will have the usual greater complication of a finite-dimensional system, or randomly connected with each bond present with a probability C IN, in which case there results an analogue of the Viana-Bray [13] spin-glass. In each of these cases the explicit factors involving N in the {hj} dynamics (3) should be removed (their presence or absence being determined by the need for statistical relevance and physical scaling). Yet another generalization is to higher order interactions, for example to p-neuron ones: Hp} ({O"d) = - L Ji l , ... ,i 1'O"i 1 0"i 2 •• . 00i 1' i l, · . . ,i l' with corresponding interaction dynamics d 1 1 r-J· . -(0"' 0"' ){J} IIJ· . + -T)' . (t) dt 11 ," .t 1' N Sl'" 11' r tl , .. ·,l1' ffi Sl,· .. ,l1' or to more complex neuron types. If the symmetry-breaking fields Kij in the interaction dynamics are choosen at random, we obtain a curious theory in which we find replicas on top of replicas (the replica trick would be used to deal with the quenched disorder of the K ij , for a model in which replicas are already present. due to the coupled dynamics). Finally, our approach can in fact be generalized to any statistical mechanical system which in equilibrium is described by a Boltzmann distribution in which the Hamiltonian has (adiabatically slowly) evolving parameters. By choosing these parameters to evolve according to an appropriate Langevin process (involving the free energy of the underlying faRt system) one always arrives at a replica theory describing the coupled system in equilibrium. Acknowledgements Financial support from the U.K. Science and Engineering Research Council under grants 9130068X and GR/H26703, from the European Community under grant SISCI *915121, and from Jesus College, Oxford, is gratefully acknowledged. 454 Coolen, Penney, and Sherrington References [1] Glauber R.J. (1963) J. Math. Phys. 4 294 [2] Hebb D.O. (1949) 'The Organization of Behaviour' (Wiley, New York) [3] Shinomoto S. (1987) J. Phys. A: Math. Gen. 20 L1305 [4] Jonker II.J.J. and Cool en A.C.C. (1991) J. Phys. A: Math. Gen. 24 4219 [5] Jonker H.J.J., Coolen A.C.C. and Denier van der Gon J.J. (1993) J. Phys. A: Math. Gen. 26 2549 [6] Sherrington D. and Kirkpatrick S. (1975) Phys. Rev. Lett. 35 1792 [7] Mezard M. prit1ate communication [8] Kirkpatrick S. and Sherrington D. (1978) Phys. Rev. B 17 4384 [9] Mezard M., Parisi G. and Virasoro M.A. (1987) 'Spin Glass Theory and Beyond' (World Scientific, Singapore) [10] Sherrington D. (1980) J. Phys. A: Math. Gen. 13 637 [11] Penney R.W., Cool en A.C.C. and Sherrington D. (1993) J. Phys. A: Math. Gen. 26 3681-3695 [12] Kondor I. (198~{) J. Phys. A: Math. Gen. 16 L127 [13] Viana L. and Bray A.J. (1983) J. Phys. C 16 6817
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U sing Local Trajectory Optimizers To Speed Up Global Optimization In Dynamic Programming Christopher G. Atkeson Department of Brain and Cognitive Sciences and the Artificial Intelligence Laboratory Massachusetts Institute of Technology, NE43-771 545 Technology Square, Cambridge, MA 02139 617-253-0788, cga@ai.mit.edu Abstract Dynamic programming provides a methodology to develop planners and controllers for nonlinear systems. However, general dynamic programming is computationally intractable. We have developed procedures that allow more complex planning and control problems to be solved. We use second order local trajectory optimization to generate locally optimal plans and local models of the value function and its derivatives. We maintain global consistency of the local models of the value function, guaranteeing that our locally optimal plans are actually globally optimal, up to the resolution of our search procedures. Learning to do the right thing at each instant in situations that evolve over time is difficult, as the future cost of actions chosen now may not be obvious immediately, and may only become clear with time. Value functions are a representational tool that makes the consequences of actions explicit. Value functions are difficult to learn directly, but they can be built up from learned models of the dynamics of the world and the cost function. This paper focuses on how fast optimizers that only produce locally optimal answers can playa useful role in speeding up the process of computing or learning a globally optimal value function. Consider a system with dynamics Xk+l = f(xk, Uk) and a cost function L(Xk, Uk), 663 664 Atkeson where x is the state of the system and u is a vector of actions or controls. The subscript k serves as a time index, but will be dropped in the equations that follow. A goal of reinforcement learning and optimal control is to find a policy that minimizes the total cost, which is the sum of the costs for each time step. One approach to doing this is to construct an optimal value function, V(x). The value of this value function at a state x is the sum of all future costs, given that the system started in state x and followed the optimal policy P(x) (chose optimal actions at each time step as a function of the state). A local planner or controller can choose globally optimal actions if it knew the future cost of each action. This cost is simply the sum of the cost of taking the action right now and the future cost of the state that the action leads to, which is given by the value function. u* = arg min (L(x, u) + V(f(x, u») u (1) Value functions are difficult to learn. The environment does not provide training examples that pair states with their optimal cost (x, V(x». In fact, it seems that the optimal policy depends on the optimal value function, which in turn depends on the optimal policy. Algorithms to compute value functions typically iteratively refine a candidate value function and/or a corresponding policy (dynamic programming). These algorithms are usually expensive. We use local optimization to generate locally optimal plans and local models of the value function and its derivatives. We maintain global consistency of the local models of the value function, guaranteeing that our locally optimal plans are actually globally optimal, up to the resolution of our search procedures. 1 A SIMPLE EXAMPLE: A PENDULUM In this paper we will present a simple example to make our ideas clear. Figure 1 shows a simulated set of locally optimal trajectories in phase space for a pendulum being driven by a motor at the joint from the stable to the unstable equilibrium position. S marks the start point, where the pendulum is hanging straight down, and G marks the goal point, where the pendulum is inverted (pointing straight up). The optimization criteria quadratically penalizes deviations from the goal point and the magnitude of the torques applied. In the three locally optimal trajectories shown the pendulum either swings directly up to the goal (1), moves initially away from the goal and then swings up to the goal (2), or oscillates to pump itself and then swing to the goal (3). In what follows we describe how to find these locally optimal trajectories and also how to find the globally optimal trajectory. 2 LOCAL TRAJECTORY OPTIMIZATION We base our local optimization process on dynamic programming within a tube surrounding our current best estimate of a locally optimal trajectory (Dyer and McReynolds 1970, Jacobson and Mayne 1970). We have a local quadratic model of the cost to get to the goal (V) at each time step along the optimal trajectory (assume a time step index k in everything below unless otherwise indicated): 1 T Vex) ~ Vo + Vxx + 2x Vxxx (2) Using Local Trajectory Optimizers to Speed Up Global Optimization 665 I ~ /" • 1/// ~ e ~ / VI / III " \ \ Is \ Go \ v' ') e I~ Figure 1: Locally optimal trajectories for the pendulum swing up task. A locally optimal policy can be computed using local models of the plant (in this case local linear models) at each time step along the trajectory: Xk+l = f(x, u) ~ Ax + Bu + c (3) and local quadratic models of the one step cost at each time step along the trajectory: 1 1 L(x,u) ~ 2xT Qx+ 2uTRu+xTSu+tTu At each point along the trajectory the optimal policy is given by: u opt = -(R + BTVxxB)-1 x (BTVxxAx + ST x + BTVxxc + VxB + t) (4) One can integrate the plant dynamics forward in time based on the above policy, and then integrate the value functions and its first and second spatial derivatives backwards in time to compute an improved value function, policy, and trajectory. For a one step cost of the form: 1 T L(x, u) ~ 2(x - Xd) Q(x - Xd)+ 1 T T 2(u - Ud) R(u - Ud) + (x - Xd) S(n - Ud) the backward sweep takes the following form (in discrete time): Zx = VxA + Q(x - Xd) Zu = VxB + R(u - Ud) Zxx = ATVxxA + Q Zux = BTVxxA + S Zuu = BTVxxB + R K = Z;;: Zux VXk _ 1 = Zx - ZuK VXXk _ 1 = Zxx - ZxuK (5) (6) (7) (8) (9) (10) (11) (12) 666 Atkeson 3 STANDARD DYNAMIC PROGRAMMING A typical implementation of dynamic programming in continuous state spaces discretizes the state space into cells, and assigns a fixed control action to each cell. Larson's state increment dynamic programming (Larson 1968) is a good example of this type of approach. In Figure 2A we see the trajectory segments produced by applying the constant action in each cell, plotted on a phase space for the example problem of swinging up a pendulum. 4 USING LOCAL TRAJECTORY OPTIMIZATION WITH DP We want to minimize the number of cells used in dynamic programming by making the cells as large as possible. Combining local trajectory optimization with dynamic programming allows us to greatly reduce the resolution of the grid on which we do dynamic programming and still correctly estimate the cost to get to the goal from different parts of the space. Figure 2A shows a dynamic programming approach in which each cell contains a trajectory segment applied to the pendulum problem. Figure 2B shows our approach, which creates a set of locally optimal trajectories to the goal. By performing the local trajectory optimizations on a grid and forcing adjacent trajectories to be consistent, this local optimization process becomes a global optimization process. Forcing adjacent trajectories to be consistent means requiring that all trajectories can be generated from a single underlying policy. A trajectory can be made consistent with a neighbor by using the neighboring trajectory as an initial trajectory in the local optimization process, or by using the value function from the neighboring trajectory to generate the initial trajectory in the local optimization process. Each grid element stores the trajectory that starts at that point and achieves the lowest cost. The trajectory segments in figure 2A match the trajectories in 2B. Figures 2C and 2D are low resolution versions of the same problem. Figure 2C shows that some of the trajectory segments are no longer correct. In Figure 2D we see the locally optimal trajectories to the goal are still consistent with the trajectories in Figure 2B. Using locally optimal trajectories which go all the way to the goal as building blocks for our dynamic programming algorithm allows us to avoid the problem of correctly interpolating the cost to get to the goal function on a sparse grid. Instead, the cost to get to the goal is measured directly on the optimal trajectory from each node to the goal. We can use a much sparser grid and still converge. 5 ADAPTIVE GRIDS BASED ON CONSTANT COST CONTOURS We can limit the search by "growing" the volumes searched around the initial and goal states by gradually increasing a cost threshold Cg • We will only consider states around the goal that have a cost less than Cg to get to the goal and states around the initial state that have a cost less than Cg to get from the initial state to that state (Figure 3B). These two regions will increase in size as Cg is increased. We stop Using Local Trajectory Optimizers to Speed Up Global Optimization 667 A B c o Figure 2: Different dynamic programming techniques (see text). 668 Atkeson Figure 3: Volumes defined by a cost threshold. increasing Cg as soon as the two regions come into contact. The optimal trajectory has to be entirely within the union of these two regions, and has a cost of 2Cg . Instead of having the initial conditions of the trajectories laid out on a grid over the whole space, the initial conditions are laid out on a grid over the surface separating the inside and the outside surfaces of the volumes described above. The resolution of this grid is adaptively determined by checking whether the value function of one trajectory correctly predicts the cost of a neighboring trajectory. If it does not, additional grid points are added between the inconsistent trajectories. During this global optimization we separate the state space into a volume around the goal which has been completely solved and the rest of the state space, in which no exploration or computation has been done. Each iteration of the algorithm enlarges the completely solved volume by performing dynamic programming from a surface of slightly increased cost to the current constant cost surface. When the solved volume includes a known starting point or contacts a similar solved volume with constant cost to get to the boundary from the starting point, a globally optimal trajectory from the start to the goal has been found. 6 DP BASED ON APPROXIMATING CONSTANT COST CONTOURS Unfortunately, adaptive grids based on constant cost contours still suffer from the curse of dimensionality, having only reduced the dimensionality of the problem by 1. We are currently exploring methods to approximate constant cost contours. For example, constant cost contours can be approximated by growing "key" trajectories. Using Local Trajectory Optimizers to Speed Up Global Optimization 669 ;' / \ " Figure 4: Approximate constant cost contours based on key trajectories A version of this is illustrated in Figure 4. Here, trajectories were grown along the "bottoms" of the value function "valleys". The location of a constant cost contour can be estimated by using local quadratic models of the value function produced by the process which optimizes the trajectory. These approximate representations do not suffer from the curse of dimensionality. They require on the order of T D2, where T is the length of time the trajectory requires to get to the goal, and D is the dimensionality of the state space. 7 SUMMARY Dynamic programming provides a methodology to plan trajectories and design controllers and estimators for nonlinear systems. However, general dynamic programming is computationally intractable. We have developed procedures that allow more complex planning problems to be solved. We have modified the State Increment Dynamic Programming approach of Larson (1968) in several ways: 1. In State Increment DP, a constant action is integrated to form a trajectory segment from the center of a cell to its boundary. We use second order local trajectory optimization (Differential Dynamic Programming) to generate an optimal trajectory and form an optimal policy in a tube surrounding the optimal trajectory within a cell. The trajectory segment and local policy are globally optimal, up to the resolution of the representation of the value function on the boundary of the cell. 2. We use the optimal policy within each cell to guide the local trajectory optimization to form a globally optimal trajectory from the center of each 670 Atkeson cell all the way to the goal. This helps us avoid the accumulation of interpolation errors as one moves from cell to cell in the state space, and avoid limitations caused by limited resolution of the representation of the value function over the state space. 3. The second order trajectory optimization provides us with estimates of the value function and its first and second spatial derivatives along each trajectory. This provides a natural guide for adaptive grid approaches. 4. During the global optimization we separate the state space into a volume around the goal which has been completely solved and the rest of the state space, in which no exploration or computation has been done. The surface separating these volumes is a surface of constant cost, with respect to achieving the goal. 5. Each iteration of the algorithm enlarges the completely solved volume by performing dynamic programming from a surface of slightly increased cost to the current constant cost surface. 6. When the solved volume includes a known starting point or contacts a similar solved volume with constant cost to get to the boundary from the starting point, a globally optimal trajectory from the start to the goal has been found. No optimal trajectory will ever leave the solved volumes. This would require the trajectory to increase rather than decrease its cost to get to the goal as it progressed. 7. The surfaces of constant cost can be approximated by a representation that avoids the curse of dimensionality. 8. The true test of this approach lies ahead: Can it produce reasonable solutions to complex problems? Acknowledgenlents Support was provided under Air Force Office of Scientific Research grant AFOSR89-0500, by the Siemens Corporation, and by the ATR Human Information Processing Research Laboratories. Support for CGA was provided by a National Science Foundation Presidential Young Investigator A ward. References Bellman, R., (1957) Dynamic Programming, Princeton University Press, Princeton, NJ. Bertsekas, D.P., (1987) Dynamic Programming: Deterministic and Stochastic Models, Prentice-Hall, Englewood Cliffs, NJ. Dyer, P. and S.R. McReynolds, (1970) The Computation and Theory of Optimal Control, Academic Press, New York, NY. Jacobson, D.H. and D.Q. Mayne, (1970) Differential Dynamic Programming, Elsevier, New York, NY. Larson, R.E., (1968) State Increment Dynamic Programming, Elsevier, New York, NY.
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Connectionist Modeling and Parallel Architectures Joachim Diederich Neurocomputing Research Centre School of Computing Science Queensland University of Technology Brisbane Q 400 1 Australia Ah Chung Tsoi Department of Electrical and Computer Engineering University of Queensland St Lucia, Queensland 4072, Australia The introduction of specialized hardware platforms for connectionist modeling ("connectionist supercomputer") has created a number of research topics. Some of these issues are controversial, e.g. the efficient implementation of incremental learning techniques, the need for the dynamic reconfiguration of networks and possible programming environments for these machines. Joachim Diederich, Queensland University of Technology (Brisbane), started with a brief introduction to connectionist modeling and parallel machines. Neural network modeling can be done on various levels of abstraction. On a low level of abstraction, a simulator can support the definition and simulation of "compartmental models," chemical synapses, dendritic trees etc., i.e. explicit computational models of single neurons. These models have been built by use of SPICE (DC Berkeley) and Genesis (Caltech). On a higher level of abstraction, the Rochester Connectionist Simulator (RCS~ University of Rochester) and ICSIM (lCSI Berkeley) allow the definition of unit types and complex connectivity patterns. On a very high level of abstraction, simulators like tleam (UCSD) allow the easy realization of pre-defined network architectures (feedforward networks) and leaming algorithms such as backpropagation. Ben Gomes, International Computer Science Institute (Berkeley) introduced the Connectionist Supercomputer 1. The CNS-l is a multiprocessor system designed for moderate precision fixed point operations used extensively in connectionist network calculations. Custom VLSI digital processors employ an on-chip vector coprocessor unit tailored for neural network calculations and controlled by RISC scalar CPU. One processor and associated commercial DRAM comprise a node, which is connected in a mesh topology with other nodes to establish a MIMD array. One edge of the communications mesh is reserved for attaching various 110 devices, which connect via a custom network adaptor chip. The CNS-l operates as a compute server and one 110 port is used for connecting to a host workstation. Users with mainstream connectionist applications can use CNSim, an object-oriented, graphical high-level interface to the CNS-l environment. Those with more complicated applications can use one of several high-level programming languages (C. C++. 1178 Connectionist Modeling and Parallel Architectures 1179 Sather}, and access a complete set of hand-coded assembler subroutine libraries for connectionist applications. Simulation, debugging and profiling tools will be available to aid both types of users. Additional tools are available for the systems programmer to code at a low level for maximum perfonnance. Access to the 1I0-level processor and network functions are provided, along with the evaluation tools needed to complement the process. Urs Muller, Swiss Federal Institute of Technology (Zurich) introduced MUSIC: A high performance neural network simulation tool on a multiprocessor. MUSIC (Multiprocessor System with Intelligent Communication), a 64 processor system, runs backpropagation at a speed of 247 million connection updates per second using 32 bit floating-point precision. TIlUS the system reaches supercomputer speed (3.8 gflops peak), it still can be used as a personal desk-top computer at a researchers own disposal: The complete system consumes less than 800 Watt and fits into a 19 inch rack. Fin Martin, Intel Corporation, introduced INiI000," an REF processor which accepts 40,000 patterns per second. Input patterns of 256 dimensions by 5 bits are transferred from the host to the NilO00 and compared with the chip's "memory" of 1024 stored reference patterns, in parallel. A custom 16 bit on-chip microcontroller runs at 20 MIPS and controls all the programming and algorithm functions. RBF's are considered an advancement over traditional template matching algorithms and back propagation. Paul Murtagh and Ah Chung Tsoi, University of Queensland (St. Lucia) described a reconfigurable VLSI Systolic Array for artificial neural networks. After a brief review of some of the most common neural network architectures, e.g., multilayer perceptron, Hopfield net, Boltzmann machine, Ah Chung Tsoi showed that the training algorithms of these networks can be written in a unified manner. This unified training algoritlml is then shown to be implementable in a systolic array fashion. The individual processor can be designed accordingly. Each processor incorporates functionality reconfiguration to allow a number of neural network models to be implemented. The architecture also incorporates reconfiguration for fault tolerance and processor arrangement. Each processor occupies very little silicon area with 16 processors being able to fit onto a lOx 10 nm12 die. GUnther Palm and Franz Kurfess introduced "Neural Associative Memories." Despite having processing elements which are thousands of times faster than the neurons in the brain, modem computers still cannot match quite a few processing capabilities of the brain, many of which we even consider trivial (such as recognizing faces or voices, or following a conversation). A common principle for those capabilities lies in the use of correlations between patterns in order to identify patterns which are similar. Looking at the brain as an information processing mechanism with -probably among others -- associative processing capabilities together with the converse view of associative memories as certain types of artificial neural networks initiated a number of interesting results. These range from theoretical considerations to insights in the functioning of neurons, as well as parallel hardware implementations of neural associative memories. The talk discussed some implementation aspects and presented a few applications. Finally, Ernst Niebur, California Institute of Technology (pasadena) presented his work on biologically realistic modeling on SIMD machines (No abstract available).
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Structural and Behavioral Evolution of Recurrent Networks Gregory M. Saunders, Peter J. Angeline, and Jordan B. Pollack Laboratory for Artificial Intelligence Research Department of Computer and Information Science The Ohio State University Columbus, Ohio 43210 saunders@cis.ohio-state.edu Abstract This paper introduces GNARL, an evolutionary program which induces recurrent neural networks that are structurally unconstrained. In contrast to constructive and destructive algorithms, GNARL employs a population of networks and uses a fitness function's unsupervised feedback to guide search through network space. Annealing is used in generating both gaussian weight changes and structural modifications. Applying GNARL to a complex search and collection task demonstrates that the system is capable of inducing networks with complex internal dynamics. 1 INTRODUCTION A variety of methods to induce network architecture exist. Some start with a very simple network and incrementally add nodes and links (Hanson 1990; Fahlman & Lebiere, 1990; Fahlman 1991; Chen, et aI., 1993); others start with a large network and then prune off superfluous pieces (Mozer & Smolensky, 1989; Cun, Denker, and SolI a, 1990; Hassibi & Stork, 1993; amlin & Giles, 1993). But these constructive and destructive algorithms are monotonic extremes that ignore a more moderate solution: "dynamically add or remove pieces of architecture as needed." Moreover, by exclusively exploring either feedforward networks (e.g., Ash, 1989), fully-connected recurrent networks (e.g., Chen, et al. 1993), or some restricted middle ground (e.g., Fahlman,.1991), these algorithms allow only limited structural change. Finally, constructive and destructive algorithms are supervised methods 88 Structural and Behavioral Evolution of Recurrent Networks 89 • num-in input units Random(max-hidden) hidden units Random(max-links) links num-out output units Figure 1: Sample initial network. The number of input nodes and number of output nodes is fixed for the particular task, but the number of hidden units and the connectivity (although bounded), is random. which rely on complex predicates to determine when to add or delete pieces of network architecture (e.g., "when rate of improvement falls below threshold"). Genetic algorithms (Holland 1975), on the other hand, are unsupervised methods which can induce networks by making stochastic modifications to a population of bitstrings, each of which is interpreted as a network. Most studies, however, still assume a fixed structure for the network (e.g., Belew et aI., 1990; Jefferson, et al., 1991; see also Schaffer, et al. 1992), and those that do not allow only limited structural change (e.g., Potter, 1992, and Karunanithi et al., 1992). Evolutionary programming (Fogel, 1992) is an alternate optimization technique which, when applied to network induction, obviates the need for a bitstring-to-network mapping by mutating networks directly. Furthermore, because EP does not employ crossover (an operator of questionable efficacy on distributed representations), it is a better candidate for inducing network structures (Angeline, Saunders, and Pollack, 1993; Fogel et al., 1990). 2 THE GNARL ALGORITHM GNARL (GeNeralized Acquisition of Recurrent Links) is an evolutionary program that non-monotonically constructs recurrent networks to solve a given task. It begins with an initial population of n random individuals; a sample network N is shown in Figure 1. The number of input nodes (num-in) and number of output nodes (num-out) are fixed for a given task; the number of hidden nodes as well as the connections among them are free to vary. Self-links as well as general loops are allowed. Thus GNARL's search space is N = {N: network N has num-in input nodes and num-out output nodes}. In each epoch of search, the networks are ranked by a user-supplied fitness function f: N ~ R, where R represents the reals. Reproduction of the best n/2 individuals entails modifying both the weights and structure of each parent network N. First, the temperature T(N) is calculated: T(N) = I_f (N) f max (1) where fmax (provided by the user) is the maximum possible fitness for a given task. This 90 Saunders, Angeline, and Pollack measure of N's performance is used to anneal the structural and parametric (Barto, 1990) similarity between parent and offspring, so that networks with a high temperature are mutated severely, and those with a low temperature are mutated only slightly. This allows a coarse-grained search initially, and a finer-grained search as a network approaches a solution (cf. Kirkpatrick et aI., 1983). More concretely, parametric mutations are accomplished by perturbing each weight with gaussian noise, whose variance is T(Ny2: W f- W + Normal (0; T (N)), 'v'w E N (2) Structural mutations are accomplished by: • adding k] hidden nodes with probability Palld-node • deleting k2 hidden nodes with probability Pdelete-node • adding k3links with probability Padd-link • deleting k4 links with probability Pdelete-link where each kj is selected uniformly from a user-defined range, again annealed by T(N). When a node is added, it is initialized without connections; when a node is deleted, all its incident links are removed. All new links are initialized to O. (See also Angeline, Saunders, and Pollack, 1993.) 3 RESULTS GNARL was tested on a simple control task - the Tracker task of Jefferson, et al. (1991) and Koza (1992). In this problem, a simulated ant is placed on a two-dimensional toroidal grid and must maximize the number of pieces of food it collects in a given time period (Figure 2a). Each ant is controlled by a network with two input nodes and four output nodes (Figure 2b). At each step, the action whose corresponding output node has maximum activation is performed. Fitness is the number of grid positions cleared within 200 time steps. The experiments used a population of 100 networks. In the first run (2090 generations), GNARL found a network (Figure 3b) that cleared 81 grid positions within the 200 time steps. Figure 4 shows the state of the output units of the network over three different sets of inputs. Each point is a triple of the form (move, right, left). (No-op is not shown because it was never used in the final network.) Figure 4a shows the result of supplying to the network 200 "food" inputs - a fixed point that executes "Move." Figure 4b shows the sequence of states reached when 200 "no food" signals are supplied to the network - a collection of points describing a limit cycle of length 5 that repeatedly executes the sequence "Right, Right, Right, Right, Move." These two attractors determine the response of the network to the task (Figure 4c,d); the additional points in Figure 4c are transients encountered as the network alternates between these attractors. However, not all evolved network behaviors are so simple as to approximate an FSA (Pollack, 1991). In a second run (1595 generations) GNARL induced a network that cleared 82 grid points within the 200 time steps. Figure 5 demonstrates the behavior of this network. Once again, the "food" attractor, shown in Figure 5a, is a single point in the space that always executes "Move." The "no food" behavior, however, is not an FSA; instead, it is a Structural and Behavioral Evolution of Recurrent Networks 91 " rp1 , Move Turn left Tum right N o-op @ 6) Food No food (a) (b) Figure 2: The ant problem. (a) The trail is connected initially, but becomes progressively more difficult to follow. The underlying 2-d grid is toroidal, so that position "P" is the first break in the trail. The ellipse indicates the 7 pieces of food that the network of the second run failed to reach. (b) The semantics of the I/O units for the ant network. The first input node denotes the presence of food in the square directly in front of the ant; the second denotes the absence of food in this same square. No-op, from Jefferson, allows the network to stay in one position while activation flows through recurrent links. This particular network "eats" 42 pieces of food before spinning endlessly in place at position P, illustrating a very deep local minimum in the search space. quasiperiodic trajectory of points shaped like a "D" in output space (Figure Sb). The placement of the "D" is in the "Move / Right" corner of the space and encodes a complex alternation between these two operations (Figure Sd). 4 CONCLUSIONS Artificial architectural constraints (such as "feedforwardness") close the door on entire classes of behavior; forced liberties (such as assumed full recurrence) may unnecessarily increase structural complexity or learning time. By relying on a simple stochastic process, GNARL strikes a middle ground between these two, allowing the network's complexity and behavior to emerge in response to the demands of the task. Acknowledgments The research reported in this paper has been partially supported by Office of Naval Research grants NOOO14-93-1-00S9 and NOOO14-92-J-119S. We are indebted to all those who read and reviewed this work, especially John Kolen, Ed Large, and Barbara Becker. 92 Saunders, Angeline, and Pollack • , . , , , ,.\ i. 1 \ , 1 \ , I \ , 1 \ I 1 \ , 1 \ , " \ I " . \ : 1 \ I I \ I ./ \~ ~"". • (a) Move Left Right No-op ~: Food No food (c) ----------------------(b) Figure 3: The Tracker Task, first run. (a) The best network in the initial population. Nodes o & 1 are input, nodes 5-8 are output, and nodes 2-4 are hidden nodes. (b) Network induced by GNARL after 2090 generations. Forward links are dashed; bidirectional links & loops are solid. The light gray connection between nodes 8 and 13 is the sole backlink. This network clears the trail in 319 epochs. (c) Jefferson et al.'s fixed network structure for the Tracker task. References Angeline, P., Saunders, G., Pollack, J. (1993). An evolutionary algorithm that constructs recurrent neural networks. LAIR Technical Report 93-PA-GNARL, The Ohio State University, Columbus Ohio. To be published in IEEE Transactions on Neural Networks. 1 1 (a) 1 1 (c) Structural and Behavioral Evolution of Recurrent Networks 93 .~ 400 ..... . ;;) o 0.. >< 1 (b) 1 1 Figure 4: Limit behavior of the network that clears the trail in 319 steps. Graphs show the state of the output units Move, Right, Left. (a) Fixed point attractor that results for sequence of 200 "food" signals; (b) Limit cycle attractor that results when a sequence of 200 "no food" signals is given to network; (c) All states visited while traversing the trail; (d) The x position of the ant over time when run on an empty grid. Ash, T. (1989). "Dynamic node creation in backpropagation networks," Connection Science, 1 :365-375. Barto, A. G. (1990). Connectionist learning for control. In Miller, W. T. III, Sutton, R. S., and Werbos, P. J., editors, Neural Networksfor Control. Chapter 1, pages 5-58. MIT Press, Cambridge. Belew, R. K., McInerney, J., and Schraudolf, N. N. (1990). Evolving networks: Using the genetic algorithm with connectionist learning. Technical Report CS90-174, University of California, San Diego. 94 Saunders, Angeline, and Pollack 1 (a) 1 1 1 (c) § 3000 .... . -g 2000 o 0.. >< 1 (b) 3 Figure 5: Limit behavior of the network of the second run. Graphs show the state of the output units Move, Right, Left. (a) Fixed point attractor that results for sequence of 500 "food" signals; (b) Limit cycle attractor that results when a sequence of 500 "no food" signals is given to network; (c) All states visited while traversing the trail; (d) The x position of the ant over time when run on an empty grid. Chen, D., Giles, C., Sun, G., Chen, H., Less, Y., and Goudreau, M. (1993). Constructive learning of recurrent neural networks. IEEE International Conference on Neural Networks, 3:1196-1201. Cun, Y.L., Denker, J., and SoIIa, S. (1990). Optimal brain damage. In Touretzky, D., editor, Advances in Neural Information Processing Systems 2. Morgan Kaufmann. Fahlman, S. and Lebiere, C. (1990). The cascade-correlation architecture. In Touretzky, D. S., editor, Advances in Neural Information Processing Structures 2, pages 524-532. Morgan Kaufmann. Fahlman, S. (1991). The recurrent cascade-correlation architecture. In Lippmann, R., Structural and Behavioral Evolution of Recurrent Networks 95 Moody, J., and Touretzky, D., editors, Advances in Neural Information Processing Systems 3, pages 190-196. Morgan Kaufmann, San Mateo. Fogel, D. (1992). Evolving Artificial Intelligence. Ph.D. thesis, University of California, San Diego. Fogel, D., Fogel, L., and Porto, V. W. (1990). Evolving neural networks. Biological Cybernetics. 63:487~93. Hanson, S. J. (1990). Meiosis networks. In Touretzky, D., editor,Advances in NeuralInformation Processing Systems 2, pages 533-541. Morgan Kaufmann, San Mateo. Hassibi, B. and Stork, D. G. (1993). Second order derivatives for network pruning: Optimal brain surgeon. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 164-171. Morgan Kaufmann. Holland, J. (1975). Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor, MI. Jefferson, D., Collins, R, Cooper, C., Dyer, M., Flowers, M., Korf, R, Taylor, C., and Wang, A. (1991). Evolution as a theme in artificial life: The genesys/tracker system. In Langton, C. G., Taylor, C., Farmer, J. D., and Rasmussen, S., editors, Artificial Life II: Proceedings of the Workshop on Artificial Life. pages 549-577 . Addison-Wesley. Karunanithi, N., Das, R, and Whitley, D. (1992). Genetic cascade learning for neural networks. In Proceedings of COGANN-92 International Workshop on Combinations of Genetic Algorithms and Neural Networks. Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220:671-680. Koza, J. (1992). Genetic evolution and co-evolution of computer programs. In Christopher G. Langton, Charles Taylor, J. D. F. and Rasmussen, S., editors, Artificial Life II. Addison Wesley Publishing Company, Reading Mass. Mozer, M. and Smolensky, P. (1989). Skeletonization: A technique for trimming the fat from a network via relevance assessment. In Touretzky, D., editor, Advances in Neural Information Processing Systems 1, pages 107-115. Morgan Kaufmann, San Mateo. Omlin, C. W. and Giles, C. L. (April 1993). Pruning recurrent neural networks for improved generalization performance. Technical Report Tech Report No 93-6, Computer Science Department, Rensselaer Polytechnic Institute. Pollack, J. B. (1991). The induction of dynamical recognizer. Machine Learning. 7:227252. Potter, M. A. (1992). A genetic cascade-correlation learning algorithm. In Proceedings of COGANN-92 International Workshop on Combinations of Genetic Algorithms and Neural Networks. Schaffer, J. D., Whitley, D., and Eshelman, L. J. (1992). Combinations of genetic algorithms and neural networks: A survey of the state of the art. In Proceedings of COGANN92 International Workshop on Combinations of Genetic Algorithms and Neural Networks.
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How to Describe Neuronal Activity: Spikes, Rates, or Assemblies? Wulfram Gerstner and J. Leo van Hemmen Physik-Department der TU Miinchen D-85748 Garching bei Miinchen, Germany Abstract What is the 'correct' theoretical description of neuronal activity? The analysis of the dynamics of a globally connected network of spiking neurons (the Spike Response Model) shows that a description by mean firing rates is possible only if active neurons fire incoherently. If firing occurs coherently or with spatio-temporal correlations, the spike structure of the neural code becomes relevant. Alternatively, neurons can be gathered into local or distributed ensembles or 'assemblies'. A description based on the mean ensemble activity is, in principle, possible but the interaction between different assemblies becomes highly nonlinear. A description with spikes should therefore be preferred. 1 INTRODUCTION Neurons communicate by sequences of short pulses, the so-called action potentials or spikes. One of the most important problems in theoretical neuroscience concerns the question of how information on the environment is encoded in such spike trains: Is the exact timing of spikes with relation to earlier spikes relevant (spike or interval code (MacKay and McCulloch 1952) or does the mean firing rate averaged over several spikes contain all important information (rate code; see, e.g., Stein 1967)? Are spikes of single neurons important or do we have to consider ensembles of equivalent neurons (ensemble code)? If so, can we find local ensembles (e.g., columns; Hubel and Wiesel 1962) or do neurons form 'assemblies' (Hebb 1949) distributed all over the network? 463 464 Gerstner and van Hemmen 2 SPIKE RESPONSE MODEL We consider a globally connected network of N neurons with 1 ~ i ~ N. A neuron i fires, if its membrane potential passes a threshold (). A spike at time t{ is described by a 6-pulse; thus Sf (t) = L:~=1 6(t - t{) is the spike train of neuron i. Spikes are labelled such that tt is the most recent spike and tf is the Fth spike going back in time. In the Spike Response Model, short SRM, (Gerstner 1990, Gerstner and van Hemmen 1992) a neuron is characterized by two different response junctions, f and "1ref . Spikes which neuron i receives from other neurons evoke a synaptic potential (1) where the response kernel { 0 for s < Ll tr f(S) = ,,_atr (,,_a tr ) CAt -::-r- exp - -lor s > u r T. T, (2) describes a typical excitatory or inhibitory postsynaptic potential; see Fig. 1. The weight Jij is the synaptic efficacy of a connection from j to i, Ll tr is the axonal (and synaptic) transmission time, and T" is a time constant of the postsynaptic neuron. The origin S = 0 in (2) denotes the firing time of a presynaptic spike. In simulations we usually assume T" = 2 ms and for Lltr a value between 1 and 4 ms Similarly, spike emission induces refractoriness immediately after spiking. This is modelled by a refractory potential with a refractory function ref () { -00 "1 s = "1o/(s _ ,ref) for S ~ ,ref for S > ,ref. (3) (4) For 0 ~ s ~ ,ref the neuron is in the absolute refractory period and cannot spike at all whereas for s > ,ref spiking is possible but difficult (relative refractory period). To put it differently, () "1ref (s) describes an increased threshold immediately after spiking; cf. Fig. 1. In simulations, ,ref is taken to be 4 ms. Note that, for the sake of simplicity, we assume that only the most recent spike Sf induces refractoriness whereas all past spikes Sf contribute to the synaptic potential; cf., Eqs. (1) and (3). How to Describe Neuronal Activity: Spikes, Rates, or Assemblies? 465 Fig 1 Response functions. 9-n(s) w f 0.5 Immediately after firing at 8 = o the effective threshold is increased to (J - TIre! (8) (dashed). The form of an excitatory postsynaptic potential (EPSP) is described by the response function f( 8) (solid). It is delayed by a time ~ tr. The arrow denotes CD 0.0 '-........o...-_Ll------'-_L-..--'---.----l---=::::t=~ 0.0 5.0 10.0 15.0 20.0 the period Tosc of coherent os5 [m5] cillations; d. Section 5. The total membrane potential is the sum of both parts, i.e. hi(t) = h~ef (t) + h:yn(t). Noise is included by introduction of a firing probability PF(h; 6t) = r- 1 (h) 6t. (5) (6) where 6t is an infinitesimal time interval and r(h) is a time constant which depends on the momentary value of the membrane potential in relation to the threshold (). In analogy to the chemical reaction constant we assume r(h) = ro exp[-,B(h - (})], (7) where ro is the response time at threshold. The parameter ,B determines the amount of noise in the system. For,B --+ 00 we recover the noise-free behavior, i.e., a neuron fires immediately, if h > () (r --+ 0), but it cannot fire, if h < () (r --+ (0). Eqs. (1), (3), (5), and (6) define the spiking dynamics in a network of SRM-neurons. 3 FIRING STATISTICS We start our considerations with a large ensemble of identical neurons driven by the same arbitrary synaptic potential h3yn(t). We assume that all neurons have fired a first spike at t = t{ . Thus the total membrane potential is h(t) = hsyn(t) + 7]ref (tto. If h(t) slowly approaches (), some of the neurons will fire again. We now ask for the probability that a neuron which has fired at time t{ will fire again at a later time t. The conditional probability p~2\tlt{) that the next spike of a given neuron occurs at time t > t{ is p~2)(tlt{) = r-l[h(t)] exp { -1; r- 1[h(S')]dS'} . (8) The exponential factor is the portion of neurons that have survived from time t{ to time t without firing again and the prefactor r- 1 [h(t)] is the instantaneous firing probability (6) at time t. Since the refractory potential is reset after each spike, the spiking statistics does not depend on earlier spikes, in other words, it is fully described by p~2)(tlt{). This will be used below; cf. Eq. (14) . 466 Gerstner and van Hemmen As a special case, we may consider constant synaptic input h3yn = hO• In this case, (8) yields the distribution of inter-spike intervals in a spike train of a neuron driven by constant input hO• The mean firing rate at an input level hO is defined as the inverse of the mean inter-spike interval. Integration by parts yields I[ho] = {J.;dt(t-t{lP~2)(tlt{l} -I = {J.oodsexp{-lT-I[hO+~"f (s'l]ds'} } -I (9) Thus both firing rate and interval distribution can be calculated for arbitrary inputs. 4 ASSEMBLY FORMATION AND NETWORK DYNAMICS We now turn to a large, but structured network. Structure is induced by the formation of different assemblies in the system. Each neuronal assembly aP. (Hebb 1949) consists of neurons which have the tendency to be active at the same time. Following the traditional interpretation, active means an elevated mean firing rate during some reasonable period of time. Later, in Section 5.3, we will deal with a different interpretation where active means a spike within a time window of a few ms. In any case, the notion of simultaneous activity allows to define an activity pattern {~r, 1 :::; i :::; N} with ~r = 1 if i E aP. and ~r = 0 otherwise. Each neuron may belong to different assemblies 1 :::; I-l :::; q. The vector ei = (a, ... ,~n is the 'identity card' of neuron i, e.g., ei = (1,0,0,1,0) says that neuron i belongs to assembly 1 and 4 but not to assembly 2,3, and 5. Note that, in general, there are many neurons with the same identity card. This can be used to define ensembles (or sublattices) L(x) of equivalent neurons, i.e., L(x) = {ilei = x} (van Hemmen and Kiihn 1991). In general, the number of neurons IL(x)1 in an ensemble L(x) goes to infinity if N --;. 00, and we write IL(x)1 = p(x)N. The mean activity of an ensemble L(x) can be defined by I t+at A(x, t) = lim lim IL(x)I- 1 L S[ (t)dt. at--+o N--+oo t iEL(X) (10) In the following we assume that the synaptic efficacies have been adjusted according to some Hebbian learning rule in a way that allows to stabilize the different activity patterns or assemblies ap.. To be specific, we assume J q q Jij = ~ L L Qp.vpost(~r)pre(~j) (11) p.=lv=l where post(x) and pre(x) are some arbitrary functions characterizing the pre- and postsynaptic part of synaptic learning. Note that for Qp.v = fJp.v and post(x) and pre(x) linear, Eq. (11) can be reduced to the usual Hebb rule. With the above definitions we can write the synaptic potential of a neuron i E L(x) in the following form q q (>0 h3yn(x, t) = Jo L L Qp.vpost(xp.) Lpre(zV) 10 f(s')p(z)A(z, t - s')ds'. (12) p.=lv=l z 0 How to Describe Neuronal Activity: Spikes, Rates, or Assemblies? 467 We note that the index i and j has disappeared and there remains a dependence upon x and z only. The activity of a typical ensemble is given by (Gerstner and van Hemmen 1993, 1994) A(x, t) = 1 00 p?)(tlt - s)A(x, t - s)ds (13) where p~2)(tlt-s) = r- 1 [h',yn(x, t)+7]ref (s)] exp {-13r- 1 [h3yn(x, t - s+s' )+7]ref (s')]ds' } (14) is the conditional probability (8) that a neuron i E L(x) which has fired at time t-s fires again at time t. Equations (12) - (14) define the ensemble dynamics of the network. 5 DISCUSSION 5.1 ENSEMBLE CODE Equations. (12) - (14) show that in a large network a description by mean ensemble activities is, in principle, possible. A couple of things, however, should be noted. First, the interaction between the activity of different ensembles is highly nonlinear. It involves three integrations over the past and one exponentiation; cf. (12) - (14). If we had started theoretical modeling with an approach based on mean activities, it would have been hard to find the correct interaction term. Second, L(x) defines an ensemble of equivalent neurons which is a subset of a given assembly al-'. A reduction of (12) to pure assembly activities is, in general, not possible. Finally, equivalent neurons that form an ensemble L(x) are not necessarily situated next to each other. In fact, they may be distributed all over the network; cf. Fig. 2. In this case a local ensemble average yields meaningless results. A theoretical model based on local ensemble averaging is useful only if we know that neighboring neurons have the same 'identity card'. a) activity ~': t l 100 150 200 b) time [ms] 0) rate [Hz] 30 ••• .. .. .. .. .. .. .. .. .. .. .. .... 30 r:::::: .....• : .. : .. :': ' .... : .. :.: .. : .. : ': ': -: \ , :J _ 20 20 .. .. .... .. .. .. .. .. .. .. .. .. .. ~ .. -. -... -. -. .. I •• :. -............ . ~ 10 ••• .- .- ..... - .... -. ill. I .- .- .- .-.. 10 .. .. .. .. .. .. .... .. .. .. .. .. .. .. .. ::> .. " ...... ... -.- .................... " .. " .. " .:::I o···~····-······· 0 100 150 200 0 100 200 time [ms] rate [Hz] 5.2 RATE CODE Fig. 2 Stationary activity (incoherent firing). In this case a description by firing rates is possible. (a) Ensemble averaged activity A(x, t). (b) Spike raster of 30 neurons out of a network of 4000. (c) Time-averaged mean firing rate f. We have two different assemblies, one of them active (dtr = 2 ms, f3 = 5). Can the system of Eqs. (12) -(14) be transformed into a rate description? In general, this is not the case but if we assume that the ensemble activities are constant in 468 Gerstner and van Hemmen 1.0 .---~--~----~--~--------~--~---.----~--~----~--, O.B O.B 0.4 ~.x-2 ~.)(-3.5 0.2 0.0 o 100 200 300 400 500 BOO Zeit [rn5] Fig. 3 Stability of stationary states.The postsynaptic potential h~yn is plotted as a function of time. Every 100 ms the delay Lltr has been increased by 0.5 ms. In the stationary state (Lltr = 1.5 ms and Lltr = 3.5 ms), active neurons fire regularly with rate T;l = 1/5.5 ms. For a delay Ll tr > 3.5 ms, oscillations with period Wl = 27r /Tp build up rapidly. For intermediate delays 2 ~ Ll tr ~ 2.5 small-amplitude oscillations with twice the frequency occur. Higher harmonics are suppressed by noise (/3 = 20). time, i.e., A(x, t) = A(x), then an exact reduction is possible. The result IS a fixed-point equation (Gerstner and van Hemmen 1992) q q A(x) = f[Jo L L Q~lIpost(X~) L pre(zll)p(z)A(z)] (15) ~=lll=l z where f[h,yn] = {J.oo dsexp{-1.' r- 1[h,yn + ~"J(8')]ds'}} -1 (16) is the mean firing rate (9) of a typical neuron stimulated by a synaptic input h3yn. Constant activities correspond to incoherent, stationary firing and in this case a rate code is sufficient; cf. Fig. 2. Two points should, however, be kept in mind. First, a stationary state of incoherent firing is not necessarily stable. In fact, in a noise-free system the stationary state is always unstable and oscillations build up (Gerstner and van Hemmen 1993). In a system with noise, the stability depends on the noise level f3 and the delay Ll tr of axonal and synaptic transmission (Gerstner and van Hemmen 1994). This is shown in Fig. 3 where the delay Lltr has been increased every 100 ms. The frequency of the small-amplitude oscillation around the stationary state is approximately equal to the mean firing rate (16) in the stationary state or higher harmonics thereof. A small-amplitude oscillation corresponds to partially synchronized activity. Note that for Ll tr = 4 ms a large-amplitude oscillation builds up. Here all neurons fire in nearly perfect synchrony; cf. Fig. 4. In the noiseless case f3 00, the oscillations period of such a collective or 'locked' oscillation can be found from the threshold condition T", = inf {s I 0 = ~"J (8) + Jo ~ f(nS)} . (17) In most cases the contribution with n = 1 is dominant which allows a simple graphical solution. The first intersection of the effective threshold () TJref (s) with the How to Describe Neuronal Activity: Spikes. Rates. or Assemblies? 469 weighted EPSP JOf( s) yields the oscillation period; cf. Fig 1. An analytical argument shows that locking is stable only if ;" dTooc > 0 (Gerstner and van Hemmen 1993). a) b) activity ~:lliHHHlUHHHj 100 1~ 200 time [ms] ~ ................. . I f S S S S SIS ) S S S S \ 'a \ \ ,., 20 . . . . . . . . . . . . . . . . . . ~ ·.\'\\'111".·.·'1111 ! 10 \ \ \ \ \ , \ I 1 \ •••• , \ \ \ \ \ . . . . . . . . . . . . . . . . . . "1'111 \\1\\1 "' .. 1 0) rate [Hz] ] 10 t==:::;;= o •••••••••••••••• •• 100 150 200 0 0 100 200 time [ms] rate [Hz] Fig. 4 Oscillatory activity (coherent firing). In this case a description by firing rates must be combined with a description by ensemble activities. (a) Ensemble averaged activity A(x, t). (b) Spike raster of 30 neurons out of a network of 4000. (c) Timeaveraged mean firing rate f. In this simulation, we have used Ll tr = 4 ms and f3 = 8. Second, even if the incoherent state is stable and attractive, there is always a transition time before the stationary state is assumed. During this time, a rate description is insufficient and we have to go back to the full dynamic equations (12) - (14). Similarly, if neurons are subject to a fast time-dependent external stimulus, a rate code fails. 5.3 SPIKE CODE A superficial inspection of Eqs. (12) - (14) gives the impression that all information about neuronal spiking has disappeared. This is, however, false. The term A(x, t-s) in (13) denotes all neurons with 'identity card' x that have fired at time t-s. The integration kernel in (13) is the conditional probability that one of these neurons fires again at time t. Keeping t - s fixed and varying t we get the distribution of inter-spike intervals for neurons in L(x). Thus information on both spikes and intervals is contained in (13) and (14). We can make use of this fact, if we consider network states where in every time step a different assembly is active. This leads to a spatia-temporal spike pattern as shown in Fig. 5. To transform a specific spike pattern into a stable state of the network we can use a Hebbian learning rule. However, in contrast to the standard rule, a synapse is strenthened only if pre- and postsynaptic activity occurs simultaneously within a time window of a few ms (Gerstner et al. 1993). Note that in this case, averaging over time or space spoils the information contained in the spike pattern. 5.4 CONCLUSIONS Equations. (12) - (14) show that in our large and fully connected network an ensemble code with an appropriately chosen ensemble is sufficient. If, however, the efficacies (11) and the connection scheme become more involved, the construction of appropriate ensembles becomes more and more difficult. Also, in a finite network we cannot make use of the law of large number in defining the activities (10). Thus, in general, we should always start with a network model of spiking neurons. 470 Gerstner and van Hemmen a) b) activity ~~C =: ] 100 150 200 time [ms] 0) rata [Hz] 30 30 n-"~----' .. o 0 .. 20 g !5 • ·0 .. • ! 10 0 • .. . . .... 20 10 ••• e. o '--__ o-".'___---:-~---"'----~ 0 ,-,-. (---''---'. 1()0 200 0100 200 rata [Hz] Fig. 5 Spatio-temporal spike pattern. In this case, neither firing rates nor locally averaged activities contain enough information to describe the state of the network. (a) Ensemble averaged activity A(t). (b) Spike raster of 30 neurons out of a network of 4000. ( c) Time-averaged mean firing rate f. Acknowledgements: This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under grant No. He 1729/2-1. References Gerstner W (1990) Associative memory in a network of 'biological' neurons. In: Advances in Neural Information Processing Systems 3, edited by R.P. Lippmann, J .E. Moody, and D.S. Touretzky (Morgan Kaufmann, San Mateo, CA) pp 84-90 Gerstner Wand van Hemmen JL (1992a) Associative memory in a network of 'spiking' neurons. Network 3:139-164 Gerstner W, van Hemmen JL (1993) Coherence and incoherence in a globally coupled ensemble of pulse-emitting units. Phys. Rev. Lett. 71:312-315 Gerstner W, Ritz R, van Hemmen JL (1993b) Why spikes? Hebbian learning and retrieval of time-resolved excitation patterns. BioI. Cybern. 69:503-515 Gerstner Wand van Hemmen JL (1994) Coding and Information processing in neural systems. In: Models of neural networks, Vol. 2, edited by E. Domany, J .L. van Hemmen and K. Schulten (Springer-Verlag, Berlin, Heidelberg, New York) pp Iff Hebb DO (1949) The Organization of Behavior. Wiley, New York van Hemmen JL and Kiihn R(1991) Collective phenomena in neural networks. In: Models of neural networks, edited by E. Domany, J .L. van Hemmen and K. Schulten (Springer-Verlag, Berlin, Heidelberg, New York) pp Iff Hubel DH, Wiesel TN (1962) Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. J. Neurophysiol. 28:215-243 MacKay DM, McCulloch WS (1952) The limiting information capacity of a neuronal link. Bull. of Mathm. Biophysics 14:127-135 Stein RB (1967) The information capacity of nerve cells using a frequency code. Biophys. J. 7:797-826
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Two-Dimensional Object Localization by Coarse-to-Fine Correlation Matching Chien-Ping Lu and Eric Mjolsness Department of Computer Science Yale University New Haven, CT 06520-8285 Abstract We present a Mean Field Theory method for locating twodimensional objects that have undergone rigid transformations. The resulting algorithm is a form of coarse-to-fine correlation matching. We first consider problems of matching synthetic point data, and derive a point matching objective function. A tractable line segment matching objective function is derived by considering each line segment as a dense collection of points, and approximating it by a sum of Gaussians. The algorithm is tested on real images from which line segments are extracted and matched. 1 Introduction Assume that an object in a scene can be viewed as an instance of the model placed in space by some spatial transformation, and object recognition is achieved by discovering an instance of the model in the scene. Two tightly coupled subproblems need to be solved for locating and recognizing the model: the correspondence problem (how are scene features put into correspondence with model features?), and the localization problem (what is the transformation that acceptably relates the model features to the scene features?). If the correspondence is known, the transformation can be determined easily by least squares procedures. Similarly, for known transformation, the correspondence can be found by aligning the model with the scene, or the problem becomes an assignment problem if the scene feature locations are jittered by noise. 985 986 Lu and Mjolsness Several approaches have been proposed to solve this problem. Some tree-pruning methods [1, 3] make hypotheses concerning the correspondence by searching over a tree in which each node represents a partial match. Each partial match is then evaluated through the pose that best fits it. In the generalized Hough transform or equivalently template matching approach [7, 3], optimal transformation parameters are computed for each possible pairing of a model feature and a scene feature, and these "optimal" parameters then "vote" for the closest candidate in the discretized transformation space. By contrast with the tree-pruning methods and the generalized Hough transform, we propose to formulate the problem as an objective function and optimize it directly by using Mean Field Theory (MFT) techniques from statistical physics, adapted as necessary to produce effective algorithms in the form of analog neural networks. 2 Point Matching Consider the problem of locating a two-dimensional "model" object that is believed to appear in the "scene". Assume first that both the model and the scene are represented by a set of "points" respectively, {xd and {Ya}. The problem is to recover the actual transformation (translation and rotation) that relates the two sets of points. It can be solved by minimizing the following objective function Ematch(Mia, 0, t) = L Miallxi - ReYa - tll 2 ia (1) where {Mia} = M is a Ofl-valued "match matrix" representing the unknown correspondence, Re is a rotation matrix with rotation angle 0, and t is a translation vector. 2.1 Constraints on match variables We need to enforce some constraints on correspondence (match) variables Mia; otherwise all Mia = a in (1). Here, we use the following constraint LMia = N, 'iMia ~ 0; (2) ia implying that there are exactly N matches among all possible matches, where N is the number of the model features. Summing over permutation matrices obeying this constraint, the effective objective function is approximately [5]: F(O, t, (3) = -.!. L e-.8l1 x .-R8y .. -tIl 2 , 13 ia which has the same fixed points as 1 Epenalty(M, 0, t) = Ematch(M, 0, t) + - L Mia (log Mia - 1), 13 ia (3) (4) where Mia is treated as a continuous variable and is subject to the penalty function x(logx-l). Two-Dimensional Object Localization by Coarse-ta-Fine Correlation Matching 987 Figure 1: Assume that there is only translation between the model and the scene, each containing 20 points. The objective functions at at different temperatures (,8- 1): 0.0512 (top left), 0.0128 (top right) , 0.0032 (bottom left) and 0.0008 (bottom right), are plotted as energy surfaces of x and y components of translation. Now, let j3 = 1/2u2 and write Epoint(O, t) = L e-~lIx,-R8y(1-tIl2. (5) ia The problem then becomes that of maximizing Epoint , which in turn can be interpretated as minimizing the Euclidean distance between two Gaussian-blurred images containing the scene points Xi and a transformed version of the model points Ya. Tracking the local maximum of the objective function from large u to small u, as in deterministic annealing and other continuation methods, corresponds to a coarseto-fine correlation matching. See Figure 1 for a demonstration of a simpler case in which only translation is applied to the model. 2.2 The descent dynamics A gradient descent dynamics for finding the saddle point of the effective objective function F is ia o -I\, L mia(Xi - R 9Ya t)t(R9+~Ya) , (6) ia 988 Lu and Mjolsness where mia = (Mia}/3 = e-/3llx.-R 8y,,-tIl 2 is the "soft correspondence" associated with Mia. Instead of updating t by descent dynamics, we can also solve for t directly. 3 The Vernier Network Though the effective objective is non-convex over translation at low temperatures, its dependence on rotation is non-convex even at relatively high temperatures. 3.1 Hierachical representation of variables We propose overcoming this problem by applying Mean Field Theory (M FT) to a hierachical representation of rotation resulting from the change of variables [4] B-1 o L Xb(Ob + (h), (h E [-te, te], (7) b=O where te = 7r /2B, Ob = (b + l)~ are the constant centers of the intervals, and (h are fine-scale "vernier" variabfes. The Xb'S are binary variables (so Xb E {O, I}) that satisfy the winner-take-all (WTA) constraint Lb Xb = 1. The essential reason that this hierarchical representation of 0 has fewer spurious local minima than the conventional analog representation is that the change of variables also changes the connectivity of the network's state space: big jumps in 0 can be achieved by local variations of X. 3.2 Vernier optimization dynamics Epoint can be transformed as (see [6, 4]) 1 Epoint(O, t) ~vbl ~ E(LXb(Ob +Ob),2:XVt b) b b LXbE(Ov + Ob, tb) b 1 Notation: Coordinate descent with 2-phase clock 'IlIa(t): a • EB for clocked sum • x for a clamped variable • x A for a set of variables to be optimized analytically • (v, u)H for Hopfield/Grossberg dynamics (8) • E(x, y)fJJ for coordinate descent/ascent on x, then y, iterated if necessary. Nested angle brackets correspond to nested loops. Two-Dimensional Object Localization by Coarse-to-Fine Correlation Matching 989 • . e.. . , , , . , " • • I • •• •• I •• '. , , . . 0.·· -. • 0 0 •••• • • 0 •• 0 0• <f> () 0 , o • 0 ,0 -0' o COO . . ' ...... . , . 00· • o· . o '. '0,4), • c;P~ I 00 .• 0 o· : 0 •• -. Q:)q •• • .•••• 0 , ~ 0 ~ ~ , . Q, • • e, I •• ~Q . .. [) ~f) 0' •• €) -•• Figure 2: Shown here is an example of matching a 20-point model to a scene with 66.7% spurious outliers. The model is represented by circles. The set of square dots is an instance of the model in the scene. All other dots are outliers. From left to right are configurations at the annealing steps 1, 10, and 51, respectively. MFT ~ [ ~ A 1 ~ sinh(tub) ~ XbE(th + Vb, tb) + ,8 ~(UbVb -log t ) b b + WTA(x,,8)] (((v, u)H, t A), XA)$ (9) Each bin-specific rotation angle Vb can be found by the following fixed point equations a ia (10) The algorithm is illustrated in Figure 2. 4 Line Segment Matching In many vision problems, representation of images by line segments has the advantage of compactness and subpixel accuracy along the direction transverse to the line. However, such a representation of an object may vary substantially from image to image due to occlusions and different illumination conditions. 4.1 Indexing points on line segements The problem of matching line segments can be thought of as a point matching problem in which each line segment is treated as a dense collection of points. Assume now that both the scene and the model are represented by a set of line segments respectively, {sil and {rna}. Both the model and the scene line segments are 990 Lu and Mjolsness '! 'J .... ./ \ o 1!io ( " f \ D.' 1.2\ 1 S -e .lS Figure 3: Approximating e(t) by a sum of 3 Gaussians. represented by their endpoints as Si = (pi, p~) and rna = (qa, q~), where Pi, p~, and qa, q~ are the endpoints of the ith scene segment and the ath model segment, respectively. The locations of the points on each scene segment and model segments can be parameterized as Xi = Si(U) = Ya = IDa(v) = Pi + u(p~ - Pi), U E [0,1] and qa + v(q~ - qa), v E [0,1]. (ll) (12) N ow the model points and the scene points can be though of as indexed by i = (i, u) and a = (a, v). Using this indexing, we have Li ex Li Ii Jol du and La ex La1aJoi dv, where Ii = Ilpi-P~II andla = IIqa-q~ll· The point matching objective function (5) can be specialized to line segment matching as [5] Eseg((}, t) = L hla t (I e- ~IIS.(u)-Rem,,(v)-tIl2 du dv. ia Jo Jo (13) As a special case of point matching objective function, (13) can readily be transformed to the vernier network previously developed for point matching problem. 4.2 Gaussian sum approximation Note that, as in Figure 3 and [5], e(t) -_ {I if t E [0: 1] ~ 1 (Ck - t)2 o otherWIse ~ ~ Ak exp -"2 (72 k~I k (14) where by numerical minimization of the Euclidean distance between these two functions of t, the parameters may be chosen as Al = A3 = 0.800673, A2 = 1.09862, (71 = (73 = 0.0929032, (72 = 0.237033, C1 = 1 C3 = 0.1l6807, and C2 = 0.5. Using this approximation, each finite double integral in (13) can be replaced by 3 1+00 1+00 __ 1_(Ck_U)2 1 (cr-v)2 1 R t 2 k~l AkAl -00 -00 e 2"'~ e ~ e- 2,;2l1s.(u)+ em,,(v)- II du dv. (15) Each of these nine Gaussian integrals can be done exactly. Defining Viakl = Si(Ck) Rema(cl) t Pi = pi - Pi, qa = Re(q~ - qa), (16) (17) Two-Dimensional Object Localization by Coarse-to-Fine Correlation Matching 991 Figure 4: The model line segments, which are transformed with the optimal parameter found by the matching algorithm, are overlayed on the scene image. The algorithm has successfully located the model object in the scene. (15) becomes 1 vlaklu2 + (Viakl X pd2u~ + (Viakl X Qa)2uf X exp -"2 (u2 + f>;un(u2 + Q~uf) U~U;(f>i . Qa)2 (18) as was calculated by Garrett [2, 5]. From the Gaussian sum approximation, we get a closed form objective function which can be readily optimized to give a solution to the line segment matching problem. 5 Results and Discussion The line segment matching algorithm described in this paper was tested on scenes captured by a CCD camera producing 640 x 480 images, which were then processed by an edge detector. Line segments were extracted using a polygonal approximation to the edge images. The model line segments were extracted from a scene containing a canonically positioned model object (Figure 4 left). They were then matched to that extracted from a scene containing differently positioned and partially occluded model object (Figure 4 nght). The result of matching is shown in Figure 5. Our approach is based on a scale-space continuation scheme derived from an application of Mean Field Theory to the match variables. It provides a means to avoid trapping by local extrema and is more efficient than stochastic searches such as simulated annealing. The estimation of location parameters based on continuously improved "soft correspondences" and scale-space is often more robust than that based on crisp (but usually inaccurate) correspondences. The vernier optimization dynamics arises from an application of Mean Field Theory to a hierarchical representation of the rotation, which turns the original unconstrained optimization problem over rotation e into several constrained optimization problems over smaller e intervals. Such a transformation results in a Hopfield-style 992 Lu and Mjolsness Figure 5: Shows how the model line segments (gray) and the scene segments (black) are matched. The model line segments, which are transformed with the optimal parameter found by the matching algorithm, are overlayed on the scene line segments with which they are matched. Most of the the endpoints and the lengths of the line segments are different. Furthermore, one long segment frequently corresponds to several short ones. However, the matching algorithm is robust enough to uncover the underlying rigid transformation from the incomplete and ambiguous data. dynamics on rotation 0, which effectively coordinates the dynamics of rotation and translation during the optimization. The algorithm tends to find a roughly correct translation first, and then tunes up the rotation. 6 Acknowledgements This work was supported under grant NOOOl4-92-J-4048 from ONRjDARPA. References [1] H. S. Baird. Model-Based Image Matching Using Location. The MIT Press, Cambridge, Massachusetts, first edition, 84. [2] C. Garrett, 1990. Private communication to Eric Mjolsness. [3] W. E. L. Grimson and T. Lozano-Perez. Localizing overlapping parts by searching the interpretation tree. IEEE Transaction on Pattern Analysis and Machine Int elligence, 9 :469-482, 1987. [4] C.-P. Lu and E. Mjolsness. Mean field point matching by vernier network and by generalized Hough transform. In World Congress on Neural Networks, pages 674-684, 1993. [5] E. Mjolsness. Bayesian inference on visual grammars by neural nets that optimize. In SPIE Science of Artificial Neural Networks, pages 63-85, April 1992. [6] E. Mjolsness and W. L. Miranker. Greedy Lagrangians for neural networks: Three levels of optimization in relaxation dynamics. Technical Report YALEUjDCSjTR-945, Yale Computer Science Department, January 1993. [7] G. Stockman. Object recognition and localization via pose clustering. Computer Vision, Graphics, and Image Processing, (40), 1987.
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WATTLE: A Trainable Gain Analogue VLSI Neural Network Richard Coggins and Marwan Jabri Systems Engineering and Design Automation Laboratory Department of Electrical Engineering J03, University of Sydney, 2006. Australia. Email: richardc@sedal.su.oz.au marwan@sedal.su.oz.au Abstract This paper describes a low power analogue VLSI neural network called Wattle. Wattle is a 10:6:4 three layer perceptron with multiplying DAC synapses and on chip switched capacitor neurons fabricated in 1.2um CMOS. The on chip neurons facillitate variable gain per neuron and lower energy/connection than for previous designs. The intended application of this chip is Intra Cardiac Electrogram classification as part of an implantable pacemaker / defibrillator system. Measurements of t.he chip indicate that 10pJ per connection is achievable as part of an integrated system. Wattle has been successfully trained in loop on parity 4 and ICEG morphology classification problems. 1 INTRODUCTION A three layer analogue VLSI perceptron has been previously developed by [Leong and Jabri, 1993]. This chip named Kakadu uses 6 bit digital weight storage, multiplying DACs in the synapses and fixed value off chip resistive neurons. The chip described in this paper called Wattle has the same synapse arrays as Kakadu, however, has the neurons implemented as switched capacitors on chip. For both Kakadu and Wattle, analogue techniques have been favoured as they offer greater opportunity to achieve a low energy and small area design over standard digital 874 WATfLE: A Trainable Gain Analogue VLSI Neural Network 875 00 II I ---------------------lout+ ODd Jout- 0UlpUt &_-.,..- -=~-~ c:oonoct ..... , , , : NEURON CIRCUIT ------------------------------------ -_.-- --------------------, WEIGHT STORAGE ----------------------1 HII mAC ~ , , , , I , ~---------------------, : I I I L ______________________________________________________________ J Figure 1: Wattle Synapse Circuit Diagram SYNAPSE CIRCUIT techniques since the transistor count for the synapse can be much lower and the circuits may be biased in subthreshold. Some work has been done in the low energy digital area using subthreshold and optimised threshold techniques, however no large scale circuits have been reported so far. [Burr and Peterson, 1991] The cost of using analogue techniques is however, increased design complexity, sensitivity to noise, offsets and component tolerances. In this paper we demonstrate that difficult nonlinear problems and real world problems can be trained despite these effects. At present, commercially available pacemakers and defibrillators use timing decision trees implemented on CMOS microprocessors for cardiac arrythmia detection via peak detection on a single ventricular lead. Even when atrial leads are used, Intra Cardiac Electrogram (ICEG) morphology classification is required to separate some potentially fatal rhythms from harmless ones. [Leong and J abri, 1992] The requirements of such a morphology classifier are: • Adaptable to differing morphology within and across patients. • Very low power consumption. ie. minimum energy used per classification. • Small size and high reliability. This paper demonstrates how this morphology classification may be done using a neural network architecture and thereby meet the constraints of the implantable arrythmia classification system. In addition, in loop training results will also be given for parity 4, another difficult nonlinear training problem. 876 Coggins and Jabri reset clock Vdd s s ~ ~_c_lk_0 __ -+ __________ ~ ______ ~ __________ ~ t------------------t---------'co='-p -lL----> fan outto charging clock '" synapse row COM CIP connects , 1i::)>----_CIU _____________ _______ -----' Figure 2: Wattle Neuron Circuit Diagram '--Column Addrus ~ ~ Row Addrus 10x6 Synapse Array I hi muHlplexor I 6x4Synapse Array neuron. Indkclemux I DD DO DO DO neurons D D .----.Dar"'O'ta---=Re,.-g-.I,....., .. :-e-r -""1 Buffers Figure 3: Wattle Floor Plan ; next layer WATTLE: A Trainable Gain Analogue VLSI Neural Network 877 .. 611"',. Figure 4: Photomicrograph of Wattle 2 ARCHITECTURE Switched capacitors were chosen for the neurons on Wattle after a test chip was fabricated to evaluate three neuron designs. [Coggins and Jabri, 1993] The switched capacitor design was chosen as it allowed flexible gain control of each neuron, investigation of gain optimisation during limited precision in loop training and the realisation of very high effective resistances. The wide gain range of the switched capacitor neurons and the fact that they are implemented on chip has allowed Wattle to operate over a very wide range of bias currents from 1 pA LSB DAC current to 10nA LSB DAC current. Signalling on Wattle is fully differential to reduce the effect of common mode noise. The synapse is a multiplying digital to analogue convertor with six bit weights. The synapse is shown in figure L This is identical to the synapse used on the Kakadu chip [Leong and Jabri, 1993]. The MDAC synapses use a weighted current source to generate the current references for the weights. The neuron circuit is shown in figure 2. The neuron requires reset and charging clocks. The period of the charging clock determines the gain. Buffers are used to drive the neuron outputs off chip to avoid the effects of stray pad capacitances. Figure 3 shows a floor plan of the wattle chip. The address and data for the weights access is serial and is implemented by the shift registers on the boundary of the chip. The hidden layer multiplexor allows access to the hidden layer neuron outputs. The neuron demultiplexor switches the neuron clocks between the hidden and output layers. Figure 4 shows a photomicrograph of the wattle die. 3 ELECTRICAL CHARACTERISTICS Tests have been performed to verify the operation of the weighted current source for the MDAC synapse arrays, the synapses, the neurons and the buffers driving the neuron voltages off chip. The influences of noise, offsets, crosstalk and bandwidth of these different elements have been measured. In particular, the system level noise measurement showed that the signal to noise ratio was 40dB. A summary of the electrical characteristics appears in table L 878 Coggins and Jabri Table 1: Electrical Characteristics and Specifications Parameter Value Comment Area 2.2 x 2.2mm~ Technology 1.2um Nwell CMOS 2M2P standard process Resolution weights 6bit, gains 7bit weights on chip, gains off Energy per connection 43pJ all weights maximum LSB DAC current 200pA typical Feedforward delay 1.5ms @200pA, 3V supply Synapse Offset 5mV typical maximum Gain cross talk delta 20% maxImum A gain cross talk effect between the neurons was discovered during the electrical testing. The mechanism for this cross talk was found to be transients induced on the current source reference lines going to all the synapses as individual neuron gains timed out. The worst case cross talk coupled to a hidden layer neuron was found to be a 20% deviation from the singularly activated value. However, the training results of the chip do not appear to suffer significantly from this effect. A related effect is the length of time for the precharging of the current summation lines feeding each neuron due to the same transients being coupled onto the current source when each neuron is active. The implication of this is an increase in energy per classification for the network due to the transient decay time. However, one of the current reference lines was available on an outside pin, so the operation of the network free of these transients could also be measured. For this design, including the transient conditions, an energy per connection of 43pJ can be achieved. This may be reduced to 10pJ by modifying the current source to reduce transients and neglecting the energy of the buffers. This is to be compared with typical digital lOnJ per connection and analogue of 60pJ per connection appearing in the literature. [Delcorso et. al., 1993], Table 1. 4 TRAINING BOTH GAINS AND WEIGHTS A diagram of the system used to train the chip is shown in figure 5. The training software is part of a package called MUME [J abri et. al., 1992], which is a multi module neural network simulation environment. Wattle is interfaced to the work station by Jiggle, a general purpose analogue and digital chip tester developed by SEDAL. Wattle, along with gain counter circuitry, is mounted on a separate daughter board which plugs into Jiggle. This provides a software configurable testing environment for Wattle. In loop training then proceeds via a hardware specific module in MUME which writes the weights and reads back the analogue output of the chip. Wattle can then be trained by a wide variety of algorithms available in MUME. Wattle has been trained in loop using a variation on the Combined Search Algorithm (CSA) for limited precision training. [Xie and Jabri, 1992] (Combination of weight perturbation and axial random search). The variation consists of training the gains
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Neurobiology, Psychophysics, and Computational Models of Visual Attention Ernst Niebur Computation and Neural Systems California Institute of Technology Pasadena, CA 91125, USA Bruno A. Olshausen Department of Anatomy and Neurobiology Washington University School of Medicine St. Louis, MO 63110 The purpose of this workshop was to discuss both recent experimental findings and computational models of the neurobiological implementation of selective attention. Recent experimental results were presented in two of the four presentations given (C.E. Connor, Washington University and B.C. Motter, SUNY and V.A. Medical Center, Syracuse), while the other two talks were devoted to computational models (E. Niebur, Caltech, and B. Olshausen, Washington University). Connor presented the results of an experiment in which the receptive field profiles of V 4 neurons were mapped during different states of attention in an awake, behaving monkey. The attentional focus was manipulated in this experiment by altering the position of a behaviorally relevant ring-shaped stimulus. The animal's task was to judge the size of the ring when compared to a reference ring (i.e., same or different). In order to map the receptive field profile, a behaviorally irrelevant bar stimulus was flashed at one of several positions inside and outside the classical receptive field (CRF). It was found that shifts of attention produced alterations in receptive field profiles for over half the cells studied. In most cases the receptive field center of gravity translated towards attentional foci in or near the CRF. Changes in width and shape of the receptive field profile were also observed, but responsive regions were not typically limited to the location of the attended ring stimulus. Attentionrelated effects often included enhanced responses at certain locations as well as diminished responses at other locations. Motter studied the basic mechanisms of visual search as manifested in the single unit activity of rhesus monkeys. The animals were trained to select a bar stimulus among others based on the color or luminance of the target stimulus. The majority of neurons were selectively activated when the color or luminance of the stimulus in the receptive field matched the color or luminance of the cue, whereas the activity was attenuated when there was no match. Since a cell responds differently to the same stimulus depending on the color or luminance of the cue (which is given far away from the stimulus by the color or luminance of the fixation spot), the activity of the neurons reflect a selection based on the cued feature and not simply the physical color or luminance of the receptive field stimulus. Motter showed that the 1167 1168 Niebur and Olshausen selection can also be based on memory by switching off the cue in the course of the experiment. The monkey could then perform the task only by relying on his memory and the pattern of V4 activity. In the memory-based task as well as in the experiments with the stimulus continuously present, the differential activation was independent of spatial location and offers therefore a physiological correlate to psychophysical studies suggesting that stimuli can be preferentially selected in parallel across the visual field. Niebur presented a model for the neuronal implementation of selective visual attention based on temporal correlation among groups of neurons. In the model, neurons in primary visual cortex respond to visual stimuli with a Poisson distributed spike train with an appropriate, stimulus-dependent mean firing rate. The spike trains of neurons whose receptive fields do not overlap with the "focus of attention" are distributed according to homogeneous (time-independent) Poisson process with no correlation between action potentials of different neurons. In contrast, spike trains of neurons with receptive fields within the focus of attention are distributed according to non-homogeneous (time-dependent) Poisson processes. Since the short-term average spike rates of all neurons with receptive fields in the focus of attention covary, correlations between these spike trains are introduced which are detected by inhibitory interneurons in V 4. These cells, modeled as modified integrate-and-fire neurons, function as coincidence detectors and suppress the response of V 4 cells associated with non-attended visual stimuli. The model reproduces quantitatively experimental data obtained in cortical area V 4 of monkey. The model presented by Olshausen proposed that attentional gating takes place via an explicit control process, without relying on temporal correlation. This model is designed to serve as a possible explanation for how the visual cortex forms position and scale invariant representations of objects. Control neurons dynamically modify the synaptic strengths of intracortical connections so that information from a windowed region of primary visual cortex is selectively routed to higher cortical areas, preserving spatial relationships. The control signals for setting the position and size ofthe attentional window are hypothesized to originate from neurons in the pulvinar and in the deep layers of visual cortex. The dynamics of these control neurons are governed by simple differential equations that can be realized by neurobiologically plausible circuits. In pre-attentive mode, the control neurons receive their input from a low-level "saliency map" representing potentially interesting regions of a scene. During the pattern recognition phase, control neurons are driven by the interaction between top-down (memory) and bottom-up (retinal input) sources. The model predicts that the receptive fields of cortical neurons should shift with attention, as found in Connor's experiments, although the predicted shifts are somewhat larger than those found to date. Acknowledgement The work of EN and BAO was supported by the Office of Naval Research. EN was additionally supported by the National Science Foundation.
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Bayesian Backpropagation Over 1-0 Functions Rather Than Weights David H. Wolpert The Santa Fe Institute 1660 Old Pecos Trail Santa Fe, NM 87501 Abstract The conventional Bayesian justification of backprop is that it finds the MAP weight vector. As this paper shows, to find the MAP i-o function instead one must add a correction tenn to backprop. That tenn biases one towards i-o functions with small description lengths, and in particular favors (some kinds of) feature-selection, pruning, and weight-sharing. 1 INTRODUCTION In the conventional Bayesian view ofbackpropagation (BP) (Buntine and Weigend, 1991; Nowlan and Hinton,1994; MacKay,I992; Wolpert, 1993), one starts with the "likelihood" conditional distribution P(training set = t I weight vector w) and the "prior" distribution P(w). As an example, in regression one might have a "Gaussian likelihood", P(t I w) oc: exp[-x2(w, t)] == I1i exp [-(net(w, tx(i» - ty(i) )2/2c?] for some constant CJ. (tx(i) and ty(i) are the successive input and output values in the training set respectively, and net(w, .) is the function, induced by w, taking input neuron values to output neuron values.) As another example, the "weight decay" (Gaussian) prior is P(w) oc: eXp(-a(w2» for some constant a. Bayes' theorem tells us that P(w I t) oc P(t I w) P(w). Accordingly, the most probable weight given the data - the "maximum a posteriori" (MAP) w - is the mode over w of P(t I w) P(w), which equals the mode over w of the "cost function" L(w, t) == In[P(t I w)] + In[P(w)]. So for example with the Gaussian likelihood and weight decay prior, the most probable w given the data is the w minimizing X2(w, t) + aw2. Accordingly BP with weight decay can be viewed as a scheme for trying to find the function from input neuron values to output neuron values (i-o function) induced by the MAP w. 200 Bayesian Backpropagation over 1-0 Functions Rather Than Weights 201 One peculiar aspect of this justification of weight-decay BP is the fact that rather than the i-o function induced by the most probable weight vector. in practice one would usually prefer to know the most probable i-o function. (In few situations would one care more about a weight vector than about what that weight vector parameterizes.) Unfortunately. the difference between these two i-o functions can be large; in general it is not true that "the most probable output corresponds to the most probable parameterU (Denker and LeCun. 1991). This paper shows that to fmd the MAP i-o function rather than the MAP w one adds a "correction termU to conventional BP. That term biases one towards i-o functions with small description lengths. and in particular favors feature-selection. pruning and weight-sharing. In this that term constitutes a theoretical justification for those techniques. Although cast in terms of neural nets. this paper·s analysis applies to any case where convention is to use the MAP value of a parameter encoding Z to estimate the value of Z. 2 BACKPROP OVER 1-0 FUNCTIONS Assume the nee s architecture is fixed. and that weight vectors w live in a Euclidean vector space W of dimension IWI. Let X be the set of vectors x which can be loaded on the input neurons. and 0 the set of vectors 0 which can be read off the output neurons. Assume that the number of elements in X (lXI) is finite. This is always the case in the real world. where measuring devices have finite accuracy. and where the computers used to emulate neural nets are finite state machines. For similar reasons 0 is also finite in practice. However for now assume that 0 is very large and "fine-grained". and approximate it as a Euclidean vector space of dimension 101. (This assumption usually holds with neural nets. where output values are treated as real-valued vectors.) This assumption will be relaxed later. Indicate the set of functions taking X to 0 by cl>. (net(w •. ) is an element of cl>.) Any cI» E cl> is an (lXI x 101)-dimensional Euclidean vector. Accordingly. densities over W are related to densities over cl> by the usual rules for transforming densities between IWI-dimensional and (IXI x IOI)-dimensional Euclidean vector spaces. There are three cases to consider: 1) IWI < IXIIOL In general. as one varies over all w's the corresponding i-o functions net(w, .) map out a sub-manifold of cl> having lower dimension than cl>. 2) IWI > IXIIOL There are an infinite number of w's corresponding to each cI». 3) IWI = IXIIOI. This is the easiest case to analyze in detail. Accordingly I will deal with it first, deferring discussion of cases one and two until later. With some abuse of notation, let capital letters indicate random variables and lower case letters indicate values of random variables. So for example w is a value of the weight vector random variable W. Use 'p' to indicate probability densities. So for example P<l>IT<cI» I t) is the density of the i-o function random variable cl>, conditioned on the training set random variable T, and evaluated at the values cl> = cI» and T = t. In general, any i-o function not expressible as net(w, .) for some w has zero probability. For the other i-o functions, with S(.) being the multivariable Dirac delta function, p<l>(net(w •. » = jdw' Pw(w') S(net(w', .) - net(w, .». (1) When the mapping cl> = net(W, .) is one-to-one, we can evaluate equation (1) to get p4>lT<net(w, .) I t) = Pwrr(w I t) / J<I>.W<w), (2) where J<I> w(w) is the Jacobian of the W ~ cl> mapping: • 202 Wolpert J<I>,W(w) == I det[ ()<l>i / dWj lew) I = I det[ d net(w, ')i / dWj ] I. (3) "net(w, .)t means the i'th component of the i-o function net(w, .). "net(w, x)" means the vector 0 mapped by net(w, .) from the input x, and "net(w, X)k" is the k'th component of o. So the "i" in "net(w, .)( refers to a pair of values {x, k}. Each matrix value a~ / dWj is the partial derivative of net(w, x>t with respect to some weight, for some x and k. J<I>,w(w) can be rewritten as detla [gij(W)], where gij(w) == ~ [(d<!>t / dWi) (d~ / dWj)] is the metric of the W ~ ct» mapping. This form of J~ w(w) is usually more difficult to evaluate though. , Unfortunately, cI» = net(w, .) is not one-to-one; where J<I>,w(w) * 0 the mapping is locally one-to-one, but there are global symmetries which ensure that more than one w corresponds to each cI». (Such symmetries arise from things like permuting the hidden neurons or changing the sign of all weights leading into a hidden neuron - see (Fefferman, 1993) and references therein.) To circumvent this difficulty we must make a pair of assumptions. To begin, restrict attention to Winj, those values w of the variable W for which the Jacobian is non-zero. This ensures local injectivity of the map between W and ct». Given a particular w E W inj' let k be the number of w' E W inj such that net(w, .) = net(w', .). (Since net(w,.) = net(w, .), k ~ 1.) Such a set ofk vectors form an equivalence class, {w}. The first assumption is that for all w E W inj the size of (w) (i.e., k) is the same. This will be the case if we exclude degenerate w (e.g .• w's with all first layer weights set to 0). The second assumption is that for all w' and w in the same equivalence class, PWID (w I d) = PWID (w' I d). This is usually the case. (For example, start with w' and relabel hidden neurons to get a new WE (w'). If we assume the Gaussian likelihood and prior, then since neither differs for the two w's the weight-posterior is also the same for the two w's.) Given these assumptions, p<l>IT(net(w, .) I t) = k pWlnw I t) / J<I>,w(w). So rather than minimize the usual cost function, L(w, t), to find the MAP ct» BP should minimize L'(w. t) == L(w, t) + In[ J~W<w)]. The In[ J~w(w)] term constitutes a correction term to conventional BP. , , One should not confuse the correction term with the other quantities in the neural net literature which involve partial derivative matrices. As an example, one way to characterize the "quality" of a local peak w' of a cost function involves the Hessian of that cost function (Buntine and Weigend, 1991). The correction term doesn't directly concern the validity of such a Hessian-based quality measure. However it does concern the validity of some implementations of such a measure. In particular. the correction term changes the location of the peak w'. It also suggests that a peak's quality be measured by the Hessian of L'(w', t) with respect to cI», rather than by the Hessian of L(w', t) with respect to w. (As an aside on the subject of Hessians, note that some workers incorrectly use Hessians when they attempt to evaluate quantities like output-variances. See (Wolpert, 1994).) If we stipulate that the pcI>ln cI» I t) one encounters in the real world is independent of how one chooses to parameterize ct», then the probability density of our parameter must depend on how it gets mapped to ct». This is the basis of the correction term. As this suggests, the correction term won't arise if we use non-pcI>lncl» I t)-based estimators, like maximum-likelihood estimators. (This is a basic difference between such estimators and MAP estimators with a uniform prior.) The correction term is also irrelevant if it we use an MAP estimate but J~ w(w) is independent of w (as when net (w •. ) depends linearly on w). And for non, linear net(w, .), the correction term has no effect for some non-MAP-based ways to apply Bayesianism to neural nets, like guessing the posterior average ct» (Neal, 1993): Bayesian Backpropagation over 1-0 Functions Rather Than Weights 203 E(ct» It) == Idct> Pcl>lnct> It) ct> = Idw PWIT(w I t) net(w, .), (4) so one can calculate E(ct» I t) by working in W, without any concern for a correction tenn. (Loosely speaking, the Jacobian associated with changing integration variables cancels the Jacobian associated with changing the argument of the probability density. A formal derivation - applicable even when IWI-:/: IXI x 101 - is in the appendix of (Wolpert, 1994).) One might think that since it's independent of 1, the correction term can be absorbed into Pw(w). Ironically, it is precisely because quantities like E(ct» I t) aren't affected by the correction tenn that this is impossible: Absorb the correction term into the prior, giving a new prior P*w(w) == d x Pw(w) x J<I) w(w) (asterisks refers to new densities, and d is a normal, ization constant). Then p*<I)IT(net(w, .) I t) = pwrr(w I t). So by redefining what we call the prior we can justify use of conventional uncorrected BP; the (new) MAP ct> corresponds to the w minimizing L(w, t). However such a redefinition changes E(ct» I t) (amongst other things): Idct> P*<I)IT(ct> I t) ct> = Idw P*Wlnw I t) net(w, .) -:/: Idw Pwrr(w I t) net(w, .) = Idct> Pcl>lnct> I t) ct>. So one can either modify BP (by adding in the correction term) and leave E(ct» I t) alone, or leave BP alone but change E(ct» I t); one can not leave both unchanged. Moreover, some procedures involve both prior-based modes and prior-based integrals, and therefore are affected by the correction tenn no matter how Pw(w) is redefined. For example, in the evidence procedure (Wolpert, 1993; MacKay, 1992) one fixes the value of a hyperparameter r (e.g., ex from the introduction) to the value 1 maximizing Pr IT(11 t). Next one find the value s' maximizing PSIT,r (s' I t, 1) for some variable S. Finally, one guesses the <I> associated with s'. Now it's hard to see why one should use this procedure with S = W (as is conventional) rather than with S = ct». But with S = ct» rather than W, one must factor in the correction term when calculating PSIT,r (s I t, 1), and therefore the guessed ct> is different from when S = W. If one tries to avoid this change in the guessed ct> by absorbing the correction tenn into the prior PWIr(w I y), then Pn nY I t) - which is given by an integral involving that prior - changes. This in turn changes 1, and therefore the guessed ct> again is different. So presuming one is more directly interested in ct» rather than W, one can't avoid having the correction term affect the evidence procedure. It should be noted that calculating the correction tenn can be laborious in large nets. One should bear in mind the determinant-evaluation tricks mentioned in (Buntine and Weigend, 1991), as well as others like the identity In[ J<I),w(w) ] = Tr(ln[ ~ / dwj ]) == Tr(ln*[ d$i / dwj n, where In*(.) is In(.) evaluated to several orders. 3 EFFECTS OF THE CORRECTION TERM To illustrate the effects of the correction term, consider a perceptron with a single output neuron, N input neurons and a unary input space: 0 = tanh(w . x), and x always consist of a single one and N - 1 zeroes. For this scenario d<l>i / dwj is an N x N diagonal matrix, and In[ J<I),w(w)] = -2 ~~=l In[ COSh(Wk)]. Assume the Gaussian prior and likelihood of the introduction, and for simplicity take 2cr2 = 1. Both L(w, t) and L'(w, t) are sums of terms each of which only concerns one weight and the corresponding input neuron. Accordingly, it suffices to consider just the i' th weight and the corresponding input neuron. Let xCi) be the input vector which has its 1 in neuron i. Let ojCi) be the output of the j'th of the pairs in the training set with input x(i), and mi the number of such pairs. With ex = 0 204 Wolpert (no weight decay), L(w, t) = X2(1, w), which is minimized by W'i = tanh-l [ l:j~l oj(i) / mil. If we instead try to minimize X2(t, w) + Jw.MW) though, then for low enough mi (e.g., mi . = 1), we find that there is no minimum. The correction term pushes waway from 0, and for low enough mi the likelihood isn't strong enough to counteract this push. -o o -, -, o -, o Figures 1 through 3: Train using unmodified BP on training set 1, and feed input x into the resultant net. The horizontal axis gives the output you get If t and x were still used but training had been with modified BP, the output would have been the value on the vertical axis. In succession, the three figures have a = .6, .4, .4, and m = 1,4, 1. I _ .. _ ..... ...................................... o I ·1 _ ...... _ .... ......... ............... . ... I I I -, o Figure 3. ao o Figure 4: The horizontal axis is IWil. The top curve depicts the weight decay regularizer, aw?, and the bottom curve shows that regularizer modified by the correction term. a = .2. When weight-decay is used though, modified BP finds a solution, just like unmodified BP does. Since the correction term "pushes out" w, and since tanh(.) grows with its argument, a <I> found by modified BP has larger (in magnitude) values of 0 than does the corresponding <I> found by unmodified BP. In addition, unlike unmodified BP, modified BP has multiple extrema over certain regimes. All of this is illustrated in figures (1) through (3), which graph the value of 0 resulting from using modified BP with a particular training set t and input value x vs. the value of 0 resulting from using unmodified BP with t and x. Figure (4) depicts the wi-dependences of the weight decay term and of the weight-decay term plus the correction term. (When there's no data, BP searches for minima of those curves.) Now consider multi-layer nets, possibly with non-unary X. Denote a vector of the compoBayesian Backpropagation over 1-0 Functions Rather Than Weights 20S nents of w which lead from the input layer into hidden neuron K by w[K)' Let x· be the input vector consisting of all O's. Then a tanh(W[K) . x') I aWj = 0 for any j, w, and K, and for any w, there is a row of ~I awj which is all zeroes. This in tum means that Jw,w(w) = o for any w, which means that Winj is empty, and PWIT(' I t) is independent of the data t. (Intuitively, this problem arises since the 0 corresponding to x· can't vary with w, and therefore the dimension of ct> is less than IWI) So we must forbid such an all-zeroes x'. The easiest way to do this is to require that one input neuron always be on, i.e., introduce a bias unit. An alternative is to redefine ct> to be the functions from the set {X - (0, 0, ... , O)} to 0 rather than from the set X to O. Another alternative, appropriate when the original X is the set of all input neuron vectors consisting of O's and 1 's, is to instead have input neuron values E {z * 0, I}. (In general z * -1 though; due to the symmetries of the tanh, for many architectures z = -1 means that two rows of a<l>i I awj are identical up to an overall sign, which means that Jw,w(w) = 0.) This is the solution implicitly assumed from now on. Jw,w(w) will be small - and therefore Pw(net(w, .» will be large - whenever one can make large changes to w without affecting, = net(w, .) much. In other words, pw(net(w, .» will be large whenever we don't need to specify w very accurately. So the correction factor favors those w which can be expressed with few bits. In other words, the correction factor enforces a sort of automatic MDL (Rissanen, 1986; Nowlan and Hinton, 1994). More generally, for any multi-layer architecture there are many "singular weights" w sin ~ W inj such that Jw.w(w sin) is not just small but equals zero exactly. Pw(w) must compensate for these singularities, or the peaks of PCI)rr<, I t) won't depend on t. So we need to have pw(w) ~ 0 as w ~ wsin' Sometimes this happens automatically. For example often Wsin includes infinite-valued w's, since tanh'(oo) = O. Because Pw(oo) = 0 for the weightdecay prior, that prior compensates for the infmite-w singularities in the correction term. For other w sin there is no such automatic compensation, and we have to explicitly modify pwCw) to avoid singularities. In doing so though it seems reasonable to maintain a "bias" towards the wsin, that Pw(w) goes to zero slowly enough so that the values pw(net(w, .» are "enhanced" for w near wsin' Although a full characterization of such enhanced w is not in hand, it's easy to see that they include certain kinds of pruned nets (Hassibi and Stork, 1992), weight-shared nets (Nowlan and Hinton, 1994), and feature-selected nets. To see that (some kinds of) pruned nets have singular weights, let w* be a weight vector with a zero-valued weight coming out of hidden neuron K. By (1) Pw(net(w*, .» = Jdw' Pw(w,) S(net(w', .) - net(w*, .». Since we can vary the value of each weight w*i leading into neuron K without affecting net(w*, .), the integral diverges. So w* is singUlar; removing a hidden neuron results in an enhanced probability. This constitutes an a priori argument in favor of trying to remove hidden neurons during training. This argument does not apply to weights leading into a hidden neuron; Jw,w(w) treats weights in different layers differently. This fact suggests that however pw(w) compensates for the singularities in Jw.w(w), weights in different layers should be treated differently by Pw(w). This is in accord with the advice given in (MacKay, 1992). To see that some kinds of weight-shared nets have singular weights, let w' be a weight vector such that for any two hidden neurons K and K' the weight from input neuron i to K equals the weight from i to K', for all input neurons i. In other words, w is such that all hid206 Wolpert den neurons compute identical functions of x. (For some architectures we'll actually only need a single pair of hidden neurons to be identical.) Usually for such a situation there is a pair of columns of the matrix ~ / awj which are exactly proportional to one another. (For example, in a 3-2-1 architecture, with X = {z, I} 3, IWI = IXI x 101 = 8, and there are four such pairs of columns.) This means that JfI>,W(w') = 0; w' has an enhanced probability, and we have an a priori argument in favor of trying to equate hidden neurons during training. The argument that feature-selected nets have singular weights is architecture-dependent, and there might be reasonable architectures for which it fails. To illustrate the argument, consider the 3-2-1 architecture. Let xl(k) and x2(k) with k = {I, 2,3) designate three distinct pairs of input vectors. For each k have xl (k) and x2(k) be identical for all input neurons except neuron A, for which they differ. (Note there are four pairs of input vectors with this property, one for each of the four possible patterns over input neurons B and C.) Let w' be a weight vector such that both weights leaving A equal zero. For this situation net(w', xl(k» = net(w', x2(k» for all k. In addition a net(w, xl(k» / awj = a net(w, x2(k» / awj for all weights Wj except the two which lead out of A. So k = 1 gives us a pair of rows of the matrix a~ / awj which are identical in all but two entries (one row for Xl (k) and one for x2(k». We get another such pair of rows, differing from each other in the exact same two entries, for k = 2, and yet another pair for k = 3. So there is a linear combination of these six rows which is all zeroes. This means that JfI>, w(w') = O. This constitutes an a priori argument in favor of trying to remove input neurons during training. Since it doesn't favor any Pw(w), the analysis of this paper doesn't favor any pfl>( <1». However when combined with empirical knowledge it suggests certain pfl>(cj). For example, there are functions g(w) which empirically are known to be good choices for pfl>(net(w, .» (e.g., g(w) oc:exp[awl]). There are usually problems with such choices of Pfl>(cj) though. For example, these g(w) usually make more sense as a prior over W than as a prior over <1>, which would imply pfl>(net(w, .» = g(w) / J<I>W(w). Moreover it's empirically true that , enhanced w should be favored over other w, as advised by the correction term. So it makes sense to choose a compromise between g(w) and g(w) / J<I>W(w). An example is pfl>(cj) oc: , g(w) / [A} + tanh(~ x JfI>,w(w»] for two hyperparameters A} > 0 and ~ > O. 4 BEYOND THE CASE OF BACKPROP WITH IWI = IXIIOI When 0 does not approximate a Euclidean vector space, elements of <1> have probabilities rather than probability densities, and P(cj) It) = jdw PWl'r(w I t) S(net(w, .), cj), (0(., .) being a Kronecker delta function). Moreover, if 0 is a Euclidean vector space but WI > IXI 101, then again one must evaluate a difficult integral; <1> = net(W, .) is not one-to-one so one must use equation (1) rather than (2). Fortunately these two situations are relatively rare. The final case to consider is IWI < IXIIOI (see section two). Let Sew) be the surface in <1> which is the image (under net(W, .» ofW. For all <I> PfI>(cj) is either zero (when cj) Ii': S(W» or infinite (when cj) E S(W». So as conventionally defined, "MAP cj)" is not meaningful. One way to deal with this case is to embed the net in a larger net, where that larger net's output is relatively insensitive to the values of the newly added weights. An alternative that is applicable when IWI / 101 is an integer is to reduce X by removing "uninteresting" x's. A third alternative is to consider surface densities over Sew), Ps(W)(cj), instead of volBayesian Backpropagation over 1-0 Functions Rather Than Weights 207 ume densities over <%». P«l>(e!»). Such surface densities are given by equation (2). if one uses the metric form of J«l>,w(w). (Buntine has emphasized that the Jacobian form is not even defined for IWI < IXIIOI. since ()cj)i / aWj is not square then (personal communication).) As an aside, note that restricting P«l>(e!») to Sew) is an example of the common theoretical assumption that "target functions" come from a pre-chosen "concept class". In practice such an assumption is usually ludicrous - whenever it is made there is an implicit hope that it constitutes a valid approximation to a more reasonable P«l>(e!»). When decision theory is incorporated into Bayesian analysis. only rarely does it advise us to evaluate an MAP quantity (Le.. use BP). Instead Bayesian decision theory usually advises us to evaluate quantities like E(<%» I t) (Wolpert. 1994). Just as it does for the use of MAP estimators. the analysis of this paper has implications for the use of such E(<%» I t) estimators. In particular. one way to evaluate E(<%»I t) = jdw PwIT(w I t) net(w •. ) is to expand net(w •. ) to low order and then approximate PWlnw I t) as a sum of Gaussians (Buntine and Weigend. 1991). Equation (4) suggests that instead we write E(<%» I t) as jde!» P«l>lne!» I t) e!» and approximate P«l>IT(e!» I t) as a sum of Gaussians. Since fewer approximations are used (no low order expansion of net(w •. », this might be more accurate. Acknowledgements Thanks to David Rosen and Wray Buntine for stimulating discussion. and to TXN and the SF! for funding. This paper is a condensed version of (Wolpert 1994). References Buntine. W .• Weigend. A. (1991). Bayesian back-propagation. Complex Systems. S.p. 603. Denker. J., LeCun, Y. (1991). Transforming neural-net output levels to probability distributions. In Neural Information Processing Systems 3, R. Lippman et al. (Eds). Fefferman, C. (1993). Reconstructing a neural net from its output. Sarnoff Research Center TR 93-01. Hassibi. B., and Stork, D. (1992). Second order derivatives for network pruning: optimal brain surgeon. Ricoh Tech Report CRC-TR-9214. MacKay, D. (1992). Bayesian Interpolation, and A Practical Framework for Backpropagation Networks. Neural Computation. 4. pp. 415 and 448. Neal, R. (1993). Bayesian learning via stochastic dynamics. In Neural Information Processing Systems 5, S. Hanson et al. (Eds). Morgan Kaufmann. Nowlan, S., and Hinton. G. (1994). Simplifying Neural Networks by Soft Weight-Sharing. In Theories of Induction: Proceedings of the SFIICNLS Workshop on Formal Approaches to Supervised Learning, D. Wolpert (Ed.). Addison-Wesley, to appear. Rissanen, J. (1986). Stochastic complexity and modeling. Ann. Stat .• 14, p. 1080. Wolpert, D. (1993). On the use of evidence in neural networks. In Neural I nformation Processing Systems 5, S. Hanson et aI. (Eds). Morgan-Kauffman. Wolpert, D. (1994). Bayesian back-propagation over i-o functions rather than weights. SF! tech. report. ftp'ablefrom archive.cis.ohio-state.edu, as pub/neuroprose/wolpert.nips.93.Z.
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Fast Non-Linear Dimension Reduction Nanda Kambhatla and Todd K. Leen Department of Computer Science and Engineering Oregon Graduate Institute of Science & Technology P.O. Box 91000 Portland, OR 97291-1000 Abstract We present a fast algorithm for non-linear dimension reduction. The algorithm builds a local linear model of the data by merging PCA with clustering based on a new distortion measure. Experiments with speech and image data indicate that the local linear algorithm produces encodings with lower distortion than those built by five layer auto-associative networks. The local linear algorithm is also more than an order of magnitude faster to train. 1 Introduction Feature sets can be more compact than the data they represent. Dimension reduction provides compact representations for storage, transmission, and classification. Dimension reduction algorithms operate by identifying and eliminating statistical redundancies in the data. The optimal linear technique for dimension reduction is principal component analysis (PCA). PCA performs dimension reduction by projecting the original ndimensional data onto the m < n dimensional linear subspace spanned by the leading eigenvectors of the data's covariance matrix. Thus PCA builds a global linear model of the data (an m dimensional hyperplane). Since PCA is sensitive only to correlations, it fails to detect higher-order statistical redundancies. One expects non-linear techniques to provide better performance; i.e. more compact representations with lower distortion. This paper introduces a local linear technique for non-linear dimension reduction. We demonstrate its superiority to a recently proposed global non-linear technique, 152 Fast Non-Linear Dimension Reduction 153 and show that both non-linear algorithms provide better performance than PCA for speech and image data. 2 Global Non-Linear Dimension Reduction Several researchers (e.g. Cottrell and Metcalfe 1991) have used layered feedforward auto-associative networks with a bottle-neck middle layer to perform dimension reduction. It is well known that auto-associative nets with a single hidden layer cannot provide lower distortion than PCA (Bourlard and Kamp, 1988). Recent work (e.g. Oja 1991) shows that five layer auto-associative networks can improve on PCA. These networks have three hidden layers (see Figure l(a)). The first and third hidden layers have non-linear response, and are referred to as the mapping layers. The m < n nodes of the middle or representation layer provide the encoded signal. The first two layers of weights produce a projection from Rn to Rm. The last two layers of weights produce an immersion from R minto R n. If these two maps are well chosen, then the complete mapping from input to output will approximate the identity for the training data. If the data requires the projection and immersion to be non-linear to achieve a good fit, then the network can in principal find such functions. J (a) x ----,. Low Dimensional Encoding Original High «--- Dimensional Representation (b) 1 Figure 1: (a) A five layer feedforward auto-associative network. This network can perform a non-linear dimension reduction from n to m dimensions. (b) Global curvilinear coordinates built by a five layer network for data distributed on the surface of a hemisphere. When the activations of the representation layer are swept, the outputs trace out the curvilinear coordinates shown by the solid lines. The activities of the nodes in the representation layer form global curvilinear coordinates on a submanifold of the input space (see Figure l(b)). We thus refer to five layer auto-associative networks as a global, nonlinear dimension reduction technique. 154 Kambhatla and Leen 3 Locally Linear Dimension Reduction Five layer networks have drawbacks; they can be very slow to train and they are prone to becoming trapped in poor local optima. Furthermore, it may not be possible to accurately fit global, low dimensional, curvilinear coordinates to the data. We propose an alternative that does not suffer from these problems. Our algorithm pieces together local linear coordinate patches. The local regions are defined by the partition of the input space induced by a vector quantizer (VQ). The orientation of the local coordinates is determined by PCA (see Figure 2). In this section, we present two ways to obtain the partition. First we describe an approach that uses Euclidean distance, then we describe a new distortion measure which is optimal for our task (local PCA). -1 -.5 o .;.;.54r-----__ ~ .5 r==---~~~~........25 o -.25 1 Figure 2: Local coordinates built by our algorithm (dubbed VQPCA) for data distributed on the surface of a hemisphere. The solid lines represent the two principal eigen-directions in each Voronoi cell. The region covered by one Voronoi cell is shown shaded. 3.1 Euclidean partitioning Here, we do a clustering (with Euclidean distance) followed by PCA in each of the local regions. The hybrid algorithm, dubbed VQPCA, proceeds in three steps: 1. Using competitive learning, train a VQ (with Euclidean distance) with Q reference vectors (weights) (rl' r2, ... ,rQ). 2. Perform a local PCA within each Voronoi cell of the VQ. For each cell, compute the local covariance matrix for the data with respect to the corresponding reference vector (centroid) rc. Next compute the eigenvectors (e1, ... ,e~) of each covariance matrix. 3. Choose a target dimension m and project each data vector x onto the leading m eigenvectors to obtain the local linear coordinates z = (e1 . (x - rc), ... , e~ . (x - rc)). Fast Non-Linear Dimension Reduction 155 The encoding of x consists of the index c of the reference cell closest (Euclidean distance) to x, together with the m < n component vector z. The decoding is given by (1) i=l where r c is the reference vector (centroid) for the cell c, and ei are the leading eigenvectors of the covariance matrix of the cell c. The mean squared reconstruction error incurred by VQPCA is m (2) i=l where E[·] denotes an expectation with respect to x, and x is defined in (1). Training the VQ and performing the local PCA are very fast relative to training a five layer network. The training time is dominated by the distance computations for the competitive learning. This computation can be speeded up significantly by using a multi-stage architecture for the VQ (Gray 1984). 3.2 Projection partitioning The VQPCA algorithm as described above is not optimal because the clustering is done independently of the PCA projection. The goal is to minimize the expected error in reconstruction (2). We can realize this by using the expected reconstruction error as the distortion measure for the design of the VQ. The reconstruction error for VQPCA (Erecon defined in (2)) can be written in matrix form as Erecon = E[ (x - ref P; Pc(X - rc)] , (3) where Pc is an m x n matrix whose rows are the orthonormal trailing eigenvectors of the covariance matrix for the cell c. This is the mean squared Euclidean distance between the data and the local hyperplane. The expression for the VQPCA error in (2) suggests the distortion measure d(x, rc) = (x - rc)T P; Pc(x - rc) . (4) We call this the reconstruction distance. The reconstruction distance is the error incurred in approximating x using only m local PCA coefficients. It is the squared projection of the difference vector x - r c on the trailing eigenvectors of the covariance matrix for the cell c. Clustering with respect to the reconstruction distance directly minimizes the expected reconstruction error Erecon. The modified VQPCA algorithm is: 1. Partition the input space using a VQ with the reconstruction distance measure 1 in (4). 2. Perform a local PCA (same as in steps 2 and 3 of the algorithm as described in section 3.1). IThe VQ is trained using the (batch mode) generalized Lloyd's algorithm (Gersho and Gray, 1992) rather than an on-line competitive learning. This avoids recomputing the matrix Pc (which depends on Tc) for each input vector. 156 Kambhatla and Leen 4 Experimental Results We apply PCA, five layer networks (5LNs), and VQPCA to dimension reduction of speech and images. We compare the algorithms using two performance criteria: training time and the distortion in the reconstructed signal. The distortion measure is the normalized reconstruction error: £norm 4.1 Model Construction £recon E[ IIx1l 2 ] E[llx-xI12] E [ IIxll2 ] The 5LNs were trained using three optimization techniques: conjugate gradient descent (CGD), the BFGS algorithm (a quasi-Newton method (Press et al1987)), and stochastic gradient descent (SGD). In order to limit the space of architectures, the 5LNs have the same number of nodes in both of the mapping (second and fourth) layers. For the VQPCA with Euclidean distance, clustering was implemented using standard VQ (VQPCA-Eucl) and multistage quantization (VQPCA-MS-E). The multistage architecture reduces the number of distance calculations and hence the training time for VQPCA (Gray 1984). 4.2 Dimension Reduction of Speech We used examples of the twelve monothongal vowels extracted from continuous speech drawn from the TIMIT database (Fisher and Doddington 1986). Each input vector consists of 32 DFT coefficients (spanning the frequency range 0-4kHz), timeaveraged over the central third of the utterance. We divided the data set into a training set containing 1200 vectors, a validation set containing 408 vectors and a test set containing 408 vectors. The validation set was used for architecture selection (e.g the number of nodes in the mapping layers for the five layer nets). The test set utterances are from speakers not represented in the training set or the validation set. Motivated by the desire to capture formant structure in the vowel encodings, we reduced the data from 32 to 2 dimensions. (Experiments on reduction to 3 dimensions gave similar results to those reported here (Kambhatla and Leen 1993).) Table 1 gives the test set reconstruction errors and the training times. The VQPCA encodings have significantly lower reconstruction error than the global PCA or five layer nets. The best 5LNs have slightly lower reconstruction error than PC A, but are very slow to train. Using the multistage search, VQPCA trains more than two orders of magnitude faster than the best 5LN, and achieves an error about 0.7 times as great. The modified VQPCA algorithm (with the reconstruction distance measure used for clustering) provides the least reconstruction error among all the architectures tried. Fast Non-Linear Dimension Reduction 157 Table 1: Speech data test set reconstruction errors and training times. Architectures represented here are from experiments with the lowest validation set error over the parameter ranges explored. The numbers in the parentheses are the values of the free parameters for the algorithm represented (e.g 5LN-CGD (5) indicates a network with 5 nodes in both the mapping (2nd and 4th) layers, while VQPCA-Eucl (50) indicates a clustering into 50 Voronoi cells). ALGORITHM PCA 5LN-CGD (5) 5LN-BFGS (30) 5LN-SGD (25) VQPCA-Eucl (50) VQPCA-MS-E (9x9) VQPCA-Recon (45) i norm 0.0060 0.0069 0.0057 0.0055 0.0037 0.0036 0.0031 TRAINING TIME (in seconds) 11 956 28,391 94,903 1,454 142 931 Table 2: Reconstruction errors and training times for a 50 to 5 dimension reduction of images. Architectures represented here are from experiments with the lowest validation set error over the parameter ranges explored. ALGORITHM PCA 5LN-CGD (40) 5LN-BFGS (20) 5LN-SGD (25) VQPCA-Eucl (20) VQPCA-MS-E (8x8) VQPCA-Recon (25) i norm 0.458 0.298 0.052 0.350 0.140 0.176 0.099 4.3 Dimension Reduction of Images TRAINING TIME (in seconds) 5 3,141 10,389 15,486 163 118 108 The data consists of 160 images of the faces of 20 people. Each is a 64x64, 8-bit/pixel grayscale image. We extracted the first 50 principal components of each image and use these as our experimental data. This is the same data and preparation that DeMers and Cottrell used in their study of dimension reduction with five layer auto-associative nets (DeMers and Cottrell 1993). They trained auto-associators to reduce the 50 principal components to 5 dimensions. We divided the data into a training set containing 120 images, a validation set (for architecture selection) containing 20 images and a test set containing 20 images. We reduced the images to 5 dimensions using PCA, 5LNs2 and VQPCA. Table 2 2We used 5LNs with a configuration of 50-n-5-n-50, n varying from 10 to 40 in increments of 5. The BFGS algorithm posed prohibitive memory and time requirements for n > 20 for this task. 158 Kambhatla and Leen Table 3: Reconstruction errors and training times for a SO to S dimension reduction of images (training with all the data). Architectures represented here are from experiments with the lowest error over the parameter ranges explored. ALGORITHM [norm TRAINING TIME (in seconds) PCA 0.40S4 7 SLN-SGD (30) 0.1034 2S,306 SLN-SGD (40) 0.0729 31,980 VQPCA-Eucl (SO) 0.0009 90S VQPCA-Recon (SO) 0.0017 216 summarizes the results. We notice that a five layer net obtains the encoding with the least error for this data, but it takes a long time to train. Presumably more training data would improve the best VQPCA results. ~.- . '~'-··f-···: .,.. -.•. ~ .. >':~ .. ~.:' . ... _:.I T ~1-' ,., . ~ ' .. '(;' .... '-".'.'~ ..~ Figure 3: Two representative images: Left to right - Original SO-PC image, reconstruction from S-D encodings: PCA, SLN-SGD(40), VQPCA(lO), and VQPCA(SO). For comparison with DeMers and Cottrell's (DeMers and Cottrell 1993) work, we also conducted experiments training with all the data. The results are summarized3 in Table 3 and Figure 3 shows two sample faces. Both non-linear techniques produce encodings with lower error than PCA, indicating significant non-linear structure in the data. With the same data, and with a SLN with 30 nodes in each mapping layer, DeMers (DeMers and Cottrell 1993) obtains a reconstruction error [norm 0.13174 . We note that the VQPCA algorithms achieve an order of magnitude improvement over five layer nets both in terms of speed of training and the accuracy of encodings. 3For 5LNs, we only show results with SGD in order to compare with the experimental results of DeMers. For this data, 5LN-CGD gave encodings with a higher error and 5LNBFGS posed prohibitive memory and computational requirements. 4DeMers reports half the MSE per output node, E = (1/2) * (1/50) * MSE = 0.00l. This corresponds to [norm = 0.1317 Fast Non-Linear Dimension Reduction 159 5 Summary We have presented a local linear algorithm for dimension reduction. We propose a new distance measure which is optimal for the task of local PCA. Our results with speech and image data indicate that the nonlinear techniques provide more accurate encodings than PCA. Our local linear algorithm produces more accurate encodings (except for one simulation with image data), and trains much faster than five layer auto-associative networks. Acknowledgments This work was supported by grants from the Air Force Office of Scientific Research (F49620-93-1-0253) and Electric Power Research Institute (RP8015-2). The authors are grateful to Gary Cottrell and David DeMers for providing their image database and clarifying their experimental results. We also thank our colleagues in the Center for Spoken Language Understanding at OGI for providing speech data. References H. Bourlard and Y. Kamp. (1988) Auto-association by multilayer perceptrons and singular value decomposition. Biological Cybernetics, 59:291-294. G. Cottrell and J. Metcalfe. (1991) EMPATH: Face, emotion, and gender recognition using holons. In R. Lippmann, John Moody and D. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 564-571. Morgan Kauffmann. D. DeMers and G. Cottrell. (1993) Non-linear dimensionality reduction. In Giles, Hanson, and Cowan, editors, Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann. W. M. Fisher and G. R. Doddington. (1986) The DARPA speech recognition research database: specification and status. In Proceedings of the DARPA Speech Recognition Workshop, pages 93-99, Palo Alto, CA. A. Gersho and R. M. Gray. (1992) Vector Quantization and Signal Compression. Kluwer academic publishers. R. M. Gray. (1984) Vector quantization. IEEE ASSP Magazine, pages 4-29. N. Kambhatla and T. K. Leen. (1993) Fast non-linear dimension reduction. In IEEE International Conference on Neural Networks, Vol. 3, pages 1213-1218. IEEE. E. Oja. (1991) Data compression, feature extraction, and autoassociation in feedforward neural networks. In Artificial Neural Networks, pages 737-745. Elsevier Science Publishers B. V. (N orth-Holland) . 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/New York.
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Observability of Neural Network Behavior Max Garzon Fernanda Botelho sarzonmOherme •. msci.mem.t.edu botelhoflherme •. msci.mem.t.edu Institute for Intelligent Systems Department of Mathematical Sciences Memphis State University Memphis, TN 38152 U.S.A. Abstract We prove that except possibly for small exceptional sets, discretetime analog neural nets are globally observable, i.e. all their corrupted pseudo-orbits on computer simulations actually reflect the true dynamical behavior of the network. Locally finite discrete (boolean) neural networks are observable without exception. 1 INTRODUCTION We address some aspects of the general problem of implementation and robustness of (mainly recurrent) autonomous discrete-time neural networks with continuous activation (herein referred to as analog networks) and discrete activation (herein, boolean networks). There are three main sources of perturbations from ideal operation in a neural network. First, the network's parameters may have been contaminated with noise from external sources. Second, the network is being implemented in optics or electronics (digital or analog) and inherent measurement limitations preclude the use of perfect information on the network's parameters. Third, as has been the most common practice so far, neural networks are simulated or implemented on a digital device, with the consequent limitations on precision to which net parameters can be represented. Finally, for these or other reasons, the activation functions (e.g. sigmoids) of the network are not known precisely or cannot be evaluated properly. Although perhaps negligible in a single iteration, these perturbations are likely to accumulate under iteration, even in feedforward nets. Eventually, they may, in fact, distort the results of the implementation to the point of making the simulation useless, if not misleading. 455 456 Garzon and Botelho There is, therefore, an important difference between the intended operation of an idealized neural network and its observable behavior. This is a classical problem in systems theory and it has been addressed in several ways. First, the classical notions of distinguishability and observability in control theory (Sontag, 1990) which roughly state that every pair of system's states are distinguishable by different outputs over evolution in finite time. This is thus a notion of local state observability. More recently, several results have established more global notions of identifiability of discrete-time feedfoward (Sussmann, 1992; Chen, Lu, Hecht-Nelson, 1993) and continuous-time recurrent neural nets (Albertini and Sontag, 1993a,b), which roughly state that for given odd activation functions (such as tanh), the weights of a neural network are essentially uniquely determined (up to permutation and cell redundancies) by the input/output behavior of the network. These notions do assume error-free inputs, weights, and activation functions. In general, a computer simulation of an orbit of a given dynamical system in the continuuum (known as a pseudo-orbit) is, in fact, far from the orbit in the ideal system. Motivated by this problem, Birkhoff introduced the so-called shadowing property. A system satisfies the shadowing propertyif all pseudo-orbits are uniformly approximated by actual orbits so that the former capture the long-term behavior of the system. Bowen showed that sufficiently hyperbolic systems in real euclidean spaces do have the shadowing property (Bowen, 1978). However, it appears difficult even to give a characterization of exactly which maps on the interval possess the property -see e.g. (Coven, Kan, Yorke, 1988). Precise definitions of all terms can be found in section 2. By comparison to state observability and identifiability, the shadowing property is a type of global observability of a system through its dynamical behavior. Since autonomous recurrent networks can be seen as dynamical systems, it is natural to investigate this property. Thus, a neural net is observable in the sense that its behavior (i.e. the sequence of its ideal actions on given initial conditions) can be observed on computer simulations or discrete implementations, despite inevitable concomitant approximations and errors. The purpose of this paper is to explore this property as a deterministic model for perturbations of neural network behavior in the presence of arbitrary small errors from various sources. The model includes both discrete and analog networks. In section 4 we sketch a proof that locally finite boolean neural networks (even with an infinite number of neurons)' are all observable in this sense. This is not true in general for analog networks, and section 3 is devoted to sketching necessary and sufficient conditions for the relatively few analog exceptions for the most common transfer functions: hard-thresholds, a variety of sigmoids (hyperbolic tangent, logistic, etc.) and saturated linear maps. Finally, section 5 discusses the results and poses some other problems worthy of further research. 2 DEFINITIONS AND MAIN RESULTS This section contains notation and precise definitions in a general setting, so as to include discrete-time networks both with discrete and continuous activations. Let f : X ~ X be a continuous map of a compact metric space with metric 1 *, * I. Observability of Neural Network Behavior 457 The orbit of x E X is the sequence {x, f(x), ... , fk(x) ... }, i.e. a sequence of points {xkh~o for which Xk+1 = f(xk ), for all k ~ o. Given a number 6 > 0, a 6-pseudoorbit is a sequence {xk} so that the distances If(xk), xk+11 < 6 for all k ~ o. Pseudo-orbits arise as trajectories of ideal dynamical processes contaminated by errors and noise. In such cases, especially when errors propagate exponentially, it is important to know when the numerical process is actually representing some meaningful trajectory of the real process. Definition 2.1 The map f on a metric space X is (globally) observable (equivalently] has the shadowing property] or is traceable) if and only if for every f > 0 there exists a 6 > 0 so that any 6 -pseudo-orbit {xk} is f-approximated by the orbit] under f] of some point z E X] i.e. Ixk, fk(z) I < f for all k > o. One might observe that computer simulations only run for finite time. On compact spaces (as is the case below)' observability can be shown to be equivalent to a similar property of shadowing finite pseudo-orbits. 'Analog neural network' here means a finite number n of units (or cells), each of which is characterized by an activation (sometimes called output) function Ui : R -+ R, and weight matrix W of synaptic strengths between the various units. Units can assume real-valued activations Xi, which are updated synchronously and simultaneously at discrete instants of time, according to the equation Xi(t + 1) udL wi,ixi(t)]. (1) i The total activation of the network at any time is hence given by a vector x in euclidean space Rn, and the entire network is characterized by a global dynamics T(x) u[W x], (2) where W x denotes ordinary product and u is the map acting as Ui along the ith component. This component in a vector x is denoted Xi (as opposed to xk, the kth term of a sequence). The unit hypercube in Rn is denoted In. An analog network is then defined as a dynamical system in a finite-dimensional euclidean space and one may then call a neural network (globally) observable if its global dynamics is an observable dynamical system. Likewise for boolean networks, which will be defined precisely in section 4. We end this section with some background facts about observability on the continuum. It is perhaps surprising but a trivial remark that the identity map of the real interval is not observable in this sense, since orbits remain fixed, but pseudo-orbits may drift away from the original state and can, in fact, be dense in the interval. Likewise, common activation functions of neural networks (such as hard thresholds and logistic maps) are not observable. For linear maps, observability has long been known to be equivalent to hyperbolicity (all eigenvalues>. have 1>'1 =f:. 1). Composition of observable maps is usually not observable (take, for instance, a hyperbolic homeomorphism and its inverse). In contrast, composition of linear and nonobservable activation functions in neural networks are, nevertheless, observable. The main take-home message can be loosely summarized as follows. Theorem 2.1 Except for a negligible fraction of exceptions, discrete-time analog neural nets are observable. All discrete (boolean) neural networks are observable. 458 Garzon and Botelho 3 ANALOG NEURAL NETWORKS This section contains (a sketch) of necessary and sufficient conditions for analog networks to be observable for common types of activations functions. 3.1 HARD-THRESHOLD ACTIVATION FUNCTIONS It is not hard to give necessary and sufficient conditions for observability of nets with discrete activation functions of the type .- { ~ where Oi is a theshold characterizing cell i. if 1.£ ~ Oi else. Lemma 3.1 A map 1 : Rn -+ Rn with finite range is observable il and only il it is continuous at each point of its range. PROOF. The condition is clearly sufficient. If 1 is continuous at every point of its range, small enough perturbations Xk +1 of an image I(xk ) have the same image I(xk+ l ) = l(f(xk)) and hence, for 8 small enough, every 8-pseudo-orbit is traced by the first element of the pseudo-orbit. Conversely, assume 1 is not continuous at a poin t of its range 1 (XO). Let xl, x2, ... be a sequence constant under 1 whose image does not converge to 1(I(xO)) (such a sequence can always be so chosen because the range is discrete). Let c= ~ min I/(x), f(y)l. 2.z,yER ... For a given 8 > 0 the pseudo-orbit xo, xk, f(xk), 12(xk), ... is not traceable for k large enough. Indeed, for any z within €-distance of xO, either f(z) =I f(xO), in which case this distance is at least €, or else they coincide, in which case 1/2(z), l(xk)1 > € anyway by the choice of xk. 0 Now we can apply Lemma 3.1 to obtain the following characterization. Theorem 3.1 A discrete-time neural net T with weight matrix W := (Wij) and threshold vector 0 is observable if and only ~f for every y in the range 01 T, (W Y)i =I OJ for every i (1 ::; i ::; n). 3.2 SIGMOIDAL ACTIVATION FUNCTIONS In this section, we establish the observability of arbitrary neural nets with a fairly general type of sigmoidal activation functions, as defined next. Definition 3.1 A map (j : R -+ R is sigmoidal if it is strictly increasing, bounded (above and below), and continuously differentiable. Important examples are the logistic map 1 a·(1.£) ---:--~ , 1 + exp( -J.L1.£) , Observability of Neural Network Behavior 459 the arctan and the hyperbolic tangent maps adu) = arctan(J.tu) exp(u) - exp(-u) , adu) =tanh(u) = () (). exp u + exp -u Note that, in particular, the range of a sigmoidal map is an open and bounded interval, which without loss of generality, can be assumed to be the unit interval I. Indeed, if a neural net has weight matrix Wand activation function a which is conjugate to an activation function a' by a conjugacy ~, then a 0 W --.J a' ~W ~-1 where --.J denotes conjugacy. One can, moreover, assume that the gain factors in the sigmoid functions are all J.t = 1 (multiply the rows of W). Theorem 3.2 Every neural networks with a sigmoidal activation function has a strong attractor, and in particular, it is observable. PROOF. Let a neural net with n cells have weight matrix Wand sigmoidal a. Consider a parametrized family {T,L}", of nets with sigmoidals given by a", := J.ta. It is easy to see that each T", (J.t > 0) is conjugate to T. However, W needs to be replaced by a suitable conjugation with a homeomorphism ~w By Brouwer's fixed point theorem, T,L has a fixed point p* in In. The key idea in the proof is the fact that the dynamics of the network admits a Lyapunov function given by the distance from p*. Indeed, II T",(x) - T,L(P*) II~ sup I JT", I II x - p* II, y where J denotes jacobian. Using the chain rule and the fact that the derivatives of ~'" and aiL are bounded (say, below by b and above by B), the Jacobian satisfies IJT,L(Y)I ~ J.tn(bB)nIWI, where IW I denotes the determinant of W. Therefore we can choose J.t small enough that the right-hand side of this expression is less than 1 for arbitrary y, so that T", is a contraction. Thus, the orbit of the first element in any €-pseudo-orbit €-traces the orbit. 0 3.3 SATURATED-LINEAR ACTIVATION FUNCTIONS The case of the nondifferentiable saturated linear sigmoid given by the piecewise linear map { 0, u, I, for u < 0 for 0 ~ u ~ 1 for u > 1 (3) presents some difficulties. First, we establish a general necessary condition for observability, which follows easily for linear maps since shadowing is then equivalent to hyperbolicity. Theorem 3.3 If T leaves a segment of positive length pointwise fixed, then T lS not observable. 460 Garzon and Botelho Although easy to see in the case of one-dimensional systems due to the fact that the identity map is not observable, a proof in higher dimensions requires showing that a dense pseudo-orbit in the fixed segment is not traceable by points outside the segment. The proof makes use of an auxiliary result. Lemma 3.2 A linear map L : Rn Rn, acts along the orbit of a point x in the unit hypercube either as an attractor to 0, a repellor to infinity, or else as a rigid rotation or reflection. PROOF. By passing to the complexification L' : en - en of L and then to a conjugate, assume without loss of generality that L has a matrix in Jordan canonical form with blocks either diagonal or diagonal with the first upper minor diagonal of Is. It suffices to show the claim for each block, since the map is a cartesian product of the restrictions to the subspaces corresponding to the blocks. First, consider the diagonal case. If the eigenvalues P,I < 1 (P,I > I, respectively), clearly the orbit Lk(x) _ 0 (II Lk(x) 11- 00). If P,I = I, L acts as a rotation. In the nondiagonal case, it is easy to see that the iterates of x = (XlJ .. " xm ) are given by t t-l Lt(x) .- L (k) ).t-k Xk+1 + L (k) ).t-k Xk+2 + ... + ).txm' (4) k=O k=O The previous argument for the diagonal block still applies for 1).1 =I 1. If 1).1 = 1 and if at least two components of x E In are nonzero, then they are positive and again II L(x) 11- 00. In the remaining case, L acts as a rotation since it reduces to multiplication of a single coordinate of x by).. 0 PROOF OF THEOREM 3.3. Assume that T = u 0 Land T leaves invariant a segment xy positive length. Suppose first that L leaves invariant the same segment as well. By Lemma 3.2, a pseudo-orbit in the interior of the hypercube In cannot be traced by the orbit of a point in the hypercube. If L moves the segment xy invariant under T, we can aSsume without loss of generality it lies entirely on a hyperplane face F of In and the action of u on L(xy) is just a projection over F. But in that case, the action of T on the segment is a (composition of two) linear map(s) and the same argument applies. 0 We point out that, in particular, T may not be observable even if W is hyperbolic. The condition in Theorem 3.3 is, in fact, sufficient. The proof is more involved and is given in detail in (Garzon & Botelho, 1994). WIth Theorem 3.3 one can then determine relatively simple necessary and sufficient conditions for observability (in terms of the eigenvalues and determinants of a finite number of linear maps). They establish Theorem 2.1 for saturated-linear activation functions. 4 BOOLEAN NETWORKS This section contains precise definitions of discrete (boolean) neural networks and a sketch of the proof that they are observable in general. Discrete neural networks have a finite number of activations and their state sets are endowed with an addition and multiplication. The activation function OJ (typically Observability of Neural Network Behavior 461 a threshold function) can be given by an arbitrary boolean table, which may vary from cell to cell. They can, moreover, have an infinite number of cells (the only case of interest here, since finite booolean networks are trivially observable). However, since the activation set if is finite, it only makes sense to consider locally finite networks, for which every cell i only receives input from finitely many others. A total state is now usually called a configuration. A configuration is best thought of as a bi-infinite sequence x := XIX2X3 .•• consisting of the activations of all cells listed in some fixed order. The space of all configurations is a compact metric space if endowed with any of a number of equivalent metrics, such as lx, YI := 2;'" where m = inf{i : Xi =1= Yd. In this metric, a small perturbation of a configuration is obtained by changing the values of x at pixels far away from Xl. The simplest question about observability in a general space concerns the shadowing of the identity function. Observability of the identity happens to be a property characteristic of configuration spaces. Recall that a totally disconnected topological space is one in which the connected component of every element is itself. Theorem 4.1 The idenh'ty map id of a compact metric space X is observable iff X is totally disconnected. The first step in the proof of Theorem 4.3 below is to characterize observability of linear boolean networks (i.e. those obeying the superposition principle). Theorem 4.2 Every linear continuous map has the shadowing property. For the other step we use a global decomposition T = F 0 L of the global dynamics of a discrete network as a continuous transformation of configuration space due to (Garzon & Franklin, 1990). The reader is referred to (Garzon & Botelho, 1992) for a detailed proof of all the results in this section. Theorem 4.3 Every discrete (boolean) neural network is observable. 5 CONCLUSION AND OPEN PROBLEMS It has been shown that the particular combination of a linear map with an activation function is usually globally observable, despite the fact that neither of them is observable and the fact that, ordinarily, composition destroys observability. Intuitively, this means that observing the input/output behavior of a neural network will eventually give away the true nature of the network's behavior, even if the network perturbs its behavior slighlty at each step of its evolution. In simple terms, such a network cannot fool all the people all of the time. The results are valid for virtually every type of autonomous first-order network that evolves in discrete-time, whether the activations are boolean or continuous. Several results follow from this characterization. For example, in all likelihood there exist observable universal neural nets, despite the consequent undecidability of their computational behavior. Also, neural nets are thus a very natural set of primitives for approximation and implementation of more general dynamical systems. These and other consequences will be explored elsewhere (Botelho & Garzon, 1994). 462 Garzon and Botelho Natural questions arise from these results. First, whether observability is a general property of most analog networks evolving in continuous time as well. Second, what other type of combinations of non observable systems of more general types creates observability, i.e. to what extent neural networks are peculiar in this regard. For example, are higher-order neural networks observable? Those with sigma-pi units? Finally, there is the broader question of robustness of neural network implementations, which bring about inevitable errors in input and/or weights. The results in this paper give a deeper explanation for the touted robustness and fault-tolerance of neural network solutions. But, further, they also seem to indicate that it may be possible to require that neural net solutions have observable behavior as well, without a tradeoff in the quality of the solution. An exact formulation of this question is worthy of further research. Acknow ledgements The work of the first author was partially done while on support from NSF grant CCR-9010985 and CNRS-France. References F. Albertini and E.D. Sontag. (1993) Identifiability of discrete-time neural networks. In Proc. European Control Conference, 460-465. Groningen, The Netherlands: Springer-Verlag. F. Albertini and E.D. Sontag. (1993) For neural networks, function determines form. Neural Networks 6(7): 975-990. F. Botelho and M. Garzon. (1992) Boolean Neural Nets are Observable, Memphis State University: Technical Report 92-18. F. Botelho and M. Garzon. (1994) Generalized Shadowing Properties. J. Random and Computat£onal Dynamics, in print. R. Bowen. (1978) On Axiom A diffeomorphisms. In CBMS Regional Conference Ser£es £n Math. 35. Providence, Rhode Island: American Math. Society. A.M. Chen, H. Lu, and R. Hecht-Nielsen, (1993) On the Geometry of Feedforward Neural Network Error Surfaces. Neural Computat£on 5(6): 910-927. E. Coven, 1. Kan, and J. Yorke. (1988) Pseudo-orbit shadowing in the family of tent maps. Trans. AMS 308: 227-241. M. Garzon and S. P. Franklin. (1990) Global dynamics in neural networks II. Complex Systems 4(5): 509-518. M. Garzon and F. Botelho. (1994) Observability of Discrete-time Analog Networks, preprint. E.D. Sontag. (1990) Mathemat~·cal Control Theory: Deterministic Fin£teDimens£onal Dynam£cal Systems. New York: Springer-Verlag. H. Sussmann. (1992) Uniqueness of the Weights for Minimal Feedforward Nets with a Given Input-Output Map. Neural Networks 5(4): 589-593.
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Optimal signalling in Attractor Neural Networks Isaac Meilijson Eytan Ruppin ... School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, 69978 Tel-Aviv, Israel. Abstract In [Meilijson and Ruppin, 1993] we presented a methodological framework describing the two-iteration performance of Hopfieldlike attractor neural networks with history-dependent, Bayesian dynamics. We now extend this analysis in a number of directions: input patterns applied to small subsets of neurons, general connectivity architectures and more efficient use of history. We show that the optimal signal (activation) function has a slanted sigmQidal shape, and provide an intuitive account of activation functions with a non-monotone shape. This function endows the model with some properties characteristic of cortical neurons' firing. 1 Introduction It is well known that a given cortical neuron can respond with a different firing pattern for the same synaptic input, depending on its firing history and on the effects of modulator transmitters (see [Connors and Gutnick, 1990] for a review). The time span of different channel conductances is very broad, and the influence of some ionic currents varies with the history of the membrane potential [Lytton, 1991]. Motivated by the history-dependent nature of neuronal firing, we continue .our previous investigation [Meilijson and Ruppin, 1993] (henceforth, M & R) describing the performance of Hopfield-like attract or neural networks (ANN) [Hopfield, 1982] with history-dependent dynamics. ·Currently in the Dept. of Computer science, University of Maryland 485 486 Meilijson and Ruppin Building upon the findings presented in M & R, we now study a more general framework: • We differentiate between 'input' neurons receiving the initial input signal with high fidelity and 'background' neurons that receive it with low fidelity. • Dynamics now depend on the neuron's history of firing, in addition to its history of input fields. • The dependence of ANN performance on the network architecture can be explicitly expressed. In particular, this enables the investigation of corticallike architectures, where neurons are randomly connected to other neurons, with higher probability of connections formed between spatially proximal neurons [Braitenberg and Schuz, 1991]. Our goal is twofold: first, to search for the computationally most efficient historydependent neuronal signal (firing) function, and study its performance with relation to memoryless dynamics. As we shall show, optimal history-dependent dynamics are indeed much more efficient than memory less ones. Second, to examine the optimal signal function from a biological perspective. As will shall see, it shares some basic properties with the firing of cortical neurons. 2 The model Our framework is an ANN storing m + 1 memory patterns ~1, e, ... ,~m+l, each an N-dimensional vector. The network is composed of N neurons, each of which is randomly connected to K other neurons. The (m + l)N memory entries are independent with equally likely ±1 values. The initial pattern X, signalled by L(~ N) initially active neurons, is a vector of ±l's, randomly generated from one of the memory patterns (say ~ = em+!) such that P(Xi = ~i) = Ii! for each of the L initially active neurons and P(Xi = ~i) = li6 for each initially quiescent (non-active) neuron. Although f,6 E [0,1) are arbitrary, it is useful to think of f as being 0.5 (corresponding to an initial similarity of 75%) and of 6 as being zero - a quiescent neuron has no prior preference for any given sign. Let al = mlnl denote the initial memory load, where nl = LK I N is the average number of signals received by each neuron. The notion of 'iteration' is viewed as an abstraction of the overall dynamics for some length of time, during which some continuous input/output signal function (such as the conventional sigmoidal function) governs the firing rate of the neuron. We follow a Bayesian approach under which the neuron's signalling and activation decisions are based on the a-posteriori probabilities assigned to its two possible true memory states, ±1. Initially, neuron i is assigned a prior probability Ai(O) = P(~ = 11Xi, 1/1» = l~! or 1±2 6 which is conveniently expressed as Ai(O) = 1 (0)' where, letting get) = 1+e-2g j ! log l±1 2 l-t' if i is active if i is silent Optimal Signalling in Attractor Neural Networks 487 The input field observed by neuron i as a result of the initial activity is N loCI) - ~ '" w, . .J.. 1·(1) X ' J 1 LJ IJ IJ ] ] nl j=1 (1) where 1/1) = 0, 1 indicates whether neuron j has fired in the first iteration, lij = 0,1 indicates whether a connection exists from neuron j to neuron i and Wij denotes its magnitude, given by the Hopfield prescription m+l Wij = L: eJJ ieJJ j . (2) JJ=1 As a result of observing the input field fi(I), which is approximately normally distributed (given ei, Xi and 1/1»), neuron i changes its opinion about {ei = I} from Ai(O) to A·(1) - P (e· - llX· J.(1) t .(I») _ 1 1 1 I , 1 ,I 1 + e- 2gi (1) , (3) expressed in terms of the ( additive) generalized field 9i( 1) = gi(O) + :1 f/l). We now get to the second iteration, in which, as in the first iteration, some of the neurons become active and signal to the network. We model the signal function neuron i emits as h(9i(1), Xi, li(l»). The field observed by neuron i (with n2 updating neurons per neuron) is N 1.(2) - 2- "'W, .. I .. h(g.(I) X' / ·(1») JI LJ '] I] ] , ], J , n2 . 1 ]= (4) on the basis of which neuron i computes its posterior belief Ai(2) p(ei = llXi, li(I), f/1), f/2») and expresses its final choice of sign as Xi(2) = sign(A/2) 0.5). The two-iteration performance of the network is measured by the final similarity 1 1 ",N X (2)c Sf = 1 + ff = p(X/2) = ei) = + N L-j=1 j '-oj 2 2 (5) 3 Analytical results The goals of our analysis have been: A. To present an expression for the performance under arbitrary architecture and activity parameters, for general signal functions ho and hi. B. Use this expression to find the best choice of signal functions which maximize performance. We show the following: The neuron's final decision is given by Xi(2) = Sign [(Ao + Boli(1»)Xi + Al fi(l) + A 2 fi(2)] for some constants AD, Bo, At and A2 • (6) 488 Meilijson and Ruppin The performance achieved is where, for some A3 > 0 '" m m a =n'" nl +mA3 ' (Q"'~)(x, t) = 1; t ~ (x + g~») + 1 2 t ~ (x _ g~») and ~ is the standard normal cumulative distribution function. The optimal analog signal function, illustrated in figure 1, is ho = h(g/1),+1,0) = R(gj(l),O) hI = h(gj(1), +1,1) = R(gi(l), €) - 1 where, for some A4 > ° and A5 > 0, R(s, t) = A4 tanh(s) - A5(S - g(t». (b) Silent neurons - - - - Active neurons 2.0 v (7) (8) (9) (10) I , 0.0 Signal -V ~.O , \ \ \ .... 0 L--~---'-~ __ -'---'----'-__ ~~~-----' -5.0 -3.0 -1.0 1.0 3.0 5.0 ItllUt field 1 ---~~--~~--n-~------g 1 • 4 5' Input field . . ----- -1 Figure 1: (a) A typical plot of the slanted sigmoid, Network parameters are N = 5000, K = 3000, nl = 200 and m = 50. (b) A sketch of its discretized version. The nonmonotone form of these functions, illustrated in figure 1, is clear. Neurons that have already signalled +1 in the first iteration have a lesser tendency to send positive signals than quiescent neurons. The signalling of quiescent neurons which receive no prior information (6 = 0) has a symmetric form. The optimal signal is shown to be essentially equal to the sigmoid modified by a correction term depending only on the current input field. In the limit of low memory load (f./ fol ~ 00), the best signal is simply a sigmoidal function of the generalized input field. Optimal Signalling in Attractor Neural Networks 489 To obtain a discretized version of the slanted sigmoid, we let the signal be sign(h(y)) as long as Ih(y)1 is big enough - where h is the slanted sigmoid. The resulting signal, as a function of the generalized field, is (see figure la and lb) y < {3I (j) or {34 (j) < y < {3s (j) y > {36 (j) or {32 (j) < y < (33 (j) otherwise (11) where -00 < (3l(D) < (32(O) ~ (33(O) < (34(O) < (3s(O) < (36(D) < 00 and -00 < (31(l) < (32(l) ~ (3/I) < (34(l) ~ (3S(l) < (36(1) < 00 define, respectively, the firing pattern of the neurons that were silent or active in the first iteration. To find the best such discretized version of the optimal signal, we search numerically for the activity level v which maximizes performance. Every activity level v, used as a threshold on Ih(y) I, defines the (at most) twelve parameters (3/j) (which are identified numerically via the Newton-Raphson method) as illustrated in figure lb. 4 Numerical Results 1.00 0.95 0.90 ,;;-; ;,-:..: ~ ... - ._ ....... _ ........... _-----_ .. _ .... -.. :'~' .:: Posterior-probability-bned signalling -- -- . DI_tlzad slgnalDng .... . Analog optmal signaling 1.000 0.980 I t 0.980 I ! //'--I I / , / I .... -_ ....... ..... .. -.. -..... . Discrete signalling .. ....... Analog signalUng 0.85 '-~---'-_~--"-~_-'--~~_~...J 0.940 '-~---'-_~....l.-~--:-'---~---'-_~...J 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 K K Figure 2: Two-iteration performance as a function of connectivity K. (a) Network parameters are N = 5000, nl = 200, and m = 50. All neurons receive their input state with similar initial overlap f = 6 = 0.5. (b) Network parameters are N = 5000, m = 50, ni = 200, f = 0.5 and 6 = O. Using the formulation presented in the previous section, we investigated numerically the two-iteration performance achieved in several network architectures with optimal analog signalling and its discretization. Already in small scale networks of a few hundred neurons our theoretical calculations correspond fairly accurately with 490 Meilijson and Ruppin simulation results. First we repeat the example of a cortical-like network investigated in M & R, but now with optimal analog and discretized signalling. The nearly identical marked superiority of optimal analog and discretized dynamics over the previous, posterior-probability-based signalling is evident, as shown in figure 2 (a). While low activity is enforced in the first iteration, the number of neurons allowed to become active in the second iteration is not restricted, and best performance is typically achieved when about 70% of the neurons in the network are active (both with optimal signalling and with the previous, heuristic signalling). Figure 2 (b) displays the performance achieved in the same network, when the input signal is applied only to the small fraction (4%) of neurons which are active in the first iteration (expressing possible limited resources of input information). We see that (for 1< > 1000) near perfect final similarity is achieved even when the 96% initially quiescent neurons get no initial clue as to their true memory state, if no restrictions are placed on the second iteration activity level. Next we have fixed the value of w = fit = 1, and contrasted the case (nl = 200, f = 0.5) of figure 2 (b) with (nl = 50, f = 1). The overall initial similarity under (nl = 50, f = 1) is only half its value under (nl = 200, f = 0.5). In spite of this, we have found that it achieves a slightly higher final similarity. This supports the idea that the input pattern should not be applied as the conventional uniformly distorted version of the correct memory, but rather as a less distorted pattern applied only to a small subset of the neurons. 0.970 0.950 (a) , , , , , , , ---------_ .. --.. , ..... -_ .. ",-'-' ,-, .' : DIscrete signaling , : - - - - Analog slgraliing " 1.00 0.98 f :! 0.98 ~ 0.94 (b) /7;;;;':::"?~""'~ /' /.-:,' . ~' I /./ ,.~/ ; i' . ( ! I I { I i I / "/ II Upper bound pertlnNnce df~1 !~ 1'1 ;'~ f,'I, I,Ii '" i ~II 'I I. ,I I - . 3--0 GausaIan connec1IvIty - - - 2-D GaussIan connectIvl1y - - - - Mulll-layered nelWodc _ .. '-' Lower bound perlonrence 0.11400.0 2000.0 4000.0 eooo.O 8000.0 10000.0 0.920.0 1000.0 2000.0 N K Figure 3: (a) Two-iteration performance in a full-activity network as a function of network size N. Network parameters are nl = I{ = 200, m = 40 and f = 0.5. (b) Two-iteration performance achieved with various network architectures, as a function of the network connectivity K. Network parameters are N = 5000, nl = 200, m = 50, f = 0.5 and 6 = O. Figure 3 (a) illustrates the performance when connectivity and the number of sigOptimal Signalling in Attractor Neural Networks 491 nals received by each neuron are held fixed, but the network size is increased. A region of decreased performance is evident at mid-connectivity (K ~ N /2) values, due to the increased residual variance. Hence, for neurons capable of forming K connections on the average, the network should either be fully connected or have a size N much larger than K. Since (unavoidable eventually) synaptic deletion would sharply worsen the performance of fully connected networks, cortical ANNs should indeed be sparsely connected. The final similarity achieved in the fully connected network (with N = K = 200) should be noted. In this case, the memory load (0.2) is significantly above the critical capacity of the Hopfield network, but optimal history-dependent dynamics still manage to achieve a rather high two-iterations similarity (0.975) from initial similarity 0.75. This is in agreement with the findings of [Morita, 1993, Yoshizawa et a/., 1993], who show that nonmonotone dynamics increase capacity. Figure 3 (b) illustrates the performance achieved with various network architectures, all sharing the same network parameters N, K, m and input similarity parameters nl, f, 0, but differing in the spatial organization of the neurons' synapses. As evident, even in low-activity sparse-connectivity conditions, the decrease in performance with Gaussian connectivity (in relation, say, to the upper bound) does not seem considerable. Hence, history-dependent ANNs can work well in a cortical-like architecture. 5 Summary The main results of this work are as follows: • The Bayesian framework gives rise to the slanted-sigmoid as the optimal signal function, displaying the non monotone shape proposed by [Morita, 1993]. It also offers an intuitive explanation of its form. • Martingale arguments show that similarity under Bayesian dynamics persistently increases. This makes our two-iteration results a lower bound for the final similarity achievable in ANNs. • The possibly asymmetric form of the function, where neurons that have been silent in the previous iteration have an increased tendency to fire in the next iteration versus previously active neurons, is reminiscent of the bi-threshold phenomenon observed in biological neurons [Tam, 1992]. • In the limit of low memory load the best signal is simply a sigmoidal function of the generalized input field. • In an efficient associative network, input patterns should be applied with high fidelity on a small subset of neurons, rather than spreading a given level of initial similarity as a low fidelity stimulus applied to a large subset of neurons. • If neurons have some restriction on the number of connections they may form, such that each neuron forms some K connections on the average, then efficient ANNs, converging to high final similarity within few iterations, should be sparsely connected. 492 Meilijson and Ruppin • With a properly tuned signal function, cortical-like Gaussian-connectivity ANNs perform nearly as well as randomly-connected ones . • Investigating the 0,1 (silent, firing) formulation, there seems to be an interval such that only neurons whose field values are greater than some low threshold and smaller than some high threshold should fire. This seemingly bizarre behavior may correspond well to the behavior of biological neurons; neurons with very high field values have most probably fired constantly in the previous 'iteration', and due to the effect of neural adaptation are now silenced. References [Braitenberg and Schuz, 1991] V. Braitenberg and A. Schuz. Anatomy of the Cortex: Statistics and Geometry. Springer-Verlag, 1991. [Connors and Gutnick, 1990] B.W. Connors and M.J. Gutnick. Intrinsic firing patterns of diverse neocortical neurons. TINS, 13(3):99-104, 1990. [Hopfield, 1982] J.J. Hopfield. Neural networks and physical systems with emergent collective abilities. Proc. Nat. Acad. Sci. USA, 79:2554, 1982. [Lytton, 1991] W. Lytton. Simulations of cortical pyramidal neurons synchronized by inhibitory interneurons. J. Neurophysiol., 66(3):1059-1079, 1991. [Meilijson and Ruppin, 1993] I. Meilijson and E. Ruppin. History-dependent attractor neural networks. Network, 4:1-28, 1993. [Morita, 1993] M. Morita. Associative memory with nonmonotone dynamics. Neural Networks, 6:115-126, 1993. [Tam, 1992] David C. Tam. Signal processing in multi-threshold neurons. In T. McKenna, J. Davis, and S.F. Zornetzer, editors, Single neuron computation, pages 481-501. Academic Press, 1992. [Yoshizawa et al., 1993] S. Yoshizawa, M. Morita, and S.-I. Amari. Capacity of associative memory using a nonmonotonic neuron model. Neural Networks, 6:167176, 1993.
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Neural Network Exploration Using Optimal Experiment Design David A. Cohn Dept. of Brain and Cognitive Sciences Massachusetts Inst. of Technology Cambridge, MA 02139 Abstract Consider the problem of learning input/output mappings through exploration, e.g. learning the kinematics or dynamics of a robotic manipulator. If actions are expensive and computation is cheap, then we should explore by selecting a trajectory through the input space which gives us the most amount of information in the fewest number of steps. I discuss how results from the field of optimal experiment design may be used to guide such exploration, and demonstrate its use on a simple kinematics problem. 1 Introduction Most machine learning research treats the learner as a passive receptacle for data to be processed. This approach ignores the fact that, in many situations, a learner is able, and sometimes required, to act on its environment to gather data. Learning control inherently involves being active; the controller must act in order to learn the result of its action. When training a neural network to control a robotic arm, one may explore by allowing the controller to "flail" for a length of time, moving the arm at random through coordinate space while it builds up data from which to build a model [Kuperstein, 1988]. This is not feasible, however, if actions are expensive and must be conserved. In these situations, we should choose a training trajectory that will get the most information out of a limited number of steps. Manually designing such trajectories is a slow process, and intuitively "good" trajectories often fail to sufficiently explore the state space [Armstrong, 1989]. In 679 680 Cohn this paper I discuss another alternative for exploration: automatic, incremental generation of training trajectories using results from "optimal experiment design." The study of optimal experiment design (OED) [Fedorov, 1972] is concerned with the design of experiments that are expected to minimize variances of a parameterized model. Viewing actions as experiments that move us through the state space, we can use the techniques of OED to design training trajectories. The intent of optimal experiment design is usually to maximize confidence in a given model, minimize parameter variances for system identification, or minimize the model's output variance. Armstrong [1989] used a form of OED to identify link masses and inertial moments of a robot arm, and found that automatically generated training trajectories provided a significant improvement over human-designed trajectories. Automatic exploration strategies have been tried for neural networks (e.g. [Thrun and Moller, 1992]' [Moore, 1994]), but use of OED in the neural network community has been limited. Plutowski and White [1993] successfully used it to filter a data set for maximally informative points, but its application to selecting new data has only been proposed [MacKay, 1992], not demonstrated. The following section gives a brief description of the relevant results from optimal experiment design. Section 3 describes how these results may be adapted to guide neural network exploration and Section 4 presents experimental results of implementing this adaptation. Finally, Section 5 discusses implications of the results, and logical extensions of the current experiments. 2 Optimal experiment design Optimal experiment design draws heavily on the technique of Maximum Likelihood Estimation (MLE) [Thisted, 1988]. Given a set of assumptions about the learner's architecture and sources of noise in the output, MLE provides a statistical basis for learning. Although the specific MLE techniques we use hold exactly only for linear models, making certain computational approximations allows them to be used with nonlinear systems such as neural networks. We begin with a training set of input-output pairs (Xi, Yi)i=l and a learner fw O· We define fw(x) to be the learner's output given input X and weight vector w. Under an assumption of additive Gaussian noise, the maximum likelihood estimate for the weight vector, W, is that which minimizes the sum squared error Esse = 2:7=1(JW(Xi) - Yi)2. The estimate W gives us an estimate of the output at a novel input: if = fw(x) (see e.g. Figure 1a). MLE allows us to compute the variances of our weight and output estimates. Writing the output sensitivity asgw(x) = 8fw(x)/8w, the covariances of ware where the last approximation assumes local linearity of gw(x). (For brevity, the output sensitivity will be abbreviated to g( x) in the rest of the paper.) Neural Network Exploration Using Optimal Experiment Design 681 y Figure 1: a) A set of training examples for a classification problem, and the network's best fit to the data. b) Maximum likelihood estimate of the network's output variance for the same problem. For a given reference input X r , the estimated output variance is var(xr) = g(Xr? A- 1g(xr). (1) Output variance corresponds to the model's estimate of the expected squared distance between its output fw(x) and the unknown "true" output y. Output variance then, corresponds to the model's estimate of its mean squared error (MSE) (see Figure 1 b). If the estimates are accurate then minimizing the output variance would correspond to minimizing the network's MSE. In optimal experiment design, we estimate how adding a new training example is expected to change the computed variances. Given a novel X n +1, we can use OED to predict the effect of adding Xn+1 and its as-yet-unknown Yn+1 to the training set. We make the assumption that -1 ( T)-1 An+1 ~ An + g(xn+dg(xn+d , which corresponds to assuming that our current model is already fairly good. Based on this assumption, the new parameter variances will be A~~1 = A~l A~1g(xn+d(1 + g(Xn+1? A~1g(xn+d)g(xn+t)T A~1. Combined with Equation 1, this predicts that if we take a new example at X n +1, the change in output variance at reference input Xr will be ~var(Xr ) (g(xrf A~lg(xn+l»2(1 + g(Xn+1)T A;;lg(xn+d) cov(xr, Xn+l)2(1 + var(xn+d) (2) To minimize the expected value of var(xr), we should select Xn+l so as to maximize the right side of Equation 2. For other interesting OED measures, see MacKay [1992] . 682 Cohn 3 Adapting OED to Exploration When building a world model, the learner is trying to build a mapping, e.g. from joint angles to cartesian coordinates (or from state-action pairs to next states). If it is allowed to select arbitrary joint angles (inputs) in successive time steps, then the problem is one of selecting the next "query" to make ([Cohn, 1990], [Baum and Lang, 1991]). In exploration, however, one's choices for a next input are constrained by the current input. We cannot instantaneously "teleport" to remote parts of the state space, but must choose among inputs that are available in the next time step. One approach to selecting a next input is to use selective sampling: evaluate a number of possible random inputs, choose the one with the highest expected gain. In a high-dimensional action space, this is inefficient. The approach followed here is that of gradient search, differentiating Equation 2 and hillclimbing on 8jj,var( x r )/ 8Xn +l. Note that Equation 2 gives the expected change in variance only at a single point X r , while we wish to minimize the average variance over the entire domain. Explicitly integrating over the domain is intractable, so we must make do with an approximation. MacKay [1992] proposed using a fixed set of reference points and measuring the expected change in variance over them. This produces spurious local maxima at the reference points, and has the undesirable effect of arbitrarily quantizing the input space. Our approach is to iteratively draw reference points at random (either uniformly or according to a distribution of interest), and compute a stochastic approximation of jj, var. By climbing the stochastically approximated gradient, either to convergence or to the horizon of available next inputs, we will settle on an input/action with a (locally) optimal decrease in expected variance. 4 Experimental Results In this section, I describe two sets of experiments. The first attempts to confirm that the gains predicted by optimal experiment design may actually be realized in practice, and the second studies the application of OED to a simple learning task. 4.1 Expected versus actual gain It must be emphasized that the gains predicted by OED are expected gains. These expectations are based on the relatively strong assumptions of MLE, which may not strictly hold. In order for the expected gains to materialize, two "bridges" must be crossed. First, the expected decrease in model variance must be realized as an actual decrease in variance. Second, the actual decrease in model variance must translate into an actual decrease in model MSE. 4.1.1 Expected decreases in variance --+ actual decreases in variance The translation from expected to actual changes in variance requires coordination between the exploration strategy and the learning algorithm: to predict how the variance of a weight will change with a new piece of data, the predictor must know how the weight itself (and its neighboring weights) will change. Using a black ~ ~ > ~ ... ~ 'tl Neural Network Exploration Using Optimal Experiment Design 683 0 . 012 l I 0.01 0 . 008 0 .00 6 0 . 0 04 0 . 002 XX x , , x x - - - - actual =e xpec ted x x x 0 . 002 0. 00 4 0 . 0 0 6 0 . 0 08 0 . 01 0 . 01 2 exp ec t ed d e lta var 2 .8 2 . 4 ""; , ;:: ~ 1. 6 [oJ Vl :E " ... 1. 2 ~ 'tl 0.8 0 . 4 - 0. 4 x x x ;If x x )( x X x x x i« x x x x 0 .002 0.004 0.006 0 .008 0 . 01 0 . 012 actua l d e lta var Figure 2: a) Correlations between expected change in output variance and actual change output variance b) Correlations between actual change in output variance and change in mean squared error. Correlations are plotted for a network trained on 50 examples from the arm kinematics task. box routine like backpropagation to update the weights virtually guarantees that there will be some mismatch between expected and actual decreases in variance. Experiments indicate that, in spite of this, the correlation between predicted and actual changes in variance are relatively good (Figure 2a). 4.1.2 Decreases in variance -- decreases in MSE A more troubling translation is the one from model variance to model correctness. Given the highly nonlinear nature of a neural network, local minima may leave us in situations where the model is very confident but entirely wrong. Due to high confidence, the learner may reject actions that would reduce its mean squared error and explore areas where the model is correct, but has low confidence. Evidence of this behavior is seen in the lower right corner of Figure 2b, where some actions which produce a large decrease in variance have little effect on the network's MSE. While this decreases the utility of OED, it is not crippling. We discuss one possible solution to this problem at the end of this paper. 4.2 Learning kinematics We have used the the stochastic approximation of ~var to guide exploration on several simple tasks involving classification and regression. Below, I detail the experiments involving exploration of the kinematics of a simple two-dimensional, two-joint arm. The task was to learn a forward model 8 1 x 8 2 -- X X Y through exploration, which could then be used to build a controller following Jordan [1992]. 684 Cohn The model was to be learned by a feedforward network with a sigmoid transfer function using a single hidden layer of 8 or 20 hidden units. Figure 3: Learning 2D arm kinematics with 8 hidden units. a) Geometry of the 2D, two-joint arm. b) Sample trajectory using OED-based greedy exploration. On each time step, the learner was allowed to select inputs 8 1 and 8 2 and was then given tip position x and y to incorporate into its training set. It then hillclimbed to find the next 8 1 and 8 2 within its limits of movement that would maximize the stochastic approximation of ~var . On each time step 8 1 and 8 2 were limited to change by no more than ±36° and ±18° respectively. Simulations were performed on the Xerion simulator (made available by the University of Toronto), approximating the variance gradient on each step with 100 randomly drawn points. A sample tip trajectory is illustrated in Figure 3b. We compared the performance of this one-step optimal (greedy) learner, in terms of mean squared error, with that of an identical learner which explored randomly by "flailing." Not surprisingly, the improvement of greedy exploration over random exploration is significant (Figure 4b). The asymptotic performance of the greedy learner was better than that of the random learner, and it reached its asymptote in much few steps. 5 Discussion The experiments described in this paper indicate that optimal experiment design is a promising tool for guiding neural network exploration. It requires no arbitrary discretization of state or action spaces, and is amenable to gradient search techniques. It does, however, have high computational costs and, as discussed in Section 4.1.2, may be led astray if the model settles in a local minimum. 5.1 Alternatives to greedy OED The greedy approach is prone to "boxing itself into a corner" while leaving important parts of the domain unexplored. One heuristic for avoiding local minima is to Neural Network Exploration Using Optimal Experiment Design 685 I ~\ O. 28 ~ I 0.24 I I I + w o. 201 ~ \ Ul 0 . 16V :0: 0.12-1 I I 0 . 08i 0 .04J I ~ 0.00 , .. _ --T n e : : ::~ 0.21~ \\ I . \ O. 18~ \ ~ 0 . 15~ \ . ::: \( 0 . 061 O. 0 3~ I ~~-=::::-./ o . 001-----,-, , -,---,-, I , , 20 40 60 80 100 120 o 20 40 60 80 100120140160180200 Number of steps Number of steps Figure 4: Learning 2D arm kinematics. a) MSE for a single exploration trajectory (20 hidden units). b) Plot of MSE for random and greedy exploration vs. number of training examples, averaged over 12 runs (8 hidden units). occasionally check the expected gain in other parts of the input space and move towards them if they promise much greater gain than a greedy step. The theoretically correct but computationally expensive approach is to optimize over an entire trajectory. Trajectory optimization entails starting with an initial trajectory, computing the expected gain over it, and iteratively perturbing points on the trajectory towards towards optimal expected gain (subject to other points along the trajectory being explored). Experiments are currently underway to determine how much of an improvement may be realized with trajectory optimization; it is unclear whether the improvement over the greedy approach will be worth the added computational cost. 5.2 Computational Costs The computational costs of even greedy OED are great. Selecting a next action requires computation and inversion of the hessian {)2 Eue/ ow 2 . Each time an action is selected and taken, the new data must be incorporated into the training set, and the learner retrained. In comparison, when using a flailing strategy or a fixed trajectory, the data may be gathered with little computation, and the learner trained only once on the batch. In this light, the cost of data must be much greater than the cost of computation for optimal experiment design to be a preferable strategy. There are many approximations one can make which significantly bring down the cost of OED. By only considering covariances of weights leading to the same neuron, the hessian may be reduced to a block diagonal form, with each neuron computing its own (simpler) covariances in parallel. As an extreme, one can do away with covariances entirely and rely only on individual weight variances, whose computation is simple. By the same token, one can incorporate the new examples in small batches, only retraining every 5 or so steps. While suboptimal from a data gathering perspective, they appear to still outperform random exploration, and are much cheaper than "full-blown" optimization. 686 Cohn 5.3 Alternative architectures We may be able to bring down computational costs and improve performance by using a different architecture for the learner. With a standard feedforward neural network, not only is the repeated compution of variances expensive, it sometimes fails to yield estimates suitable for use as confidence intervals (as we saw in Section 4.1.2). A solution to both of these problems may lie in selection of a more amenable architecture and learning algorithm. One such architecture, in which output variances have a direct role in estimation, is a mixture of Gaussians, which may be efficiently trained using an EM algorithm [Ghahramani and Jordan, 1994]. We expect that it is along these lines that our future research will be most fruitful. Acknowledgements I am indebted to Michael I. Jordan and David J .C. MacKay for their help in making this research possible. This work was funded by ATR Human Information Processing Laboratories, Siemens Corporate Research and NSF grant CDA-9309300. Bibliography B. Armstrong. (1989) On finding exciting trajectories for identification experiments. Int. J. of Robotics Research, 8(6):28-48. E. Baum and K. Lang. (1991) Constructing hidden units using examples and queries. In R. Lippmann et al., eds., Advances in Neural Information Processing Systems 3, Morgan Kaufmann, San Francisco, CA. D. Cohn, L. Atlas and R. Ladner. (1990) Training connectionist networks with queries and selective sampling. In D. Touretzky, editor, Advances in Neural Information Processing Systems 2, Morgan Kaufmann, San Francisco. V. Fedorov. (1972) Theory of Optimal Experiments. Academic Press, New York. Z. Ghahramani and M. Jordan. (1994) Supervised learning from incomplete data via an EM approach. In this volume. M. Jordan and D. Rumelhart. (1992) Forward models: Supervised learning with a distal teacher. Cognitive Science, 16(3):307-354. D. MacKay. (1992) Information-based objective functions for active data selection, Neural Computation 4(4): 590-604. A. Moore. (1994) The parti-game algorithm for variable resolution reinforcement learning in multidimensional state-spaces. In this volume. M. Plutowski and H. White. (1993) Selecting concise training sets from clean data. IEEE Trans. on Neural Networks, 4(2):305-318. R. Thisted. (1988) Elements of Statistical Computing. Chapman and Hall, NY. S. Thrun and K. Moller. (1992) Active Exploration in Dynamic Environments. In J. Moody et aI., editors, Advances in Neural Information Processing Systems 4. Morgan Kaufmann, San Francisco, CA.
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Directional Hearing by the Mauthner System .Audrey L. Gusik Department of Psychology University of Colorado Boulder, Co. 80309 Abstract Robert c. Eaton E. P. O. Biology University of Colorado Boulder, Co. 80309 We provide a computational description of the function of the Mauthner system. This is the brainstem circuit which initiates faststart escapes in teleost fish in response to sounds. Our simulations, using back propagation in a realistically constrained feedforward network, have generated hypotheses which are directly interpretable in terms of the activity of the auditory nerve fibers, the principle cells of the system and their associated inhibitory neurons. 1 INTRODUCTION 1.1 THE M.AUTHNER SYSTEM Much is known about the brainstem system that controls fast-start escapes in teleost fish. The most prominent feature of this network is the pair of large Mauthner cells whose axons cross the midline and descend down the spinal cord to synapse on primary motoneurons. The Mauthner system also includes inhibitory neurons, the PHP cells, which have a unique and intense field effect inhibition at the spikeinitiating zone of the Mauthner cells (Faber and Korn, 1978). The Mauthner system is part of the full brainstem escape network which also includes two pairs of cells homologous to the Mauthner cell and other populations of reticulospinal neurons. With this network fish initiate escapes only from appropriate stimuli, turn away from the offending stimulus, and do so very rapidly with a latency around 15 msec in goldfish. The Mauthner cells play an important role in these functions. Only one 574 Directional Hearing by the Mauthner System 575 fires thus controlling the direction of the initial turn, and it fires very quickly (4-5 msec). They also have high thresholds due to instrinsic membrane properties and the inhibitory inlluence of the PHP cells. (For reviews, see Eaton, et al, 1991 and Faber and Korn, 1978.) Acoustic stimuli are thought to be sufficient to trigger the response (Blader, 1981), both Mauthner cells and PHP cells receive innervation from primary auditory fibers (Faber and Korn, 1978). In addition, the Mauthner cells have been shown physiologically to be very sensitive to acoustic pressure (Canfield and Eaton, 1990). 1.2 LOCALIZING SOUNDS UNDERWATER In contrast to terrestrial vertebrates, there are several reasons for supposing that fish do not use time of arrival or intensity differences between the two ears to localize sounds: underwater sound travels over four times as fast as in air; the fish body provides no acoustic shadow; and fish use a single transducer to sense pressure which is conveyed equally to the two ears. Sound pressure is transduced into vibrations by the swim bladder which, in goldfish, is mechanically linked to the inner ear. Fish are sensitive to an additional component of the acoustic wave, the particle motion. Any particle ofthe medium taking part in the propagation of a longitudenal wave will oscillate about an equilibrium point along the axis of propagation. Fish have roughly the same density as water, and will experience these oscillations. The motion is detected by the bending of sensory hairs on auditory receptor cells by the otolith, an inertial mass suspended above the hair cells. This component of the sound will provide the axis of propagation, but there is a 180 degree ambiguity. Both pressure and particle motion are sensed by hair cells of the inner ear. In goldfish these signals may be nearly segregated. The linkage with the swim bladder impinges primarily on a boney chamber containing two of the endorgans of the inner ear: the saccule and the lagena. The utricle is a third endorgan also thought to mediate some acoustic function, without such direct input from the 3wimbladder. Using both of these components fish can localize sounds. According to the phase model (Schuijf, 1981) fish analyze the phase difference between the pressure component of the sound and the particle displacement component to calculate distance and direction. When pressure is increasing, particles will be pushed in the direction of sound propagation, and when pressure is decreasing particles will be pulled back. There will be a phase lag between pressure and particle motion which varies with frequency and distance from the sound source. This, and the separation of the pressure from the displacement signals in the ear of some species pose the greatest problems for theories of sound localization in fish. The acoustically triggered escape in goldfish is a uniquely tractable problem in underwater sound localization. First, there is the fairly good segregation of pressure from particle motion at the sensory level. Second I the escape is very rapid. The decision to turn left or right is equivalent to the firing of one or the other Mauthner cell, and this happens within about 4 msec. With transmission delay, this decision relies only on the initial 2 msec or so of the stimulus. For most salient frequencies, the phase lag will not introduce uncertainty: both the first and second derivatives of particle position and acoustic pressure will be either positive or negative. 576 Guzik and Eaton 1.3 THE XNOR MODEL A B Active pressure input p+ p+ ppLeft sound source OR---a Active displacement input Ol DR OL DR p+ -------.. pOL---a No response left Mauthner output On Off orr On Right Mauthner output Ofr On On Off 1)---- DL __ ..;:Jo.. ___ P+ --..,----p. 1)---- DR .. inhibitory 0- excitatory Figure 1 Truth table and minimal network for the XNOR model. Given the above simplification of the problem, we can see that each Mauthner cell must perform a logical operation (Guzik and Eaton, 1993j Eaton et al, 1994). The left Mauthner cell should fire when sounds are located on the left, and this occurs when either pressure is increasing and particle motion is from the left or when pressure is decreasing and particle motion is from the right. We can call displacement from the left positive for the left Mauthner cell, and immediately we Directional Hearing by the Mauthner System 577 have the logical operator exclusive-nor (or XNOR). The right Mauthner cell must solve the same problem with a redefinition of right displacement as positive. The conditions for this logic gate are shown in figure 1A for both Mauthner cells. This analysis simplifies our task of understanding the computational role of individual elements in the system. For example, a minimal network could appear as in figure lB. In this model PHP units perform a logical sub-task of the XNOR as AND gates. This model requires at least two functional classes of PHP units on each side of the brain. These PHP units will be activated for the combinations of pressure and displacement that indicate a sound coming from the wrong direction for the Mauthner cell on that side. Both Mauthner cells are activated by sufficient changes in pressure in either direction, high or low, and will be gated by the PHP cells. This minimal model emerged from explorations of the system using the connectionist paradigm, and inspired us to extend our efforts to a more realistic context. 2 THE NETWORK We used a connectionist model to explore candidate solutions to the left/right discrimination problem that include the populations of neurons known to exist and include a distributed input resembling the sort available from the hair cells of the inner ear. We were interested in generating a number of alternative solutions to be better prepared to interpret physiological recordings from live goldfish, and to look for variations of, or alternatives to, the XNOR model. 2.1 THE .ARCHITECTURE As shown in figure 2, there are four layers in the connectionist model. The input layer consists of four pools of hair cell units. These represent the sensory neurons of the inner ear. There are two pools on each side: the saccule and the utricle. Treating only the horizontal plane, we have ignored the lagena in this model. The saccule is the organ of pressure sensation and the utricle is treated as the organ of particle motion. Each pool contains 16 hair cell units maximally responsive for displacements of their sensory hairs in one particular direction. They are activated as the eosine of the difference between their preferred direction and the stimulus dellection. All other units use sigmoidal activation functions. The next layer consists of units representing the auditory fibers of the VIIIth nerve. Each pool receives inputs from only one pool of hair cell units, as nerve fibers have not been seen to innervate more than one endorgan. There are 10 units per fiber pool. The fiber units provide input to both the inhibitory PHP units, and to the Mauthner units. There are four pools of PHP units, two on each side of the fish. One set on each side represents the collateral PHP eells, and the other set represents the commissural PHP cells (Faber and Korn, 1978). Both types receive inputs from the auditory fibers. The collaterals project only to the Mauthner cell on the same side. The commissurals project to both Mauthner cells. There are five units per PHP pool. 578 Guzik and Eaton The Mauthner cell units receive inputs from saecular and utricular fibers on their same side only, as well as inputs from a single collateral PHP population and both commissural PHP populations. Left Saccule Left Utricle Right Utricle Right Saccule Hair Cells Auditory Nerve Fiber Pools PHPs Left Mauthner Right Mautlll1er Figure 2 The architecture. Weights from the PHP units are all constrained to be negative, while all others are constrained to be positive. The weights are implemented using the function below, positive or negative depending on the polarity of the weight. f(w) = 1/2 (w + In cosh(w) + In 2) The function asymptotes to zero for negative values, and to the identity function for values above 2. This function vastly improved learning compared with the simpler, but highly nonlinear exponential function used in earlier versions of the model. 2.2 TRAINING We used a total of 240 training examples. We began with a set of 24 directions for particle motion, evenly distributed around 360 degrees. These each appeared twice, once with increasing pressure and once with decreasing pressure, making a base set of 48 examples. Pressure was introduced as a deflection across saccular hair cells of either 0 degrees for low pressure, or 180 degrees for high pressure. These should be thought of as reflecting the expansion or compression of the swim bladder. Targets for the Mauthner cells were either 0 or 1 depending upon the conditions as described in the XNOR model, in figure lA. Directional Hearing by the Mauthner System 579 N ext by randomly perturbing the activations of the hair cells for these 48 patterns, we generated 144 noisy examples. These were randomly increased or decreased up to 10%. An additional 48 examples were generated by dividing the hair cell adivity by two to represent sub-threshold stimuli. These last 48 targets were set to zero. The network was trained in batch mode with backpropagation to minimize a crossentropy error measure, using conjugate gradient search. Unassisted backpropagation was unsuccessful at finding solutions. For the eight solutions discussed here, two parameters were varied at the inputs. In some solutions the utride was stimulated with a vedor sum of the displacement and the pressure components, or a "mixed" input. In some solutions the hair cells in the utride are not distributed uniformly, but in a gaussian manner with the mean tuning of 45 degrees to the right or left, in the two ears respedively. This approximates the actual distribution of hair cells in the goldfish utride (Platt, 1977). 3 RESULTS Analyzing the activation of the hidden units as a fundion of input pattern we found activity consistent with known physiology, nothing inconsistent with our knowledge of the system, and some predidions to be evaluated during intracellular recordings from PHP cells and auditory afFerents. First, many PHP cells were found exhibiting a logical fUndion, which is consistent with our minimal model described above. These tended to projed only to one Mauthner cell unit, which suggests that primarily the collateral PHP cells will demonstrate logical properties. Most logical PHP units were NAND gates with very large weights to one Mauthner cell. An example is a unit which is on for all stimuli except those having displacements anywhere on the left when pressure is high. Second, saccular fibers tended to be either sensitive to high or low pressure, consistent with recordings of Furukawa and Ishii (1967). In addition there were a dass which looked like threshold fibers, highly active for all supra-threshold stimuli, and inactive for all sub-threshold stimuli. There were some fibers with no obvious seledivity, as well. Third, utricular fibers often demonstrate sensitivity for displacements exclusively from one side ofthe fish, consistent with our minimal model. Right and left utricular fibers have not yet been demonstrated in the real system. Utricular fibers also demonstrated more coarsely tuned, less interpretable receptive fields. All solutions that included a mixed input to the utrieie, for example, produced fibers that seemed to be "not 180 degree" ,or "not 0 degree", countering the pressure vedors. We interpret these fibers as doing dean-up given the absence of negative weights at that layer. Fourth, sub-threshold behavior of units is not always consistent with their suprathreshold behavior. At sub-threshold levels of stimulation the adivity of units may not refted their computational role in the behavior. Thus, intracellular recordings should explore stimulus ranges known to elicit the behavior. 580 Guzik and Eaton Fifth, Mauthner units usually receive very strong inputs from pressure fibers. This is consistent with physiological recordings which suggest that the Mauthner cells in goldfish are more sensitive to sound pressure than displacement (Canfield and Eaton, 1990). Sixth, Mauthner cells always acquired rdatively equal high negative biases. This is consistent with the known low input resistance of the real Mauthner eells, giving them a high threshold (Faber and Korn, 1978). Seventh, PHP cells that maintain substantial bilateral connections tend to be tonically active. These contribute additional negative bias to the Mauthner cells. The relative sizes of the connections are often assymetric. This suggests that the commissural PHP cells serve primarily to regulate Mauthner threshold, ensure behavioral response only to intense stimuli, consistent with Faber and Korn (1978). These cells could only contribute to a partial solution of the XNOR problem. Eighth, all solutions consistently used logic gate PHP units for only 50% to 75% of the training examples. Probably distributed solutions relying on the direct connections of auditory nerve fibers to Mauthner cells were more easily learned, and logic gate units only developed to handle the unsolved eases. Cases solved without logic gate units were solved by assymetric projections to the Mauthner cells of one polarity of pressure and one class of direction fibers, left or right. Curiously, most of these eases involved a preferential projection from high pressure fibers to the Mauthner units, along with directional fibers encoding displacements from each Mauthner unit's positive direction. This means the logic gate units tended to handle the low pressure eases. This may be a result of the presence of the assymetric distributions of utricular hair cells in 6 out of the 8 solutions. 4 CONCLUSIONS \Ve have generated predictions for the behavior of neurons in the Mauthner system under different conditions of acoustic stimulation. The predictions generated with our connectionist model are consistent with our interpretation of the phase model for underwater sound localization in fishes as a logical operator. The results are also consistent with previously described properties of the Mauthner system. Though perhaps based on the characteristics more of the training procedure, our solutions suggest that we may find a mixed solution in the fish. Direct projections to the Mauthner cells from the auditory nerve perhaps handle many of the commonly encountered acoustic threats. The results of Blaxter (1981) support the idea that fish do escape from stimuli regardless of the polarity of the initial pressure change. Without significant nonlinear processing at the Mauthner cell itsdf, or more complex processing in the auditory fibers, direct connections could not handle all of these eases. These possibilities deserve exploration. We propose different computational roles for the two classes of inhibitory PHP neurons. We expect only unilaterally-projecting PHP cells to demonstrate some logical function of pressure and particle motion. We believe that some elements of the Mauthner system must be found to demonstrate such minimal logical functions if the phase modd is an explanation for left-right discrimination by the Mauthner system. Directional Hearing by the Mauthner System 581 We are currently preparing to deliver controlled acoustic stimuli to goldfish during acute intracellular recording procedures from the PHP neurons, the afferent fibers and the Mauthner cells. Our insights from this model will greatly assist us in designing the stimulus regimen, and in interpreting our experimental results. Plans for future computational work are of a dynamic model that will include the results of these physiological investigations, as well as a more realistic version of the Mauthner cell . .Acknowledgements We are grateful for the technical assistance of members of the Boulder Connectionist Research Group, especially Don Mathis for help in debugging and optimizing the original code. We thank P.L. Edds-Walton for crucial discussions. This work was supported by a grant to RCE from the National Institutes of Health (ROI NS22621). References Blader, J.H.S., J.A.B. Gray, and E.J. Denton (1981) Sound and startle responses in herring shoals. J. Mar. BioI. Assoc. UK, 61: 851-869 Canfield, J.G. and R.C. Eaton (1990) Swimbladder acoustic pressure transduction intiates Mauthner-mediated escape. Nature, 3~7: 760-762 Eaton, R.C., J.G. Canfield and A.L. Guzik (1994) Left-right discrimination of sound onset by the Mauthner system. Brain Behav. Evol., in pre66 Eaton, R.C., R. DiDomenico and J. Nissanov (1991) Role of the Mauthner cell in sensorimotor integration by the brain stem escape network. Brain Behav. Evol., 37: 272-285 Faber, D.S. and H. Korn (1978) Electrophysiology of the Mauthner cell: Basic properties, synaptic mechanisms and associated networks. In Neurobiology of the Mauthner Cell, D.S. Faber and H. Korn (eds) , Raven Press, NY, pp. 47-131 Fay, R.R.(1984) The goldfish ear codes the axis of acoustic particle motion in three dimensions. Science, 225: 951-954 Furukawa, T. and Y. Ishii (1967) Effects of static bending of sensory hairs on sound reception in the goldfish. Japanese J. Physiol., 17: 572-588 Guzik, A.L. and R.C. Eaton (1993) The XNOR model for directional hearing by the Mauthner system. Soc. Neurosci. Abstr. PIaU, C. (1977) Hair cell distribution and orientation in goldfish otolith organs. J. Compo Neurol., 172: 283-298 Schuijf, A. (1981) Models of acoustic localization. In Hearing and Sound Communication in Fishes, W.N. Tavolga, A.N. Popper and R.R. Fay (eds.), Springer, New York,. pp. 267-310
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Asynchronous Dynamics of Continuous Time Neural Networks Xin Wang Computer Science Department University of California at Los Angeles Los Angeles, CA 90024 Qingnan Li Department of Mathematics University of Southern California Los Angeles, CA 90089-1113 Edward K. Blum Department of Mathematics University of Southern California Los Angeles, CA 90089-1113 ABSTRACT Motivated by mathematical modeling, analog implementation and distributed simulation of neural networks, we present a definition of asynchronous dynamics of general CT dynamical systems defined by ordinary differential equations, based on notions of local times and communication times. We provide some preliminary results on globally asymptotical convergence of asynchronous dynamics for contractive and monotone CT dynamical systems. When applying the results to neural networks, we obtain some conditions that ensure additive-type neural networks to be asynchronizable. 1 INTRODUCTION Neural networks are massively distributed computing systems. A major issue in parallel and distributed computation is synchronization versus asynchronization (Bertsekas and Tsitsiklis, 1989). To fix our idea, we consider a much studied additive-type model (Cohen and Grossberg, 1983; Hopfield, 1984; Hirsch, 1989) of a continuoustime (CT) neural network of n neurons, whose dynamics is governed by n Xi(t) = -ajXi(t) + L WijO'j (Jlj Xj (t)) + Ii, i = 1,2, ... , n, j=1 (1) 493 494 Wang. Li. and Blum with neuron states Xi (t) at time t, constant decay rates ai, external inputs h, gains JJj, neuron activation functions Uj and synaptic connection weights Wij. Simulation and implementation of idealized models of neural networks such as (1) on centralized computers not only limit the size of networks, but more importantly preclude exploiting the inherent massive parallelism in network computations. A truly faithful analog implementation or simulation of neural networks defined by (1) over a distributed network requires that neurons follow a global clock t, communicate timed states Xj(t) to all others instantaneously and synchronize global dynamics precisely all the time (e.g., the same Xj(t) should be used in evolution of all Xi(t) at time t). Clearly, hardware and software realities make it very hard and sometimes impossible to fulfill these requirements; any mechanism used to enforce such synchronization may have an important effect on performance of the network. Moreover, absolutely insisting on synchronization contradicts the biological manifestation of inherent asynchrony caused by delays in nerve signal propagation, variability of neuron parameters such as refractory periods and adaptive neuron gains. On the other hand, introduction of asynchrony may change network dynamics, for example, from convergent to oscillatory. Therefore, validity of asynchronous dynamics of neural networks must be assessed in order to ensure desirable dynamics in a distributed environment. Motivated by the above issues, we study asynchronous dynamics of general CT dynamical systems with neural networks in particular. Asynchronous dynamics has been thoroughly studied in the context of iterative maps or discrete-time (DT) dynamical systems; see, e.g., (Bertsekas and Tsitsiklis, 1989) and references therein. Among other results are that P-contractive maps on Rn (Baudet, 1978) and continuous maps on partially ordered sets (Wang and Parker, 1992) are asynchronizable, i.e., any asynchronous iterations of these maps will converge to the fixed points under synchronous (or parallel) iterations. The synchronization issue has also been addressed in the context of neural networks. In fact, the celebrated DT Hopfield model (Hopfield, 1982) adopts a special kind of asynchronous dynamics: only one randomly chosen neuron is allowed to update its state at each iterative step. The issue is also studied in (Barhen and Gulati, 1989) for CT neural networks. The approach there is, however, to convert the additive model (1) into a DT version through the Euler discretization and then to apply the existing result for contractive mappings in (Baudet, 1978) to ensure the discretized system to be asynchronizable. Overall, studies for asynchronous dynamics of CT dynamical systems are still lacking; there are even no reasonable definitions for what it means, at least to our knowledge. In this paper, we continue our studies on relationships between CT and DT dynamical systems and neural networks (Wang and Blum, 1992; Wang, Blum and Li, 1993) and concentrate on their asynchronous dynamics. We first extend a concept of asynchronous dynamics of DT systems to CT systems, by identifying the distinction between synchronous and asynchronous dynamics as (i) presence or absence of a common global clock used to synchronize the dynamics of the different neurons and (ii) exclusion or inclusion of delay times in communication between neurons, and present some preliminary results for asynchronous dynamics of contractive and monotone CT systems. Asynchronous Dynamics of Continuous Time Neural Networks 495 2 MATHEMATICAL FORMULATION To be general, we consider a CT dynamical system defined by an n-dimensional system of ordinary differential equations, (2) where Ii : Rn --+ R are continuously differentiable and x(t) E Rn for all t in R+ (the set of all nonnegative real numbers). In contrast to the asynchronous dynamics given below, dynamics of this system will be called synchronous. An asynchronous scheme consists of two families of functions Ci : R+ --+ R+ and rj : R+ --+ R+, i, j = 1, ... , n, satisfying the following constraints: for any t > 0, (i) Initiation: Ci(t) ~ 0 and rJ(t) ~ 0; (ii) Non-starvation: Ci'S are differentiable and l\(t) > 0; (iii) Liveness: limt_oo Ci(t) = 00 and limt_oo rJ(t) = 00; (iv) Accessibility: rj(t) ~ Cj(t). Given an asynchronous scheme ({cd, {rJ}), the associated asynchronous dynamics of the system (2) is the solution of the following parametrized system: (3) We shall call this system an asynchronized system of the original one (2). The functions Ci(t) should be viewed as respective "local" times (or clocks) of components i, as compared to the "global" time (or clock) t. As each component i evolves its state according to its local time Ci(t), no shared global time t is needed explicitly; t only occurs implicitly. The functions rj(t) should be considered as time instants at which corresponding values Xi of components j are used by component i; hence the differences (ci(t) - rj(t» ~ 0 can be interprated as delay times in communication between the components j and i. Constraint (i) reflects the fact that we are interested in the system dynamics after some global time instance, say 0; constraint (ii) states that the functions Ci are monotone increasing and hence the local times evolve only forward; constraint (iii) characterizes the live ness property of the components and communication channels between components; and, finally, constraint (iv) precludes the possibility that component i accesses states x j ahead of the local times Cj(t) of components j which have not yet been generated. Notice that, under the assumption on monotonicity of Ci(t), the inverses C;l(t) exist and the asynchronized system (3) can be transformed into (4) by letting Yi(t) = Xi( Ci(t» and y} (t) = Xj (rJ(t» = Yj (c;l (rJ(t» for i, j = 1,2, ... , n. The vector form of (4) can be given by iJ = Cf F[Y] (5) 496 Wang, Li, and Blum where yet) = [Yl (t), "" Yn(t)]T, C' = diag(dcl (t)/dt, "" dcn(t)/dt) , F = [/1, "" fn]T, y = [Y;] and [ /1 cYi(t) , yHt), "" y~(t)) 1 _ hcYr(t), y~(t), "" y~(t)) F[Y] = , ' fn (i/'l (t), y~(t), "" y~(t)) Notice that the complication in the way F applies to Y ~imply means ,that every component i will use possibly different "global" states [Yi(t) , y2(t) , "" y~(t)] , This peculiarity makes the equation (5) fit into none ofthe categories of general functional differential equations (Hale, 1977), However, if rJ(t) for i = 1, "., n are equal, all the components will use a same global state y = [yHt) , y~(t), .. " y~(t)] and the asynchronized system (5) assumes a form of retarded functional differential equations, iJ = c' FcY), (6) We shall call this case uniformly-delayed, which will be a main consideration in the next section where we discuss asynchronizable systems, The system (5) includes some special cases. In a no communication delay situation, rj(t) = Cj(t) for all i and the system (5) reduces to iJ = C' F(y), This includes the simplest case where the local times Ci(t) are taken as constant-time scalings cit of the global time t; specially, when all Ci(t) = t the system goes back to the original one (2), If, on the other hand, all the local time~ are identi~al to the global time t and the communication times take the form of rJ(t) = t - OJ(t) one obtains a most general delayed system (7) where the state Yj(t) of component j may have different delay times O)(t) for different other components i. Finally, we should point out that the above definitions of asynchronous schemes and dynamics are analogues of their counterparts for DT dynamical systems (Bertsekas and Tsitsiklis, 1989; Blum, 1990), Usually, an asynchronous scheme for a DT system defined by a map f : X -+ X, where X = Xl X X2 X '" X X n , consists of a family {Ti ~ N I i = 1, , .. , n} of subset~ of discrete times (N) at which components i update their states and a family {rJ : N -+ N I i = 1,2"", n} of communication times, Asynchronous dynamics (or chaotic iteration, relaxation) is then given by X.(t + 1) = { fi(xl(rt(t)), "', xn(r~(t))) if t E ~ I Xi(t) otherwise. Notice that the sets Ti can be interpreted as local times of components i. In fact, one can define local time functions Ci : N -+ N as Ci(O) = 0 and Ci(t + 1) = Ci(t) + 1 if t E 11 and Ci(t) otherwise. The asynchronous dynamics can then be defined by Xi(t + 1) - Xi(t) = (Ci(t + 1) - ci(t))(fi(xl(rf(t)), ... ,Xn(r~(t))) - Xi(t)), which is analogous to the definition given in (4). Asynchronous Dynamics of Continuous Time Neural Networks 497 3 ASYNCHRONIZABLE SYSTEMS In general, we consider a CT dynamical system as asynchronizable ifits synchronous dynamics (limit sets and their asymptotic stability) persists for some set of asynchronous schemes. In many cases, asynchronous dynamics of an arbitrary CT system will be different from its synchronous dynamics, especially when delay times in communication are present. An example can be given for the network (1) with symmetric matrix W. It is well-known that (synchronous) dynamics of such networks is quasi-convergent, namely, all trajectories approach a set of fixed points (Hirsch, 1989). But when delay times are taken into consideration, the networks may have sustained oscillation when the delays exceed some threshold (Marcus and Westervelt, 1989). A more careful analysis on oscillation induced by delays is given in (Wu, 1993) for the networks with symmetric circulant weight matrices. Here, we focus on asynchronizable systems. We consider CT dynamical systems on Rn of the following general form Ax(t) = -x(t) + F(x(t» (8) where x(t) ERn, A = diag(a1,a2, ... ,an) with aj > 0 and F = [Ji] E G1(Rn). It is easy to see that a point x E Rn is a fixed point of (8) if and only if x is a fixed point of the map F. Without loss of generality, we assume that 0 is a fixed point of the map F. According to (5), the asynchronized version of (8) for an arbitrary asynchronous scheme ({ cd, { rj}) is Ay = G'( -y + F[Y]), where jj = (jjtct), jj~(t), ... , y~(t)]. 3.1 Contractive Systems (9) Our first effort attempts to obtain a result similar to the one for P-contractive maps in (Baudet, 1978). We call the system (8) strongly P-contractive if there is a symmetric and invertible matrix S such that IS- 1 F(Sx)1 < Ixl for all x E Rn and IS- 1 F(Sx)1 = Ixl only for x = 0; here Ixl denotes the vector with components Ixil and < is component-wise. Theorem 1 If the system (8) is strongly P-contractive, then it is asynchronizable for any asynchronous schemes without self time delays (i. e., rf (t) = Ci(t) for all i=1,2, ... ,n). Proof. It is not hard to see that synchronous dynamics of a strongly P-contractive system is globally convergent to the fixed point O. Now, consider the transformation z = A- 1y and the system for z Ai = G'( -z + S-1 F[SZ]) = G'( -z + G[Z]), where G[Z] = S-1 FS[Z]. This system has the same type of dynamics as (9). Define a function E : R+ x Rn --+ R+ by E(t) = z T (t)Az(t)j2, whose derivative with respect to t is E = z T G' (-z + G(Z» < IIG'II (-z T z + IzlT IG(Z)!) < IIG'II( -z T z + IzlT Izl) ::; O. 498 Wang, Li, and Blum Hence E is an energy function and the asynchronous dynamics converges to the fixed point O. 0 Our second result is for asynchronous dynamics of contractive systems with no communication delay. The system (8) is called contractive if there is a real constant o ~ a < 1 such that IIF(x) - F(y)1I ~ allz - yll for all x, y E Rn; here II . II denotes the usual Euclidean norm on Rn. Theorem 2 If the system (8) is contractive, then it is asynchronizable for asynchronous schemes with no communication delay. Proof. The synchronous dynamics of contractive systems is known to be globally convergent to a unique fixed point (Kelly, 1990). For an asynchronous scheme with no communication delay, the system (8) is simplified to Ali = G'( -y + F(y». We consider again the function E = y T Ay/2, which is an energy function as shown below. E = Y T G' (-y + F(y» ~ IIG/II( -lIyll2 + lIyIlIlF(y)ID < O. Therefore, the asynchronous dynamics converges to the fixed point O. For the additive-type neural networks (1), we have o Corollary 1 Let the network (1) have neuron activation functions Ui of sigmoidal type with 0 < uHz) ~ SUPzER ui(z) = 1. If it satisfies the condition (10) where M = diag(J-ll, ... , J-ln), then it is asynchronizable for any asynchronous schemes with no communication delay. Proof. The condition (10) ensures the map F(x) = A-I Wu(M x) + A- 1 I to be contractive. 0 Notice that the condition (10) is equivalent to many existing ones on globally asymptotical stability based on various norms of matrix W, especially the contraction condition given in (Kelly, 1990) and some very recent ones in (Matsuoka, 1992). The condition (10) is also related very closely to the condition in (Barhen and Gulati, 1989) for asynchronous dynamics of a discretized version of (1) and the condition in (Marcus and Westervelt, 1989) for the networks with delay. We should emphasize that the results in Theorem 2 and Corollary 1 do not directly follow from the result in (Kelly, 1990); this is because local times Ci(t) are allowed to be much more general functions than linear ones Ci t. 3.2 Monotone Systems A binary relation ~ on Rn is called a partial order if it satisfies that, for all x, y, z E Rn, (i) x ~ x; (ii) x ~ y and y ~ x imply x = y; and (iii) x -< y and y -< z imply x -< z. For a partial order ~ on Rn, define ~ on Rn by x ~ y iff x < y and Xi # Yi for all i = 1, .. " n. A map F : Rn -I- Rn is monotone if x ~ y implies Asynchronous Dynamics of Continuous Time Neural Networks 499 F(x) -< F(y). A CT dynamical system of the form (2) is monotone if Xl ~ X2 implies the trajectories Xl(t), X2(t) with Xl(O) = Xl and X2(0) = X2 satisfy Xl(t) ::5 X2(t) for all t ~ 0 (Hirsch, 1988). Theorem 3 If the map F in (8) is monotone, then the system (8) is asynchronizable for uniformly-delayed asynchronous schemes, provided that all orbits x(t) have compact orbit closure and there is a to > 0 with x(to) ~ x(O) or x(to) ~ x(O). Proof. This is an application of a Henry's theorem (see Hirsch, 1988) that implies that the asynchronized system (9) in the no communication delay situation is monotone and Hirsch's theorem (Hirsch, 1988) that guarantees the asymptotic convergence of monotone systems to fixed points. 0 Corollary 2 If the additive-type neural network (1) with sigmoidal activation functions is cooperative (i.e., Wij > 0 for i # j (Hirsch, 1988 and 1989)), then it is asynchronizable for uniformly-delayed asynchronous schemes, provided that there is a to > 0 with x(to) ~ x(O) or x(to) ~ x(O). Proof. According to (Hirsch, 1988), cooperative systems are monotone. As the network has only bounded dynamics, the result follows from the above theorem. 0 4 CONCLUSION By incorporating the concepts of local times and communication times, we have provided a mathematical formulation of asynchronous dynamics of continuous-time dynamical systems. Asynchronized systems in the most general form haven't been studied in theories of dynamical systems and functional differential equations. For contractive and monotone systems, we have shown that for some asynchronous schemes, the systems are asynchronizable, namely, their asynchronizations preserve convergent dynamics of the original (synchronous) systems. When applying these results to the additive-type neural networks, we have obtained some special conditions for the networks to be asynchronizable. We are currently investigating more general results for asynchronizable dynamical systems, with a main interest in oscillatory dynamics. References G. M. Baudet (1978). Asynchronous iterative methods for multiprocessors. Journal of the Association for Computing Machinery, 25:226-244. J. Barhen and S. Gulati (1989). "Chaotic relaxation" in concurrently asynchronous neurodynamics. In Proceedings of International Conference on Neural Networks, volume I, pages 619-626, San Diego, California. Bertsekas and Tsitsiklis (1989). Parallel and Distributed Computation: Numerical Methods. Englewood Cliffs, N J: Prentice Hall. E. K. Blum (1990). Mathematical aspects of outer-product asynchronous contentaddressable memories. Biological Cybernetics, 62:337-348, 1990. 500 Wang, Li, and Blum E. K. Blum and X. Wang (1992). Stability of fixed-points and periodic orbits, and bifurcations in analog neural networks. Neural Networks, 5:577-587. J. Hale (1977). Theory of Functional Differential Equations. New York: SpringerVerlag. M. W. Hirsch (1988). Stability and convergence in strongly monotone dynamical systems. J. reine angew. Math., 383:1-53. M. W. Hirsch (1989). Convergent activation dynamics in continuous time networks. Neural Networks, 2:331-349. J. Hopfield (1982). Neural networks and physical systems with emergent computational abilities. Proc. Nat. Acad. Sci. USA, 79:2554-2558. J. Hopfield (1984). Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. USA, 81:30883092. D. G. Kelly (1990). Stability in contractive nonlinear neural networks. IEEE Trans. Biomedi. Eng., 37:231-242. Q. Li (1993). Mathematical and Numerical Analysis of Biological Neural Networks. Unpublished Ph.D. Thesis, Mathematics Department, University of Southern California. C. M. Marcus and R. M. Westervelt (1989). Stability of analog neural networks with delay. Physical Review A, 39(1):347-359. K. Matsuoka (1992). Stability conditions for nonlinear continuous neural networks with asymmetric connection weights. Neural Networks, 5:495-500. J. M. Ortega and W. C. Rheinboldt (1970). Iterative solution of nonlinear equations in several variables. New York: Academic Press. X. Wang, E. K. Blum, and Q. Li (1993). Consistency on Local Dynamics and Bifurcation of Continuous-Time Dynamical Systems and Their Discretizations. To appear in the AMS proceedings of Symposia in Applied Mathematics, Mathematics of Computation 1943 - 1993, Vancouver, BC, August, 1993, edited by W. Gautschi. X. Wang and E. K. Blum (1992). Discrete-time versus continuous-time neural networks. Computer and System Sciences, 49:1-17. X. Wang and D. S. Parker (1992). Computing least fixed points by asynchronous iterations and random iterations. Technical Report CSD-920025, Computer Science Department, UCLA. J .-H. Wu (1993). Delay-Induced Discrete Waves of Large Amplitudes in Neural Networks with Circulant Connection Matrices. Preprint, Department of Mathematics and Statistics, York University.
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896 Digital Boltzmann VLSI for constraint satisfaction and learning Michael Murray t Ming-Tak Leung t Kan Boonyanit t Kong Kritayakirana t James B. Burrt* Gregory J. Wolff+ Takahiro Watanabe+ Edward Schwartz+ David G. Storktt Allen M. Petersont t Department of Electrical Engineering Stanford University Stanford, CA 94305-4055 +Ricoh California Research Center 2882 Sand Hill Road Suite 115 Menlo Park, CA 94025-7022 and * . Sun Mlcrosystems 2550 Garcia Ave., MTV-29, room 203 Mountain View, CA 94043 Abstract We built a high-speed, digital mean-field Boltzmann chip and SBus board for general problems in constraint satjsfaction and learning. Each chip has 32 neural processors and 4 weight update processors, supporting an arbitrary topology of up to 160 functional neurons. On-chip learning is at a theoretical maximum rate of 3.5 x 108 connection updates/sec; recall is 12000 patterns/sec for typical conditions. The chip's high speed is due to parallel computation of inner products, limited (but adequate) precision for weights and activations (5 bits), fast clock (125 MHz), and several design insights. Digital Boltzmann VLSI for Constraint Satisfaction and Learning 897 1 INTRODUCTION A vast number of important problems can be cast into a form of constraint satisfaction. A crucial difficulty when solving such problems is the fact that there are local minima in the solution space, and hence simple gradient descent methods rarely suffice. Simulated annealing via the Boltzmann algorithm (BA) is attractive because it can avoid local minima better than many other methods (Aarts and Korst, 1989). It is well known that the problem of learning also generally has local minima in weight (parameter) space; a Boltzmann algorithm has been developed for learning which is effective at avoiding local minima (Ackley and Hinton, 1985). The BA has not received extensive attention, however, in part because of its slow operation which is due to the annealing stages in which the network is allowed to slowly relax into a state of low error. Consequently there is a great need for fast and efficient special purpose VLSI hardware for implementing the algorithm. Analog Boltzmann chips have been described by Alspector, Jayakumar and Luna (1992) and by Arima et al. (1990); both implement stochastic BA. Our digital chip is the first to implement the deterministic mean field BA algorithm (Hinton, 1989), and although its raw throughput is somewhat lower than the analog chips just mentioned, ours has unique benefits in capacity, ease of interfacing and scalability (Burr, 1991, 1992). 2 BOLTZMANN THEORY The problems of constraint satisfaction and of learning are unified through the Boltzmann learning algorithm. Given a partial pattern and a set of constraints, the BA completes the pattern by means of annealing (gradually lowering a computational "temperature" until the lowest energy state is found) an example of constraint satisfaction. Over a set of training patterns, the learning algorithm modifies the constraints to model the relationships in the data. 2.1 CONSTRAINT SATISFACTION A general constraint satisfaction problem over variables Xi (e.g., neural activations) is to find the set Xi that minimize a global energy function E = -~ Lij WijXiXj, where Wij are the (symmetric) connection weights between neurons i and j and represent the problem constraints. There are two versions of the BA approach to minimizing E. In one version the stochastic BA each binary neuron Xi E {-I, I} is polled randomly, independently and repeatedly, and its state is given a candidate perturbation. The probability of acceptance of this perturbation depends upon the amount of the energy change and the temperature. Early in the annealing schedule (Le., at high temperature) the probability of acceptance is nearly independent of the change in energy; late in annealing (Le., at low temperature), candidate changes that lead to lower energy are accepted with higher probability. In the deterministic mean field BA, each continuous valued neuron (-1 < Xi ::; 1) is updated simultaneously and in parallel, its new activation is set to Xi = I(Lj WijXj), where 10 is a monotonic non-linearity, typically a sigmoid which corresponds to a stochastic unit at a given temperature (assuming independent 898 Murray, Leung, Boonyanit, Kritayakirana, Burr, Wolff, Watanabe, Schwartz, Stork, and Peterson inputs). The inverse slope of the non-linearity is proportional to the temperature; at the end of the anneal the slope is very high and f ( .) is effectively a step function. It has been shown that if certain non-restrictive assump'tions hold, and if the annealing schedule is sufficiently slow, then the final binary states (at 0 temperature) will be those of minimum E (Hinton, 1989, Peterson and Hartman, 1989). 2.2 LEARNING The problem of Boltzmann learning is the following: given a network topology of input and output neurons, interconnected by hidden neurons, and given a set of training patterns (input and desired output), find a set of weights that leads to high probability of a desired output activations for the corresponding input activations. In the Boltzmann algorithm such learning is achieved using two main phases the Teacher phase and the Student phase followed by the actual Weight update. During the Teacher phase the network is annealed with the inputs and outputs clamped (held at the values provided by the omniscient teacher). During the anneal of the Student phase, only the inputs are clamped the outputs are allowed to vary. The weights are updated according to: D..Wij = €( (x!x;) - (x:xj)) (1) where € is a learning rate and (x~x;) the coactivations of neurons i and j at the end of the Teacher phase and (x:xj) in at the end of the Student phase (Ackley and Hinton, 1985). Hinton (1989) has shown that Eq. 1 effectively performs gradient descent on the cross-entropy distance between the probability of a state in the Teacher (clamped) and the Student (free-running) phases. Recent simulations by Galland (1993) have shown limitations of the deterministic BA for learning in networks having hidden units directly connected to other hidden units. While his results do not cast doubt on the deterministic BA for constraint satisfaction, they do imply that the deterministic BA for learning is most successful in networks of a single hidden layer. Fortunately, with enough hidden units this topology has the expressive power to represent all but the most pathological inputoutput mappings. 3 FUNCTIONAL DESIGN AND CHIP OPERATION Figure 1 shows the functional block diagram of our chip. The most important units are the Weight memory, Neural processors, Weight update processors, Sigmoid and Rotating Activation Storage (RAS), and their operation are best explained in terms of constraint satisfaction and learning. 3.1 CONSTRAINT SATISFACTION For constraint satisfaction, the weights (constraints) are loaded into the Weight memory, the form of the transfer function is loaded into the Sigmoid Unit, and the values and duration of the annealing temperatures (the annealing schedule) are loaded into the Temperature Unit. Then an input pattern is loaded into a bank of the RAS to be annealed. Such an anneal occurs as follows: At an initial high Digital Boltzmann VLSI for Constraint Satisfaction and Learning 899 temperature, the 32 Neural processors compute Xi = Lj WijXj in parallel for the hidden units. A 4 x multiplexing here permits networks of up to 128 neurons to be annealed, with the remaining 32 neurons used as (non-annealed) inputs. Thus our chip supports networks of up to 160 neurons total. These activations are then stored in the Neural Processor Latch and then passed sequentially to the Sigmoid unit, where they are multiplied by the reciprocal of the instantaneous temperature. This Sigmoid unit employs a lookup table to convert the inputs to neural outputs by means of non-linearity f(·). These outputs are sequentially loaded back into the activation store. The temperature is lowered (according to the annealing schedule), and the new activations are calculated as before, and so on. The final set of activations Xi (i.e., at the lowest temperature) represent the solution. Rotating Activation 1 Sigmoid r-----t .... 4 weight update processors weight update cache Weight memory 32 Neural Processors (NP) Figure 1: Boltzmann VLSI block diagram. The rotating activation storage (black) consists of three banks, which for learning problems contain the last pattern (already annealed), the current pattern (being annealed) and the next pattern (to be annealed) read onto the chip through the external interface. 3.2 LEARNING When the chip is used for learning, the weight memory is initialized with random weights and the first, second and third training patterns are loaded into the RAS. The three-bank RAS is crucial for our chip's speed because it allows a three-fold 900 Murray, Leung, Boonyanit, Kritayaldrana, Burr, Wolff, Watanabe, Schwartz, Stork, and Peterson concurrency: 1) a current pattern of activations is annealed, while 2) the annealed last pattern is used to update the weights, while 3) the next pattern is being loaded from off-chip. The three banks form a circular buffer, each with a Student and a Teacher activation store. During the Teacher anneal phase (for the current pattern), activations of the input and output neurons are held at the values given by the teacher, and the values of the hidden units found by annealing (as described in the previous subsection). After the last such annealling step (Le., at the lowest temperature), the final activations are left in the Teacher activation store the Teacher phase is then complete. The annealing schedule is then reset to its initial temperature, and the above process is then repeated for the Student phase; here only the input activations are clamped to their values and the outputs are free to vary. At the end of this Student anneal, the final activations are left in the Student activation storage. In steady state, the MUX then rotates the storage banks of the RAS such that the next, current, and last banks are now called the current, last, and next, respectively. To update the weights, the activations in the Student and Teacher storage bank for the pattern just annealed (now called the "last" pattern) are sent to the four Weight update processors, along with the weights themselves. The Weight update processors compute the updated weights according to Eq. 1, and write them back to the Weight memory. While such weight update is occuring for the last pattern, the current pattern is annealing and the next pattern is being loaded from off chip. After the chip has been trained with all of the patterns, it is ready for use in recall. During recall, a test pattern is loaded to the input units of an activation bank (Student side), the machine performs a Student anneal and the final output activations are placed in the Student activation store, then read off the chip to the host computer as the result. In a constraint satisfaction problem, we merely download the weights (constraints) and perform a Student anneal. 4 HARDWARE IMPLEMENTATION Figure 2 shows the chip die. The four main blocks of the Weight memory are at the top, surrounded by 32 Neural processors (above and below this memory), and four Weight update processors (between the memory banks). The three banks of the Rotating Activation Store are at the bottom of the chip. The Sigmoid processor is at the lower left, and instruction cache and external interface at the lower right. Most of the rest of the chip consists of clocking and control circuitry. 4.1 VLSI The chip mixes dynamic and static memory on the same die. The Activation and Temperature memories are static RAM (which needs no refresh circuitry) while the Weight memory is dynamic (for area efficiency). The system clock is distributed to various local clock drivers in order to reduce the global clock capacitance and to selectively disable the clocks in inactive subsystems for reducing power consumption. Each functional block has its own finite state machine control which communicates Digital Boltzmann VLSI for Constraint Satisfaction and Learning 901 .. " ._ • ...-. .. .... • . - , "o.t ' . . IM .... . . '7 ","", Figure 2: Boltzmann VLSI chip die. asynchronously. For diagnostic purposes, the State Machines and counters are observable through the External Interface. There is a Single Step mode which has been very useful in verifying sub-system performance. Figure 3 shows the power dissipation throughout a range of frequencies. Note that the power is less than 2 Watts throughout. Extensive testing of the first silicon revealed two main classes of chip error: electrical and circuit. Most of the electrical problems can be traced to fast edge rates on the DRAM sense-amp equalization control signals, which cause inductive voltage transients on the power supply rails of roughly 1 Volt. This appears to be at least partly responsible for the occasional loss of data in dynamic storage nodes. There also seems to be insufficient latchup protection in the pads, which is aggravated by the on-chip voltage surges. The circuit problems can be traced to having to modify the circuits used in the layout for full chip simulation. In light of these problems, we have simulated the circuit in great detail in order to explore possible corrective steps. We have modified the design to provide improved electrical isolation, resized drivers and reduced the logic depth in several components. These corrections solve the problems in simulation, and give us confidence that the next fab run will yield a fully working chip. 4.2 BOARD AND SBus INTERFACE An SBus interface board was developed to allow the Boltzmann chip to be used with a SparcStation host. The registers and memory in the chip can be memory mapped so that they are directly accessible to user software. The board can support 902 Murray, Leung, Boonyanit, Kritayakirana, Burr, Wolff, Watanabe, Schwartz, Stork, and Peterson Table 1: Boltzmann VLSI chip specifications Architecture Size Neurons Weight memory Activation store Technology Transistors Pins Clock I/O rate Learning rate Recall rate Power dissipation n-Iayer, arbitrary intercoItnnections 9.5 mm x 9.8 mm 32 processors --+ 160 virtual 20,480 5-bit weights (on chip) 3 banks, 160 teacher & 160 student values in each 1. 2 11m CMOS 400,000 84 125 MHz (on chip) 3 x 107 activations/sec (sustained) 3.5 x 108 connection updates/sec (on chip) 12000 patterns/sec :::;2 Watts (see Figure 3) 20-bit transfers to the chip at a sustained rate in excess of 8 Mbytes/second. The board uses reconfigurable Xilinx FPGAs (field-programmable gate arrays) to allow flexibility for testing with and without the chip installed. 4.3 SOFTWARE The chip control program is written in C (roughly 1,500 lines of code) and communicates to the Boltzmann interface card through the virtual memory. The user can read/write to all activation and weight memory locations and all functions of the chip (learning, recall, annealing, etc.) can thus be specified in software. 5 CONCLUSIONS AND FUTURE WORK The chip was designed so that interchip communications could be easily incorporated by means of high-speed parallel busses. The SBus board, interface and software described above will require only minor changes to incorporate a multi-chip module (MCM) containing several such chips (for instance 16). There is minimal 2 1. 75 til 1.5 .w .w 1. 25 111 2: 1 ~ 0.75 Q) ~ 0.5 0 0. 0.25 0 i I ---i ,--f--T i , i ; I , i , ! i i I i I i , I 50 60 70 80 90 100 110 frequency, MHz Figure 3: Power dissipation of the chip during full operation at 5 Volts. Digital Boltzmann VLSI for Constraint Satisfaction and Learning 903 inter chip communication delay « 3% overhead), and thus MCM versions of our system promise to be extremely powerful learning systems for large neural network problems (Murrayet al., 1992). Acknowledgements Thanks to Martin Boliek and Donald Wynn for assistance in design and construction of the SBus board. Research support by NASA through grant NAGW419 is gratefully acknowledged; VLSI fabrication by MOSIS. Send reprint requests to Dr. Stor k: stor k@crc.ricoh.com. References E. Aarts & J. Korst. (1989) Simulated Annealing and Boltzmann Machines: A stochastic approach to combinatorial optimization and neural computing. New York: Wiley. D. H. Ackley & G. E. Hinton. (1985) A learning algorithm for Boltzmann machines. Cognitive Science 9, 147-169. J. Alspector, A. Jayakumar & S. Luna. (1992) ExpeJimental evaluation of learning in a neural microsystem. Advances in Neural Information Processing Systems-4, J. E. Moody, S. J. Hanson & R. P. Lippmann (eds.), San Mateo, CA: Morgan Kaufmann, 871-878. Y. Arima, K. Mashiko, K. Okada, T. Yamada, A. Maeda, H. Kondoh & S. Kayano. (1990) A self-learning neural network chip with 125 neurons and 10K selforganization synapses. In Symposium on VLSI Circuits, Solid State Circuits Council Staff, Los Alamitos, CA: IEEE Press, 63-64. J. B. Burr. (1991) Digital Neural Network Implementations. Neural Networks: Concepts, Applications, and Implementations, Volume 2, P. Antognetti & V. Milutinovic (eds.) 237-285, Englewood Cliffs, NJ: Prentice Hall. J. B. Burr. (1992) Digital Neurochip Design. Digital Parallel Implementations of Neural Networks. K. Wojtek Przytula & Viktor K. Prasanna (eds.), Englewood Cliffs, N J: Prentice Hall. C. C. Galland. (1993) The limitations of deterministic Boltzmann machine learning. Network 4, 355-379. G. E. Hinton. (1989) Deterministic Boltzmann learning performs steepest descent in weight-space. Neural Computation 1, 143-150. C. Peterson & E. Hartman. (1989) Explorations of the mean field theory learning algorithm. Neural Networks 2, 475-494. M. Murray, J. B. Burr, D. G. Stork, M.-T. Leung, K. Boonyanit, G. J. Wolff & A. M. Peterson. (1992) Deterministic Boltzmann machine VLSI can be scaled using multi-chip modules. Proc. of the International Conference on Application Specific Array Processors. Berkeley, CA (August 4-7) Los Alamitos, CA: IEEE Press, 206-217.
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Development of Orientation and Ocular Dominance Columns in Infant Macaques Klaus Obermayer Howard Hughes Medical Institute Salk-Institute La Jolla, CA 92037 Lynne Kiorpes Center for Neural Science New York University New York, NY 10003 Gary G. Blasdel Department of Neurobiology Harvard Medical School Boston, MA 02115 Abstract Maps of orientation preference and ocular dominance were recorded optically from the cortices of 5 infant macaque monkeys, ranging in age from 3.5 to 14 weeks. In agreement with previous observations, we found that basic features of orientation and ocular dominance maps, as well as correlations between them, are present and robust by 3.5 weeks of age. We did observe changes in the strength of ocular dominance signals, as well as in the spacing of ocular dominance bands, both of which increased steadily between 3.5 and 14 weeks of age. The latter finding suggests that the adult spacing of ocular dominance bands depends on cortical growth in neonatal animals. Since we found no corresponding increase in the spacing of orientation preferences, however, there is a possibility that the orientation preferences of some cells change as the cortical surface expands. Since correlations between the patterns of orientation selectivity and ocular dominance are present at an age, when the visual system is still immature, it seems more likely that their development may be an innate process and may not require extensive visual experience. 543 544 Obennayer, Kiorpes, and Blasdel 1 INTRODUCTION Over the past years, high-resolution images of the simultaneous representation of orientation selectivity and ocular dominance have been obtained in large areas of macaque striate cortex using optical techniques [3, 4, 5, 6, 12, 18]. These studies confirmed that ocular dominance and orientation preference are organized in large parts in slabs. While optical recordings of ocular dominance are in accordance with previous findings, it turned out that iso-orientation slabs are much shorter than expected, and that the orientation map contains several other important elements of organization - singularities, fractures, and saddle-points. A comparison between maps of orientation preference and ocular dominance, which were derived from the same region of adult monkey striate cortex, showed a pronounced relationship between both patterns [5, 12, 13, 15, 17]. Fourier analyses, for example, reveal that orientation preferences repeat at closer intervals along the ocular dominance slabs than they do across them. Singularities were found to align with the centers of ocular dominance bands, and the iso-orientation bands, which connect them, intersect the borders of ocular dominance bands preferably at angles close to 90°. Given the fact that these relationships between the maps of orientation and ocular dominance are present in all maps recorded from adult macaques, one naturally wonders how this organization matures. If the ocular dominance slabs were to emerge initially, for example, the narrower slabs of iso-orientation might later develop in between. This might seem likely given the anatomical segregation which is apparent for ocular dominance but not for orientation [9]. However, this possibility is contradicted by physiological studies that show normal, adult-like sequences of orientation preference in the early postnatal weeks in macaque when ocular dominance slabs are still immature [19]. The latter findings suggest a different developmental hypothesis; that the organization into regions of different orientation preferences may precede or even guide ocular dominance formation. A third possibility, consistent with both previous results, is that orientation and ocular dominance maps form independently and align in later stages of development. In order to provide evidence for one or the other hypothesis, we investigated the relationship between ocular dominance and orientation preference in very young macaque monkeys. Results are presented in the remainder of this paper. Section 2 contains an overview about the experimental data, and section 3 relates the data to previous modelling efforts. 2 ORIENTATION AND OCULAR DOMINANCE COLUMNS IN INFANT MACAQUES 2.1 THE OVERALL STRUCTURE Figure 1 shows the map of orientation preference (Fig. 1a) and ocular dominance (Fig. 1 b) recorded from area 17 of a 3.5 week old macaque. 1 Both maps look similar 1 For all animals orientation and ocular dominance were recorded from a region close to the border to area 18 and close to midline. Development of Orientation and Ocular Dominance Columns in Infant Macaques 545 a b C Figure 1: Spatial pattern of orientation preference and ocular dominance recorded from area 17 of a macaque, 3.5 weeks of age. Figures (a) and (b) show orientation preferences and ocular dominance bands within the same 3.1 mm x 4.3 mm large region of striate cortex. Brightness values in Fig. (a) indicate orientation preferences, where the interval of 180° is represented by the progression in colors from black to white. Brightness values in Fig. (b) indicate ocular dominance, where bright and dark denote ipsi- and contralateral eye-preference. respectively. The data was recorded from a region close to the border to area 18 and close to midline. Figure (c) shows an enlarged section of this map in the preference (left) and the in contour plot (right) representations. Iso-orientation lines on the right indicate intervals of 11.25°. Letters indicate linear zones (L), saddle points (H), singularities (S), and fractures (F). to maps which have been recorded from adults. The orientation map exhibits all of the local elements which have been described [12, 13]: linear zones, saddle points, 546 Obermayer, Kiorpes, and Blasdel a ')..r = 741pm ')..7; = 612pm c ::: • 00 .... A. = 724f.1m 1.0 -----------, c: o -.:::: ...-. 0-0 c: 41) 0.5 .a.~ cCU .2 E 100 c: 0.0 41)... ... o o -0.5 o 200 400 600 800 distance [~m] b -0 41) .~ (ij E .... 0 c: .... 41) ~ 0 a. 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 5 spatial frequency [l/mm] Figure 2: Fourieranalysis of the orientation map shown in Figure la. (a) Complex 2D-Fouriertransform. Each pixel corresponds to one Fouriermode and its blackness indicates the corresponding energy. A distance of one pixel corresponds to O.23/mm. (b) Power as a function of radial spatial frequency. (c) Autocorrelations Sij as a function of distance. The indices i, j E {3,4} denote the two cartesian coordinates of the orientation preference vector. For details on the calculation see [13, 15]. singularities, and fractures (Fig. lc). The ocular dominance map shows its typical pattern of alternating bands. Figure 2a shows the result of a complex 2D Fourier transform of the orientation map shown in Figure la. Like for maps recorded from adult monkeys [13] the spectrum is characterized by a slightly elliptical band of modes which is centered at the origin. The major axis approximately aligns with the axis parallel to the border to area 18 as well as with the ocular dominance bands. Therefore, like in the adults, the orientatiQn map is stretched perpendicular to the ocular dominance bands, apparently to adjust to the wider spacing. When one neglects the slight anisotropy of the Fourier spectra one can estimate a power spectrum by averaging the squared Fourier amplitudes for similar frequencies. The result is a pronounced peak whose location is given by the characteristic frequency of the orientation map (Fig. 2b). As a consequence, autocorrelation functions have a Mexican-hat shape (Fig. 2c), much like it has been reported for adults [13, 15]. In summary, the basic features of the patterns of orientation and ocular dominance are established as early as 3.5 weeks of age. Data which were recorded from four Development of Orientation and Ocular Dominance Columns in Infant Macaques 547 Table 1: Characteristic wavelengths (AOD) and signal strengths «TOD) for the ocular dominance pattern, as well as characteristic wavelengths p.op), density of +180°singularities (p+), density of -180°-singularities (p_), total density of singularities (p), and percentage of area covered by linear zones (alin) for the orientation pattern as a function of age. age UOD >'OD >'OP p+ pP alin (weeks) (}jm) (}jm) (mm- 2) (mm- 2) (mm- 2) (% area) 3.5 0.92 686 660 3.9 3.9 7.8 47 5.5 0.96 730 714 3.7 3.7 7.4 49 7.5 0.66 870 615 4.5 4.5 9.0 45 14 1.23 917 700 3.9 3.8 7.7 36 adult 1.36 950 768 3.9 3.8 7.7 43 other infants ranging from 5.5 to 14 weeks (not shown) confirm the above findings. 2.2 CHARACTERISTIC WAVELENGTHS AND SIGNAL STRENGTH A more detailled analysis of the recorded patterns, however, reveals changes of certain features with age. Table 1 shows the changes in the typical wavelength of the orientation and ocular dominance patterns as well as the (normalized) ocular dominance signal strength with age. The strength of the ocular dominance signal increases by a factor of 1.5 between 3.5 weeks and adulthood, a fact, which could be explained by the still ongoing segregation of fibers within layer IV c. The spacing of the ocular dominance columns increases by approximately 30% between 3.5 weeks and adulthood. This change in spacing would be consistent with the growth of cortical surface area during this period [16] if one assumes that cortex grows anisotropically in the direction perpendicular to the ocular dominance bands. Interestingly, the characteristic wavelengths of the orientation patterns do not exhibit such an increase. The wavelengths for the patterns recorded from the different infants are close to the "adult" values. More evidence for a stable orientation pattern is provided by the fact, that the density of'singularities is approximately constant with age2 and that the percentage of cortical area covered by linear zones does neither increase nor decrease. Hence we are left with the puzzle that at least the pattern of orientation does not follow cortical growth. 2.3 CORRELATIONS BETWEEN THE ORIENTATION AND OCULAR DOMINANCE MAPS Figure 3 shows a contour plot representation of the pattern of orientation preference in overlay with the borders of the ocular dominance bands for the 3.5 week old animal. Iso-orientation contours (thin lines) indicate intervals of 15°. Thick lines indicate the border of the ocular dominance bands. From visual inspection it is 2Note that both types of singularities appear in equal numbers. S48 Obennayer. Kiorpes. and Blasdel Figure 3: Contour plot representation of t.he orient.a.t.ion map shown in Figure la in overlay with the borders of the ocular dominance bands taken from Figure 1 b. Iso-orient.ation lines (thin lines) indicate intervals of 15°. The borders of the ocular dominance bands are indicated by thick lines. already apparent that singularities have a strong tendency to align with the center of the ocular dominance bands (arrow 1) and that in the linear zones (arrow 2), where iso-orientation bands exist, these bands intersect ocular dominance bands at angles close to 90° most of the time. Table 2 shows a quantitative analysis of the local intersection angle. Percentage of area covered by linear zones (cf. [12] for details of the calculation) is given for regions, where orientation bands intersect ocular dominance bands within 18° of perpendicular, and regions where they intersect within 18° of parallel. For all of the animals investigated the percentages are two to four times higher for regions, where orientation bands intersect ocular dominance bands at angles close to 90°, much like it has been observed in adults [12]. In particular there is no consistent trend with age: the correlations between the orientation and ocular dominance maps are established as early as 3.5 weeks of age. age perp par Table 2: Percentage of area covered by aUn a'in linear zones as a function of age for re(weeks) (%area) (% area) gions, where orientation bands inter3.5 15.9 4.1 sect ocular dominance bands within 18° of perpendicular (af/::'P) , and re5.5 12.2 6.8 7.5 13.3 6.2 gions where they intersect within 18° 14 12.4 3.7 of parallel (afi~) (cf. [12] for details of adult 18.0 2.7 the calculation). Development of Orientation and Ocular Dominance Columns in Infant Macaques 549 3 CONCLUSIONS AND RELATION TO MODELLING In summary, our results provide evidence that the pattern of orientation is established at a time when the pattern of ocular dominance is still developing. However, they provide also evidence for the fact that the pattern of orientation is not linked to cortical growth. This latter finding still needs to be firmly established in studies where the development of orientation is followed in one and the same animal. But if it is taken seriously the consequence would be that orientation preferences may shift. and that pairs of singularities are formed. The early presence of strong correlations between both maps indicate that the development of orientation and ocular dominance are not independent processes. Both patterns have to adjust. to each other while cortex is growing. It, therefore, seems as if the third hypothesis is true (see Introduction) which states that both patterns develop independently and adjust to each other in the late stages of development. As has been shown in [13, 15] and is suggested in [7, 14] these processes are certainly in the realm of models based on Hebbian learning. Many features of the orientation and ocular dominance maps are present at an age when the visual system of the monkey is still immature [8, 11]. In particular, they are present at a time when spatial vision is strongly impared. Consequently, it seems unlikely that the development of these features as well as of the correlations between both patterns requires high acuity form vision, and models which try to predict the structure of these maps from the structure of visual images [1, 2, 10] have to take this fact into account. The early development of orientation preference and its correlations with ocular dominance make it also seem more likely that their development may me an innate process and may not require extensive visua.l experience. Further experiments, however, are needed to settle these questions. Acknowledgements This work was funded in part by the Klingenstein Foundation, the McKnight Foundation, the New England Primate Research Center (P51RR0168-31), the Seaver Institute, and the Howard Hughes Medical Institute. We thank Terry Sejnowski, Peter Dayan, and Rich Zemel for useful comments on the manuscript. Linda Ascomb, Jaqueline Mack, and Gina Quinn provided excellent technical assistance. References [1] H. G. Barrow and A. J. Bray. Activity induced color blob formation . In I. Alexander and J. Taylor, editors, Artificial Neural Networks II, pages 5-9. Elsevier Publishers, 1992. [2] H. G. Barrow and A. J. Bray. A model of the adaptive development of complex cortical cells. In I. Alexander and J. Taylor, editors, A rtificial Neural Networks II, pages 1-4. Elsevier Publishers, 1992. [3] E. Bartfeld and A. Grinvald. Relationships between orientation-preference pinwheels, cytochrome oxidase blobs, and ocular-dominance columns in primate striate cortex. Proc. Nail. Acad. Sci. USA, 89:11905-11909, 1992. 550 Obennayer, Kiorpes, and Blasdel [4] G. G. Blasdel. Differential imaging of ocular dominance and orientat.ion selectivity in monkey striate cortex. J. Neurosci., 12:3117-31:>8, 1992. [5] G. G. Blasdel. Orientation selectivity, preference, and continuity in monkey striate cortex. J. Neuroscl., 12:3139-3161, 1992. [6] G. G. Blasdel and G. Salama. Voltage sensitive dyes reveal a modular organization in monkey striate cortex. Nature, 321:579-585, 1986. [7] R. Durbin and G. Mitchison. A dimension reduction framework for understanding cortical maps. Nature, 343:644-647, 1990. [8] L. Kiorpes and T. Movshon. Behavioural analysis of visual development. In J. R. Coleman, editor, Development of Sensory Systems m Mammals, pages 125-154. John Wiley, 1990. [9] S. LeVay, D. H. Hubel, and T. N. Wiesel. The development of ocular dominance columns in normal and visually deprived monkeys. J. Compo Neurol., 191:1-51, 1980. [10] Y. Liu and H. Shouval. Principal component analysis of natural images - an analytic solution. Preprint. [11] T. Movshon and L. Kiorpes. Biological limits on visual development. in primates. In K. Simon, editor, Handbook of Infant Vision. Oxford University Press, 1993. in press. [12] K. Obermayer and G. G. Blasdel. Geometry of orientation and ocular dominance columns in monkey striate cortex. J. Neurosci., 13:4114-4129, 1993. [13] K. Obermayer, G. G. Blasdel, and K. Schulten. A statistical mechanical a.nalysis of self-organization and pattern formation during the development of visual maps. Phys. Rev. A15, 45:7568-7589, 1992. [14] K. Obermayer, H. Ritter, and K. Schulten. A principle for the forma.t.ion of the spatial structure of cortical feature maps. Proc. Natl. Acad. Sci. USA, 87:8345-8349, 1990. [15] K. Obermayer, K. Schulten, and G. G. Blasdel. A comparison of a neural network model for the formation of brain maps with experimental data. In D. S. Touretzky and R. Lippman, editors, Advances in Neural Information Processing Systems ./, pages 83-90. Morgan Kaufmann Publishers, 1992. [16] D. Purves and A. LaMantia. Development of blobs in the visual cortex of macaques. J. Compo Neurol., 332:1-7,1993. [17] N. Swindale. A model for the coordinated development of columnar systems in primate striate cortex. Bioi. Cybern., 66:217-230, 1992. [18] D. Y. Tso, R. D. Frostig, E. E. Lieke, and A. Grinvald. Functional organization of primate visual cortex revealed by high resolution optical ima.ging. Science, 249:417-420, 1990. [19] T. N. Wiesel and D. H. Rubel. Ordered arrangement of orientation columns in monkeys lacking visual experience. J. Compo Neurol., 158:307-318, 1974.
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Counting function theorem for multi-layer networks Adam Kowalczyk Telecom Australia, Research Laboratories 770 Blackburn Road, Clayton, Vic. 3168, Australia (a.kowalczyk@trl.oz.au) Abstract We show that a randomly selected N-tuple x of points ofRn with probability> 0 is such that any multi-layer percept ron with the first hidden layer composed of hi threshold logic units can implement exactly 2 2:~~~ ( Nil) different dichotomies of x. If N > hin then such a perceptron must have all units of the first hidden layer fully connected to inputs. This implies the maximal capacities (in the sense of Cover) of 2n input patterns per hidden unit and 2 input patterns per synaptic weight of such networks (both capacities are achieved by networks with single hidden layer and are the same as for a single neuron). Comparing these results with recent estimates of VC-dimension we find that in contrast to the single neuron case, for sufficiently large nand hl, the VC-dimension exceeds Cover's capacity. 1 Introduction In the course of theoretical justification of many of the claims made about neural networks regarding their ability to learn a set of patterns and their ability to generalise, various concepts of maximal storage capacity were developed. In particular Cover's capacity [4] and VC-dimension [12] are two expressions of this notion and are of special interest here. We should stress that both capacities are not easy to compute and are presen tly known in a few particular cases of feedforward networks only. VC-dimension, in spite of being introduced much later, has been far 375 376 Kowalczyk more researched, perhaps due to its significance expressed by a well known relation between generalisation and learning errors [12, 3]. Another reason why Cover's capacity gains less attention, perhaps, is that for the single neuron case it is twice higher than VC-dimension. Thus if one would hypothesise a similar relation to be true for other feedforward networks, he would judge Cover's capacity to be quite an unattractive parameter for generalisation estimates, where VC-dimension is believed to be unrealistically big. One of the aims of this paper is to show that this last hypothesis is not true, at least for some feedforward networks with sufficiently large number of hidden units. In the following we will always consider multilayer perceptrons with n continuously-valued inputs, a single binary output, and one or more hidden layers, the first of which is made up of threshold logic units only. The derivation of Cover's capacity for a single neuron in [4] is based on the so-called Function Counting Theorem, proved for the linear function in the sixties (c.f. [4]), which states that for an N -tuple i of points in general position one can implement C( N, n) deC 2 2::~=o (Nil) different dichotomies of i. Extension of this result to the multilayer case is still an open problem (c.f. T. Cover's address at NIPS'92). One of the complications arising there is that in contrast to the single neuron case even for perceptrons with two hidden units the number of implementable dichotomies may be different for different N -tuples in general position [8]. Our first main result states that this dependence on i is relatively weak, that for a multilayer perceptron the number of implementable dichotomies (counting function) is constant on each of a finite number of connected components into which the space of N-tuples in general position can be decomposed. Then we show that for one of these components C(N, nh1 ) different dichotomies can be implemented, where hl is the number of hidden units in the first hidden layer (all assumed to be linear threshold logic units). This leads to an upper bound on Cover's capacity of 2n input patterns per (hidden) neuron and 2 patterns per adjustable synaptic weight, the same as for a single neuron. Comparing this result with a recent lower bound on VC-dimension of multilayer perceptrons [10] we find that for for sufficiently large nand hl the VC-dimension is higher than Cover's capacity (by a factor log2(h1)). The paper extends some results announced in [5] and is an abbreviated version of a forthcoming paper [6J. 2 Results 2.1 Standing assumptions and basic notation We recall that in this paper a multilayer perceptron means a layered feedforward network with one or more hidden layers, and the first hidden layer built exclusively from threshold logic units. A dichotomy of an N-tuple i = (Xl, ... , XN) E (Rn)N is a function 6: {Xl, ... , XN} {0,1}. For a multilayer perceptron F : Rn {O,l} let i ~ CF(i) denote the number of different dichotomies of i which can be implemented for all possible selections of synaptic weights and biases. We shail call CF(i) a counting function following the terminology used in [4]. Counting Function Theorem for Multi-Layer Networks 377 Example 1. C¢(x) = C(N, n) def 2 :E?=o (Nil) for a single threshold logic unit ¢ : R n -+ {O, 1} [4]. 0 Points of an N-tuple x E (Rn)N are said to be in general po&ition if there does not exist an 1 ~r min(N, n - l)-dimensional affine hyperplane in R n containing (l + 2) of them. We use a symbol gP(n, N) C (Rn)N to denote that set of all N-tuples x in general position. Throughout this paper we assume to be given a probability measure dlJ def f dx on Rn such that the density f : Rn -+ R is a continuous function. 2.2 Counting function is locally constant We start with a basic characterisations of the subset gP(n, N) C (Rn)N. Theorem 1 (i) gP(n, N) is an open and dense subset of (Rn)N with a finite number of connected components. (ii) Any of these components is unbounded, has an infinite Lebesgue measure and has a positive probability measure. Proof outline. (i) The key point to observe is that gP(n, N) = {x : p(x) =I- O}, where p : (Rn)N -+ R is a polynomial on (Rn)N. This implies immediately that gP( n, N) is open and dense in (R n)N. The finite number of connected components follows from the results of Milnor [7] (c.f. [2]). (ii) This follows from an observation that each of the connected components Ci has the property that if (Xl, ... , XN) E Ci and a > 0, then (ax!, ... ,axN) E C,. 0 As Example 1 shows, for a single neuron the counting function is constant on gP(n, N). However, this may not be the case even for perceptrons with two hidden units and two inputs (c.f. [8, 6] for such examples and Corollary 8). Our first main result states that this dependence on x is relatively weak. Theorem 2 CF(X) is constant on connected components ofgP(n, N). Proof outline. The basic heuristic behind the proof of this theorem is quite simple. If we have an N-tuple x E (Rn)N which is split into two parts by a hyperplane, then this split is preserved for any sufficiently small perturbation Y E (R n)N of x, and vice versa, any split of y corresponds to a split of X. The crux is to show that if x is in general position, then a minute perturbation y of x cannot allow a bigger number of splits than is possible for x. We refer to [6] for details. 0 The following corollary outlines the main impact of Theorem 2 on the rest of the paper. It reduces the problem of investigation of the function CF(X) on gP(n, N) to a consideration of a set of individual, special cases of N-tuples which, in particular, are amenable to be solved analytically. Corollary 3 If x E gP(n, N), then CF(X) = CF(f) for a randomly &elected Ntuple f E (Rn)N with a probability> O. 378 Kowalczyk 2.3 A case of special component of gP( n, N) The following theorem is the crux of the paper. Theorem 4 There exists a connected component CC C gP( n , N) C (R n)N such that h1n (N -1) CF(i) = C(N, nh1 ) = 2 t; i (for i E CC) with equality iff the input and first hidden layer are fully connected. The synaptic weights to units not in the first hidden layer can be constant. U sing now Corollary 3 we obtain: Corollary 5 CF(i) = C(N, nh1) for i E (Rn)N with a probability> O. The component CC C gP(n, N) in Theorem 4 is defined as the connected component containing (1) where c : R __ Rn is the curve defined as c(t) de! (t, t2, ... ,tn) for t E Rand o < tt < t2 < ... < tN are some numbers (this example has been considered previously in [11]). The essential part of the proof of Theorem 4 is showing the basic properties of the N-tuple PN which will be described by the Lemma below. Any dichotomy h of the N-tuple fiN (c.f. 1) is uniquely defined by its value at C(tl) (2 options) and the set of indices 1 :s; il < i2 < ... < ile < N of all transitional pairs (C(ti;), C(ti;+I)), i.e. all indices i j such that h(C(ti;)) =f: h(C(ti;+I)), where j = 1, "'1 k, (additional (N;l) options). Thus it is easily seen that there exist altogether 2 (N;I) different dichotomies of PN for any given number k of transitional pairs, where 0 5 k < N. Lemma 6 Given integers n, N, h > 0, k ~ 0 and a dichotomy h of PN with k transitional pairs. (i) If k 5 nh, then there exist hyperplanes H(Wi,bi)' (Wi, bd E Rn x R, such that .(pj) = 9 (bo + t,.,9(W'. P; + b,») , (2) (3) for i = 1, ... , hand j = 1, "', N; here Vi de! 1 ifn is even and Vi del (_l)i ifn is odd, bo de! -0.5 if n is odd, h is even and h(po) = 1, and bo de! 0.5, otherwise. (ii) If k = nh, then Wij =f: 0 for j = 1, ... , nand i = 1, "'1 h, where Wi = (Will Wi2, ... ,Win)' (iii) If k > nh, then (2) and (3) cannot be satisfied. The proof of Lemma 6 relies on usage of the Vandermonde determinant and its derivatives. It is quite technical and thus not included here (c.f. [6] for details). 10 2 Counting Function Theorem for Multi-Layer Networks 379 Theorem 7 (Mitchison & Durbin [lO]f../·· Huang & Huang [6] .' ... Baum [2]. Sakurai [11] .... ......... 1 -t--~~-.I--~~~I~I~~~I~~~-.I--~.~ 1 2 5 1 0 102 1 03 104 Number of hidden units(h1) Figure 1: Some estimates of capacity. 3 Discussion 3.1 An upper bound on Cover's capacity The Cover's capacity (or just capacity) of a neural network F : R n -+ {O,1}, G ap( F), is defined as the maximal N such that for a randomly selected N -tuple i = (Xl, ... ,XN) E (Rn)N of points of Rn, the network can implement 1/2 of all dichotomies of x with probability 1 [4, 8]. Corollary 5 implies that Gap(F) is not greater than maximal N such that Gp(PN )/2N = G(N, nhl) ~ 1/2. (4) since any property which holds with probability 1 on (Rn)N must also hold probability 1 on GG (c.f Theorem 4). The left-hand-side of the above equation is just the sum of the binomial expansion of (1/2 + 1/2)N-l up to hln-th term, so, using the symmetry argument, we find that it is ~ 1/2 if and only if it has at least half of the all terms, i.e. when N - 1 + 1 ::; 2(hln + 1). Thus the 2(hln + 1) is the maximal value of N satisfying (4). 1 Now let us recall that a multilayer perceptron as in this paper can implement any dichotomy of any N-tuple x in general position if N < nhl + 1 [I, 11]. This leads to the following result: Theorem 7 nhl + 1 ~ Gap(F) ::; 2(nhl + 1). lNote that for large N the choice of cutoff value 1/2 is not critical, since the probability of a dichotomy being implementable drops rapidly as hi n a.pproa.ches 2N /2. 380 Kowalczyk N I #w 10 I/) dVC<F)/#w ~ 0) (Sakurai [11]) ·iii :it .2 8 i5.. co c >.. I/) 6 0 ... CD .a E ~ Z 4 ..... CIl ..---(Cap(F)/#W ) E g 15 (Theorem 7) c.. 2 "5 c.. .s 0 Figure 2: Comparison of estimates of the ratios of Cover's capacity per synaptic weight (Cap(F)/#w) and VC-dimension per synaptic weight (dvc(F)/#w). (Note that the upper bound for VC-dimension has so far been proved for low number of hidden layers [9,10].) for any multilayer perceptron F : R n -+ {O, I} with the first hidden layer built from the hi threshold logic units. For the most efficient networks in this class, with a single hidden layer, we thus obtain the following result: 1 - O(I/nhl) ::; Cap(F)/#w ::; 2, where #w denotes the number of synaptk weights and biases. 3.2 A relation to VC-dimension The VC-dimension, dvc(F), is defined as the largest N such that there exists an N-tuple i = (Xl, ... ,XN) E (Rn)N for which the network can implement all possible 2N dichotomies. Recent results of Sakurai [10] imply (5) For sufficiently large nand hl this estimate exceeds 2(nhl + 1) which is an upper bound on Cap(F). Thus, in contrast to the single threshold logic unit case we have the following (c.f. Fig. 3): Corollary 8 Cap(F) < dvc(F) if hi » 1. 3.3 Memorisation ability of multilayer perceptron Corollary 8 combined with Theorem 7 and Figure 2 imply that for some cases of patters in general position multilayer perceptron can memorise and reliably retrieve Counting Function Theorem for Multi-Layer Networks 381 (even with 100% accuracy) much more (~ log2(h1 ) times more) than 2 patterns per connection, as is the case for a single neuron [4]. This proves that co-operation between hidden units can significantly improve the storage efficiency of neural networks. 3.4 A relation to PAC learning Vapnik's estimate of generalisation error [12] (an error rate on independent test set) EG(F) ~ EL(F) + D(N, dvc(F), EL, '1) (6) holds for N > dvc(F) with probability larger that (1 - '1). It contains two terms: (i) learning error E L( F) and (ii) confidence interval D(p, dvc, EL, '1) del 2W(p, dvc, '1) [1 + ,,11 + EL!W(p, dvc, '1)] , where 2N dvc In '1 w(N, dvc, 11) = (In dvc + 1) 2N - N· The ability of obtaining small learning error EL(F) is, in a sense, controlled by Cap(F), while the size of the confidence interval D is controlled by both dvc(F) and Cap(F) (through EL(F)). For a multilayer perceptron as in Theorem 7 when dvc(F) » Cap(F) (Fig. 2) it can turn out that actually the capacity rather than the VC-dimension is the most critical factor in obtaining low generalisation error EG(F). This obviously warrants further research into the relation between capacity and generalisation. The theoretical estimates of generalisation error based on VC-dimension are believed to be too pessimistic in comparison with some experiments. One may hypothesise that this is caused by too high values of dvc(F) used in estimates such as (6). Since Cover's capacity in the case multilayer perceptron with hl » 1 turned up to be much lower than VC-dimension, one may hope that more realistic estimates could be achieved with generalisation estimates linked directly to capacity. This subject will obviously require further research. Note that some results along these lines can be found in Cover's paper [4]. 3.5 Some open problems Theorem 7 gives estimates of capacity per variable connection for a network with the minimal number of neurons in the first hidden layer showing that these neurons have to be fully connected. The natural question arises at this point as to whether a network with a bigger number but not fully connected neurons in the first hidden layer can achieve a better capacity (per adjustable synaptic weight). The values of the counting function i f-t Cp(i) are provided in this paper for the particular class of points in general position, for i E CC C (Rn)N. The natural question is whether they may be by chance a lower or upper bound for the counting function for the general case of i E (Rn)N ? The results of Sakurai [11] seem to point to the former case: in his case, the sequences PN = (p!, ... , PN) turned out to be "the hardest" in terms of hidden units required to implement 100% of 382 Kowalczyk dichotomies. Corollary 8 and Figure 1 also support this lower bound hypothesis. They imply in particular that there exists a N'-tuple Y = (Yl, Yl, ... , YN') E (Rfl.)N', where N' deC VC-dimension > N, such that CF(Y) = 2N' » 2N > CF(PN) for sufficiently large nand h. 4 Acknowledgement The permission of Managing Director, Research and Information Technology, Telecom Australia, to publish this paper is gratefully acknowledged. References [1] E. Baum. On the capabilities of multilayer perceptrons. Journal of Complezity, 4:193-215, 1988. [2] S. Ben-David and M. Lindenbaum. Localization VS. identification of semialgebraic sets. In Proceedings of the Sizth Annual Workshop on Computational Learning Theory (to appear), 1993. [3] A. Blumer, A. Ehrenfeucht, D. Haussler, and M.K. Warmuth. Learnability and the Vapnik-Chernovenkis dimensions. Journal of the ACM, 36:929-965, (Oct. 1989). [4] T.M. Cover. Geometrical and statistical properties of linear inequalities with applications to pattern recognition. IEEE Trans. Elec. Comp., EC-14:326334, 1965. [5) A. Kowalczyk. Some estimates of necessary number of connections and hidden units for feed-forward networks. In S.l. Hanson et al., editor, Advances in Neural Information Processing Systems, volume 5. Morgan Kaufman Publishers, Inc., 1992. [6] A. Kowalczyk. Estimates of storage capacity of multi-layer perceptron with threshold logic hidden units. In preparation, 1994. [7) J. Milnor. On Betti numbers of real varieties. Proceedings of AMS, 15:275-280, 1964. [8] G.J. Mitchison and R.M. Durbin. Bounds on the learning capacity of some multi-layer networks. Biological Cybernetics, 60:345-356, (1989). [9] A. Sakurai. On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions. Manuscript, Advanced Research Laboratory, Hitachi Ltd., 1993. [10] A. Sakurai. Tighter bounds of the VC-dimension of three-layer networks. In WCNN93, 1993. [11] A. Sakurai. n-h-1 networks store no less n· h + 1 examples but sometimes no more. In Proceedings of IJCNN9~, pages 111-936-111-941. IEEE, June 1992. [12] V. Vapnik. Estimation of Dependences Based on Empirical Data. SpringerVerlag, 1982.
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Supervised Learning with Growing Cell Structures Bernd Fritzke Institut fiir Neuroinformatik Ruhr-U niversitat Bochum Germany Abstract We present a new incremental radial basis function network suitable for classification and regression problems. Center positions are continuously updated through soft competitive learning. The width of the radial basis functions is derived from the distance to topological neighbors. During the training the observed error is accumulated locally and used to determine where to insert the next unit. This leads (in case of classification problems) to the placement of units near class borders rather than near frequency peaks as is done by most existing methods. The resulting networks need few training epochs and seem to generalize very well. This is demonstrated by examples. 1 INTRODUCTION Feed-forward networks of localized (e.g., Gaussian) units are an interesting alternative to the more frequently used networks of global (e.g., sigmoidal) units. It has been shown that with localized units one hidden layer suffices in principle to approximate any continuous function, whereas with sigmoidal units two layers are necessary. In the following we are considering radial basis function networks similar to those proposed by Moody & Darken (1989) or Poggio & Girosi (1990). Such networks consist of one layer L of Gaussian units. Each unit eEL has an associated vector We E Rn indicating the position of the Gaussian in input vector space and a standard 255 256 Fritzke deviation U c . For a given input datum e E Rn the activation of unit c is described by D (C) _ (_ lie - wcll2) c '" exp 2· Uc (1) On top of the layer L of Gaussian units there are m single layer percepirons. Thereby, m is the output dimensionality of the problem which is given by a number of input/output pairsl (e, () E (Rn x Rm). Each of the single layer perceptrons computes a weighted sum of the activations in L: Oi(e) = L Wij Dj (0 iE{1, ... ,m} (2) jEL With Wij we denote the weighted connection from local unit j to output unit i. Training of a single layer perceptron to minimize square error is a very well understood problem which can be solved incrementally by the delta rule or directly by linear algebra techniques (Moore-Penrose inverse). Therefore, the only (but severe) difficulty when using radial basis function networks is choosing the number of local units and their respective parameters, namely center position wand width u. One extreme approach is to use one unit per data points and to position the units directly at the data points. If one chooses the width of the Gaussians sufficiently small it is possible to construct a network which correctly classifies the training data, no matter how complicated the task is (Fritzke, 1994). However, the network size is very large and might even be infinite in the case of a continuous stream of non-repeating stochastic input data. Moreover, such a network can be expected to generalize poorly. Moody & Darken (1989), in contrast, propose to use a fixed number of local units (which is usually considerably smaller than the total number of data points). These units are first distributed by an unsupervised clustering method (e.g., k-means). Thereafter, the weights to the output units are determined by gradient descent. Although good results are reported for this method it is rather easy to come up with examples where it would not perform well: k-means positions the units based on the density of the training data, specifically near density peaks. However, to approximate the optimal Bayesian a posteriori classifier it would be better to position units near class borders. Class borders, however, often lie in regions with a particularly low data density. Therefore, all methods based on k-means-like unsupervised placement of the Gaussians are in danger to perform poorly with a fixed number of units or - similarly undesirable - to need a huge number of units to achieve decent performance. From this one can conclude that - in the case of radial basis function networks - it is essential to use the class labels not only for the training of the connection weights but also for the placement of the local units. Doing this forms the core of the method proposed below. IThroughout this article we assume a classification problem and use the corresponding terminology. However, the described method is suitable for regression problems as well. Supervised Learning with Growing Cell Structures 257 2 SUPERVISED GROWING CELL STRUCTURES In the following we present an incremental radial basis function network which is able to simultaneously determine a suitable number of local units, their center positions and widths as well as the connection weights to the output units. The basic idea is a very simple one: O. Start with a very small radial basis function network. 1. Train the current network with some I/O-pairs from the training data. 2. Use the observed accumulated error to determine where in input vector space to insert new units. 3. If network does not perform well enough goto 1. One should note that during the training phase (Step 1.) error is accumulated over several data items and this accumulated error is used to determine where to insert new units (Step 2.). This is different from the approach of Platt (1991) where insertions are based on single poorly mapped patterns. In both cases, however, the goal is to position new units in regions where the current network does not perform well rather than in regions where many data items stem from. In our model the center positions of new units are interpolated from the positions of existing units. Specifically, after some adaptation steps we determine the unit q which has accumulated the maximum error and insert a new unit in between q and one of its neighbors in input vector space. The interpolation procedure makes it necessary to allow the center positions of existing units to change. Otherwise, all new units would be restricted to the convex hull of the centers of the initial network. We do not necessarily insert a new unit in between q and its nearest neighbor. Rather we like to choose one of the units with adjacent Voronoi regions2 . In the two-dimensional case these are the direct neighbors of q in the Delaunay triangulation (Delaunay-neighbors) induced by all center positions. In higher-dimensional spaces there exists an equivalent based on hypertetrahedrons which, however, is very hard to compute. For this reason, we arrange our units in a certain topological structure (see below) which has the property that if two units are direct neighbors in that structure they are mostly Delaunay-neighbors. By this we get with very little computational effort an approximate subset of the Delaunay-neighbors which seems to be sufficient for practical purposes. 2.1 NETWORK STRUCTURE The structure of our network is very similar to standard radial basis function networks. The only difference is that we arrange the local units in a k-dimensional topological structure consisting of connected simplices3 (lines for k = 1, triangles 2The Voronoi region of a unit c denotes the part of the input vector space which consists of points for which c is the nearest unit. 3 A historical reason for this specific approach is the fact that the model was developed from an unsupervised network (see Fritzke, 1993) where the k-dimensional neighborhood was needed to reduce dimensionality. We currently investigate an alternative (and more 258 Fritzke for k = 2, tetrahedrons for k = 3 and hypertetrahedrons for larger k). This arrangement is done to facilitate the interpolation and adaptation steps described below. The initial network consists of one k-dimensional simplex (k + 1 local units fully connected with each other). The neighborhood connections are not weighted and do not directly influence the behavior of the network. They are, however, used to determine the width of the Gaussian functions associated with the units. Let for each Gaussian unit c denote Ne the set of direct topological neighbors in the topological structure. Then the width of c is defined as (je = (1/INe l) L: Ilwe - wdl12 (3) dENc which is the mean distance to the topological neighbors. If topological neighbors have similar center positions (which will be ensured by the way adaptation and insertion is done) then this leads to a covering ofthe input vector space with partially overlapping Gaussian functions. 2.2 ADAPTATION It was mentioned above that several adaptation steps are done before a new unit is inserted. One single adaptation step is done as follows (see fig. 1): • Chose an I/O-pair (e,(),e E Rn,( E Rm) from the training data. • determine the unit s closest to e (the so-called best-matching unit). • Move the centers of s and its direct topological neighbors towards e. dWe = en (e - we) for all c E N~ eb and en are small constants with eb > > en. • Compute for each local unit eEL the activation De(e) (see eqn. 1) • Compute for each output unit i the activation Oi (see eqn. 2) • Compute the square error by m SE = L:«(i - Oi)2 i=l • Accumulate error at best-matching unit s: derrs = SE • Make Delta-rule step for the weights (a denotes the learning rate): iE{1, ... ,m},jEL Since together with the best-matching unit always its direct topological neighbors are adapted, neighboring units tend to have similar center positions. This property can be used to determine suitable center positions for new units as will be demonstrated in the following. Supervised Learning with Growing Cell Structures 259 a) Before ... b) during, and ... c) ... after adaptation Figure 1: One adaptation step. The center positions of the current network are shown and the change caused by a single input signal. The observed error SE for this pattern is added to the local error variable of the best-matching unit. f a) Before ... b) ... and after insertion Figure 2: Insertion of a new unit. The dotted lines indicate the Voronoi fields. The unit q has accumulated the most error and, therefore, a new unit is inserted between q and one of its direct neighbors. 2.3 INSERTION OF NEW UNITS After a constant number A of adaptation steps a new unit is inserted. For this purpose the unit q with maximum accumulated error is determined. Obviously, q lies in a region of the input vector space where many misclassifications occur. One possible reason for this is that the gradient descent procedure is unable to find suitable weights for the current network. This again might be caused by the coarse resolution at this region of the input vector space: if data items from different classes are covered by the same local unit and activate this unit to about the same degree then it might be the case that their vectors of local unit activations are nearly identical which makes it hard for the following single layer perceptrons to distinguish among them. Moreover, even if the activation vectors are sufficiently different they still might be not linearly separable. accurate) approximation of the Delaunay triangulation which is based on the "Neural-Gas" method proposed by Martinetz & Schulten (1991). 260 Fritzke o 0 • . . o 0 0 0 o •••• • • o • • • 00000 • 0 • o. 00 0 • o • 0 • g °0 • 0 • .o.o:~~oo.o.o .o·.~"~·o o -. •• 0 • o -.... 0 0 o • . . . . o 0 0 . . a) two spiral problem: 194 points in two classes b) decision regions for CascadeCorrelation (reprinted with permission from Fahlman & Lebiere, 1990) Figure 3: Two spiral problem and learning results of a constructive network. The insertion of a new local unit near q is likely to improve the situation: This unit will probablY be activated to a different degree by the data items in this region and will, therefore, make the problem easier for the single layer perceptrons. What exactly are we doing? We choose one of the direct topological neighbors of q, say a unit f (see also fig. 2). Currently this is the neighbor with the maximum accumulated error. Other choices, however, have shown good results as well, e.g., the neighbor with the most distant center position or even a randomly picked neighbor. We insert a new unit r in between q and f and initialize its center by (4) We connect the new unit with q and f and with all common neighbors of q and f. The original connection between q and f is removed. By this we get a structure of k-dimensional simplices again. The new unit gets weights to the output units which are interpolated from the weights of its neighbors. The same is done for the initial error variable which is linearly interpolated from the variables of the neighbors of r. After the interpolation all the weights of r and its neighbors and the error variables of these units are multiplied by a factor INrl/(INrl + 1)1. This is done to disturb the output of the network as less as possible4 • However, the by far most important 4The redistribution of the error variable is again a relict from the unsupervised version (Fritzke, 1993). There we count signals rather than accumulate error. An elaborate scheme for redistributing the signal counters is necessary to get good local estimates of the probability density. For the supervised version this redistribution is harder to justify since the insertion of a new unit in general makes previous error information void. However, even though there is still some room for simplification, the described scheme does work very well already in its present form. Supervised Learning with Growing Cell Structures 261 o a) final network with 145 cells b) decision regions Figure 4: Performance of the Growing Cell Structures on the two spiral benchmark. decision seems to be to insert the new unit near the unit with maximum error. The weights and the error variables adjust quickly after some learning steps. 2.4 SIMULATION RESULTS Simulations with the two spiral problem (fig. 3a) have been performed. This classification benchmark has been widely used before so that results for comparison are readily available.Figure 3b) shows the result of another constructive algorithm. The data consist of 194 points arranged on two interlaced spirals in the plane. Each spiral corresponds to one class. Due to the high nonlinearity of the task it is particular difficult for networks consisting of global units (e.g., multi-layer perceptrons). However, the varying density of data points (which is higher in the center of the spirals) makes it also a challenge for networks of local units. As for most learning problems the interesting aspect is not learning the training examples but rather the performance on new data which is often denoted as generalization. Baum & Lang (1991) defined a test set of 576 points for this problem consisting of three equidistant test points between each pair of adjacent same-class training points. They reported for their best network 29 errors on the test set in the mean. In figure 4 a typical network generated by our method can be seen as well as the corresponding decision regions. No errors on the test set of Baum and Lang are made. Table 1 shows the necessary training cycles for several algorithms. The new growing network uses far less cycles than the other networks. Other experiments have been performed with a vowel recognition problem (Fritzke, 1993). In all simulations we obtained significantly better generalization results than Robinson (1989) who in his thesis investigated the performance of several connectionist and conventional algorithms on the same problem. The necessary 262 Fritzke Table 1: Training epochs necessary for the two spiral problem network model epochs test error reported in Backpropagation 20000 yes Lang & Witbrock (1989) Cross Entropy BP 10000 yes Lang & Witbrock (1989) Cascade-Correlation 1700 yes Fahlman & Lebiere (1990) Growmg Cell Structures 180 no Fntzke ( 1993) number of training cycles for our method was lower by a factor of about 37 than the numbers reported by Robinson (1993, personal communication). REFERENCES Baum, E. B. & K. E. Lang [1991]' "Constructing hidden units using examples and queries," in Advances in Neural Information Processing Systems 3, R.P. Lippmann, J.E. Moody & D.S. Touretzky, eds., Morgan Kaufmann Publishers, San Mateo, 904910. Fahlman, S. E. & C. Lebiere [1990], "The Cascade-Correlation Learning Architecture," in Advances in Neural Information Processing Systems 2, D.S. Touretzky, ed., Morgan Kaufmann Publishers, San Mateo, 524-532. Fritzke, B. [1993], "Growing Cell Structures - a self-organizing network for unsupervised and supervised learning," International Computer Science Institute, TR-93-026, Berkeley. Fritzke, B. [1994], "Making hard problems linearly separable - incremental radial basis function approaches," (submitted to ICANN'94: International Conference on Artificial Neural Networks), Sorrento, Italy. Lang, K. J. & M. J. Witbrock [1989], "Learning to tell two spirals apart," in Proceedings of the 1988 Connectionist Models Summer School, D. Touretzky, G. Hinton & T . Sejnowski, eds., Morgan Kaufmann, San Mateo, 52-59. Martinetz, T. M. & K. J. Schulten [1991]' "A "neural-gas" network learns topologies," in Artificial Neural Networks, T. Kohonen, K. Makisara, O. Simula & J. Kangas, eds., North-Holland, Amsterdam, 397-402. Moody, J. & C. Darken [1989], "Learning with Localized Receptive Fields," in Proceedings of the 1988 Connectionist Models Summer School, D. Touretzky, G. Hinton & T. Sejnowski, eds., Morgan Kaufmann, San Mateo, 133-143. Platt, J. C. [1991], "A Resource-Allocating Network for Function Interpolation," Neural Computation 3, 213-225. Poggio, T. & F. Girosi [1990], "Regularization Algorithms for Learning That Are Equivalent to Multilayer Networks," Science 247, 978-982. Robinson, A. J. [1989], "Dynamic Error Propagation Networks," Cambridge University, PhD Thesis, Cambridge.
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A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics Gert Cauwenberghs* California Institute of Technology Department of Electrical Engineering 128-95 Caltech, Pasadena, CA 91125 E-mail: gertalcco. cal tech. edu Abstract We present experimental results on supervised learning of dynamical features in an analog VLSI neural network chip. The recurrent network, containing six continuous-time analog neurons and 42 free parameters (connection strengths and thresholds), is trained to generate time-varying outputs approximating given periodic signals presented to the network. The chip implements a stochastic perturbative algorithm, which observes the error gradient along random directions in the parameter space for error-descent learning. In addition to the integrated learning functions and the generation of pseudo-random perturbations, the chip provides for teacher forcing and long-term storage of the volatile parameters. The network learns a 1 kHz circular trajectory in 100 sec. The chip occupies 2mm x 2mm in a 2JLm CMOS process, and dissipates 1.2 m W. 1 Introduction Exact gradient-descent algorithms for supervised learning in dynamic recurrent networks [1-3] are fairly complex and do not provide for a scalable implementation in a standard 2-D VLSI process. We have implemented a fairly simple and scalable ·Present address: Johns Hopkins University, ECE Dept., Baltimore MD 21218-2686. 858 A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics 859 learning architecture in an analog VLSI recurrent network, based on a stochastic perturbative algorithm which avoids calculation of the gradient based on an explicit model of the network, but instead probes the dependence of the network error on the parameters directly [4]. As a demonstration of principle, we have trained a small network, integrated with the learning circuitry on a CMOS chip, to generate outputs following a prescribed periodic trajectory. The chip can be extended, with minor modifications to the internal structure of the cells, to accommodate applications with larger size recurrent networks. 2 System Architecture The network contains six fully interconnected recurrent neurons with continuoustime dynamics, d 6 T dtXi = -Xi + L Wij U(Xj - (Jj) + Yi , j=l (1) with Xi(t) the neuron states representing the outputs of the network, Yi(t) the external inputs to the network, and u(.) a sigmoidal activation function. The 36 connection strengths Wij and 6 thresholds (Jj constitute the free parameters to be learned, and the time constant T is kept fixed and identical for all neurons. Below, the parameters Wij and (Jj are denoted as components of a single vector p. The network is trained with target output signals x[(t) and xf(t) for the first two neuron outputs. Learning consists of minimizing the time-averaged error 1 jT 2 £(p) = lim 2T L Ixf(t) - Xk(t)IVdt , T-+oo -T k=l (2) using a distance metric with norm v. The learning algorithm [4] iteratively specifies incremental updates in the parameter vector p as p(k+l) = p(k) _ J1, t(k) 7r(k) (3) with the perturbed error t(k) = ~ (£(p(k) + 7r(k») _ £(p(k) _ 7r(k»)) (4) obtained from a two-sided parallel activation of fixed-amplitude random perturbations '1ri(k) onto the parameters p/k); '1ri(k) = ±u with equal probabilities for both polarities. The algorithm basically performs random-direction descent of the error as a multi-dimensional extension to the Kiefer-Wolfowitz stochastic approximation method [5], and several related variants have recently been proposed for optimization [6,7] and hardware learning [8-10]. To facilitate learning, a teacher forcing signal is initially applied to the external input y according to Yi(t) = .x ,(xi(t) - Xi(t)) , i = 1,2 (5) providing a feedback mechanism that forces the network outputs towards the targets [3]. A symmetrical and monotonically increasing "squashing" function for ,(.) serves this purpose. The teacher forcing amplitude .x needs to be attenuated along the learning process, as to suppress the bias in the network outputs at convergence that might result from residual errors. 860 Cauwenberghs 3 Analog VLSI Implementation The network and learning circuitry are implemented on a single analog CMOS chip, which uses a transconductance current-mode approach for continuous-time operation. Through dedicated transconductance circuitry, a wide linear dynamic range for the voltages is achieved at relatively low levels of power dissipation (experimentally 1.2 m W while either learning or refreshing). While most learning functions, including generation of the pseudo-random perturbations, are integrated on-chip in conjunction with the network, some global and higher-level learning functions of low dimensionality, such as the evaluation of the error (2) and construction of the perturbed error (4), are performed outside the chip for greater flexibility in tailoring the learning process. The structure and functionality of the implemented circuitry are illustrated in Figures 1 to 3, and a more detailed description follows below. 3.1 Network Circuitry Figure 1 shows the schematics of the synapse and neuron circuitry. A synapse cell of single polarity is shown in Figure 1 (a). A high output impedance triode multiplier, using an adjustable regulated casco de [11], provides a constant current Iij linear in the voltage Wij over a wide range. The synaptic current Iij feeds into a differential pair, injecting a differential current hj a(xj - OJ) into the diode-connected Id:.t and I;;"t output lines. The double-stack transistor configuration of the differential pair offers an expanded linear sigmoid range. The summed output currents Itut and I;;;"t of a row of synapses are collected in the output cell, Figure 1 (b), which also subtracts the reference currents I;"c and I;;c obtained from a reference rOw of "dummy" synapses defining the "zero-point" synaptic strength Wolf for bipolar operation. The thus established current corresponds to the summed synaptic contributions in (1). Wherever appropriate (i = 1,2), a differential transconductance element with inputs Xi and xT is added to supply an external input current for forced teacher action in accordance with (5). I~U~ ~ 1",,/ Xi f-Vc (a) (b) Figure 1 Schematics of synapse and neuron circuitry. (a) Synapse of single polarity. (b) Output cell with current-to-voltage converter. The output current is converted to the neuron output voltage Xi, through an active resistive element using the same regulated high output impedance triode circuitry as used in the synaptic current source. The feedback delay parameter T in (1) corresponds to the RC A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics 861 product of the regulated triode active resistance value and the capacitance Gout. With Gout = 5 pF, the delay ranges between 20 and 200jLsec, adjustable by the control voltage of the regulated cascode. Figure 2 shows the measured static characteristics of the synapse and neuron functions for different values of Wij and ()j ( i = j = 1), obtained by disabling the neuron feedback and driving the synapse inputs externally. ~ 0.0 '~ -0.2 CII ~ .... -0.4 0 ;> .... ~ -0.6 .& <5 -0.8 -1.0 -0.5 0.0 Input Voltage x j (a) O.OV - 0.8V 0.5 1.0 (V) ~ .~ CII ~ .... 0 ;> .... ~ <5 -1.0 -0.5 0.0 0.5 Input Voltage x j (V) (b) 1.0 Figure 2 Measured static synapse and neuron characteristics, for various values of (a) the connection strength Wij, and (b) the threshold ()j. 3.2 Learning Circuitry Figure 3 (a) shows the simplified schematics of the learning and storage circuitry, replicated locally for every parameter (connection strength or threshold) in the network. Most of the variables relating to the operation of the cells are local, with exception of a few global signals communicating to all cells. Global signals include the sign and the amplitude of the perturbed error t and predefined control signals. The stored parameter and its binary perturbation are strictly local to the cell, in that they do not need to communicate explicitly to outside circuitry (except trivially through the neural network it drives), which simplifies the structural organization and interconnection of the learning cells. The parameter voltage Pi is stored on the capacitor Gstore, which furthermore couples to capacitor G pert for activation of the perturbation. The perturbation bit 7ri selects either of two complementary signals V+<T and V-<T with corresponding polarity. With the specific shape of the waveforms V+<T and V-<T depicted in Figure 3 (b), the proper sequence of perturbation activations is established for observation of the complementary error terms in (4). The obtained global value for t is then used, in conjunction with the local perturbation bit 7ri, to update the parameter value Pi according to (3). A fineresolution charge-pump, shown in the dashed-line inset of Figure 3 (a), is used for this purpose. The charge pump dumps either of a positive or negative update current, of equal amplitude, onto the storage capacitor whenever it is activated by means of an EN_UPD high pulse, effecting either of a given increment or decrement on the parameter value Pi respectively. The update currents are supplied by two complementary transistors, and are switched by driving the source voltages of the transistors rather than their gate voltages in order to avoid typical clock feed-through effects. The amplitude of the incremental update, set proportionally to Itl, is controlled by the VUPD nand VUPD p gate voltage levels, operated in the sub-threshold region. The polarity of the increment or decrement action is determined by the control signal DECR/INCR, obtained from the polarities of 862 Cauwenberghs the perturbed error t and the perturbation bit 11"; through an exclusive-or operation. The learning cycle is completed by activating the update by a high pulse on EN_UPD. The next learning cycle then starts with a new random bit value for the perturbation 11";. I I 1t; X i i X -E>O El'CUPD : : :: rL I I I I v+o I I I I I I -"1t TI I v-(J I -2+ III II --1_ T I I I I I_ I;;/" E(p) E(p + It) E(p - It) (a) (b) Figure 3 Learning cell circuitry. (a) Simplified schematics. (b) Waveform and timing diagram. The random bit stream 1I";(k) is generated on-chip by means of a set of linear feedback shift registers [12]. For optimal performance, the perturbations need to satisfy certain statistical orthogonality conditions, and a rigorous but elaborate method to generate a set of uncorrelated bit streams in VLSI has been derived [13]. To preserve the scalability of the learning architecture and the local nature of the perturbations, we have chosen a simplified scheme which does not affect the learning performance to first order, as verified experimentally. The array of perturbation bits, configured in a two-dimensional arrangement as prompted by the location of the parameters in the network, is constructed by an outer-product exclusive-or operation from two generating linear sets of uncorrelated row and column bits on lines running horizontally and vertically across the network array. In the present implementation the evaluation of the error functional (2) is performed externally with discrete analog components, leaving some flexibility to experiment with different formulations of error functionals that otherwise would have been hardwired. A mean absolute difference (/I = 1) norm is used for the metric distance, and the timeaveraging of the error is achieved by a fourth-order Butterworth low-pass filter. The cut-off frequency is tuned to accommodate an AC ripple smaller than 0.1 %, giving rise to a filter settling time extending 20 periods of the training signal. 3.3 Long-Term Volatile Storage After learning, it is desirable to retain ("freeze") the learned information, in principle for an infinite period of time. The volatile storage of the parameter values on capacitors undergoes a spontaneous decay due to junction leakage and other drift phenomena, and needs to be refreshed periodically. For eight effective bits of resolution, a refresh rate of 10 Hz is sufficient. Incidentally, the charge pump used for the learning updates provides for refresh of the parameter values as well. To that purpose, probing and multiplexing circuitry (not shown) are added to the learning cell of Figure 3 (a) for sequential refresh. In the experiment conducted here, the parameters are stored externally and refreshed sequentially by activating the corresponding charge pump with a DECR/INCR bit defined by the polarity of the observed deviation between internally probed and externally stored A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics 863 values. The parameter refresh is performed in the background with a 100 msec cycle, and does not interfere with the continuous-time network operation. A simple internal analog storage method obliterating the need of external storage is described in [14], and is supported by the chip architecture. 4 Learning Experiment As a proof of principle, the network is trained with a circular target trajectory defined by the quadrature-phase oscillator { xi (t) xr(t) A cos (27rft) A sin (27rft) (6) with A = o.SV and f = 1kHz. In principle a recurrent network of two neurons suffices to generate quadrature-phase oscillations, and the extra neurons in the network serve to accommodate the particular amplitUde and frequency requirements and assist in reducing the nonlinear harmonic distortion. Clearly the initial conditions for the parameter values distinguish a trivial learning problem from a hard one, and training an arbitrarily initialized network may lead to unpredictable results of poor generality. Incidentally, we found that the majority of randomly initialized learning sessions fail to generate oscillatory behavior at convergence, the network being trapped in a local minimum defined by a strong point attractor. Even with strong teacher forcing these local minima persist. In contrast, we obtained consistent and satisfactory results with the following initialization of network parameters: strong positive diagonal connection strengths W ii = 1, zero off-diagonal terms Wij = 0 ; i f. j and zero thresholds (}i = O. The positive diagonal connections Wii repel the neuron outputs from the point attractor at the origin, counteracting the spontaneous decay term -Xi in (1). Applying non-zero initial values for the cross connections Wij ; i f. j would introduce a bias in the dynamics due to coupling between neurons. With zero initial cross coupling, and under strong initial teacher forcing, fairly fast and robust learning is achieved. Figure 4 shows recorded error sequences under training of the network with the target oscillator (6), for five different sessions of 1, 500 learning iterations each starting from the above initial conditions. The learning iterations span 60 msec each, for a total of 100 sec per session. The teacher forcing amplitude .A is set initially to 3 V, and thereafter decays logarithmically over one order of magnitude towards the end of the sessions. Fixed values of the learning rate and the perturbation amplitude are used throughout the sessions, with J.L = 25.6 V-I and (J' = 12.5 m V. All five sessions show a rapid initial decrease in the error under stimulus of the strong teacher forcing, and thereafter undergo a region of persistent flat error slowly tapering off towards convergence as the teacher forcing is gradually released. Notice that this flat region does not imply slow learning; instead the learning constantly removes error as additional error is adiabatically injected by the relaxation of the teacher forcing. 864 Cau wenberghs 3.0 ~ 25 ... 2.0 0 t: ~ 15 ::I 0... 1.0 8 05 0.0 0 20 40 60 Time (sec) -1 Jl = 25.6 V (J = 12.5 mV 80 100 Figure 4 Recorded evolution of the error during learning, for five different sessions on the network. Near convergence, the bias in the network error due to the residual teacher forcing becomes negligible. Figure 5 shows the network outputs and target signals at convergence, with the learning halted and the parameter refresh activated, illustrating the minor effect of the residual teacher forcing signal on the network dynamics. The oscillogram of Figure 5 (a) is obtained under a weak teacher forcing signal, and that of Figure 5 (b) is obtained with the same network parameters but with the teacher forcing signal disabled. In both cases the oscilloscope is triggered on the network output signals. Obviously, in absence of teacher forcing the network does no longer run synchronously with the target signal. However, the discrepancy in frequency, amplitude and shape between either of the free-running and forced oscillatory output waveforms and the target signal waveforms is evidently small. (a) (b) Figure 5 Oscillograms of the network outputs and target signals after learning, (a) under weak teacher forcing, and (b) with teacher forcing disabled. Top traces: Xl(t) and Xl T(t). Bottom traces: X2(t) and X2T(t). A Learning Analog Neural Network Chip with Continuous-Time Recurrent Dynamics 865 5 Conclusion We implemented a small-size learning recurrent neural network in an analog VLSI chip, and verified its learning performance in a continuous-time setting with a simple dynamic test (learning of a quadrature-phase oscillator). By virtue of its scalable architecture, with constant requirements on interconnectivity and limited global communication, the network structure with embedded learning functions can be freely expanded in a two-dimensional arrangement to accommodate applications of recurrent dynamical networks requiring larger dimensionality. A present limitation of the implemented learning model is the requirement of periodicity on the input and target signals during the learning process, which is needed to allow a repetitive and consistent evaluation of the network error for the parameter updates. Acknowledgments Fabrication of the CMOS chip was provided through the DARPA/NSF MOSIS service. Financial support by the NIPS Foundation largely covered the expenses of attending the conference. References [1] B.A. Pearlmutter, "Learning State Space Trajectories in Recurrent Neural Networks," Neural Computation, vol. 1 (2), pp 263-269, 1989. [2] RJ. Williams and D. Zipser, "A Learning Algorithm for Continually Running Fully Recurrent Neural Networks," Neural Computation, vol. 1 (2), pp 270-280, 1989. [3] N .B. Toomarian, and J. Barhen, "Learning a Trajectory using Adjoint Functions and Teacher Forcing," Neural Networks, vol. 5 (3), pp 473-484, 1992. [4] 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, vol. 5, pp 244-251, 1993. [5] H.J. Kushner, and D.S. Clark, "Stochastic Approximation Methods for Constrained and Unconstrained Systems," New York, NY: Springer-Verlag, 1978. [6] M.A. Styblinski, and T.-S. Tang, "Experiments in Nonconvex Optimization: Stochastic Approximation with Function Smoothing and Simulated Annealing," Neural Networks, vol. 3 (4), pp 467-483, 1990. [7] J.C. Spall, "Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation," IEEE Trans. Automatic Control, vol. 37 (3), pp 332-341, 1992. [8] J. Alspector, R. Meir, B. Yuhas, and A. Jayakumar, "A Parallel Gradient Descent Method for Learning in Analog VLSI Neural Networks," in Advances in Neural Information Processing Systems, San Mateo, CA: Morgan Kaufman, vol. 5, pp 836-844, 1993. [9] 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, vol. 5, pp 212-219, 1993. [10] 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, vol. 5, pp 789-796, 1993. [11] J.W. Fattaruso, S. Kiriaki, G. Warwar, and M. de Wit, "Self-Calibration Techniques for a Second-Order Multibit Sigma-Delta Modulator," in ISSCC Technical Digest, IEEE Press, vol. 36, pp 228-229, 1993. [12] S.W. Golomb, "Shift Register Sequences," San Francisco, CA: Holden-Day, 1967. [13] 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 T. Circuits and Systems, 38 (1), pp 109-123, 1991. [14] G. Cauwenberghs, and A. Yariv, "Method and Apparatus for Long-Term Multi-Valued Storage in Dynamic Analog Memory," U.s. Patent pending, filed 1993.
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Functional Models of Selective Attention and Context Dependency Thomas H. Hildebrandt Department of Electrical Engineering and Computer Science Room 304 Packard Laboratory 19 Memorial Drive West Lehigh University Bethlehem PA 18015-3084 thildebr@aragorn.eecs.lehigh.edu Scope This workshop reviewed and classified the various models which have emerged from the general concept of selective attention and context dependency, and sought to identify their commonalities. It was concluded that the motivation and mechanism of these functional models are "efficiency" and ''factoring'', respectively. The workshop focused on computational models of selective attention and context dependency within the realm of neural networks. We treated only ''functional'' models; computational models of biological neural systems, and symbolic or rule-based systems were omitted from the discussion. Presentations Thomas H. Hildebrandt presented the results of his recent survey of the literature on functional models of selective attention and context dependency. He set forth the notions that selective attention and context dependency are equivalent, that the goal of these methods is to reduce computational requirements, and that this goal is achieved by what amounts to factoring or a divide-and-conquer technique which takes advantage of nonlinearities in the problem. Daniel S. Levine (University of Texas at Arlington) showed how the gated dipole structure often used in the ART models can be used to account for time-dependent phenomena such as habituation and overcompensation. His adjusted model appropriately modelled the public's adverse reaction to "New Coke". Lev Goldfarb (University of New Brunswick) presented a formal model for inductive learning based on symbolic transformation systems and parametric distance functions as an alternative to the commonly used algebraic transformation system and Euclidean distance function. The drawbacks of the latter system were briefly discussed, and it was shown how this new formal system can give rise to learning models which overcome these problems. 1180 Functional Models of Selective Attention and Context Dependency 1181 Chalapathy Neti (IBM, Boca Raton) presented a model which he has used to increase signal-to-noise ratio (SNR) in noisy speech signals. The model is based on adaptive filtering of frequency bands with a constant frequency to bandwidth ratio. This thresholding in the wavelet domain gives results which are superior to similar methods operating in the Adaptive Fourier domain. Several types of signal could be detected with SNRs close to Odb. Paul N. Refenes (University of London Business School) demonstrated the need to take advantage of contextual information in attempting to model the capital markets. There exist some fundamental economic formulae, but they hold only in the long term. The desire to model events on a finer time scale requires reference to significant factors within a smaller window. To do this effectively requires the identification of appropriate short-term indicators, as mere overparameterization has been shown to lead to negative results. Jonathan A. Marshall (University of North Carolina) reviewed the EXIN model, which correctly encodes partially overlapping patterns as distinct activations in the output layer, while allowing the simultaneous appearance of nonoverlapping patterns to give rise to multiple activations in the output layer. The model thus produces a factored representation of complex scenes. Albert Nigrin (American University) presented a model, similar in concept to the EXIN model. It correctly handles synonymous inputs by means of cross-inhibition of the links connecting the synonyms to the target node. Thomas H. Hildebrandt also presented a model for adaptive classification based on decision feedback equalization. The model shifts the decision boundaries of the underlying classifier to compensate shifts in the statistics of the input. On handwritten character classification, it outperformed an identical classifier which used only static decision boundaries. Summary According to Hildebrandt's first talk, the concepts underlying selective attention are quite broad and generally applicable. Large nonlinearities in the problem permit the use of problem subdivision or factoring (by analogy with the factoring of a Boolean equation). Factoring is a good method for reducing the complexity of nonlinear systems. The talks by Levine and Refenes showed that context enters naturally into the description, formulation, and solution ofreal-world modelling problems. Those by Neti and Hildebrandt showed that specific reference to temporal context can result in immediate performance gains. The presentations by Marshall and Nigrin provided models for appropriately encoding contexts involving overlapping and synonymous patterns, respectively. The talk by Goldfarb indicates that abandoning assumptions regarding linearity ab initio may lead to more powerful learning systems. Refer to [1] for further information. References [1] Hildebrandt, Thomas H. Neural Network Models for Selective Attention and Context Dependency. Submitted to Neural Networks, December 1993.
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Learning Classification with Unlabeled Data Virginia R. de Sa desa@cs.rochester.edu Department of Computer Science University of Rochester Rochester, NY 14627 Abstract One of the advantages of supervised learning is that the final error metric is available during training. For classifiers, the algorithm can directly reduce the number of misclassifications on the training set. Unfortunately, when modeling human learning or constructing classifiers for autonomous robots, supervisory labels are often not available or too expensive. In this paper we show that we can substitute for the labels by making use of structure between the pattern distributions to different sensory modalities. We show that minimizing the disagreement between the outputs of networks processing patterns from these different modalities is a sensible approximation to minimizing the number of misclassifications in each modality, and leads to similar results. Using the Peterson-Barney vowel dataset we show that the algorithm performs well in finding appropriate placement for the codebook vectors particularly when the confuseable classes are different for the two modalities. 1 INTRODUCTION This paper addresses the question of how a human or autonomous robot can learn to classify new objects without experience with previous labeled examples. We represent objects with n-dimensional pattern vectors and consider piecewise-linear classifiers consisting of a collection of (labeled) codebook vectors in the space of the input patterns (See Figure 1). The classification boundaries are gi ven by the voronoi tessellation of the codebook vectors. Patterns are said to belong to the class (given by the label) of the codebook vector to which they are closest. 112 • o o • XB o Learning Classification with Unlabeled Data 113 • 0 XB 0 o Figure 1: A piecewise-linear classifier in a 2-Dimensional input space. The circles represent data samples from two classes (filled (A) and not filled (B)). The X's represent codebook vectors (They are labeled according to their class A and B). Future patterns are classified according to the label of the closest codebook vector. In [de Sa and Ballard, 1993] we showed that the supervised algorithm LVQ2.1[Kohonen, 1990] moves the codebook vectors to minimize the number of misclassified patterns. The power of this algorithm lies in the fact that it directly minimizes its final error measure (on the training set). The positions of the codebook vectors are placed not to approximate the probability distributions but to decrease the number of misclassifications. Unfortunately in many situations labeled training patterns are either unavailable or expensive. The classifier can not measure its classification performance while learning (and hence not directly maximize it). One such unsupervised algorithm, Competitive Learning[Grossberg, 1976; Kohonen, 1982; Rumelhart and Zipser, 1986], has unlabeled codebook vectors that move to minimize a measure of the reconstruction cost. Even with subsequent labeling of the codebook vectors, they are not well suited for classification because they have not been positioned to induce optimal borders. Supervised Unsupervised Self-Supervised - implausible label -limited power - derives label from a co-occuring input to "COW" another modality Target ~ \I 000 000 O~ 0 0 600 • • • • • • • • • • • • o O{}OO o O{}OO o O{}OO o O{}OO ~ ~ ~ Input 2 moo Figure 2: The idea behind the algorithm This paper presents a new measure for piecewise-linear classifiers receiving unlabeled patterns from two or more sensory modalities. Minimizing the new measure is an approximation to minimizing the number of misclassifications directly. It takes advantage of the structure available in natural environments which results in sensations to different sensory modalities (and sub-modalities) that are correlated. For example, hearing "mooing" and 114 de Sa p 0.5 0.4 0 . 3 p 0.5 0 . 4 1\ 0 . 3 I \ P(CB)P(,,*~) \ 0.2 I I \ I \ I \ Figure 3: This figure shows an example world as sensed by two different modalities. If modality A receives a pattern from its Class A distribution, modality 2 receives a pattern from its own class A distribution (and the same for Class B). Without receiving information about which class the patterns came from, they must try to determine appropriate placement of the boundaries b l and b2• P(C;) is the prior probability of Class i and p(xjIC;) is the conditional density of Class i for modality j seeing cows tend to occur together. So, although the sight of a cow does not come with an internal homuncular "cow" label it does co-occur with an instance of a "moo". The key is to process the "moo" sound to obtain a self-supervised label for the network processing the visual image of the cow and vice-versa. See Figure 2. 2 USING MULTI-MODALITY INFORMATION One way to make use of the cross-modality structure is to derive labels for the codebook vectors (after they have been positioned either by random initialization or an unsupervised algorithm). The labels can be learnt with a competitive learning algorithm using a network such as that shown in Figure 4. In this network the hidden layer competitive neurons represent the codebook vectors. Their weights from the input neurons represent their positions in the respective input spaces. Presentation of the paired patterns results in activation of the closest codebook vectors in each modality (and D's elsewhere). Co-occurring codebook vectors will then increase their weights to the same competitive output neuron. After several iterations the codebook vectors are given the (arbitrary) label of the output neuron to which they have the strongest weight. We will refer to this as the "labeling algorithm". 2.1 MINIMIZING DISAGREEMENT A more powerful use of the extra information is for better placement of the codebook vectors themselves. In [de Sa, 1994] we derive an algorithm that minimizesl the disagreement between the outputs of two modalities. The algorithm is originally derived not as a piecewise-linear classifier but as a method of moving boundaries for the case of two classes and an agent with two I-Dimensional sensing modalities as shown in Figure 3. Each class has a particular pro babili ty distri buti on for the sensation received by each modality. If modality 1 experiences a sensation from its pattern A distribution, modality 2 experiences a sensation from its own pattern A distribution. That is, the world presents patterns Ithe goal is actually to find a non-trivial local minimum (for details see [de Sa, 1994]) ModaiitylNetwork 1 Learning Classification with Unlabeled Data 115 Output (Class) 000 Hidden Layer Code book Vectors (W) Input (X) ModalitylNetwork 2 Figure 4: This figure shows a network for learning the labels of the codebook vectors. The weight vectors of the hidden layer neurons represent the codebook vectors while the weight vectors of the connections from the hidden layer neuron!; to the output neurons represent the output class that each codebook vector currently represents. In this example there are 3 output classes and two modalities each of which has 2-D input patterns and 5 codebook vectors. from the 2-D joint distribution shown in Figure 5a) but each modality can only sample its 1-D marginal distribution (shown in Figure 3 and Figure 5a). We show [de Sa, 1994] that minimizing the disagreement error the proportion of pairs of patterns for which the two modalities output different labels E(b), b2) = Pr{x) < b) & X2 > bJ} + Pr{x) > b) & X2 < b2} (1) (2) (where f(x). X2) = P(CA)p(xtICA)P(X2ICA) + P(CB)p(x1ICB)p(x2ICB) is the joint probability density for the two modalities) in the above problem results in an algorithm that corresponds to the optimal supervised algorithm except that the "label" for each modality's pattern is the hypothesized output of the other modality. Consider the example illustrated in Figure 5. In the supervised case (Figure 5a») the labels are given allowing sampling of the actual marginal distributions. For each modality, the number of misclassifications can be minimized by setting the boundaries for each modality at the crossing points of their marginal distributions. However in the self-supervised system, the labels are not available. Instead we are given the output of the other modality. Consider the system from the point of view of modality 2. Its patterns are labeled according to the outputs of modality 1. This labels the patterns in Class A as shown in Figure 5b). Thus from the actual Class A patterns, the second modality sees the "labeled" distributions shown. Letting a be the fraction of misclassified patterns from Class A, the resulting distributions are given by (1 - a)P(CA)P(X2ICA) and (a)P(CA)P(X2ICA). Similarly Figure 5c) shows the effect on the patterns in class B. Letting b be the fraction of Class B patterns misclassified, the distributions are given by (1 - b)P( CB)P(X2ICB) 116 de Sa and (b)P( CB)p(X2ICB). Combining the effects on both classes results in the "labeled" distributions shown in Figure 5d). The "apparent Class ~' distribution is given by (1 - a)P(CA)P(X2ICA) + (b)P(CB)p(X2ICB and the "apparent Class B" distribution by (a)P(CA)P(X2ICA) + (1-b)P(CB)p(x2ICB). Notice that even though the approximated distributions may be discrepant, if a:::: b, the crossing point will be close. Simultaneously the second modality is labeling the patterns to the first modality. At each iteration of the algorithm both borders move according to the samples from the "apparent" marginal distributions. - P(CA)p(x1ICA) - P(CB)p(x1ICB) - (a)P(CA}p(x2ICA) - (1-a)P(CA)p(x2ICA) a) Figure 5: This figure shows an example of the joint and marginal distributions (For better visualization the scale of the joint distribution is twice that of the marginal distributions) for the example problem introduced in Figure 3. The darker gray represents patterns labeled "N', while the lighter gray are labeled "B". The dark and light curves are the corresponding marginal distributions with bold and regular labels respectively. a) shows the labeling for the supervised case. b),c) and d) reflect the labels given by modality 1 and the corresponding marginal distributions seen by modality 2. See text for more details 2.2 Self-Supervised Piecewise-Linear Classifier The above ideas have been extended[de Sa, 1994] to rules for moving the codebook vectors in a piecewise-linear classifier. Codebook vectors are initially chosen randomly from the data patterns. In order to complete the algorithm idea, the codebook vectors need to be given initial labels (The derivation assumes that the current labels are correct). In LVQ2.1 Learning Classification with Unlabeled Data 117 the initial codebook vectors are chosen from among the data patterns that are consistent with their neighbours (according to a k-nearest neighbour algorithm); their labels are then taken as the labels of the data patterns. In order to keep our algorithm unsupervised the "labeling algorithm" mentioned earlier is used to derive labels for the initial codebook vectors. Also due to the fact that the codebook vectors may cross borders or may not be accurately labeled in the initialization stage, they are updated throughout the algorithm by increasing the weight to the output class hypothesized by the other modality, from the neuron representing the closest codebook vector. The final algorithm is given in Figure 6 1. Randomly choose initial codebook vectors from data vectors 2. Initialize labels of codebook vectors using the labeling algorithm described in text 3. Repeat for each presentation of input patterns XI(n) and X2(n) to their respective modalities • find the two nearest codebook vectors in modality 1 -- wl.i; , Wl.i;, and modality 2 -- W2,k;, W2,k; to the respective input patterns • Find the hypothesized output class (CA , CB) in each modality (as given by the label of the closest codebook vector) • For each modality update the weights according to the following rules (Only the rules for modality 1 are given) If neither or both Wli', WI;' have the same label as w2,k' or XI(n) does , 1 ' 2 1 not lie within c(n) of the border between them no updates are done, otherwise () *( 1) )(XI(n)-wv(n-l)) wi,i' n =WI,i n +a(n IIXI(n)-wV(n-I)1I * (XI(n)-wIJ,(n-I)) WIJ* (n) = wi/n - 1) - a(n) IIXI (n) _ w~J(n -1)11 where WI,i' is the codebook vector wi th the same label, and WIJ' is the codebook vector with another label. • update the labeling weights Figure 6: The Self-Supervised piecewise-linear classifier algorithm 3 EXPERIMENTS The following experiments were all performed using the Peterson and Barney vowel formant data 2. The dataset consists of the first and second formants for ten vowels in a /h V d/ context from 75 speakers (32 males, 28 females, 15 children) who repeated each vowel twice 3. To enable performance comparisons, each modality received patterns from the same dataset. This is because the final classification performance within a modality depends 20 btained from Steven Nowlan 33 speakers were missing one vowel and the raw data was linearly transformed to have zero mean and fall within the range [-3, 3] in both components 118 de Sa Table 1: Tabulation of performance figures (mean percent correct and sample standard deviation over 60 trials and 2 modalities). The heading i - j refers to performance measured after the lh step during the ilh iteration. (Note Step 1 is not repeated during the multi-iteration runs). same-paired vowels random pairing not only on the difficulty of the measured modality but also on that of the other "labeling" modality. Accuracy was measured individually (on the training set) for both modalities and averaged. These results were then averaged over 60 runs. The results described below are also tabulated in Table 1 In the first experiment, the classes were paired so that the modalities received patterns from the same vowel class. If modality 1 received an [a] vowel, so did modality 2 and likewise for all the vowel classes (i.e. p(xt!Cj ) = p(x2ICj) for all j). After the labeling algorithm stage, the accuracy was 60±5% as the initial random placement of the codebook vectors does not induce a good classifier. After application of the third step in Figure 6 (the minimizing-disagreement part of the algorithm) the accuracy was 75 ±4%. At this point the codebook vectors are much better suited to defining appropriate classification boundaries. It was discovered that all stages of the algorithm tended to produce better results on the runs that started with better random initial configurations. Thus, for each run, steps 2 and 3 were repeated with the final codebook vectors. Average performance improved (73±4% after step 2 and 76±4% after step 3). Steps 2 and 3 were repeated several more times with no further significant increase in performance. The power of using the cross-modality information to move the codebook vectors can be seen by comparing these results to those obtained with unsupervised competitive learning within modalities followed by an optimal supervised labeling algorithm which gave a performance of 72 %. One of the features of multi-modality information is that classes that are easily confuseable in one modality may be well separated in another. This should improve the performance of the algorithm as the "labeling" signal for separating the overlapping classes will be more reliable. In order to demonstrate this, more tests were conducted with random pairing of the vowels for each run. For example presentation of [a] vowels to one modality would be paired with presentation of [i] vowels to the other. That is p(xIICj ) = p(x2ICaj) for a random permutation aI, a2 .. alO. For the labeling stage the performance was as before (60 ± 4%) as the difficulty within each modality has not changed. However after the minimizingdisagreement algorithm the results were better as expected. After 1 and 2 iterations of the algorithm, 77 ± 3% and 79 ± 2% were classified correctly. These results are close to those obtained with the related supervised algorithm LVQ2.1 of 80%. 4 DISCUSSION In summary, appropriate classification borders can be learnt without an explicit external labeling or supervisory signal. For the particular vowel recognition problem, the performance of this "self-supervised" algorithm is almost as good as that achieved with superLearning Classification with Unlabeled Data 119 vised algorithms. This algorithm would be ideal for tasks in which signals for two or more modalities are available, but labels are either not available or expensive to obtain. One specific task is learning to classify speech sounds from images of the lips and the acoustic signal. Stork et. al. [1992] performed this task with a supervised algorithm but one of the main limitations for data collection was the manual labeling of the patterns [David Stork, personal communication, 1993]. This task also has the feature that the speech sounds that are confuseable are not confuseable visually and vice-versa [Stork et ai., 1992]. This complementarity helps the performance of this classifier as the other modality provides more reliable labeling where it is needed most. The algorithm could also be used for learning to classify signals to a single modality where the signal to the other "modality" is a temporally close sample. As the world changes slowly over time, signals close in time are likely from the same class. This approach should be more powerful than that of [FOldiak, 1991] as signals close in time need not be mapped to the same codebook vector but the closest codebook vector of the same class. Acknowledgements I would like to thank Steve Nowlan for making the vowel formant data available to me. Many thanks also to Dana Ballard, Geoff Hinton and Jeff Schneider for their helpful conversations and suggestions. A preliminary version of parts of this work appears in greater depth in [de Sa, 1994]. References [de Sa, 1994] Virginia R. de Sa, "Minimizing disagreement for self-supervised classification," In M.C. Mozer, P. Smolensky, D.S. Touretzky, J.L. Elman, and A.S. Weigend, editors, Proceedings of the 1993 Connectionist Models Summer School, pages 300-307. Erlbaum Associates, 1994. [de Sa and Ballard, 1993] Virginia R. de Sa and Dana H. Ballard, "a note on learning vector quantization," In c.L. Giles, SJ.Hanson, and J.D. Cowan, editors, Advances in Neural Information Processing Systems 5, pages 220-227. Morgan Kaufmann, 1993. [Foldiak, 1991] Peter FOldiak, "Learning Invariance from Transformation Sequences," Neural Computation, 3(2):194-200, 1991. [Grossberg, 1976] Stephen Grossberg, "Adaptive Pattern Classification and Universal Recoding: I. Parallel Development and Coding of Neural Feature Detectors," Biological Cybernetics, 23: 121134, 1976. [Kohonen, 1982] Teuvo Kohonen, "Self-Organized Formation of Topologically Correct Feature Maps," Biological Cybernetics, 43:59-69, 1982. [Kohonen, 1990] Teuvo Kohonen, "Improved Versions of Learning Vector Quantization," In IJCNN International Joint Conference on Neural Networks, volume 1, pages 1-545-1-550, 1990. [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. [Stork et at., 1992] David G. Stork, Greg Wolff, and Earl Levine, "Neural network lipreading system for improved speech recognition," In IJCNN International Joint Conference on Neural Networks, volume 2, pages 11-286-11-295, 1992.
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Temporal Difference Learning of Position Evaluation in the Game of Go Nicol N. Schraudolph Peter Dayan Terrence J. Sejnowski schraudo~salk.edu dayan~salk.edu terry~salk.edu Computational Neurobiology Laboratory The Salk Institute for Biological Studies San Diego, CA 92186-5800 Abstract The game of Go has a high branching factor that defeats the tree search approach used in computer chess, and long-range spatiotemporal interactions that make position evaluation extremely difficult. Development of conventional Go programs is hampered by their knowledge-intensive nature. We demonstrate a viable alternative by training networks to evaluate Go positions via temporal difference (TD) learning. Our approach is based on network architectures that reflect the spatial organization of both input and reinforcement signals on the Go board, and training protocols that provide exposure to competent (though unlabelled) play. These techniques yield far better performance than undifferentiated networks trained by selfplay alone. A network with less than 500 weights learned within 3,000 games of 9x9 Go a position evaluation function that enables a primitive one-ply search to defeat a commercial Go program at a low playing level. 1 INTRODUCTION Go was developed three to four millenia ago in China; it is the oldest and one of the most popular board games in the world. Like chess, it is a deterministic, perfect information, zero-sum game of strategy between two players. They alternate in 817 818 Schraudolph, Dayan, and Sejnowski placing black and white stones on the intersections of a 19x19 grid (smaller for beginners) with the objective of surrounding more board area (territory) with their stones than the opponent. Adjacent stones of the same color form groups; an empty intersection adjacent to a group is called a liberty of that group. A group is captured and removed from the board when its last liberty is occupied by the opponent. To prevent loops, it is illegal to make a move which recreates a prior board position. A player may pass at any time; the game ends when both players pass in succession. Unlike most other games of strategy, Go has remained an elusive skill for com puters to acquire indeed it has been recognized as a /I grand challenge" of Artificial Intelligence (Rivest, 1993). The game tree search approach used extensively in computer chess is infeasible: the game tree of Go has an average branching factor of around 200, but even beginners may routinely look ahead up to 60 plies in some situations. Humans appear to rely mostly on static evaluation of board positions, aided by highly selective yet deep local lookahead. Conventional Go programs are carefully (and protractedly) tuned expert systems (Fotland, 1993). They are fundamentally limited by their need for human assistance in compiling and integrating domain knowledge, and still play barely above the level of a human beginner a machine learning approach may thus offer considerable advantages. (Brugmann, 1993) has shown that a knowledge-free optimization approach to Go can work in principle: he obtained respectable (though inefficient) play by selecting moves through simulated annealing (Kirkpatrick et al., 1983) over possible continuations of the game. The pattern recognition component inherent in Go is amenable to connectionist methods. Supervised backpropagation networks have been applied to the game (Stoutamire, 1991; Enderton, 1991) but face a bottleneck in the scarcity of handlabelled training data. We propose an alternative approach based on the TD(A) predictive learning algorithm (Sutton, 1984; Sutton, 1988; Barto et al., 1983), which has been successfully applied to the game of backgammon by (Tesauro, 1992). His TD-Gammon program uses a backpropagation network to map preselected features of the board position to an output reflecting the probability that the player to move would win. It was trained by TD(O) while playing only itself, yet learned an evaluation function that coupled with a full two-ply lookahead to pick the estimated best move made it competitive with the best human players in the world (Robertie, 1992; Tesauro, 1994). In an early experiment we investigated a straightforward adaptation of Tesauro's approach to the Go domain. We trained a fully connected 82-40-1 backpropagation network by randomized! self-play on a 9x9 Go board (a standard didactic size for humans). The output learned to predict the margin of victory or defeat for black. This undifferentiated network did learn to squeak past Wally, a weak public domain program (Newman, 1988), but it took 659,000 games of training to do so. We have found that the efficiency of learning can be vastly improved through appropriately structured network architectures and training strategies, and these are the focus of the next two sections. 1 Unlike backgammon, Go is a deterministic game, so we had to generate moves stochastically to ensure sufficient exploration of the state space. This was done by Gibbs sampling (Geman and Geman, 1984) over values obtained from single-ply search, annealing the temperature parameter from random towards best-predicted play. Temporal Difference Learning of Position Evaluation in the Game of Go 819 E evaluation :;;;:::-:- :--s;; ~ ___ ~~ ____ -'~ processed --',----:.> / features t C ____ c_o_n_s_t_r_a_in_t--::-s_a_t_is_f_a_c_t_io_n ____ ) .. t raw feature maps J connectivity map ~~-----------------~ --~ r 1 r ~/"-... [::::J [::::) --- ~~~~:try[::J [::J --~ Go board ii?Q1 ~ Figure 1: A modular network architecture that takes advantage of board symmetries, translation invariance and localized reinforcement to evaluate Go positions. Also shown is the planned connectivity prediction mechanism (see Discussion). 2 NETWORK ARCHITECTURE One of the particular advantages of Go for predictive learning is that there is much richer information available at the end of the game than just who won. Unlike chess, checkers or backgammon, in which pieces are taken away from the board until there are few or none left, Go stones generally remain where they are placed. This makes the final state of the board richly informative with respect to the course of play; indeed the game is scored by summing contributions from each point on the board. We make this spatial credit assignment accessible to the network by having it predict the fate of every point on the board rather than just the overall score, and evaluate whole positions accordingly. This bears some similarity with the Successor Representation (Dayan, 1993) which also integrates over vector rather than scalar destinies.2 Given the knowledge-based approach of existing Go programs, there is an embarrassment of input features that one might adopt for Go: Wally already uses about 30 of them, stronger programs disproportionately more. In order to demonstrate reinforcement learning as a viable alternative to the conventional approach, however, we require our networks to learn whatever set of features they might need. The complexity of this task can be significantly reduced by exploiting a number 2Sharing information within the network across multiple outputs restricts us to A = 0 for efficient implementation of TD( A). Note that although (Tesauro, 1992) did not have this constraint, he nevertheless found A = 0 to be optimal. 820 Schraudolph, Dayan, and Sejnowski of constraints that hold a priori in this domain. Specifically, patterns of Go stones retain their properties under color reversal, reflection and rotation of the board, and modulo the considerable influence of the board edges translation. Each of these invariances is reflected in our network architecture: Color reversal invariance implies that changing the color of every stone in a Go position, and the player whose tum it is to move, yields an equivalent position from the other player's perspective. We build this constraint directly into our networks by using antisymmetric input values (+1 for black, -1 for white) and squashing functions throughout, and negating the bias input when it is white's tum to move. Go positions are also invariant with respect to the eightfold (reflection x rotation) symmetry of the square. We provided mechanisms for constraining the network to obey this invariance by appropriate weight sharing and summing of derivatives (Le Cun et al., 1989). Although this is clearly beneficial during the evaluation of the network against its opponents, it appears to impede the course of learning.3 To account for translation invariance we use convolution with a weight kernel rather than multiplication by a weight matrix as the basic mapping operation in our network, whose layers are thus feature maps produced by scanning a fixed receptive field across the input. One particular advantage of this technique is the easy transfer of learned weight kernels to different Go board sizes. It must be noted, however, that Go is not translation-invariant: the edge of the board not only affects local play but modulates other aspects of the game, and indeed forms the basis of opening strategy. We currently account for this by allowing each node in our network to have its own bias weight, giving it one degree of freedom from its neighbors. This enables the network to encode absolute position at a modest increse in the number of adjustable parameters. Furthermore, we provide additional redundancy around the board edges by selective use of convolution kernels twice as wide as the input. Figure 1 illustrates the modular architecture suggested by these deliberations. In the experiments described below we implement all the features shown except for the connectivity map and lateral constraint satisfaction, which are the subject of future work. 3 TRAINING STRATEGIES Tern poral difference learning teaches the network to predict the consequences of following particular strategies on the basis of the play they produce. The question arises as to which strategies should be used to generate the large number of Go games needed for training. We have identified three criteria by which we compare alternative training strategies: • the computational efficiency of move generation, • the quality of generated play, and • reasonable coverage of plausible Go positions. 3We are investigating possible causes and cures for this phenomenon. Temporal Difference Learning of Position Evaluation in the Game of Go 821 Tesauro trained TD-Gammon by self-play ie. the network's own position evaluation was used in training to pick both players' moves. This technique does not require any external source of expertise beyond the rules of the game: the network is its own teacher. Since Go is a deterministic game, we cannot always pick the estimated best move when training by self-play without running the risk of trapping the network in some suboptimal fixed state. Theoretically, this should not happen the network playing white would be able to predict the idiosyncrasies of the network playing black, take advantage of them thus changing the outcome, and forcing black's predictions to change commensurately- but in practice it is a concern. We therefore pick moves stochastically by Gibbs sampling (Geman and Geman, 1984), in which the probability of a given move is exponentially related to the predicted value of the position it leads to through a "temperature" parameter that controls the degree of randomness. We found self-play alone to be rather cumbersome for two reasons: firstly, the single-ply search used to evaluate all legal moves is com putationally intensive and although we are investigating faster ways to accomplish it, we expect move evaluation to remain a computational burden. Secondly, learning from self-play is sluggish as the network must bootstrap itself out of ignorance without the benefit of exposure to skilled opponents. However, there is nothing to keep us from training the network on moves that are not based on its own predictions for instance, it can learn by playing against a conventional Go program, or even by just observing games between human players. We use three computer opponents to train our networks: a random move generator, the public-domain program Wally (Newman, 1988), and the commercial program The Many Faces of Go (Fotland, 1993). The random move generator naturally doesn't play Go very we1l4, but it does have the advantages of high speed and ergodicity a few thousand games of random Go proved an effective way to prime our networks at the start of training. The two conventional Go programs, by contrast, are rather slow and deterministic, and thus not suitable generators of training data when playing among themselves. However, they do make good opponents for the network, which can provide the required variety of play through its Gibbs sam pIer. When training on games played between such dissimilar players, we must match their strength so as to prevent trivial predictions of the outcome. Against Many Faces we use standard Go handicaps for this purpose; Wally we modified to intersperse its play with random moves. The proportion of random moves is reduced adaptively as the network improves, providing us with an on-line performance measure. Since, in all cases, the strategies of both players are intimately intertwined in the predictions, one would never expect them to be correct overall when the network is playing a real opponent. This is a particular problem when the strategy for choosing moves during learning is different from the policy adopted for 'optimal' network play. (Samuel, 1959) found it inadvisable to let his checker program learn from games which it won against an opponent, since its predictions might otherwise reflect poor as well as good play. This is a particularly pernicious form of over-fitting the network can learn to predict one strategy in exquisite detail, without being able to play well in general. 4In order to ensure a minimum of stability in the endgame, it does refuse to fill in its own eyes a particular, locally recognizable type of suicidal move. 822 Schraudolph, Dayan, and Sejnowski hO-+reinf " board-+hO hl-+reinf archi tecture ;0 q-q--<t-....... . ~...---.-........,~~ I I r rein; 11 r value I ~~~ I ff ....... ~ ........ ~ I lr hO Hr hl~ ~~"-'-'I It:><t 4H-H ............ -+-+-* ...... +-++-I 11'" board Iinurn I board -+ reinf turn -+ reinf Figure 2: A small network that learned to play 9x9 Go. Boxes in the architecture panel represent 9x9 layers of units, except for turn which is a single bias unit. Arrows indicate convolutions with the corresponding weight kernels. Black disks represent excitatory, white ones inhibitory weights; within each matrix, disk area is proportional to weight magnitude. 4 RESULTS In exploring this domain, we trained many networks by a variety of methods. A small sample network that learned to beat Many Faces (at low playing level) in 9x9 Go within 3,000 games of training is shown in Figure 2. This network was grown during training by adding hidden layers one at a time; although it was trained without the (reflection x rotation) symmetry constraint, many of the weight kernels learned approximately symmetric features. The direct projection from board to reinforcement layer has an interesting structure: the negative central weight within a positive surround stems from the fact that a placed stone occupies (thus loses) a point of territory even while securing nearby areas. Note that the wide 17x17 projections from the hidden layers have considerable fringes ostensibly a trick the network uses to incorporate edge effects, which are also prominent in the bias projections from the turn unit. We compared training this architecture by self-play versus play against Wally. The initial rate of learning is similar, but soon the latter starts to outperform the former (measured against both Wally and Many Faces), demonstrating the advantage of having a skilled opponent. After about 2000 games, however, it starts to overfit to Wally and consequently worsens against Many Faces. Switching training partner to Many Faces at this point produced (after a further 1,000 games) a network that could reliably beat this opponent. Although less capable, the self-play network did manage to edge past Wally after 3,000 games; this compares very favorably with Temporal Difference Learning of Position Evaluation in the Game of Go 823 the undifferentiated network described in the Introduction. Furthermore, we have verified that weights learned from 9x9 Go offer a suitable basis for further training on the full-size (19x19) board. 5 DISCUSSION In general our networks appear more competent in the opening than further into the game. This suggests that although reinforcement information is indeed propagating all the way back from the final position, it is hard for the network to capture the multiplicity of mid-game situations and the complex combinatorics characteristic of the endgame. These strengths and weaknesses partially complement those of symbolic systems, suggesting that hybrid approaches might be rewarding. We plan to further improve network performance in a number of ways: It is possible to augment the input representation of the network in such a way that its task becomes fully translation-invariant. We intend to do this by adding an extra input layer whose nodes are active when the corresponding points on the Go board are empty, and inactive when they are occupied (regardless of color). Such an explicit representation of liberties makes the three possible states of a point on the board (black stone, white stone, or empty) linearly separable to the network, and eliminates the need for special treatment of the board edges. The use of limited receptive field sizes raises the problem of how to account for long-ranging spatial interactions on the board. In Go, the distance at which groups of stones interact is a function of their arrangement in context; an important subproblem of position evaluation is therefore to compute the connectivity of groups of stones. We intend to model connectivity explicitly by training the network to predict the correlation pattern of local reinforcement from a given position. This information can then be used to control the lateral propagation of local features in the hidden layer through a constraint satisfaction mechanism. Finally, we can train networks on recorded games between human players, which the Internet Go Server provides in steady quantities and machine-readable format. We are only beginning to explore this promising supply of instantaneous (since prerecorded), high-quality Go play for training. The main obstacle encountered so far has been the human practice of abandoning the game once both players agree on the outcome typically well before a position that could be scored mechanically is reached. We address this issue by eliminating early resignations from our training set, and using Wally to bring the remaining games to completion. We have shown that with sufficient attention to network architecture and training procedures, a connectionist system trained by temporal difference learning alone can achieve significant levels of performance in this knowledge-intensive domain. Acknowledgements We are grateful to Patrice Simard and Gerry Tesauro for helpful discussions, to Tim Casey for the plethora of game records from the Internet Go Server, and to Geoff Hinton for tniterations. Support was provided by the McDonnell-Pew Center for Cognitive Neuroscience, SERC, NSERC and the Howard Hughes Medical Institute. 824 Schraudolph, Dayan, and Sejnowski References Barto, A., Sutton, R, and Anderson, C. (1983). Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics, 13. Brugmann, B. (1993). Monte Carlo Go. Manuscript available by Internet anonymous file transfer from bsdserver.ucsf.edu, file Go/comp/mcgo.tex.Z. Dayan, P. (1993). Improving generalization for temporal difference learning: The successor representation. Neural Computation, 5(4):613-624. Enderton, H. D. (1991). The Golem Go program. Technical Report CMU-CS-92101, Carnegie Mellon University. Report available by Internet anonymous file transfer from bsdserver.ucsf.edu, file Go/comp/golem.sh.Z. Fotland, D. (1993). Knowledge representation in the Many Faces of Go. Manuscript available by Internet anonymous file transfer from bsdserver.ucsf.edu, file Go/comp/mfg.Z. Geman, S. and Geman, D. (1984). Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6. Kirkpatrick, S., GelattJr., C. D., and Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220:671-680. Le Cun, Y., Boser, B., Denker, J., Henderson, D., Howard, R, Hubbard, W., and Jackel, L. (1989). Backpropagation applied to handwritten zip code recognition. Neural Computation, 1:541-55l. Newman, W. H. (1988). Wally, a Go playing program. Shareware C program available by Internet anonymous file transfer from bsdserver.ucsf.edu, file Go/comp/wally.sh.Z. Rivest, R (1993). MIT Press, forthcoming. Invited talk: Computational Learning Theory and Natural Learning Systems, Provincetown, MA. Robertie, B. (1992). Carbon versus silicon: Matching wits with TD-Gammon. Inside Backgammon, 2(2):14-22. Samuel, A. L. (1959). Some studies in machine learning using the game of checkers. IBM Journal of Research and Development,3:211-229. Stoutamire, D. (1991). Machine learning applied to Go. Master's thesis, Case Western Reserve University. Reprint available by Internet anonymous file transfer from bsdserver.ucsf.edu, file Go/comp/report.ps.Z. Sutton, R (1984). Temporal Credit Assignment in Reinforcement Learning. PhD thesis, University of Massachusetts, Amherst. Sutton, R (1988). Learning to predict by the methods of temporal differences. Machine Learning, 3:9-44. Tesauro, G. (1992). Practical issues in temporal difference learning. Machine Learning, 8:257-278. Tesauro, G. (1994). TD-Gammon, a self-teaching backgammon program, achieves master-level play. Neural Computation, 6(2):215-219.
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Recovering a Feed-Forward Net From Its Output Charles Fefferman * and Scott Markel David Sarnoff Research Center CN5300 Princeton, N J 08543-5300 e-mail: cf9imath.princeton .edu smarkel@sarnoff.com ABSTRACT We study feed-forward nets with arbitrarily many layers, using the standard sigmoid, tanh x. Aside from technicalities, our theorems are: 1. Complete knowledge of the output of a neural net for arbitrary inputs uniquely specifies the architecture, weights and thresholds; and 2. There are only finitely many critical points on the error surface for a generic training problem. Neural nets were originally introduced as highly simplified models of the nervous system. Today they are widely used in technology and studied theoretically by scientists from several disciplines. However, they remain little understood. Mathematically, a (feed-forward) neural net consists of: (1) A finite sequence of positive integers (Do, D 1 , ... , D£); (2) A family of real numbers (wJ d defined for 1 :5 e 5: L, 1 5: j 5: Dl , 1 5: k :5 Dl-l ; and (3) A family of real numbers (OJ) defined for 15: f 5: L, 15: j 5: Dl. The sequence (Do, D1 , .. " DL ) is called the architecture of the neural net, while the W]k are called weights and the OJ thresholds. Neural nets are used to compute non-linear maps from }R.N to }R.M by the following construction. vVe begin by fixing a nonlinear function 0-( x) of one variable. Analogy with the nervous system suggests that we take o-(x) asymptotic to constants as x tends to ±oo; a standard choice, which we adopt throughout this paper, is o-(.r) = * Alternate address: Dept. of Mathematics. Princeton University, Princeton, NJ 08544-1000. 335 336 Fefferman and Markel tanh ax). Given an "input" (tl , ... ,tDo) E JR Do , we define real numbers x; for Os l S L, 1 S j S De by the following induction on l . ( 4) If l = 0 then x; = t j . (5) If the x~-l are known with l fixed (1 SlS L), then we set for ISjSDe. Here xf , ... , Xhl are interpreted as the outputs of Di "neurons" in the lth "layer" of the net. The output map of the net is defined as the map In practical applications, one tries to pick the neural net [(Do, Dl"'" DL), (W]k)' (OJ)] so that the output map <I> approximates a given map about which we have only imperfect information. The main result of this paper is that under generic conditions, perfect knowledge of the output map <I> uniquely specifies the architecture, the weights and the thresholds of a neural net, up to obvious symmetries. ~Iore precisely, the obvious symmetries are as follows . Let C1o, 11, . .. , ~(L) be permutations, with 11.= {I, ... , De} -T {I, . . . , De}; and let {e]: Os f. S L, IS j 50 De} be a collection of ± 1 'so Assume that Ii = (identity) and e] = + 1 whenever l = 0 or £ = L. Then one checks easily that the neural nets (7) [(Do, D 1 , .. . , DL), (wh), (eJ)] and (8) [(Do , D 1,.·. , DL), (W]k)' (O'J)] have the same output map if we set (9) and This reflects the facts that the neurons in layer l are interchangeable (1 50 f. 50 L - 1) , and that the function 0'( x) is odd. The nets (7) and (8) will be called isomorphtc if they are related by (9). Note in particular that isomorphic neural nets have the same architecture. Our main theorem asserts that, under generic conditions, any two neural nets with the same output map are isomorphic. \Ve discuss the generic conditions which we impose on neural nets. \Ve have to avoid obvious counterexamples such as: (10) Suppose all the weights W]k are zero. Then the output map <I> is constant. The architecture and thresholds of the neural net are clearly not uniquely determined by <I>. (11) Fix lo, JI, h with IS fo S L 1 and Isil < h 50 Dio ' Suppose we have elo = O~o and w~o = w~o for all k. Then (5) gi ves x~o = x~o Therefore the 11 J2 11k 12k Jl J2' , Recovering a Feed-Forward Net from Its Output 337 output depends on ;,J~j~l and wJj;l only through the sum i. .. ;Jj~l + wJr;-l. So the output map does not uniquely determine the weights. Our hypotheses are more than adequate to exclude these counterexamples. Specifically, we assume that (12) OJ 1= 0 and :0;1 1= I£1J/I for j 1= j'. (13) wh 1= 0; and for j 1= j', the ratio WJdW]lk is not equal to any fraction of the form pi q with p, q integers and 1 ~ q ~ 100 DlEvidently, these conditions hold for generic neural nets. The precise statement of our main theorem is as follows. If two neural nets satisfy (12), (13) and ha've the same output, then the nets are isomorphic. It would be interesting to replace (12), (13) by minimal hypotheses. and to study functions O'(x) other than tanh (~x). \Ve now sketch the proof of our main result . sacrificing accuracy for simplicity. After a trivial reduction. we may assume Do = DL = 1. Thus, the outputs of the nodes xJ(t) are functions of one variable, and the output map of the neural net is t ~ xf (t). The key idea is to continue the xJ (t) analytically to complex values of t, and to read off the structure of the net from the set of singularities of the xJ, ~ote that 0'( x) = tanh Ox) is meromorphic, with poles at the points of an arithmetic progression {(2m + l);ri: mE £:}. This leads to two crucial observations. (14) When P. = 1, the poles of X] (t) form an arithmetic progression II;. and (15) 'Vhen e. > 1, every pole of any xi-1(t) is an accumulation point of poles of any X] (t). In fact, (14) is immediate from the formula x;(t) = O'(WJlt + O}), which is merely the special case Do = 1 of (5). \Ve obtain (16) 1 _ {(2m + l);ri OJ . } IIj 1 . mE 2 wjl To see (15), fix e., j, 'It, and assume for simplicity that X~-l(t) has a simple pole at to, while xi- 1(t) (k 1= t:) is analytic in a neighborhood of to. Then (17) t. 1 A xr.- (t) = t _ to + /(t), with / analytic in a neighborhood of to. From (17) and (5), we obtain (18) xJ(t) = O'(W;t-;A(t - to)-1 + g(t», with (19) g(t) = wJtcf(t) + LWJkX~-I(t) + £1J analytic in a neighborhood of to. k;c~ Thus, in a neighborhood of to, the poles of X] (1) are the solutions tm of the equation (20) mE:: . 338 Fefferman and Markel There are infinitely many solutions of (20), accumulating at to. Hence. to is an accumulation point of poles of xJ(t), which completes the proof of (15). In view of (14), (15), it is natural to make the following definitions. The natural domain of a neural net is the largest open subset of the complex plane to which the output map t ........ xf(t) can be analytically continued. For l? 0 we define the lth singular set Singe C) by setting Sing(O) = complement of the natural domain in C, and Singe e + 1) = the set of all accumulation points of Singe f). These definitions are made entirely in terms of the output map, without reference to the structure of the given neural net. On the other hand, the sets Sing( £) contain nearly complete information on the architecture, weights and thresholds of the net. This will allow us to read off the structure of a neural net from the analytic continuation of its output map. To see how the sets Sing(f) reflect the structure of the net, we reason as follows. From (14) and (15) we expect that (21) For 1 $f $ L, Sing(L -l) is the union over j = 1, ... , Dl of the set of poles of xJ(t), together with their accumulation points (which we ignore here), and (22) For f? L, Sing(l) is empty. Immediately, then, we can read off the "depth"' L of the neural net; it is simply the smallest e for which Sing(l) is empty. vVe need to solve for Dt , wh, OJ. We proceed by induction on l. When f = 1, (14) and (21) show that Sing(L - 1) is the union of arithmetic progressions IT}, j == 1, ... , D 1 . Therefore, from Sing(L - 1) we can read off Dl and the IT]. (vVe will return to this point later in the introduction.) In view of (16), IT] determines the weights and thresholds at layer 1. modulo signs. Thus. we have found D I , W}k' g}. When l > 1, we may assume that (23) The D l " wJ~, Of are already known, for 1 ~ l' < f. Our task is to find De, W]k' gJ. In view of (23), we can find a pole to of xk-1(t) for our favorite k. Assume for simplicity that to is a simple pole of x~-I(tL and that the X~-l(t) (k ::j:. ~) are analytic in a neighborhood of to. Then X~-I(t) is given by (17) in a neighborhood of to, with A already known by virtue of (23). Let U be a small neighborhood of to. We will look at the image Y of U n Singe L - l) under the map t ........ t:to' Since A, to and Sing(L - e) are already known, so is Y. On the other hand, we can relate Y to De. WJk' OJ as follows. From (21) we see that Y is the union over j = 1,. ", Dl of (24) Yj = image of U n { Poles of xJ (t)} under t f---> tt:to)' Recovering a Feed-Forward Net from Its Output 339 For fixed j, the poles of xJ(t) in a neighborhood of to are the lm given by (20). \Ve write (25) Equation (20) shows that the first expression in brackets in (25) is equal to (2m + 1 )'7ri. Also, since tm -+ to as Iml 00 and 9 is analytic in a neighborhood of to, the second expression in brackets in (25) tends to zero. Hence, W~ leA _) = (2m+1)7ri-g(to)+o(1) forlargem. tm - to Comparing this with the definition (24), \':e see that Yj is asymptotic to the arithmetic progression (26) ITl _ {(2m + 1)7ri - g(to). ~} ]l .mEtL.. . Wjt. Thus, the known set Y is the union over j = 1 ... " Dl of sets Yj, with Yj asymptotic to the arithmetic progression IT~ . From Y, we can therefore read off Dl and the II~ . (\Ve will return to this point in a moment.) \Ve see at once from (26) that wJ ~ is determined up to sign by II]. Thus, we have found Dl and who \Vith more work, we can also find the OJ, completing the induction on t. The above induction shows that the structure of a neural net may be read off from the analytic continuation of its output map. \Ve believe that the analytic continuation of the output map will lead to further consequences in the study of neural nets. Let us touch briefly on a few points which we glossed over above. First of all, suppose we are given a set Y C C, and we know that Y is the union of sets Yl , ... , Y D, with Yj asymptotic to an arithmetic progression ITj . vVe assumed above that III, ... , ITD are uniquely determined by Y. In fact, without some further hypothesis on the IT j, this need not be true. For instance, we cannot distinguish IT 1 U IT 2 from II3 if II 1 = {odd integers}, II:! = {even integers}. II3 = {all integers} . On the other hand, we can clearly recognize ITl = {all integers} and IT2 = {mj2 : m an integer} from their union ITI U II 2 . Thus, irrational numbers enter the picture. The role of our generic hypothesis (13) is to control the arithmetic progressions that arise in our proof. Secondly, suppose xk(t) has a pole at to. We assumed for simplicity that xt(t) is analytic in a neighborhood of to for k -::j:. k. However, one of the xk(t) (k -::j:. ft) may also have a pole at to. In that case, the X~+l (t) may all be analytic in a neighborhood of to, because the contributions of the singularities of the xf in (J" (~WJtlxt + OJ+l) may cancel. Thus, the singularity at to may disappear from the output map. \Vhile this circumstance is hardly generic, it is not ruled out by our hypotheses (12), (13). 340 Feffennan and Markel Because singularities can disappear, we have to make technical changes in our description of Sing(f). For example, in the discussion following (23), Y need not be the union of the sets rj. Rather, Y is their "approximate union". (See [FD, Next, we should point out that the signs of the weights and thresholds require some attention, even though we have some freedom to change signs by applying isomorphisms. (See (9).) Finally, in the definition of the natural domain, we have assumed that there is a unique maximal open set to which the output map continues analytically. This need not be true of a general real-analytic function on the line - for instance. take f(t) = (1 + t2)1/2. Fortunately, the natural domain is well-defined for any function that continues analytically to the complement of a countable set. The defining formula (5) lets us check easily that the output map continues to the complement of a countable set, so the natural domain makes sense. This concludes our overview of the proof of our main theorem. The full proof of our results will appear in [F]. Both the uniqueness problem and the use of analytic continuation have already appeared in the neural net literature. In particular, it was R. Hecht-Nielson who pointed out the role of isomorphisms and posed the uniqueness problem. His paper with Chen and Lu [CLH] on "equioutput transformations" on the space of all neural nets influenced our work. E. Sontag [So] and H. Sussman [Su] proved sharp uniqueness theorems for one hidden layer. The proof in [So] uses complex variables. Acknow ledgements Fefferman is grateful to R. Crane, S. j\Iarkel, J. Pearson, E. Sontag, R. Sverdlove, and N. vVinarsky for introducing him to the study of neural nets. This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract F49620-92-C-0072. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. This work was also supported by the National Science Foundation. The following posters, presented at XIPS 93, may clarify our uniqueness theorem. References [CLH] R. Hecht-Nielson, et al., On the geometry of feedforward neural network error surfaces. (to appear). [F] C. Fefferman, Reconstructing a neural network from its output, Re\'ista Mathematica Iberoamericana. (to appear). [So] F. Albertini and E. Sontag, Uniqueness of weights for neural networks. (to appear). [Su] H. Sussman, Uniqueness of the weights JOT minimal feedforward nets u'ith a given input-output map, Neural Networks 5 (1992), pp. 589-593. "......,.---.... Recovering a Feed-Forward Net from Its 0 utput Charles Fefferman David Sarnoff Research Center and Princeton Univarsity Princeton. N_ Jersey PI_edt., Scot! A. Markel David Sarnoff Research Center Princeton, N_ Jersey ", .......... The Output Map of a Neural Network Fix a feed-forward neural network with the standard sigmoid CI (x) = tanh x. ~ ~ Y, y. y. The map that carries input vectors (XI' •••• x.J to outputvectors (YI' ••.• Y,,) is called the OUTPUT MAP of the neural network. Obvious Examples of Two Neural Networks with the Same Output Map Start with a neural network N. Thene~her 1. permlte the nodes in a hidden layer. or 2. fix a hidden node. and change the sign d evefY weight (Including the bias weght) that involves that node This yields a new neural n~ork with the same output map as N. Recovering a Feed-Forward Net from Its Output 341 Suppose an unknown neural netwOf1t i. placed in a black box. You aren't allowed to look in the box, blA you are IIIlowed to observe the outputs produced by the network for arbitrary inputs. Then. in principle, you have enough information to determine the network architecture (number d layers and number of nodes in each layer) and the unique values for a. the weghts. The Key Question When can two neural networks have the same output map? Unlquene .. Theorem Let N and N' be neural networks that satisfy generic conditions described below. " N and N' have the same output map. then they differ only by sign changes and permutations of hidden node •• 342 Fefferman and Markel ............ Generic CondKlon. We essume thet • aI _ighl. ere non-zero • bias weight. within each layer have distinct ebsollte values • the ralio of weighl. from node i in layer I to nodes j and k in layer (1+ 1) is not equal to any fraction of the form ~q with p. q Integers and 1~q~100'(number of nodes in layer I) Some such assumptions are needed to avoid obvious counterexamples. .......... .c .... .. ......, .... Reduction to • Network wtth Single Input and Output Node • • focus attention on a single output node, ignoring the others • study only input data w~h a single non-zero entry Determining the Network Architecture from the Picture • three kinds 01 singularities (small dots, smaH squares. large dots) => thr_ layers of sigmoids, i.e. two hidden layers and an output layer • thr_ 'spiral arms' of small squares accumulate at each large dot => three nodes in the second hidden layer • two 'spiral arms' of small dots accumulate at each smaa square => two nodes in the first hidden layer .--.,.-Outline of the Proof • it's enough to con.ider networks with one input node and one output node (see below) • al node output. are nt:IW functions of a .ingle. real variable t (the network input) • analytically continue the network output to a function f of a .ingle. cofl1llex varillble t • the qualitative geometry of the pole. of the function f determines the network architecture (_ belCM') • the asymptotica of the function f near its singularities determine the weights Geometric Description of the Pole. '~'Y •• ) ... : ... , • ;)/' <.~' ••• .J.' •• . . . -. . ....: 0 •• ,) •• . ...... . ... . . . . . . • • • poles (smell dols) accumulate al essenllal singularities (smell squares) • essential singularities (small squares) accumulate at more complicated essentlal slngularitles (large dots) .. c_ Detennlnlng the Network Architecture from the Picture (conl'd) • from the network reduction we know thai there is one input node and one output node • therefore. the network architech.e is es pictured
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Optimal Stochastic Search and Adaptive Momentum Todd K. Leen and Genevieve B. Orr Oregon Graduate Institute of Science and Technology Department of Computer Science and Engineering P.O.Box 91000, Portland, Oregon 97291-1000 Abstract Stochastic optimization algorithms typically use learning rate schedules that behave asymptotically as J.t(t) = J.to/t. The ensemble dynamics (Leen and Moody, 1993) for such algorithms provides an easy path to results on mean squared weight error and asymptotic normality. We apply this approach to stochastic gradient algorithms with momentum. We show that at late times, learning is governed by an effective learning rate J.tejJ = J.to/(l - f3) where f3 is the momentum parameter. We describe the behavior of the asymptotic weight error and give conditions on J.tejJ that insure optimal convergence speed. Finally, we use the results to develop an adaptive form of momentum that achieves optimal convergence speed independent of J.to. 1 Introduction The rate of convergence for gradient descent algorithms, both batch and stochastic, can be improved by including in the weight update a "momentum" term proportional to the previous weight update. Several authors (Tugay and Tanik, 1989; Shynk and Roy, 1988) give conditions for convergence of the mean and covariance of the weight vector for momentum LMS with constant learning rate. However stochastic algorithms require that the learning rate decay over time in order to achieve true convergence of the weight (in probability, in mean square, or with probability one). 477 478 Leen and Orr This paper uses our previous work on weight space probabilities (Leen and Moody, 1993; Orr and Leen, 1993) to study the convergence of stochastic gradient algorithms with annealed learning rates of the form Jl = Jlo/t, both with and without momentum. The approach provides simple derivations of previously known results and their extension to stochastic descent with momentum. Specifically, we show that the mean squared weight misadjustment drops off at the maximal rate ex 1/ t only if the effective learning rate JlejJ = Jlo/(1 - (3) is greater than a critical value which is determined by the Hessian. These results suggest a new algorithm that automatically adjusts the momentum coefficient to achieve the optimal convergence rate. This algorithm is simpler than previous approaches that either estimate the curvature directly during the descent (Venter, 1967) or measure an auxilliary statistic not directly involved in the optimization (Darken and Moody, 1992). 2 Density Evolution and Asymptotics We consider stochastic optimization algorithms with weight w E RN. We confine attention to a neighborhood of a local optimum w* and express the dynamics in terms of the weight error v = w - w*. For simplicity we treat the continuous time algorithm 1 d~~t) = Jl(t) H[ v(t), x(t)] (1) where Jl(t) is the learning rate at time t, H is the weight update function and x(t) is the data fed to the algorithm at time t. For stochastic gradient algorithms H = - \7 v £(v, x(t)), minus the gradient of the instantaneous cost function. Convergence (in mean square) to w* is characterized by the average squared norm of the weight error E [ 1 V 12] = Trace C where C - J dNv vvT P(v,t) (2) is the weight error correlation matrix and P(v, t) is the probability density at v and time t. In (Leen and Moody, 1993) we show that the probability density evolves according to the Kramers-Moyal expansion ap(v, t) at 00 2: i=l ( _l)i ., 2. N ai 2: aVil aV12 ... aVj; il,···j;=l 1 Although algorithms are executed in discrete time, continuous time formulations are often advantagous for analysis. The passage from discrete to continuous time is treated in various ways depending on the needs of the theoretical exposition. Kushner and Clark (1978) define continous time functions that interpolate the discrete time process in order to establish an equivalence between the asymptotic behavior of the discrete time stochastic process, and solutions of an associated deterministic differential equation. Heskes et ai. (1992) draws on the results of Bedeaux et ai. (1971) that link (discrete time) random walk trajectories to the solution of a (continuous time) master equation. Heskes' master equation is equivalent to our Kramers-Moyal expansion (3). Optimal Stochastic Search and Adaptive Momentum 479 where H j " denotes the jlh component of the N-component vector H, and ( .. ')x denotes averaging over the density of inputs. Differentiating (2) with respect to time, u~ing (3) and integrating by parts, we obtain the equation of motion for the weight error correlation dd~ J-L(t) J dN V P(v, t) [v (H(v, xf)x + (H(v, x))x vTJ + J-L(t)2 J dN V P(v, t) (H(v, x) H(v, x)T)x (4) 2.1 Asymptotics of the Weight Error Correlation Convergence of v can be understood by studying the late time behavior of (4). Since the update function H(v, x) is in general non-linear in v, the time evolution of the correlation matrix Cij is coupled to higher moments E [Vi Vj Vk ••• ] of the weight error. However, the learning rate is assumed to follow a schedule J-L(t) that satisfies the requirements for convergence in mean square to a local optimum. Thus at late times the density becomes sharply peaked about v = 02 . This suggests that we expand H(v, x) in a power series about v = 0 and retain the lowest order non-trivial terms in (4) leaving: dC dt = - J-L(t) [ (R C) + (C RT) ] + J-L(t)2 D , (5) where R is the Hessian of the average cost function (E) x' and D = (H(O,x)H(O,xf)x (6) is the diffusion matrix, both evaluated at the local optimum w*. (Note that RT = R.) We use (5) with the understanding that it is valid for large t. The solution to (5) is C(t) = U(t,to)C(to)UT(t,to) + t d7 J-L(7)2 U(t,7) D UT(t,7) (7) ito where the evolution operator U(t2' td is U(t2, t1) = exp [ -R 1:' dr tt(r) ] (8) We assume, without loss of generality, that the coordinates are chosen so that R is diagonal (D won't be) with eigenvalues Ai, i = 1 ... N. Then with J-L(t) = J-Lo/t we obtain E[lvI 2] = Trace [C(t)] N { ( ttO ) 21-1-0 Ai t; Cii (to) [ 1 1 (to) 21-1-0 A, 1 } . (9) t to t 2In general the density will have nonzero components outside the basin of w* . We are neglecting these, for the purpose of calculating the second moment of the the local density in the vicinity of w*. 480 Leen and Orr We define 1 J.lerit == --2 Amin (10) and identify two regimes for which the behavior of (9) is fundamentally different: 1. J.lo > J.lcri( E [lvI 2 ] drops off asymptotically as lit. 2. J.lo < J.lerit: E [lvI 2 ] drops off asymptotically as ( t ) (2 ~o ATnin ) i.e. more slowly than lit. Figure 1 shows results from simulations of an ensemble of 2000 networks trained by LMS, and the prediction from (9). For the simulations, input data were drawn from a gaussian with zero mean and variance R = 1.0. The targets were generated by a noisy teacher neuron (i.e. targets =w*x +~, where (~) = 0 and (e) = (72). The upper two curves in each plot (dotted) depict the behavior for J.lo < J.lerit = 0.5. The remaining curves (solid) show the behavior for J.lo > J.lerit. 0 0 ,... ,... I I ..... C\I C\I i.CII i.CII ~' ~ I W WC') CIt? ClI 0 0 oJ oJ "r 'of I U? U? 100 1000 5000 50000 100 1000 5000 50000 Fig.1: LEFT - Simulation results from an ensemble of 2000 one-dimensional LMS algorithms with R = 1.0, (72 = 1.0 and /-L = /-Lo/t. RIGHT - Theoretical predictions from equation (9). Curves correspond to (top to bottom) /-Lo = 0.2, 0.4, 0.6, 0.8, 1.0, 1.5 . By minimizing the coefficient of lit in (9), the optimal learning rate is found to be J.lopt = 11 Amin. This formalism also yields asymptotic normality rather simply (Orr and Leen, 1994). These conditions for "optimal" (Le. lit) convergence of the weight error correlation and the related results on asymptotic normality have been previously discussed in the stochastic approximation literature (Darken and Moody, 1992; Goldstein, 1987; White, 1989; and references therein) . The present formal structure provides the results with relative ease and facilitates the extension to stochastic gradient descent with momentum. 3 Stochastic Search with Constant Momentum The discrete time algorithm for stochastic optimization with momentum is: v(t + 1) = v(t) + J.l(t) H[v(t), x(t)] + f3 f!(t) (11) Optimal Stochastic Search and Adaptive Momentum 481 n(t + 1) v(t + 1) - v(t) n(t) + /1(t) H[v(t), x(t)] + ((3 - 1) n(t), (12) or in continuous time, dv(t) /1(t) H[v(t), x(t)] + (3 n(t) (13) dt dn(t) /1(t) H[v(t), x(t)] + ((3 - 1) n(t). (14) dt As before, we are interested in the late time behavior of E [lvI 2 ]. To this end, we define the 2N-dimensional variable Z = (v, nf and, following the arguments of the previous sections, expand H[v(t), x(t)] in a power series about v = 0 retaining the lowest order non-trivial terms. In this approximation the correlation matrix C _ E[ZZT] evolves according to with dC T 2dt = KC + CK + /1(t) D (15) (16) I is the N x N identity matrix, and Rand D are defined as before. The evolution operator is now U(t2' ttl = exp [t' dr K(r)] (17) and the solution to (15) is C = U(t, to) C(to) U T (t, to) + t dr /12(r) U(t, r) D U T (t, r) (18) ltD The squared norm of the weight error is the sum of first N diagonal elements of C. In coordinates for which R is diagonal and with /1(t) = /10 It, we find that for t » to E[lvI2] '" t, {c,,(to) (t;) 'i~~' + This reduces to (9) when (3 = O. Equation (19) defines two regimes of interest: 1. /10/(1 - (3) > /1cri( E[lvI2] drops off asymptotically as lit. 2. /10/(1 - (3) < /1cri( E[lvI2] drops off asymptotically as 21-'Q'xmjn (~) 1 ~ I.e. more slowly than lit. 482 Leen and Orr The form of (19) and the conditions following it show that the asymptotics of gradient descent with momentum are governed by the effective learning rate _ M MejJ = 1 - {3 . Figure 2 compares simulations with the predictions of (19) for fixed Mo and various {3. The simulations were performed on an ensemble of 2000 networks trained by LMS as described previously but with an additional momentum term of the form given in (11). The upper three curves (dotted) show the behavior of E[lvI 2] for MejJ < Merit· The solid curves show the behavior for MejJ > Merit· The derivation of asymptotic normality proceeds similarly to the case without momentum. Again the reader is referred to (Orr and Leen, 1994) for details . .... .... , , ...... N ...... N (\I' (\I' < ~ "> > -C') ur' -C') ur' C) C) 0'<:1" ...J, 0'<:1" ...J, III III , , 100 1000 5000 50000 100 1000 SOOO 50000 t t Fig.2: LEFT - Simulation results from an ensemble of 2000 one-dimensional LMS algorithms with mome~tum with R = 1.0, (12 = 1.0, and /10 0.2. RIGHTTheoretical predictions from equation (19). Curves correspond to (top to bottom) {3 = 0.0, 004, 0.5, 0.6, 0.7, 0.8 . 4 Adaptive Momentum Insures Optimal Convergence The optimal constant momentum parameter is obtained by minimizing the coefficient of lit in (19). Imposing the restriction that this parameter is positive3 gives (3opt = max(O, 1 MOAmin). (20) As with Mopt, this result is not of practical use because, in general, Amin is unknown. For I-dimensional linear networks, an alternative is to use the instantaneous estimate of A, :\(t) = x 2(t) where x(t) is the network input at time t. We thus define the adaptive momentum parameter to be (3adapt = max(O, 1 MOX2 ) (I-dimension). (21) An algorithm based on (21) insures that the late time convergence is optimally fast. An alternative route to achieving the same goal is to dispense with the momentum term and adaptively adjust the learning rate. Vetner (1967) proposed an algorithm 3 E[lvI 2 ] diverges for 1{31 > 1. For -1 < {3 < 0, E[lvI 2] appears to converge but oscillations are observed. Additional study is required to determine whether {3 in this range might be useful for improving learning. Optimal Stochastic Search and Adaptive Momentum 483 that iteratively estimates A for 1-D algorithms and uses the estimate to adjust J.Lo. Darken and Moody (1992) propose measuring an auxiliary statistic they call "drift" that is used to determine whether or not J.Lo > J.Lcrit. The adaptive momentum scheme generalizes to multiple dimensions more easily than Vetner's algorithm, and, unlike Darken and Moody's scheme, does not involve calculating an auxiliary statistic not directly involved with the minimization. A natural extension to N dimensions is to define a matrix of momentum coefficients, 'Y = I - J.Lo X xT, where I is the N x N identity matrix. By zeroing out the negative eigenvalues of 'Y, we obtain the adaptive momentum matrix (3adapt = I - ex xT, where e = min(J.Lo, 1/(xT x)). (22) =1.5 -1+_~_--::=====-_~~~ L 1 2 3 °9(t) Fig.3: Simulations of 2-D LMS with 1000 networks initialized at Vo = (.2, .3) and with (72 = 1, ).1 = .4, ).2 = 4, and /-Lcrit = 1.25. LEFT- {3 = 0, RIGHT - {3 = (3adapt. Dashed curves correspond to adaptive momentum. Figure 3 shows that our adaptive momentum not only achieves the optimal convergence rate independent of the learning rate parameter J.Lo but that the value of log(E[lvI2]) at late times is nearly independent of J.Lo and smaller than when momentum is not used. The left graph displays simulation results without momentum. Here, convergence rates clearly depend on J.Lo and are optimal for J.Lo > J.Lcrit = 1.25. When J.Lo is large there is initially significant spreading in v so that the increased convergence rate does not result in lower log(E[lvI2]) until very late times (t ~ 105 ). The graph on the right shows simulations with adaptive momentum. Initially, the spreading is even greater than with no momentum, but log(E[lvI2]) quickly decreases to reach a much smaller value. In addition, for t ~ 300, the optimal convergence rate (slope=-l) is achieved for all three values of J.Lo and the curves themselves lie almost on top of one another. In other words, at late times (t ;::: 300), the value of log(E[lvI2]) is independent of J.Lo when adaptive momentum is used. 5 Summary We have used the dynamics of the weight space probabilities to derive the asymptotic behavior of the weight error correlation for annealed stochastic gradient algorithms with momentum. The late time behavior is governed by the effective learning rate J.Lejj = J.Lo/(l - (3) . For learning rate schedules J.Lolt, if J.Leff > 1/(2 Arnin) , then the squared norm of the weight error v - w - w* falls off as lit. From these results we have developed a form of momentum that adapts to obtain optimal convergence rates independent of the learning rate parameter. 484 Leen and Orr Acknowledgments This work was supported by grants from the Air Force Office of Scientific Research (F49620-93-1-0253) and the Electric Power Research Institute (RP8015-2). References D. Bedeaux, K. Laktos-Lindenberg, and K. Shuler. (1971) On the Relation Between Master Equations and Random Walks and their Solutions. Journal of Mathematical Physics, 12:2116-2123. Christian Darken and John Moody. (1992) Towards Faster Stochastic Gradient Search. In J.E. Moody, S.J. Hanson, and R.P. Lipmann (eds.) Advances in Neural Information Processing Systems, vol. 4. Morgan Kaufmann Publishers, San Mateo, CA, 1009-1016. Larry Goldstein. (1987) Mean Square Optimality in the Continuous Time Robbins Monro Procedure. Technical Report DRB-306, Dept. of Mathematics, University of Southern California, LA. H.J. Kushner and D.S. Clark. (1978) Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer-Verlag, New York. Tom M. Heskes, Eddy T.P. Slijpen, and Bert Kappen. (1992) Learning in Neural Networks with Local Minima. Physical Review A, 46(8):5221-5231. Todd K. Leen and John E. Moody. (1993) Weight Space Probability Densities in Stochastic Learning: 1. Dynamics and Equilibria. In Giles, Hanson, and Cowan (eds.), Advances in Neural Information Processing Systems, vol. 5, Morgan Kaufmann Publishers, San Mateo, CA, 451-458. G. B. Orr and T. K. Leen. (1993) Weight Space Probability Densities in Stochastic Learning: II. Transients and Basin Hopping Times. In Giles, Hanson, and Cowan (eds.), Advances in Neural Information Processing Systems, vol. 5, Morgan Kaufmann Publishers, San Mateo, CA, 507-514. G. B. Orr and T. K. Leen. (1994) Momentum and Optimal Stochastic Search. In M. C. Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman, and A. S. Weigend (eds.), Proceedings of the 1993 Connectionist Models Summer School, 351-357. John J. Shynk and Sumit Roy. (1988) The LMS Algorithm with Momentum Updating. Proceedings of the IEEE International Symposium on Circuits and Systems, 2651-2654. Mehmet Ali Tugay and Yal<;in Tanik. (1989) Properties of the Momentum LMS Algorithm. Signal Processing, 18:117-127. J. H. Venter. (1967) An Extension of the Robbins-Monro Procedure. Annals of Mathematical Statistics, 38:181-190. Halbert White. (1989) Learning in Artificial Neural Networks: A Statistical Perspective. Neural Computation, 1:425-464.
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The Parti-game Algorithm for Variable Resolution Reinforcement Learning in Multidimensional State-spaces Andrew W. Moore School of Computer Science Carnegie-Mellon University Pittsburgh, PA 15213 Abstract Parti-game is a new algorithm for learning from delayed rewards in high dimensional real-valued state-spaces. In high dimensions it is essential that learning does not explore or plan over state space uniformly. Part i-game maintains a decision-tree partitioning of state-space and applies game-theory and computational geometry techniques to efficiently and reactively concentrate high resolution only on critical areas. Many simulated problems have been tested, ranging from 2-dimensional to 9-dimensional state-spaces, including mazes, path planning, non-linear dynamics, and uncurling snake robots in restricted spaces. In all cases, a good solution is found in less than twenty trials and a few minutes. 1 REINFORCEMENT LEARNING Reinforcement learning [Samuel, 1959, Sutton, 1984, Watkins, 1989, Barto et al., 1991] is a promising method for control systems to program and improve themselves. This paper addresses its biggest stumbling block: the curse of dimensionality [Bellman, 1957], in which costs increase exponentially with the number of state variables. Some earlier work [Simons et al., 1982, Moore, 1991, Chapman and Kaelbling, 1991, Dayan and Hinton, 1993] has considered recursively partitioning state-space while learning from delayed rewards. The new ideas in the parti-game algorithm in711 712 Moore clude (i) a game-theoretic splitting criterion to robustly choose spatial resolution (ii) real-time incremental maintenance and planning with a database of all previous experIences, and (iii) using local greedy controllers for high-level "funneling" actions. 2 ASSUMPTIONS The parti-game algorithm applies to difficult learning control problems in which: 1. State and action spaces are continuous and multidimensional. 2. "Greedy" and hill-dim bing techniques would become stuck, never attaining the goal. 3. Random exploration would be hopelessly time-consuming. 4. The system dynamics and control laws can have discontinuities and are unknown: they must be learned. The experiments reported later all have properties 1-4. However, the initial algorithm, described and tested here, has the following restrictions: 5. Dynamics are deterministic. 6. The task is specified by a goal, not an arbitrary reward function. 7. The goal state is known. 8. A "good" solution is required, not necessarily the optimal path. This nation of goodness can be formalized as "the optimal path to within a given resolution of state space". 9. A local greedy controller is available, which we can ask to move greedily towards any desired state. There is no guarantee that a request to the greedy controller will succeed. For example, in a maze a greedy path to the goal would quickly hit a wall. Future developments may include relatively straightforward additions to the algorithm that would remove the need for restrictions 6-9. Restriction 5 is harder to remove. 3 ESSENTIALS OF THE PARTI-GAME ALGORITHM The state space is broken into partitions by a kd-tree [Friedman et al., 1977]. The controller can always sense its current (continuous valued) state, and can cheaply compute which partition it is in. The space of actions is also discretized so that in a partition with N neighboring partitions, there are N high-level actions. Each high level action corresponds to a local greedy controller, aiming for the center of the corresponding neighboring partition. Each partition keeps records of all the occasions on which the system state has passed through it. Along with each record is a memory of which high level action was used (i.e. which neighbor was aimed for) and what the outcome was. Figure 1 provides an illustration. Given this database of (partition, high-level-action, outcome) triplets, and our knowledge of the partition containing the goal state, we can try to compute the The Parti-Game Algorithm for Variable Resolution Reinforcement Learning 713 Partition I Partition 2 Figure 1: Three trajectories starting ................... in partition 1, using high-level action "Aim at partition 2". Partition 1 remembers three outcomes. (Part 1, Aim 2 --+ Part 2) (Part 1, Aim 2 --+ Part 1) , (Part 1, Aim 2 --+ Part 3) I Partition 3 I I best route to the goal. The standard approach would be to model the system as a Markov Decision Task in which we empirically estimate the partition transition probabilities. However, the probabilistic interpretation of coarse resolution partitions can lead to policies which get stuck. Instead, we use a game-theoretic approach, in which we imagine an adversary. This adversary sees our choice of high-level action, and is allowed to select any of the observed previous outcomes of the action in this partition. Partitions are scored by minimaxing: the adversary plays to delay or prevent us getting to the goal and we play to get to the goal as quickly as possible. Whenever the system's continuous state passes between partitions, the database of state transitions is updated and, if necessary, the minimax scores of all partitions are updated. If real-time constraints do not permit full recomputation, the updates take place incrementally in a manner similar to prioritized sweeping [Moore and Atkeson, 1993]. As well as being robust to coarseness, the game-theoretic approach also tells us where we should increase the resolution. Whenever we compute that we are in a losing partition we perform resolution increase. We first compute the complete set of connected partitions which are also losing partitions. We then find the subset of these partitions which border some non-losing region. We increase the resolution of all these border states by splitting them along their longest axes1 . 4 INITIAL EXPERIMENTS Figure 2 shows a 2-d continuous maze. Figure 3 shows the performance of the robot during the very first trial. It begins with intense exploration to find a route out of the almost entirely enclosed start region. Having eventually reached a sufficiently high resolution, it discovers the gap and proceeds greedily towards the goal, only to be stopped by the goal's barrier region. The next barrier is traversed at a much lower resolution, mainly because the gap is larger. Figure 4 shows the second trial, started from a slightly different position. The policy derived from the first trial gets us to the goal without further exploration. The trajectory has unnecessary bends. This is because the controller is discretized according to the current partitioning. If necessary, a local optimizer could be used 1 More intelligent splitting criteria are under investigation. 714 Moore Start I· Figure 2: A 2-d maze problem. The point robot must find a path from start to goal without crossing any of the barrier lines. Remember that initially it does not know where any obstacles are, and must discover them by finding impassable states. to refine this trajectory2. Figure 3: The path taken during the entire first trial. See text for explanation. The system does not explore unnecessary areas. The barrier in the top left remains at low resolution because the system has had no need to visit there. Figures 5 and 6 show what happens when we now start the system inside this barrier. Figure 7 shows a 3-d state space problem. If a standard grid were used, this would need an enormous number of states because the solution requires detailed threepoint-turns. Parti-game's total exploration took 18 times as much movement as one run of the final path obtained. Figure 8 shows a 4-d problem in which a ball rolls around a tray with steep edges. The goal is on the other side of a ridge. The maximum permissible force is low, and so greedy strategies, or globally linear control rules, get stuck in a limit cycle. Parti-game's solution runs to the other end of the tray, to build up enough velocity to make it over the ridge. The exploration-length versus final-path-Iength ratio is 24. Figure 9 shows a 9-joint snake-like robot manipulator which must move to a specified configuration on the other side of a barrier. Again, no initial model is given: the controller must learn it as it explores. It takes seven trials before fixing on the solution shown. The exploration-length versus final-path-length ratio is 60. 2 Another method is to increase the resolution along the trajectory [Moore, 1991]. The Parti-Game Algorithm for Variable Resolution Reinforcement Learning 715 (') 1/ ,. rn c-fV v'\ 1 r ~ I ,r-Il 1 II j 11 ~ /-"f-I- L~ IT'" l.J J 1-+' 1 "/ r---./ 1-1 f-H r---./ 1-1 f-H ~ .f"" /'-/--. -I-~ J ) ) -1 --1 Figure 4: The second trial. Figure 5: Starting inside the top left barrier. Figure 6: that. The trial after Figure 7: A problem with a planar rod being guided past obstacles. The state space is three-dimensional: two values specify the position of the rod's center, and the third specifies the rod's angle from the horizontal. The angle is constrained so that the pole's dotted end must always be below the other end. The pole's center may be moved a short distance (up to 1/40 of the diagram width) and its angle may be altered by up to 5 degrees, provided it does not hit a barrier in the process. Parti-game converged to the path shown below after two trials. The partitioning lines on the solution diagram only show a 2-d slice of the full kd-tree. Trials 10 Steps no Partitions 149 149 149 Change 716 Moore Figure 8: A puck sliding over a hilly surface (hills shown by contours below: the surface is bowl shaped, with the lowest points nearest the center, rising steeply at the edges). The state space is four-dimensional: two position and two velocity variables. The controls consist of a force which may be applied in any direction, but with bounded magnitude. Convergence time was two trials. Trials 1 2 Steps 2609 115 Partitions 13 13 3 no change lu..&.I.I.WI'U'I.· •••• 10 Figure 9: A nine-degree-of-freedom planar robot must move from the shown start configuration to the goal. The solution entails curling, rotating and then uncurling. It may not intersect with any of the barriers, the edge of the workspace, or itself. Convergence occurred after seven trials. Trials 1 2 Steps 1090 430 Partitions 41 66 f-Fixed base 3 4 353 330 67 69 5 6 7 8 739 200 52 78 85 85 The Parti-Game Algorithm for Variable Resolution Reinforcement Learning 717 5 DISCUSSION Possible extensions include: • Splitting criteria that lay down splits between trajectories with spatially distinct outcomes. • Allowing humans to provide hints by permitting user-specified controllers ("behaviors") as extra high-level actions. • Coalescing neighboring partitions that mutually agree. We finish by noting a promising sign involving a series of snake robot experiments with different numbers of links (but fixed total length). Intuitively, the problem should get easier with more links, but the curse of dimensionality would mean that (in the absence of prior knowledge) it becomes exponentially harder. This is borne out by the observation that random exploration with the three-link arm will stumble on the goal eventually, whereas the nine link robot cannot be expected to do so in tractable time. However, Figure 10 indicates that as the dimensionality rises, the amount of exploration (and hence computation) used by parti-game does not rise exponentially. Real-world tasks may often have the same property as the snake example: the complexity of the ultimate task remains roughly constant as the number of degrees of freedom increases. If so, we may have uncovered the Achilles' heel of the curse of dimensionality. ~ ell "'" 180 ~ ~ = 160 Q CJ 140 ~ "'" 120 ~ ~ 100 .CI ~ 80 ~ ~ 60 e fI.l 40 = Q .... ... 20 := 0 "'" ~ ~ References I 3 4 5 6 7 8 9 Dimensionality Figure 10: The number of partitions finally created against degrees of freedom for a set of snakelike robots. The kd-trees built were all highly non-uniform, typically having maximum depth nodes of twice the dimensionality. The relation between exploration time and dimensionality (not shown) had a similar shape. [Barto et ai., 1991] A. G. Barto, S. J. Bradtke, and S. P. Singh. Real-time Learning and Control using Asynchronous Dynamic Programming. Technical Report 91-57, University of Massachusetts at Amherst, August 1991. [Bellman, 1957] R. E. Bellman. Dynamic Programming. Princeton University Press, Princeton, N J, 1957. [Chapman and Kaelbling, 1991) D. Chapman and L. P. Kaelbling. Learning from Delayed Reinforcement In a Complex Domain. Technical Report, Teleos Research, 1991. 718 Moore [Dayan and Hinton, 1993] P. Dayan and G. E. Hinton. Feudal Reinforcement Learning. In S. J. Hanson, J. D Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems 5. Morgan Kaufmann, 1993. [Friedman et al., 1977) J. H. Friedman, J. L. Bentley, and R. A. Finkel. An Algorithm for Finding Best Matches in Logarithmic Expected Time. ACM Trans. on Mathematical Software, 3(3):209-226, September 1977. [Moore and Atkeson, 1993] A. W. Moore and C. G. Atkeson. Prioritized Sweeping: Reinforcement Learning with Less Data and Less Real Time. Machine Learning, 13, 1993. [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: Proceedings of the Eighth International Workshop. Morgan Kaufman, June 1991. [Samuel, 1959] A. L. Samuel. Some Studies in Machine Learning using the Game of Checkers. IBM Journal on Research and Development, 3, 1959. Reprinted in E. A. Feigenbaum and J. Feldman, editors, Computers and Thought, McGraw-Hill, 1963. [Simons et al., 1982) J. Simons, H. Van Brussel, J. De Schutter, and J. Verhaert. A SelfLearning Automaton with Variable Resolution for High Precision Assembly by Industrial Robots. IEEE Trans. on Automatic Control, 27(5):1109-1113, October 1982. [Singh, 1993] S. Singh. Personal Communication. ,1993. [Sutton, 1984) R. S. Sutton. Temporal Credit Assignment in Reinforcement Learning. Phd. thesis, University of Massachusetts, Amherst, 1984. [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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Feature Densities are Required for Computing Feature Correspondences Subutai Ahmad Interval Research Corporation 1801-C Page Mill Road, Palo Alto, CA 94304 E-mail: ahmadCDinterval.com Abstract The feature correspondence problem is a classic hurdle in visual object-recognition concerned with determining the correct mapping between the features measured from the image and the features expected by the model. In this paper we show that determining good correspondences requires information about the joint probability density over the image features. We propose "likelihood based correspondence matching" as a general principle for selecting optimal correspondences. The approach is applicable to non-rigid models, allows nonlinear perspective transformations, and can optimally deal with occlusions and missing features. Experiments with rigid and non-rigid 3D hand gesture recognition support the theory. The likelihood based techniques show almost no decrease in classification performance when compared to performance with perfect correspondence knowledge. 1 INTRODUCTION The ability to deal with missing information is crucial in model-based vision systems. The feature correspondence problem is an example where the correct mapping between image features and model features is unknown at recognition time. For example, imagine a network trained to map fingertip locations to hand gestures. Given features extracted from an image, it becomes important to determine which features correspond to the thumb, to the index finger, etc. so we know which input units to clamp with which numbers. Success at the correspondence matching step 961 962 Ahmad I / Class bouudary Class 2 Class 1 • P2 o L-________ ~ ______________ _ o I Xl Figure 1: An example 2D feature space. Shaded regions denote high probability. Given measured values of 0.2 and 0.9, the points PI and P2 denote possible instantiations but PI is much more likely. is vital for correct classification. There has been much previous work on this topic (Connell and Brady 1987; Segen 1989; Huttenlocher and Ullman 1990; Pope and Lowe 1993) but a general solution has eluded the vision community. In this paper we propose a novel approach based on maximizing the probability of a set of models generating the given data. We show that neural networks trained to estimate the joint density between image features can be successfully used to recover the optimal correspondence. Unlike other techniques, the likelihood based approach is applicable to non-rigid models, allows perspective 3D transformations, and includes a principled method for dealing with occlusions and missing features. 1.1 A SIMPLE EXAMPLE Consider the idealized example depicted in Figure 1. The distribution of features is highly non-uniform (this is typical of non-rigid objects). The classification boundary is in general completely unrelated to the feature distribution. In this case, the class (posterior) probability approaches 1 as feature Xl approaches 0, and 0 as it approaches 1. Now suppose that two feature values 0.2 and 0.9 are measured from an image. The task is to decide which value gets assigned to X I and which value gets assigned to X2. A common strategy is to select the correspondence which gives the maximal network output (i.e. maximal posterior probability). In this example (and in general) such a strategy will pick point P2, the wrong correspondence. This is because the classifer output represents the probability of a class given a specific feature assignment and specific values. The correspondence problem however, is something completely different: it deals with the probability of getting the feature assignments and values in the first place. Feature Densities Are Required for Computing Feature Correspondences 963 2 LIKELIHOOD BASED CORRESPONDENCE MATCHING We can formalize the intuitive arguments in the previous section. Let C denote the set of classes under consideration. Let X denote the list of features measured from the image with correspondences unknown. Let A be the set of assignments of the measured values to the model features. Each assignment a E A reflects a particular choice of feature correspondences. \Ve consider two different problems: the task of choosing the best assignment a and the task of classifying the object given X. Selecting the best correspondence is equivalent to selecting the permutation that maximizes p(aIX, C). This can be re-written as: ( I X C) = p(Xla, C)p(aIC) p a - , p(XIC) (1) p(XIC) is a normalization factor that is constant across all a and can be ignored. Let Xa denote a specific feature vector constructed by applying permutation a to X. Then (1) is equivalent to maximizing: p(aIX, C) = p(xaIC)p(aIC) (2) p(aIC) denotes our prior knowledge about possible correspondences. (For example the knowledge that edge features cannot be matched to color features.) When no prior knowledge is available this term is constant. We denote the assignment that maximizes (2) the maximum likelihood correspondence match. Such a correspondence maximizes the probability that a set of visual models generated a given set of image features and will be the optimal correspondence in a Bayesian sense. 2.1 CLASSIFICATION In addition to computing correspondences, we would like to classify a model from the measured image features, i.e. compute p( CdX, C). The maximal-output based solution is equivalent to selecting the class Ci that maximizes p(Cilxa, C) over all assignments a and all classes Ci. It is easy to see that the optimal strategy is actually to compute the following weighted estimate over all candidate assignments: (C.IX C) = Lap(CiIX, a, C)p(Xla, C)p(aIC) p , , p(XIC) (3) Classification based on (3) is equivalent to selecting the class that maximizes: (4) a Note that the network output based solution represents quite a degraded estimate of (4). It does not consider the input density nor perform a weighting over possible 964 Ahmad correspondences. A reasonable approximation is to select the maximum likelihood correspondence according to (2) and then use this feature vector in the classification network. This is suboptimal since the weighting is not done but in our experience it yields results that are very close to those obtained with (4). 3 COMPUTING CORRESPONDENCES WITH GBF NETWORKS In order to compute (2) and (4) we consider networks of normalized Gaussian basis functions (GBF networks). The i'th output unit is computed as: (5) with: Here each basis function j is characterized by a mean vector f.lj and by oJ, a vector representing the diagonal covariance matrix. Wji represents the weight from the j'th Gaussian to the i'th output. 7rj is a weight attached to each basis function. Such networks have been popular recently and have proven to be useful in a number of applications (e.g. (Roscheisen et al. 1992; Poggio and Edelman 1990). For our current purpose, these networks have a number of advantages. Under certain training regimes such as EM or "soft clustering" (Dempster et al. 1977; Nowlan 1990) or an approximation such as K-11eans (Neal and Hinton 1993), the basis functions adapt to represent local probability densities. In particular p(xaIC) :::::: E j bj(xa). If standard error gradient training is used to set the weights Wij then Yi(Xa ) :::::: p( Cilxa, C) Thus both (2) and (4) can be easilty computed.(Ahmad and Tresp 1993) showed that such networks can effectively learn feature density information for complex visual problems. (Poggio and Edelman 1990) have also shown that similar networks (with a different training regime) can learn to approximate the complex mappings that arise in 3D recognition. 3.1 OPTIMAL CORRESPONDENCE MATCHING WITH OCCLUSION An additional advantage of G BF networks trained in this way is that it is possible to obtain closed form solutions to the optimal classifier in the presence of missing or noisy features. It is also possible to correctly compute the probability of feature vectors containing missing dimensions. The solution consists of projecting each Gaussian onto the non-missing dimensions and evaluating the resulting network. Note that it is incorrect to simply substitute zero or any other single value for the missing dimensions. (For lack of space we refer the reader to (Ahmad and Tresp Feature Densities Are Required for Computing Feature Correspondences 965 '"five" "four" '1hree" "two" "one" .. tlumbs _up " )Jointing ,. Figure 2: Classifiers were trained to recognize these 7 gestures. a 3D computer model of the hand is used to generate images of the hand in various poses. For each training example, we randomly choose a 3D orientation and depth, compute the 3D positions of the fingertips and project them onto 2D. There were 5 features yielding a lOD input space. 1993) for further details.) Thus likelihood based approaches using GBF networks can simultaneously optimally deal with occlusions and the correspondence problem. 4 EXPERIMENTAL RESULTS We have used the task of 3D gesture recognition to compare likelihood based methods to the network output based technique. (Figure 2 describes the task.) "\rVe considered both rigid and non-rigid gesture recognition tasks. We used a GBF network with 10 inputs, 1050 basis functions and 7 output units. For comparision we also trained a standard backpropagation network (BP) with 60 hidden units on the task. For this task we assume that during training all feature correspondences are known and that during training no feature values are noisy or missing. For this task we assume that during training all feature correspondences are known and that during training no feature values are noisy or missing. Classification performance with full correspondence information on an independent test set is about 92% for the GBF network and 93% for the BP network. (For other results see (\Villiams et al. 1993) who have also used the rigid version of this task as a benchmark.) 4.1 EXPERIMENTS WITH RIGID HAND POSES Table 1 plots the ability of the various methods to select the correct correspondence. Random patterns were selected from the test set and all 5! = 120 possible combinations were tried. MLCM denotes the percentage of times the maximum likelihood method (equation (2)) selected the correct feature correspondence. GBFM and BP-M denotes how often the maximal output method chooses the correct correspondence using GBF nets and BP. "Random" denotes the percentage if correspondences are chosen randomly. The substantially better performance of MLCM suggests that, at least for this task, density information is crucial. It is also interesting to examine the errors made by MLCM. A common error is to switch the features for the pinky and the adjacent finger for gestures "one", "two", "thumbs-up" and "pointing". These two fingertips often project very close to one another in many poses; such a mistake usually do not affect subsequent classification. 966 Ahmad Selection Method Percentage Correct Random 1.2% GBF-NI 8.8% BP-M 10.3% MLCM 62.0% Table 1: Percentage of correspondences selected correctly. Classifier Classification Performance BP-Random 28.0% BP-11ax 39.2% GBF-Max 47.3% GBF-vVLC 86.2% GBF-Known 91.8% Table 2: Classification without correspondence information. Table 2 shows classification performance when the correspondence is unknown. GBF-WLC denotes weighted likelihood classification using GBF networks to compute the feature densities and the posterior probabilities. Performance with the output based techniques are denoted with GBF-M and BP-M. BP-R denotes performance with random correspondences using the back propagation network. GBFknown plots the performance of the G BF network when all correspondences are known. The results are quite encouraging in that performance is only slightly degraded with WLC even though there is substantially less information present when correspondences are unknown. Although not shown, results with MLCM (i.e. not doing the weighting step but just choosing the correspondence with highest probability) are about 1 % less than vVLC. This supports the theory that many of the errors of MLCM in Table 1 are inconsequential. 4.1.1 Missing Features and No Correspondences Figure 3 shows error as a function of the number of missing dimensions. (The missing dimensions were randomly selected from the test set.) Figure 3 plots the average number of classes that are assigned higher probability than the correct class. The network output method and weighted likelihood classification is compared to the case where all correspondences are known. In all cases the basis functions were projected onto the non-missing dimensions to approximate the Bayes-optimal condition. As before, the likelihood based method outperforms the output based method. Surprisingly, even with 4 of the 10 dimensions missing and with correspondences unknown, \VLC assigns highest probability to the correct class on average (performance score < 1.0). Feature Densities Are Required for Computing Feature Correspondences 967 Error vs missing features without correspondence 3.5 GBF-M +-3 WLC -eG BF -Known +-2.5 2 Error 1.5 1 0.5 rL-_-~-::ro L-______ ~ ________ ~ ________ _L ________ ~ ______ ~ o 1 2 3 4 5 No. of missing features Figure 3: Error with mIssmg features when no correspondence information is present. The y-axis denotes the average number of classes that are assigned higher probability than the correct class. 4.2 EXPERIMENTS WITH NON-RIGID HAND POSES In the previous experiments the hand configuration for each gesture remained rigid. Correspondence selection with non-rigid gestures was also tried out. As before a training set consisting of examples of each gesture was constructed. However, in this case, for each sample, a random perturbation (within 20 degrees) was added to each finger joint. The orientation of each sample was allowed to randomly vary by 45 degrees around the x, y, and z axes. When viewed on a screen the samples give the appearance of a hand wiggling around. Surprisingly, GBF networks with 210 hidden units consistently selected the correct correspondences with a performance of 94.9%. (The performance is actually better than the rigid case. This is because in this training set all possible 3D orientations were not allowed.) 5 DISCUSSION We have shown that estimates of joint feature densities can be used to successfully deal with lack of correspondence information even when some input features are missing. We have dealt mainly with the rather severe case where no prior information about correspondences is available. In this particular case to get the optimal correspondece, all n! possibilities must be considered. However this is usually not necessary. Useful techniques exist for reducing the number of possible correspondences. For example, (Huttenlocher and Ullman 1990) have argued that three fea968 Ahmad ture correspondences are enough to constrain the pose of rigid objects. In this case only O(n3) matches need to be tested. In addition features usually fall into incompatible sets (e.g. edge features, corner features: etc.) further reducing the nWllber of potential matches. Finally: with ullage sequences one can use correspondence ulformation from the previous frame to constraul the set of correspondences in the current frame. \\llatever the situation, a likelihood based approach is a prulcipled method for evaluatulg the set of available matches. Acknowledgements 1'luch of this research was conducted at Siemens Central Research in :Munich, Germany. I would like to thank Volker Tresp at Siemens for many interesting discussions and Brigitte \Virtz for providulg the hand model. References Ahmad, S. and V. Tresp (1993). Some solutions to the missing feature problem Ul vision. In S. Hanson, J. Cowan: and C. Giles (Eds.), Advances in Neural Information Processing Systems 5, pp. 393 400. 110rgan Kaufmann Publishers. Connell, J. and 1'1. Brady (1987). Generating and generalizing models of visual objects. A1·tificial Intelligence 31: 159 183. Dempster, A.: N. Laird: and D. Rubin (1977). 1tlaximwu-likelihood fromulcomplete data via the E1'1 algorithm. J. Royal Statistical Soc. Ser. B 39, 1 38. Huttenlocher: D. and S. Ullman (1990). Recognizing solid objects by alignment with an ullage. International Journal of Computer Vision 5(2), 195 212. Neal, R. and G. HiIlton (1993). A new view of the E1tl algorithm that justifies incremental and other variants. Biometrika, submitted. Nowlan: S. (1990). 1/1aximwll likelihood competitive learnulg. III D. Touretzky (Ed.), Advances in Neural Information Processing Systems 2: pp. 574 582. San 1tlateo: CA: 1tlorgan Kaufmann Publishers. Poggio, T. and S. Edelman (1990). A network that learns to recognize threediIllensional objects. Nature 343(6225), 1 3. Pope: A. and D. Lowe (1993: 1tIay). Learuulg object recognition models from llllages. In Fourth International Confe1'ence on Computer Vision, Berlin. IEEE Computer Society Press. Roscheisen: ~I.: R. Hofmann, and V. Tresp (1992). Neural control for rolling mills: Incorporating domain theories to overcome data deficiency. In 1·1. J., H. S.J.: and L. R. (Eds.), Advances in Neural Information Processing Systems 4. 1'Iorgan Kaufman. Segen: J. (1989). 1:1odel learning and recognition of nonrigid objects. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition: San Diego: CA. \Villiams: C. K.: R. S. Zemel: and ~I. C. 1/10zer (1993). Unsupervised learning of object models. In AAAI Fall 1993 Symposium on Machine Lea·ming in Computer Vision: pp. 20 24. Proceedings available as AAAI Tech Report FSS-93-04.
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Complexity Issues in Neural Computation and Learning V. P. Roychowdhnry School of Electrical Engineering Purdue University West Lafayette, IN 47907 Email: vwani@ecn.purdue.edu K.-Y. Sin Dept.. of Electrical & Compo Engr. U ni versit.y of California at Irvine Irvine, CA 92717 Email: siu@balboa.eng.uci.edu The general goal of this workshop was to bring t.ogether researchers working toward developing a theoretical framework for the analysis and design of neural networks. The t.echnical focus of the workshop was to address recent. developments in understanding the capabilities and limitations of variolls modds for neural computation and learning. The primary topics addressed the following three areas: 1) Computational complexity issues in neural networks, 2) Complexity issues in learning, and 3) Convergence and numerical properties of learning algorit.hms. Other topics included experiment.al/simulat.ion results on neural llet.works, which seemed to pose some open problems in the areas of learning and generalizat.ion properties of feedforward networks. The presentat.ions and discussions at the workshop highlighted the int.erdisciplinary nature of research in neural net.works. For example, several of the present.at.ions discussed recent contributions which have applied complexity-theoretic techniques to characterize the computing power of neural net.works, t.o design efficient neural networks, and t.o compare the computational capabilit.ies of neural net.works wit.h that. of convent.ional models for comput.ation. Such st.udies, in t.urn, have generated considerable research interest. among computer scient.ists, as evidenced by a significant number of research publications on related topics. A similar development can be observed in t.he area of learning as well: Techniques primarily developed in the classical theory of learning are being applied to understand t.he generalization and learning characteristics of neural networks. In [1, 2] attempts have been made to integrate concept.s from different areas and present a unifie(i treatment of the various results on the complexity of neural computation ancllearning. In fact, contributions from several part.icipants in the workshop are included in [2], and interested readers could find det.ailed discussions of many of the n-~sults IHesented at t.he workshop in [2] . Following is a brief descriptioll of the present.ations, along with the Hames and email addresses of the speakers. W. Maass (maa.~.~@igi . tu-gT·(Jz.(!(" . at) and A. Sakurai (sakllmi@hadgw92.lwd.hitachi.co.,ip) made preseutatiol1s Oll tlw VC-dimension and t.he comput.ational power of feedforwarcl neural net.works. Many neural net.s of depth 3 (or larger) with linear threshold gat.es have a VC-dimf'usion t.hat. is superlinear in t.he number of weights of the net. The talks presPllted llPW results which establish 1161 1162 Roychowdhury and Siu effective upper bounds and almost. t.ight lower boun(ls on t.he VC-dimension of feedforward networks with various activation functions including linear threshold and sigmoidal functions. Such nonlinear lower bounds on t.he VC-dimension were also discussed for networks with bot.h integer and rea.l weights. A presentation by G. Turan (@VM.CC.PURDUE.EDU:Ul1557@UICVM) discussed new result.s on proving lower bounds on t.he size of circuits for comput.ing specific Boolean functions where each gate comput.es a real-valued function. In particular the results provide a lower bound for t.he size of formulas (i.e., circuit.s wit.h fan-out 1) of polynomial gates, computing Boolean func.t.ions in t.he sensp. of sign-representation. The presentations on learning addressed both sample allli algorithmic complexity. The t.alk by V. Cast.elli (vittor·io@i81.stanford.edu) and T. Cover st.udip.d the role of labeled and unlabeled samples in pat.tern recognit.ion. Let. samples be chosen from two populations whose distribut.ions are known, and ld the proport.ion (mixing parameter) of the two classes be unknown. Assume t.hat a t.raining set composed of independent observations from the t.wo classes is given, where part. of the samples are classified and part are not. The talk present.ed new rt~sults which investigate the relative value of the labeled and unlabeled samples in reducing the probability of error of the classifier. In particular, it was shown that. uuder the above hypotheses t.he relative value of labeled and unlabeled samples is proportional t.o the (Fisher) Informat.ion they carry about, the unknown mixing parameter. B. Dasgupta (dasgupta@cs.umn.ed1l), on the othE'r hand, addressed tlw issue of the trad.ability of the t.raining problem of neural net.works. New rp.sults showing tha.t. the training problem remains NP-complete when the act.iva.t.ion functions are piecewise linear were presented. The talk by B. Hassibi (hassibi@msCClls.stan/oni.uill.) provided a minimax interpretation of instant.aneous-gradient-based learning algorit.hms such as LMS and backpropagation. When t.he underlying model is linear, it was shown t.hat the LMS algorithm minimizes the worst C3.<;e ratio of pl'f~ clicted error energy to disturbance energy. When the model is nonlinear, which arises in t.hE' contp.xt. of neural net.works, it was shown that t.he backpropagation algorithm performs this minimizat.ion in a local sense. These results provide theoretical justificat.ioll for the widely observed excellent robustness properties of the LMS and backpropagatioll algorithms. The last. t.alk by R. Caruana (car·ltana@GS79.SP.Ch'.CMU.EDU) presented a set. of int.eresting empirical results on the learning properties of neural networks of different sizes. Some of the issues (based on empirical evidence) raised during the talk are: 1) If cross-validation is used to prevent overt.raining, excess capacity rarely reduces the generalization performance of fully connected feed-forward backpropagation net.works. 2) Moreover, too little capacity usn ally hurt.s generalization performance more than too much capacit.y. References [1] K.-Y . Siu, V. P. Roychowdhnry, and T. Kailath. Di.H:r'fi(; Nfllml Computation: A Theordical Foundation. Englewood Cliffs, N.1: Prent.ice-H all , 1994. [2] V. P. Roychowdhury, K.-Y. Siu, and A. Orlitsky, edit.ors. ThwT'(;tical Advances in N(;uT'ai Compltiation and LUlT'Tl.ing. Bost.on: Kluwer Academic Publishers, 1994.
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Stability and Observability Max Garzon Fernanda Botelho garzonmGhermea.maci.memat.edu botelhofGhermea.maci.memat.edu Institute for Intelligent Systems Department of Mathematical Sciences Memphis State University Memphis, TN 38152 U.S.A. The theme was the effect of perturbations of the defining parameters of a neural network due to: 1) mea"urement" (particularly with analog networks); 2) di"cretization due to a) digital implementation of analog nets; b) bounded-precision implementation of digital networks; or c) inaccurate evaluation of the transfer function(s}; 3) noise in or incomplete input and/or output of the net or individual cells (particularly with analog networks). The workshop presentations address these problems in various ways. Some develop models to understand the influence of errors/perturbation in the output, learning and general behavior of the net (probabilistic in Piche and TresPi optimisation in Rojas; dynamical systems in Botelho k Garson). Others attempt to identify desirable properties that are to be preserved by neural network solutions (equilibria under faster convergence in Peterfreund & Baram; decision regions in Cohen). Of particular interest is to develop networks that compute robustly, in the sense that small perturbations of their parameters do not affect their dynamical and observable behavior (stability in biological networks in Chauvet & Chauvet; oscillation stability in learning in Rojas; hysterectic finite-state machine simulation in Casey). In particular, understand how biological networks cope with uncertainty and errors (Chauvet & Chauvet) through the type of stability that they exhibit. QUESTIONS AND ANSWERS Some questions served to focus the presentations and discussion. Some were (partially) answered, and others were barely touched: <> What are the mod "ignificant error" in defining parameter" with re"pect to output behavior? By evidence presented, i/o and weights seem to be the most sensitive. <> Is there an essential difference between perturbations in weights (long-term memory) and inputs (short-memory)? They seem to playa symmetric role in feedforward and, to some extent, recurrent nets. But evidence is not conclusive. <> How can the effects of perturbation" be kept under control or eliminated altogether'! If one is only interested in dynamical qualitative features, small enough errors of any kind (as incurred in digital implementations for example) are not relevant for most nets (What you see on the screen is what should be happening). <> Are they architecture (in)dependentf On the other hand, they spread rapidly under iteration and exact quantification varies with the architecture. <> Are stability and implementation based on dynamical features the only ways to 1171 1172 Garzon and Botelho cope with error!/perturbatiofU f The difficulty to quantify (perhaps due to lack of research) seems to indicate so. Stability worth a closer look for its own sake. <> Doe, requiring robud computation really redrict the capabilitie, of neural network, f Apparently not, since in all likelihood there exist universal neural nets which tolerate small errors (see talk by Botelho & Garlon). Wide open. TALKS AND SHORT ABSTRACTS • TraJ~tory Control of Convergent Networks, Natan Peterfreund and Y. Baram. We present a class of feedback control functions which accelerate convergence rates of autonomous nonlinear dynamical systems such as neural network models, without affecting the basic convergence properties (e.g. equilibrium points). natanOtx.technion.ac.il • Sensitivity of Neural Network to Errors, Steven Piche. Using stochastic models, analytic expressions for the effects of such errors are derived for arbitrary feedforward neural networks. Both, the degree of nonlinearity and the relationship between input correlation and the weight vectors, are found to be important in determining the effects of errors. picheOlllcc. COm • Stability of Learning in Neural Networks, Raul Roja!. Finding optimal combinations of learning and momentum rates for the standard backpropagation involves difficult tradeoffs across fractal boundaries. We show that statistic preprocessing can bring error functions under control. rOjaaOinf. fu-berlin.de • Stability of Purklnje Cells in Cerebellar Cortex, Gilbert Ohauvet and Pierre Ohauvet. The cerebellar cortex (involved in learning and retrieving) is a hierarchical functional unit built around a Purkinje cell, which has its own functional properties. We have shown experimentally that Purkinje dynamical systems have a unique solution, which is asymptotically stable. It seems possible to give a general explanation of stability in biological systems. chauvetOibt. uni v-angers. fr. • Recall and Learning with Deficient Data, Volker Tresp, Subutai Ahmad, Ralph Neuneier. Mean values and maximum likelihood estimators are not the best ways to cope with noisy data. See their LA:5 poster summary in these proceedings for an extended abstract. treapOzfe. aiemena. de • Computation Dynamics in Discrete-Time Recurrent Networks, Mike Oasey. We consider training recurrent higher-order neural networks to recognize regular languages, using the cycles in their diagrams for hysterectic simulation of finite state machines. The latter suggests a general logical approach to solving the 'neural code' problem for living organisms, necessary for understanding information processing in the nervous system. mcaseyOsdcc. ucsd. edu • Synthesis of Decision Regions in Dynamical Systems, Mike Oohen. As a first step toward a representation theory of decision functions via neural nets, he presented a method which enables the construction of a system of differential equations exhibiting a given finite set of decision regions and equilibria with a very large class of indices consistent with the Morse inequalities. mikeOpark. bu. edu • Observability of Discrete and Analog Networks, F. Botelho and M. Garzon. We show that most networks (with finitely many analog or infinitely many boolean neurons) are observable (i.e., all their corrupted pseudo-orbits actually reflect true orbits). See their DS:2 poster summary in these proceedings.
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The Role of MT Neuron Receptive Field Surrounds in Computing Object Shape from Velocity Fields G.T.Buracas & T.D.Albright Vision Center Laboratory, The Salk Institute, P.O.Box 85800, San Diego, California 92138-9216 Abstract The goal of this work was to investigate the role of primate MT neurons in solving the structure from motion (SFM) problem. Three types of receptive field (RF) surrounds found in area MT neurons (K.Tanaka et al.,1986; Allman et al.,1985) correspond, as our analysis suggests, to the oth, pt and 2nd order fuzzy space-differential operators. The large surround/center radius ratio (;::: 7) allows both differentiation of smooth velocity fields and discontinuity detection at boundaries of objects. The model is in agreement with recent psychophysical data on surface interpolation involvement in SFM. We suggest that area MT partially segregates information about object shape from information about spatial relations necessary for navigation and manipulation. 1 INTRODUCTION Both neurophysiological investigations [8] and lesioned human patients' data show that the Middle Temporal (MT) cortical area is crucial to perceiving three-dimensional shape in moving stimuli. On the other hand, 969 970 Buracas and Albright a solid body of data (e.g. [1]) has been gathered about functional properties of neurons in the area MT. Hoever, the relation between our ability to perceive structure in stimuli, simulating 3-D objects, and neuronal properties has not been addressed up to date. Here we discuss a possibility, that area MT RF surrounds might be involved in shape-frommotion perception. We introduce a simplifying model of MT neurons and analyse the implications to SFM problem solving. 2 REDEFINING THE SFM PROBLEM 2.1 RELATIVE MOTION AS A CUE FOR RELATIVE DEPTH Since Helmholtz motion parallax is known to be a powerful cue providing information about both the structure of the surrounding environment and the direction of self-motion. On the other hand, moving objects also induce velocity fields allowing judgement about their shapes. We can capture both cases by assuming that an observer is tracking a point on a surface of interest. The velocity field of an object then is (fig. 1): V = t z + W x (R - Ro) =-tz+wxz, where w=[wx,wy,O] is an effective rotation vector of a surface z=[x,y,z(x,y)]; Ro=[O,O,zo] is a positional vector of the fixation point; tz is a translational component along Z axis. z Fig.l: The coordinate system assumed in this paper. The origin is set at the fixation point. The observer is at Zo distance from a surface. The Role of MT Neuron Receptive Field Surrounds in Computing Object Shape 971 The component velocities of a retinal velocity field under perspective projection can be calculated from: 2 -xtz - WxXY+WyX (Zo + Z)2 WxZ V=-"-Zo +Z 2 -ytz +WyXY-W xy (Zo + Z)2 In natural viewing conditions the distance to the surface Zo is usually much larger than variation in distance on the surface z : zo»z. In such the second term in the above equations vanishes. In the case of translation tangential to the ground, to which we confine our analysis, w=[O,wy,O] = [O,w,O], and the retinal velocity reduces to u = -wz/(zo+z) ::::: -wz/zo ' v=O (1). The latter relation allows the assumption of orthographic projection, which approximates the retinal velocity field rather well within the central 20 deg of the visual field. 2.2 SFM PERCEPTION INVOLVES SURFACE INTERPOLATION Human SFM perception is characterized by an interesting peculiarity -surface interpolation [7]. This fact supports the hypothesis that an assumption of surface continuity is embedded in visual system. Thus, we can redefine the SFM problem as a problem of characterizing the interpolating surfaces. The principal normal curvatures are a local measure of surface invariant with respect to translation and rotation of the coordinate system. The orientation of the surface (normal vector) and its distance to the observer provide the information essential for navigation and object manipulation. The first and second order differentials of a surface function allow recovery of both surface curvature and orientation. 3 MODEL OF AREA MT RECEPTIVE FIELD SURROUNDS 3.1 THREE TYPES OF RECEPTIVE FIELD SURROUNDS The Middle Temporal (MT) area of monkeys is specialized for the systematic representation of direction and velocity of visual motion [1,2]. MT neurons are known to posess large, silent (RFS, the "nonclassical RF". Born and Tootell [4] have very recently reported that the RF surrounds of neurons in owl monkey MT can be divided into antagonistic and synergistic types (Fig.2a). 972 Buracas and Albright 25 ~2O ~ 15 ~10 c. 5 en a) o~~----~------~ o 10 20 Fig.2: Top left (a): an example of a synergistic RF surround, redrawn from [4] (no velocity tuning known). Bottom left (b): a typical V-shaped tuning curve for RF surround The horizontal axis represents the logarithmic scale of ratio between stimulus speeds in the RF center and surround, redrawn from [9]. Bottom (c,d): monotonically increasing and decreasing tuning curves for RF surrounds, redrawn from [9]. AnnlJus diameter deg b) c) c:t 1 V 1 J 1 III Ql 0.8 Ql Ql 0.8 Q8 ~ > (II > til 06 ,- c: 06 ''; ~ 0.6 .. c . £I Q. 4 £I Q. 04 Gi (II O. Qj til 0.4 a: 2! 0.2 a: 2! 0.2 02 0 0 0 0.1 1 10 0.1 1 10 01 1 10 R otio of CIS speeds R otIo of CIS speeds RatootCS speeds About 44% of the owl monkey neuron RF8s recorded by Allman et al. [3] showed antagonistic properties. Approximately 33% of these demonstrated V(or U)-shaped (Fig.2b), and 66% - quasi-linear velocity tuning curves (Fig.2c,d). One half of Macaca fuscata neurons with antagonistic RF8 found by Tanaka et al [9] have had V(U)-shaped velocity tuning curves, and 50% monotonically increasing or decreasing velocity tuning curves. The RF8 were tested for symmetry [9] and no asymmetrical surrounds were found in primate MT. 3.2 CONSTRUCTING IDEALIZED MT FILTERS The surround (8) and center (C) responses seem to be largely independent (except for the requirement that the velocity in the center must be nonzero) and seem to combine in an additive fashion [5]. This property allows us to combine C and 8 components in our model independently. The resulting filters can be reduced to three types, described below. 3.2.1 Discrete Filters The essential properties of the three types of RF8s in area MT can be captured by the following difference equations. We choose the slopes of velocity tuning curves in the center to be equal to the ones in the surround; this is essential for obtaining the desired properties for 12 but not 10, The 0order (or low-pass) and the 2nd order (or band-pass) filters are defined by: The Role of MT Neuron Receptive Field Surrounds in Computing Object Shape 973 i j i j where g is gain, Wij =1, ije [-r,r] (r = radius of integration). Speed scalars u(iJ) at points [ij] replace the velocity vectors V due to eq. (1). Constants correspond to spontaneous activity levels. In order to achieve the V(U) -shaped tuning for the surround in Fig.2b, a nonlinearity has to be introduced: II = gl L L (ue Us (i,j))2 + Constl. (3) i j The responses of 11 and 12 filters to standard mapping stimuli used in [3,9] are plotted together with their biological correlates in Fig.3. 3.2.2 Continuous analogues of MT filters We now develop continuous, more biologicaly plausible, versions of our three MT filters. We assume that synaptic weights for both center and surround regions fall off with distance from the RF center as a Gaussian function G(x,y,O'), and 0' is different for center and surround: O'c 7; O's. Then, by convolving with Gaussians equation (2) can be rewritten: Lo (i,j) = u(i, j)* G( 0' e) + u(i,j)* G( 0' s ), L~ (i, j ) = ± [u ( i , j ) * G ( 0' e ) - U ( i, j) * G ( 0' s )]. The continuous nonlinear Ll filter can be defined if equivalence to 11 (eq. 3) is observed only up to the second order term of power series for u(ij): LI (i, j) = U 2 (i, j ) * G ( 0' e ) + U 2 (i, j ) * G ( 0' s ) - C . [ u ( i , j ) * G ( 0' e )]. [u ( i , j ) * G ( 0' s )]; u2(ij) corresponds to full-wave rectification and seems to be common in area VI complex neurons; C = 2IErf2(nl2112) is a constant, and Erf() is an error function. 3.3 THE ROLE OF MT NEURONS IN SFM PERCEPTION. Expanding z(x,y) function in (1) into power series around an arbitrary point and truncating above the second order term yields: u(x,y)=w(ax2+by2+cxy+dx+ey+Olzo, where a,b,c,d,e,f are expansion coefficients. We assume that w is known (from proprioceptive input) and =1. Then Zo remans an unresolved scaling factor and we omit it for simplicity. 974 Buracas and Albright DATA MODEL 0.5 V J L, 0 0.5 J / L+, 0 0.5 ~ 0 ~ L2 1/4 112 I 2 4 1/4 112 I 2 4 Surround/Center speed ratio -15 -10 -5 o 5 10 15 Fig. 3: The comparison between data [9] and model velocity tuning curves for RF surrounds. The standard mapping stimuli (optimaly moving bar in the center of RF, an annulus of random dots with varying speed) were applied to L1 and L2 filters. Thee output of the filters was passed through a sigmoid transfer function to accout for a logarithmic compresion in the data. Fig. 4: Below, left: the response profile of the L1 filter in orientation space (x and y axes represent the components of normal vector). Right: the response profile of the L2 filter in curvature space. x and y axes represent the two normal principal curvatures. L2 response in curvature space -15 -10 ·5 o 5 10 15 Applying Lo on u(x,y), high spatial frequency information is filtered out, but otherwise u(x,y) does not change, i.e. Lo*u covaries with lower frequencies ofu(x,y). L2 applied on u(x,y) yields: L2 * U = (2 a + 2 b ) C 2 (0' ~ - 0'; ) = C 2 ( 0' ~ - 0'; ) V 2 U , (4) that is, L2 shows properties of the second order space-differential operator Laplacian; C2(O'c2 - 0'82) is a constant depending only on the widths of the center and surround Gaussians. Note that L2*u == 1<:1 + 1<:2 ' (1<:12 are principal normal curvatures) at singular points of surface z(x,y). ' The Role of MT Neuron Receptive Field Surrounds in Computing Object Shape 975 When applied on planar stimuli up(x,y) = d x + e y, L1 has properties of a squared first order differential operator: ~ * up = (d 2 +e2 )C, (a~ -a;) = C, (a~ -a; >( (!)2 +( ~)2 )up, (5) where C2(O'e2 - O's2) is a function of O'e and O's only. Thus the output of L1 is monotonically related to the norm of gradient vector. It is straightforward to calculate the generic second order surface based on outputs of three Lo, four L1 and one L2 filters. Plotting the responses of L1 and L2 filters in orientation and curvature space can help to estimate the role they play in solving the SFM problem (FigA). The iso-response lines in the plot reflect the ambiguity of MT filter responses. However, these responses covary with useful geometric properties of surfaces -- norm of gradient (L1) and mean curvature (L2). 3.4 EXTRACTING VECTOR QUANTITIES Equations (4) and (5) show, that only averaged scalar quantities can be extracted by our MT operators. The second order directional derivatives for estimating vectorial quantities can be computed using an oriented RFs with the following profile: 02=G(x,O's) [G(y,O's) - G(y'O'e)). 01 then can be defined by the center - surround relationship of L1 filter. The outputs of MT filters L1 and L2 might be indispensible in normalizing responses of oriented filters. The normal surface curvature can be readily extracted using combinations of MT and hypothetical ° filters. The oriented spatial differential operators have not been found in primate area MT so far. However, preliminary data from our lab indicate that elongated RFs may be present in areas FST or MST [6). 3.5 L2: LAPLACIAN VS. NAKAYAMA'S CONVEXITY OPERATOR The physiologically tested ratio of standard deviations for center and surround Gaussians O'/O'e ;::: 7. Thus, besides performing the second order differentiation in the low frequency domain, L2 can detect discontinuities in optic flow. 4. CONCLUSIONS We propose that the RF surrounds in MT may enable the neurons to function as differential operators. The described operators can be thought of as providing a continuous interpolation of cortically represented surfaces. Our model predicts that elongated RFs with flanking surrounds will be found (possibly in areas FST or MST [6]). These RFs would allow extraction 976 Buracas and Albright of the directional derivatives necessary to estimate the principal curvatures and the normal vector of surfaces. From velocity fields, area MT extracts information relevant to both the "where" stream (motion trajectory, spatial orientation and relative distance of surfaces) and the "what" stream (curvature of surfaces). Acknowledgements Many thanks to George Carman, Lisa Croner, and Kechen Zhang for stimulating discussions and Jurate Bausyte for helpful comments on the poster. This project was sponsored by a grant from the National Eye Institute to TDA and by a scholarship from the Lithuanian Foundation to GTB. The presentation was supported by a travel grant from the NIPS foundation. References [1] Albright, T.D. (1984) Direction and orientation selectivity of neurons in visual area MT of the macaque. J. Neurophysiol., 52: 1106-1130. [2] Albright, T.D., R.Desimone. (1987) Local precision of visuotopic organization in the middle temporal area (MT) of the macaque. Exp.Brain Res., 65, 582-592. [3] Allman, J., Miezin, F., McGuinnes. (1985) Stimulus specific responses from beyond the classical receptive field. Ann.Rev.Neurosci., 8, 407-430. [4] Born R.T. & Tootell R.B.H. (1992) Segregation of global and local motion processing in primate middle temporal visual area. Nature, 357, 497-499. [5] Born R.T. & Tootell R.B.H. (1993) Center - surround interactions in direction - selective neurons of primate visual area MT. Neurosci. Abstr., 19,315.5. [6] Carman G.J., unpublished results. [7] Hussain M., Treue S. & Andersen R.A. (1989) Surface interpolation in three-dimensional Structure-from-Motion perception. Neural Computation, 1,324-333. [8] Siegel, R.M. and R.A. Andersen. (1987) Motion perceptual deficits following ibotenic acid lesions of the middle temporal area in the behaving rhesus monkey. Soc.Neurosci.Abstr., 12, 1183. [9]Tanaka, K., Hikosaka, K., Saito, H.-A., Yukie, M., Fukada, Y., Iwai, E. (1986) Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey. J.Neurosci., 6,134-144.
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Estimating analogical similarity by dot-products of Holographic Reduced Representations. Tony A. Plate Department of Computer Science, University of Toronto Toronto, Ontario, Canada M5S 1A4 email: tap@ai.utoronto.ca Abstract Models of analog retrieval require a computationally cheap method of estimating similarity between a probe and the candidates in a large pool of memory items. The vector dot-product operation would be ideal for this purpose if it were possible to encode complex structures as vector representations in such a way that the superficial similarity of vector representations reflected underlying structural similarity. This paper describes how such an encoding is provided by Holographic Reduced Representations (HRRs), which are a method for encoding nested relational structures as fixed-width distributed representations. The conditions under which structural similarity is reflected in the dot-product rankings of HRRs are discussed. 1 INTRODUCTION Gentner and Markman (1992) suggested that the ability to deal with analogy will be a "Watershed or Waterloo" for connectionist models. They identified "structural alignment" as the central aspect of analogy making. They noted the apparent ease with which people can perform structural alignment in a wide variety of tasks and were pessimistic about the prospects for the development of a distributed connectionist model that could be useful in performing structural alignment. In this paper I describe how Holographic Reduced Representations (HRRs) (Plate, 1991; Plate, 1994), a fixed-width distributed representation for nested structures, can be used to obtain fast estimates of analogical similarity. A HRR is a high dimensional vector, 1109 1110 Plate and the vector dot-product of two HRRs is an efficiently computable estimate of the overall similarity between the two structures represented. This estimate reflects both surface similarity and some aspects of structural similarity,l even though alignments are not explicitly calculated. I also describe contextualization, an enrichment ofHRRs designed to make dot-product comparisons of HRRs more sensitive to structural similarity. 2 STRUCTURAL ALIGNMENT & ANALOGICAL REMINDING People appear to perform structural alignment in a wide variety of tasks, including perception, problem solving, and memory recall (Gentner and Markman, 1992; Markman, Gentner and Wisniewski, 1993). One task many researchers have investigated is analog recall. A subject is shown a number of stories and later is shown a probe story. The task is to recall stories that are similar to the probe story (and sometimes evaluate the degree of similarity and perform analogical reasoning). MACIFAC, a computer models of this process, has two stages(Gentner and Forbus, 1991). The first stage selects a few likely analogs from a large number of potential analogs. The second stage searches for an optimal (or at least good) mapping between each selected story and the probe story and outputs those with the best mappings. Two stages are necessary because it is too computationally expensive to search for an optimal mapping between the probe and all stories in memory. An important requirement for a first stage is that its performance scale well with both the size and number of episodes in long-term memory. This prevents the first stage of MACIFAC from considering any structural features. Large pool of items in memory Probe 0000 0 0 00 000 analogies 3 00800 0 00 0 0000 0 0 0 0 0 00 000 00 0 00 o:J 0 0 0 0 Cheap filtering process o 0 ~ ~ 0 based on surface features Good analogies o o Expensive selection process based on structural features Figure 1: General architecture of a two-stage retrieval model. While it is indisputable that people take structural correspondences into account when evaluating and using analogies (Gentner, Rattermann and Forbus, 1993), it is less certain whether structural similarity influences access to long term memory (i.e., the first-stage reminding process). Some studies have found little effect of analogical similarity on reminding (Gentner and Forbus, 1991; Gentner, Rattermann and Forbus, 1993), while others have found some effect (Wharton et aI., 1994). l"Surface features" of stories are the features of the entities and relations involved, and "structural features" are the relationships among the relations and entities. Estimating Analogical Similarity by Dot-Products of Holographic Reduced Representations 1111 In any case, surface features appear to influence the likelihood of a reminding far more than do structural features. Studies that have found an effect of structural similarity on reminding seem to indicate the effect only exists, or is greater, in the presence of surface similarity (Gentner and Forbus, 1991; Gentner, Rattermann and Forbus, 1993; Thagard et al., 1990). 2.1 EXAMPLES OF ANALOGY BETWEEN NESTED STRUCTURES. To test how well the HRR dot-product works as an estimate of analogical similarity between nested relational structures I used the following set of simple episodes (see Plate (1993) for the full set). The memorized episodes are similar in different ways to the probe. These examples are adapted from (Thagard et al., 1990). Probe: Spot bit Jane, causing Jane to flee from Spot. Episodes in long-term memory: El (L8) Fido bit John, causing John to flee from Fido. E2 (ANcm) Fred bit Rover, causing Rover to flee from Fred. E3 (AN) Felix bit Mort, causing Mort to flee from Felix. E6 (88) John fled from Fido, causing Fido to bite John. E7 (FA) Mort bit Felix, causing Mort to flee from Felix. In these episodes Jane, John, and Fred are people, Spot, Fido and Rover are dogs, Felix is a cat, and Mort is a mouse. All of these are objects, represented by token vectors. Tokens of the same type are considered to be similar to each other, but not to tokens of other types. Bite, flee, and cause are relations. The argument structure of the cause relation, and the patterns in which objects fill multiple roles constitutes the higher-order structure. The second column classifies the relationship between each episode and the probe using Gentner et aI's types of similarity: LS (Literal Similarity) shares relations, object features, and higher-order structure; AN (Analogy, also called True Analogy) shares relations and higher-order structure, but not object features; SS (Surface Similarity, also called Mere Appearance) shares relations and object features, but not higher-order structure; FA (False Analogy) shares relations only. ANcm denotes a cross-mapped analogy - it involves the same types of objects as the probe, but the types of corresponding objects are swapped. 2.2 MACIFAC PERFORMANCE ON TEST EXAMPLES The first stage of MACIFAC (the "Many Are Called" stage) only inspects object features and relations. It uses a vector representation of surface features. Each location in the vector corresponds to a surface feature of an object, relation or function, and the value in the location is the number of times the feature occurs in the structure. The first-stage estimate of the similarity between two structures is the dot-product of their feature-count vectors. A threshold is used to select likely analogies. It would give El (L8), E2 (ANcm), and E6 (88) equal and highest scores, i.e., (L8, ANcm, 88) > (AN, FA) The Structure Mapping Engine (SME) (Falkenhainer, Forbus and Gentner, 1989) is used as the second stage of MACIFAC (the "Few Are Chosen" stage). The rules of SME are that mapped relations must match, all the arguments of mapped relations must be mapped consistently, and mapping of objects must be one-to-one. SME would detect structural correspondences between each episode and the probe and give the literally similar and analogous episodes the highest rankings, i.e., LS > AN > (SS, FA). 1112 Plate A simplified view of the overall similarity scores from MAC and the full MACIFAC is shown in Table 1. There are four conditions - the two structures being compared can be similar in structure and/or in object attributes. In all four conditions, the structures are assumed to involve similar relations - only structural and object attribute similarities are varied. Ideally, the responses to the mixed conditions should be flexible, and controlled by which aspects of similarity are currently considered important. Only the relative values of the scores are important, the absolute values do not matter. Structural Object Attribute Similarity Structural Object Attribute Similarity Similarity YES NO Similarity YES NO YES (LS) High (AN) Low YES (LS) High (AN) tMed-High NO (SS) High (FA) Low NO (SS) :J:Med-Low (FA) Low (a) Scores from MAC. (b) Ideal similarity scores. Table 1: (a) Scores from the fast (MAC) similarity estimator in MACIFAC. (b) Scores from an ideal structure-sensitive similarity estimator, e.g., SME as used in MACIFAC. In the remainder of this paper I describe how HRRs can be used to compute fast similarity estimates that are more like ratings in Table 1 b, i.e., estimates that are flexible and sensitive to structure. 3 HOLOGRAPHIC REDUCED REPRESENTATIONS A distributed representation for nested relational structures requires a solution to the binding problem. The representation of a relation such as bite (spot, jane) ("Spot bit Jane.") must bind 'Spot' to the agent role and 'Jane' to the object role. In order to represent nested structures it must also be possible to bind a relation to a role, e.g., bite (spot, jane) and the antecedent role of the cause relation. n-l Zi = 2.:= XkYj-k k=O (Subscript are modulo-n) (a) (b) Zo = XoYo + X2Yl + XIY2 Zl = XIYO + XOYI + X2Y2 Z2 = X2YO + XIYl + XOY2 Figure 2: (a) Circular convolution. (b) Circular convolution illustrated as a compressed outer product for n = 3. Each of the small circles represents an element of the outer product of x and Y, e.g., the middle bottom one is X2Yl. The elements of the circular convolution of x and yare the sums of the outer product elements along the wrapped diagonal lines. Holographic Reduced Representations (HRRs) (Plate, 1994) use circular convolution to solve the binding problem. Circular convolution (Figure 2a) is an operation that maps two n-dimensional vectors onto one n-dimensional vector. It can be viewed as a compressed outer product, as shown in Figure 2b. Algebraically, circular convolution behaves like multiplication - it is commutative, associative, and distributes over addition. Circular Estimating Analogical Similarity by Dot-rroducts of Holographic Reduced Representations 1113 convolution is similarity preserving: if ~ ~ ~' then ~ ® b ~ ~' ® b. Associations can be decoded using a stable approximate inverse: ~ * ® (~ ® b) ~ b (provided that the vector elements are normally distributed with mean zero and variance lin). The approximate inverse is a permutation of vector elements: ar = an-i. The dot-product of two vectors, a similarity measure, is: ~. b = L~:Ol aibi. High dimensional vectors (n in the low thousands) must be used to ensure reliable encoding and decoding. The HRR for bi te (spot, jane) is: F = < bite + biteagt ® spot + biteobj ® jane>, where < . > is a normalization operation « ~ >= ~I V!! . ~). Multiple associations are superimposed in one vector and the representations for the objects (spot and jane) can also be added into the HRR in order to make it similar to other HRRs involving Spot and Jane. The HRR for a relation is the same size as the representation for an object and can be used as the filler for a role in another relation. 4 EXPT. 1: HRR DOT-PRODUCT SIMILARITY ESTIMATES Experiment 1 illustrates the ways in which the dot-products of ordinary HRRs reflect, and fail to reflect, the similarity of the underlying structure of the episodes. Base vectors Token vectors person, dog, cat, mouse bite, flee, cause biteagt, fleeagt, causeantc biteobj, flee from, causecnsq jane =< person + idjane > spot =< dog + idspot > john =< person + idjohn > fido =< dog + idfido > fred =< person + idfred > rover =< dog + idrover > mort =< mouse + idntort > felix =< cat + id felix> The set of base and tokens vectors used in Experiments 1, 2 and 3 is shown above. All base and id vectors had elements independently chosen from a zero-mean normal distribution with variance lin. The HRR for the probe is constructed as follows. and the HRRs for the other episodes are constructed in the same manner. Pbite =< bite + biteagt ® spot + biteobj ® jane> P flee =< flee + fleeagt ® jane + flee from ® spot> P objects =< jane + spot> P =< cause + P objects + Pbite + P flee + causeantc ® Pbite + causecnsq ® P flee> Experiment 1 was run 100 times, each time with a new choice of random base vectors. The vector dimension was 2048. The means and standard deviations of the HRR dot-products of the probe and each episode are shown in Table 2. Dot-product with probe Probe: Spot bit Jane. causing Jane to flee from Spot. Exptl Expt2 Expt3 Episodes in long-term memory: Avg Sd El LS Fido bit John, causing John to flee from Fido. 0.70 0.016 0.63 0.81 E2 ANCnt Fred bit Rover, causing Rover to flee from Fred. 0.47 0.022 0.47 0.69 E3 AN Felix bit Mort, causing Mort to flee from Felix. 0.39 0.024 0.39 0.61 E6 SS John fled from Fido, causing Fido to bite John. 0.47 0.018 0.44 0.53 E7 FA Mort bit Felix, causing Mort to flee from Felix. 0.39 0.024 0.39 0.39 Table 2: Results of Experiments 1,2 and 3. In 94 out of 100 runs, the ranking of the HRR dot-products was consistent with LS > (ANcm, SS) > (FA, AN) 1114 Plate (where the ordering within the parenthesis varies). The order violations are due to "random" fluctuations of dot-products, whose variance decreases as the vector dimension increases. When the experiment was rerun with vector dimension 4096 there was only one violation of this order out of 100 runs. These results represent an improvement over the first stage of MACIFAC - the HRR dotproduct distinguishes between literal and surface similarity. However, when the episodes do not share object attributes, the HRR dot-product is not affected by structural similarity and the scores do not distinguish analogy from false analogy or superficial similarity. 5 EXPERIMENTS 2 AND 3: CONTEXTUALIZED HRRS Dot-product comparisons ofHRRs are not sensitive to structural similarity in the absence of similar objects. This is because the way in which objects fill multiple roles is not expressed as a surface feature in HRRs. Consequently, the analogous episodes E2 (ANcm) and E3 (AN) do not receive higher scores than the non analogous episodes E6 (SS) and E7 (FA). We can force role structure to become a surface feature by "contextualizing" the representations of fillers. Contextualization involves incorporating information about what other roles an object fills in the representation of a filler. This is like thinking of Spot (in the probe) as an entity that bites (a biter) and an entity that is fled from (a "fled-from"). In ordinary HRRs the filler alone is convolved with the role. In contextualized HRRs a blend of the filler and its context is convolved with the role. The representation for the context of object in a role is the typical fillers of the other roles the object fills. The context for Spot in the flee relation is represented by typ~~; and the context in the bite relation is represented by typ~~eo:n (where typ~~; = bite ® bite~gt and typ~~:em = flee ® fleejrom). The degree of contextualization is governed by the mixing proportions ""0 (object) and ""c (context). The contextualized HRR for the probe is constructed as follows: Pbite =< bite + biteagt ® (X:ospot + X:ctyp~~:eTn) + biteobj ® (X:ojane + X:ctyp!~~e) > P flee =< flee + fleeagt ® (X:ojane + X:ctyp~tn + fleefroTn ® (X:ospot + X:ctyp~~n > P objects =< jane + spot> P =< cause + P objects + P bite + P flee + causeantc ® Pbite + causecnsq ® P flee> A useful similarity estimator must be flexible and able to adjust salience of different aspects of similarity according to context or command. The degree to which role-alignment affects the HRR dot-product can be adjusted by changing the degree of contextualization in just one episode of a pair. Hence, the items in memory can be encoded with a fixed ,.., values (,..,-: and ,..,;;-) and the salience of role alignment can be changed by altering the degree of contextualization in the probe (,..,~ and ,..,n. This is fortunate as it would be impractical to recode all items in memory in order to alter the salience of role alignment in a particular comparison. The same technique can be used to adjust the importance of other features. Two experiments were performed with contextualized HRRs, with the same episodes as used in Experiment 1. In Experiment 2 the probe was non-contextualized (,..,~ = 1, ,..,~ = 0), and in Experiment 3 the probe was contextualized (,..,~ = 1/~,,..,~ = 1/~). For both Experiments 2 and 3 the episodes in memory were encoded with the same degree of contextualization (,..,-: = 1/~,,..,;;- = 1/ ~). As before, each set of comparisons was run 100 times, and the vector dimension was 2048. The results are shown in Table 2. Estimating Analogical Similarity by Dot-Products of Holographic Reduced Representations 1115 The scores in Experiment 2 (non-contextualized probe) were consistent (in 95 out of 100 runs) with the same order as given for Experiment 1: L8 > (ANcm ,88) > (FA, AN) The scores in Experiment 3 (contextualized probe) were consistent (in all 100 runs) with an ordering that ranks analogous episodes as strictly more similar than non-analogous ones: L8 > ANcm > AN > 88 > FA 6 DISCUSSION The dot-product of HRRs provides a fast estimate of the degree of analogical match and is sensitive to various structural aspects of the match. It is not intended to be a model of complex or creative analogy making, but it could be a useful first stage in a model of analogical reminding. Structural Object Attribute Similarity Structural Object Attribute Similarity Similarity YES NO Similarity YES NO YES (LS) High (AN) Low YES (LS) High (AN) tMed-High NO (SS) Med (FA) Low NO (SS) tMed-Low (FA) Low (a) Ordinary-HRR dot-products. (b) Contextualized-HRR dot-products. Table 3: Similarity scores from ordinary and contextualized HRR dot-product comparisons. The flexibility comes adjusting the weights of various components in the probe. The dot-product of ordinary HRRs is sensitive to some aspects of structural similarity. It improves on the existing fast similarity matcher in MACIFAC in that it discriminates the first column of Table 3 - it ranks literally similar (LS) episodes higher than superficially similar (88) episodes. However, it is insensitive to structural similarity when corresponding objects are not similar. Consequently, it ranks both analogies (AN) and false analogies (FA) lower than superficially similar (S8) episodes. The dot-product of contextualized HRRs is sensitive to structural similarity even when corresponding objects are not similar. It ranks the given examples in the same order as would the full MACIFAC or ARCS system. Contextualization does not cause all relational structure to be expressed as surface features in the HRR vector. It only suffices to distinguish analogous from non-analogous structures when no two entities fill the same set of roles. Sometimes, the distinguishing context for an object is more than the other roles that the object fills. Consider the situation where two boys are bitten by two dogs, and each flees from the dog that did not bite him. With contextualization as described above it is impossible to distinguish this from the situation where each boy flees from the dog that did bite him. HRR dot-products are flexible - the salience of various aspects of similarity can be adjusted by changing the weights of various components in the probe. This is true for both ordinary and contextualized HRRs. HRRs retain many of the advantages of ordinary distributed representations: (a) There is a simple and computationally efficient measure of similarity between two representations 1116 Plate the vector dot-product. Similar items can be represented by similar vectors. (b) Items are represented in a continuous space. ( c) Information is distributed and redundant. Hummel and Biederman (1992) discussed the binding problem and identified two main problems faced by conjunctive coding approaches such as Tensor Products (Smolensky, 1990). These are exponential growth of the size of the representation with the number of associated objects (or attributes), and insensitivity to attribute structure. HRRs have much in common with conjunctive coding approaches (they can be viewed as a compressed conjunctive code), but do not suffer from these problems. The size of HRRs remains constant with increasing numbers of associated objects, and sensitivity to attribute structure has been demonstrated in this paper. The HRR dot-product is not without its drawbacks. Firstly, examples for which it will produce counter-intuitive rankings can be constructed. Secondly, the scaling with the size of episodes could be a problem - the sum of structural-feature matches becomes a less appropriate measure of similarity as the episodes get larger. A possible solution to this problem is to construct a spreading activation network of HRRs in which each episode is represented as a number of chunks, and each chunk is represented by a node in the network. The software used for the HRR calculations is available from the author. References Falkenhainer, B., Forbus, K. D., and Gentner, D. (1989). The Structure-Mapping Engine: Algorithm and examples. Artificial Intelligence, 41: 1-63. Gentner, D. and Forbus, K. D. (1991). MAC/FAC: A model of similarity-based retrieval. In Proceedings of the Thirteenth Annual Cognitive Science Society Conference, pages 504-509, Hillsdale, NJ. Erlbaum. Gentner, D. and Markman, A. B. (1992). Analogy - Watershed or Waterloo? Structural alignment and the development of connectionist models of analogy. In Giles, C. L., Hanson, S. J., and Cowan, J. D., editors, Advances in Neural Information Processing Systems 5 (NIPS*92), pages 855-862, San Mateo, CA. Morgan Kaufmann. Gentner, D., Rattermann, M. J., and Forbus, K. D. (1993). The roles of similarity in transfer: Separating retrievability from inferential soundness. Cognitive Psychology, 25:431-467. Hummel, J. E. and Biederman, I. (1992). Dynamic binding in a neural network for shape recognition. Psychological Review, 99(3):480-517. Markman, A. B., Gentner, D., and Wisniewski, E. J. (1993). Comparison and cognition: Implications of structure-sensitive processing for connectionist models. Unpublished manuscript. Plate, T. A. (1991). Holographic Reduced Representations: Convolution algebra for compositional distributed representations. In Mylopoulos, J. and Reiter, R., editors, Proceedings of the 12th International loint Conference on Artificial Intelligence, pages 30-35, San Mateo, CA. Morgan Kaufmann. Plate, T. A. (1993). Estimating analogical similarity by vector dot-products of Holographic Reduced Representations. Unpublished manuscript. Plate, T. A. (1994). Holographic reduced representations. IEEE Transactions on Neural Networks. To appear. Smolensky, P. (1990). Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artificial Intelligence, 46(1-2):159-216. Thagard, P., Holyoak, K. J., Nelson, G., and Gochfeld, D. (1990). Analog Retrieval by Constraint Satisfaction. Artificial Intelligence, 46:259-310. Wharton, C. M., Holyoak, K. J., Downing, P. E., Lange, T. E., Wickens, T. D., and Melz, E. R. (1994). Below the surface: Analogical similarity and retrieval competition in reminding. Cognitive Psychology. To appear.
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Segmental Neural Net Optimization for Continuous Speech Recognition Ymg Zhao Richard Schwartz John Makhoul George Zavaliagkos BBN System and Technologies 70 Fawcett Street Cambridge MA 02138 Abstract Previously, we had developed the concept of a Segmental Neural Net (SNN) for phonetic modeling in continuous speech recognition (CSR). This kind of neural network technology advanced the state-of-the-art of large-vocabulary CSR, which employs Hidden Marlcov Models (HMM), for the ARPA 1oo0-word Resource Management corpus. More Recently, we started porting the neural net system to a larger, more challenging corpus - the ARPA 20,Ooo-word Wall Street Journal (WSJ) corpus. During the porting, we explored the following research directions to refine the system: i) training context-dependent models with a regularization method; ii) training SNN with projection pursuit; and ii) combining different models into a hybrid system. When tested on both a development set and an independent test set, the resulting neural net system alone yielded a perfonnance at the level of the HMM system, and the hybrid SNN/HMM system achieved a consistent 10-15% word error reduction over the HMM system. This paper describes our hybrid system, with emphasis on the optimization methods employed. 1 INTRODUCTION Hidden Martov Models (HMM) represent the state-of-the-art for large-vocabulary continuous speech recognition (CSR). Recently, neural network technology has been shown to advance the state-of-the-art for CSR by integrating neural nets and HMMs [1,2]. In principle, the advance is based on the fact that neural network modeling can avoid some limitations of the HMM modeling, for example, the conditional-independence assumption of HMMs and the fact that segmental features are hard to incorporate. Our work has been based on the concept of a Segmental Neural Net (SNN) [2]. 1059 1060 Zhao, Schwartz, Makhoul, and Zavaliagkos A segmental neural network is a neural network that attempts to recognize a complete phoneme segment as a single unit. Its basic structure is shown in Figure 1. The input to the network is a fixed length representation of the speech segment, which is obtained from the warping (quasi-linear sampling) of a variable length phoneme segment. If the network is trained to minimize a least squares error or a cross entropy distortion measure, the output of the network can be shown to be an estimate of the posterior probability of the phoneme class given the input segment [3,4]. warping phonetic .egment .core neural network Figure 1: The SNN model samples the frames and produces a single segment score. Our inith1 SNN system comprised a set of one-layer sigmoidal nets. This system is trained to minimize a cross entropy distortion measure by a quasi-Newton error minimization algorithm. A vanable length segment is warped into a fixed length of 5 input frames. Since each frame includes 16 feature~, 14 mel cepstrum, power and difference of power, an input to the neural network forms a 16 x 5 = 80 dimensional vector. Previously, our experimental domain was the ARPA 1000-word Resource Management (RM) Corpus, where we used 53 output phoneme classes. When tested on three independent evaluation sets (Oct 89, Feb 91 and Sep 92), our system achieved a consistent 10-20% word error rate reduction over the state-of-the-art HMM system [2]. 2 THE WALL STREET JOURNAL CORPUS After the final September 92 RM corpus evaluation, we ported our neural network system to a larger corpus the Wall Street Journal (WSJ) Corpus. The WSJ corpus consists primarily of read speech, with a 5,000- to 20,000-word vocabulary. It is the current ARPA speech recognition research corpus. Compared to the RM corpus, it is a more challenging corpus for the neural net system due to the greater length of WSJ utterances and the higher perplexity of the WSJ task. So we would expect greater difficulty in improving perfOITnClllCe on the WSJ corpus. Segmental Neural Net Optimization for Continuous Speech Recognition 1061 3 TRAINING CONTEXT-DEPENDENT MODELS WITH REGULARIZATION 3.1 WHY REGULARIZATION In contrast to the context-independent modeling for the RM corpus, we are concentrating on context-dependent modeling for the WSJ corpus. In context-dependent modeling, instead of using a single neural net to recognize phonemes in all contexts, different neural networks are used to recognize phonemes in different contexts. Because of the paucity of training data for some context models, we found that we had an overfitting problem. Regularization provides a class of smoothing techniques to ameliorate the overfitting problem [5]. We started using regularization in our initial one-layer sigmoidal neural network system. The regularization tenn added here is to regulate how far the context-dependent parameters can move away from their initial estimates, which are context-independent parameters. TIus is different from the usual weight decay technique in neural net literature, and it is designed specifically for our problem. The objective function is shown below: - ~ d ~ [~IOg(1 - J.) + ~ log f} :'IIW ~ .WolI~ (I) , J Regulanzatton Tenn v Distortion measure Er(W) where Ii is the net output for class i, II W II is the Euclidean nonn of all weights in all the networks, IIWol1 is the initial estimate of weights from a context-independent neural network, Nd is the number of data points. ). is the regularization parameter which controls the tradeoff between the "smoothness" of the solution, as measured by IIW - Wo11 2, and the deviation from the data as measured by the distortion. The optimal )., wllich gives the best generalization to a test set, can be estimated by generalized cross-validation [5]. If the distortion measure as shown in (2) (2) is a qu:.:-ctratic function in tenns of network weights W, the optimal ). is that which gives the minin,um of a generalized cross-validation index V().) [6]: Nt IIA()') - bW V(),) = d I ~d tr(A()')) (3) where A(>.) = A(AT A+Nd)'I)AT . V()') is an easily calculated function based on singular value decomposition (SVD): (4) where A = U DVT , singular decomposition of A, z = UT b. Figure 2 shows an example plot of V().). A typical optimal ). has an inverse relation to the number of samples in each class, indicating that ). is gradually reduced with the presence of more data. 1062 Zhao, Schwartz, Makhoul, and Zavaliagkos CI: ~ ~------------------------------------------. o 8 o 8 § ~ I I R ~ I o 5*10"-7 2.5*1011-6 lambda Figure 2: A typical V(A) Just as the linear least squares method can be generalized to a nonlinear least squares problem by an iterative procedure, so selecting the optimal value of the regularization parameter in a quadratic error criterion can be generalized to a non-quadratic error criterion iteratively. We developed an iterative procedure to apply the cross-validation technique to a non-..]uadratic error function, for example, the cross-entropy criterion Er(W) in (1) as follows: 1. Compute distortion Er(Wn ) for an estimate Wn • 2. Compute gradient gn and Hessian Hn of the distortion Er(Wn). 3. Compute the singular value decomposition of Hn = V! Dn Vn. Set Zn = v"gn. 4. Evaluate a generalized cross-validation index Vn(A) similar to (2) as follows, for a range of A'S and select the An that gives the minimum Vn • Segmental Neural Net Optimization for Continuous Speech Recognition 1063 N [E TifT) " !dj+Nd.\ 2] d r(Hn L.Jj (dj+Nd.\)2Zn VnC\) = 2 [Nd - Lj dj:1vd'\] (5) 5. Set Wn+l = Wn - (Hn + NdAn)-lgn. 6. Go to 1 and iterate. Note that A is adjusted at each iteration. The final value of An is taken as the optimal A. Iterative regularization parameter selection shows that A converges, for example, to 1~~' from one of our experiments. 3.2 A TWO-LAYER NEURAL NETWORK SYSTEM WITH REGULARIZATION We then extended our regularization work from the one-layer sigmoidal network system to a two-layer sigmoidal network system. The first layer of the network works as a feature extractor and is shared by all phonetic classes. Theoretically, in order to benefit from its larger capability of representing phonetic segments, the number of hidden units of a two-layer network should be much greater than the number of input dimensions. However, a large number of hidden units can cause serious overfitting problems when the number of training ~amples is less than the number of parameters for some context models. Therefore, regularization is more useful here. Because the second layer can be trained as a one-layer net, the regularization techniques we developed for a one-layer net can be applied here to train the second layer. In our implementation, a weighted least squares error measure was used at the output layer. First, the weights for the two-layer system were initialized with random numbers between -1 and 1. Fixing the weights for the second layer, we trained the first layer by using gradient descent; then fixing the weights for the first layer, we trained the second layer by linear least squares with a re gularization term, without the sigmoidal function at the output. We stopped after one iteration for our initial experiment. 4 TRAINING SNN WITH PROJECTION PURSUIT 4.1 WHY PROJECTION PURSUIT As we described in the previous section, regularization is especially useful in training the second layer of a two-layer network. In order to take advantage of the two-layer layer structure, we want to train the first layer as well. However, once the number of the hidden units is large, the number of weights in the first layer is huge, which makes the first layer very difficult to train. Projection pursuit presents a !lseful technique to use a large hidden layer but still keep the number of weights in the first layer as small as possible. The original pJojection PU13Uit is a nonparametric statistical technique to find interesting low dimensional projections of high dimensional data sets [7]. The parametric version of it, a projection pursuit learning network. (pPLN) has a structure very similar to a two-layer sigmoidal network network [7]. In a traditional two-layer neural network, the weights in the first layer can be viewed as hypetplanes in the input space. It has been proposed that a special function of the first layer is to partition the input space into cells through these hyperplanes [8]. The second layer groups these cells together to form decision regions. 1064 Zhao, Schwartz, Makhoul, and Zavaliagkos The accuracy or resolution of the decision regions is completely specified by the size and density of the cells which is detennined by the number and placement of the first layer hyperplanes in the input space. In a two-layer neural net, since the weights in the first layer can go anywhere, there are no restrictions on the placement of these hyperplanes. In contrast, a projection pursuit learning network. restricts these hyperplanes in some major "interesting" directions. In other words, hidden units are grouped into several distinct directions. Of course, with this grouping, the number of cells in the input space is reduced somewhat. However, the interesting point here is that this resoiction does not reduce the number of cells asymptotically [7]. In other words, grouping hidden units does not affect the number of cells much. Consequently, for a fixed number of hidden units, the number of parameters in the first layer in a projection pursuit learning network. is much less than in a traditional neural network. Therefore, a projection pursuit learning network is easier to train and generalizes better. 4.2 HOW TO TRAIN A PPLN In our implementation, the distinct projection directions were shared by all contextdependent models, and they were trained context-independently. We then trained these direction parameters with back-propagation. The second layer was trained with regularization. Iterations can go back and forth between the two layers. 5 COMBINATIONS OF DIFFERENT MODELS In the last two sections, we talked about using regularization and projection pursuit to optimize our neural network system. In this section, we will discuss another optimization method, combining different models into a hybrid system. The combining method is based on the N-best rescoring paradigm [2]. The N-best rescoring paradigm is a mechanism that allows us to build a hybrid system by combining different knowledge sources. For example, in the RM corpus, we successfully combined the HMM: system, th~ SNN system and word-pair grammar into a single hybrid system which achieved the state-of-the-art. We have been using this N-best rescoring paradigm to combine different models in the WSJ corpus as well. These different models include SNN left context, right context, and diphone models, HMM models, and a language model known as statistical grammar. We will show how to obtain a reasonable combination of different systems from Bayes rule. The goal is to compute P(SIX), the probability of the sentence 5 given the observation sequence X. From Bayes rule, P(SIX)SNN = P(S)P(XIS) P(X) ~ P(S) II P(xIS) P(x) x ~ P(S) II P(xlp, c) P(x) x ~ P(S) II P(Plx, c) P(plc) :r: Segmental Neural Net Optimization for Continuous Speech Recognition 1065 where X is a sequence of acoustic features x in each phonetic segment; p and c is the phoneme class and context for the segment, respectively. The following three approximations are used here: • P(XIS) = Ox P(xIS). • P(:z:IS) = P(:z:lp, c). • P(clx) = P(c). Therefol'e, in a SNN system, we use the following approximation from Bayes rule: P(SIX)N N ~ P(S) II P~~~;) x where P(S): Word grammar score. Ox P(plx, c): Neural net score. Ox P(plc): Phone grammar score. These three scores together with HMM scores are combined in the SNN/HMM hybrid system. 6 EXPERIMENTAL RESULTS Development Set Nov92 Test HMM 11.0 8.5 Baseline SNN 11. 7 RegtIlarization and P!'ojection Pursuit SNN 11.2 9.1 Baseline SNN/HMM 10.3 7.7 ~~~~~=-~~==~~~------~----------~~--Regularization and Projection Pursuit SNN/HMM 9.5 7.2 Table 1: Word Error Rates for 5K, Bigram Grammar Development Set Nov93 Test 14.4 14.0 Regularization and Projection Pursuit SNN 14.6 Regularization and Projection Pursuit SNN/HMM 13.0 12.3 Table 2: Word Error Rates for 20K, Trigram Grammar Speaker-independent CSR tests were performed on the 5,000-word (5K) and 20,000-word (20K) ARPA Wall Street Journal corpus. Bigram and trigram statistical grammars were used. The basic neural network structure consists of 80 inputs, 500 hidden units and 46 outputs. There are 125 projection directions in the first layer. Context models consist of 1066 Zhao, Schwartz, Makhoul, and Zavaliagkos right context models and left diphone models. In the right context models, we used 46 different networks to recognize each phoneme in each of the different right contexts. In the left diphone models, a segment input consisted of the first half segment of the current phone plus the second half segment of the previous phone. Word error rates are shown in Tables 1 and 2. Comparing the first two rows of Table 1 and Table 2, we can see that the two-layer neural network system alone is at the level of state-of-the-art HMM systems. Shown in Row 3 and 5 of Table 1, regularization and projection pursuit improve the performance of neural net system. The hybrid SNN/HMM system reduces the word error rate 10%-15% over the HMM system in both tables. 7 CONCLUSIONS Neural net te':hnology is useful in advancing the state-of-the-art in continuous speech recognition system. Optimization methods, like regularization and projection pursuit, improve the performance of the neural net syst£:m. Our hybrid SNN/HMM system reduces the word error rate 10%-15% over the HMM system on 5,000-word and 20,000-word WSJ corpus. Acknowledgments This work was funded by the Advanced Research Projects Agency of the Department of Defense. References [1] M. Cohen, H. Franco, N. Morgan, D. Rumelhart and V. Abrash, "Context-Dependent Multiple Distribution Phonetic Modeling with :MLPS", in em Advances in Neural Information Processing Systems 5, eds. S. J. Hanson, J. D. Cowan and C. L. Giles. Morgan Kaufmann Publishers, San Mateo, 1993. [2] G. Zavaliagkos, Y. Zhao, R. Schwartz and J. Makhoul, " A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition", in em Advances in Neural Information Processing Systems 5, eds. S. J. Hanson, J. D. Cowan and C. L. Giles. Morgan Kaufinann Publishers, San Mateo, 1993. [3] A. Barron, "Statistical properties of artificial neural networks," IEEE Cont Decision and Control, Tampa, FL, pp. 280-285, 1989. [4] H. Gish, "A probabilistic approach to the understanding and training of neural network classifiers," IEEE Int. Cont. Acoust .• Speech. Signal Processing, April 1990. [5] G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, 1990. [6] D. M. Bates, M. J. Lindstrom, G. Wahba and B. S. Yandell, "GCVPACK-Routines for Generalized Cross Validation", Comm. Statist.- Simula., 16(4), 1247-1253 (1987). [7] Y. Zhao and C. G. Atkeson, "Implementing Projection Pursuit Learning", to appear in Neural Computation, in preparation. [8] J. Makhoul, A. El-Jaroudi and R. Schwartz, "Partitioning Capabilities of 1\vo-Iayer Neural Networks", IEEE Transactions on Signal Processing, 39, pp. 1435-1440, 1991. PART X COGNITIVE SCIENCE
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Grammatical Inference by Attentional Control of Synchronization in an Oscillating Elman Network Bill Baird Dept Mathematics, U.C.Berkeley, Berkeley, Ca. 94720, baird@math.berkeley.edu Todd Troyer Dept of Phys., U.C.San Francisco, 513 Parnassus Ave. San Francisco, Ca. 94143, todd@phy.ucsf.edu Abstract Frank Eeckman Lawrence Livermore National Laboratory, P.O. Box 808 (L-270), Livermore, Ca. 94550, eeckman@.llnl.gov We show how an "Elman" network architecture, constructed from recurrently connected oscillatory associative memory network modules, can employ selective "attentional" control of synchronization to direct the flow of communication and computation within the architecture to solve a grammatical inference problem. Previously we have shown how the discrete time "Elman" network algorithm can be implemented in a network completely described by continuous ordinary differential equations. The time steps (machine cycles) of the system are implemented by rhythmic variation (clocking) of a bifurcation parameter. In this architecture, oscillation amplitude codes the information content or activity of a module (unit), whereas phase and frequency are used to "softwire" the network. Only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules contributes incoherent crosstalk noise. Attentional control is modeled as a special subset of the hidden modules with ouputs which affect the resonant frequencies of other hidden modules. They control synchrony among the other modules and direct the flow of computation (attention) to effect transitions between two subgraphs of a thirteen state automaton which the system emulates to generate a Reber grammar. The internal crosstalk noise is used to drive the required random transitions of the automaton. 67 68 Baird, Troyer, and Eeckman 1 Introduction Recordings of local field potentials have revealed 40 to 80 Hz oscillation in vertebrate cortex [Freeman and Baird, 1987, Gray and Singer, 1987]. The amplitude patterns of such oscillations have been shown to predict the olfactory and visual pattern recognition responses of a trained animal. There is further evidence that although the oscillatory activity appears to be roughly periodic, it is actually chaotic when examined in detail. This preliminary evidence suggests that oscillatory or chaotic network modules may form the cortical substrate for many of the sensory, motor, and cognitive functions now studied in static networks. It remains be shown how networks with more complex dynamics can performs these operations and what possible advantages are to be gained by such complexity. 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 views cortex as a set of coupled oscillatory associative memories, and is guided by the principle that attractors must be used by macroscopic systems for reliable computation in the presence of noise. This system must function reliably in the midst of noise generated by crosstalk from it's own activity. 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". In a similar manner, the network we have constructed is a symbol processing system, but with analog input and oscillatory subsymbolic representations. The architecture operates as a thirteen state finite automaton that generates the symbol strings of a Reber grammar. It is designed to demonstrate and study the following issues and principles of neural computation: (1) Sequential computation with coupled associative memories. (2) Computation with attractors for reliable operation in the presence of noise. (3) Discrete time and state symbol processing arising from continuum dynamics by bifurcations of attractors. (4) Attention as selective synchronization controling communication and temporal program flow. (5) chaotic dynamics in some network modules driving randomn choice of attractors in other network modules. The first three issues have been fully addressed in a previous paper [Baird et. al., 1993], and are only briefly reviewed. ".le focus here on the last two. 1.1 Attentional Processing 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 irrelevant to the present computation are contributing crosstalk noise. This is smilar to the problem of coding messages in a computer architecture like the Connection Machine so that they can be picked up from the common communication buss line by the proper receiving module. Periodic or nearly periodic (chaotic) variation of a signal introduces additional degrees of freedom that can be exploited in a computational architecture. We investigate the principle that selective control of synchronization, which we hypopthesize to be a model of "attention", can be used to solve this coding problem and control communication and program flow in an architecture with dynamic attractors. The architecture illust.rates the notion that synchronization not only "binds" senGrammatical Inference by Attentional Control of Synchronization 69 sory inputs into "objects" [Gray and Singer, 1987], but binds the activity of selected cortical areas into a functional whole that directs behavior. It is a model of "attended activity" as that subset which has been included in the processing of the moment by synchronization. This is both a spatial and temporal binding. Only the inputs which are synchronized to the internal oscillatory activity of a module can effect previously learned transitions of at tractors within it. For example, consider two objects in the visual field separately bound in primary visual cortex by synchronization of their components at different phases or frequencies. One object may be selectively attended to by its entrainment to oscillatory processing at higher levels such as V4 or IT. These in turn are in synchrony with oscillatory activity in motor areas to select the attractors there which are directing motor output. In the architecture presented here, we have constrained the network dynamics so that there exist well defined notions of amplitude, phase, and frequency. The network has been designed so that amplitude codes the information content or activity of a module, whereas phase and frequency are used to "softwire" the network. An oscillatory network module has a passband outside of which it will not synchronize with an oscillatory input. Modules can therefore easily be de synchronized by perturbing their resonant frequencies. Furthermore, only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules contributes incoherant crosstalk or noise. The flow of communication between modules can thus be controled by controlling synchrony. By changing the intrinsic frequency of modules in a patterned way, the effective connectivity of the network is changed. The same hardware and connection matrix can thus subserve many different computations and patterns of interaction between modules without crosstalk problems. The crosstalk noise is actually essential to the function of the system. It serves as the noise source for making random choices of output symbols and automaton state transitions in this architecture, as we discuss later. In cortex there is an issue as to what may constitute a source of randomness of sufficient magnitude to perturb the large ensemble behavior of neural activity at the cortical network level. It does not seem likely that the well known molecular fluctuations which are easily averaged within one or a few neurons can do the job. The architecture here models the hypothesis that deterministic chaos in the macroscopic dynamics of a network of neurons, which is the same order of magnitude as the coherant activity, can serve this purpose. In a set of modules which is desynchronized by perturbing the resonant frequencies of the group, coherance is lost and "random" phase relations result. The character of the model time traces is irregular as seen in real neural ensemble activity. The behavior of the time traces in different modules of the architecture is similar to the temporary appearance and switching of synchronization between cortical areas seen in observations of cortical processing during sensory/motor tasks in monkeys and humans [Bressler and Nakamura, 1993]. The structure of this apparently chaotic signal and its use in network learning and operation are currently under investigation. 2 Normal Form Associative Memory Modules The mathematical foundation for the construction of network modules is contained in the normal form projection algorithm [Baird and Eeckman, 1993]. This is a learning algorithm for recurrent analog neural networks which allows associative memory storage of analog patterns, continuous periodic sequences, and chaotic 70 Baird, Troyer, and Eeckman attractors in the same network. An N node module can be shown to function as an associative memory for up to N /2 oscillatory, or N /3 chaotic memory attractors [Baird and Eeckman, 1993]. 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]. The network modules of this architecture were developed previously as models of olfactory cortex with distributed patterns of activity like those observed experimentally [Baird, 1990, Freeman and Baird, 1987]. Such a biological network is dynamically equivalent to a network in normal form and may easily be designed, simulated, and theoretically evaluated in these coordinates. When the intramodule competition is high, they are "memory" or winner-take-all cordinates where attractors have one oscillator at maximum amplitude, with the other amplitudes near zero. In figure two, the input and output modules are demonstrating a distributed amplitude pattern ( the symbol "T"), and the hidden and context modules are two-attractor modules in normal form coordinates showing either a right or left side active. In this paper all networks are discussed in normal form coordinates. By analyzing the network in these coordinates, the amplitude and phase dynamics have a particularly simple interaction. When the input to a module is synchronized with its intrinsic oscillation, the amplitude of the periodic activity may be considered separately from the phase rotation. The module may then be viewed as a static network with these amplitudes as its activity. To illustrate the behavior of individ ualnetwork modules, we examine a binary (twoattractor) module; the behavior of modules with more than two attractors is similar. Such a unit is defined in polar normal form coordinates by the following equations of the Hopf normal form: rli 1l.irli - Cdi + (d - bsin(wclockt))rlir5i + L wtlj cos(Oj - Oli) j rOi 1l.jr Oi - crgi + (d - bsin(wclockt))roirii + L wijlj cos(Oj - OOi) j Oli Wi + L wt(Ij /1·li) sin(Oj - Oli) j OOi Wi + L wij(Ij/rOi) sin(Oj - OOi) j The clocked parameter bsin(wclockt) is used to implement the discrete time machine cycle of the Elman architecture as discussed later. It has lower frequency (1/10) than the intrinsic frequency of the unit Wi. Examination of the phase equations shows that a unit has a strong tendency to synchronize with an input of similar frequency. Define the phase difference cp = 00 OJ = 00 - wJt between a unit 00 and it's input OJ. For either side of a unit driven by an input of the same frequency, WJ = Wo, There is an attractor at zero phase difference cp = 00 OJ = ° and a repellor at cp = 180 degrees. In simulations, the interconnected network of these units described below synchronizes robustly within a few cycles following a perturbation. If the frequencies of some modules of the architecture are randomly dispersed by a significant amount, WJ - Wo #- 0, phase-lags appear first, then synchronization is lost in those units. An oscillating module therefore acts as a band pass filter for oscillatory inputs. Grammatical Inference by Attentional Control of Synchronization 71 When the oscillators are sychronized with the input, OJ - Oli = 0, the phase terms cos(Oj - Oli) = cos(O) = 1 dissappear. This leaves the amplitude equations rli and rOi with static inputs Ej wt;Ij and E j wijlj. Thus we have network modules which emulate static network units in their amplitude activity when fully phaselocked to their input. Amplitude information is transmitted between modules, with an oscillatory carrier. For fixed values of the competition, in a completely synchronized system, the internal amplitude dynamics define a gradient dynamical system for a fourth order energy fUllction. External inputs that are phase-locked to the module's intrinsic oscillation simply add a linear tilt to the landscape. 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 along one axis over the other. Thinking of Tli as the "acitivity" of the unit, this acitivity becomes a monotonically increasing function of input. The module behaves as an analog connectionist unit whose transfer function can be approximated by a sigmoid. We refer to this as the "analog" mode of operation of the module. With high levels of competition, the unit will behave as a binary (bistable) digital flip-flop element. There are two deep potential wells, one on each axis. Hence the module performs a winner-take-all choice on the coordinates of its initial state and maintains that choice "clamped" and independent of external input. This is the "digital" or "quantized" mode of operation of a module. We think of one attractor within the unit as representing "1" (the right side in figure two) and the other as representing "0" . 3 Elman Network of Oscillating Associative Memories 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. Previously we cons structed a system which functions as the six Figure 1. state finite automaton that perfectly recognizes or generates the set of strings defined by the Reber grammar described in Cleeremans et. al. [Cleeremans et al., 1989]. We found the connections for this network by using the backpropagation algorithm in a static network that approximates the behavior of the amplitudes of oscillation in a fully synchronized dynamic network [Baird et al., 1993]. Here we construct a system that emulates the larger 13 state automata similar (less one state) to the one studied by Cleermans, et al in the second part of their paper. The graph of this automaton consists of two subgraph branches each of which has the graph structure of the automaton learned as above, but with different assignments of transition output symbols (see fig. 1). T 72 Baird, Troyer, and Beckman We use two types of modules in implementing the Elman network architecture shown in figure two below. The input and output layer each consist of a single associative memory module with six oscillatory attractors (six competing oscillatory modes), one for each of the six symbols in the grammar. The hidden and context layers consist of the binary "units" above composed of a two oscillatory attractors. The architecture consists of 14 binary modules ill the hidden and context layers - three of which are special frequency control modules. The hidden and context layers are divided into four groups: the first three correspond to each of the two subgraphs plus the start state, and the fourth group consists of three special control modules, each of which has only a special control output that perturbs the resonant frequencies of the modules (by changing their values in the program) of a particular state coding group when it is at the zero attractor, as illustrated by the dotted control lines in figure two. This figure shows control unit two is at the one attractor (right side of the square active) and the hidden units coding for states of subgraph two are in synchrony with the input and output modules. Activity levels oscillate up and down through the plane of the paper. Here in midcycle, competition is high in all modules. Figure 2. OSCILLATING ELMAN NETWORK OUTPUT INPUT The discrete machine cycle of the Elman algorithm is implemented by the sinusoidal variation (clocking) of the bifurcation parameter in the normal form equations that determines the level of intramodule competition [Baird et al., 1993]. At the beginning of a machine cycle, when a network is generating strings, the input and context layers are at high competition and their activity is clamped at the bottom of deep basins of attraction. The hidden and output modules are at low competition and therefore behave as a traditional feedforward network free to take on analog values. In this analog mode, a real valued error can be defined for the hidden and output units and standard learning algorithms like backpropagation can be used to train the connections. Then the situation reverses. For a Reber grammar there are always two equally possible next symbols being activated in the output layer, and we let the crosstalk noise Grammatical Inference by Attentional Control of Synchronization 73 break this symmetry so that the winner-take-all dynamics of the output module can chose one. High competition has now also "quantized" and clamped the activity in the hidden layer to a fixed binary vector. Meanwhile, competition is lowered in the input and context layers, freeing these modules from their attractors. An identity mapping from hidden to context loads the binarized activity of the hidden layer into the context layer for the next cycle, and an additional identity mapping from the output to input module places the chosen output symbol into the input layer to begin the next cycle. 4 Attentional control of Synchrony We introduce a model of attention as control of program flow by selective synchronization. The attentional controler itself is modeled in this architecture as a special set of three hidden modules with ouputs that affect the resonant frequencies of the other corresponding three subsets of hidden modules. Varying levels of intramodule competition control the large scale direction of information flow between layers of the architecture. To direct information flow on a finer scale, the attention mechanism selects a subset of modules within each layer whose output is effective in driving the state transition behavior of the system. By controling the patterns of synchronization within the network we are able to generate the grammar obtained from an automaton consisting of two subgraphs connected by a single transition state (figure 1). During training we enforce a segregation of the hidden layer code for the states of the separate subgraph branches of the automaton so that different sets of synchronized modules learn to code for each subgraph of the automaton. Then the entire automaton is hand constructed with an additional hidden module for the start state between the branches. Transitions in the system from states in one subgraph of the automaton to the other are made by "attending" to the corresponding set of nodes in the hidden and context layers. This switching of the focus of attention is accomplished by changing the patterns of synchronization within the network which changes the flow of communication between modules. Each control module modulates the intrinsic frequency of the units coding for the states a single su bgraph or the unit representing the start state. The control modules respond to a particular input symbol and context to set the intrinsic frequency of the proper subset of hidden units to be equal to the input layer frequency. As described earlier, modules can easily be desynchronized by perturbing their resonant frequencies. By perturbing the frequencies of the remaining modules away from the input frequency, these modules are no longer communicating with the rest of the network. Thus coherent information flows from input to output only through one of three channels. Viewing the automata as a behavioral program, the control of synchrony constitutes a control of the program flow into its subprograms (the subgraphs of the automaton). When either exit state of a subgraph is reached, the "B" (begin) symbol is then emitted and fed back to the input where it is connected through the first to second layer weight matrix to the attention control modules. It turns off the synchrony of the hidden states of the subgraph and allows entrainment of the start state to begin a new string of symbols. This state in turn activates both a "T" and a "P' in the output module. The symbol selected by the crosstalk noise and fed back to the input module is now connected to the control modules through the weight matrix. It desynchronizes the start state module, synchronizes in the subset of hidden units 74 Baird. Troyer. and Eeckman coding for the states of the appropriate subgraph, and establishes there the start state pattern for that subgraph. Future work will investigate the possibilities for self-organization of the patterns of synchrony and spatially segregated coding in the hidden layer during learning. The weights for entire automata, including the special attention control hidden units, should be learned at once. 4.1 Acknowledgments 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, 1990] Baird, B. (1990). 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 and Eeckman, 1993] Baird, B. and Eeckman, F. H. (1993). A normal form projection algorithm for associative memory. In Hassoun, M. H., editor, Associative Neural Memories: Theory and Implementation, New York, NY. Oxford University Press. [Baird et al., 1993] Baird, B., Troyer, T., and Eeckman, F. H. (1993). Synchronization and gramatical inference in an oscillating elman network. In Hanson, S., Cowan, J., and Giles, C., editors, Advances in Neural Information Processing Systems S, pages 236-244. Morgan Kaufman. [Bressler and Nakamura, 1993] Bressler, S. and Nakamura. (1993). Interarea synchronization in Macaque neocortex during a visual discrimination task. In Eeckman,F. H., and Bower, J., editors, Computation and Neural Systems, page 515. Kluwer. [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, 1987] Freeman, W. and Baird, B. (1987). Relation of olfactory EEG to behavior: Spatial analysis. Behavioral Neuroscience, 101:393-408. [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. ADAPTIVE KNOT PLACEMENT FOR NONPARAMETRIC REGRESSION Hossein L. Najafi* Department of Computer Science University of Wisconsin River Falls, WI 54022 Vladinlil' Cherkas sky Department of Electrical Engineering University of Minnesota Minneapolis, Minnesota 55455 Abstract Performance of many nonparametric methods critically depends on the strategy for positioning knots along the regression surface. Constrained Topological Mapping algorithm is a novel method that achieves adaptive knot placement by using a neural network based on Kohonen's self-organizing maps. We present a modification to the original algorithm that provides knot placement according to the estimated second derivative of the regression surface. 1 INTRODUCTION Here we consider regression problems. Using mathematical notation, we seek to find a function f of N - 1 predictor variables (denoted by vector X) from a given set of n data points, or measurements, Zi = (Xi , Yi ) (i = 1, ... , n) in N-dimensional sample space: Y = f(X) + error (l) where error is unknown (but zero mean) and its distribution may depend on X. The distribution of points in the training set can be arbitrary, but uniform distribution in the domain of X is often used. • Responsible for correspondence, Telephone (715) 425-3769, e-mail hosseiu.najafi@uwrf.edu. 247 248 Najafi and Cherkassky The goal of this paper is to show how statistical considerations can be used to improve the performance of a novel neural network algorithm for regression [eN91], in order to achieve adaptive positioning of knots along the regression surface. By estimating and employing the second derivative of the underlying function, the modified algorithm is made more flexible around the regions with large second derivative. Through empirical investigation, we show that this modified algorithm allocates more units around the regions where the second derivative is large. This increase in the local knot density introduces more flexibility into the model (around the regions with large second derivative) and makes the model less biased around these regions. However, no over-fitting is observed around these regions. 2 THE PROBLEM OF KNOT LOCATION One of the most challenging problems in practical implementations of adaptive methods for regression is adaptive positioning of knots along the regression surface. Typical1y, knot positions in the domain of X are chosen as a subset of the training data set, or knots are uniformly distributed in X. Once X-locations are fixed, commonly used data-driven methods can be applied to determine the number of knots. However, de Boor [dB78] showed that a polynomial spline with unequally spaced knots can approximate an arbitrary function much better than a spline with equally spaced knots. Unfortunately, the minimization problem involved in determination of the optimal placement of knots is highly nonlinear and the solution space is not convex [FS89). Hence, t.he performance of many recent algorit.hms that include adaptive knot placement (e.g. MARS) is difficult to evaluate analytically. In addition, it is well-known that when data points are uniform, more knots should be located where the second derivative of the function is large. However, it is difficult to extend these results for non-uniform data in conjunction with data-dependent noise. Also, estimating the second derivative of a true function is necessary for optimal knot placement. Yet, the function itself is unknown and its estimation depends on the good placement of knots. This suggests the need for some iterative procedure that alternates between function estimation(smoothing) and knot posit.ioning steps. Many ANN methods effectively try to solve the problem of adaptive knot location using ad hoc strategies that are not statistically optimal. For example, local adaptive methods [Che92) are generalizat.ion of kernel smoothers where the kernel functions and kernel centers are determined from the data by some adaptive algorithm. Examples of local adaptive methods include several recently proposed ANN models known as radial basis function (RBF) networks, regularization networks, networks with locally tuned units etc [BL88, MD89, PG90). When applied to regression problems, all these methods seek to find regression estimate in the (most general) form 2::=1 biHi(X, Ci) where X is the vector of predictor variable, Ci is the coordinates of the i-th 'center' or 'bump', Hi is the response function of the kernel type (the kernel width may be different for each center i), bi are linear coefficients to be determined, and k is the total number of knots or 'centers'. Whereas the general formulat.ion above assumes global opt.imizat.ion of an error measure for the training set with respect. to all parameters, i.e. center locations, kernel width and linear coefficients, this is not practically feasible because the error surface is generally non-convex and may have local minima [PG90, MD89). Hence most Adaptive Knot Placement for Nonparametric Regression 249 practical approaches first solve the problem of center(knot) location and assume identical kernel functions. Then the remaining problem of finding linear coefficients bi is solved by using familiar methods of Linear Algebra [PG90] or gradient-descent techniques [MD89]. It appears that the problem of center locations is the most critical one for the local neural network techniques. Unfortunately, heuristics used for center location are not based on any statistical considerations, and empirical results are too sketchy [PG90, MD89]. In statistical methods knot locations are typically viewed as free parameters of the model, and hence the number of knots directly controls the model complexity. Alternatively, one can impose local regularization constraints on adjacent knot locations, so that neighboring knots cannot move independently. Such an approach is effectively implemented in the model of self-organization known as Kohonen's Self-Organizing Maps (SOM) [Koh84]. This model uses a set of units ("knots") with neighborhood relations between units defined according to a fixed topological structure (typically 1 D or 2D grid). During training or self-organization, data points are presented to the map iteratively, one at a time, and the unit closest to the data moves towards it, also pulling along its topological neighbors. 3 MODIFIED CTM ALGORITHM FOR ADAPTIVE KNOT PLACEMENT The SOM model has been applied to nonparametric regression by Cherkassky and Najafi [CN9I] in order to achieve adaptive positioning of knots along the regression surface. Their technique, called Constrained Topological Mapping (CTM), is a modification of Kohonen's self-organization suitable for regression problems. CTM interprets the units of the Kohonen map as movable knots of a regression surface. Correspondingly, the problem of finding regression estimate can be stated as the problem of forming an M dimensional topological map using a set of samples from N-dimensional sample space (where AI ~ N - 1) . Unfortunately, straightforward application of the Kohonen Algorithm to regression problem does not work well [CN9I]. Because, the presence of noise in the training data can fool the algorithm to produce a map that is a multiple-valued function of independent variables in the regression problem (1). This problem is overcome in the CTM algorithm, where the nearest neighbor is found in the subspace of predictor variables, rather than in the input(sample) space [CN9I]. We present next a concise description of the CTM algorithm. Using standard formulation (1) for regression, the training data are N-dimensional vectors Zi = (Xi , Yi), where Y i is a noisy observation of an unknown function of N - 1 predictor variables given by vector Xi. The CTM algorithm constructs an M - dimensional topological map in N-dimensional sample space (M ~ N - 1) as follows: o. Initialize the M - dimensional t.opological map in N-dimensional sample space. 1. Given an input vector Z in N-dimensional sample space, find the closest (best matching) unit i in the subspace of independent val·iables: II Z*(k) - Wi II = Minj{IIZ* - W; II} Vj E [I, ... ,L] 250 Najafi and Cherkassky where Z· is the projection of the input vector onto the subspace of independent variables, Wi is the projection of the weight vector of unit j, and k is the discrete time step. 2. Adjust the units' weights according to the following and return to 1: 'Vi (2) where /3( k) is the learning rate and Cj (k) is the neighborhood for unit i at iteration k and are given by: (k: .. ) 1 /3(k) = /30 x (~~) ,Cj(k) = -----~~ o 5 ( IIi - ill ) exp' /3(k) x So (3) where kmax is the final value of the time step (kmax is equal to the product of the training set size by the number of times it was recycled), /30 is the initial learning rate, and /3/ is the final learning rate (/30 = 1.0 and /3/ = 0.05 were used in all of our experiments), Iii - ill is the topological distance between the unit i and the best matched unit i and So is the initial size of the map (i.e., the number of units per dimension) . Note that CTM method achieves placement of units (knots) in X-space according to density of training data. This is due to the fact that X-coordinates of CTM units during training follow the standard Kohonen self-organization algorithm [Koh84], which is known to achieve faithful approximation of an unknown distribution. However, existing CTM method does not place more knots where the underlying function changes rapidly. The improved strategy for CTM knot placement in X-space takes into account estimated second derivative of a function as is described next. The problem with estimating second derivative is that the function itself is unknown. This suggests using an iterative strategy for building a model, i.e., start with a crude model, estimate the second derivative based on this crude model, use the estimated second derivative to refine the model, etc. This strategy can be easily incorporated into the CTM algorithm due to its iterative nature. Specifically, in CTM method the map of knots(i.e., the model) becomes closer and closer to the final regression model as the training proceeds. Therefore, at each iteration, the modified algorithm estimates the second derivative at the best matching unit (closest to the presented data point in X-space), and allows additional movement of knots proportional to this estimate. Estimating the second derivative from the map (instead of using the training data) makes sense due to smoothing properties of CTM. The modified CTM algorithm can be summarized as follows: 1. Present training sample Zi = (Xi, Yi) to the map and find the closest (best matching) unit i in the su bspace of independent variables to this data point. (same as in the original CTM) 2. Move the the map (i.e., the best matching unit and all its neighbors) toward the presented data point (same as in the original CTM) Adaptive Knot Placement for Nonparametric Regression 251 3. Estimate average second derivative of the function at the best matching unit based on the current positions of the map units. 4. Normalize this average second derivative to an interval of [0,1]. 5. Move the map toward the presented data point at a rate proportional to the estimated normalizes average second derivative and iterate. For multivariate functions only gradients along directions given by the topological structure ofthe map can be estimated in step 4. For example, given a 2-dimensional mesh that approximates function I(XI, X2), every unit of the map (except the border units for which there will be only one neighbor) has two neighboring units along each topological dimension. These neighboring units can be used to approximate the function's gradients along the corresponding topological dimension of the map. These values along each dimension can then be averaged to provide a local gradient estimate at a given knot. In step 5, estimated average second derivative I" is normalized to [0,1] range using 1/Ji = 1 - exp(lf"ll tan(T)) This is done because the value of second derivative is used as the learning rate. In step 6, the map is modified according to the following equation: 'Vj (4) It is this second movement of the map that allows for more flexibility around the region of the map where the second derivative is large. The process described by equation (4) is equivalent to pulling all units towards the data, with the learning rate proportional to estimated second derivative at the best matched unit. Note that the influence of the second derivative is gradually increased during the process of self-organization by the factor (1-,B( k)). This factor account for the fact that the map becomes closer and closer to the underlying function during self-orga.nization; hence, providing a more reliable estimate of second deriva.tive. 4 EMPIRICAL COMPARISON Performance of the two algorithms (original and modified CTM) was compared for several low-dimensional problems. In all experiments the two algorithms used the same training set of 100 data points for the univariate problems and 400 data points for the 2-variable problems. The training samples (Xi, Yi) were generated according to (1), with Xi randomly drawn from a uniform distribution in the closed interval [-1,1]' and the error drawn from the normal distribution N(O, (0.1)2). Regression estimates produced by the self-organized maps were tested on a different set of n = 200 samples (test set) generated in the same manner as the training set. We used the Average Residual, AR = j ~ L~=l [Yi - I(Xd]2, as the performance measure on the test set. Here, I(X) is the piecewise linear estimate of the function with knot locations provided by coordinates of the units of trained CTM. The Aver252 Najafi and Cherkassky age Residual gives an indication of standard deviation of the overall generalization error. 1.2 1 0.8 -. 0.6 >( t::;' 0.4 0.2 True function Original CTM ~-. Modified CTM -+-0 ~~~~~ ................ : ............... ~. ~~~~~ -0.2 -0.8 -0.6 -0.4 -0.2 o 0.2 0.4 0.6 0.8 x Figure 1: A 50 unit map formed by the original and modified algorithm for the Gaussian function. 1.2 1 0.8 Q 0.6 t::;' 0.4 0.2 True function Original CTM ~- . Modified CTM -+-o ~~~_00!6I~~~;..J ............................................................ . 1 -0.2 I....-_ ........ __ ........ __ --'-__ ......... __ -'-__ ..L.-__ .L..-_........IL.-_---' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Figure 2: A 50 unit map formed by the original and modified algorithm for the step function. We used a gaussian function (f(x) = exp-64X 2 ) and a step function for our first set of experiments. Figure 1 and 2 show the actual maps formed by the original and modified algorithm for these functions. It is clear from these figures that the modified algorithm allocates more units around the regions where the second derivative is large. This increase in the local knot density has introduced more flexibility into the model around the regions with large second derivatives. As a result of this the 1 Adaptive Knot Placement for Nonparametric Regression 253 model is less biased around these regions. However, there is no over-fitting in the regions where the second derivative is large. 0.29 r-----:"-___r---..,..---~---~---r__--___r--___, 0.28 0.27 0.26 0.25 c.::: 0.24 ~ 0.23 0.22 0.21 0.2 0.19 ',~-~ ~---- '"'+--__ ..L --aoj- __ +-- ---r- ____ + __ + __ _ Original CTM ~ Modified CTM +_. -L ...... ---y... +---+ 0.18 ""'-__ --L. ___ .......... __ ----L ___ .......... ___ '---__ ......L.. __ ----' o 10 20 30 40 50 60 # of units per dimension Figure 3: Average Residual error as a function of the size of the map for the 3dimensional Step function 0.55 0.5 0.45 c.::: 0.4 ~ 0.35 0.3 0.25 0.2 0 10 Original CTM ~ Modified CTM +_. ----~---+ +++----+-----+--+--+--+ 20 30 40 50 60 # of units per dimension Figure 4: Average Residual error as a function of the size of the map for the 3dimensional Sine function To compare the behavior of the two algorithms in their predictability of structureless data, we trained them on a constant function I(x) = a with eTT01' = N(O, (0.1)2). This problem is known as smoothing pure noise in regression analysis. It has been shown [CN9l] that the original algorithm handles this problem well and quality of CTM smoothing is independent of the number of units in the map. Our experiments 70 70 254 Najafi and Cherkas sky show that the modified algorithm performs as good as the original one in this respect. Finally, we used the following two-variable functions (step, and sine) to see how well the modified algorithm performs in higher dimensional settings. Ste : f(x x) = {I for ((x~ < 0.5) 1\ (X2 < 0.5)) V ((Xl ~ 0.5) 1\ (X2 ~ 0.5)) PI, 2 0 otherwise Sine: f(XI, X2) = sin (27rJ(xt)2 + (X2)2) The results of these experiments are summarized in Figure 3 and 4. Again we see that the modified algorithm outperforms the original algorithm. Note that the above example of a two-variable step function can be easily handled by recursive partitioning techniques such as CART [BFOS84]. However, recursive methods are sensitive to coordinate rotation. On the other hand, CTM is a coordinate-independent method, i.e. its performance is independent of any affine transformation in X-space. References [BFOS84] 1. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone. Classification and Regression Trees. Wadswordth, Belmont, CA, 1984. [BL88] [Che92] [CN91] [dB78] [FS89] D.S. Broomhead and D. Lowe. Multivariable functional interpolation and adaptive networks. Complex Systems, 2:321-355, 1988. V. Cherkassky. Neural networks and nonparametric regression. In S.Y. Kung, F. Fallside, J .Aa. Sorenson, and C.A. Kamm, editors, Neural Networks for Signal Processing, volume II. IEEEE, Piscataway, N J, 1992. V. Cherkassky and H.L. Najafi. Constrained topological mapping for nonparametric regression analysis. Neural Networks, 4:27-40, 1991. C. de Boor. A Practical Guide to Splines. Springer-Verlag, 1978. J .H. Friedman and B.W. Silverman. Flexible parsimonious smoothing and additive modeling. Technometrics, 31(1):3-21, 1989. [Koh84] T. Kohonen. Self-Organization and Associative Memory. SpringerVerlag, third edition, 1984. [MD89] J. Moody and C.J. Darken. Fast learning in networks of locally tuned processing units. Neural Computation, 1:281, 1989. [PG90] T. Poggio and F. Girosi. Networks for approximation and learning. Proceedings of the IEEE, 78(9):1481-1497, 1990.
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Bayesian Modeling and Classification of Neural Signals 590 Michael S. Lewicki Computation and Neural Systems Program California Institute of Technology 216-76 Pasadena, CA 91125 lewickiOcns.caltech.edu Abstract Signal processing and classification algorithms often have limited applicability resulting from an inaccurate model of the signal's underlying structure. We present here an efficient, Bayesian algorithm for modeling a signal composed of the superposition of brief, Poisson-distributed functions. This methodology is applied to the specific problem of modeling and classifying extracellular neural waveforms which are composed of a superposition of an unknown number of action potentials CAPs). Previous approaches have had limited success due largely to the problems of determining the spike shapes, deciding how many are shapes distinct, and decomposing overlapping APs. A Bayesian solution to each of these problems is obtained by inferring a probabilistic model of the waveform. This approach quantifies the uncertainty of the form and number of the inferred AP shapes and is used to obtain an efficient method for decomposing complex overlaps. This algorithm can extract many times more information than previous methods and facilitates the extracellular investigation of neuronal classes and of interactions within neuronal circuits. Bayesian Modeling and Classification of Neural Signals 591 1 INTRODUCTION Extracellular electrodes typically record the activity of several neurons in the vicinity of the electrode tip (figure 1). Most electrophysiological data is collected by isolating action potentials (APs) from a single neuron by using a level detector or window discriminator. Methods for extracting APs from multiple neurons can, in addition to the obvious advantage of providing more data, provide the means to investigate local neuronal interactions and response properties of neuronal populations. Determining from the voltage waveform what cell fired when is a difficult, ill-posed problem which is compounded by the fact that cells frequently fire simultaneously resulting in large variations in the observed shapes. There are three major difficulties in identifying and classifying action potentials (APs) in a neuron waveform. The first is determining the AP shapes, the second is deciding the number of distinct shapes, and the third is decomposing overlapping spikes into their component parts. In general, these problems cannot be solved independently, since the solution of one will affect the solution of the others. 2: rn_Cl. Figure 1: Each neuron generates a stereotyped action potential (AP) which is observed through the electrode as a voltage fluctuation. This shape is primarily a function of the position of a neuron relative to the tip. The extracellular waveform shows several different APs generated by an unknown number of neurons. Note the frequent presence of overlapping APs which can completely obscure individual spikes. The approach summarized here is to model the waveform directly to obtain a probabilistic description of each action potential and, in turn, of the whole waveform. This method allows us to compute the class conditional probabilities of each AP. In addition, it is possible to quantify the certainty of both the form and number of spike shapes. Finally, we can use this description to decompose overlapping APs efficiently and assign probabilities to alternative spike sequences. 2 MODELING SINGLE ACTION POTENTIALS The data from the event observed (at time zero) is modeled as resulting from a fixed underlying spike function, s(t), plus noise: (1) 592 Lewicki where v is the parameter vector that defines the spike function. The noise, 1], is modeled as Gaussian with zero mean and standard deviation u1]' From the Bayesian perspective, the task is to infer the posterior distribution of the spike function parameters (assuming, for the moment, that u1] and Uw are known): P( ID M) - P(Dlv, 0"'1' M) P(vluw, M) v 'O"1]'O"w, P(DIO"1],O"w,M) . (2) The two terms specifying the posterior distribution of v are 1) the probability of the data given the model: (3) and 2) the prior assumptions of the structure of s(t) which are assumed to be of the form: (4) The superscript (m) denotes differentiation which for these demonstrations we assumed to be m = 1 corresponding to linear splines. The smoothness of s(t) is controlled through Uw with small values of Uw penalizing large fluctuations. The final step in determining the posterior distribution is to eliminate the dependence of P(vID, 0"1]' O"w, M) on 0"1] and O"w. Here, we use the approximation: (5) The most probable values of 0"1] and O"w were obtained using the methods of MacKay (1992) in which reestimation formulas are obtained from a Gaussian approximation of the posterior distribution for 0"1] and O"w, P(O"1] , O"wID, M). Correct inference of O"w prevents the spike function from overfitting the data. 3 MODELING MULTIPLE ACTION POTENTIALS When a waveform contains multiple types of APs, determining the component spike shapes is more difficult because the classes are not known a priori. The uncertainty of which class an event belongs to can be incorporated with a mixture distribution. The probability of a particular event, D n , given all spike models, M 1:K , is K P(Dnlvl:K' 1r, 0"1]' M1 :K) = L 1I"k P(Dnlvk, 0"'1' Mk), k=l (6) where 1I"k is the a priori probability that a spike will be an instance of Mk, and E 1I"k = l. As before, the objective is to determine the posterior distribution for the parameters defining a set of spike models, P(V 1 :K, 1rID 1:N , 0"1]1 trw, M 1:K) which is obtained again using Bayes' rule. Bayesian Modeling and Classification of Neural Signals 593 Finding the conditions satisfied at a posterior maximum leads to the equation: (7) where 'Tn is the inferred occurrence time (typically to sub-sample period accuracy) of the event Dn. This equation is solved iteratively to obtain the most probable values of V l :K • Note that the error for each event, D n , is weighted by P(Mk IOn, Vk, 1r, 0''7) which is the probability that the event is an instance of the kth spike model. This is a soft clustering procedure, since the events are not explicitly assigned to particular classes. Maximizing the posterior yields accurate estimates of the spike functions even when the clusters are highly overlapping. The techniques described in the previous section are used to determine the most probable values for 0''7 and rTw and, in turn, the most probable values of V l :K and 1r. 4 DETERMINING THE NUMBER OF SPIKE SHAPES Choosing a set of spike models that best fit the data, would result eventually in a model for each event in the waveform. Heuristics might indicate whether two spike models are identical or distinct, but ad hoc criteria are notoriously dependent on particular circumstances, and it is difficult to state precisely what information the rules take into account. To determine the most probable number of spike models, we apply probability theory. Let Sj = {MHJ} denote a set of spike models and H denote information known a priori. The probability of Sj, conditioned only on H and the data, is obtained using Bayes' rule: (8) The only data-dependent term is P(OI:NISj, H) which is the evidence for Sj (MacKay, 1992). With the assumption that all hypotheses SI:3 are equally probable a priori, P(D l :NISj, H) ranks alternative spike sets in terms of their probability. The evidence term P(OI:N[Sj, H) is convenient because it is the normalizing constant for the posterior distribution of the parameters defining the spike set. Although calculation of P(O I :N I Sj ,H) is analytically intractable, it is often wellapproximated with a Gaussian integral which was the approximation used for these demonstrations. A convenient way of collapsing the spike set is to compare spike models pairwise. Two models in the spike set are selected along with a sampled set of events fit by each model. We then evaluate P(DISl) and P(D[S2)' S1 is the hypothesis that the data is modeled by a single spike shape, S2 says there are two spike shapes. If P(D[S1) > P(D[S2), we replace both models in S2 by the one in S1. The procedure terminates when no more pairs can be combined to increase the evidence. 594 Lewicki 5 DECOMPOSING OVERLAPPING SPIKES Overlaps must be decomposed into their component spikes for accurate inference of the spike functions and accurate classification of the events. Determining the best-fitting decomposition is difficult becaus(~ of the enormous number of possible spike sequences, not only all possible model combinations for each event but also all possible event times. A brute-force approach to this problem is to perform an exhaustive search of the space of overlapping spike functions and event times to find the sequence with maximum probability. This approach was used by Atiya (1992) in the case of two overlapping spikes with the times optimized to one sample period. Unfortunately, this is often computationally too demanding even for off-line analysis. We make this search efficient utilizing dynamic programming and k-dimensional trees (Friedman et al., 1977). Once the best-fitting decomposition can be obtained, however, it may not be optimal, since adding more spike shapes can overfit the data. This problem is minimized by evaluating the probability for alternative decompositions to determine the most probable spike sequence (figure 2) . a .. ,,' . b' c Figure 2: Many spike function sequences can account for the same region of data. The thick lines show the data, thin lines show individual spike functions. In this case, the bestfitting overlap solution is not the most probable: the sequence with 4 spike functions is more than 8 time& more probable than the other solutions, even though these have smaller mean squared error. Using the best-fitting overlap solution may increase the classification error. Classification error is minimized by using t he overlap solution that is most probable. 6 PERFORMANCE The algorithm was tested on 40 seconds of neurophysiological data. The task is to determine the form and number of spike ~hapes in a waveform and to infer the occurrence times of each spike shape. The output of the algorithm is shown in figure 3. The uniformity of the residual error indicates that the six inferred spike shapes account for the entire 40 seconds of data. The spike functions M2 and M3 appear similar by eye, but the probabilities calculated with the methods in section 4 indicate that the two functions are significantly different. When plotted against each Bayesian Modeling and Classification of Neural Signals 595 "Xl +----IHf----+-----+---+----j .~ ... . .. ...• ;: ~ . . ,'" ,." ,...-.:.' ... \"f' ·m+----+~~-+_---+---+----1 y Tlme(rTS) '(Xl +_---iIIIr---+_-- -+---+---_j '~l,~ . "J~ ~'~~'k'! .. ", . \.<" . m +_---+--'W', .~ .. -.+_----+---~---_j -bI !~>,"1:~.~iJ;o:~~··· >' .. ;;,:; ·"t',,' '",:.',1' ;~ \' .• ~. . .::\t" ,~ ;-•. i .. '::"~a\f ~":''';' lf'. . ;";': .. ~:~";':-~. ,~; ,'; :;'~::~ ';R~:' . ..... ~:._:.,l.., :., . . ',~.?" .. ".: :-.. ! :-t'r.i"~'~. TlrTl {rT15} M5 "Xl +----+---+----+----+-----j ·m +_---+---+-----+---+- --_j ' 300 +-----+-~~_+__---+--... +-----l ., TIIT.'mI) ~~-- --r----r_--_r----r_--~ . .... ,: ..... v~· · .... ''' .. '., '. ~;: T .... ( ... l ""+_---+---+----;----+-------1 . "" l:..,E:· .. ~~~ .. . .~. " .:.(, :.:. , ;. :: .••.. .,,,,, +_---;----+-----+-----t--.', : , .... i';', . ··:.h'i II" ... .' ',." .' , , , .~ r"~ •..• ' c. ':""". ', . .,.. . "-".'.,'. :. Tme,,,..) ,(Xl +_- --+----I----4---.-+------l ."" .':~ ' • • L'tI" ... 'IJ ... ' \.. ,.,~~.:.~~ •• ~.,,:< .. , '::;', ",,~.: .:-:;,\'., ,'. ,:,;\ ... ~: ,"" ..... , , -~, r-' ..... .' .-:.'"'' . :'~:: ". d' ., ; ...... :, .~;.:~. -\. " ~ ' 4~ '" . 'Ii", ~:",'~ ' , ...•. ;/;~~~: ... : ''':~ ; . .. <,.h/., 6-' ... , .. ~{:~:..r.J,G:, '~' ... ,.,' .,".' " ,:, :" "" ":' ,:" .-:'''i.-.c . : • :'.:r:-':""-':." .. ', "". '.'\ Figure 3: The solid lines are the inferred spike models. The data overlying each model is a sample of at most 40 events with overlapping spikes subtracted out. The residual errors are plotted below each model. This spike set was obtained after three iterations of the algorithm, decomposing overlaps and determining the most probable number of spike functions after each iteration. The whole inference procedure used 3 minutes of CPU time on a Sparc IPX. Once the spike set is infe! red, classification of the same 40 second waveform takes about 10 seconds. 596 Lewicki other, the two populations of APs are distinctly separated in the region around the peak with M3 being wider than M 2 • The accuracy of the algorithm was tested by generating an artificial data set composed of the six inferred shapes shown in figure 3. The event times were Poisson distributed with frequency equal the inferred firing rate of the real data set. Gaussian noise was then added with standard deviation equal to 0"'1. The classification results are summarized in the tables below. Table 1: Results of the spike model inference algorithm on the synthesized data set. I Model /I 1 I 2 I 3 I 4 I 5 I 6 II I b.max/O"fJ II 0.44 I 0.36 I 1.07 I 0.78 I 0.84 I 0.40 II The number of spike models was correctly determined by the algorithm with the six-model spike set was preferred over the most probable five-model spike set byexp(34) : 1 and over the most probable seven-model spike set by exp(19) : 1. The inferred shapes were accurate to within a maximum error of 1.0717'1. The row elements show the maximum absolute difference, normalized by 17'1' between each true spike function and the corresponding inferred function. Table 2: Classification results for the synthesized data set (non-overlapping events). True Inferred Models Missed Total Models 1 2 3 4 5 6 Events Events 1 17 0 0 0 0 0 0 17 2 0 25 1 0 0 0 0 26 3 0 0 15 0 0 0 0 15 4 0 0 0 116 0 0 1 117 5 0 0 0 0 56 0 17 73 6 0 0 0 0 0 393 254 647 Table 3: Classification results for the synthesized data set (overlapping events). True Inferred Models Missed Total Models 1 2 3 4 5 6 Events Events 1 22 0 0 0 0 0 0 22 2 0 36 1 0 0 0 0 37 3 0 0 20 0 0 0 0 20 4 0 1 0 116 0 1 3 121 5 0 0 0 1 61 1 19 82 6 0 0 0 3 2 243 160 408 Tables 2 and 3: Each matrix component indicates the number of times true model i was classified as inferred model j. Events were missed if the true spikes were not detected in an overlap sequence or if all sample values for the spike fell below the event detection threshold (417'1). There was 1 false positive for Ms and 7 for M 6 • Bayesian Modeling and Classification of Neural Signals 597 7 DISCUSSION Formulating the task as having to infer a probabilistic model made clear what was necessary to obtain accurate spike models. The soft clustering procedure accurately determines the spike shapes even when the true underlying shapes are similar. U nless the spike shapes are well-separated, commonly used hard clustering procedures will lead to inaccurate estimates. Probability theory also allowed for an objective means of determining the number of spike models which is an essential reason for the success of this algorithm. With the wrong number of spike models overlap decomposition becomes especially difficult. The evidence has proved to be a sensitive indicator of when two classes are distinct. Probability theory is also essential to accurate overlap decomposition. Simply fitting data with compositions of spike models leads to the same overfitting problem encountered in determining the number of spike models and in determining the spike shapes. Previous approaches have been able to handle only a limited class of overlaps, mainly due to the difficultly in making the fit efficient. The algorithm used here can fit an overlap sequence of virtually arbitrary complexity in milliseconds. In practice, the algorithm extracts many times more information from a neural waveform than previous methods. Moreover, this information is qualitatively different from a simple list of spike times. Having reliable estimates of the action potential shapes makes it possible to study the properties of these classes, since distinct neuronal types can have distinct neuronal spikes. Finally, accurate overlap decomposition makes it possible to investigate interactions among local neurons which were previously very difficult to observe. Acknowledgements I thank David MacKay for helpful discussions and Jamie Mazer for many conversations and extensive help with the development of the software. This work was supported by Caltech fellowships and an NIH Research Training Grant. References A.F. Atiya. (1992) Recognition of multiunit neural signals. IEEE Transactions on Biomedical Engineering 39(7):723-729. J .H. Friedman, J.L. Bently, and R.A. Finkel. (1977) An algorithm for finding best matches in logarithmic expected time. ACM Trans. Math. Software 3(3):209-226. D. J. C. MacKay. (1992) Bayesian interpolation. Neural Computation 4(3):415-445.
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Supervised learning from incomplete data via an EM approach Zoubin Ghahramani and Michael I. Jordan Department of Brain & Cognitive Sciences Massachusett.s Institute of Technology Cambridge, MA 02139 Abstract Real-world learning tasks may involve high-dimensional data sets with arbitrary patterns of missing data. In this paper we present a framework based on maximum likelihood density estimation for learning from such data set.s. VVe use mixture models for the density estimates and make two distinct appeals to the ExpectationMaximization (EM) principle (Dempster et al., 1977) in deriving a learning algorithm-EM is used both for the estimation of mixture components and for coping wit.h missing dat.a. The resulting algorithm is applicable t.o a wide range of supervised as well as unsupervised learning problems. Result.s from a classification benchmark-t.he iris data set-are presented. 1 Introduction Adaptive systems generally operate in environments t.hat are fraught with imperfections; nonet.heless they must cope with these imperfections and learn to extract as much relevant information as needed for their part.icular goals. One form of imperfection is incomplet.eness in sensing information. Incompleteness can arise extrinsically from the data generation process and intrinsically from failures of the system's sensors. For example, an object recognition system must be able to learn to classify images with occlusions, and a robotic controller must be able to integrate multiple sensors even when only a fraction may operate at any given time. In this paper we present a. fra.mework-derived from parametric statistics-for learn120 Supervised Learning from Incomplete Data via an EM Approach 121 ing from data sets with arbitrary patterns of incompleteness. Learning in this framework is a classical estimation problem requiring an explicit probabilistic model and an algorithm for estimating the parameters of the model. A possible disadvantage of parametric methods is their lack of flexibility when compared with nonparametric methods. This problem, however, can be largely circumvented by the use of mixture models (McLachlan and Basford, 1988). Mixture models combine much of the flexibility of nonparametric methods with certain of the analytic advantages of parametric methods. Mixture models have been utilized recently for supervised learning problems in the form of the "mixtures of experts" architecture (Jacobs et al., 1991; Jordan and Jacobs, 1994). This architecture is a parametric regression model with a modular structure similar to the nonparametric decision tree and adaptive spline models (Breiman et al., 1984; Friedman, 1991). The approach presented here differs from these regression-based approaches in that the goal of learning is to estimate the density of the data. No distinction is made between input and output variables; the joint density is estimated and this estimate is then used to form an input/output map. Similar approaches have been discussed by Specht (1991) and Tresp et al. (1993). To estimate the vector function y = I(x) the joint density P(x, y) is estimated and, given a particular input x, the conditional density P(ylx) is formed. To obtain a single estimate of y rather than the full conditional density one can evaluate y = E(ylx), the expectation of y given x. The density-based approach to learning can be exploited in several ways. First, having an estimate of the joint density allows for the representation of any relation between the variables. From P(x, y), we can estimate y = I(x), the inverse x = 1-1 (y), or any other relation between two subsets of the elements of the concatenated vector (x, y). Second, this density-based approach is applicable both to supervised learning and unsupervised learning in exactly the same way. The only distinction between supervised and unsupervised learning in this framework is whether some portion of the data vector is denoted as "input" and another portion as "target". Third, as we discuss in this paper, the density-based approach deals naturally with incomplete data, i.e. missing values in the data set. This is because the problem of estimating mixture densities can itself be viewed as a missing data problem (the "labels" for the component densities are missing) and an Expectation-Maximization (EM) algorithm (Dempster et al., 1977) can be developed to handle both kinds of missing data. 2 Density estimation using EM This section outlines the basic learning algorithm for finding the maximum likelihood parameters of a mixture model (Dempster et al., 1977; Duda and Hart, 1973; Nowlan, 1991). \IVe assume that. t.he data . .:t' = {Xl, ... , XN} are generated independently from a mixture density 1\1 P(Xi) = LP(XiIWj;(}j)P(Wj), (1) ;=1 122 Ghahramani and Jordan where each component of the mixture is denoted Wj and parametrized by (}j. From equation (1) and the independence assumption we see that the log likelihood of the parameters given the data set is N M l((}IX) = LlogLP(xilwj;Oj)P(Wj). (2) i=1 j=1 By the maximum likelihood principle the best model of the data has parameters that maximize l(OIX). This function, however, is not easily maximized numerically because it involves the log of a sum. Intuitively, there is a "credit-assignment" problem: it is not clear which component of the mixture generated a given data point and thus which parameters to adjust to fit that data point. The EM algorithm for mixture models is an iterative method for solving this credit-assignment problem. The intuition is that if one had access to a "hidden" random variable z that indicated which data point was genera.ted by which component, then the maximization problem would decouple into a set of simple maximizations. Using the indicator variable z, a "complete-data" log likelihood function can be written N M lc((}IX, Z) = L L Zij log P(XdZi; O)P(Zi; (}), (3) ;=1 j=1 which does not involve a log of a summation. Since Z is unknown lc cannot be utilized directly, so we instead work with its expectation, denoted by Q(OI(}k)' As shown by (Dempster et aI., 1977), l(OIX) can be maximized by iterating the following two steps: Estep: Q(OI(}k) E[lc(OIX,Z)IX,(}k] M step: (}k+l argmax Q((}IOk)' (4) o The E (Expectation) step computes the expected complete data log likelihood and the M (Maximization) step finds the parameters that maximize this likelihood. These two steps form the basis of the EM algorithm; in the next two sections we will outline how they can be used for real and discrete density estimation. 2.1 Real-valued data: Inixture of Gaussians Real-valued data can be modeled as a mixture of Gaussians. For this model the E-step simplifies to computing hij = E[Zijlxi,Ok], the probability that Gaussian j, as defined by the parameters estimated at time step k, generated data point i. Itj 1- 1/ 2 exp{ -~ (Xi - itj)Tt;l,k(Xi - itj)} h .. = (5) I} L~1 IEfl-l/2exp{-~(Xi - it7)TE,I,k(Xi - it7)}' The M-step re-estimates the data set weighted by the hii= ) ~ k+l _ L~l hijXi a I-Lj N ' Li=1 hij means and covariances of the Gaussians1 using the 1 Though this derivation assumes equal priors for the Gaussians, if the priors arc viewed as mixing parameters they can also be learned in the maximization step. Supervised Learning from Incomplete Data via an EM Approach 123 2.2 Discrete-valued data: Inixture of Bernoullis D-dimensional binary data x = (Xl, . .. ,Xd, . .. XD), Xd E {O, 1}, can be modeled as a mixture of !II Bernoulli densities. That is, M D P(xIO) = L P(Wj) IT /-ljd(1 - /-ljd)(l-Xd). (7) For this model the E-step involves computing nD pX,ld (1 _ p. )(1-Xld) h .. d=l}d }d I) - 'Ef'!l nf=l P7J d (1 - Pld)(1-xld) , (8) and the M-step again re-estimates the parameters by ~ k+l _ 'E~l hijXi ttj N . 'Ei=l hij (9) More generally, discrete or categorical data can be modeled as generated by a mixture of multinomial densities and similar derivations for the learning algorithm can be applied. Finally, the extension to data with mixed real, binary. and categorical dimensions can be readily derived by assuming a joint density with mixed components of the three types. 3 Learning from inco111plete data In the previous section we presented one aspect of the EM algorithm: learning mixture models. Another important application of EM is to learning from data sets with missing values (Little and Rubin, 1987; Dempster et aI., 1977). This application has been pursued in the statistics literature for non-mixture density estimation problems; in this paper we combine this application of EM with that of learning mixture parameters. We assume that. the data set ,l:' = {Xl •.. . , XN} is divided into an observed component ,yo and a missing component ;t'm. Similarly, each data vector Xi is divided into (xi, xi) where each data vector can have different missing components-this would be denoted by superscript Dli and OJ. but we have simplified the notation for the sake of clarity. To handle missing data we rewrite the EM algorithm as follows Estep: M step: E[ic( fJl,t'°, ;t'm , Z) I;t'°. Ok] argmax Q(fJlfJk). o (10) Comparing to equation (4) we see that aside from t.he indicator variables Z we have added a second form of incomplete data, ;t'm , corresponding to the missing values in the data set. The E-step of the algorithm estimates both these forms of missing information; in essence it uses the current estimate of the data density to complete the missing values. 124 Ghahramani and Jordan 3.1 Real-valued data: mixture of Gaussians We start by writing the log likelihood of the complete data, N M N M ic(OIXO, xm, Z) = L L Zij log P(xdzj, 0) + L L Zij log P(zdO). (11) j j We can ignore the second term since we will only be estimating the parameters of the P(XdZi, 0). Using equation (11) for the mixture of Gaussians we not.e that if only the indicator variables Zi are missing, the E step can be reduced to estimating E[ Zij lXi, 0]. For the case we are interested in, with two types of missing data Zi and xi, we expand equation (11) using m and 0 superscripts to denote subvectors and submatrices of the parameters matching the missing and observed components of the data, N M Ic(OIXO, xm, Z) = L L Zij[n log27r + ! log IEj 1- !(xi -l1-jf E;l,OO(xi -l1-j) ..22 2 I J ( 0 o)T~-l,Om( m m) 1( m m)T~-l,mm( m m)] Xi I1-j L...j Xi I1-j - 2 Xi I1-j L...j Xi I1-j • Note that after taking the expectation, the sufficient statistics for the parameters involve three unknown terms, Zij, ZijXi, and zijxixiT. Thus we must compute: E[Zijlx?,Ok]' E[Zijxilx?,Ok], and E[ZijxixinTlx?,Ok]. One intuitive approach to dealing with missing data is to use the current estimate of the data density to compute the expectat.ion of the missing data in an E-step, complete the data with these expectations, and then use this completed data to reestimate parameters in an M-step. However, this intuition fails even when dealing with a single two-dimensional Gaussian; the expectation of the missing data always lies along a line, which biases the estimate of the covariance. On the other hand, the approach arising from application of the EM algorithm specifies that one should use the current density estimate to compute the expectation of whatever incomplete terms appear in the likelihood maximization. For the mixture of Gaussians these incomplete terms involve interactions between the indicator variable :;ij and the first and second moments of xi. Thus, simply computing the expectation of the missing data Zi and xi from our model and substituting those values into the M step is not sufficient to guarantee an increase in the likelihood of the parameters. The above terms can be computed as follows: E[ Zij lxi, Ok] is again hij, the probability as defined in (5) measured only on the observed dimensions of Xi, and E[Zijxilxi, Ok] = hijE[xilzij = 1, xi, Od = hij(l1-j + EjOEjO-l (xi -Il.'}», (12) Defining xi] = E[xi IZij = 1, xi, Ok], the regression of xi on xi using Gaussian j, E[ .. m mTI ° 0 ] _ h .. (~mm ~mo~oo-l ~moT ~ m ~ mT) Z'Jxi Xi xi' k 'J L...j L...j ~j L...j + XijXij . (13) The M-step uses these expectations substituted into equations (6)a and (6)b to re-estimate the means and covariances. To re-estimate the mean vector, I1-j' we substitute the values E[xilzij = 1, xi, Ok] for the missing components of Xi in equation (6)a. To re-estimate the covariance matrix we substitute t.he values E[xixiTlzij = 1, xi, Ok] for the outer product matrices involving the missing components of Xi in equation (6)b. Supervised Learning from Incomplete Data via an EM Approach 125 3.2 Discrete-valued data: Inixture of Bernoullis For the Bernoulli mixture the sufficient statistics for the M-step involve t he incomplete terms E[Zij Ix?, Ok] and E[ Zij xi Ix~, Ok]. The first is equal to hij calculated over the observed subvector of Xi. The second, since we assume that within a class the individual dimensions of the Bernoulli variable are independent., is simply hijl-Lj. The M-step uses these expectations substituted into equation (9). 4 Supervised learning If each vector Xi in the data set is composed of an "input" subvector, x}, and a "target" or output subvector, x?, then learning the joint density of the input and target is a form of supervised learning. In supervised learning we generally wish to predict the output variables from the input variables. In this section we will outline how this is achieved using the estimated density. 4.1 Function approximation For real-valued function approximation we have assumed that the densit.y is estimated using a mixture of Gaussians. Given an input vector x~ we ext ract all the relevant information from the density p(xi, XO) by conditionalizing t.o p(xOlxD. For a single Gaussian this conditional densit.y is normal, and, since P(x 1 , XO) is a mixture of Gaussians so is P(xolxi ). In principle, this conditional density is the final output of the density estimator. That is, given a particular input the network returns the complete conditional density of t.he output. However, since many applications require a single estimate of the output, we note three ways to obtain estimates x of XO = f(x~): the least squares estimate (LSE), which takes XO(xi) = E(xOlxi); stochastic sampling (STOCH), which samples according to the distribution xO(xD "" P(xOlxi); single component LSE (SLSE), which takes xO(xD = E(xOlxLwj) where j = argmaxk P(zklx~). For a given input, SLSE picks the Gaussian with highest posterior and approximates the out.put with the LSE estimator given by that Gaussian alone. The conditional expectation or LSE estimator for a Gaussian mixt.ure is (14) which is a convex sum of linear approximations, where the weights hij vary nonlinearly according to equation (14) over the input space. The LSE estimator on a Gaussian mixture has interesting relations to algorithms such as CART (Breiman et al., 1984), MARS (Friedman, 1991), and mixtures of experts (Jacobs <.'t al., 1991; Jordan and Jacobs, 1994), in that the mixture of Gaussians competit.ively partitions the input space, and learns a linear regression surface on each part-it.ion. This similarity has also been noted by Tresp et al. (1993) . The stochastic estimator (STOCH) and the single component estimator (SLSE) are better suited than any least squares method for learning non-convex ill verse maps, where the mean of several solutions to an inverse might not be a solut ion. These 126 Ghahramani and Jordan Figure 1: Classification of the iris data set. 100 data points were used for training and 50 for testing. Each data point consisted of 4 real-valued attributes and one of three class labels. The figure shows classification performance ± 1 standard error (11 = 5) as a function of proportion missing features for the EM algorithm and for mean imputation (MI), a common heuristic where the missing values are replaced with their unconditional means. Classification with missing inputs 100 ~" 0-1---~, ! -t EM U , ;.;:: \ '" '" !! 60 U ... U II) .. o 40 U ~ 20 o , , , \ , 'l,_ 20 40 60 80 % missing features -'-'tI MI 100 estimators take advantage of the explicit representat.ion of the input/output density by selecting one of the several solutions to the inverse. 4.2 Classification Classification problems involve learning a mapping from an input space into a set of discrete class labels. The density estimat.ion framework presented in this paper lends itself to solving classification problems by estimating the joint density of the input and class label using a mixture model. For example, if the inputs have realvalued attributes and there are D class labels, a mixture model with Gaussian and multinomial components will be used: AI 1 ~ ~jd P(x, e = dlO) = ~ P(Wj) (27r)n/2IEj 11/2 exp{ -"2 (x - I-tj fEj1 (x I-'j n, (15) denoting the joint probability that the data point. is x and belongs to class d, where the ~j d are the parameters for the multinomial. Once this density has been estimated, the maximum likelihood label for a particular input x may be obtained by computing P(C = dlx, 0). Similarly, the class conditional densities can be derived by evaluating P( x Ie = d, 0). Condi tionalizing over classes in this way yields class conditional densities which are in turn mixtures of Gaussians. Figure 1 shows the performance of the EM algorithm on an example classification problem with varying proportions of missing features. We have also applied these algorithms to the problems of clustering 35-dimensional greyscale images and approximating the kinematics of a three-joint planar arm from incomplete data. 5 Discussion Densit.y estimation in high dimensions is generally considered to be more difficultrequiring more parameters-than function approximation. The density-estimationbased approach to learning, however, has two advantages. First, it permits ready incorporation of results from the statistical literature on missing data to yield flexible supervised and unsupervised learning architectures. This is achieved by combining two branches of application of the EM algorithm yielding a set of learning rules for mixtures under incomplete sampling. Supervised Learning from Incomplete Data via an EM Approach 127 Second, estimating the density explicitly enables us to represent any relation between the variables. Density estimation is fundamentally more general than function approximation and this generality is needed for a large class of learning problems arising from inverting causal systems (Ghahramani, 1994). These problems cannot be solved easily by traditional function approximation techniques since the data is not generated from noisy samples of a function, but rather of a relation. Acknow ledgmuents Thanks to D. M. Titterington and David Cohn for helpful comments. This project was supported in part by grants from the McDonnell-Pew Foundation, ATR Auditory and Visual Perception Research Laboratories, Siemens Corporation, the N ational Science Foundation, and the Office of Naval Research. The iris data set was obtained from the VCI Repository of Machine Learning Databases. References Breiman, L., Friedman, J. H., Olshen, R. A., and Stone, C. J. (1984). Classification and Regression Trees. Wadsworth International Group, Belmont, CA. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood fwm incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1-38. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. Wiley, New York. Friedman, J. H. (1991). Multivariate adaptive regression splines. The Annols of Statistics, 19:1-141. Ghahramani, Z. (1994). Solving inverse problems using an EM approach to density estimation. In Proceedings of the 1993 Connectionist Models Summer School. Erlbaum, Hillsdale, NJ. Jacobs, R., Jordan, M., Nowlan, S., and Hinton, G. (1991). Adaptive mixture of local experts. Neural Computation, 3:79-87. Jordan, M. and Jacobs, R. (1994). Hierarchical mixtures of experts ano the EM algorithm. Neural Computation, 6:181-214. Little, R. J. A. and Rubin, D. B. (1987). Statistical Analysis with Mis.'ling Data. Wiley, New York. McLachlan, G. and Basford, K. (1988). Mixture models: Inference and applications to clustering. Marcel Dekkel'. Nowlan, S. J. (1991). Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. CMV-CS-91-126, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA. Specht, D. F. (1991). A general I'egression neural network. IEEE Trans. Neural Networks, 2(6):568-576. Tresp, V., Hollatz, J., and Ahmad, S. (1993). Network structuring and training using rule-based knowledge. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5. Morgan Kaufman Publishers, San Mateo, CA.
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Correlation Functions in a Large Stochastic Neural Network Iris Ginzburg School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University Tel-Aviv 69978, Israel Haim Sompolinsky Racah Institute of Physics and Center for Neural Computation Hebrew University Jerusalem 91904, Israel Abstract Most theoretical investigations of large recurrent networks focus on the properties of the macroscopic order parameters such as population averaged activities or average overlaps with memories. However, the statistics of the fluctuations in the local activities may be an important testing ground for comparison between models and observed cortical dynamics. We evaluated the neuronal correlation functions in a stochastic network comprising of excitatory and inhibitory populations. We show that when the network is in a stationary state, the cross-correlations are relatively weak, i.e., their amplitude relative to that of the auto-correlations are of order of 1/ N, N being the size of the interacting population. This holds except in the neighborhoods of bifurcations to nonstationary states. As a bifurcation point is approached the amplitude of the cross-correlations grows and becomes of order 1 and the decay timeconstant diverges. This behavior is analogous to the phenomenon of critical slowing down in systems at thermal equilibrium near a critical point. Near a Hopf bifurcation the cross-correlations exhibit damped oscillations. 471 472 Ginzburg and Sompolinsky 1 INTRODUCTION In recent years there has been a growing interest in the study of cross-correlations between the activities of pairs of neurons in the cortex. In many cases the crosscorrelations between the activities of cortical neurons are approximately symmetric about zero time delay. These have been taken as an indication of the presence of "functional connectivity" between the correlated neurons (Fetz, Toyama and Smith 1991, Abeles 1991). However, a quantitative comparison between the observed cross-correlations and those expected to exist between neurons that are part of a large assembly of interacting population has been lacking. Most of the theoretical studies of recurrent neural network models consider only time averaged firing rates, which are usually given as solutions of mean-field equations. They do not account for the fluctuations about these averages, the study of which requires going beyond the mean-field approximations. In this work we perform a theoretical study of the fluctuations in the neuronal activities and their correlations, in a large stochastic network of excitatory and inhibitory neurons. Depending on the model parameters, this system can exhibit coherent undamped oscillations. Here we focus on parameter regimes where the system is in a statistically stationary state, which is more appropriate for modeling non oscillatory neuronal activity in cortex. Our results for the magnitudes and the time-dependence of the correlation functions can provide a basis for comparison with physiological data on neuronal correlation functions. 2 THE NEURAL NETWORK MODEL We study the correlations in the activities of neurons in a fully connected recurrent network consisting of excitatory and inhibitory populations. The excitatory connections between all pairs of excitatory neurons are assumed to be equal to J / N where N denotes the number of excitatory neurons in the network. The excitatory connections from each of the excitatory neurons to each of the inhibitory neurons are J' / N. The inhibitory coupling of each of the inhibitory neurons onto each of the excitatory neurons is K / M where M denotes the number of inhibitory neurons. Finally, the inhibitory connections between pairs of inhibitory neurons are ](' / M. The values of these parameters are in units of the amplitude of the local noise (see below). Each neuron has two possible states, denoted by Si = ±1 and Ui = ±1 for the i-th excitatory and inhibitory neurons, respectively. The value -1 denotes a quiet state. The value + 1 denotes an active state that corresponds to a state with high firing rate. The neurons are assumed to be exposed to local noise resulting in stochastic dynamics of their states. This dynamics is specified by transition probabilities between the -1 and + 1 states that are sigmoidal functions of their local fields. The local fields of the i-th excitatory neuron, Ei and the i-th inhibitory neuron, Ii, at time t, are Ei(t) = J s(t) K u(t) () (1) J's(t) K' u(t) () (2) Correlation Functions in a Large Stochastic Neural Network 473 where () represents the local threshold and sand 0' are the population-averaged activities s(t) = l/N"'£j Sj(t), and O'(t) = l/M"'£j O'j(t) of the excitatory and inhibitory neurons, respectively. 3 AVERAGE FIRING RATES The macroscopic state of the network is characterized by the dynamics of s(t) and O'(t). To leading order in l/N and l/M, they obey the following well known equations ds TO dt = -s + tanh(Js - J{O' - 0) (3) dO' (I -,I ) TO- = -0' + tanh J s - K 0' - 0 dt (4) where TO is the microscopic time constant of the system. Equations of this form for the two population dynamics have been studied extensively by Wilson and Cowan (Wilson and Cowan 1972) and others (Schuster and Wagner 1990, Grannan, Kleinfeld and Sompolinsky 1992) Depending on the various parameters the stable solutions of these equations are either fixed-points or limit cycles. The fixed-point solutions represent a stationary state of the network in which the popUlation-averaged activities are almost constant in time. The limit-cycle solutions represent nonstationary states in which there is a coherent oscillatory activity. Obviously in the latter case there are strong oscillatory correlations among the neurons. Here we focus on the fixed-point case. It is described by the following equations So = tanh ( J So - K 0'0 - 0) 0'0 = tanh (J' So - K'O'o - 0) (5) (6) where So and 0'0 are the fixed-point values of sand O'. Our aim is to estimate the magnitude of the correlations between the temporal fluctuations in the activities of neurons in this statistically stationary state. 4 CORRELATION FUNCTIONS There are two types of auto-correlation functions, for the two different populations. For the excitatory neurons we define the auto-correlations as: (7) where 6si(t) = Sj(t)-so and < ... >t means average over time t. A similar definition holds for the auto-correlations of the inhibitory neurons. In our network there are three different cross-correlations: excitatory-excitatory, inhibitory- inhibitory, and inhibitory-excitatory. The excitatory-excitatory correlations are Cij(T) = {8si(t)8sj(t + T)}t Similar definitions hold for the other functions. (8) 474 Ginzburg and Sompolinsky We have evaluated these correlation functions by solving the equations for the correlations of 6Si(t) in the limit of large Nand M. We find the following forms for the correlations: 1 3 Gii(T) ~ (1- s~)exp(-A1T) + N La,exp(-AIT) '=1 (9) 1 3 Gij(T)~ NLb,exP(-A,T) . 1=1 (10) The coefficients a, and b, are in general of order 1. The three A, represent three inverse time-constants in our system, where Re(AI) ~ Re(A2) ~ Re(A3)' The first inverse time constant equals simply to Al = liTo, and corresponds to a purely local mode of fluctuations. The values of A2 and A3 depend on the parameters of the system. They represent two collective modes of fluctuations that are coherent across the populations. An important outcome of our analysis is that A2 and A3 are exactly the eigenvalues of the stability matrix obtained by linearizing Eqs. (3) and (4) about the fixed-point Eqs. (5) and (6) . The above equations imply two differences between the auto-correlations and the cross-correlations. First, Gii are of order 1 whereas in general Gij is of 0(1/ N). Secondly, the time-dependence of Gii is dominated by the local, fast time constant TO, whereas Gij may be dominated by the slower, collective time-constants. The conclusion that the cross-correlations are small relative to the auto-correlations might break down if the coefficients b, take anomalously large values. To check these possibility we have studied in detail the behavior of the correlations near bifurcation points, at which the fixed point solutions become unstable. For concreteness we will discuss here the case of Hopfbifurcations. (Similar results hold for other bifurcations as well). Near a Hopf bifurcation A2 and A3 can be written as A± ~ € ± iw, where € > 0 and vanishes at the bifurcation point. In this parameter regime, the amplitudes b1 « b2, b3 and b2 ~ b3 ~ ~. Similar results hold for a2 and a3. Thus, near the bifurcation, we have Gii( T) ~ (1 s~) exp( -T /ro)cos(wr) B Gij(r) ~ N€ exp(-€r)cos(wr) . (11) (12) Note that near a bifurcation point € is linear in the difference between any of the parameters and their value at the bifurcation. The above expressions hold for €« 1 but large compared to l/N.When € ~ liN the cross-correlation becomes of order 1, and remains so throughout the bifurcation. Figures 1 and 2 summarize the results of Eqs. (9) and (10) near the Hopf bifurcation point at J,J',K,K',O = 225,65, 161,422,2.4. The population sizes are N = 10000, M = 1000. We have chosen a parameter range so that the fixed point values of So and lTo will represent a state with low firing rate resembling the spontaneous activity levels in the cortex. For the above parameters the rates relative to the saturation rates are 0.01 and 0.03 for the excitatory and inhibitory populations respectively. Correlation Functions in a Large Stochastic Neural Network 475 0.45 04 035 0.3 025 02 015 01 005 O~~==C==C==~~--~~--~~ 180 185 190 195 200 205 210 215 220 225 J FIG URE 1. The equal-time cross-correlations between a pair of excitatory neurons, and the real part of its inverse time-constant,f, vs. the excitatory coupling parameter J. The values of Cij (0) and of the real-part of the inverse-time constants of Cij are plotted (Fig. 1) as a function of the parameter J holding the rest of the parameters fixed at their values at the bifurcation point. Thus in this case f a(225 - J). The Figure shows the growth of Cij and the vanishing of the inverse time constant as the bifurcation point is approached. 0.15 ..-----r--..-----,---.,....---,---r----,----,--.,.----, 0.1 0.05 o -0.05 -0.1 -0 .15 L-_....l.-_--L_---l. __ .L..-_-'--_-'-_~_-.-I __ ~_-' o 5 10 15 20 25 30 35 40 45 50 delay The time-dependence of the cross-correlations near the bifurcation (J = 215) is shown in Fig. 2. Time is plotted in units of TO. The pronounced damped oscillations are, according to our theory, characteristic of the behavior of the correlations near but below a Hopf bifurcation. 476 Ginzburg and Sompolinsky 5 CONCLUSION Most theoretical investigations of large recurrent networks focus on the properties of the macroscopic order parameters such as population averaged activity or average overlap with memories. However, the statistics of the fluctuations in the activities may be an important testing ground for comparison between models and observed cortical dynamics. We have studied the properties of the correlation functions in a stochastic network comprising of excitatory and inhibitory populations. We have shown that the cross-correlations are relatively weak in stationary states, except in the neighborhoods of bifurcations to nonstationary states. The growth of the amplitude of these correlations is coupled to a growth in the correlation time-constant. This divergence of the correlation time is analogous to the phenomenon of critical slowing down in systems at thermal equilibrium near a critical point. Our analysis can be extended to stochastic networks consisting of a small number of interacting homogeneous populations. Detailed comparison between the model's results and experimental values of autoand cross- correlograms of extracellularly measured spike trains in the neocortex have been carried out (Abeles, Ginzburg and Sompolinsky). The tentative conclusion of this study is that the magnitude of the observed correlations and their time-dependence are inconsistent with the expected ones for a system in a stationary state. They therefore indicate that cortical neuronal assemblies are in a nonstationary (but aperiodic) dynamic state. Acknowledgements: We thank M. Abeles for most helpful discussions. This work is partially supported by the USA-Israel Binational Science Foundation. REFERENCES Abeles M., 1991. Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge University Press. Abeles M., Ginzburg I. & Sompolinsky H. Neuronal Cross-Correlations and Organized Dynamics in the Neocortex. to appear Fetz E., Toyama K. & Smith W., 1991. Synaptic Interactions Between Cortical Neurons. Cerebral Cortex, edited by A. Peters & G. Jones Plenum Press,NY. Vol 9. 1-43. Grannan E., Kleinfeld D. & Sompolinsky H., 1992. Stimulus Dependent Synchronization of Neuronal Assemblies. Neural Computation 4,550-559. Schuster H. G. & Wagner P., 1990. BioI. Cybern. 64, 77. Wilson H. R. & Cowan J. D., 1972. Excitatory and Inhibitory Interactions m Localized Populations of Model Neurons. Biophy. J. 12, 1-23.
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Clustering with a Domain-Specific Distance Measure Steven Gold, Eric Mjolsness and Anand Rangarajan Department of Computer Science Yale University New Haven, CT 06520-8285 Abstract With a point matching distance measure which is invariant under translation, rotation and permutation, we learn 2-D point-set objects, by clustering noisy point-set images. Unlike traditional clustering methods which use distance measures that operate on feature vectors - a representation common to most problem domains - this object-based clustering technique employs a distance measure specific to a type of object within a problem domain. Formulating the clustering problem as two nested objective functions, we derive optimization dynamics similar to the Expectation-Maximization algorithm used in mixture models. 1 Introduction Clustering and related unsupervised learning techniques such as competitive learning and self-organizing maps have traditionally relied on measures of distance, like Euclidean or Mahalanobis distance, which are generic across most problem domains. Consequently, when working in complex domains like vision, extensive preprocessing is required to produce feature sets which reflect properties critical to the domain, such as invariance to translation and rotation. Not only does such preprocessing increase the architectural complexity of these systems but it may fail to preserve some properties inherent in the domain. For example in vision, while Fourier decomposition may be adequate to handle reconstructions invariant under translation and rotation, it is unlikely that distortion invariance will be as amenable to this technique (von der Malsburg, 1988). 96 Clustering with a Domain-Specific Distance Measure 97 These problems may be avoided with the help of more powerful, domain-specific distance measures, including some which have been applied successfully to visual recognition tasks (Simard, Le Cun, and Denker, 1993; Huttenlocher et ai., 1993). Such measures can contain domain critical properties; for example, the distance measure used here to cluster 2-D point images is invariant under translation, rotation and labeling permutation. Moreover, new distance measures may constructed, as this was, using Bayesian inference on a model of the visual domain given by a probabilistic grammar (Mjolsness, 1992). Distortion invariant or graph matching measures, so formulated, can then be applied to other domains which may not be amenable to description in terms of features. Objective functions can describe the distance measures constructed from a probabilistic grammar, as well as learning problems that use them. The clustering problem in the present paper is formulated as two nested objective functions: the inner objective computes the distance measures and the outer objective computes the cluster centers and cluster memberships. A clocked objective function is used, with separate optimizations occurring in distinct clock phases (Mjolsness and Miranker, 1993). The optimization is carried out with coordinate ascent/descent and deterministic annealing and the resulting dynamics is a generalization of the ExpectationMaximization (EM) algorithm commonly used in mixture models. 2 Theory 2.1 The Distance Measure Our distance measure quantifies the degree of similarity between two unlabeled 2-D point images, irrespective of their position and orientation. It is calculated with an objective that can be used in an image registration problem. Given two sets of points {Xj} and {Yk }, one can minimize the following objective to find the translation, rotation and permutation which best maps Y onto X : Ereg(m, t, 0) = L mjkllXj - t - R(0) . Yk l1 2 jk with constraints: 'Vj L:k mjk = 1 , 'Vk L:j mjk = l. Such a registration permits the matching of two sparse feature images in the presence of noise (Lu and Mjolsness, 1994). In the above objective, m is a permutation matrix which matches one point in one image with a corresponding point in the other image. The constraints on m ensure that each point in each image corresponds to one and only one point in the other image (though note later remarks regarding fuzziness). Then given two sets of points {Xj} and {Yk } the distance between them is defined as: D({Xj}, {Yk}) = min(Ereg(m,t,0) I constraints on m) . (1) m,t,e This measure is an example of a more general image distance measure derived in (Mjolsness, 1992): d(x, y) = mind(x, T(y)) E [0,00) T where T is a set of transformation parameters introduced by a visual grammar. In (1) translation, rotation and permutation are the transformations, however scaling 98 Gold, Mjolsness, and Rangarajan or distortion could also have been included, with consequent changes in the objective function. The constraints are enforced by applying the Potts glass mean field theory approximations (Peterson and Soderberg,1989) and then using an equivalent form of the resulting objective, which employs Lagrange multipliers and an x log x barrier function (as in Yuille and Kosowsky, 1991): Ereg(m, t, 8) L: mjkllXj - t - R(8) · YkW + f31 L: mjk(logmjk -1) jk jk +L:J.tj(L:mjk-1)+L:vk(L:mjk-1). (2) j k k j In this objective we are looking for a saddle point. (2) is minimized with respect to m, t, and 8, which are the correspondence matrix, translation,and rotation, and is maximized with respect to J.t and v, the Lagrange multipliers that enforce the row and column constraints for m. 2.2 The Clustering Objective The learning problem is formulated as follows: Given a set of I images, {Xd, with each image consisting of J points, find a set of A cluster centers {Ya } and match variables {Mia} defined as M. - {I if Xi is in Ya's cluster la 0 otherwise, such that each image is in only one cluster, and the total distance of all the images from their respective cluster centers is minimized. To find {Ya} and {Mia} minimize the cost function, Ec/U8ter(Y, M) = L: MiaD(Xi, Ya) , ia with the constraint that 'Vi l:a Mia = 1. D(Xi, Ya), the distance function, is defined by (1). The constraints on M are enforced in a manner similar to that described for the distance measure, except that now only the rows of the matrix M need to add to one, instead of both the rows and the columns. The Potts glass mean field theory method is applied and an equivalent form of the resulting objective is used: 1 Ec/u8ter(Y, M) = ~ MiaD(Xi, Ya) + f3 ~ Mia (log Mia - 1) + ~ Ai(L: Mia -1) ta za z a (3) Replacing the distance measure by (2), we derive: Ec/u8ter(Y, M, t, 8, m) = L:Mia L: miajkllXij - tia - R(8ia) . Ya k11 2+ ia jk ~[f3~ ~k miajk(logmiajk - 1) + ~ J.tiaj(L:k miajk - 1) + za J J L:Viak(L:miajk -1)]+ -;- L:Mia(logMia -1)+ L: Ai(L: Mia -1) k j M ia i a Clustering with a Domain-Specific Distance Measure 99 A saddle point is required. The objective is minimized with respect to Y, M, m, t, 0, which are respectively the cluster centers, the cluster membership matrix, the correspondence matrices, the rotations, and the translations. It is maximized with respect to A, which enforces the row constraint for M, and J..l and v which enforce the column and row constraints for m. M is a cluster membership matrix indicating for each image i, which cluster a it falls within, and mia is a permutation matrix which assigns to each point in cluster center Ya a corresponding point in image Xi. 0ia gives the rotation between image i and cluster center a. Both M and mare fuzzy, so a given image may partially fall within several clusters, with the degree of fuzziness depending upon 13m and 13M. Therefore, given a set of images, X, we construct Ecltuter and upon finding the appropriate saddle point of that objective, we will have Y, their cluster centers, and M, their cluster memberships. 3 The Algorithm 3.1 Overview - A Clocked Objective Function The algorithm to minimize the above objective consists of two loops - an inner loop to minimize the distance measure objective (2) and an outer loop to minimize the clustering objective (3). Using coordinate descent in the outer loop results in dynamics similar to the EM algorithm for clustering (Hathaway, 1986). (The EM algorithm has been similarly used in supervised learning [Jordan and Jacobs, 1993].) All variables occurring in the distance measure objective are held fixed during this phase. The inner loop uses coordinate ascent/descent which results in repeated row and column projections for m. The minimization of m, t and 0 occurs in an incremental fashion, that is their values are saved after each inner loop call from within the outer loop and are then used as initial values for the next call to the inner loop. This tracking of the values of m, t, and 0 in the inner loop is essential to the efficiency of the algorithm since it greatly speeds up each inner loop optimization. Each coordinate ascent/descent phase can be computed analytically, further speeding up the algorithm. Local minima are avoided, by deterministic annealing in both the outer and inner loops. The resulting dynamics can be concisely expressed by formulating the objective as a clocked objective function, which is optimized over distinct sets of variables in phases, Ecloc1ced = Ecl'luter( (((J..l, m)A , (v, m)A)$' 0 A, tA)$, (A, M)A, yA)$ with this special notation employed recursively: E{x, Y)$ : coordinate descent on x, then y, iterated (if necessary) x A : use analytic solution for x phase The algorithm can be expressed less concisely in English, as follows: Initialize t, 0 to zero, Y to random values Begin Outer Loop Begin Inner Loop Initialize t, 0 with previous values 100 Gold, Mjolsness, and Rangarajan Find m, t, e for each ia pair: Find m by softmax, projecting across j, then k, iteratively Find e by coordinate descent Find t by coordinate descent End Inner Loop If first time through outer loop i 13m and repeat inner loop Find M ,Y using fixed values of m, t, e determined in inner loop: Find M by soft max, across i Find Y by coordinate descent i 13M, 13m End Outer Loop When the distances are calculated for all the X - Y pairs the first time time through the outer loop, annealing is needed to minimize the objectives accurately. However on each succeeding iteration, since good initial estimates are available for t and e (the values from the previous iteration of the outer loop) annealing is unnecessary and the minimization is much faster. The speed of the above algorithm is increased by not recalculating the X - Y distance for a given ia pair when its Mia membership variable drops below a threshold. 3.2 Inner Loop The inner loop proceeds in three phases. In phase one, while t and e are held fixed, m is initialized with the softmax function and then iteratively projected across its rows and columns until the procedure converges. In phases two and three, t and e are updated using coordinate descent. Then 13m is increased and the loop repeats. In phase one m is updated with softmax: exp( -13m "Xij tia - R(eia ) . Yak 112) miajk = Lk' exp( -13m IIXij - tia - R(eia) . Yak/112) Then m is iteratively normalized across j and k until Ljk t:t.miajk < f : miajk miajk = =-~- '1\'., m· .I k L.JJ ,aJ Using coordinate descent e is calculated in phase two: And t in phase three: Finally 13m is increased and the loop repeats. Clustering with a Domain-Specific Distance Measure 101 By setting the partial derivatives of (2) to zero and initializing I-lJ and v2 to zero, the algorithm for phase one may be derived. Phases two and three may be derived by taking the partial derivative of (2) with respect to 0, setting it to zero, solving for 0, and then solving for the fixed point of the vector (tl, t2). Beginning with a small 13m allows minimization over a fuzzy correspondence matrix m, for which a global minimum is easier to find. Raising 13m drives the m's closer to 0 or 1, as the algorithm approaches a saddle point. 3.3 Outer Loop The outer loop also proceeds in three phases: (1) distances are calculated by calling the inner loop, (2) M is projected across a using the softmaxfunction, (3) coordinate descent is used to update Y . Therefore, using softmax M is updated in phase two: exp( -13M Ljk miajkllXij - tia - R(0ia) . Yak112) Mia = ~----------~------~----------~~~----~7 La' exp( -13M Ljk mia' jk IIXij - tia, - R(0ia,) . Ya, k 112) Y, in phase three is calculated using coordinate descent: Li Mia Lj miajk( cos 0 ia (Xij 1 - tiad + sin 0ia(Xij2 - tia2)) Li Mia Lj miaj k Li Mia Lj miajk( - sin 0ia(Xijl - tiad + cos 0ia(Xij2 - tia2)) Yak2 Li Mia Ej miajk Then 13M is increased and the loop repeats. 4 Methods and Experimental Results In two experiments (Figures la and Ib) 16 and 100 randomly generated images of 15 and 20 points each are clustered into 4 and 10 clusters, respectively. A stochastic model, formulated with essentially the same visual grammar used to derive the clustering algorithm (Mjolsness, 1992), generated the experimental data. That model begins with the cluster centers and then applies probabilistic transformations according to the rules laid out in the grammar to produce the images. These transformations are then inverted to recover cluster centers from a starting set of images. Therefore, to test the algorithm, the same transformations are applied to produce a set of images, and then the algorithm is run in order to see if it can recover the set of cluster centers, from which the images were produced. First, n = 10 points are selected using a uniform distribution across a normalized square. For each of the n = 10 points a model prototype (cluster center) is created by generating a set of k = 20 points uniformly distributed across a normalized square centered at each orginal point. Then, m = 10 new images consisting of k = 20 points each are generated from each model prototype by displacing all k model points by a random global translation, rotating all k points by a random global rotation within a 54° arc, and then adding independent noise to each of the translated and rotated points with a Gaussian distribution of variance (1"2. 102 Gold, Mjolsness, and Rangarajan 10 0.2 0.' 0.6 0.1 j t t 1 . 2 1.' ,. 10 t j I Figure 1: (a): 16 images, 15 points each (b):100 images, 20 points each t The p = n x m = 100 images so generated is the input to the algorithm. The algorithm, which is initially ignorant of cluster membership information, computes n = 10 cluster centers as well as n x p = 1000 match variables determining the cluster membership of each point image. u is varied and for each u the average distance of the computed cluster centers to the theoretical cluster centers (i.e. the original n = 10 model prototypes) is plotted. Data (Figure 1a) is generated with 20 random seeds with constants of n = 4, k = 15, m = 4, p = 16, varying u from .02 to .14 by increments of .02 for each seed. This produces 80 model prototype-computed cluster center distances for each value of u which are then averaged and plotted, along with an error bar representing the standard deviation of each set. 15 random seeds (Figure 1 b) with constants of n = 10, k = 20, m = 10, p = 100, u varied from .02 to .16 by increments of .02 for each seed, produce 150 model prototype-computed cluster center distances for each value of u. The straight line plotted on each graph shows the expected model prototype-cluster center distances, b = ku / vn, which would be obtained if there were no translation or rotation for each generated image, and if the cluster memberships were known. It can be considered a lower bound for the reconstruction performance of our algorithm. Figures 1a and 1 b together summarize the results of 280 separate clustering experiments. For each set of images the algorithm was run four times, varying the initial randomly selected starting cluster centers each time and then selecting the run with the lowest energy for the results. The annealing rate for 13M and 13m was a constant factor of 1.031. Each run of the algorithm averaged ten minutes on an Indigo SGI workstation for the 16 image test, and four hours for the 100 image test. The running time of the algorithm is O(pnk2). Parallelization, as well as hierarchical and attentional mechanisms, all currently under investigation, can reduce these times. 5 Summary By incorporating a domain-specific distance measure instead of the typical generic distance measures, the new method of unsupervised learning substantially reduces the amount of ad-hoc pre-processing required in conventional techniques. Critical features of a domain (such as invariance under translation, rotation, and permuClustering with a Domain-Specific Distance Measure 103 tation) are captured within the clustering procedure, rather than reflected in the properties of feature sets created prior to clustering. The distance measure and learning problem are formally described as nested objective functions. We derive an efficient algorithm by using optimization techniques that allow us to divide up the objective function into parts which may be minimized in distinct phases. The algorithm has accurately recreated 10 prototypes from a randomly generated sample database of 100 images consisting of 20 points each in 120 experiments. Finally, by incorporating permutation invariance in our distance measure, we have a technique that we may be able to apply to the clustering of graphs. Our goal is to develop measures which will enable the learning of objects with shape or structure. Acknowledgements This work has been supported by AFOSR grant F49620-92-J-0465 and ONR/DARPA grant N00014-92-J-4048. References R. Hathaway. (1986) Another interpretation of the EM algorithm for mixture distributions. Statistics and Probability Letters 4:53:56. D. Huttenlocher, G. Klanderman and W. Rucklidge. (1993) Comparing images using the Hausdorff Distance. Pattern Analysis and Machine Intelligence 15(9):850:863. A. L. Yuille and J.J. Kosowsky. (1992). Statistical physics algorithms that converge. Technical Report 92-7, Harvard Robotics Laboratory. M.l. Jordan and R.A. Jacobs. (1993). Hierarchical mixtures of experts and the EM algorithm. Technical Report 9301, MIT Computational Cognitive Science. C. P. Lu and E. Mjolsness. (1994). Two-dimensional object localization by coarseto-fine correlation matching. In this volume, NIPS 6 . C. von der Malsburg. (1988) . Pattern recognition by labeled graph matching. Neural Networks,1:141:148. E. Mjolsness and W. Miranker. (1993). Greedy Lagrangians for neural networks: three levels of optimization in relaxation dynamics. Technical Report 945, Yale University, Department of Computer Science. E. Mjolsness. Visual grammars and their neural networks. (1992) SPIE Conference on the Science of Artificial Neural Networks, 1710:63:85. C. Peterson and B. Soderberg. A new method for mapping optimization problems onto neural networks. (1989) International Journal of Neural Systems,I(1):3:22. P. Simard, Y. Le Cun, and J. Denker. Efficient pattern recognition using a new transformation distance. (1993). In S. Hanson, J . Cowan, and C. Giles, (eds.), NIPS 5 . Morgan Kaufmann, San Mateo CA.
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Emergence of Global Structure from Local Associations Thea B. Ghiselli-Crippa Department of Infonnation Science University of Pittsburgh Pittsburgh PA 15260 Paul W. Munro Department of Infonnation Science University of Pittsburgh Pittsburgh PA 15260 ABSTRACT A variant of the encoder architecture, where units at the input and output layers represent nodes on a graph. is applied to the task of mapping locations to sets of neighboring locations. The degree to which the resuIting internal (i.e. hidden unit) representations reflect global properties of the environment depends upon several parameters of the learning procedure. Architectural bottlenecks. noise. and incremental learning of landmarks are shown to be important factors in maintaining topographic relationships at a global scale. 1 INTRODUCTION The acquisition of spatial knowledge by exploration of an environment has been the subject of several recent experimental studies. investigating such phenomena as the relationship between distance estimation and priming (e.g. McNamara et al .• 1989) and the influence of route infonnation (McNamara et al., 1984). Clayton and Habibi (1991) have gathered data suggesting that temporal contiguity during exploration is an important factor in detennining associations between spatially distinct sites. This data supports the notion that spatial associations are built by a temporal process that is active during exploration and by extension supports Hebb's (1949) neurophysiological postulate that temporal associations underlie mechanisms of synaptic learning. Local spatial infonnation acquired during the exploration process is continuously integrated into a global representation of the environment (cognitive map). which is typically arrived at by also considering global constraints. such as low dimensionality. not explicitly represented in the local relationships. 1101 1102 Ghiselli-Crippa and Munro 2 NETWORK ARCHITECTURE AND TRAINING The goal of this network design is to reveal structure among the internal representations that emerges solely from integration of local spatial associations; in other words. to show how a network trained to learn only local spatial associations characteristic of an environment can develop internal representations which capture global spatial properties. A variant of the encoder architecture (Ackley et al .• 1985) is used to associate each node on a 2D graph with the set of its neighboring nodes. as defined by the arcs in the graph. This 2D neighborhood mapping task is similar to the I-D task explored by Wiles (1993) using an N-2-N architecture. which can be characterized in terms of a graph environment as a circular chain with broad neighborhoods. In the neighborhood mapping experiments described in the following, the graph nodes are visited at random: at each iteration, a training pair (node-neighborhood) is selected at random from the training set. As in the standard encoder task, the input patterns are all or-' thogonal. so that there is no structure in the input domain that the network could exploit in constructing the internal representations; the only information about the structure of the environment comes from the local associations that the network is shown during training. 2.1 N·H·N NETWORKS The neighborhood mapping task was first studied using a strictly layered feed-forward NH-N architecture, where N is the number of input and output units. corresponding to the number of nodes in the environment, and H is the number of units in the single hidden layer. Experiments were done using square grid environments with wrap-around (toroidal) and without wrap-around (bounded) at the edges. The resulting hidden unit representations reflect global properties of the environment to the extent that distances between them correlate with distances between corresponding points on the grid. These two distance measures are plotted against one another in Figure 1 for toroidal and bounded environments. 5x5 Grid 4 Hidden Units U5·,.-----------:"'1 1.4 e:: 1.2 o _ 1.0 .!! e:: • 06 ·0 :Ie:: ~ lIS 0.6 g.~ a:c 0.4 0.2 O.ol---~----"T"""---i o 1 2 3 Grid Distance With wrap-around 5x5 Grid 4 Hidden Units 2.0-,.----------......., e:: 1.5 o iii i·1.0 .,0 !i i"~ a:c 0.5 R"2 = 0.499 : O.O ...... _,....... __ ----r--__ -...-..-4 o 234 Grid Distance 5 No wrap-around Figure 1: Scatterplots of Distances between Hidden Unit Representations vs. Distances between Corresponding Locations in the Grid Environment. Emergence of Global Structure from Local Associations 1103 2.2 N·2·H·N Networks A hidden layer with just two units forces representations into a 2-D space. which matches the dimensionality of the environment. Under this constraint. the image of the environment in the 2-D space may reflect the topological structure of the environment. This conjecture leads to a further conjecture that the 2-D representations will also reveal global relationships of the environment. Since the neighborhoods in a 2-D representation are not linearly separable regions. another layer (H-Iayer) is introduced between the two-unit layer and the output (see Figure 2). Thus. the network has a strictly layered feed-forward N-2-H-N architecture. where the N units at the input and output layers correspond to the N nodes in the environment. two units make up the topographic layer. and H is the number of units chosen for the new layer (H is estimated according to the complexity of the graph). Responses for the hidden units (in both the T- and H-layers) are computed using the hyperbolic tangent (which ranges from -1 to +1). while the standard sigmoid (0 to +1) is used for the output units. to promote orthogonality between representations (Munro. 1989). Instead of the squared error. the cross entropy function (Hinton. 1987) is used to avoid problems with low derivatives observed in early versions of the network. ~ooe@o~oo .~ <: .• ··.· ••• ·.:.:.· •. 1 .: :; .::'.,..: .............. . :' <. :>/<::>. : •. · •. 1 .•. ·•· .•• · .•. · •. :·.· ••. · •. :.· •. · •. : .••. ·.·: •.•. ·! .. · •. i: •. i.\.::.j) ' .. :.'/> .'><' f o 2 3 5 00 6 7 8 oooeooooo Figure 2: A 3x3 Environment and the Corresponding Network. When input unit 3 is activated, the network responds by activating the same unit and all its neighbors. 3 RESULTS 3.1 T·UNIT RESPONSES Neighborhood mapping experiments were done using bounded square grid environments and N-2-H-N networks. After training, the topographic unit activities corresponding to each of the N possible inputs are plotted, with connecting lines representing the arcs from 1104 Ghiselli-Crippa and Munro the environment. Each axis in Figure 3 represents the activity of one of the T-units. These maps can be readily examined to study the relationship between their global structure and the structure of the environment. The receptive fields of the T-units give an alternative representation of the same data: the response of each T-unit to all N inputs is represented by N circles arranged in the same configuration as the nodes in the grid environment. Circle size is proportional to the absolute value of the unit activity; filled circles indicate negative values, open circles indicate positive values. The receptive field represents the T-unit's sensitivity with respect to the environment. ••• ••• oCXX) • • 0 ~c8 . 0 00 000 •••• •••• • ·0 ... 1:8 26~ leoo • 00 . °8 .·0 Figure 3: Representations at the Topographic Layer. Activity plots and receptive fields for two 3x3 grids (left and middle) and a 4x4 grid(right). The two 3x3 cases shown in Figure 3 illustrate alternative solutions that are each locally consistent, but have different global structure. In the first case, it is evident how the first unit is sensitive to changes in the vertical location of the grid nodes, while the second unit is sensitive to their horizontal location. The axes are essentially rotated 45 degrees in the second case. Except for this rotation of the reference axes, both representations captured the global structure of the 3x3 environment. 3.2 NOISE IN THE HIDDEN UNITS While networks tended to fonn maps in the T -layer that reflect the global structure of the environment, in some cases the maps showed correspondences that were less obvious: i.e., the grid lines crossed, even though the network converged. A few techniques have proven valuable for promoting global correspondence between the topographic representations and the environment, including Judd and Munro's (1993) introduction of noise as pressure to separate representations. The noise is implemented as a small probability for Emergence of Global Structure from Local Associations 1105 reversing the sign of individual H-unit outputs. As reported in a previous study (Ghiselli-Crippa and Munro, 1994), the presence of noise causes the network to develop topographic representations which are more separated, and therefore more robust, so that the correct output units can be activated even if one or more of the H-units provides an incorrect output. From another point of view, the noise can be seen as causing the network to behave as if it had an effective number of hidden units which is smaller than the given number H. The introduction of noise as a means to promote robust topographic representations can be appreciated by examining Figure 4, which illustrates the representations of a 5x5 grid developed by a 25-2-20-25 network trained without noise (left) and with noise (middle) (the network was initialized with the same set of small random weights in all cases). Note that the representations developed by the network subject to noise are more separated and exhibit the same global structure as the environment. To avoid convergence problems observed with the use of noise throughout the whole training process, the noise can be introduced at the beginning of training and then gradually reduced over time. A similar technique involves the use of low-level noise injected in the T-Iayer to directly promote the formation of well-separated representations. Either Gaussian or uniform noise directly added to the T-unit outputs gives comparable results. The use of noise in either hidden layer has a beneficial influence on the formation of globally consistent representations. However. since the noise in the H-units exerts only an indirect influence on the T -unit representations, the choice of its actual value seems to be less crucial than in the case where the noise is directly applied at the T-Iayer. The drawback for the use of noise is an increase in the number of iterations required by the network to converge, that scales up with the magnitude and duration of the noise. Figure 4: Representations at the Topographic Layer. Training with no noise (left) and with noise in the hidden units (middle); training using landmarks (right). 3.3 LANDMARK LEARNING Another effective method involves the organization of training in 2 separate phases, to model the acquisition of landmark information followed by the development of route and/or survey knowledge (Hart and Moore, 1973; Siegel and White, 1975). This method is implemented by manipulating the training set during learning, using coarse spatial resolution at the outset and introducing interstitial features as learning progresses to the second phase. The first phase involves training the network only on a subset of the possible 1106 Ghiselli-Crippa and Munro N patterns (landmarks). Once the landmarks have been learned. the remaining patterns are added to the training set. In the second phase. training proceeds as usual with the full set of training patterns; the only restriction is applied to the landmark points. whose topographical representations are not allowed to change (the corresponding weights between input units and T-units are frozen). thus modeling the use of landmarks as stable reference points when learning the details of a new environment. The right pane of Figure 4 illustrates the representations developed for a 5x5 grid using landmark training; the same 25-220-25 network mentioned above was trained in 2 phases. first on a subset of 9 patterns (landmarks) and then on the full set of 25 patterns (the landmarks are indicated as white circles in the activity plot). 3.4 NOISE IN LANDMARK LEARNING The techniques described above (noise and landmark learning) can be combined together to better promote the emergence of well-structured representation spaces. In particular, noise can be used during the first phase of landmark learning to encourage a robust representation of the landmarks: Figure 5 illustrates the representations obtained for a 5x5 grid using landmark training with two different levels of noise in the H-units during the first phase. The effect of noise is evident when comparing the 4 comer landmarks in the right pane of Figure 4 (landmark learning with no noise) with those in Figure 5. With increasing levels of noise. the T-unit activities corresponding to the 4 comer landmarks approach the asymptotic values of + 1 and -1; the activity plots illustrate this effect by showing how the comer landmark representations move toward the comers of T-space, reaching a configuration which provides more resistance to noise. During the second phase of training, the landmarks function as reference points for the additional features of the environment and their positioning in the representational space therefore becomes very important. A well-fonned, robust representation of the landmarks at the end of the first phase is crucial for the fonnation of a map in T-space that reflects global structure, and the use of noise can help promote this. Figure 5: Representations at the Topographic Layer. Landmark training using noise in phase 1: low noise level (left). high noise level (right). 4 DISCUSSION Large scale constraints intrinsic to natural environments. such as low dimensionality, are not necessarily reflected in local neighborhood relations, but they constitute infonnation which is essential to the successful development of useful representations of the environEmergence of Global Structure from Local Associations 1107 ment. In our model, some of the constraints imposed on the network architecture effectively reduce the dimensionality of the representational space. Constraints have been introduced several ways: bottlenecks, noise, and landmark learning; in all cases, these constraints have had constructive influences on the emergence of globally consistent representation spaces. The approach described presents an alternative to Kohonen's (1982) scheme for capturing topography; here, topographic relations emerge in the representational space, rather than in the weights between directly connected units. The experiments described thus far have focused on how global spatial structure can emerge from the integration of local associations and how it is affected by the introduction of global constraints. As mentioned in the introduction, one additional factor influencing the process of acquisition of spatial knowledge needs to be considered: temporal contiguity during exploration. that is. how temporal associations of spatially adjacent locations can influence the representation of the environment. For example, a random type of exploration ("wandering") can be considered. where the next node to be visited is selected at random from the neighbors of the current node. Preliminary studies indicate that such temporal contiguity during training reSUlts in the fonnation of hidden unit representations with global properties qualitatively similar to those reported here. Alternatively, more directed exploration methods can be studied. with a systematic pattern guiding the choice of the next node to be visited. The main purpose of these studies will be to show how different exploration strategies can affect the formation and the characteristics of cognitive maps of the environment. Higher order effects of temporal and spatial contiguity can also be considered. However, in order to capture regularities in the training process that span several exploration steps. simple feed-forward networks may no longer be sufficient; partially recurrent networks (Elman, 1990) are a likely candidate for the study of such processes. Acknowledgements We wish to thank Stephen Hirtle, whose expertise in the area of spatial cognition greatly benefited our research. We are also grateful for the insightful comments of Janet Wiles. References D. H. Ackley. G. E. Hinton, and T. J. Sejnowski (1985) "A learning algorithm for Boltzmann machines," Cognitive Science, vol. 9. pp. 147-169. K. Clayton and A. Habibi (1991) "The contribution of temporal contiguity to the spatial priming effect," Journal of Experimental Psychology: Learning, Memory, and Cognition. vol. 17, pp. 263-27l. J. L. Elman (1990) "Finding structure in time," Cognitive Science, vol. 14, pp. 179211. T. B. Ghiselli-Crippa and P. W. Munro (1994) "Learning global spatial structures from local associations," in M. C Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman, and A. S. Weigend (Eds.), Proceedings of the 1993 Connectionist Models Summer School, Hillsdale, NJ: Erlbaum. 1108 Ghiselli-Crippa and Munro R. A. Hart and G. T. Moore (1973) "The development of spatial cognition: A review," in R. M. Downs and Stea (Eds.), Image and Environment, Chicago, IL: Aldine. D. O. Hebb (1949) The Organization of Behavior, New York, NY: Wiley. G. E. Hinton (1987) "Connectionist learning procedures," Technical Report CMU-CS87-115, version 2, Pittsburgh, PA: Carnegie-Mellon University, Computer Science Department. S. Judd and P. W. Munro (1993) "Nets with unreliable hidden nodes learn error-correcting codes," in C. L. Giles, S. J. Hanson, and J. D. Cowan, Advances in Neural Information Processing Systems 5, San Mateo, CA: Morgan Kaufmann. T. Kohonen (1982) "Self-organized fonnation of topological correct feature maps," Biological Cybernetics, vol. 43, pp. 59-69. T. P. McNamara, J. K. Hardy, and S. C. Hirtle (1989) "Subjective hierarchies in spatial memory," Journal of Experimental Psychology: Learning, Memory, and Cognition, vol. 15, pp. 211-227. T. P. McNamara, R. Ratcliff, and G. McKoon (1984) "The mental representation of knowledge acquired from maps," Journal of Experimental Psychology: Learning, Memory, and Cognition, vol. 10, pp. 723-732. P. W. Munro (1989) "Conjectures on representations in backpropagation networks," Technical Report TR-89-035, Berkeley, CA: International Computer Science Institute. A. W. Siegel and S. H. White (1975) "The development of spatial representations of large-scale environments," in H. W. Reese (Ed.), Advances in Child Development and Behavior, New York, NY: Academic Press. J. Wiles (1993) "Representation of variables and their values in neural networks," in Proceedings of the Fifteenth Annual Conference of the Cognitive Science Society, Hillsdale, NJ: Erlbaum.
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SPEAKER RECOGNITION USING NEURAL TREE NETWORKS Kevin R. Farrell and Richard J. Marnrnone CAIP Center, Rutgers University Core Building, Frelinghuysen Road Piscataway, New Jersey 08855 Abstract A new classifier is presented for text-independent speaker recognition. The new classifier is called the modified neural tree network (MNTN). The NTN is a hierarchical classifier that combines the properties of decision trees and feed-forward neural networks. The MNTN differs from the standard NTN in that a new learning rule based on discriminant learning is used, which minimizes the classification error as opposed to a norm of the approximation error. The MNTN also uses leaf probability measures in addition to the class labels. The MNTN is evaluated for several speaker identification experiments and is compared to multilayer perceptrons (MLPs), decision trees, and vector quantization (VQ) classifiers. The VQ classifier and MNTN demonstrate comparable performance and perform significantly better than the other classifiers for this task. Additionally, the MNTN provides a logarithmic saving in retrieval time over that of the VQ classifier. The MNTN and VQ classifiers are also compared for several speaker verification experiments where the MNTN is found to outperform the VQ classifier. 1 INTRODUCTION Automatic speaker recognition consists of having a machine recognize a person based on his or her voice. Automatic speaker recognition is comprised of two categories: speaker identification and speaker verification. The objective of speaker identification is to identify a person within a fixed population based on a test utterance from that person. This is contrasted to speaker verification where the objective is to verify a person's claimed identity based on the test utterance. 1035 1036 Farrell and Mammone Speaker recognition systems can be either text dependent or text independent. Text-dependent speaker recognition systems require that the speaker utter a specific phrase or a given password. Text-independent speaker identification systems identify the speaker regardless of the utterance. This paper focuses on text-independent speaker identification and speaker verification tasks. A new classifier is introduced and evaluated for speaker recognition. The new classifier is the modified neural tree network (MNTN). The MNTN incorporates modifications to the learning rule of the original NTN [1] and also uses leaf probability measures in addition to the class labels. Also, vector quantization (VQ) classifiers, multilayer perceptrons (MLPs), and decision trees are evaluated for comparison. This paper is organized as follows. Section 2 reviews the neural tree network and discusses the modifications. Section 3 discusses the feature extraction and classification phases used here for text-independent speaker recognition. Section 4 describes the database used and provides the experimental results. The summary and conclusions of the paper are given in Section 5. 2 THE MODIFIED NEURAL TREE NETWORK The NTN [1] is a hierarchical classifier that uses a tree architecture to implement a sequential linear decision strategy. Each node at every level of the NTN divides the input training vectors into a number of exclusive subsets of this data. The leaf nodes of the NTN partition the feature space into homogeneous subsets, i.e., a single class at each leaf node. The NTN is recursively trained as follows. Given a set of training data at a particular node, if all data within that node belongs to the same class, the node becomes a leaf. Otherwise, the data is split into several subsets, which become the children of this node. This procedure is repeated until all the data is completely uniform at the leaf nodes. For each node the NTN computes the inner product of a weight vector wand an input feature vector x, which should be approximately equal to the the output label y E {O,1}. Traditional learning algorithms minimize a norm of the error € = (y- < w, x », such as the L2 or L1 norm. The splitting algorithm of the modified NTN is based on discriminant learning [2]. Discriminant learning uses a cost function that minimizes the classification error. For an M class NTN, the discriminant learning approach first defines a misclassification measure d( x) as [2]: 1 d( x) = - < Wi, X > + { M ~ 1 I) < Wj, x > t } n , jf;i (1) where n is a predetermined smoothing constant. If x belongs to class i, then d(x) will be negative, and if x does not belong to class i, d( x) will be positive. The misclassification measure d( x) is then applied to a sigmoid to yield: 1 g[d(x)] = d( ). (2) 1 + ex The cost function in equation (2) is approximately zero for correct classifications and one for misclassifications. Hence, minimizing this cost function will tend to o~/ ............. LABEL=O CONFIDENCE c 1.0 .--;r~ 1 \ o ... 1 .... o 1 0 \ 1 '. '. LABEL= 1 CONFIDENCE = 0.6 Speaker Recognition Using Neural Tree Networks 1037 , \··,~,o 0 0 \ 0 1 0./ , 1 ~/ .', ..... LABEL = 0 CONFIDENCE c 0.8 \1 ... 0 '. 1 LABEL = 1 ... 1 " ... 1 \\. CONFIDENCE = 0.7 Figure 1: Forward Pruning and Confidence Measures mmlmize the classification error. It is noted that for binary NTNs, the weight updates obtained by the discriminant learning approach and the Ll norm of the error are equivalent [3]. The NTN training algorithm described above constructs a tree having 100% performance on the training set. However, an NTN trained to this level may not have optimal generalization due to overtraining. The generalization can be improved by reducing the number of nodes in a tree, which is known as pruning. A technique known as backward pruning was recently proposed [1] for the NTN. Given a fully grown NTN, i.e., 100% performance on the training set, the backward pruning method uses a Lagrangian cost function to minimize the classification error and the number of leaves in the tree. The method used here prunes the tree during its growth, hence it is called forward pruning. The forward pruning algorithm consists of simply truncating the growth of the tree beyond a certain level. For the leaves at the truncated level, a vote is taken and the leaf is assigned the label of the majority. In addition to a label, the leaf is also assigned a confidence. The confidence is computed as the ratio of the number of elements for the vote winner to the total number of elements. The confidence provides a measure of confusion for the different regions of feature space. The concept of forward pruning is illustrated in Figure 1. 3 FEATURE EXTRACTION AND CLASSIFICATION The process of feature extraction consists of obtaining characteristic parameters of a signal to be used to classify the signal. The extraction of salient features is a key step in solving any pattern recognition problem. For speaker recognition, the features extracted from a speech signal should be invariant with regard to the desired 1038 Farrell and Mammone Speaker 1 yl i ,----.. (NTN, VQ r--=--. Codebook) Speaker Feature Xi Speaker 2 y2 i Identity (NTN, VQ ..:.-=-. Decision Vector Codebook) or • Authenticity • • Speaker N yN i ~ (NTN, VQ r:----=-Codebook) Figure 2: Classifier Structure for Speaker Recognition speaker while exhibiting a large distance to any imposter. Cepstral coefficients are commonly used for speaker recognition [4] and shall be considered here to evaluate the classifiers. The classification stage of text-independent speaker recognition is typically implemented by modeling each speaker with an individual classifier. The classifier structure for speaker recognition is illustrated in Figure 2. Given a specific feature vector, each speaker model associates a number corresponding to the degree of match with that speaker. The stream of numbers obtained for a set of feature vectors can be used to obtain a likelihood score for each speaker model. For speaker identification, the feature vectors for the test utterance are applied to all speaker models and the corresponding likelihood scores are computed. The speaker is selected as having the largest score. For speaker verification, the feature vectors are applied only to the speaker model for the speaker to be verified. If the likelihood score exceeds a threshold, the speaker is verified or else is rejected. The classifiers for the individual speaker models are trained using either supervised or unsupervised training methods. For supervised training methods the classifier for each speaker model is presented with the data for all speakers. Here, the extracted feature vectors for that speaker are labeled as "one" and the extracted feature vectors for everyone else are labeled as "zero" . The supervised classifiers considered here are the multilayer perceptron (MLP), decision trees, and modified neural tree network (MNTN). For unsupervised training methods each speaker model is presented with only the extracted feature vectors for that speaker. This data can then be clustered to determine a set of centroids that are representative of that speaker. The unsupervised classifiers evaluated here are the full-search and treestructure vector quantization classifiers, henceforth denoted as FSVQ and TSVQ. Speaker models based on supervised training capture the differences of that speaker to other speakers, whereas models based on unsupervised training use a similarity measure. Specifically, a trained NTN can be applied to speaker recognition as follows. Given a sequence of feature vectors x from the test utterance and a trained NTN for Speaker Recognition Using Neural Tree Networks 1039 speaker Si, the corresponding speaker score is found as the "hit" ratio: (3) Here. M is the number of vectors classified as "one" and N is the number of vectors classified as "zero" . The modified NTN computes a hit ratio weighed by the confidence scores: ",M 1 L..Jj=l Cj PMNTN(xISi) = ",N 0 ",M l' L..JJ=l Cj + L..JJ=l cj (4) where c1 and cO are the confidence scores for the speaker and antispeaker, respectively. These scores can be used for decisions regarding identification or verification. 4 EXPERIMENTAL RESULTS 4.1 Database The database considered for the speaker identification and verification experiments is a subset of the DARPA TIMIT database. This set represents 38 speakers of the same (New England) dialect. The preprocessing of the TIMIT speech data consists of several steps. First, the speech is downsampled from 16KHz to 8 KHz sampling frequency. The downsampling is performed to obtain a toll quality signal. The speech data is processed by a silence removing algorithm followed by the application of a pre-emphasis filter H(z) = 1-0.95z-1 . A 30 ms Hamming window is applied to the speech every 10 ms. A twelfth order linear predictive (LP) analysis is performed for each speech frame. The features consist of the twelve cepstral coefficients derived from this LP polynomial. There are 10 utterances for each speaker in the selected set. Five of the utterances are concatenated and used for training. The remaining five sentences are used individually for testing. The duration of the training data ranges from 7 to 13 seconds per speaker and the duration of each test utterance ranges from 0.7 to 3.2 seconds. 4.2 Speaker Identification The first experiment is for closed set speaker identification using 10 and 20 speakers from the TIMIT New England dialect. The identification is closed set in that the speaker is assumed to be one of the 10 or 20 speakers, i.e., no "none of the above" option. The NTN, MLP [5], and VQ [4] classifiers are each evaluated on this data in addition to the ID3 [6], C4 [7], CART [8], and Bayesian [9] decision trees. The VQ classifier is trained using a K-means algorithm and tested for codebook sizes varying from 16 to 128 centroids. The MNTN used here is pruned at levels ranging from the fourth through seventh. The MLP is trained using the backpropagation algorithm [10] for architectures having 16, 32, and 64 hidden nodes (within one hidden layer). The results are summarized in Table 1. The * denotes that the CART tree could not be grown for the 20 speaker experiment due to memory limitations. 1040 Farrell and Mammone Classifier ID3 CART C4 Table 1: Speaker Identification Experiments 4.3 Speaker Verification The FSVQ classifier and MNTN are evaluated next for speaker verification. The first speaker verification experiment consists of 10 speakers and 10 imposters (i.e., people not used in the training set). The second speaker verification experiment uses 20 enrolled speakers and 18 imposters. The MNTN is pruned at the seventh level (128 leaves) and the FSVQ classifier has a codebook size of 128 entries. Speaker verification performance can be enhanced by using a technique known as cohort normalization [11]. Traditional verification systems accept a speaker if: p(XII) > T(I), (5) where p( X I I) is the likelihood that the sequence of feature vectors X was generated by speaker I and T( I) is the corresponding likelihood threshold. Instead of using the fixed threshold criteria in equation (5), an adaptive threshold can be used via the likelihood measure: P(XII) T(I). P(XII) > (6) Here, the speaker score is first normalized by the probability that the feature vectors X were generated by a speaker other than I. The likelihood p(XII) can be estimated with the scores of the speaker models that are closest to I, denoted as 1's cohorts [11]. This estimate can consist of a maximum, minimum, average, etc., depending on the classifier used. The threshold for the VQ and MNTN likelihood scores are varied from the point of 0% false acceptance to 0% false rejection to yield the operating curves shown in Figures 3 and 4 for the 10 and 20 speaker populations, respectively. Note that all operating curves presented in this section for speaker verification represent the posterior performance of the classifiers, given the speaker and imposter scores. Here it can be seen that the MNTN and VQ classifiers are both improved by the cohort normalized scores. The equal error rates for the MNTN and VQ classifier are summarized in Table 2. For both experiments (10 and 20 speakers), the MNTN provides better performance than the VQ classifier, both with and without cohort normalization, for most of the operating curve. Speaker Recognition Using Neural Tree Networks 1041 MNTN 10 Speakers 20 Speakers (]) Vl Table 2: Equal error rates for speaker verification Speaker Verification (10 speakers) 0.35r----~-----r---_r_---..___--___,---_, • • I • • - -_ .......... : ................ : ................ : ................ ; ................ ; ............. . · . ~ -+ va ~ · . · : : -. va with cohort . . .. . , ... ..... ~ ....... -... _ ... ' :' ................ : .............. .. : ................ ~ .. " .......... . · . . . . . . . .. . . . . . . . ~ 0.15 CL 0.1 0.05 0.01 0.02 0.03 0.04 0.05 0.06 P(Falsa Accept} Figure 3: Speaker Verification (10 Speakers) Speaker Verification (20 speakers) 0.45,------r----.---_r_---..------,-----, 0.4 ... ........ ... : .......... __ .... ; ............... ; .... .. ..... .. . -. ~ - .- .... ........ : .......... .... . · . . . . · . . . . , . . . . . , . . , . " . . . . 0.35 ........... .... ;. .. .. .... ... .... : ... ..... ....... , .. .... ·· ···· ····:· ····· ··· ······· i······ ······ ·· : ~ : -+ va ~ : : : : - va with cohort : 0.3 i' .... ·· ...... : .......... · .. · .. : ........ · .. · .. ·:·· .. ............ f .............. ·~ .. · .. ·· ....... . n. . . . -. MNTN , . 10.25 ~: .. .. ········j· .. ······ ...... ·f .. · .... ·· .. ·· .. ~· ::·MNI~·1i!~ .. ~~.9n · · ·i ·· · ··· ...... ··· · . . . : : : 0.1 0.05 ... .... --.~ ........ . -_ ... , .;. . ..... . .... . .. ~ .............. . · . · . oL-----~~~~~--~--~~====~~--~ o 0.02 0.04 0.06 0.08 0.1 0.12 P(Faisa Accept) Figure 4: Speaker Verification (20 Speakers) 1042 Farrell and Mammone 5 CONCLUSION A new classifier called the modified NTN is examined for text-independent speaker recognition. The performance of the MNTN is evaluated for several speaker recognition experiments using various sized speaker sets from a 38 speaker corpus. The features used to evaluate the classifiers are the LP-derived cepstrum. The MNTN is compared to full-search and tree-structured VQ classifiers, multi-layer perceptrons, and decision trees. The FSVQ and MNTN classifiers both demonstrate equivalent performance for the speaker identification experiments and outperform the other classifiers. For speaker verification, the MNTN consistently outperforms the FSVQ classifier. In addition to performance advantages for speaker verification, the MNTN also demonstrates a logarithmic saving in retrieval time over that of the FSVQ classifier. This computational advantage can be obtained by using TSVQ, although TSVQ will reduce the performance with respect to FSVQ. 6 ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of Rome Laboratories, Contract No. F30602-91-C-OI20. The decision tree simulations utilized the IND package developed by W. Buntine of NASA. References [1] A. Sankar and R.J. Mammone. Growing and pruning neural tree networks. IEEE Transactions on Computers, C-42:221-229, March 1993. [2] S. Katagiri, B.H J uang, and A. Biemo Discriminative feature extraction. In Artificial Neural Networks for Speech and Vision Processing, edited by R.J. Mammone. Chapman and Hall, 1993. [3] K.R. Farrell. Speaker Recognition Using the Modified Neural Tree Network. PhD thesis, Rutgers University, Oct. 1993. [4] F.K. Soong, A.E. Rosenberg, L.R. Rabiner, and B.H. Juang. A vector quantization approach to speaker recognition. In Proceedings ICASSP, 1985. [5] J. Oglesby and J .S. Mason. Optimization of neural models for speaker identification. In Proceedings ICASSP, pages 261-264, 1990. [6] J. Quinlan. Induction of decision trees. Machine Learning, 1:81-106, 1986. [7] J. Quinlan. Simplifying decision trees in Knowledge Acquisition for K now ledgeBased Systems, by G. Gaines and J. Boose. Academic Press, London, 1988. [8] 1. Breiman, J .H. Friedman, R.A. Olshen, and C.J. Stone. Classification and Regression Trees. Wadsworth international group, Belmont, CA, 1984. [9] W. Buntine. Learning classification trees. Statistics and Computing, 2:63-73, 1992. [10] D.E. Rumelhart and J .L. McClelland. Parallel Distributed Processing. MIT Cambridge Press, Cambridge, Ma, 1986. [11] A.E. Rosenberg, J. Delong, C.H. Lee, B.H. Juang, and F.K. Soong. The use of cohort normalized scores for speaker recognition. In Proc. ICSLP, Oct. 1992.
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Tonal Music as a Componential Code: Learning Temporal Relationships Between and Within Pitch and Timing Components Janet Wiles Catherine Stevens Department of Psychology University of Queensland QLD 4072 Australia kates@psych.psy.uq.oz.au Depts of Psychology & Computer Science University of Queensland Abstract QLD 4072 Australia janetw@cs.uq.oz.au This study explores the extent to which a network that learns the temporal relationships within and between the component features of Western tonal music can account for music theoretic and psychological phenomena such as the tonal hierarchy and rhythmic expectancies. Predicted and generated sequences were recorded as the representation of a 153-note waltz melody was learnt by a predictive, recurrent network. The network learned transitions and relations between and within pitch and timing components: accent and duration values interacted in the development of rhythmic and metric structures and, with training, the network developed chordal expectancies in response to the activation of individual tones. Analysis of the hidden unit representation revealed that musical sequences are represented as transitions between states in hidden unit space. 1 INTRODUCTION The fundamental features of music, derivable from frequency, time and amplitude dimensions of the physical signal, can be described in terms of two systems - pitch and timing. The two systems are frequently disjoined and modeled independently of one another (e.g. Bharucha & Todd, 1989; Rosenthal, 1992). However, psychological evidence suggests that pitch and timing factors interact (Jones, 1992; Monahan, Kendall 1085 1086 Stevens and Wiles & Carterette, 1987). The pitch and timing components can be further divided into tone, octave, duration and accent which can be regarded as a quasi-componential code. The important features of a componential code are that each component feature can be viewed as systematic in its own right (Fodor & Pylyshyn, 1988). The significance of componential codes for learning devices lies in the productivity of the system - a small (polynomial) number of training examples can generalise to an exponential test set (Brousse & Smolensky, 1989; Phillips & Wiles, 1993). We call music a quasicomponential, code as there are significant interactions between, as well as within, the component features (we adopt this term from its use in describing other cognitive phenomena, such as reading, Plaut & McClelland, 1993). Connectionist models have been developed to investigate various aspects of musical behaviour including composition (Hild, Feulner & Menzel, 1992), performance (Sayegh, 1989), and perception (Bharucha & Todd, 1989). The models have had success in generating novel sequences (Mozer, 1991), or developing properties characteristic of a listener, such as tonal expectancies (Todd, 1988), or reflecting properties characteristic of musical structure, such as hierarchical organisation of notes, chords and keys (Bharucha, 1992). Clearly, the models have been designed with a specific application in mind and, although some attention has been given to the representation of musical information (e.g. Mozer, 1991; Bharucha, 1992), the models rarely explain the way in which musical representations are constructed and learned. These models typically process notes which vary in pitch but are of constant duration and, as music is inherently temporal, the temporal properties of music must be reflected in both representation and processing. There is an assumption implicit in cognitive modeling in the in/ormation processing framework that the representations used in any cognitive process are specified a priori (often assumed to be the output of a perceptual process). In the neural network framework, this view of representation has been challenged by the specification of a dual mechanism which is capable of learning representations and the processes which act upon them. Since the systematic properties of music are inherent in Western tonal music as an environment, they must also be reflected in its representation in, and processes of, a cognitive system. Neural networks provide a mechanism for learning such representations and processes, particularly with respect to temporal effects. In this paper we study how representations can be learned in the domain of Western tonal music. Specifically, we use a recurrent network trained on a musical prediction task to construct representations of context in a musical sequence (a well-known waltz), and then test the extent to which the learned representation can account for the classic phenomena of music cognition. We see the representation construction as one aspect of learning and memory in music cognition, and anticipate that additional mechanisms of music cognition would involve memory processes that utilise these representations (e.g. Stevens & Latimer, 1992), although they are beyond the scope of the present paper. For example, Mozer (1992) discusses the development of higher-order, global representations at the level of relations between phrases in musical patterns. By contrast, the present study focusses on the development of representations at the level of relations between individual musical events. We expect that the additional mechanisms would, in part, develop from the behavioural aspects of music cognition that are made explicit at this early, representation construction stage. Tonal Music as a Componential Code 1087 2 METHOD & RESULTS A simple recurrent network (Elman. 1989), consisting of 25 input and output units, and 20 hidden and context units. was trained according to Elman's prediction paradigm and used Backprop Through Time (BPlT) for one time step. The training data comprised the nrst 2 sections of The Blue Danube by Johann Strauss wherein each training pattern represented one event, or note. in the piece, coded as components of tone (12 values), rest (1 value), octave (3 values), duration (6 values) and accent (3 values). In the early stages of training, the network learned to predict the prior probability of events (see Figure 1). This type of information could be encoded in the bias of the output units alone. as it is independent of temporal changes in the input patterns. After further training, the network learned to modify its predictions based on the input event (purely feed forward information) and, later still, on the context in which the event occurred (see Figure 2). Note that an important aspect of this type of recurrent network is that the representation of context is created by the network itself (Elman, 1989) and is not specified a priori by the network designer or the environment (Jordan, 1986). In this way, the context can encode the information in the envirorunent which is relevant to the prediction task. Consequently, the network could, in principle, be adapted to other styles of music without modification of the design, or input and output representations. a. Output vector n, __ ~--'"'-_---'"~_~~_----' L- _JLl ~ 1-' _~ _ _ b. Target histogram A A# B C C# D D# E F F# G G# 4 5 6 1 2 3 4 6 8 S W V A tones octaves durations accent rest Figure 1: Comparison of output vector for the first event at Epoch 4 (a) and a histogram of the target vector averaged over all events (b). The upper grapb is the predicted output of Event 1 at Epoch 4. The lower grapb is a histogram of all the events in the piece, created by averaging all the target vectors. The comparison shows that the net learned initially to predict the mean target before learning the variations specific to each event. 1088 Stevens and Wiles ---.J1 ~ _______ --J,...'-___ --.J1 -11 ___ _ _________________ ~ ~ 1 ~ ___________ ----JI"L-r-I1 --11 ..... , ___ _ ___________ '"'-----I'L-.n ---I1--....,. __________ ---'~~._J1 ______ _ ________ ~n~ ____ __ --...lL-.-Jl ______ _ __ ~n ..... __________ _ --11..--..11'--__ _ =E~~~h~.~ _____ ~ _____ ~ ~ ---fl~_~_ ---------------~ ___ .11 -...J1'-___ _ ---------------'"'--- -.Jl~ -------------------~ _---n --.fl'--___ _ ---------------~~ --I"\-.J"1 ~ -..--J"L..r, -..I1 ___ _ ~ --..11 ___ _ ~ ---11 ____ _ =E~~=h~2~ ______ ~ ____ ~'"'___ ---l1 ----11'--__ _ ______________ ~n__ ---.J1 -11'--__ _ ----~-----~-----''"'--- ---l1 ~ _______ ~ ____ ~n__ ---J1 -1l'--__ _ ______________ ~n__ --..J'l --1'1'--__ _ -----------~~ __ J'~_n -..I1'--__ _ -------------~~ --n --11'---___ _ .. ~ --"''----T..:..;~=-= ____ ~n,--____ ---.J1 -11'--__ _ _______ --lnl.-____ ---.J1 -11 _____ _ __________ 11~ ---1L-__________ n.---. ---Il -Il ______ _ __________ fL.---1L- ---1L-__________ fL.----I"L--Il ___ _ _______ ~n'_ ___ _ ---I"L--..11 ______ _ __ --In'_ _________ _ ~"'L- -11 ______ _ 11.._ --11rL_ -.11___ _ il_ -11 __ -1L_ I"L.._ M-.-.!'L.. -1L.. ~ .. -...1'l.. ~ .. ~ .. -..I1.. --J'L.... 8 7 6 5 • 3 2 1 8 7 6 5 • 3 2 1 8 7 6 5 • 3 2 1 8 7 6 5 • 3 2 1 A M B C CI 0 Of E F F. Gat. 5 6 1 2 3 • 6 8 S W V R event tones octaves durations accent rest Figure 2: Evolution of the flrst eight events (predicted). The first block (targets) shows the correct sequence of events for the four components. In the second block (Epoch 2), the net is beginning to predict activation of strong and weak accents. In the third block (Epoch 4), the transition from one octave to another is evident. By the fourth block (Epoch 64), all four components are substantially correct. The pattern of activation across the output vector can be interpreted as a statistical description of each musical event. In psychological terms, the pattern of activation reflects the harmonic or chordal expectancies induced by each note (Bharucha, 1987) and characteristic of the tonal hierarchy (Krumhansl & Shepard, 1979). 3.1 NETWORK PERFORMANCE The performance of the network as it learned to master The Blue Danube was recorded at log steps up to 4096 epochs. One recording comprised the output predicted by the network as the correct event was recycled as input to the network (similar to a teacherforcing paradigm). The second recording was generated by feeding the best guess of the Tonal Music as a Componential Code 1089 output back as input. The accent - the lilt of the waltz - was incorporated very early in the training. Despite numerous errors in the individual events, the sequences were clearly identifiable as phrases from The Blue Danube and, for the most part, errors in the tone component were consistent with the tonality of the piece. The errors are of psychological importance and the overall performance indicated that the network learnt the typical features of The Blue Danube and the waltz genre. 3.2 INTERACTIONS BETWEEN ACCENT & DURATION Western tonal music is characterised by regularities in pitch and timing components. For example, the occurrence of particular tones and durations in a single composition is structured and regular given that only a limited number of the possible combinations occur. Therefore, one way to gauge performance of the network is to compare the regularities extracted and represented in the model with the statistical properties of components in the training composition. The expected frequencies of accent-duration pairs, such as a quarter-note coupled with a strong accent, were compared with the actual frequency of occurrence in the composition: the accent and duration couplings with the highest expected frequencies were strong quarter-note (35.7), strong half-note (10.2), weak quarter-note (59.0), and weak half-note (16.9). Scrutiny of the predicted outputs of the network over the time course of learning showed that during the initial training epochs there was a strong bias toward the event with the highest expected frequency - weak quarter-note. The output of this accent-duration combination by the network decreased gradually. Prediction of a strong quarter-note by the network reached a value close to the expected frequency of 35.7 by Epoch 2 (33) and then decreased gradually and approximated the actual composition frequency of 19 at Epoch 64. Similarly, by Epoch 64, output of the most common accent and duration pairs was very close to the actual frequency of occurrence of those pairs in the composition. 3. 3 ANALYSIS OF HIDDEN UNIT ~EPRESENT A TIONS An analysis of hidden unit space most often reveals structures such as regions, hierarchies and intersecting regions (Wiles & Bloesch, 1992). In the present network, four subspaces would be expected (tone, octave, duration, accent), with events lying at the intersection of these suo-spaces. A two-dimensional projection of hidden unit space produced from a canonical discriminant analysis (CDA) of duration-accent pairs reveals these divisions (see Figure 3). In essence, there is considerable structure in the way events are represented into clusters of regions with events located at the intersection of these regions. In Figure 3 the groups used in the CDA relate to the output values which are observable groups. An additional CDA using groups based on position in bar showed that the hidden unit space is structured around inferred variables as well as observable ones. 1090 Stevens and Wiles 21 • I, 2ft ... 2w .. .. .. 2ft .. ~ 1 J I I I I Figure 3: Two-dimensional projection of hidden unit space generated by canonical discriminant analysis of duration-accent pairs. Each note in the composition is depicted as a labelled point, and the flrst and third canonical components are represented along the abscissa and ordinate. respectively. The first canonical component divides strong from weak accents (denoted s and w). In the strong (s) accent region of the third canonical component, quarter- and half-notes are separated (denoted by 2 and 4, respectively), and the remaining right area separates rests, weak quarter- and weak half-notes. The superimposed line shows the first five notes of the opening two bars of the composition as a trajectory through hidden unit space: there is movement along the flrst canonical component depending on accent (s or w) and the second bar starts in the half-note region. 4 DISCUSSION & CONCLUSIONS The focus of this study has been the extraction of information from the envirorunentthe temporal stream of events representing The Blue Danube - and its incorporation into the static parameters of the weights and biases in the network. Evidence for the stages at which information from the environment is incorporated into the network representation is seen in the predicted output vectors (described above and illustrated in Figure 2). Different musical styles contain different kinds of infonnation in the components. For example. the accent and duration components of a waltz take complementary roles in regulating the rhythm. From the durations of events alone, the position of a note in a Tonal Music as a Componential Code 1091 bar could be predicted without error. However, if the performer or listener made a single error of duration, a rhythm system based on durations alone could not recover. By contrast, accent is not a completely reliable predictor of the bar structure, but it is effective for recovery from rhythmic errors. The interaction between these two timing components provides an efficient error correction representation for the rhythmic aspect of the system. Other musical styles are likely to have similar regulatory functions performed by different components. For example, consider the use of ornaments, such as trills and mordents, in Baroque harpsichord music which, in the absence of variations in dynamics, help to signify the beat and metric structure. Alternatively, consider the interaction between pitch and timing components with the placement of harmonicallyimportant tones at accented positions in a bar (Jones, 1992). The network described here has learned transitions and relations between and within pitch and timing musical components and not simply the components per se. The interaction between accent and duration components, for example, demonstrates the manifestation of a componential code in Western tonal music. Patterns of activation across the output vector represented statistical regularities or probabilities characteristic of the composition. Notably, the representation created by the network is reminiscent of the tonal hierarchy which reflects the regularities of tonal music and has been shown to be responsible for a number of performance and memory effects observed in both musically trained and untrained listeners (Krumhansl, 1990). The distribution of activity across the tone output units can also be interpreted as chordal or harmonic expectancies akin to those observed in human behaviour by Bharucha & Stoeckig (1986). The hidden unit activations represent the rules or grammar of the musical environment; an interesting property of the simple recurrent network is that a familiar sequence can be generated by the trained network from the hidden unit activations alone. Moreover, the intersecting regions in hidden unit space represent composite states and the musical sequence is represented by transitions between states. Finally, the course of learning in the network shows an increasing specificity of predicted events to the changing context: during the early stages of training, the default output or bias of the network is towards the average pattern of activation across the entire composition but, over time, predictions are refined and become attuned to the pattern of events in particular contexts. Acknowledgements This research was supported by an Australian Research Council Postdoctoral Fellowship granted to the first author and equipment funds to both authors from the Departments of Psychology and Computer Science, University of Queensland. The authors wish to thank Michael Mozer for providing the musical database which was adapted and used in the present simulation. The modification to McClelland & Rumelhart's (1989) bp program was developed by Paul Bakker, Department of Computer Science, University of Queensland. The comments and suggestions made by members of the Computer Science/Psychology Connectionist Research Group at the University of Queensland are acknowledged. References Bharucha. J. J. (1987). Music cognition and perceptual facilitation: A connectionist framework. Music Perception, 5, 1-30. Bharucha, J. J. (1992). Tonality and learnability. In M. R. Jones & S. Holleran (Eds.), Cognitive bases of musical communication, pp. 213-223. WaShington: American Psychological Association . 1092 Stevens and Wiles Bharucha, J. J., & Stoeckig, K. (1986). Reaction time and musical expectancy: Priming of chords. Journal of Experimental Psychology: Human Perception & Peiformance, 12, 403410. Bharucha, J., & Todd, P. M. (1989). Modeling the perception of tonal structure with neural nets. Computer Music Journal, 13, 44-53. Brousse, 0., & Smolensky, P. (1989). Virtual memories and massive generalization in connectionist combinatorial learning. In Proceedings of the 11th Annual Conference of the Cognitive Science Society, pp. 380-387. Hillsdale, NJ: Lawrence Erlbaum. Elman, J. L. (1989). Structured representations and connectionist models. (CRL Tech. Rep. No. 8901). San Diego: University of California, Center for Research in Language. Fodor, J. A., & Pylyshyn, Z. W. (1988). Connectionism and cognitive architecture: A critical analysis. Cognition, 28, 3-71. Hild, H., Feulner, J., & Menzel, W . (1992). HARMONET: A neural net for harmonizing chorales in the style of J. S. Bach. In J. E. Moody, S. J. Hanson & R. Lippmann (Eds.), Advances in Neural Information Processing Systems 4, pp. 267-274. San Mateo, CA.: Morgan Kaufmann. Jones, M. R. (1992). Attending to musical events. In M. R. Jones & S. Holleran (Eds.), Cognitive bases of musical communication, pp. 91-110. Washington: American Psychological Association. Jordan, M. I. (1986). Serial order: A parallel distributed processing approach (Tech. Rep. No. 8604). San Diego: University of California, Institute for Cognitive Science. Krumhansl, C., & Shepard, R. N. (1979). Quantification of the hierarchy of tonal functions within a diatonic context. Journal of Experimental Psychology: Human Perception & Performance, 5, 579-594. McClelland, 1. L., & Rumelhart, D. E. (1989). Explorations in parallel distributed processing: A handbook of models, programs and exercises. Cambridge, Mass.: MIT Press. Monahan, C. B., Kendall, R. A ., & Carterette, E. C. (1987). The effect of melodic and temporal contour on recognition memory for pitch change. Perception & Psychophysics, 41, 576-600. Mozer, M. C. (1991). Connectionist music composition based on melodic, stylistic, and psychophysical constraints. In P. M. Todd & D. G. Loy (Eds.), Music and connectionism, pp. 195-211. Cambridge, Mass.: MIT Press. Mozer, M. C. (1992). Induction of multiscale temporal structure. In J. E. Moody, S. J. Hanson, & R. P. Lippmann (Eds.), Advances in Neural Information Processing Systems 4, pp. 275-282. San Mateo, CA.: Morgan Kaufmann. Phillips, S., & Wiles, J. (1993). Exponential generalizations from a polynomial number of examples in a combinatorial domain. Submitted to IlCNN, Japan, 1993. Plaut, D. c., & McClelland, J. L. (1993). Generalization with componential attractors: Word and nonword reading in an attractor network. To appear in Proceedings of the 15th Annual Conference of the Cognitive Science Society. Hillsdale, NJ: Erlbaum. Rosenthal, D. (1992). Emulation of human rhythm perception. Computer Music Journal, 16, 64-76. Sayegh, S. (1989). Fingering for string instruments with the optimum path paradigm. Computer Music Journal, 13, 76-84. Stevens, c., & Latimer, C. (1991). Judgments of complexity and pleasingness in music: The effect of structure, repetition, and training. Australian Journal of Psychology, 43, 17-22. Stevens, C ., & Latimer, C . (1992). A comparison of connectionist models of music recognition and human performance. Minds and Machines, 2, 379-400. Todd, P. M. (1988). A sequential network design for musical applications. In D. Touretzky, G. Hinton & T. Sejnowski (Eds.), Proceedings of the 1988 Connectionist Models Summer School, pp. 76-84. Menlo Park, CA: Morgan Kaufmann. Wiles, J., & Bloesch, A. (1992). Operators and curried functions: Training and analysis of simple recurrent networks. In J. E. Moody, S. J. Hanson, & R . P. Lippmann (Eds.), Advances in Neural Information Processing Systems 4. San Mateo, CA: Morgan Kaufmann.
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Connectionist Models for A uditory Scene Analysis Richard o. Duda Department of Electrical Engineering San Jose State University San Jose, CA 95192 Abstract Although the visual and auditory systems share the same basic tasks of informing an organism about its environment, most connectionist work on hearing to date has been devoted to the very different problem of speech recognition. VVe believe that the most fundamental task of the auditory system is the analysis of acoustic signals into components corresponding to individual sound sources, which Bregman has called auditory scene analysis. Computational and connectionist work on auditory scene analysis is reviewed, and the outline of a general model that includes these approaches is described. 1 INTRODUCTION The primary task of any perceptual system is to tell us about the external world. The primary problem is that the sensory inputs provide too much data and too little information. A perceptual system must glean from the flood of incomplete, noisy, redundant and constantly changing streams of data those invariant properties that reveal important objects and events in the environment. For humans, the perceptual systems with the widest bandwidths are the visual system and the auditory system. There are many obvious similarities and differences between these modalities, and in addition to using them to perceive different aspects of the physical world, we also use them in quite different ways to communicate with one another. 1069 1070 Duda The earliest neural-network models for vision and hearing addressed problems in pattern recognition, with optical character recognition and isolated word recognition among the first engineering applications. However, about twenty years ago the research goals in vision and hearing began to diverge. In particular, the need for computers to perceive the external environment motivated vision researchers to seek the principles and procedures for recovering information about the physical world from visual data (Marr, 1982; Ballard and Brown, 1982). By contrast, the vast majority of work on machine audition remained focused on the communication problem of speech recognition (Morgan and Scofield, 1991; Rabiner and Juang, 1993). While this focus has produced considerable progress, the resulting systems are still not very robust, and perform poorly in uncontrolled environments. Furthermore, as Richards (1988) has noted, " ... Speech, like writing and reading, is a specialized skill of advanced animals, and understanding speech need not be the best route to understanding how we interpret the patterns of natural sounds that comprise most of the acoustic spectrum about us." In recent years, some researchers concerned with modeling audition have begun to shift their attention from speech understanding to sound understanding. The inspiration for much of this activity has come from the work of Bregman, whose book on auditory scene analysis documents experimental evidence for important gestalt principles that summarize the ways that people group elementary events in frequency /time into sound objects or streams (Bregman, 1990). In this survey paper, we briefly review this activity and consider its implications for the development of connectionist models for auditory scene analysis. 2 AUDITORY SCENE ANALYSIS In vision, Marr (1982) emphasized the importance of identifying the tasks of the visual system and developing a computational theory that is distinct from particular algorithms or implementations. The computational theory had to specify the problems to be solved, the sensory data that is available, and the additional knowledge or assumptions required to solve the problems. Among the various tasks of the visual system, Marr believed that the recovery of the three-dimensional shapes of the surfaces of objects from the sensory image data was the most fundamental. The auditory system also has basic tasks that are more primitive than the recognition of speech. These include (1) the separation of different sound sources, (2) the localization of the sources in space (3) the suppression of echoes and reverberation, (4) the decoupling of sources from the environment, (5) the characterization of the sources, and (6) the characterization of the environment. Unfortunately, the relation between physical sound sources and perceived sound streams is not a simple one-to-one correspondence. Distributed sound sources, echoes, and synthetic sounds can easily confuse auditory perception. Nevertheless, humans still do much better at these six basic tasks than any machine hearing system that exists today. From the standpoint of physics, the raw data available for performing these tasks is the pair of acoustic signals arriving at the two ears. From the standpoint of neurophysiology, the raw data is the activity on the auditory nerve. The nonlinear, mechallo-neural spectral analysis performed by the cochlea converts sound pressure fluctuations into auditory nerve firings. For better or for worse, the cochlea Connectionist Models for Auditory Scene Analysis 1071 decomposes the signal into many frequency components, transforming it into a frequency /time (or, more accurately, a place/time) spectrogram-like representation. The auditory system must find the underlying order in this dynamic flow of data. For a specific case, consider a simple musical mixture of several periodic signals. \Vithin its limits of resolution, the cochlea decomposes each individual signal into its discrete harmonic components. Yet, under ordinary circumstances, we do not hear these components as separate sounds, but rather we fuse them into a single sound having, as musicians say, its particular timbre or tone color. However, if there is something distinctive about the different signals (such as different pitch or different modulation), we do not fuse all of the sounds together, but rather hear the separate sources, each with its own timbre. What information is available to group the spectral components into sound streams? Hartmann (1988) identifies the following factors that influence grouping: (1) common onset/offset, (2) common harmonic relations, (3) common modulation, (4) common spatial origin, (5) continuity of spectral envelope, (6) duration, (7) sound pressure level, and (8) context. These properties are easier to name than to precisely specify, and it is not surprising that no current model incorporates them all. However, several auditory scene analysis systems have been built that exploit some subset of these cues (''''eintraub, 1985; Cooke, 1993; Mellinger, 1991; Brown, 1992; Brown and Cooke, 1993; Ellis, 1993). Although these are computational rather than connectionist models, most of them at least find inspiration in the structure of the mammalian auditory system. 3 NEURAL AND CONNECTIONIST MODELS The neural pathways from the cochlea through the brainstem nuclei to the auditory cortex are complex, but have been extensively investigated. Although this system is far from completely understood, neurons in the brainstem nuclei are known to be sensitive to various acoustic features onsets, offsets and modulation in the dorsal cochlear nucleus, interaural time differences (lTD's) in the medial superior olive (MSO), interaural intensity differences (IID's) in the lateral superior olive (LSO), and spatial location maps in the inferior colliculus (Pickles, 1988). Both functional and connectionist models have been developed for all of these functions. Because it is both important and relatively well understood, the cochlea has received by far the most attention (Allen, 1985). As a result of this work, we now have real-time implementations for some of these models as analog VLSI chips (Lyon and Mead, 1988; Lazzaro et al., 1993). Connectionist models for sound localization have also been extensively explored. Indeed, one of the earliest of all neural network models was Jeffress's classic crosscorrelation model (Jeffress, 1948), which was hypothesized forty years before neural crosscorrelation structures were actually found in the barn owl (Carr and Konishi, 1988). Models have subsequently been proposed for both the LSO (Reed and Blum, 1990) and the TvISO (Han and Colburn, 1991). Mathematically, both the lTD and IID cues for binaural localization are exposed by crosscorrelation. Lyon showed that cross correlation can also be used to separate as well as localize the signals (Lyon, 1983). VLSI cross correlation chips can provide this information in real time (Lazzaro and Mead, 1989; Bhadkamkar and Fowler, 1993). 1072 Duda While interaural crosscorrelation can determine the azimuth to a sound source, full three-dimensional localization also requires the determination of elevation and range. Because of a lack of symmetry in the orientation of its ears, the barn owl can actually determine azimuth from the lTD and elevation from the IID. This at least in part explains why it has been such a popular choice for connectionist modeling (Spence et al., 1990; Moiseff et al., 1991; Palmieri et al., 1991; Rosen, Rumelhart and Knudsen, 1993). Unfortunately, the localization mechanisms used by humans are more complicated. It is well known that humans use monaural, spectral shape cues to estimate elevation in the median sagittal plane (Blauert, 1983; Middlebrooks and Green, 1991), and source localization models based on this approach have been developed (Neti, Young and Schneider, 1992; Zakarauskas and Cynander, 1993). The author has shown that there are strong binaural cues for elevation at short distances away from the median plane, and has used statistical methods to estimate both azimuth and elevation accurately from IID data alone (Duda, 1994). In addition, backprop models have been developed that can estimate azimuth and elevation from IID and lTD inputs jointly (Backman and Karjalainen 93; Anderson, Gilkey and Janko, 1994). Finally, psychologists have long been aware of an important reverberationsuppression phenomenon known as the precedence effect or the law of the first wavefront (Zurek, 1987). It is usually summarized by saying that echoes of a sound source have little effect on its localization, and are not even consciously heard if they are not delayed more than the so-called echo threshold, which ranges from 5-10 ms for sharp clicks to more than 50 ms for music. It is generally believed that the precedence effect can be accounted for by contralateral inhibition in the crosscorrelation process, and Lindemann has accounted for many of the phenomena by a conceptually simple connectionist model (Lindemann, 1986). However, Clifton and her colleagues have found that the echoes are indeed heard if the timing of the echoes suddenly changes, as might happen when one moves from one acoustic environment into another one (Clifton 1987; Freyman, Clifton and Litovsky, 1991). Clifton conjectures that the auditory system is continually analyzing echo patterns to model the acoustic environment, and that the resulting expectations modify the echo threshold. This suggests that simple crosscorrelation models will not be adequate when the listener is moving, and thus that even the localization problem is still unsolved. 4 ARCHITECTURE OF AN AUDITORY MODEL If we look back at the six basic tasks for the auditory system, we see that only one (source localization) ha.s received much attention from connectionist researchers, and its solution is incomplete. In particular, current localization models cannot handle multiple sources and cannot cope with significant room echoes and reverberation. The common problem for all of the basic tasks is that of source separation, which only the a.uditory scene analysis systems have addressed. Fig. 1 shows a functional block diagram for a hypothetical auditory model that combines the computational and connectionist models and has the potential of coping with a multisource environment. The inputs to the model are the left-ear Connectionist Models for Auditory Scene Analysis 1073 and right-ear signals, and the main output is a dynamic set of streams. The system is primarily data driven, although low-bandwidth efferent paths could be added for tasks such as automatic gain control. Data flow on the left half of the diagram is monaural, and dataflow of the right half is binaural. The binaural processing is based on crosscorrelation analysis of the cochlear outputs. The author has shown that interaural differences not only effective in determining azimuth, but can also be used to determine elevation as well (Duda, 1994). V\'e have chosen to follow Slaney and Lyon (Slaney and Lyon, 1993) in basing the monaural analysis on autocorrelation analysis. Originally proposed by Licklider (1951) to explain pitch phenomena, autocorrelation is a biologically plausible operation that supports the common onset, modulation and harmonicity analysis needed for stream formation (Duda, Lyon and Slaney, 1990; Brown and Cooke, 1993). While the processes at lower levels of this diagram are relatively well understood, the process of stream formation is problematic. Bregman (1990) has posed this problem in terms of grouping the components of the "neural spectrogram" in both frequency and time. He has identified two principles that seem to be employed in stream formation: exclusive allocation (a component may not be used in more than one description at a time) and accounting (all incoming components must be assigned to some source). The various auditory scene analysis systems that we mentioned earlier provide different mechanisms for exploiting these principles to form auditory streams. Unfortunately, the principles admit of exceptions, and the existing implementations seem rather ad hoc and arbitrary. The development of a biologically plausible model for stream formation is the central unsolved problem for connectionist research in audition. Short· Term Aud~ory Memory Monaural Maps Auto-Correlatlon Analysis Stream Formation Spectral Analysis (Cochlear Model) Left Input Cross-Correiation Analysis Spectral Analysis (Cochlear Model) I Right Input Figure 1: Block diagram for a basic auditory model 1074 Duda Acknowledgements This work was supported by the National Science Foundation under NSF Grant No. IRI-9214233. This paper could not have been written without the many discussions on these topics with Al Bregman, Dick Lyon, David Mellinger, Bernard MontReynaud, John R. Pierce, Malcolm Slaney and J. Martin Tenenbaum, and from the stimulating CCRMA Hearing Seminar at Stanford University that Bernard initiated and that Malcolm has maintained and invigorated. References Allen, J. B. (1985). "Cochlear modeling," IEEE ASSP Magazine, vol. 2, pp. 3-29. Anderson, T. R., R. H. Gilkey and J. A. Janko (1994). "Using neural networks to model human sound localization," in T. Anderson and R. H. Gilkey (eds.), Binaural and Spatial Hearing. Hillsdale, NJ: Lawrence Erlbaum Associates. Backman, J. and M. Karjalainen (1993). "Modelling of human directional and spatial hearing using neural networks," ICASSP93 , pp. 1-125-1-128. (Minneapolis, MN). Bhadkamkar, N. and B. Fowler (1993). "A sound localization system based on biological analogy," 1993 IEEE International Conference on Neural Networks, pp. 1902-1907. (San Francisco, CA). Ballard, D. H. and C. M. Brown (1982). Computer Vision. Englewood Cliffs, NJ: Prentice-Hall. Blauert, J. P. (1983). Spatial Hearing. Cambridge, MA: MIT Press. Bregman, A. S. (1990). Auditory Scene Analysis. Cambridge, MA: MIT Press, 1990. Brown, G. J. (1992). "Computational auditory scene analysis: A representational approach," PhD dissertation, Department of Computer Science, University of Sheffield, Sheffield, England, UK. Brown, G. J. and Iv!. Cooke (1993). "Physiologically-motivated signal representations for computational auditory scene analysis," in M. Cooke, S. Beet and M. Crawford (eds.), Visual Representations of Speech Signals, pp. 181-188. Chichester, England: John Wiley and Sons. Carr, C. E. and M. Konishi (1988). "Axonal delay lines for time measurement in the owl's brainstem," Proc. Nat. Acad. Sci. USA, vol. 85, pp. 8311-8315. Clifton, R. K. (1987). "Breakdown of echo suppression in the precedence effect," J. Acoust. Soc. Am., vol. 82, pp. 1834-1835. Cooke, M. P. (1993). Modelling Auditory Processing and Organisation. Cambridge, UK: Cambridge University Press. Duda, R. 0., R. F. Lyon and M. Slaney (1990). "Correlograms and the separation of sounds," Proc. 24th Asilomar Conf. on Signals, Systems and Computers, pp. 457461 (Asilomar, CA). Connectionist Models for Auditory Scene Analysis 1075 Duda, R. O. (1994). "Elevation dependence of the interaural transfer function," in T. Anderson and R. H. Gilkey (eds.), Binaural and Spatial Hearing. Hillsdale, NJ: Lawrence Erlbaum Associates. Ellis, D. P. VI. (1993). "Hierarchic models of hearing for sound separation and reconstruction," 1993 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics. Freyman, R. L., R. K. Clifton and R. Y. Litovsky (1991). "Dynamic processes in the precedence effect," J. Acoust. Soc. Am., vol. 90, pp. 874-884. Han, Y. and H. S. Colburn (1991). "A neural cell model of MSO," Proc. 1991 IEEE Seventeenth Annual Northeast Bioenginering Conference, pp. 121-122 (Hartford, CT). Hartmann, \V. A. (1988). "Pitch perception and the segregation and integration of auditory entities," in G. M. Edelman, \V. E. Gail and \V. M. Cowan (eds.), Auditory Function. New York, NY: John 'Wiley and Sons, Inc. Jeffress, L. A. (1948). "A place theory of sound localization," J. Compo Physiol. Psychol., vol. 41, pp. 35-39. Lazzaro, J. and C. A. Mead (1989). "A silicon model of auditory localization," Neural Computation, vol. 1, pp. 47-57. Lazzaro, J., J. \Vawrzynek, :M. Mahowald, M. Sivilotti and D. Gillespie (1993). "Silicon auditory processors as computer peripherals," IEEE Transactions on Neural Networks, vol. 4, pp. 523-528. Licklider, J. C. R. (1951). "A duplex theory of pitch perception," Experentia, vol. 7, pp. 128-133. Lindemann, W. (1986). "Extension of a binaural cross-correlation model by contralateral inhibition. I. Simulation of lateralization for stationary signals," J. Acoust. Soc. Am., vol. 80, pp. 1608-1622; II. The law of the first wave front," J. Acoust. Soc. Am., vol. 80, pp. 1623-1630. Lyon, R. F. (1983). "A computational model of binaural localization and separation," ICASSP83 , pp. 1148-1151. (Boston, MA). Lyon, R. F. and C. Mead (1988). "An analog electronic cochlea," IEEE Trans. Acoustics, Speech and Signal Processing, vol. 36, pp. 1119-1134. Marr, D. (1982). Vision. San Francisco, CA: \V. H. Freeman and Company. Mellinger, D. K. (1991). "Event formation and separation of musical sound," PhD dissertation, Department of Music, Stanford University, Stanford, CA; Report No. STAN-M-77, Center for Computer Research in Music and Acoustics, Stanford University, Stanford, CA. Middlebrooks, J. C. and D. M. Green (1991). "Sound localization by human listeners," Annu. Rev. Psychol., vol. 42, pp. 135-159. Moiseff, A. et al. (1991). "An artificial neural network for studying binaural sound localization," Proc. 1991 IEEE Seventeenth Annual Northeast Bioengineering Conference, pp. 1-2 (Hartford, CT). 1076 Duda Morgan, D. P. and C. 1. Scofield (1991). Neural Networks and Speech Processing. Boston, MA: Kluwer Academic Publishers. Neti, C., E. D. Young and M. H. Schneider (1992). "Neural network models of sound localization based on directional filtering by the pinna," J. Acoust. Soc. Am., vol. 92, pp. 3140-3156. Palmieri, F., M. Datum, A. Shah and A. Moiseff (1991). "Sound localization with a neural network trained with the multiple extended Kalman algorithm," Proc. Int. Joint Conf. on Neural Networks, pp. 1125-1131 (Seattle, \VA). Pickles, James O. (1988). An Introduction to the Physiology of Hearing, 2nd edition. London, Academic Press, 1988. Rabiner, L. and B-H Juang (1993). Fundamentals of Speech Recognition. Engelwood Cliffs, NJ: Prentice-Hall. Reed, M. C. and J . J. Blum (1990). "A model for the computation and encoding of azimuthal information by the lateral superior olive," J. Acoust. Soc. Am., vol. 88, pp. 1442-1453. Richards, Vi. (1988). "Sound interpretation," in \TV. Richards (ed.), Natural Computation, pp. 301-308. Cambridge, MA: MIT Press. Rosen, D. , D. Rumelhart and E. Knudsen (1993). "A connectionist model of the owl's localization system," in J. D. Cowan, G. Tesauro and J. Alspector (eds.), Advances in Neural Information Processing Systems 6. San Francisco, CA: Morgan Kaufmann Publishers. Slaney, M. and R. F. Lyon (1993). "On the importance of time A temporal representation of sound," in M. Cooke, S. Beet and M. Crawford (eds.), Visual Representations of Speech Signals, pp. 95-116. Chichester, England: John \TViley and Sons. Spence, C. D. and J. C. Pearson (1990). "The computation of sound source elevation in the barn owl," in D. S. Touretzsky (ed.), Advances in Neural Information Processing Systems 2, pp. 10-17. San Mateo, CA: Morgan Kaufmann. Weintraub, M. (1985). "A theory and computational model of auditory monaural sound separation," PhD dissertation, Department of Electrical Engineering, Stanford University, Stanford, CA. Zakarauskas, P. and M. S. Cynander (1993). "A computational theory of spectral cue localization," J. Acoust. Soc. Am., vol. 94, pp. 1323-1331. Zurek, P. M. (1987). "The precedence effect," in \TV. A. Yost and G. Gourevitch (eds.) Directional Hearing, pp. 85-106. New York, NY: Springer Verlag.
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Learning in Compositional Hierarchies: Inducing the Structure of Objects from Data Joachim Utans Oregon Graduate Institute Department of Computer Science and Engineering P.O. Box 91000 Portland, OR 97291-1000 utans@cse.ogi.edu Abstract I propose a learning algorithm for learning hierarchical models for object recognition. The model architecture is a compositional hierarchy that represents part-whole relationships: parts are described in the local context of substructures of the object. The focus of this report is learning hierarchical models from data, i.e. inducing the structure of model prototypes from observed exemplars of an object. At each node in the hierarchy, a probability distribution governing its parameters must be learned. The connections between nodes reflects the structure of the object. The formulation of substructures is encouraged such that their parts become conditionally independent. The resulting model can be interpreted as a Bayesian Belief Network and also is in many respects similar to the stochastic visual grammar described by Mjolsness. 1 INTRODUCTION Model-based object recognition solves the problem of invariant recognition by relying on stored prototypes at unit scale positioned at the origin of an object-centered coordinate system. Elastic matching techniques are used to find a correspondence between features of the stored model and the data and can also compute the parameters of the transformation the observed instance has undergone relative to the stored model. An example is the TRAFFIC system (Zemel, Mozer and Hinton, 1990) or the Frameville system (Mjolsness, Gindi and 285 286 Utans i~----::;""Human I I I I I I I I I I I I I I I I I I 1. ______ 1 r--------, I r o o ' I iwi Arm i @ ~ ! L ________ J Lower Arm ~ -oo~oj 1(' Figure I: Example of a compositional hierarchy. The simple figure can be represented as hierarchical composition of parts. The hierarchy can be represented as a graph (a tree in this case). Nodes represent parts and edges represent the structural relationship. Nodes at the bottom represent individual parts of the object; nodes at higher levels denote more complex substructures. The single node at the top of the tree represents the entire object. Anandan, 1989; Gindi, Mjolsness and Anandan, 1991; Vtans, 1992). Frameville stores models as compositional hierarchies and by matching at each level in the hierarchy reduces the combinatorics of the match. The attractive feature of feed-forward neural networks for object recognition is the relative ease with which their parameters can be learned from training data. Multilayer feed-forward networks are typically trained on input/output pairs (supervised learning) and thus are tuned to recognize instances of objects as seen during training. Difficulties arise if the observed object appears at a different position in the input image, is scaled or rotated, or has been subject to distortions. Some of these problems can be overcome by suitable preprocessing or judicious choice of features. Other possibilities are weight sharing (LeCun, Boser, Denker, Henderson, Howard, Hubbard and Jackel, 1989) or invariant distance measures (Simard, LeCun and Denker, 1993). Few attempts have been reported in the neural network literature to learn the prototype models for model based recognition from data. For example, the Frameville system uses hand-designed models. However, models learned from data and reflecting the statistics of the data should be superior to the hand-designed models used previously. Segen (1988a; 1988b) reports an approach to learning structural descriptions where features are clustered to substructures using a Minimum Description Length (MDLJ criterion to obtain a sparse representation. Saund (1993) has proposed a algorithm for constructing tree presentation with multiple "causes" where observed data is accounted for by multiple substructures at higher levels in the hierarchy. Veda and Suzuki (1993) have developed an algorithm for learning models from shape contours using multiscale convex/concave structure matching to find a prototype shape typical for exemplars from a given class. 2 LEARNING COMPOSITIONAL HIERARCHIES The algorithm described here merges parts by means of grouping variables to form substructures. The model architecture is a compositional hierarchy, i.e. a part-whole hierarchy (an example is shown in Figure 1). The nodes in the graph represent parts and substructures, the arcs describe the structure of the object. At each node a probability density for part parameters is stored. A prominent advocate of such models has been Marr (1982) and models of this type are used in the Frameville system (Mjolsness et ai., 1989; Gindi et al., 1991; Vtans, 1992). The nodes in the graph represent parts and substructures, the Learning in Compositional Hierarchies: Inducing the Structure of Objects from Data 287 Figure 2: Examples of different compositional hierarchies for the same object (the digit 9 for a seven-segment LED display). One model emphasizes the parallel lines making up the square in the top part of the figure while for another model angles are chosen as intermediate substructures. The example on the right shows a hierarchy that "reuses" parts. arcs describe the structure of the object. The arcs can be regarded as "part-of" or "ina" relationships (similar to the notion used in semantic networks). At each node a probability density for part parameters such as position, size and orientation is stored. The model represents a typical prototype object at unit scale in an object-centered coordinate system. Parameters of parts are specified relative to parameters of the parent node in the hierarchy. Substructures thus provide a local context for their parts and decouple their parts from other parts and substructures in the model. The advantages of this representation are sparseness, invariance with respect to viewpoint transformations and the ability to model local deformations. In addition, the model explicitly represents the structure of an object and emphasizes the importance of structure for recognition (Cooper, 1989). Learning requires estimating the parameters of the distributions at each node (the mean and variance in the case of Gaussians) and finding the structure of model. The emphasis in this report is on learning structure from exemplars. The parameterization of substructures may be different than for the parts at the lowest level and become more complex and require more parameters as the substructures themselves become more complex. The representation as compositional hierarchy can avoid overfitting since at higher levels in the hierarchy more exemplars are available for parameter estimation due to the grouping of parts (Omohundro, 1991). 2.1 Structure and Conditional Independence: Bayesian Networks In what way should substructures be allocated? Figure 2 shows examples of different compositional hierarchies for the same object (the digit 9 for a seven-segment LED display). One model emphasizes the parallel lines making up the square in the top part of the figure while for another model angles are chosen as intermediate substructures. It is not clear which of these models to choose. The important benefit of a hierarchical representation of structure is that parts belonging to different substructures become decoupled, i.e. they are assigned to a different local context. The problem of constructing structured descriptions of data that reflect this independence relationship has been studied previously in the field of Machine Learning (see (Pearl, 1988) for a comprehensive introduction). The resulting models are Bayesian Belief Networks. Central to the idea of Bayesian Networks is the assumption that objects can be regarded as being composed of components that only sparsely interact and the network captures the probabilistic dependency of these components. The network can be represented as an interaction graph augmented with conditional probabilities. The structure of the graph represents the dependence of variables, i.e. connects them with and arc. The strength of the 288 Utans Figure 3: Bayesian Networks and conditional independence (see text). m,. 0.11 Figure 4: The model architecture. Circles denote the grouping variables ina (here a possible valid model after leaming is shown). dependence is expressed as forward conditional probability. The conditional independence is represented by the absence of an arc between two nodes and leads to the sparseness of the model. The notion of conditional independence in the context studied here manifest itself as follows. By just observing two parts in the image, one must assume that they, i.e. their parameters such as position, are dependent and must be modeled using their joint distribution. However, if one knows that these two parts are grouped to form a substructure then knowing the parameters of the substructure, the parts become conditionally independent, namely conditioned on the parameters of the substructure. Thus, the internal nodes representing the substructures summarize the interaction of their child nodes. The correlation between the child nodes is summarized in the parent node and what remains is, for example, independent noise in observed instances of the child nodes. The probability of observing an instance can be calculated from the model by starting at the root node and multiplying with the conditional probabilities of nodes traversed until the leaf nodes are reached. For example, given the graph in Figure 3, the joint distribution can be factored as P(Xl' Yl, Y2, zl, Z2, z3, Z4) = P(XdP(Yllxd P(ZllydP(ZlIYl)P(Z2IYl )P(z3IY2)P(Z4IY2) (I) (note that the hidden nodes are treatedjust like the nodes corresponding to observable parts). Note that the stochastic visual grammar described by Mjolsness (1991) is equivalent to this model. The model used there is a stochastic forward (generative) model where each level of the compositional hierarchy corresponds to a stochastic production rule that generates nodes in the next lower level. The distribution of parameters at the next lower level are conditioned on the parameters of the parent node. Thus, the model obtained from constructing a Bayesian network is equivalent to the stochastic grammar if the network is constrained to a directed acyclic graph (DAG). If all the nodes of the network correspond to observable events, techniques exist for finding the structure of the Bayesian Network and estimate its parameters (Pearl, 1988) (see also (Cooper and Herskovits, 1992)}. However, for the hierarchical models considered here, only the nodes at the lowest layer (the leaves of the tree) correspond to observable instances of parts of the object in the training data. The learning algorithm must induce hidden, unobservable substructures. That is, it is assumed that the observables are "caused" by internal nodes not directly accessible. These are represented as nodes in the network just Learning in Compositional Hierarchies: Inducing the Structure of Objects from Data 289 like the observables and their parameters must be estimated as well. See (Pearl, 1988) for an extensive discussion and examples of this idea. Learning Bayesian networks is a hard problem when the network contains hidden nodes but a construction algorithm exists if it is known that the data is in fact tree-decomposable (Pearl, 1988). The methods is based on computing the correlations p between child nodes and constraints on the correlation coefficients dictated by a particular structure. The entire tree can be constructed recursively using this method. Here, the case of Normal-distributed real-valued random variables is of interest: 1 1 (1 T -I ) p(XI, ... , Xn) = ~ Vdet'f exp --(x - p) :E (x - p) v2?r detL 2 (2) where x = (XI, X2, ... ,xn ) with mean p = E{x} and covariance matrix :E = E{(x p)(x - p)T} The method is based on a condition under which a set of random variables is star-decomposable. The question one ask is whether a set of n random variables can be represented as the marginal distribution of n + 1 variables XI, ... , X n , W such that the XI, ... , Xn are conditionally independent given w, i.e. (3) J p(XI, ... , Xn, w)dw (4) In the graph representation of the Bayesian Network w is the central node relating the XI, ... ,Xn , hence the name star-decomposable. In the general case of n variables this is hard to verify but a result by Xu and Pearl (1987) is available for 3 variables: A necessary and sufficient condition for 3 random variables with a joint normal distribution to be stardecomposable is that the pairwise correlation coefficients satisfy the triangle inequality pjk ~ PjiPik with (5) for all i, j, k E [1,2,3] and i "I j "I k. Equality holds if node w coincides with node i. For the lowest level of the hierarchy, nodes j and k represent parts and node i = w represents the common substructure. 2.2 An Objective Function for Grouping Parts The algorithm proposed here is based on "soft" grouping by means of grouping variables ina where both the grouping variables and the parameter estimates are updated concurrently. The learning algorithms described in (Pearl, 1988) incrementally construct a Bayesian network and decisions made at early stages cannot be reversed. It is hoped that the method proposed here is more robust with regard to inaccuracies of the estimates. However, if the true distribution is not a star-decomposable normal distribution it can only be approximated. Let inaij be a binary variable associated with the arc connecting node i and node j; inaij = 1 if the arc is present in the network (ina is the adjacency matrix of the graph describing the structure of the model). The model architecture is restricted to a compositional hierarchy (a departure from the more general structure of a Bayesian Network, i.e. nodes are preassigned to levels of the hierarchy (see Figure 4)). Based on the condition in equation (5) a cost 290 Utans function term for the grouping variables ina is Ep = L inawjinawk (PwjPwk - Pjk)2 w,j,kt-j (6) The term penalizes the grouping of two part nodes to the same parent if the term in parentheses is large (i and k index part nodes, w nodes at the next higher level in the hierarchy) The inawj can be regarded as assignment variables the assign child nodes j to parent nodes w. The parameters at each node and the assignment variables ina are estimated using an EM algorithm (Dempster, Laird and Rubin, 1977; Utans, 1993; Yuille, Stolorz and Utans, 1994). For the details of the implementation of grouping with match networks see (Mjolsness et at., 1989; Mjolsness, 1991; Gindi et at., 1991; Utans, 1992; Utans, 1994). At each node for each parameter a probability distribution is stored. Nodes at the lowest level of the hierarchy represent parts in the input data. For the Gaussian distributions used here for all nodes, the parameters are the mean J-t and the variance (J' and can be estimated from data. Each part node can potentially be grouped to any substructure at the next higher level in the hierarchy. The parameters of the distributions at this level are estimated from data as well but using the current value of the grouping variables inaij to weight the contribution from each part node. Because each child node j can have only one parent node i, an additional constraint for a unique assignment is Lw inawj = 1. 3 ANEXAMPLE Initial simulations of the proposed algorithm were performed using a hierarchial model for dot clusters. The training data was generated using the three-level model shown in Figure 5. Each node is parameterized by its position (x, y). The node at the top level represents the entire dot cluster. At the intermediate level nodes represent subcluster centers. The leaf nodes at the lowest level represent individual dots that are output by the model and observed in the image. The top level node represents the position of the entire cluster. At each level 1 + 1 stored offsets d!t 1 are added to the parent coordinates x~ to obtain the coordinates of the child nodes. Then, independent, zero-mean Gaussian distributed noise ( is added: xj+l = x! + d~jl + ( The training data consists of a vector of positions at the lowest level {Xj} with Xj = (Xj, Yj), j = 1 ... 9 for each exemplar. The identity of the parts in the training data is assumed known. In addition, the data consists of parts from a single object. For the simulations, the model architecture is restricted to a three-level hierarchy. Since at the top level a single node represents the entire object, only the grouping variables from the lowest to the intermediate level are unknown (the nodes at the intermediate level are implicitly grouped to the single node at the top level). In the current implementation the parameters of a parent node are defined as the average over the parameters of its child nodes: x~ = Jv Lj i~jxj+l For this problem the algorithm has recovered the structure of the model that generated the training data. Thus in this case it is possible to use the correlation coefficients to learn the structure of an object from noisy training exemplars. However, the algorithm does not recover the same parameter values x used in the generative model at the intermediate layers. These cannot uniquely specified due to the ambiguity between the parameters Xi and offsets dij (a different choice for Xi leads to different values for dij ). Learning in Compositional Hierarchies: Inducing the Structure of Objects from Data 291 0 0 0 0 • • 0 )( 0 0 • 0 0 DaIs )( Global Position • CI ustar Center 0 Dot Figure 5: The model used to generated training data. The structure of the model is a three-level hierarchy. The model parameters are chosen such that the generated dot cluster spatially overlap. On the left, an example of an instance of a dot cluster generated from the model is shown (these constitute the training data). 4 EXTENSIONS The results of the initial experiments are encouraging but more research needs to be done before the algorithm can be applied to real data. For the example used here, the training data was generated by a hierarchical model. Thus the distribution of the training exemplars could, in principle, be learned exactly using the proposed model architecture. I plan to study the effect of approximating the distribution of real-world data by applying the method to the problem of learning models for handwritten digit recognition. The model should be extended to include provisions to deal with missing data. Instead of being binary variables, inaij could be the conditional probability that part j is present in a typical instance of the object given that the parent node i itself is present (similar to the dot deletion rule described in (Mjolsness, 1991)}. These probabilities must also be estimated from data. Under this interpretation the inaij are similar to the mixture coefficients in the mixture of experts model (Jordan and Jacobs, 1993) The robustness of the algorithm can be improved when the desired locality of the model is explicitly favored via an additional constraint. E\ocal = .A L inaij inaik IXj - Xk 12 ij k In this sense, the toy problem shown here is unnecessarily difficult. Preliminary experiments indicate that including this term reduces the sensitivity to spurious correlations between parts that are far apart. As described the algorithm performs unsupervised grouping; learning the hierarchical model does not take in to account the recognition performance obtained when using the model. While the problem of learning and representing models in a hierarchical form is interesting in its own right, the final criteria for judging the model in the context of a recognition problem should be recognition performance. The assumption is that the model should pick up substructures that are specific to a particular class of objects and maximally discriminate between objects belonging to other classes. For example, after a initial model is obtained that roughly captures the structure of the training data, it can be refined on-line during the recognition stage. 292 Utans Acknowledgements Initial work on this project was performed while the author was with the International Computer Science Institute, Berkeley, CA. At OGI supported was provided in part under grant ONR N00014-92-J-4062. Discussions with S. Knerr, E. Mjolsness and S. Omohundro were helpful in preparing this work. References Cooper, G. F. and Herskovits, E. (1992), 'A bayesian method for induction of probabilistic networks from data', Machine Learning 9, 309-347. Cooper, P. R. (1989), Parallel Object Recognition from Structure (The Tinkertoy Project), PhD thesis, University of Rochester, Computer Science. also Technical Report No. 301. Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977), 'Maximum likelihood from incomplete data via the EM algorithm', J. Royal Statist. Soc. B 39, 1-39. Gindi, G., Mjolsness, E. and Anandan, P. (1991), Neural networks for model based recognition, in 'Neural Networks: Concepts, Applications and Implementations', Prentice-Hall, pp. 144-173. Jordan, M. I. and Jacobs, R. A. (1993), Hierarchical mixtures of experts and the EM algorithm, Technical Report 930 I, MIT Computational Cognitive Science. LeCun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W. and Jackel, L. D. (1989), 'Backpropagation applied to handwritten zip code recognition', Neural Computation 1,541-551. Marr, D. (1982), Vision, W. H. Freeman and Co., New York. Mjolsness, E. (1991), Bayesian inference on visual grammars by neural nets that optimize, Technical Report YALEU-DCS-TR-854, Yale University, Dept. of Computer Science. Mjolsness, E., Gindi, G. R. and Anandan, P. (1989), 'Optimization in model matching and perceptual organization', Neural Computation 1(2). Omohundro, S. M. (1991), Bumptrees for efficient function, constraint, and classification learning, in R. Lippmann, J. Moody and D. Touretzky, eds, 'Advances in Neural Information Processing 3', Morgan Kaufmann Publishers, San Mateo, CA. Pearl, J. (1988), Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann Publishers, Inc., San Mateo, CA. Saund, E. (1993), A multiple cause mixture model for unsupervised learning, Technical report, Xerox PARC, Palo Alto, CA. preprint, submitted to Neural Computation. Segen, J. (1988a), Learning graph models of shape, in 'Proceedings of the 5th International Conference on Machine Learning'. Segen, J. (1988b), 'Learning structural description of shape', Machine Vision pp. 257-269. Simard, P., LeCun, Y. and Denker, J. (1993), Efficient pattern recognition using a new transformation distance, in S. J. Hanson, J. Cowan and L. Giles, eds, 'Advances in Neural Information Processing 5', Morgan Kaufmann Publishers, San Mateo, CA. Ueda, N. and Suzuki, S. (1993), 'Learning visual models from shape contours using multiscale convex/concave structure matching' , IEEE Transactions on Pattern Analysis and Machine Intelligence 15(4), 337-352. Utans, J. (1992), Neural Networks for Object Recognition within Compositional Hierarchies, PhD thesis, Department of Electrical Engineering, Yale University, New Haven, CT 06520. Utans, J. (1993), Mixture models and the EM algorithm for object recognition within compositional hierarchies. part 1: Recognition, Technical Report TR-93-004, International Computer Science Institute, 1947 Center St., Berkeley, CA 94708. Utans, J. (1994), 'Mixture models for learning and recognition in compositional hierarchies', in preparation. Xu, L. and Pearl, J. (1987), Structuring causal tree models with continous variables, in 'Proceedings of the 3rd Workshop on Uncertainty in AI', pp. 170-179. Yuille, A., Stolorz, P. and Utans, J. (1994), 'Statistical physics, mixtures of distributions and the EM algorithm', to appear in Neural Computation. Zemel, R. S., Mozer, M. C. and Hinton, G. E. (1990), Traffic: Recognizing objects using hierarchical reference frame transformations, in D. S. Touretzky, ed., 'Advances in Neural Information Processing 2', Morgan Kaufman Pulishers, San Mateo, CA.
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VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems Andreas G. Andreou andreou@jhunix.hcf.jhu.edu Department of Electrical and Computer Engineering The Johns Hopkins University Baltimore, MD 21218 Thomas G. Edwards tedwards@src.umd.edu Department of Electrical Engineering The University of Maryland College Park, MD 20722 Abstract Recent physiological research has shown that synchronization of oscillatory responses in striate cortex may code for relationships between visual features of objects. A VLSI circuit has been designed to provide rapid phase-locking synchronization of multiple oscillators to allow for further exploration of this neural mechanism. By exploiting the intrinsic random transistor mismatch of devices operated in subthreshold, large groups of phase-locked oscillators can be readily partitioned into smaller phase-locked groups. A mUltiple target tracker for binary images is described utilizing this phase-locking architecture. A VLSI chip has been fabricated and tested to verify the architecture. The chip employs Pulse Amplitude Modulation (PAM) to encode the output at the periphery of the system. 1 Introduction In striate cortex, visual information coming from the retina (via the lateral geniculate nuclei) is processed to extract retinotopic maps of visual features. Some cells in cortex are receptive to lines of particular orientation, length, and/or movement direction (Hubel, 1988). A fundamental problem of visual processing is how to 866 VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems 867 associate certain groups of features together to form coherent representations of objects. Since there is an almost infinite number of possible feature combinations, it seems unlikely that there are dedicated "grandmother" cells which code for every possible feature combination. There probably exists a type of adaptive and transitory method to "bind" these features together. The Binding Problem (Crick, 1990) is the problem of making neural elements which are receptive to these visual features temporarily become active as a group that codes for a particular object, yet maintaining the group's specificity towards that object, even when there are several different interleaved objects in the visual field. Temporal correlation of neural response is one solution to the binding problem (von der Malsburg, 1986). Response from neurons (or neural oscillating circuits) which are receptive to a particular visual feature are required to have high temporal correlation with responses to other visual features that correspond to the same object. This would require that there is stimulus-driven oscillation in visual cortex, and that there is also a degree of oscillation synchronization between neural circuits receptive to the same object. Both of these requirements have been found in visual cortex (Gray, 1987; Gray, 1989). Furthermore, there have been several computer simulations of the synchronization phenomena and related visual processing tasks (Baldi, 1990; Eckhorn, 1990). This paper describes a phase-locking architecture for a circuit which performs a multiple-target tracking problem. It will accomplish this task by establishing a zero valued phase difference between oscillators that are receptive to those features to be "bound" together to form an object. Each object will then be recognized as a group of synchronous oscillators, and oscillators that correspond to different objects will be identified due to their lack of synchronization. We assume these oscillators have low duty-cycle pulsed outputs, and the oscillators which correspond to the same object will all pulse high at the same time. Target location will be communicated to the periphery by Pulse Amplitude Modulation (PAM). 2 The Neural Oscillator The oscillator for the target tracker must have two qualities. It needs to be capable of producing a fairly smooth phase representation so that it is easy to compare the difference between oscillator phases to allow for robust phase-locking. It is also useful to have a pulsed output present so that one group of oscillators can be easily discerned from another group of oscillators when their outputs are examined over time. The self-resetting neuron circuit (Mead, 1989) provides both of these outputs (Figure 1). Current lin provided by FET Ql charges capacitor Cl until positive feedback though the non-inverting CMOS amplifier and capacitor C2 brings Vpha.ge all the way to Vdd. This causes the output voltage to go high, which turns on Q2 thus draining charge from Cl by I reret through Q3 and lowering Vpha.ge. When the Vpha"e is brought low enough, positive feedback brings both Vpha"e and the output voltage down to Vu. This turns transistor Q2 off, and the cycle repeats. The duration of output pulses is inversely proportional to Irelet - lin, and the time between output pulses is inversely proportional to lin. Figure 2 is a plot of the pulse output voltage and Vphale vs. time. 868 Andreou and Edwards Vphase f---..... ---~ Pulse Output Figure 1: Self-Resetting Neural Oscillator Oscillator Output 5 ,..... ,..... ,..... ,..... \ \ I \ \ I \ \ I \ \ , I \ \ I \ \ \ \ \ \ \ \ o I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I / , I I I J J I I v v -1+---~--~---r--~--~----~--+---~---r--~ -1 o 1 2 3 4 5 6 7 8 9 (lO-6 sec) Figure 2: Plot of Pulse Output (line) and Phase Voltage (dashed) vs. Time for Neural Oscillator VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems 869 3 Phase Locking To achieve stable and reliable performance, the Comparator Model (Kammen, 1990) of phase-locking was used. Oscillator phase is adjusted according to aO(x,t) (1~ . ) at =w(x)+! ;f;;t0("t)-O(x,t) . Where O( x, t) is the phase of oscillator x at time t, w( x) is the intrinsic phase advance of oscillator x, n is the total number of oscillators, and ! is a sigmoid sq uashing-function. Each object in the visual field requires one averaging circuit to achieve phase-locking of its receptive oscillators. But at any time we do not know the number of objects which will be in the visual field. Therefore, instead of having a pool of monolithic averaging circuits, it is preferable to distribute the averaging function over all the oscillator cells in a way which allows partitioning of the visual field into multiple phase-locked groups of oscillators. The follower-aggregator circuit (Mead, 1989) can be used to develop the average phase information using current-mode computation. It consists of transconductance amplifiers connected as voltage-followers with all outputs tied together to form the average of all input voltages. The phase averaging circuitry can be distributed among the oscillators by placing one transconductance amplifier in each oscillator cell, and linking those oscillators to be phase-locked by a common line. The visual field can be partitioned into multiple phase-locked groups with separate average phases by using FETs to gate whether or not the averaging information can pass through an oscillator cell to its neighbors. To lock an oscillator in phase with the rest of the oscillators which are attached to the averaging line, extra current is provided to the oscillator by a transconductance amplifier to slightly speed up or slow down the oscillator to match its phase to the average phase of the oscillators in the group. Figure 3 shows t.he circuit for a complete phase-locking oscillator cell. Computer simulations of this phase-locking system were carried out using the Analog circuit simulator. Figure 4 shows the result of a simulation of two oscillators. Vgate is the voltage controlling the NFET of the transmission gate which links the phase averaging lines of the two oscillators together (the PFETs are controlled complementary). As soon as the Vgate is brought high, the oscillat.ors rapidly phase lock. 4 Target Location We will assume that the input to a visual tracking chip is a binary image projected onto the die. Phototransistors detect the brightness of each pixel, and if it is above a threshold level, the pixel control circuitry will turn the pixel's oscillator on. If a pixel oscillator is turned on, gating circuitry will allow the propagation of the phase averaging line through the pixel's oscillator cell to its nearest-neighbors. Illuminated 870 Andreou and Edwards To top neighbor v •• ~~ To left -_4 neighbor T. .--------- To right VgateP~ ,,-+--~< VgateN Vf>-1 To bottom neighbor Figure 3: Phase-Locking Oscillator Cell neighbor Pulse Output VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems 871 Sp1ke2~-...J I--------i -5 10 sec Figure 4: Phase-Locking Simulation nearest-neighbor connected pixels will thus have their oscillators turned on and will become phase-locked. The follower-aggregator circuit can be modified to determine linear position (Maher, 1989) by using voltage taps off of a resistive line as inputs to the transconductance amplifiers, and biasing the amplifiers by currents that correspond to the pulsing outputs of the oscillators (see Figure 5). During the time that a group of oscillators are spiking, the output of the tracking circuitry will yield a location corresponding to the average position of the distribution of those oscillators. There can be many different nearest-neighbor connected objects projected onto the die, and the position of the center of each object is communicated to the periphery via PAM. Thus, we can use multiplexing in time to simplify connectivity of communication with the periphery of the chip. 5 Test Chip A chip to test the Comparator Model phase-locking method and multiple-target tracking system was fabricated by the MOSIS service in 2.0 J1.m feature size CMOS. To keep this test chip simple, the oscillators were arranged in a one-dimensional chain, and voltage inputs to the chip were used to control whether or not a pixel was considered "illuminated." A polysilicon resistive line was used to provide linear position information to the tracking system. All transistors used were minimum size (6 J1.m wide and 4 J1.m long). The test chip was able to rapidly and robustly phase-lock groups of nearest-neighbor 872 Andreou and Edwards Postional Voltage Line Vpulse(n-l) Vpulse(n) Vpulse(n+l) Average Position Line Figure 5: Circuit to determine object location connected oscillators. This phase-locking could occur with oscillator frequencies set from 10 Hz to 4 KHz. A phase-locked group of oscillators would almost instantly split into two separate phase-locked groups with little temporal correlation between them when a connected chain of on oscillators was severed by turning off an oscillator in the middle of the original group. Mismatch in the transconductances of the oscillator transistors provided easy desynchronization. Position tracking was measured by examining the resistive-line aggregator output during the time a certain phase-locked group of oscillators was pulsing. When multiple phase-locked groups of oscillators were active, it was still quite easy to make out the positional PAM voltage associated with each group by triggering an oscilloscope off of the pulsing output of an oscillator in that group. While there are occasional instances of two or more groups pulsing at the same time, if the duty cycle of the spiking oscillator is kept relatively small, there is little interference on average. 6 Discussion It is becoming obvious that oscillation and synchronization phenomena in cortex may play an important role in neural information processing. In addition to striate cortex, the olfactory bulb also has oscillatory neural circuits which may be important in neural information processing (Freeman, 1988). It has been suggested that temporal correlation may be used for pattern segmentation in associative memories (Wang, 1990), and correlations between multiple oscillators may be used for storing time intervals (Miall, 1989). We have described a circuit which performs Comparator Model phase-locking. The distributed and partitionable qualities of this circuit make it attractive as a possible physiological model. The PAM representation of object position shows one way that connectivity requirements can be minimized for communication in a neuromorphic VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems 873 system. The chip has been fabricated using subthreshold CMOS technology, and thus uses little power. Acknowledgements The authors are pleased to acknowledge helpful discussion with C. Koch and J. Lazzaro. Chip fabrication was provided by the MOSIS service. References P. Baldi & R. Meir. (1990) Computing with arrays of coupled oscillators: an application to preattentive texture discrimination. Neural Computation 2, 458-47l. F. Crick & C. Koch. (1990) Towards a neurobiological theory of consciousness. Seminars in the Neurosciences 2, 263-275. R. Eckhorn, H. J. Reitboek, M. Arndt & P. Dicke. (1990) Feature linking via synchronization among distributed assemblies: simulations of results from cat visual cortex. Neural Computation 2, 293-307. W. J. Freeman, Y. Yao, & B. Burke. (1988) Central pattern generating and recognizing in olfactory bulb: a correlation learning rule. Neural Networks 1, 277-288. C. M. Gray & W. Singer. (1987) Stimulus-specific neuronal oscillations in the cat visual cortex: A cortical functional unit. Soc. Neurosci. Abstr. 13(404.3) C. M. Gray, P. Konig, A. K. Engel & W. Singer. (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature (London) 338, 334-337. D. H. Hubel. (1988) Eye, Brain, and Vision. New York, NY: Scientific American Library. D. M. Kammen, C. Koch & P. J. Holmes. (1990) Collective oscillations in the visual cortex. In D. S. Touretzky (ed.) Advances in Neural Information Processing Systems 2. San Mateo, CA: Morgan Kaufman Publishers. C. A. Mead. (1989)Analog VLSI and Neural Systems. Reading, MA: AddisonWesley. M. A. Maher, S. P. Deweerth, M. A. Mahowald & C. A. Mead. (1989) Implementing neural architectures using analog VLSI circuits. IEEE Trans. Cire. Sys. 36, 643652. C. Miall. (1989) The storage of time intervals using oscillating neurons. Neural Computation 1, 359-37l. C. von der Malsburg. (1986) A neural cocktail-party processor. Biological Cybernetics. 54,29-40. D. Wang, J. Buhmann & C. von der Malsburg. (1990) Pattern segmentation in associative memory. Neural Computation 2, 95-106.
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A Network Mechanism for the Determination of Shape-From-Texture Ko Sakai and Leif H. Finkel Department of Bioengineering and Institute of Neurological Sciences University of Pennsylvania 220 South 33rd Street, Philadelphia, PA 19104-6392 ko@ganymede.seas.upenn.edu, leif@ganymede.seas.upenn.edu Abstract We propose a computational model for how the cortex discriminates shape and depth from texture. The model consists of four stages: (1) extraction of local spatial frequency, (2) frequency characterization, (3) detection of texture compression by normalization, and (4) integration of the normalized frequency over space. The model accounts for a number of psychophysical observations including experiments based on novel random textures. These textures are generated from white noise and manipulated in Fourier domain in order to produce specific frequency spectra. Simulations with a range of stimuli, including real images, show qualitative and quantitative agreement with human perception. 1 INTRODUCTION There are several physical cues to shape and depth which arise from changes in projection as a surface curves away from view, or recedes in perspective. One major cue is the orderly change in the spatial frequency distribution of texture along the surface. In machine vision approaches, various techniques such as Fourier transformation or wavelet decomposition are used to determine spatial frequency spectra across a surface. The determination of the transformation relating these spectra is a difficult problem, and several techniques have been proposed such as an affine transformation (Super and Bovik 953 954 Sakai and Finkel 1992) or a momentum method (Krumm and Shafer 1992). We address the question of how a biological system which has access only to limited spatial frequency infonnation and has constrained computational capabilities can nonetheless accurately detennine shape and depth from texture. For example, the visual system might avoid the direct comparison of frequency spectra themselves and instead rely on a simpler characterization of the spectra such as the mean frequency, peak frequency, or the gradient of a frequency component (Sakai and Finkel 1993; Turner, Gerstein, Bajcsy 1991). In order to study what frequency infonnation is actually utilized by humans, we created novel random texture patterns and carried out psychophysical experiments with these stimuli. These patterns are generated by manipulating the frequency components of white noise stimuli in the Fourier domain so as to produce stimuli with exactly specified frequency spectra. Based on these experiments, we propose a network mechanism for the perception of shape-from-texture which takes into account physiological and anatomical constraints as well as computational considerations. 2 MODEL FOR SHAPE FROM TEXTURE The model consists of four major processes: extraction of the local spatial frequency at each orientation, frequency characterization, detennination of texture compression by frequency nonnalization, and the integration of the nonnalized frequency over space. A schematic illustration of the model is shown in figure 1. Our psychophysical experiments suggest that the visual system may use spatially averaged peak frequency for characterizing the frequency distribution. The change of surface orientation is determined from the locally aligned compression of texture which is detected by frequency normalization followed by lateral inhibition among different orientations. Depth is then computed from the integration of the normalized frequency over space. The model is implemented in feed-forward distributed networks and simulated using the NEXUS neural network simulator (Sajda, Sakai, Yen and Finkel 1993). 3 MOTIVATION FOR EACH STAGE OF THE MODEL The frequency extraction is carried out by units modeling complex cells in area VI. These units have subunits with On and Off center difference of Gaussian(DOG) masks tuned to specific frequencies and orientations. The units take local maximum of the subunits. As in energy-based models (Bergen and Adelson 1989; Malik and Perona 1990), these units accomplish some major aspects of complex cell functions in the space domain including invariance to the direction of contrast and spatial phase. The second stage of the model extracts spatially averaged peak frequency. In order to examine what frequency infonnation is actually utilized by humans, we created random texture patterns with specific frequency spectra generated by manipulating the frequency components of a white noise pattern in Fourier domain. Figure 2 shows a vertical cylinder and a tilted perspective plane constructed by this technique from white noise. We are able to see the three dimensional shape of the cylinder in (1). The stimuli were constructed by making each frequency component undergo a step change at some A Network Mechanism for the Determination of Shape-from-Texture 955 Early Vision Stage Frequency CharacterizatIOn Frequency Normalization and Lateral Inhibition Integration Figure 1. A schematic illustration of the shape-from-texture model consisting of four major stages. The early vision stage models major spatial properties of complex cells in order to decompose local spatial frequency. The second stage characterizes the frequency by the spatially averaged peak frequency. The third stage detects locally aligned texture compression by normalizing frequency and taking lateral inhibition among orientation channels. The last stage determines 3D depth by integrating the amount of texture compression - which corresponds to the local surface slant. Indices "f' and "0" denote frequency and orientation channels, respectively. max, min, ave, and LI stand for taking maximum, minimum, average, and lateral inhibition. The vertical bar indicates that the function is processed independently within each of denoted channels. 956 Sakai and Finkel position along the cylinder; higher frequencies undergo the change at positions closer to the cylinder's edges. Since the gradient of each frequency component is always either zero or infinity, this suggests that gradients of individual frequency components over space do not serve as a dominant cue for three dimensional shape perception. Similar experiments have been conducted using various stimuli with controlled frequency spectra. The results of these experiments suggest that averaged peak frequency is a strong cue for the human perception of three dimensional shape and depth. The third stage of the model normalizes local frequencies by the global lowest frequency on the surface. We assume that the region containing the global lowest frequency is the frontal plane standing vertically with respect to the viewer. One of the justifications for this assumption can be seen in simple artificial images shown in figure 3. In both (l) and (2), the bottom region looks vertical to us, and the planes above this region looks slanted, although the patterns of the center region of (1) and the lower region of (2) are identical. From a computational point of view, the normalization of frequency corresponds to an approximation of the relation between local slant and spatial frequency. Depth, Z, as a function of X (see figure 4) is given by: Z(x) = JX tan { cos-I ( Fo ) }dx = J x xo F(x) Xo eq.(l) where Fo is the global lowest frequency. Considering a boundary condition, Z(x) = 0, if F(x) = Fo, the integrand can be reasonably approximated by (F(x) - Fo) I Fo . The second stage of the model actually computes this value, and a later stage carries out the integration. Figure 2. Random texture patterns generated by manipulating the frequency components of white noise in Fourier domain. A horizontal cylinder embedded in white noise (1) , and a tilted plane (2). A Network Mechanism for the Determination of Shape-frorn-Texture 957 The second half of this stage detects the local alignment of texture compression. This local alignment is detected by taking the lateral inhibition of normalized frequencies among different orientations. Recent psychophysical experiments (Todd and Akerstrom 1987; Cumming, Johnson, and Parker 1993) show that the compression of texture in a single orientation is a cue for the perception of shape-from-texture. We can confirm this result from figure 5. Three images on the top of this figure have compression in a single orientation, but those on the bottom do not. We clearly see smooth three dimensional ellipsoids from the top images but not from the bottom images. The last stage of the model computes the integral of the nonnalized frequency in order to obtain depth. This integration begins from the region with lowest spatial frequency and follows the path of the local steepest descent in spatial frequency . ~---~~.-..~---~ .-..-. ..................... ... .-..-. ... ~.-..-... -......... ~ ... ~ ... -~ ... ~-.~-•••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• ... ~~ ............ ... ............... --... ~-... ......... .-.-...-, ...... ............ -..-. ... ~-... ~.-..-..-......... -•••••••••• • ••••••••• • ••••••••• • ••••••••• • ••••••••• Figure 3. Objects consist of three planes(left), and two planes(right). In both stimuli, the bottom regions look vertical to us, and the planes above this region look slanted, although the patterns of the center region of (1) and the lower region of (2) are identical. Depth: Z(x) Z(X-V x Xo Figure 4. The coordinate system for the equation (1). Depth, Z, is given as a function of position, X. 958 Sakai and Finkel 4 SIMULATIONS A quantitative test of the model was carried out by constructing ellipsoids with different eccentricities and texture patterns shown in figure 5. Results are plotted in figure 6. For the regular ellipsoids, there is a linear relation between real depth and that determined by the model. This linear relation agrees with psychophysical experiments (Todd and Akerstrom 1987; Biilthoff 1991) showing similar human performance for such stimuli. All of the irregular texture patterns produced little perception of depth, in agreement with human performance. Many artificial and real images have been tested with the model and show good agreement with human perception. For an example, a real image of a part of cantaloupe, and its computed depth are shown in figure 7. Real images were obtained with a CCD camera and were input to NEXUS via an Imaging Technology's S151 image processor. et!~, ..... • 1 • ••• • ••• •• ,.,.. .. :. ...... ~ •••••••••• _ ••••• ·.tr .,.. . .. , , .... ,. , .. .:". -'1' \ , ... r ... , .. ' .. ... ~ -, ... " J" ••••• • ',J./,.. ••• I,' ,II •••• ,,",t", ','..!ii.,. .. ,.,_ I ' I ..... · , , ' .. • • • ... . :'~. •• •• •• ... . .. ' '0 • •••• • ••• •• ·0' . ," .' .. ' .. ••••• •• .. -... .. Figure 5. (Top) Regular ellipsoids with eccentricities of 1,2, and 4. (Bottom) Irregular texture patterns: (left) no compression with regular density change, (middle) randomly oriented regular compression, (right) pan-orientational regular compression. A Network Mechanism for the Determination of Shape-from-Texture 959 400 , , , , ..c 300 , ..... .' , 0.. Q) Q c regular ellipsoids , " , , , , -e 200 ~ C';$ "'3 E 100 .f/.) • no compression • randomly oriented compression , • pan-orientational , , -.... Ia- -- --. ~o compression o 1 2 3 4 5 6 Eccentricity Figure 6. Depth perceived by the model as a function of actual eccentricity. The simulated depth of regular ellipsoids shows a linear relation to the actual depth. Irregular patterns produced little depth, in agreement with human perception. Figure 7. An example of the model's response to a real image. A part of cantaloupe (left), and its depth computed by the model(right). 5 CONCLUSIONS (1) We propose a biologically-based network model of shape-from-texture based on the determination of change in spatial frequency. (2) Preliminary psychophysical evidence suggests that the spatially averaged peak frequency is employed to characterize the spatial frequency distribution rather than using a frequency spectrum or each component of frequency. 960 Sakai and Finkel (3) This characterization is validated by psychophysical experiments using novel random textures with specified frequency spectra. The patterns are generated from white noise and manipulated in Fourier domain in order to realize specific frequency characteristics. (4) The model has been tested with a number of artificial stimuli and real images taken by video camera. Responses show qualitative and quantitative agreements with human perception. Acknowledgments This work is supported by grants from The Office of Naval Research (NOOOI4-90-J-1864, NOOOI4-93-1-0681), The Whitaker Foundation, and The McDonnell-Pew Program in Cognitive Neuroscience. References Super, B.J. and Bovik, A.C. (1992), Shape-from-texture by wavelet-based measurement of local spectral moments. Proc. IEEE CVPR 1992, p296-300 Krumm, J. and Shafer, S.A. (1992), Shape from periodic texture using the spectrogram. Proc. IEEE CVPR 1992, p284-289 Sakai, K. and Finkel, L.H. (1994), A cortical mechanism underlying the perception of shape-from-texture. In F.Eeckman, et al.(ed.), Computation and Neural Systems 1993 , Norwell, MA: Kluwer Academic Publisher [in press] Sajda, P., Sakai, K., Yen, S-c., and Finkel, L.H. (1993), In Skrzypek, J. (ed.), Neural Network Simulation Environments, Norwell, MA: Kluwer Academic Publisher[in press] Bergen, J.R. and Adelson, E.H. (1988), Visual texture segmentation and early vision. Nature, 333, p363-364 Malik, J. and Perona, P. (1990), Preattentive texture discrimination with early vision mechanisms. J. Opt. Soc. Am., A Vol.7, No.5, p923-932 Cumming, B.G., Johnston, E.B., and Parker, A.J. (1993), Effects of different texture cues on curved surfaces viewed stereoscopically. Vision Res. Vol.33, N05, p827-838 Todd, J. T. and Akerstrom, R.A. (1987), Perception of three-dimensional form from patterns of optical texture. Journal of Experimental Psychology, vol. I 3, No.2, p242-255, Turner, M.R., Gerstein, G.L., and Bajcsy, R. (1991), Underestimation of visual texture slant by human observers: a model. Bioi. Cybern. 65, p215-226 Btilthoff, H.H. (1991), Shape from X: Psychophysics and computation. In Landy, M.S., et al.(ed.) Computational Models of Visual Processing, Cambridge, MA: MIT press, p305-330
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Encoding Labeled Graphs by Labeling RAAM Alessandro Sperduti* Department of Computer Science Pisa University Corso Italia 40, 56125 Pisa, Italy Abstract In this paper we propose an extension to the RAAM by Pollack. This extension, the Labeling RAAM (LRAAM), can encode labeled graphs with cycles by representing pointers explicitly. Data encoded in an LRAAM can be accessed by pointer as well as by content. Direct access by content can be achieved by transforming the encoder network of the LRAAM into an analog Hopfield network with hidden units. Different access procedures can be defined depending on the access key. Sufficient conditions on the asymptotical stability of the associated Hopfield network are briefly introduced. 1 INTRODUCTION In the last few years, several researchers have tried to demonstrate how symbolic structures such as lists, trees, and stacks can be represented and manipulated in a connectionist system, while still preserving all the computational characteristics of connectionism (and extending them to the symbolic representations) (Hinton, 1990; Plate, 1991; Pollack, 1990; Smolensky, 1990; Touretzky, 1990). The goal is to highlight the potential of the connectionist approach in handling domains of structured tasks. The common background of their ideas is an attempt to achieve distal access and consequently compositionality. The RAAM model, proposed by Pollack (Pollack, 1990), is one example of how a neural network can discover compact recursive "Work partially done while at the International Computer Science Institute, Berkeley. 1125 1126 Sperduti Output Layer Hidden Layer Input Layer Label Figure 1: The network for a general LRAAM. The first layer of the network implements an encoder; the second layer, the corresponding decoder. distributed representations of trees with a fixed branching factor. This paper presents an extension of the RAAM, the Labeling RAAM (LRAAM). An LRAAM allows one to store a label for each component of the structure to be represented, so as to generate reduced representations of labeled graphs. Moreover, data encoded in an LRAAM can be accessed not only by pointer but also by content. In Section 2 we present the network and we discuss some technical aspects of the model. The possibility to access data by content is presented in Section 3. Some stability results are introduced in Section 4, and the paper is closed by discussion and conclusions in Section 5. 2 THE NETWORK The general structure of the network for an LRAAM is shown in Figure 1. The network is trained by backpropagation to learn the identity function. The idea is to obtain a compressed representation (hidden layer activation) of a node of a labeled graph by allocating a part of the input (output) of the network to represent the label (Nl units) and the remaining part to represent one or more pointers. This representation is then used as pointer to the node. To allow the recursive use of these compressed representations, the part of the input (output) layer which represents a pointer must be of the same dimension as the hidden layer (N H units). Thus, a general LRAAM is implemented by a NJ - N H - NJ feed-forward network, where NJ = Nl + nN H, and n is the number of pointer fields. Labeled graphs can be easily encoded using an LRAAM. Each node of the graph only needs to be represented as a record, with one field for the label and one field for each pointer to a connected node. The pointers only need to be logical pointers, since their actual values will be the patterns of hidden activation of the network. At the beginning of learning, their values are set at random. A graph is represented by a list of these records, and this list constitutes the initial training set for the LRAAM. During training the representations of the pointers are consistently updated according to the hidden activations. Consequently, the training set is dynamic. For example, the network for the graph shown in Figure 2 can be trained as follows: Encoding Labeled Graphs by Labeling RAAM input hidden output (Ll dn2 dn4 dn5 ) d~1 (L" d" d" d" ) 1 n2 n4 n5 (L2 dn3 dn4 nil) d~2 (L" d" d" nil") 2 n3 n4 (L3 dn6 nil nil) d~3 (L" d" nil" nil") 3 n6 (L4 dn6 dn3 nil) d~4 (L" d" d" nil") 4 n6 n3 (L5 dn4 dn6 nil) d~5 (L" d" d" nil") 5 n4 n6 (L6 nil nil nil) d~6 (L~ nil" nil" nil") where Li and dni are respectively the label and the pointer (reduced descriptor to the i-th node. For the sake of simplicity, the void pointer is represented by a single symbol, nil, but each instance of it must be considered as being different. This statement will be made clear in the next section. Once the training is complete, the patterns of activation representing pointers can be used to retrieve information. Thus, for example, if the activity of the hidden units of the network is clamped to dn1 , the output of the network becomes (Ll ,dn2 ,dn4 ,dn5 ), enabling further retrieval of information by decoding dn2 , or dn4 , or dn5 , and so on. Note that more labeled graphs can be encoded in the same LRAAM. 2.1 THE VOID POINTER PROBLEM In the RAAM model there is a termination problem in the decoding of a compressed representation: due to approximation errors introduced during decoding, it is not clear when a decoded pattern is a terminal or a nonterminal. One solution is to test for "binary-ness", which consists in checking whether all the values of a pattern are above 1 T or below T, T > 0, T « 1. However, a nonterminal may also pass the test for "binary-ness". One advantage of LRAAM over RAAM is the possibility to solve the problem by allocating one bit of the label for each pointer to represent if the pointer is void or not. This works better than fixing a particular pattern for the void pointer, such as a pattern with all the bits to 1 or 0 or -1 (if symmetrical sigmoids are used). Simulations performed with symmetrical sigmoids showed that the configurations with all bits equal to 1 or -1 were also used by non void pointers, whereas the configuration with all bits set to zero considerably reduced the rate of convergence. U sing a part of the label to solve the problem is particularly efficient, since the pointer fields are free to take on any configuration when they are void, and this increases the freedom of the system. To facilitate learning, the output activation of the void pointers in one epoch is used as an input activation in the next epoch. Experimentation showed fast convergence to different fixed points for different void Figure 2: An example of a labeled graph. 1127 1128 Sperduti pointers. For this reason, we claimed that each occurrence of the void pointer is different, and that the nil symbol can be considered as a "don't care" symbol. 2.2 REPRESENTATION OF THE TRAINING SET An important question about the way a graph is represented in the training set is which aspects of the representation itself can make the encoding task harder or easier. In (Sperduti, 1993a) we made a theoretical analysis on the constraints imposed by the representation on the set of weights of the LRAAM, under the hypotheses of perfect learning (zero total error after learning) and linear output units. Our findings were: i) pointers to nodes belonging to the same cycle of length k and represented in the same pointer field p, must be eigenvectors of the matrix (W(p))k, where W(p) is the connection matrix between the hidden layer and the output units representing the pointer field p; ii) confluent pointers, i.e., pointers to the same node represented in the same pointer field p (of different nodes), contribute to reducing the rank of the matrix W(p), the actual rank is however dependent on the constraints imposed by the label field and the other pointer fields. We have observed that different representations of the same structure can lead to very different learning performances. However, representations with roughly the same number of non void pointers for each pointer field, with cycles represented in different pointer fields and with confluent pointers seem to be more effective. 3 ACCESS BY CONTENT Retrieval of coded information is performed in RAAM through the pointers. All the terminals and nonterminals can be retrieved recursively by the pointers to the whole tree encoded in a RAAM. If direct access to a component of the tree is required, the pointer to the component must be stored and used on demand. Data encoded in an LRAAM can also be accessed directly by content. In fact, an LRAAM network can be transformed into an analog Hopfield network with one hidden layer and asymmetric connection matrix by feeding back its output into its input units. 1 Because each pattern is structured in different fields, different access 1 Experimental results have shown that there is a high correlation between elements of W(h) (the set of weights from the input to the hidden layer) and the corresponding elements in W(o)T (the set of weights from the hidden to the output layer). This is particularly true for weights corresponding to units of the label field. Such result is not a total surprise, since in the case of a static training set, the error function of a linear encoder network has been proven to have a unique global minimum corresponding to the projection onto the subspace generated by the first principal vectors of a covariance matrix associated with the training set (Baldi & Hornik, 1989). This implies that the weights matrices are transposes of each other unless there is an invertible transformation between them (see also (Bourlard & Kamp, 1988)). Encoding Labeled Graphs by Labeling RAAM 1129 n2=.-r..~= n5 =100=00=-==.=1.1 n9 nlO ~IQl R",.~.=. nl4 /n15 \ 101 •• 101.1. 1 O~.=lctJ~.= I.1 Figure 3: The labeled graph encoded in a 16-3-16 LRAAM (5450 epochs), and the labeled tree encoded in a 18-6-18 LRAAM (1719 epochs). procedures can be defined on the Hopfield network according to the type of access key. An access procedure is defined by: 1. choosing one or more fields in the input layer according to the access key(s); 2. clamping the output of such units to the the access key(s); 3. setting randomly the output of the remaining units in the network; 4. letting the remaining units of the network to relax into a stable state. A validation test of the reached stable state can be performed by: 1. unfreezing the clamped units in the input layer; 2. if the stable state is no longer stable the result of the procedure is considered wrong and another run is performed; 3. otherwise the stable state is considered a success. This validation test, however, sometimes can fail to detect an erroneous retrieval (error) because of the existence of spurious stable states that share the same known information with the desired one. The results obtained by the access procedures on an LRAAM codifying the graph and on an LRAAM codifying the tree shown in Figure 3 are reported in Table 1. For each procedure 100 trials were performed. The "mean" column in the table reports the mean number of iterations employed by the Bopfield network to converge. The access procedure by outgoing pointers was applied only for the tree. It can be seen from Table 1 that the performances of the access procedures were high for the graph (no errors and no wrong retrievals), but not so good for the tree, in particular for the access by label procedure, due to spurious memories. It is interesting to note that the access by label procedure is very efficient for the leaves of the tree. This feature can be used to build a system with two identical networks, one accessed by pointer and the other by content. The search for a label proceeds simultaneously into the two networks. The network accessed by pointer will be very fast to respond when the label is located on a node at lower levels of the tree, and the network accessed by content will be able to respond correctly and very fast "2 when the label is located on a node at higher levels of the tree. 2 Assuming an analog implementation of the Hopfield network. 1130 Sperduti GRAPH: Access by Label TREE: Access by Label key(s) success wrong error mean key success wrong error mean io 100% 0% 0% 7.35 io 0% 100% 0% 16.48 i1 100% 0% 0% 36.05 it 94% 6% 0% 14.57 i2 100% 0% 0% 6.04 i2 47% 53% 0% 16.92 i3 100% 0% 0% 3.99 i3 100% 0% 0% 18.07 i4 100% 0% 0% 23.12 i4 97% 0% 3% 32.64 15 100% 0% 0% 18.12 15 100% 0% 0% 16.03 i6 100% 0% 0% 29.26 16 49% 51% 0% 27.50 TREE: Access by Children Pointers i7 42% 58% 0% 27.10 (d1 , d2) 49% 51% 0% 6.29 is 57% 43% 0% 62.45 (d3,d4) 10% 90% 0% 8.55 i9 20% 0% 80% 14.75 (d5, d6) 40% 60% 0% 12.48 11O 100% 0% 0% 19.11 (d7 , ds) 78% 22% 0% 6.57 III 100% 0% 0% 10.83 (d9, dlO ) 9% 91% 0% 6.22 lt2 100% 0% 0% 19.12 d~~) 14% 86% 0% 14.01 it3 29% 71% 0% 23.87 (d12 ,d13 ) 14% 86% 0% 7.87 114 100% 0% 0% 12.09 ld14, d15 ) 28% 72% 0% 6.07 115 100% 0% 0% 13.11 (*) one pointer Table 1: Results obtained by the access procedures. 4 STABILITY RESULTS In the LRAAM model two stability problems are encountered. The first one arises when considering the decoding of a pointer along a cycle of the encoded structures. Since the decoding process suffers, in general, of approximation errors, it may happen that the decoding diverges from the correct representations of the pointers belonging to the cycle. Thus, it is fundamental to discover under which conditions the representations obtained for the pointers are asymptotically stable with respect to the pointer transformation. In fact, if the representations are asymptotically stable, the errors introduced by the decoding function are automatically corrected. The following theorem can be proven (Sperduti, 1993b): Theorem 1 A decoding sequence l(i;+I) = F(p';)(l(iJ»), j = 0, .. . ,L (1) with l(iL+d = l(to) , satisfying m L Ibikl < 1, i = 1, ... ,m (2) k=l for some index Pi'l' q = 0, ... , L, is asymptotically stable, where btk is the (i, k) th element of a matrix B, given by B = J(P"I) (l( i'l) )J(P"I-l ) (l( i'l_ J)) ... J(p'{J) (l( io) )J(p, L \ l(iL») ... J(P"I+l ) (d (i'l+d). In the statement of the theorem, F(p;) (l) = F(D(p; )l+~;») is the transformation of the reduced descriptor (pointer) d by the pointer field Pj, and J(pJ)(l) is its Encoding Labeled Graphs by Labeling RAAM 1131 Jacobian matrix. As a corollary of this theorem we have that if at least one pointer belonging to the cycle has saturated components, then the cycle is asymptotically stable with respect to the decoding process. Moreover, the theorem can be applied with a few modifications to the stability analysis of the fixed points of the associated Hopfield network. The second stability problem consists into the discovering of sufficient conditions under which the property of asymptotical stability of a fixed point in one particular constrained version of the associated Hopfield network, i.e., an access procedure, can be extended to related fixed points of different constrained versions of it, i.e., access procedures with more information or different information. The result of Theorem 1 was used to derive three theorems regarding this issue (see (Sperduti, 1993b) ). 5 DISCUSSION AND CONCLUSIONS The LRAAM model can be seen from various perspectives. It can be considered as an extension of the RAAM model, which allows one to encode not only trees with information on the leaves, but also labeled graphs with cycles. On the other hand, it can be seen as an approximate method to build analog Hopfield networks with a hidden layer. An LRAAM is probably somewhere in between. In fact, although it extends the representational capabilities of the RAAM model, it doesn't possess the same synthetic capabilities as the RAAM, since it explicitly uses the concept of pointer. Different subsets of units are thus used to codify labels and pointers. In the RAAM model, using the same set of units to codify labels and reduced representations is a more natural way of integrating a previously developed reduced descriptor as a component of a new structure. In fact, this ability was Pollack's original rationale behind the RAAM model, since with this ability it is possible to fill a linguistic role with the reduced descriptor of a complex sentence. In the LRAAM model the same target can be reached, but less naturally. There are two possible solutions. One is to store the pointer of some complex sentence (or structure, in general), which was previously developed, in the label of a new structure. The other solution would be to have a particular label value which tells us that the information we are looking for can be retrieved using one conventional or particular pointer among the current ones. An issue strictly correlated with this is that, even if in an LRAAM it is possible to encode a cycle, what we get from the LRAAM is not an explicit reduced representation of the cycle, but several pointers to the components of the cycle forged in such a way that the information on the cycle is only represented implicitly in each of them. However, the ability to synthesize reduced descriptors for structures with cycles is what makes the difference between the LRAAM and the RAAM. The only system that we know of which is able to represent labeled graphs is the DUAL system proposed by Dyer (Dyer, 1991). It is able to encode small labeled graphs representing relationships among entities. However, the DUAL system cannot be considered as being on the same level as the LRAAM, since it devises a reduced representation of a set of functions relating the components of the graph rather than a reduced representation for the graph. Potentially also Holographic Reduced Representations (Plate, 1991) are able to encode cyclic graphs. 1132 Sperduti The LRAAM model can also be seen as an extension of the Hopfield networks philosophy. A relevant aspect of the use of the Hopfield network associated with an LRAAM, is that the access procedures defined on it can efficiently exploit subsets of the weights. In fact, their use corresponds to generating several smaller networks from a large network, one for each kind of access procedure, each specialized on a particular feature of the stored data. Thus, by training a single network, we get several useful smaller networks. In conclusion an LRAAM has several advantages over a standard RAAM. Firstly, it is more powerful, since it allows to encode directed graphs where each node has a bounded number of outgoing arcs. Secondly, an LRAAM allows direct access to the components of the encoded structure not only by pointer, but also by content. Concerning the applications where LRAAMs can be exploited, we believe there are at least three possibilities: in knowledge representation, by encoding Conceptual Graphs (Sowa, 1984); in unification, by representing terms in restricted domains (Knight, 1989); in image coding, by storing Quadtrees (Samet, 1984); References P. Baldi & K. Hornik. (1989) Neural networks and principal component analysis: Learning from examples without local minima. Neural Networks, 2:53-58. H. Bourlard & Y. Kamp. (1988) Auto-association by multilayer perceptrons and singular value decomposition. Biological Cybernetics, 59:291-294. M. G. Dyer. (1991) Symbolic NeuroEngineering for Natural Language Processing: A Multilevel Research Approach., volume 1 of Advances in Connectionist and Neural Computation Theory, pages 32-86. Ablex. G. E. Hinton. (1990) Mapping part-whole hierarchies into connectionist networks. A rtificial Intelligence, 46:47-75. K. Knight. (1989) Unification: A multidisciplinary survey. A CM Computing Surveys, 21:93-124. T. Plate. (1991) Holographic reduced representations. Technical Report CRG-TR-91-1, Department of Computer Science, University of Toronto. J. B. Pollack. (1990) Recursive distributed representations. Artificial Intelligence, 46(12):77-106. H. Samet. (1984) The quadtree and related hierarchical data structures. A CM Computing Surveys, 16:187-260. P. Smolensky. (1990) Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artificial Intelligence, 46:159-216. J.F. Sowa. (1984) Conceptual Structures: Information Processing in Mind and Machine. Addison-Wesley. A. Sperduti. (1993a) Labeling RAAM. TR 93-029, ICSI, Berkeley. A. Sperduti. (1993b) On some stability properties of the LRAAM model. TR 93-031, ICSI, Berkeley. D. S. Touretzky. (1990) Boltzcons: Dynamic symbol structures in a connectionist network. A rtificial Intelligence, 46:5-46. PART XI ADDENDA TO NIPS 5
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Learning Mackey-Glass from 25 examples, Plus or Minus 2 Mark Plutowski· Garrison Cottrell· Halbert White·· Institute for Neural Computation *Department of Computer Science and Engineering **Department of Economics University of California, San Diego La J oHa, CA 92093 Abstract We apply active exemplar selection (Plutowski &. White, 1991; 1993) to predicting a chaotic time series. Given a fixed set of examples, the method chooses a concise subset for training. Fitting these exemplars results in the entire set being fit as well as desired. The algorithm incorporates a method for regulating network complexity, automatically adding exempla.rs and hidden units as needed. Fitting examples generated from the Mackey-Glass equation with fractal dimension 2.1 to an rmse of 0.01 required about 25 exemplars and 3 to 6 hidden units. The method requires an order of magnitude fewer floating point operations than training on the entire set of examples, is significantly cheaper than two contending exemplar selection techniques, and suggests a simpler active selection technique that performs comparably. 1 Introduction Plutowski &. White (1991; 1993), have developed a method of active selection of training exemplars for network learning. Active selection uses information about the state of the network when choosing new exemplars. The approach uses the statistical sampling criterion Integrated Squared Bias (ISB) to derive a greedy selection method that picks the training example maximizing the decrement in this measure. (ISB is a special case of the more familiar Integrated Mean Squared Error in the case that noise variance is zero.) We refer to this method as A.ISB. The method automatically regulates network complexity by growing the network as necessary 1135 1136 Plutowski, Cottrell, and White to fit the selected exemplars, and terminates when the model fits the entire set of available examples to the desired accuracy. Hence the method is a nonparametric regression technique. In this paper we show that the method is practical by applying it to the Mackey-Glass time series prediction task. We compare AISB with the method of training on all the examples. AIS8 consistently learns the time series from a small subset of the available examples, finding solutions equivalent to solutions obtained using all of the examples. The networks obtained by AISB consistently perform better on test data for single step prediction, and do at least as well at iterated prediction, but are trained at much lower cost. Having demonstra.ted that this particular type of exemplar selection is worthwhile, we compare AISE with three other exemplar selection methods which are easier to code and cost less to compute. We compare the total cost of training, as well as the size of the exemplar sets selected. One of the three contending methods was suggested by the AISB algorithm, and is also an active selection technique, as its calculation involves the network state. Among the four exemplar selection methods, we find that the two active selection methods provide the greatest computational savings and select the most concise training sets. 2 The Method We are provided with a set of N "candidate" examples of the form (Zi, g(zd). Given g, we can denote this as x N . Let 1(·, w) denote the network function parameterized by weights w. For a particular subset of the examples denoted x n , let Wn = Wn (zn) minimize Let w· be the "best" set of weights, which minimizes where IJ is the distribution over the inputs. Our objective is to select a subset zn of zN such that n < N, while minimizing J(/(z, wn ) - I(z, w·))21J(dz). Thus, we desire a subset representative of the whole set. We choose the zn C zN giving weights Wn that minimize the Integrated Squared Bias (ISB): (1) We generate zn incrementally. Given a candidate example Zn+l, let zn+l = (zn, Zn+l). Selecting Zl optimally with respect to (1) is straightforward. Then given zn minimizing ISB(zn), we opt to select Zn+l E zN maximizing ISB(zn)ISB(xn+1). Note that using this property for Zn+1 will not necessarily deliver the globally optimal solution. Nevertheless, this approach permits a computationally feasible and attractive method for sequential selection of training examples. Learning Mackey-Glass from 25 Examples, Plus or Minus 2 1137 Choosing Zn+l to maximize this decrement directly is expensive. We use the following simple approximation (see Plutowski &- White, 1991) for justification): Given zn, select Zn+l E argmaxzn+l~ISB(xn+llzn), where N 6ISB(xn+llzn) = 6Wn+l' L V w!(Zi, wn)(g(zd - !(Zi, wn», i=l and n H(zn ,wn ) = LV w!(Zj, wn)Vw!(Zi, wn }' . i=l In practice we approximate H appropriately for the task at hand. Although we arrive at this criterion by making use of approximations valid for large n, this criterion has an appealing interpretation as picking the single example having individual error gradient most highly correlated with the average error gradient of the entire set of examples. Learning with this example is therefore likely to be especially informative. The 6ISB criterion thus possesses heuristic appeal in training sets of any size. 3 The Algorithm Before presenting the algorithm we first explain certain implementation details. We integrated the ~I SB criterion with a straightforward method for regulating network complexity. We begin with a small network and an initial training set composed of a single exemplar. When a new exemplar is added, if training stalls, we randomize the network weights and restart training. After 5 stalls, we grow the network by adding another unit to each hidden layer. Before we can select a new exemplar, we require that the network fit the current training set "sufficiently well." Let en(zm) measure the rmse (root mean squared error) network fit over m arbitrary examples zm when trained on xn. Let Fn E ~+ denote the rmse fit we require over the current set of n exemplars before selecting a new one. Let FN E ~+ denote the rmse fit desired over all N examples. (Our goal is en(zN) < FN.) It typically suffices to set Fn = FN, that is, to train to a fit over the exemplars which is at least as stringent as the fit desired over the entire set (normalized for the number of exemplars.) However,. active selection sometimes chooses a new exemplar "too close" to previously selected exemplars even when this is the case. This is easy to detect, and in this case we reject the new exemplar and continue with training. We use an "exemplar spacing" parameter d to detect when a new exemplar is too close to a previous selection. Two examples Xi and Xi are "close" in this sense if they are within Euclidean distance d, and if additionally Ig(Zi) - g(xi)1 < FN. The additional condition allows the new exemplar to be accepted even when it is close to a previous selection in input space, provided it is sufficiently far away in the output space. In our experiments, the input and output space are of the same scale, so we set d = FN. When a new selection is too close to a current exemplar, we reject the 1138 Plutowski, Cottrell, and White new selection, reduce Fn by 20%, and continue training, resetting Fn = FN when a subsequent selection is appended to the current training set. We now outline the algorithm: Initialize: • Specify user-set parameters: initial network size, the desired fit FN, the exemplar spacing parameter, and the maximum number of restarts . • Select the first training set, xl = {xd. Set n = 1 and Fn = FN. Train the network on xl until en{x 1) 5 Fn. While(en(xN) > FN) { Select a new exemplar, Zn+l E x N , maximizing 6.ISB. H (Zn+l is "too close" to any Z E zn) { Reject Zn+l Reduce Fn by 20%. } Else { Append Zn+l to zn. Increment n. Set Fn = FN . } While(en(zn} > Fn) { Train the network on the current training set zn, restarting and growing as necessary. }} 4 The Problem We generated the data from the Mackey-Glass equation (Mackey &, Glass, 1977), with T = 17, a = 0.2, and b = 0.1. We integrated the equation using fourth order Runge-Kutta with step size 0.1, and the history initialized to 0.5. We generated two data sets. We iterated the equation for 100 time steps before beginning sampling; this marks t = O. The next 1000 time steps comprise Data Set 1. We generated Data Set 2 from the 2000 examples following t = 5000. We used the standard feed-forward network architecture with [0, 1] sigmoids and one or two hidden layers. Denoting the time series as z(t}, the inputs were z(t), x(t 6}, z(t - 12), z(t - 18), and the desired output is z(t + 6) (Lapedes &, Farber, 1987). We used conjugate gradient optimization for all of the training runs. The line search routine typically required 5 to 7 passes through the data set for each downhill step, and was restricted to use no more than 10. Initially, the single hidden layer network has a single hidden unit, and the 2 hidden layer network has 2 units per hidden layer. A unit is added to each hidden layer when growing either architecture. All methods use the same growing procedure. Thus, other exemplar selection techniques are implemented by modifying how the next training set is obtained at the beginning of the outer while loop. The method of using all the training examples uses only the inner while loop. In preliminary experiments we evaluated sensitivity of 6.ISB to the calculation of H. We compared two ways of estimating H, in terms of the number of exemplars Learning Mackey-Glass from 25 Examples, Plus or Minus 2 1139 selected and the total cost of training. The first approach uses the diagonal terms of H (Plutowski &. White, 1993). The second approach replaces H with the identity matrix. Evaluated over 10 separate runs, fitting 500 examples to an rmse of 0.01, ~ISB gave similar results for both approaches, in terms of total computation used and the number of exemplars selected. Here, we used the second approach. 5 The Comparisons We performed a number of experiments, each comparing the ~ISB algorithm with competing training methods. The competing methods include the conventional method of using all the examples, henceforth referred to as "the strawman," as well as three other data selection techniques. In each comparison we denote the cost as the total number of floating point multiplies (the number of adds and divides is always proportional to this count). For each comparison we ran two sets of experiments. The first compares the total cost of the competing methods as the fit requirement is varied between 0.02, 0.015, and 0.01, using the first 500 examples from Data Set 1. The second compares the cost as the size of the "candidate" set (the set of available examples) is varied using the first 500, 625, 750, 875, and 1000 examples of Data Set I, and a tolerance of 0.01. To ensure that each method is achieving a comparable fit over novel data, we evaluated each network over a test set. The generalization tests also looked at the iterated prediction error (IPE) over the candidate set and test set (Lapedes &. Farber, 1987). Here we start the network on the first example from the set, and feed the output back into the network to obtain predictions in multiples of 6 time steps. Finally, for each of these we compare the final network sizes. Each data point reported is an average of five runs. For brevity, we only report results from the two hidden layer networks. 6 Comparison With Using All the Examples We first compare ~ISB with the conventional method of using all the available examples, which we will refer to as "the strawman." For this test, we used the first 500 examples of Data Set 1. For the two hidden layer architecture, each method required 2 units per hidden layer for a fit of 0.02 and 0.015 rmse, and from 3 to 4 (typically 3) units per hidden layer for a fit of 0.01 rmse. While both methods did quite well on the generalization tests, ~ISB clearly did better. Whereas the strawman networks do slightly worse on the test set than on the candidate set, networks trained by ~ISB tended to give test set fits close to the desired (training) fit. This is partially due to the control flow of the algorithm, which often fits the candidate set better than necessary. However, we also observed ~ISB networks exhibited a test set fit better than the candidate set fit 7 times over these 15 training runs. This never occurred over any of the strawman runs. Overall, ~ISB networks performed at least as well as the strawman with respect to IPE. Figure 1a shows the second half of Data Set 1, which is novel to this network, plotted along with the iterated prediction of a ~ISB network to a fit of 0.01, giving an IPE of 0.081 rmse, the median IPE observed for this set of five runs. Figure 1b shows the iterated prediction over the first 500 time steps of Data Set 2, which is 1140 Plutowski, Cottrell, and White 4500 time steps later than the training set. The IPE is 0.086 rmse, only slightly worse than over the "nearer" test set. This fit required 22 exemplars. Generalization tests were excellent for both methods, although t1ISB was again better overall. t1ISB networks performed better on Data Set 2 than they did on the candidate set 9 times out of the 25 runs; this never occurred for the strawman. These effects demand closer study before using them to infer that data selection can introduce a beneficial bias. However, they do indicate that the t1ISB networks performed at least as well as the strawman, ensuring the validity of our cost comparisons. Figure 1: Itell.ted prediction for a 2 hidden layer network trained to 0.01 rmae over the first 500 time steps of Data Set 1. The dotted line gives the network prediction; the solid line is the target time series. Figure la, on the left, is over the next (consecutive) 500 time steps of Data Set 1, with IPE = 0.081 rmse. Figure Ib, on the right, is over the first 500 steps of Data Set 2, with IPE = 0.086 rmse. This network was typical, being the median IPE of 5 runs. Figure 2a shows the average total cost versus required fit FN for each method. The strawman required 109, 115, and 4740 million multiplies for the respective tolerances, whereas t1ISB required 8, 28, and 219 million multiplies, respectively. The strawman is severely penalized by a tighter fit because growing the network to fit requires expensive restarts using all of the examples. Figure 2b shows the average total cost versus the candidate set sizes. One reason for the difference is that t1ISB tended to select smaller networks. For candidate sets of size 500, 625, 750 and 875, each method typically required 3 units per hidden layer, occasionally 4. Given 1000 examples, the strawman selected networks larger than 3 hidden units per layer over twice as often as t1ISB. t1ISB also never required more than 4 hidden units per layer, while the strawman sometimes required 6. This suggests that the growing technique is more likely to fit the data with a smaller network when exemplar selection is used. Cost 7000 6000 5000 4000 3000 2000 1000 0.02 Cost 35000 30000 25000 20000 15000 10000 5000 Figure 2: Cost (in millions of multiplies) oftraining t1ISB, compared to the Strawman. Figure 2a on the left gives total cost versus the desired fit, and Figure 2b on the right gives total cost versus the number of ca.ndidate examples. Each point is the average of 5 runs; the error bars are equal in width to twice the standard deviation. Learning Mackey-Glass from 25 Examples, Plus or Minus 2 1141 7 Contending Data Selection Techniques The results above clearly demonstrate that exemplar selection can cut the cost of training dramatically. In what follows we compare ~ISB with three other exemplar selection techniques. Each of these is easier to code and cheaper to compute, and are considerably more challenging contenders than the strawman. In addition to comparing the overall training cost we will also evaluate their data compression ability by comparing the size of the exemplar sets each one selects. We proceed in the same manner as with ~ISB, sequentially growing the training set as necessary, until the candidate set fit is as desired. Two of these contending techniques do not depend upon the state of the network, and are therefore are not "Active Selection" methods. Random Selection selects an exampk randomly from the candidate set, without replacement, and appends it to the current exemplar set. Uniform Grid exploits the time series representation of our data set to select training sets composed of exemplars evenly spaced at regular intervals in time. Note that Uniform Grid does. not append a single exemplar to the training set, rather it selects an entirely new set of exemplars each time the training set is grown. Note further that this technique relies heavily upon the time series representation. The problem of selecting exemplars uniformly spaced in the 4 dimensional input space would be much more difficult to compute. The third method, "Maximum Error," was suggested by the ~ISB algorithm, and is also an Active Selection technique, since it uses the network in selecting new exemplars. Note that the error between the network and the desired value is a component of the tiISB criterion. ~ISB need not select an exemplar for which network error is maximum, due to the presence of terms involving the gradient of the network function. In comparison, the Maximum Error method selects an exemplar maximizing network error, ignoring gradient information entirely. It is cheaper to compute, typically requiring an order of magnitude fewer multiplies in overhead cost as compared to tiISB. This comparison will test, for this particular learning task, whether the gradient information is worth its additional overhead. 7.1 Comparison with Random Selection Random Selection fared the worst among the four contenders. However, it still performed better overall than the strawman method. This is probably because the cost due to growing is cheaper, since early on restarts are performed over small training sets. As the network fit improves, the likelihood of randomly selecting an informative exemplar decreases, and Random Selection typically reaches a point where it adds exemplars in rapid succession, often doubling the size of the exemplar set in order to attain a slightly better fit. Random Selection also had a very high variance in cost and number of exemplars selected. 7.2 Comparison with Uniform Grid and Maximum Error Uniform Grid and Maximum Error are comparable with tiISB in cost as well as in the size of the selected exemplar sets. Overall, tiISB and Maximum Error performed about the same, with Uniform Grid finishing respectably in third place. Maximum Error was comparable to ~ISB in generalization also, doing better on the test set than on the candidate set 10 times out of 40, whereas tiISB did so a 1142 Plutowski, Cottrell, and White total of 16 times. This occurred only 3 times out of 40 for Uniform Grid. Figure 3a shows that Uniform Grid requires more exemplars at all three tolerances, whereas ~ISB and Maximum Errorselect about the same number. Figure 3b shows that Uniform Grid typically requires about twice as many exemplars as the other two. Maximum Error and ~ISB selected about the same number of exemplars, typically selecting about 25 exemplars, plus or minus two. 50 40 30 20 10 n De ta ISB 0.02 0.015 n 60 Uniforrr 50 40 30 Max Er 20 10 0.02 RInse 1-ri Uniforn I -t Delt .. ISB * ~ Max Error ~ ~ !, 500 625 750 875 1000 N Figure 3: Number of examples selected by three contending selection techniques: Uniform, ~ISB (diamonds) and Max Error (triangles.) Figure 3a on the left gives number of examples selected versus the desired fit, and Figure 3b on the right is versus the number of candidate examples. The two Active Selection techniques selected about 25 exemplars, ±2. Each point is the average of 5 runSi the error bars are equal in width to twice the standard deviation. The datapoints for ~I SB and Max Error are shifted slightly in the graph to make them easier to distinguish. 8 Conclusions These results clearly demonstrate that exemplar selection can dramatically lower the cost of training. This particular learning task also showed that Active Selection methods are better overall than two contending exemplar selection techniques. ~I S B and Maximum Error consistently selected concise sets of exemplars, reducing the total cost of training despite the overhead associated with exemplar selection. This particular learning task did not provide a clear distinction between the two Active Selection techniques. Maximum Error is more attractive on problems of this scope even though we have not justified it analytically, as it performs about as well as ~ISB but is easier to code and cheaper to compute. Acknowledgements This work was supported by NSF grant IRI 92-03532. References Lapedes, Alan, and Robert Farber. 1987. "Nonlinear signal processing using neural networks. Prediction and system modelling." Los Alamos technical report LA-UR-87-2662. Mackey, M.C., and L. Glass. 1977. "Oscillation and chaos in physiological control systems." Science 197, 287. Plutowski, Mark E., and Halbert White. 1991. "Active selection of training examples for network learning in noiseless environments." Technical Report No. CS91-180, CSE Dept., UCSD, La Jolla, California. Plutowski, Mark E., and Halbert White. 1993. "Selecting concise training sets from clean data." To appear, IEEE Transactions on Neural Networks. 3, 1.
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Structured Machine Learning For 'Soft' Classification with Smoothing Spline ANOVA and Stacked Tuning, Testing and Evaluation Grace Wahba Dept of Statistics University of Wisconsin Madison, WI 53706 Yuedong Wang Dept of Statistics University of Wisconsin Madison, WI 53706 Chong Gu Dept of Statistics Purdue University West Lafayette, IN 47907 Ronald Klein, MD Dept of Ophthalmalogy University of Wisconsin Madison, WI 53706 Barbara Klein, MD Dept of Ophthalmalogy University of Wisconsin Madison, WI 53706 Abstract We describe the use of smoothing spline analysis of variance (SSANOVA) in the penalized log likelihood context, for learning (estimating) the probability p of a '1' outcome, given a training set with attribute vectors and outcomes. p is of the form pet) = eJ(t) /(1 + eJ(t)), where, if t is a vector of attributes, f is learned as a sum of smooth functions of one attribute plus a sum of smooth functions of two attributes, etc. The smoothing parameters governing f are obtained by an iterative unbiased risk or iterative GCV method. Confidence intervals for these estimates are available. 1. Introduction to 'soft' classification and the bias-variance tradeoff. In medical risk factor analysis records of attribute vectors and outcomes (0 or 1) for each example (patient) for n examples are available as training data. Based on the training data, it is desired to estimate the probability p of the 1 outcome for any 415 416 Wahba, Wang, Gu, Klein, and Klein new examples in the future, given their attribute vectors. In 'soft' classification, the estimate p of p is of particular interest, and might be used, say, by a physician to tell a patient that if he reduces his cholesterol from t to t', then he will reduce his risk of a heart attack from p(t) to p(t'). We assume here that p varies 'smoothly' with any continuous attribute (predictor variable). It is long known that smoothness penalties and Bayes estimates are intimately related (see e.g. Kimeldorf and Wahba(1970, 1971), Wahba(1990) and references there). Our philosophy with regard to the use of priors in Bayes estimates is to use them to generate families of reasonable estimates (or families of penalty functionals) indexed by those smoothing or regularization parameters which are most relevant to controlling the generalization error. (See Wahba(1990) Chapter 3, also Wahba(1992)). Then use cross-validation, generalized cross validation (GCV), unbiased risk estimation or some other performance oriented method to choose these parameter(s) to minimize a computable proxy for the generalization error. A person who believed the relevant prior might use maximum likelihood (ML) to choose the parameters, but ML may not be robust against an unrealistic prior (that is, ML may not do very well from the generalization point of view if the prior is off), see Wahba(1985). One could assign a hyperprior to these parameters. However, except in cases where real prior information is available, there is no reason to believe that the use of hyperpriors will beat out a performance oriented criterion based on a good proxy for the generalization error, assuming, of course, that low generalization error is the true goal. O'Sullivan et al(1986) proposed a penalized log likelihood estimate of I, this work was extended to the SS-ANOVA context in Wahba, Gu, Wang and Chappell(1993), where numerous other relevant references are cited. This paper is available by ftp from ftp. stat. wise. edu, cd pub/wahba in the file soft-class. ps. Z. An extended bibliography is available in the same directory as ml-bib. ps. The SSANOVA allows a variety of interpretable structures for the possible relationships between the predictor variables and the outcome, and reduces to simple relations in some of the attributes, or even, to a two-layer neural net, when the data suggest that such a representation is adequate. 2. Soft classification and penalized log likelihood risk factor estimation To describe our 'worldview', let t be a vector of attributes, tEn E T, where n is some region of interest in attribute space T. Our 'world' consists of an arbitrarily large population of potential examples, whose attribute vectors are distributed in some way over n and, considering all members of this 'world' with attribute vectors in a small neighborhood about t, the fraction of them that are l's is p(t). Our training set is assumed to be a random sample of n examples from this population, whose outcomes are known, and our goal is to estimate p(t) for any tEO. In 'soft' classification, we do not expect one outcome or the other to be a 'sure thing', that is we do not expect p(t) to be 0 or 1 for large portions of n. Next, we review penalized log likelihood risk estimates. Let the training data be {Yi, t(i), i = 1, ... n} where Yi has the value 1 or 0 according to the classification of example i, and t(i) is the attribute vector for example i. If the n examples are a random sample from our 'world', then the likelihood function of this data, given "Soft" Classification with Smoothing Spline ANOVA 417 p( .), is likelihood{y, p} = II~=lP(t(i))Yi (1 - p(t(i) ))l-Yi, (1) which is the product of n Bernoulli likelihoods. Define the logit f(t) by f(t) = 10g[P(t)/(I- p(t))], then p(t) = eJ(t) 1(1 + eJ(t)). Substituting in f and taking logs gIves n -log likelihood{y, f} = £(y, f) = L log(1 + eJ(t(i))) - Yif(t(i)). (2) i=l We estimate f assuming that it is in some space 1l of smooth functions. (Technically, 1l is a reproducing kernel Hilbert space, see Wahba(1990), but you don't need to know what this is to read on). The fact that f is assumed 'smooth' makes the methods here very suitable for medical data analysis. The penalized log likelihood estimate f>.. of f will be obtained as the minimizer in 1l of n £(y, f) + "2)"J(J) (3) where J(J) is a suitable 'smoothness' penalty. A simple example is, T = [0,1] and J(J) = Jo1 (J(m) (t))2dt, in which case f>.. is a polynomial spline of degree 2m - 1. If (4) then f>.. is a thin plate spline. The thin plate spline is a linear combination of polynomials of degree m or less in d variables, and certain radial basis functions. For more details and other penalty functionals which result in rbf's, see Wahba(1980, 1990, 1992). The likelihood function £(y, f) will be maximized if p(t(i)) is 1 or ° according as Yi is 1 or 0. Thus, in the (full-rank) spline case, as ).. -+ 0, 1>.. tends to +00 or -00 at the data points. Therefore, by letting).. be small, we can come close to fitting the data points exactly, but unless the 1 's and O's are well separated in attribute space, f>.. will be a very 'wiggly' function and the generalization error (not precisely defined yet) may be large. The choice of ).. represents a tradeoff between overfitting and underfitting the data (bias-variance tradeoff). It is important in practice good value of )... We now define what we mean by a good value of )... Given the family PA,).. > 0, we want to choose ).. so that PA is close to the 'true' but unknown p so that, if new examples arrive with attribute vector in a neighborhood of t, PA (t) will be a good estimate of the fraction of them that are 1 'so 'Closeness' can be defined in various reasonable ways. We use the Kullbach-Leibler (K L) distance (not a real distance!). The K L distance between two probability measures (g, g) is defined as K L(g, g) = Eg [log (g 1 g)], where Eg means expectation given g is the true distribution. If v(t) is some probability measure on T, (say, a proxy for the distribution ofthe attributes in the population), then define K Lv (p, PA) (for Bernoulli random variables) with respect to v as K Lv(p, PA) = J [P(t)log (;(~l)) + (1 - p(t)) log (11 ~ :A(~l)) ] dv(t). (5) 418 Wahba, Wang, Gu, Klein, and Klein Since K Lv is not computable from the data, it is necessary to develop a computable proxy for it, By a computable proxy is meant a function of), that can be calculated from the training set which has the property that its minimizer is a good estimate of the minimizer of K Lv, By letting p>.(t) = e!>.(t) /(1 + e!>.(t») it is seen that to minimize K Lv, it is only necessary to minimize J [log(l + e!>.(t») - p(t)f>.(t)]dv(t) (6) over). since (5) and (6) differ by something that does not depend on )., Leavingout-half cross validation (!CV) is one conceptually simple and generally defensible (albeit possibly wasteful) way of choosing). to minimize a proxy for K Lv(p, P>.), The n examples are randomly divided in half and the first n/2 examples are used to compute P>. for a series of trial values of )., Then, the remaining n/2 examples are used to compute KLl.cv ().) = ~ ~ [log(l + e!>.(t(i») - Yif>.(t(i))] (7) ~ n ~ i::~+l for the trial values of )., Since the expected value of Yi is p(t(i)), (7) is, for each), an unbiased estimate of (6) with dv the sampling distribution of the {tel), ,." t(n/2)}, ). would then be chosen by minimizing (7) over the trial values. It is inappropriate to just evaluate (7) using the same data that was used to obtain f>., as that would lead to overfitting the data, Variations on (7) are obtained by successively leaving out groups of data. Leaving-out-one versions of (7) may be defined, but the computation may be prohibitive. 3. Newton-Raphson Iteration and the Unbiased Risk estimate of A. We use the unbiased risk estimate given in Craven and Wahba(1979) for smoothing spline estimation with Gaussian errors, which has been adapted by Gu(1992a) for the Bernoulli case, To describe the estimate we need to describe the NewtonRaphson iteration for minimizing (3). Let b(J) = log(l + ef ), then Ley, f) = E?::db(J(t(i)) - Yif(t(i))], It is easy to show that Ey; = f(t(i)) = b'(f(t(i)) and var Yi = p(t(i))(l - p(t(i)) = b"(f(t(i)). Represent f either exactly by using a basis for the (known) n-space of functions containing the solution, or approximately by suitable approximating basis functions, to get N f ~ L CkBk· (8) k=l Then we need to find C = (C1' . ' . , C N)' to minimize n N N 1>.(c) = L beL CkBk(t(i))) - Yi(L CkBk(t(i))) + ~ ).c'~c, (9) ;=1 k=l k=l where E is the necessarily non-negative definite matrix determined by J (Ek Ck Bk) = c'Ec. The gradient \l 1>. and the Hessian \l2l.x of l.x are given by = X' (Pc - y) + n).~c, (10) "Soft" Classification with Smoothing Spline ANOVA 419 = X' WcX + nXE, (11) where X is the matrix with ijth entry Bj(t(i)), Pc is the vector with ith entry Pc (t(i)) given by Pc (t(i)) = (1~:c/~g~:») where fcO = 2::=1 ekBk(·), and Wc is the diagonal matrix with iith entry Pc(t(i))(I-Pc(t(i))). Given the ith Newton-Raphson iterate eCl), e(l+1) is given by e(l+1) = eel) - (X'WC<l) X + nA~)-l(X'(pc(l) - y) + nA~e(l)) and e( l+ 1) is the minimizer of Iil\e) = IIz(l) Wcl(~~ Xell 2 + nAe'~e. where z(l), the so-called pseudo-data, is given by z(l) = Wc(l~/2(y - Pdl») + W:(~~XeCl). (12) (13) (14) The 'predicted' value z(l) = W:(~~ X e, where e is the minimizer of (13), is related to the pseudo-data z(l) by Z(l) = A(l)(A)Z(l), where A(l)(A) is the smoother matrix given by A(l)(A) = W:(~~ X(X'Wc(l)X + nA~)-l X'W:(~~. (15) (16) In Wahba(1990), Section 9.2 1, it was proposed to obtain a GCV score for A in (9) as follows: For fixed A, iterate (12) to convergence. Define VCl)(A) = ~II(I - A(l) (A))z(l) 112 /(~tr(I - A(l) (A)))2 . Letting L be the converged value of i, compute VCL)(A) = ~II(I - A(L) (A))z(L) 112 ,...., ~IIW:clr(Y - pC<L»)1I2 (~tr(I - A(L)(A)))2 (~tr(I - A(L)(A)))2 (17) and minimize VeL) with respect to A. Gu(1992a) showed that (since the variance is known once the mean is known here) that the unbiased risk estimate U (A) in Craven and Wahba can also be adapted to this problem as U(l)(A) = .!.IIW(l~/2(y - Pc(l»)112 + ~tr A(l)(A). (18) n c n He also proposed an alternating iteration, different than that described in Wahba(1990), namely, given eCl) = e(l)(A(l»), find A = ACl+l) to minimize (18). Given A(l+!) , do a Newton step to get eCl+1), get A(l+2) by minimizing (18), continue until convergence. He showed that the alternating iteration gave better estimates of A using V than the iteration in Wahba(1990), as measured by the [( L-distance. His results (with the alternating iteration) suggested U had somewhat of an advantage over V, and that is what we are using in the present work. Zhao et aI, this volume, have used V successfully with the alternating iteration. lThe definition of A there differs from the definition here by a factor of n/2. Please note the typographical error in (9.2.18) there where A should be 2A. 420 Wahba, Wang, Gu, Klein, and Klein 4. Smoothing spline analysis of variance (SS-ANOVA) In SS-ANOVA, /(t) = l(t1, ... , td) is decomposed as I(t) = I-' + L /a(ta) + L /a/3(ta, t/3) + ... (19) a a</3 where the terms in the expansion are uniquely determined by side conditions which generalize the side conditions ofthe usual ANOVA decompositions. Let the logit/(t) be of the form (19) where the terms are summed over Ct EM, Ct, f3 E M, etc. where M indexes terms which are chosen to be retained in the model after a model selection procedure. Then 1>..,8, an estimate of I, is obtained as the minimizer of £(y, 1>.,8) + )"J8 (I) (20) where J8(1) = L (J~lJa(fa) + L (J;JJa/3(fa/3) +... (21) aEM a,{3EM The Ja, Ja/3, ... are quadratic 'smoothness' penalty functionals, and the (J's satisfy a single constraint. For certain spline-like smoothness penalties, the minimizer of (20) is known to be in the span of a certain set of n functions, and the vector c of coefficients of these functions can (for fixed ().., (J)) be chosen by the Newton Raphson iteration. Both)" and the (J's are estimated by the unbiased risk estimate of Gu using RKPACK( available from netlibClresearch. att. com) as a subroutine at each Newton iteration. Details of smoothing spline ANOVA decompositions may be found in Wahba(1990) and in Gu and Wahba(1993) (also available by ftp to ftp.stat.wisc.edu, cd to pub/wahba , in the file ssanova.ps.Z). In Wahba et al(1993) op cit, we estimate the risk of diabetes given some of the attributes in the Pima-Indian data base. There M was chosen partly by a screening process using paramteric GLIM models and partly by a leaving out approximately 1/3 procedure. Continuing work involves development of confidence intervals based on Gu(1992b), development of numerical methods suitable for very large data sets based on Girard's(1991) randomized trace estimation, and further model selection issues. In the Figures we provide some preliminary analyses of data from the Wisconsin Epidemiological Study of Diabetic Retinopathy (WESDR, Klein et al 1988). The data used here is from people with early onset diabetes participating in the WESDR study. Figure 1(left) gives a plot of body mass index (bmi) (a measure of obesity) vs age (age) for 669 instances (subjects) in the WESDR study that had no diabetic retinopathy or non proliferative retinopathy at the start of the study. Those subjects who had (progressed) retinopathy four years later, are marked as * and those with no progression are marked as '. The contours are lines of estimated posterior standard deviation of the estimate p of the probability of progression. These contours are used to delineate a region in which p is deemed to be reliable. Glycosylated hemoglobin (gly), a measure of blood sugar control. was also used in the estimation of p. A model of the form p = eJ /(1 + eJ ), I(age, gly, bmi) = I-' + h(age) + b· gly + h(bmi) + ha(age, bmi) was selected using some of the screening procedures described in Wahba et al(1993), along with an examination of the estimated multiple smoothing parameters, which indicated that the linear term in gly was sufficient to describe the (quite strong) dependence on gly. Figure l(right) shows the estimated probability of progression "Soft" Classification with Smoothing Spline ANOVA 421 given by this model. Figure 2(left) gives cross sections of the fitted model of Figure 1(right), and Figure 2(right) gives another cross section, along with its confidence interval. Interesting observations can be made, for example, persons in their late 20's with higher gly and bmi are at greatest risk for progression of the disease . 10 20 30 40 50 60 age (yr) ...•. ... - . -.. ........ : -... ............ : ............. . ' . ..... ": .... : • : · · · • · · · · · · Figure 1: Left: Data and contours of constant posterior standard deviation at the median gly, as a function of age and bmi. Right: Estimated probability of progression at the median gly, as a function of age and bmi. q CD o o o q1 bmi q2bmi q3bmi q4bmi gy.q2 gy-q3 10 20 30 40 50 60 age (yr) 10 20 30 40 50 60 age (yr) l:jI,,-••• <:AJian bmi-median 10 20 30 40 50 60 age (yr) Figure 2: Left: Eight cross sections of the right panel of Figure 1, Estimated probability of progression as a function of age, at four levels of bmi by two of gly. q1, ... q4 are the quartiles at .125, .375, .625 and .875. Right: Cross section of the right panel of Figure 1 for bmi and gly at their medians, as a function of age, with Bayesian 'condifidence interval' (shaded) which generalizes Gu(1992b) to the multivariate case. 422 Wahba, Wang, Gu, Klein, and Klein Acknowledgements Supported by NSF DMS-9121003 and DMS-9301511, and NEI-NIH EY09946 and EY03083 References Craven, P. & Wahba, G. (1979), 'Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized crossvalidation', Numer. Math. 31,377-403. Girard, D. (1991), 'Asymptotic optimality of the fast randomized versions of GCV and C L in ridge regression and regularization', Ann. Statist. 19, 1950-1963. Gu, C. (1992a), 'Cross-validating non-Gaussian data', J. Comput. Graph. Stats. 1,169-179. Gu, C. (1992b), 'Penalized likelihood regression: a Bayesian analysis', Statistica Sinica 2,255-264. Gu, C. & Wahba, G. (1993), 'Smoothing spline ANOVA with component-wise Bayesian "confidence intervals''', J. Computational and Graphical Statistics 2, 1-2l. Kimeldorf, G. & Wahba, G. (1970), 'A correspondence between Bayesian estimation of stochastic processes and smoothing by splines', Ann. Math. Statist. 41,495-502. Klein, R., Klein, B., Moss, S. Davis, M., & DeMets, D. (1988), Glycosylated hemoglobin predicts the incidence and progression of diabetic retinopathy, JAMA 260, 2864-287l. O'Sullivan, F., Yandell, B. & Raynor, W. (1986), 'Automatic smoothing of regression functions in generalized linear models', J. Am. Stat. Soc. 81, 96-103. Wahba, G. (1980), Spline bases, regularization, and generalized cross validation for solving approximation problems with large quantities of noisy data, in W. Cheney, ed., 'Approximation Theory III', Academic Press, pp. 905-912. Wahba, G. (1985), 'A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem', Ann. Statist. 13, 13781402. Wahba, G. (1990), Spline Models for Observational Data, SIAM. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59. Wahba, G. (1992), Multivariate function and operator estimation, based on smoothing splines and reproducing kernels, in M. Casdagli & S. Eubank, eds, 'Nonlinear Modeling and Forecasting, SFI Studies in the Sciences of Complexity, Proc. Vol XII' , Addison-Wesley, pp. 95-112. Wahba, G., Gu, C., Wang, Y. & Chappell, R. (1993), Soft classification, a. k. a. risk estimation, via penalized log likelihood and smoothing spline analysis of variance, to appear, Proc. Santa Fe Workshop on Supervised Machine Learning, D. Wolpert and A. Lapedes, eds, and Proc. CLNL92, T. Petsche, ed, with permission of all eds.
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Robot Learning: Exploration and Continuous Domains David A. Cohn MIT Dept. of Brain and Cognitive Sciences Cambridge, MA 02139 The goal of this workshop was to discuss two major issues: efficient exploration of a learner's state space, and learning in continuous domains. The common themes that emerged in presentations and in discussion were the importance of choosing one's domain assumptions carefully, mixing controllers/strategies, avoidance of catastrophic failure, new approaches with difficulties with reinforcement learning, and the importance of task transfer. 1 Domain assumptions Andrew Moore (CMU) discussed the problem of standardizing and making explicit the set of assumptions that researcher makes about his/her domain. He suggested that neither "fewer assumptions are better" nor "more assumptions are better" is a tenable position, and that we should strive to find and use standard sets of assumptions. With no such commonality, comparison of techniques and results is meaningless. Under Moore's guidance, the group discussed the possibility of designing an algorithm which used a number of well-chosen assumption sets and switched between them according to their empirical validity. Suggestions were made to draw on the AI approach of truth maintenance systems. This theme of detecting failure of an assumption set/strategy was echoed in the discussion on mixing controllers and avoiding failure (described below). 2 Mixing controllers and strategies Consensus appeared to be against using single monolithic approaches, and in favor of mixing controllers. Spatial mixing resulted in local models, as advocated by Stefan Schaal (MIT) using locally weighted regression. Controllers could also be mixed over the entire domain. Jeff Schneider (Rochester) discussed mixing a na'ive feedback controller with a "coaching signal." Combining the coached controller with an un coached one further improved performance. During the main conference, Satinder Singh (MIT) described a controller that learned by reinforcement to mix the strategies of two "safe" but suboptimal controllers, thus avoiding unpleasant surprises and catastrophic failure. 1169 1170 Cohn 3 Avoiding failure The issue of det,ecting impending failure and avoiding catastrophic failure clarified differences in several approaches. A learning strategy that learns in few trials is useless on a real robot if the initial trials break the robot by crashing it into walls. Singh's approach has implicit failure avoidance, but may be hampered by an unnecessarily large margin of safety. Terry Sanger (JPL) discussed a trajectory extension algorithm by which a controller could smoothly "push the limits" of its performance, and detect impending failure of the control strategy. 4 Reinforcement learning Reinforcement learning seems to have come into its own, with people realizing the diverse ways in which it may be applied to problems. Long-Ji Lin (Siemens) described his group's unusual but successful application of reinforcement learning in landmark-based navigation. The Siemens RatBot uses reinforcement to select landmarks on the basis of their recognizability and their value to the eventual precision of position estimation. The reinforcement signal is simply the cost and the robot's final position error after it has used a set of landmarks. Jose del R. Millan (JRC) presented an approach similar to Schneider's, but training a neural controller with reinforcement learning. As the controller's performance improves, it supplants the mobile robot's reactive "instincts," which are designed to prevent catastrophic failure. With new applications, however, come new pitfalls. Leemon Baird (WPAFB) showed how standard reinforcement learning approaches can fail when adapted to exploration in continuous time. He then described the "advantage updating" algorithm which was designed to work in noisy domains with continuous or small time steps. The issue of exploration in continuous space, especially with noise, has not been as easily addressed. Jiirgen Schmidhuber (TUM) described the approach one should take if interested solely in exploration: use prior information gain as a reinforcement signal to decide on an "optimal" action. The ensuing discussion centered on the age-old and intractable tradeoff between exploration and exploitation. Final consensus was that we, as a group, should become more familiar with the literature on dual control, which addresses exactly this issue. 5 Task transfer An unorchestrated theme that emerged from the discussion was the need to address, or even define, task transfer. As with last year's workshop on Robot Learning, it was generally agreed that "one-task learning" is not a suitable goal when designing a learning robot. During the discussion, Long-Ji Lin (Siemens) and Lori Pratt (CSM) described several types of task transfer that are considered in the literature. These included model learning for multiple tasks, hierarchical control and learning, and concept (or bias) sharing across tasks.
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Lower Boundaries of Motoneuron Desynchronization via Renshaw Interneurons Mitchell Gil Maltenfort It Dept. of Biomedical Engineering Northwestern University Evanston, IT.. 60201 c. J. Heckman V. A. Research Service Lakeside Hospital and Dept. of Physiology Northwestern University Chicago, IT.. 60611 Abstract Robert E. Druzinsky Dept. of Physiology Northwestern University Chicago, IT.. 60611 w. Zev Rymer Dept. of Physiology and Biomedical Engineering Northwestern University Chicago, IT.. 60611 Using a quasi-realistic model of the feedback inhibition ofmotoneurons (MNs) by Renshaw cells, we show that weak inhibition is sufficient to maximally desynchronize MNs, with negligible effects on total MN activity. MN synchrony can produce a 20 - 30 Hz peak in the force power spectrum, which may cause instability in feedback loops. 1 INTRODUCTION The structure of the recurrent inhibitory connections from Renshaw cells (RCs) onto motoneurons (MNs) (Figure 1) suggests that the RC forms a simple negative feedback * send mail to: Mitchell G. Maltenfort, SMPP room 1406, Rehabilitation Insitute of Chicago, 345 East Superior Street, Chicago, IT.. 60611. Email address is mgm@nwu.edu 535 536 Maltenfort, Druzinsky, Heckman, and Rymer loop. Past theoretical work has examined possible roles of this feedback in smoothing or gain regulation of motor output (e.g., Bullock and Contreras-Vidal, 1991; Graham and Redman, 1993), but has assumed relatively strong inhibitory effects from the RC. Experimental observations (Granit et al.,1961) show that maximal RC activity can only reduce MN frring rates by a few impulses per second. although this weak inhibition is sufficient to affect the timing of MN fuings, reducing the probability that any two MNs will fire simultaneously (Adam et al., 1978; Windhorst et al., 1978). In this study, simulations were used to examine the impact of RC inhibition on MN frring synchrony and to predict the effects of such synchrony on force output. + Figure 1: Simplified Schematic of Recurrent Inhibition 2 CONSTRUCTION OF THE MODEL 2.1 MODELING OF INDIVIDUAL NEURONS The integrate-and-fIre neuron model of MacGregor (1987) adequately mimics specific frring patterns. Coupled first-order differential equations govern membrane potential and afterhyperpolarization (AHP) based on injected current and synaptic inputs. A spike is frred when the membrane potential crosses a threshold. The model was modified to include a membrane resistance in order to model MN s of varying current thresholds. Membrane resistance and time constants of model MNs were set to match published data (Gustaffson and Pinter, 1984). The parameters governing AHPs were adjusted to agree with observations from single action potentials and steady-state current-rate plOts (Heckman and Binder, 1991). Realistic frring behavior could be generated for MNs with current thresholds of 4 - 40 nA. Although there are no direct measurements of RC membrane properties available, appropriate parameters were estimated by extrapolation from the MN parameter set. The simulated RC has a 30 ms AHP and a current-rate plot matching that reported by Hultborn and Pierrot-Deseilligny (1979). Spontaneous frring of 8 pps is produced in the model by setting the RC firing threshold to 0.01 mV below resting potential; in vivo Lower Boundaries of Motoneuron Desynchronization via Renshaw Intemeurons 537 this fuing is likely due to descending inputs (Hamm et al., 1987a), but there is no quantitative description of such inputs. The RCs are assumed to be homgeneous. 2.2 CONNECTIVITY OF THE POOL Simulated neurons were arranged along a 16 by 16 grid. The network consists of 256 MNs and 64 RCs, with the RCs ordered on even-numbered rows and columns; as a result, the MN - RC connections are inhomogeneous along the pool. For each trial, MN pools were randomly generated following the distribution of MN current thresholds for a model of the cat medial gastrocnemius motor pool (Heckman and Binder, 1991). Communication between neurons is mediated by synaptic conductances which open when a presynaptic cell fues, then decay exponentially. MN excitation of RCs was set to produce RC fuing rates S; 190 pps (Cleveland et aI., 1981) which linearly increase with MN activity (Cleveland and Ross, 1977). MN activation of RCs scales inversely with MN current threshold (Hultbom et al., 1988). Connectivity is based on observations that synapses from RCs to MNs have a longer spatial range than the reverse (reviewed in Windhorst, 1990). The IPSPs produced by single MN fIrings are 4 - 6 times larger than those produced by single RC fIrings (Hamm et al., 1987b; van Kuelen, 1981). In the model, each MN excites RCs within one column or row of itself, and each RC inhibits MNs up to two rows or columns away; thus, each MN excites 1 - 4 RCs (mean 2.25) and receives feedback from 4 - 9 RCs (mean 6.25). 2.3 ACTIVATION OF THE POOL The MNs are activated by applied step currents. Although this is not realistic, it is computationally efficient. An option in the simulation program allows for the addition of bandlimited noise to the activation current, to simulate a synchronizing common synaptic input. This signal has an rms value of 3% of the mean applied current and is lOW-pass mtered with a cutoff of 30 Hz. This allows us look at the effects due purely to RC activity and to establish which effects persist when the MN pool is being actively synchronized. 3 EFFECTS OF RC STRENGTH ON MN SYNCHRONY 3.1 DEFINITION OF SYNCHRONY COEFFICIENT Consider the total number of spikes frred by the MN pool as a time series. During synchronous firing, the MN spikes will clump together and the time series will have regions of very many or very few MN spikes. When the MNs are de synchronized, the range of spike counts in each time bin will contract towards the mean. It follows that a simple measure of MN synchrony is the the coeffIcient of variation (c. v. = s.d. I mean) of the time series formed by the summed MN activity. Figure 2 shows typical MN pool fIring before and after RC feedback inhibition is added; the changes in "clumping" described above are quite visible in the two plots. 538 Maltenfort, Druzinsky, Heckman, and Rymer 3.2 "PLATEAU" OF DESYNCHRONIZATION The magnitude of the synaptic conductance from RCs onto MNs was changed from zero to twice physiological in order to compare the effects of 'weak' and 'strong' recurrent inhibition. At activation levels sufficient to recruit at least 70% of MNs in the pool (mean tiling rate ~ 15 pps), a surprising plateau effect was seen. The synchrony coefficient fell off with RC synaptic conductance until the physiological level was reached, and then no further de synchronization was seen. The effect persisted when synchronizing noise was added (Figure 3). At activation levels sufficient to show this plateau, this "comer" inhibition level was always the same. Synchronized Firing (no RC inhibition) 50 O~~~u-~~~~~~~~~~~~~~~~~~ o 50 100 150 200 Desynchronized Firing (RC inhibition added) 20 O~----~--~--------~~~----~u-~~~~ o 50 100 150 200 Time (ms) Figure 2: Comparison of Synchronous and Asynchronous MN Firing At this comer level, the decrease in mean MN firing rate was ~ 1 pps and not statistically significant. There was also no discernible change in the percentage of the MN pool active. The c.v. of the interspike interval of single MN filings during constant activation is ~ 2.5 % even with RCs active - this implies that the RC system finds an optimal arrangement of the MN fuings and then performs few if any further shifts. When synchronizing noise is added, the RC effect on the interspike interval is swamped by the effect of the synchronizing random input. Figure 4 shows the effect of increasing MN activation on the synchrony coefficients before and after RC inhibition is added. The change is statistically significant at all levels, but is only large at higber levels as discussed above. As activation of the MNs Lower Boundaries of Motoneuron Desynchronization via Renshaw Intemeurons 539 increases, the "before" level of synchrony increases while the "after" level seems to move asymptotically towards a minimum level of about 0.35. This minimum level of MN synchrony, as well as the dependence of the effect on the activation level of the pool, suggests that a certain amount of synchrony becomes inevitable as more MNs are activated and fIre at higher rates. 1.4 * 1.2 1 synchronizing noise added 0.8 0.6 0.4 o 0.002 0.004 0.006 0.008 0.01 RC Synaptic Conductance ijJ.Siemens) Figure 3: MN Firing Synchrony vs. RC Strength 4 EFFECTS OF MN SYNCHRONY ON MUSCLE FORCE 4.1 MODELING OF FORCE OUTPUT Single twitches of motor units are modeled with a second-order model, f(t) = Be-tit, t where the amplitude F and time constant t are matched to MN current threshold according to the model of Heckman and Binder (1991). A rate-based gain factor adapted from Fuglevand (1989) produces fused tetanus at high fuing rates. The tenfold difference in current thresholds maps to a fifty-fold difference in twitch forces. Twitch time constants range 30-90 ms. 540 Maltenfort, Druzinsky, Heckman, and Rymer 4.1 EFFECTS OF RECURRENT INHmITION ON FORCE The force model sharply low-pass fllters the neural input signal (S 5 Hz). As a result, the c.v. of the force output is much lower than that of the associated MN input (S 0.01). Although the plot of force c.v. vs. RC strength during constant activation follows the curve in Figure 3, adding synchronizing noise removes any correlation between force .c.v. and magnitude of recurrent inhibition. The effect of recurrent inhibition on mean force is similar to that on the mean firing rate: small (S5 % decrease) and generally not statistically significant 1.8 0 1.6 1.4 ~ c 1.2 ~ u before recurrent inhibition ~ 1 ~ bO :5 0.8 ~ ~ 0.6 0.4 0.2 5 10 15 20 25 30 35 Activation Current (nA) Figure 4: Effects of MN Activation on Synchrony Before and After Recurrent Inhibition When the change in synchrony due to RCs is large, a peak appears in the force power spectrum in the range 20 - 30 Hz. This peak is reduced by RCs even when the MN pool is being actively synchronized (Figure 5). Peaks in the force spectrum match peaks in the spectrum of pooled MN activity, suggesting the effect is due to synchronous MN ruing. Although the magnitude of this peak is small (S 0.5% of mean force), its relatively high frequency suggests that in derivative feedback - where spectral components are multiplied by 21t times their frequency - its impact could be substantial. The feedback loop which measures muscle stretch contains such a derivative component (Hook and Rymer, 1981). Lower Boundaries of Motoneuron Desynchronization via Renshaw Intemeurons 541 5 DISCUSSION The preceding shows that the ostensibly weak recurrent inhibition is sufficient to sharply reduce the maximum number of synchronous Irrings of a neuron population, while having a negligible effect on the total population activity. This has a broad implication for neural networks in that it suggests the existence of a "switching mechanism" which forces the peaks in the output of an ensemble of neurons to remain below a threshold level without significantly suppressing the total ensemble activity. One possible role for such a mechanism would be in the accommodation to a step or ramp increase in a stimulus. The initial increase synchronizes the neural signal from the receptor, which is then desyncbronized by the recurrent inhibition. The synchronized ruing phase would be sufficient to excite a target neuron past its ruing threshold, but after that, the desyncbronized neural signal would remain well below the target's threshold. 0.2 0.15 0.1 0.05 O~--------~--------~----------~------~ o 10 20 30 40 Frequency (Hz) Figure 5: Recurrent Inhibition Reduces Spectral Peak. 95% confidence limit of means plotted, solid lines before recurrent inhibition and dashed lines after. Acknowledgments The authors are indebted to Dr. Tom Buchanan for use of his IBM RS/6000 workstation. This work was supported by NIH grants NS28076-02 and NS30295-01. 542 Maltenfort, Druzinsky, Heckman, and Rymer References Adam D, Windhorst U, Inbar GF: The effects of recurrent inhibition on the crosscorrelated flring patterns of motoneurons (and their relation to signal transmission in the spinal cord-muscle channel). Bioi. Cybern., 29: 229-235, 1978. Bullock D, Contreras-Vidal J: How spinal neural networks reduce discrepancies between motor intention and motor realization. Tech.Report CAS/CNS-91-023, Boston U., 1991. Cleveland S, Kuschmierz A, Ross H-G: Static input-output relations in the spinal recurrent inhibitory pathway. Bioi. Cybern., 40: 223-231, 1981. Cleveland S, Ross H-G: Dynamic properties of Renshaw cells: Frequency response characteristics. Bioi. Cybem., 27: 175-184, 1977. Fuglevand AJ: A motor unit pool model: relationship of neural control properties to isometric muscle tension and the electromyogram. Ph.D. Thesis, U. of Waterloo, 1989. Graham BP, Redman SJ: Dynamic behaviour of a model of the muscle stretch reflex. Neural Networks, 6: 947-962, 1993. Granit R, Haase J, Rutledge L T: Recurrent inhibition in relation to frequency of flring and limitation of discharge rate of extensor motoneurons. J. Physiol., 158: 461-475, 1961. Gustaffson B, Pinter MJ: An investigation of threshold properties among cat spinal amotoneurons. J. Physiol., 357: 453-483, 1984. Hamm 1M, Sasaki S, Stuart DG, Windhorst U, Yuan C-S: Distribution of single-axon recurrent inhibitory post-synaptic potentials in the cat. J. Physiol., 388: 631-651,1987a. Hamm 1M, Sasaki S, Stuart 00, Windhorst U, Yuan C-S: The measurement of single motor-axon recurrent inhibitory post-synaptic potentials in a single spinal motor nucleus in the cat. J. Physiol., 388: 653-664, 1987b . Heckman CJ, Binder MD: Computer simulation of the steady-state input-output function of the cat medial gastrocnemius motoneuron pool. J. Neurophysiol., 65: 952-967, 1991. Houk JC, Rymer WZ: Chapter 8: Neural control of muscle length and tension. In Handbook of Physiology: the Nervous System II pt. I, ed.VB Brooks. Am. Physiol. Soc., Bethesda, MD, 1981. Hultborn H, Peirrot-Deseillgny E: Input-output relations in the pathway of recurrent inhibition to motoneurons in the cat. J. Physiol., 297: 267-287, 1979. MacGregor RJ: Neural and Brain Modeling. Academic Press, San Diego, 1987. Van Kuelen LCM: Autogenetic recurrent inhibition of individual spinal motoneurons of the cat. Neurosci. Lett., 21: 297-300, 1981. Windhorst U: Activation of Renshaw cells. Prog. in Neurobiology, 35: 135-179, 1990. Windhorst U, Adam D, Inbar GF: The effects of recurrent inhibitory feedback in shaping discharge patterns of motoneurones excited by phasic muscle stretches. Bioi. Cybem., 29: 221-227, 1978.
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Catastrophic interference in connectionist networks: Can it be predicted, can it be prevented? Robert M. French Computer Science Department Willamette University Salem, Oregon 97301 french@willamette.edu 1 OVERVIEW Catastrophic forgetting occurs when connectionist networks learn new information, and by so doing, forget all previously learned information. This workshop focused primarily on the causes of catastrophic interference, the techniques that have been developed to reduce it, the effect of these techniques on the networks' ability to generalize, and the degree to which prediction of catastrophic forgetting is possible. The speakers were Robert French, Phil Hetherington (Psychology Department, McGill University, het@blaise.psych.mcgill.ca), and Stephan Lewandowsky (Psychology Department, University of Oklahoma, lewan@constellation.ecn.uoknor.edu). 2 PROTOTYPE BIASING AND FORCED SEPARATION OF HIDDEN-LAYER REPRESENTATIONS French indicated that catastrophic forgetting is at its worst when high representation overlap at the hidden layer combines with significant teacher-output error. He showed that techniques to reduce this overlap tended to decrease catastrophic forgetting. Activation sharpening, a technique that produces representations having a few highly active nodes and many low-activation nodes, was shown to be effective because it reduced representation overlap. However, this technique was ineffective for large data sets because creating localized representations reduced the number of possible hidden-layer representations. Hidden layer representations that were more distributed but still highly separated were needed. French introduced prototype biasing, a technique that uses a separate network to learn a prototype for each teacher pattern. Hidden-layer representations of new items are made to resemble their prototypes. Each representation is also "separated" from the representation of the previously encountered pattern according to the difference between the respective teachers. This technique produced hidden-layer representations that 1176 Catastrophic Interference in Connectionist Networks 1177 were both distributed and well separated. The result was a significant decrease in catastrophic forgetting. 3 ELIMINATING CATASTROPHIC INTERFERENCE BY PRETRAINING Hetherington presented a technique that consisted of prior training of the network on a large body of items of the same type as the new items in the sequential learning task. Hetherington measured the degree of actual forgetting, as did all of the authors, by the method of savings, i.e., by determining how long the network takes to relearn the original data set that has been "erased" by learning the new data. He showed that when networks are given the benefit of relevant prior knowledge, the representations of the new items are constrained naturally and interference may be virtually eliminated. The previously encoded knowledge causes new items to be encoded in more orthogonal manner (i.e., with less mutual overlap) than in a naive (Le., non-pretrained) network. The resulting decrease in representation overlap produced the virtual elimination of catastrophic forgetting. Hetherington also presented another technique that substantially reduced catastrophic interference in the sequential learning task. Learning of new items takes place in a windowed, or overlapping fashion. In other words, as new items are learned the network continues learning on the most recently presented items. 4 THE RELATIONSHIP BETWEEN INTERFERENCE AND GENERALIZATION Lewandowsky examined the hypothesis that generalization is compromised in networks that had been "manipulated" to decrease catastrophic interference by creating semi-distributed (i.e., only partially overlapping) representations at the hidden layer. He gave a theoretical analysis of the relationship between interference and generalization and then presented results from several different simulations using semi-distributed representations. His conclusions were that semi-distributed representations can significantly reduce catastrophic interference in backpropagation networks without diminishing their generalization abilities. This was only true, however, for techniques (e.g., activation sharpening) that reduced interference by creating a more robust final weight pattern but that did not change the activation surfaces of the hidden units. On the other hand, in models where interference is reduced by eliminating overlap between receptive fields of static hidden units (i.e., by altering their response surface), generalization abilities are impaired. In addition, Lewandowsky presented a technique that relied on orthogonalizing the input vectors to a standard backpropagation network by converting standard asymmetric input vectors (each node at 0 or 1) to symmetric input vectors (each input node at -lor 1). This technique was also found to significantly reduce catastrophic interference.
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Signature Verification using a "Siamese" Time Delay Neural Network Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Sickinger and Roopak Shah AT&T Bell Laboratories Holmdel, N J 07733 jbromley@big.att.com Copyrighte, 1994, American Telephone and Telegraph Company used by permission. Abstract This paper describes an algorithm for verification of signatures written on a pen-input tablet. The algorithm is based on a novel, artificial neural network, called a "Siamese" neural network. This network consists of two identical sub-networks joined at their outputs. During training the two sub-networks extract features from two signatures, while the joining neuron measures the distance between the two feature vectors. Verification consists of comparing an extracted feature vector ~ith a stored feature vector for the signer. Signatures closer to this stored representation than a chosen threshold are accepted, all other signatures are rejected as forgeries. 1 INTRODUCTION The aim of the project was to make a signature verification system based on the NCR 5990 Signature Capture Device (a pen-input tablet) and to use 80 bytes or less for signature feature storage in order that the features can be stored on the magnetic strip of a credit-card. Verification using a digitizer such as the 5990, which generates spatial coordinates as a function of time, is known as dynamic verification. Much research has been carried out on signature verification. Function-based methods, which fit a function to the pen trajectory, have been found to lead to higher performance while parameter-based methods, which extract some number of parameters from a signa737 738 Bromley, Guyon, Le Cun, Sackinger, and Shah ture, make a lower requirement on memory space for signature storage (see Lorette and Plamondon (1990) for comments). We chose to use the complete time extent of the signature, with the preprocessing described below, as input to a neural network, and to allow the network to compress the information. We believe that it is more robust to provide the network with low level features and to allow it to learn higher order features during the training process, rather than making heuristic decisions e.g. such as segmentation into balistic strokes. We have had success with this method previously (Guyon et al., 1990) as have other authors (Yoshimura and Yoshimura, 1992). 2 DATA COLLECTION All signature data was collected using 5990 Signature Capture Devices. They consist of an LCD overlayed with a transparent digitizer. As a guide for signing, a 1 inch by 3 inches box was displayed on the LCD. However all data captured both inside and outside this box, from first pen down to last pen up, was returned by the device. The 5990 provides the trajectory of the signature in Cartesian coordinates as a function of time. Both the trajectory of the pen on the pad and of the pen above the pad (within a certain proximity of the pad) are recorded. It also uses a pen pressure measurement to report whether the pen is touching the writing screen or is in the air. Forgers usually copy the shape of a signature. Using such a tablet for signature entry means that a forger must copy both dynamic information and the trajectory of the pen in the air. Neither of these are easily available to a forger and it is hoped that capturing such information from signatures will make the task of a forger much harder. Strangio (1976), Herbst and Liu (1977b) have reported that pen up trajectory is hard to imitate, but also less repeatable for the signer. The spatial resolution of signatures from the 5990 is about 300 dots per inch, the time resolution 200 samples per second and the pad's surface is 5.5 inches by 3.5 inches. Performance was also measured using the same data treated to have a lower resolution of 100 dots per inch. This had essentially no effect on the results. Data was collected in a university and at Bell Laboratories and NCR cafeterias. Signature donors were asked to sign their signature as consistently as possible or to make forgeries. When producing forgeries, the signer was shown an example of the genuine signature on a computer screen. The amount of effort made in producing forgeries varied. Some people practiced or signed the signature of people they knew, others made little effort. Hence, forgeries varied from undetectable to obviously different. Skilled forgeries are the most difficult to detect, but in real life a range of forgeries occur from skilled ones to the signatures of the forger themselves. Except at Bell Labs., the data collection was not closely monitored so it was no surprise when the data was found to be quite noisy. It was cleaned up according to the following rules: • Genuine signatures must have between 80% and 120% of the strokes of the first signature signed and, if readable, be of the same name as that typed into the data collection system. (The majority of the signatures were donated by residents of North America, and, typical for such signatures, were readable.) The aim of this was to remove signatures for which only Signature Verification Using a "Siamese" Time Delay Neural Network 739 some part of the signature was present or where people had signed another name e.g. Mickey Mouse. • Forgeries must be an attempt to copy the genuine signature. The aim of this was to remove examples where people had signed completely different names. They must also have 80% to 120% of the strokes of the signature . • A person must have signed at least 6 genuine signatures or forgeries. In total, 219 people signed between 10 and 20 signatures each, 145 signed genuines, 74 signed forgeries. 3 PREPROCESSING A signature from the 5990 is typically 800 sets of z, y and pen up-down points. z(t) and y(t) were originally in absolute position coordinates. By calculating the linear estimates for the z and y trajectories as a function of time and subtracting this from the original z and y values, they were converted to a form which is invariant to the position and slope of the signature. Then, dividing by the y standard deviation provided some size normalization (a person may sign their signature in a variety of sizes, this method would normalize them). The next preprocessing step was to resample, using linear interpolation, all signatures to be the same length of 200 points as the neural network requires a fixed input size. Next, further features were computed for input to the network and all input values were scaled so that the majority fell between + 1 and -1. Ten different features could be calculated, but a subset of eight were used in different experiments: feature 1 pen up = -1 i pen down = +1, (pud) feature 2 x position, as a difference from the linear estimate for x(t), normalized using the standard deviation of 1/, (x) feature 3 y position, as a difference from the linear estimate for y(t), normalized using the standard deviation of 1/, (y) feature 4 speed at each point, (spd) feature 5 centripetal acceleration, (ace-c) feature 6 tangential acceleration, (acc-t) feature 7 the direction cosine of the tangent to the trajectory at each point, (cosS) feature 8 the direction sine of the tangent to the trajectory at each point, (sinS) feature 9 cosine of the local curvature of the trajectory at each point, (cost/J) feature 10 sine of the local curvature of the trajectory at each point, (sint/J) In contrast to the features chosen for character recognition with a neural network (Guyon et al., 1990), where we wanted to eliminate writer specific information, the features such as speed and acceleration were chosen to carry information that aids the discrimination between genuine signatures and forgeries. At the same time we still needed to have some information about shape to prevent a forger from breaking the system by just imitating the rhythm of a signature, so positional, directional amd curvature features were also used. The resampling of the signatures was such as to preserve the regular spacing in time between points. This method penalizes forgers who do not write at the correct speed. 740 Bromley, Guyon, Le Cun, Sackinger, and Shah TARGET ..... t ?---------' :01 fltt be11 . 2OOu .... • -.... ~ ~ ... beUli Figure 1: Architecture 1 consists of two identical time delay neural networks. Each network has an input of 8 by 200 units, first layer of 12 by 64 units with receptive fields for each unit being 8 by 11 and a second layer of 16 by 19 units with receptive fields 12 by 10. 4 NETWORK ARCHITECTURE AND TRAINING The Siamese network has two input fields to compare two patterns and one output whose state value corresponds to the similarity between the two patterns. Two separate sub-networks based on Time Delay Neural Networks (Lang and Hinton, 1988, Guyon et al. 1990) act on each input pattern to extract features, then the cosine of the angle between two feature vectors is calculated and this represents the distance value. Results for two different subnetworks are reported here. Architecture 1 is shown in Fig 1. Architecture 2 differs in the number and size of layers. The input is 8 by 200 units, the first convolutional layer is 6 by 192 units with each unit's receptive field covering 8 by 9 units of the input. The first averaging layer is 6 by 64 units, the second convolution layer is 4 by 57 with 6 by 8 receptive fields and the second averaging layer is 4 by 19. To achieve compression in the time dimension, architecture 1 uses a sub-sampling step of 3, while architecture 2 uses averaging. A similar Siamese architecture was independently proposed for fingerprint identification by Baldi and Chauvin (1992). Training was carried out using a modified version of back propagation (LeCun, 1989). All weights could be learnt, but the two sub-networks were constrained to have identical weights. The desired output for a pair of genuine signatures was for a small angle (we used cosine=l.O) between the two feature vectors and a large angle Signature Verification Using a "Siamese" Time Delay Neural Network 741 Table 1: Summary of the Training. Note: GA is the percentage of genuine signature pairs with output greater than 0, FR the percentage of genuine:forgery signature pairs for which the output was less than O. The aim of removing all pen up points for Network 2 was to investigate whether the pen up trajectories were too variable to be helpful in verification. For Network 4 the training simulation crashed after the 42nd iteration and was not restarted. Performance may have improved if training had continued past this point. 2, arc 1 3, arc 1 pu acc-c acc-t sp cosH sinS cos'" sin~ 4, arc 1 same as network 3, but a larger training set 5, arc 2 same as 4, except architecture 2 was used (we used cosine= -0.9 and -1.0) if one of the signatures was a forgery. The training set consisted of 982 genuine signatures from 108 signers and 402 forgeries of about 40 of these signers. We used up to 7,701 signature pairsj 50% genuine:genuine pairs, 40% genuine:forgery pairs and 10% genuine:zero-effort pairs. 1 The validation set consisted of 960 signature pairs in the same proportions as the training set. The network used for verification was that with the lowest error rate on the validation set. See Table 1 for a summary of the experiments. Training took a few days on a SPARe 1+. 5 TESTING When used for verification, only one sub-network is evaluated. The output of this is the feature vector for the signature. The feature vectors for the last six signatures signed by each person were used to make a multivariate normal density model of the person's signature (see pp. 22-27 of Pattern Classification and Scene Analysis by Duda and Hart for a fuller description of this). For simplicity, we assume that the features are statistically independent, and that each feature has the same variance. Verification consists of comparing a feature vector with the model of the signature. The probability density that a test signature is genuine, p-yes, is found by evaluating 1 zero-effort forgeries, also known as random forgeries, are those for which the forger makes no effort to copy the genuine signature, we used genuine signatures from other signers to simulate such forgeries. 742 Bromley, Guyon, Le Cun, Sackinger, and Shah 100 10 I 10 70 c! J 10 50 15 • 40 f 30 20 10 0 I lie lie a. SI2 90 sa sa ... 82 80 Percentage 01 Genuine Signatures Accepted Figure 2: Results for Networks 4 (open circles) and 5 (closed circles). The training of Network 4 was essentially the same as for Network 3 except that more data was used in training and it had been cleaned of noise. They were both based on architecture 1. Network 5 was based on architecture 2. The signature feature vector from this architecture is just 4 by 19 in size. the normal density function. The probability density that a test signature is a forgery, p-no, is assumed, for simplicity, to be a constant value over the range of interest. An estimate for this value was found by averaging the p-yes values for all forgeries. Then the probability that a test signature is genuine is p-yesj(p-yes + pno). Signatures closer than a chosen threshold to this stored representation are accepted, all other signatures are rejected as forgeries. Networks 1, 2 and 3, all based on architecture I, were tested using a set of 63 genuine signatures and 63 forgeries for 18 different people. There were about 4 genuine test signatures for each of the 18 people, and 10 forgeries for 6 of these people. Networks 1 and 2 had identical training except Network 2 was trained without pen up points. Network 1 gave the better results. However, with such a small test set, this difference may be hardly significant. The training of Network 3 was identical to that of Network I, except that x and y were used as input features, rather than acc-c and acc-t. It had somewhat improved performance. No study was made to find out whether the performance improvement came from using x and y or from leaving out acc-c and acc-t. Plamondon and Parizeau (1988) have shown that acceleration is not as reliable as other functions. Figure 2 shows the results for Networks 4 and 5. They were tested using a set of 532 genuine signatures and 424 forgeries for 43 different people. There were about 12 genuine test signatures for each person, and 30 forgeries for 14 of the people. This graph compares the performance of the two different architectures. It takes 2 to 3 minutes on a Sun SPARC2 workstation to preprocess 6 signatures, Signature Verification Using a "Siamese" Time Delay Neural Network 743 collect the 6 outputs from the sub-network and build the normal density model. 6 RESULTS Best performance was obtained with Network 4. With the threshold set to detect 80% of forgeries, 95.5% of genuine signatures were detected (24 signatures rejected). Performance could be improved to 97.0% genuine signatures detected (13 rejected) by removing all first and second signature from the test set 2. For 9 of the remaining 13 rejected signatures pen up trajectories differed from the person's typical signature. This agrees with other reports (Strangio, 1976 Herbst and Liu, 1977b) that pen up trajectory is hard to imitate, but also a less repeatable signature feature. However, removing pen up trajectories from training and test sets did not lead to any improvement (Networks 1 and 2 had similar performance), leading us to believe that pen up trajectories are useful in some cases. Using an elastic matching method for measuring distance may help. Another cause of error came from a few people who seemed unable to sign consistently and would miss out letters or add new strokes to their signature. The requirement to represent a model of a signature in 80 bytes means that the signature feature vector must be encodable in 80 bytes. Architecture 2 was specifically designed with this requirement in mind. Its signature feature vector has 76 dimensions. When testing Network 5, which was based on this architecture, 50% of the outputs were found (surprisingly) to be redundant and the signature could be represented by a 38 dimensional vector with no loss of performance. One explanation for this redundancy is that, by reducing the dimension of the output (by not using some outputs), it is easier for the neural network to satisfy the constraint that genuine and forgery vectors have a cosine distance of -1 (equivalent to the outputs from two such signatures pointing in opposite directions). These results were gathered on a Sun SPARC2 workstation where the 38 values were each represented with 4 bytes. A test was made representing each value in one byte. This had no detrimental effect on the performance. Using one byte per value allows the signature feature vector to be coded in 38 bytes, which is well within the size constraint. It may be possible to represent a signature feature vector with even less resolution, but this was not investigated. For a model to be updatable (a requirement of this project), the total of all the squares for each component of the signature feature vectors must also be available. This is another 38 dimensional vector. From these two vectors the variance can be calculated and a test signature verified. These two vectors can be stored in 80 bytes. 7 CONCLUSIONS This paper describes an algorithm for signature verification. A model of a person's signature can easily fit in 80 bytes and the model can be updated and become more accurate with each successful use of the credit card (surely an incentive for people to use their credit card as frequently as possible). Other beneficial aspects of this verification algorithm are that it is more resistant to forgeries for people who sign 2people commented that they needed to sign a few time to get accustomed to the pad 744 Bromley, Guyon, Le Cun, Sackinger, and Shah consistently, the algorithm is independent of the general direction of signing and is insensitive to changes in size and slope. As a result of this project, a demonstration system incorporating the neural network signature verification algorithm was developed. It has been used in demonstrations at Bell Laboratories where it worked equally well for American, European and Chinese signatures. This has been shown to commercial customers. We hope that a field trial can be run in order to test this technology in the real world. Acknowledgements All the neural network training and testing was carried out using SN2.6, a neural network simulator package developed by Neuristique. We would like to thank Bernhard Boser, John Denker, Donnie Henderson, Vic Nalwa and the members of the Interactive Systems department at AT&T Bell Laboratories, and Cliff Moore at NCR Corporation, for their help and encouragement. Finally, we thank all the people who took time to donate signatures for this project. References P. Baldi and Y. Chauvin, "Neural Networks for Fingerprint Recognition", Neural Computation,5 (1993). R. Duda and P. Hart, Pattern Classification and Scene Analysis, John Wiley and Sons, Inc., 1973. I. Guyon, P. Albrecht, Y. LeCun, J. S. Denker and W. Hubbard, "A Time Delay Neural Network Character Recognizer for a Touch Terminal", Pattern Recognition, (1990). N. M. Herbst and C. N. Liu, "Automatic signature verification based on accelerometry", IBM J. Re,. Develop., 21 (1977)245-253. K. J. Lang and G. E. Hinton, "A Time Delay Neural Network Architecture for Speech Recognition", Technical Report CMU-cs-88-152, Carnegie-Mellon University, Pittsburgh, PA,1988. Y. LeCun, "Generalization and Network Design Strategies", Technical Report CRG-TR89-4 University of Toronto Connectionist Research Group, Canada, 1989. G. Lorette and R. Plamondon, "Dynamic approaches to handwritten signature verification", in Computer processing of handwriting, Eds. R. Plamondon and C. G. Leedham, World Scientific, 1990. R. Plamondon and M. Parizeau, "Signature verification from position, velocity and acceleration signals: a comparative study", in Pro<;. 9th Int. Con. on Pattern Recognition, Rome, Italy, 1988, pp 260-265. C. E. Strangio, "Numerical comparison of similarly structured data perturbed by random variations, as found in handwritten signatures", Technical Report, Dept. of Elect. Eng., 1976. I. Yoshimura and M. Yoshimura, "On-line signature verification incorporating the direction of pen movement - an experimental examination of the effectiveness", in From pixel, to features III: frontiers in Handwriting recognition, Eds. S. Impedova and J. C. Simon, Elsevier, 1992.
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Hoeffding Races: Accelerating Model Selection Search for Classification and Function Approximation Oded Maron Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Andrew W. Moore Robotics Institute School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Selecting a good model of a set of input points by cross validation is a computationally intensive process, especially if the number of possible models or the number of training points is high. Techniques such as gradient descent are helpful in searching through the space of models, but problems such as local minima, and more importantly, lack of a distance metric between various models reduce the applicability of these search methods. Hoeffding Races is a technique for finding a good model for the data by quickly discarding bad models, and concentrating the computational effort at differentiating between the better ones. This paper focuses on the special case of leave-one-out cross validation applied to memorybased learning algorithms, but we also argue that it is applicable to any class of model selection problems. 1 Introduction Model selection addresses "high level" decisions about how best to tune learning algorithm architectures for particular tasks. Such decisions include which function approximator to use, how to trade smoothness for goodness of fit and which features are relevant. The problem of automatically selecting a good model has been variously described as fitting a curve, learning a function, or trying to predict future 59 60 Maron and Moore 0.22 '-' e 0.2 '-' ~ = = 0.18 ';: ~ 7a 0.16 ;;>~ 0.14 e U 0.12 1 3 5 7 9 k Nearest Neigh bors Used Figure 1: A space of models consisting of local-weighted-regression models with different numbers of nearest neighbors used. The global minimum is at one-nearestneighbor, but a gradient descent algorithm would get stuck in local minima unless it happened to start in in a model where k < 4. instances of the problem. One can think of this as a search through the space of possible models with some criterion of "goodness" such as prediction accuracy, complexity of the model, or smoothness. In this paper, this criterion will be prediction accuracy. Let us examine two common ways of measuring accuracy: using a test set and leave-one-out cross validation (Wahba and Wold, 1975) . • The test set method arbitrarily divides the data into a training set and a test set. The learner is trained on the training set, and is then queried with just the input vectors of the test set. The error for a particular point is the difference between the learner's prediction and the actual output vector. • Leave-one-out cross validation trains the learner N times (where N is the number of points), each time omitting a different point. We attempt to predict each omitted point. The error for a particular point is the difference between the learner's prediction and the actual output vector. The total error of either method is computed by averaging all the error instances. The obvious method of searching through a space of models, the brute force approach, finds the accuracy of every model and picks the best one. The time to find the accuracy (error rate) of a particular model is proportional to the size of the test set IT EST!, or the size of the training set in the case of cross validation. Suppose that the model space is discretized into a finite number of models IMODELSI then the amount of work required is O(IMODELSI x ITEST!), which is expensive. A popular way of dealing with this problem is gradient descent. This method can be applied to find the parameters (or weights) of a model. However, it cannot be used to find the structure (or architecture) of the modeL There are two reasons for Hoeffding Races: Accelerating Model Selection 61 this. First, we have empirically noted many occasions on which the search space is peppered with local minima (Figure 1). Second, at the highest level we are selecting from a set of entirely distinct models, with no numeric parameters over which to hill-climb. For example, is a neural net with 100 hidden units closer to a neural net with 50 hiden units or to a memory-based model which uses 3 nearest neighbors? There is no viable answer to this question since we cannot impose a viable metric on this model space. The algorithm we describe in this paper, Hoeffding Races, combines the robustness of brute force and the computational feasibility of hill climbing. We instantiated the algorithm by specifying the set of models to be memory-based algorithms (Stanfill and Waltz, 1986) (Atkeson and Reinkensmeyer, 1989) (Moore, 1992) and the method of finding the error to be leave-one-out cross validation. We will discuss how to extend the algorithm to any set of models and to the test set method in the full paper. We chose memory-based algorithms since they go hand in hand with cross validation. Training is very cheap - simply keep all the points in memory, and all the algorithms of the various models can use the same memory. Finding the leave-one-out cross validation error at a point is cheap as making a prediction: simply "cover up" that point in memory, then predict its value using the current model. For a discussion of how to generate various memory-based models, see (Moore et al., 1992). 2 Hoeffding Races The algorithm was inspired by ideas from (Haussler, 1992) and (Kaelbling, 1990) and a similar idea appears in (Greiner and Jurisica, 1992). It derives its name from Hoeffding's formula (Hoeffding, 1963), which concerns our confidence in the sample mean of n independently drawn points Xl, ••. , X n . The probability of the estimated mean Ee3t = ~ 2::l<i<n Xi being more than epsilon far away from the true mean Etrue after n independently drawn points is bounded by: where B bounds the possible spread of point values. We would like to say that with confidence 1 - 8, our estimate of the mean is within € of the true mean; or in other words, Pr(IEtrue - Ee3tl > f) < 8. Combining the two equations and solving for € gives us a bound on how close the estimated mean is to the true mean after n points with confidence 1 - 8: _ j B 2 1og(2/6) € 2n The algorithm starts with a collection of learning boxes. We call each model a learning box since we are treating the models as if they were black boxes. We are not looking at how complex or time-consuming each prediction is, just at the input and output of the box. Associated with each learning box are two pieces of information: a current estimate of its error rate and the number of points it has been tested upon so far. The algorithm also starts with a test set of size N. For leave-one-out cross validation, the test set is simply the training set. 62 Maron and Moore ERROR ---------- ----------;; Uppez Bound I o ~------r_----_+------~----~~----_r------+_----~------------learning box #0 learning box 411 learning box 112 learning box 413 learning box 114 learning box lIS learning box 116 Figure 2: An example where the best upper bound of learning box #2 eliminates learning boxes #1 and #5. The size of f varies since each learning box has its own upper bound on its error range, B. At each point in the algorithm, we randomly select a point from the test set. We compute the error at that point for all learning boxes, and update each learning box's estimate of its own total error rate. In addition, we use Hoeffding's bound to calculate how close the current estimate is to the true error for each learning box. We then eliminate those learning boxes whose best possible error (their lower bound) is still greater than the worst error of the best learning box (its upper bound); see Figure 2. The intervals get smaller as more points are tested, thereby "racing" the good learning boxes, and eliminating the bad ones. We repeat the algorithm until we are left with just one learning box, or until we run out of points. The algorithm can also be stopped once f has reached a certain threshhold. The algorithm returns a set of learning boxes whose error rates are insignificantly (to within f) different after N test points. 3 Proof of Correctness The careful reader would have noticed that the confidence {; given in the previous section is incorrect. In order to prove that the algorithm indeed returns a set of learning boxes which includes the best one, we'll need a more rigorous approach. We denote by ~ the probability that the algorithm eliminates what would have been the best learning box. The difference between ~ and {; which was glossed over in the previous section is that 1 ~ is the confidence for the success of the entire algrithm, while 1 {; is the confidence in Hoeffding's bound for one learning box Hoeffding Races: Accelerating Model Selection 63 during one iteration of the algorithm. We would like to make a formal connection between Ll and {;. In order to do that, let us make the requirement of a correct algorithm more stringent. We'll say that the algorithm is correct if every learning box is within f of its true error at every iteration of the algorithm. This requirement encompasses the weaker requirement that we don't eliminate the best learning box. An algorithm is correct with confidence Ll if Pr{ all learning boxes are within f on all iterations} :2: 1 - Ll. We'll now derive the relationship between {; and Ll by using the disjunctive probability inequality which states that Pr{A V B} ~ Pr{A} + Pr{B}. Let's assume that we have n iterations (we have n points in our test set), and that we have m learning boxes (LBl .. ·LBm). By Hoeffding's inequality, we know that Pr{ a particular LB is within f on a particular iteration} :2: 1 {; Flipping that around we get: Pr{ a particular LB is wrong on a particular iteration} < {; Using the disjunctive inequality we can say Pr{ a particular LB is wrong on iteration 1 V a particular LB is wrong on iteration 2 V a particular LB is wrong on iteration n} ~ {; . n Let's rewrite this as: Pr{ a particular LB is wrong on any iteration} ~ {; . n N ow we do the same thing for all learning boxes: Pr{ LBl is wrong on any iteration V LB2 is wrong on any iteration V LBm is wrong on any iteration} ~ {; . n . m or in other words: Pr{ some LB is wrong in some iteration} ~ {; . n . m We flip this to get: Pr{ all LBs are within f on all iterations} :2: 1 - {; . n . m Which is exactly what we meant by a correct algorithm with some confidence. Therefore, {; = n~m. When we plug this into our expression for f from the previous section, we find that we have only increased it by a constant factor. In other words, by pumping up f, we have managed to ensure the correctness of this algorithm with confidence Ll. The new f is expressed as: f = V~B-~-(l-Og-(-2-nm-n-)--I-O-g(-~-)-) 64 Maron and Moore Problem ROBOT PROTEIN ENERGY POWER POOL DISCONT Table 1: Test problems DescrIption 10 input attributes, 5 outputs. Given an initial and a final description of a robot arm, learn the control needed in order to make the robot perform devil-sticking (Schaal and Atkeson, 1993). 3 inputs, output is a classification into one of three classes. This is the famous protein secondary structure database, with some preprocessing (Zhang et al., 1992). Given solar radiation sensing, predict the cooling load for a building. This is taken from the Building Energy Predictor Shootout. Market data for electricity generation pricing period class for the new United Kingdom Power Market. The visually perceived mapping from pool table configurations to shot outcome for two-ball collisions (Moore, 1992). An artificially constructed set of points with many discontinuities. Local models should outperform global ones. Clearly this is an extremely pessimistic bound and tighter proofs are possible (Omohundro, 1993). 4 Results We ran Hoeffding Races on a wide variety of learning and prediction problems. Table 1 describes the problems, and Table 2 summarizes the results and compares them to brute force search. For Table 2, all ofthe experiments were run using Ll = .01. The initial set of possible models was constructed from various memory based algorithms: combinations of different numbers of nearest neighbors, different smoothing kernels, and locally constant vs. locally weighted regression. We compare the algorithms relative to the number of queries made, where a query is one learning box finding its error at one point. The brute force method makes ITESTI x ILEARNING BOXESI queries. Hoeffding Races eliminates bad learning boxes quickly, so it should make fewer querIes. 5 Discussion Hoeffding Races never does worse than brute force. It is least effective when all models perform equally well. For example, in the POOL problem, where there were 75 learning boxes left at the end of the race, the number of queries is only slightly smaller for Hoeffding Races than for brute force. In the ROBOT problem, where there were only 6 learning boxes left, a significant reduction in the number of queries can be seen. Therefore, Hoeffding Races is most effective when there exists a subset of clear winners within the initial set of models. We can then search over a very broad set of models without much concern about the computational expense Hoeffding Races: Accelerating Model Selection 65 Table 2: Results of Brute Force vs. Hoeffding Races. Initial # queries queries learning with with Problem points learning Brute Hoeffding boxes boxes Force Races left ROBOT 972 95 92340 15637 6 PROTEIN 4965 95 471675 349405 60 ENERGY 2444 189 461916 121400 40 POWER 210 95 19950 13119 48 POOL 259 95 24605 22095 75 DISCONT 500 95 47500 25144 29 60000 60000 400 00 :;';0000 Figure 3: The x-axis is the size of a set of initial learning boxes (chosen randomly) and the y-axis is the number of queries to find a good model for the ROBOT problem. The bottom line shows performance by the Hoeffding Race algorithm) and the top line by brute force. 66 Maron and Moore of a large initial set. Figure 3 demonstrates this. In all the cases we have tested, the learning box chosen by brute force is also contained by the set returned from Hoeffding Races. Therefore, there is no loss of performance accuracy. The results described here show the performance improvement with relatively small problems. Preliminary results indicate that performance improvements will increase as the problems scale up. In other words, as the number of test points and the number of learning boxes increase, the ratio of the number of queries made by brute force to the number of queries made by Hoeffding Races becomes larger. However, the cost of each query then becomes the main computational expense. Acknowledgements Thanks go to Chris Atkeson, Marina Meila, Greg Galperin, Holly Yanco, and Stephen Omohundro for helpful and stimulating discussions. References [Atkeson and Reinkensmeyer, 1989] C. G. Atkeson and D. J. Reinkensmeyer. Using associative content-addressable memories to control robots. In W. T. Miller, R. S. Sutton, and P. J. Werbos, editors, Neural Networks for Control. MIT Press, 1989. [Greiner and Jurisica, 1992] R. Greiner and I. Jurisica. A statistical approach to solving the EBL utility problem. In Proceedings of the Tenth International conference on Artificial Intelligence (AAAI-92). MIT Press, 1992. [Haussler, 1992] D. Haussler. Decision theoretic generalizations of the pac model for neural net and other learning applications. Information and Computation, 100:78-150, 1992. [Hoeffding, 1963] Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13-30, 1963. [Kaelbling, 1990] 1. P. Kaelbling. Learning in Embedded Systems. PhD. Thesis; Technical Report No. TR-90-04, Stanford University, Department of Computer Science, June 1990. [Moore et al., 1992] A. W. Moore, D. J. Hill, and M. P. Johnson. An empirical investigation of brute force to choose features, smoothers and function approximators. In S. Hanson, S. Judd, and T. Petsche, editors, Computational Learning Theory and Natural Learning Systems, Volume 9. MIT Press, 1992. [Moore, 1992] A. W. Moore. Fast, robust adaptive control by learning only forward models. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4. Morgan Kaufmann, April 1992. [Omohundro, 1993] Stephen Omohundro. Private communication, 1993. [Pollard, 1984] David Pollard. Convergence of Stochastic Processes. Springer-Verlag, 1984. [Schaal and Atkeson, 1993] S. Schaal and C. G. Atkeson. Open loop stable control strategies for robot juggling. In Proceedings of IEEE conference on Robotics and Automation, May 1993. [Stanfill and Waltz, 1986] C. Stanfill and D. Waltz. Towards memory-based reasoning. Communications of the A CM, 29(12):1213-1228, December 1986. [Wahba and Wold, 1975] G. Wahba and S. Wold. A completely automatic french curve: Fitting spline functions by cross-validation. Communications in Statistics, 4(1), 1975. [Zhang et al., 1992] X. Zhang, J.P. Mesirov, and D.L. Waltz. Hybrid system for protein secondary structure prediction. Journal of Molecular Biology, 225: 1 049-1 063, 1992.
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Connectionism for Music and Audition Andreas S. Weigend Department of Computer Science and Institute of Cognitive Science University of Colorado Boulder, CO 80309-0430 Abstract This workshop explored machine learning approaches to 3 topics: (1) finding structure in music (analysis, continuation, and completion of an unfinished piece), (2) modeling perception of time (extraction of musical meter, explanation of human data on timing), and (3) interpolation in timbre space. In recent years, NIPS has heard neural networks generate tunes and harmonize chorales. With a large amount of music becoming available in computer readable form, real data can be used to train connectionist models. At the beginning of this workshop, Andreas Weigend focused on architectures to capture structure on multiple time scales. J. S. Bach's last (unfinished) fugue from Die Kunst der Fuge served as an example (Dirst & Weigend, 1994).1 The prediction approach to continuation and completion, as well as to modeling expectations, can be characterized by the question "What's next?". Moving to time as the primary medium of musical communication, the inquiry in music perception and cognition shifted to the question "When next?" . In other words, so far we have considered patterns in time. They assume prior identification and subsequent processing of events. Bob Port, coming from the speech community, considered patterns of time, discussing timing in linguistic polyrhythms (e.g., hot cup of tea). He also drew parallels between timing in Japanese language and timing in music, supporting the hypothesis that perceptional rhythms entrain attentional rhythms. As a mechanism for entrainment, Devin McAuley presented adaptive oscillators: the oscillators adapt their frequencies such that their "firing" coincides with the beat of the music (McAuley, 1994). As the beat can be viewed as entrainment of an individual oscillator, the meter can be viewed as entrainment of multiple oscillators. Ed Large described human perception of metrical structure in analogy to two pendulum clocks that synchronize their motions by hanging on the same wall. An advantage of these entrainment 1 This fugue is available via anonymous ftp from ftp. santafe. edu as data set F. dat of the Santa Fe Time Series Analysis and Prediction Competition. 1163 1164 Weigend approaches (which focus on time as time) over traditional approaches (which focus on music notation and treat time symbolically) is their ability to model phenomena in music performance, such as expressive timing. Taking a Gibsonian perspective, Fred Cummins emphasized the relevance of ecological constraints on audition: perceptually relevant features are not easily spotted in the wave form or the spectrum. Among the questions he posed were: what "higher-order" features might be useful for audition, and whether recurrent networks could be useful to extract such features. The last contribution also addressed the issue of representation, but with sound synthesis in mind: wouldn't a musician like to control sound in a perceptually relevant space, rather than fiddling with non-intuitive coefficients of an FM-algorithm? Such a space was constructed with human input: subjects were asked to similarityjudge sounds from different instruments (normalized in pitch, duration and volume). Multidimensional scaling was used to define a low-dimensional sub-space keeping the distance relations. Michael Lee first trained a network to find a map from timbre space to the space of the first 33 harmonics (Lee, 1994). He then used the network to generate rich new sounds by interpolating in this perceptually relevant space, through physical gestures, such as from a data glove, or through an interface musicians might be comfortable with, such as a cello. The discussion turned to the importance of working with perceptually adequate, "ecologically sound" representations (e.g., by using a cochlea model as pre-processor, or a speech model as post-processor for sonification applications). Finally, to probe human cognition, we discussed synthetic sounds, designed to reveal fundamental characteristics of the auditory system, independent of our daily experience. Returning to the title, the workshop turned out to be problem driven: people presented a problem or a finding and searched for a solution-connectionist or otherwise-rather than applying canned connectionist ideas to music and cognition. I thank the speakers, Fred Cummins (fcummins@indiana.edu), Ed Large (large@cis.ohio-state.edu), Michael Lee (lee@cnmat.berkeley.edu), Devin McAuley (mcauley@cs.indiana.edu), Robert Port (port@indiana.edu), as well as all participants. I also thank Tom Ngo (ngo@interval.com) for sending me the notes he took at the workshop, and Eckhard Kahle (kahle@ircam.fr) for discussing this summary. References Dirst, M., and A. S. Weigend (1994) "Baroque Forecasting: On Completing J. S. Bach's Last Fugue." In Time Series Prediction: Forecasting the Future and Understanding the Past, edited by A. S. Weigend and N. A. Gershenfeld, pp. 151172. Addison-Wesley. Lee, M., and D. Wessel (1992) "Connectionist Models for Real-Time Control of Synthesis and Compositional Algorithms." In Proceedings of the International Computer Music Conference, pp. 277-280. San Francisco, CA: International Computer Music Association. McAuley, J. D. (1994) "Finding metrical structure in time." In Proceedings of the 1993 Connectionist Models Summer School, edited by M. C. Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman and A. S. Weigend, pp. 219-227. Lawrence Erlbaum.
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Central and Pairwise Data Clustering by Competitive Neural Networks Joachim Buhmann & Thomas Hofmann Rheinische Friedrich-Wilhelms-U niversiHit Institut fiir Informatik II, RomerstraBe 164 D-53117 Bonn, Fed. Rep. Germany Abstract Data clustering amounts to a combinatorial optimization problem to reduce the complexity of a data representation and to increase its precision. Central and pairwise data clustering are studied in the maximum entropy framework. For central clustering we derive a set of reestimation equations and a minimization procedure which yields an optimal number of clusters, their centers and their cluster probabilities. A meanfield approximation for pairwise clustering is used to estimate assignment probabilities. A se1fconsistent solution to multidimensional scaling and pairwise clustering is derived which yields an optimal embedding and clustering of data points in a d-dimensional Euclidian space. 1 Introduction A central problem in information processing is the reduction of the data complexity with minimal loss in precision to discard noise and to reveal basic structure of data sets. Data clustering addresses this tradeoff by optimizing a cost function which preserves the original data as complete as possible and which simultaneously favors prototypes with minimal complexity (Linde et aI., 1980; Gray, 1984; Chou et aI., 1989; Rose et ai., 1990). We discuss an objective function for the joint optimization of distortion errors and the complexity of a reduced data representation. A maximum entropy estimation of the cluster assignments yields a unifying framework for clustering algorithms with a number of different distortion and complexity measures. The close analogy of complexity optimized clustering with winner-take-all neural networks suggests a neural-like implementation resembling topological feature maps (see Fig. 1). 104 Central and Pairwise Data Clustering by Competitive Neural Networks lOS X· 1 Pyli Figure 1: Architecture of a three layer competitive neural network for central data clustering with d neurons in the input layer, K neurons in the clustering layer with activity (Mia) and G neurons in the classification layer. The output neurons estimate the conditional probability Pl'li of data point i being in class 1. Given is a set of data points which are characterized either by coordinates {Xi I Xi E ~d; i = 1, ... , N} or by pairwise distances {Dikli, k = 1, ... , N}. The goal of data clustering is to determine a partitioning of a data set which either minimizes the average distance of data points to their cluster centers or the average distance between data points of the same cluster. The two cases are refered to as central or pairwise clustering. Solutions to central clustering are represented by a set of data prototypes {y alY a E ~d; a = 1, ... ,K}, and the size K of that set. The assignments {Miala = 1, ... ,K; i = 1, ... ,N}, Mia E {O, I} denote that data point i is uniquely assigned to cluster a (Lv Miv = 1). Rate distortion theory specifies the optimal choice of Ya being the cluster centroids, i.e., l:i Mia a~ 0 Dia (Xi, Ya) = O. Given only a set of distances or dissimilarities the solution to pairwise clustering is characterized by the expected assignment variables (Mia). The complexity {Cal a = 1, ... , K} of a clustering solution depends on the specific information processing application at hand, in particular, we assume that Ca is only a function of the cluster probability Pa = L~1 Mia/N. We propose the central clustering cost function N K c~( {Miv}) = L L Miv (DiV(Xi, Y v) + ACv(Pv)) (1) i=1 v=1 and the pairwise clustering cost function N KIN Ck( {Miv}) = L LMiv (2 N L MkvDik + ACv(Pv)). i=1 v=1 Pv k=1 (2) The distortion and complexity costs are adjusted in size by the weighting parameter A. The cost functions (1,2) have to be optimized in an iterative fashion: (i) vary the assignment variables Mia for a fixed number K of clusters such that the costs c,;,pc ( {Mia} ) decrease; (ii) increment the number of clusters K ~ K + 1 and optimize Mia again. Complexity costs which penalize small, sparsely populated clusters, i.e., Ca = l/p~, s = 1,2 .... , favor equal cluster probabilities, thereby emphasizing the hardware aspect of a clustering solution. The special case s = 1 with constant costs per cluster corresponds to K -means clustering. An alternative complexity measure which estimates encoding costs for data compression and data transmission is the Shannon entropy of a cluster set (C) - LvPvCv = -l:v Pv logpv. 106 Buhmann and Hofmann The most common choice for the distortion measure are distances Via = IIXi - YaW" which preserve the permutation symmetry of (1) with respect to the cluster index /J. A data partitioning scheme without permutation invariance of cluster indices is described by the cost function E1 = L L Mil/ ( ((Vi 1/ )) + .\C(PI/) ) . (3) t 1/ The generalized distortion error ((Via)) - 2:')' To.')' Vi,), (Xi , Y')') between data point Xi and cluster center Yo. quantifies the intrinsic quantization errors Vh(Xi, Y')') and the additional errors due to transitions To.')' from index ry to a. Such transitions might be caused by noise in communication channels. These index transitions impose a topological order on the set of indices {a I a = 1, ... , K} which establishes a connection to self-organizing feature maps (Kohonen, 1984; Ritter et al., 1992) in the case of nearest neighbor transitions in a d-dimensional index space. We refer to such a partitioning of the data space as topology preserving clustering. 2 Maximum Entropy Estimation of Central Clustering Different combinations of complexity terms, distortion measures and topology constraints define a variety of central clustering algorithms which are relevant in very different information processing contexts. To derive robust, preferably parallel algorithms for these data clustering cases, we study the clustering optimization problem in the probabilistic framework of maximum entropy estimation. The resulting Gibbs distribution proved to be the most stable distribution with respect to changes in expected clustering costs (Tikochinsky et al., 1984) and, therefore, has to be considered optimal in the sense of robust statistics. Statistical physics (see e.g. (Amit, 1989; Rose et at., 1990)) states that maximizing the entropy at a fixed temperature T = 1/ f3 is equivalent to minimizing the free energy -TlnZ = -Tln( L exp(-f3EK)) {M", } -.\N L PI/ 2 ~c: - ~ 2;= log (L exp [-f3( ((Vil/)) + .\c~)]) 1/ P t 1/ (4) with respect to the variables PI/, Y 1/' The effective complexity costs are C~ - a (Pl/CI/) / api/' For a derivation of (4) see (Buhmann, Kiihnel, 1993b). The resulting re-estimation equations for the expected cluster probabilities and the expected centroid positions are necessary conditions of :F K being minimal, i.e. Po. (5) t o 1 a N LLT')'a(Mh)aVia(Xi,Ya), . Yo. t ')' (6) exp [-f3( ((Via)) + .\C~)] K (7) Lexp[-f3(((Vil/)) + .\C~)] 1/=1 Central and Pairwise Data Clustering by Competitive Neural Networks 107 The expectation value (Mia) of the assignment variable Mia can be interpreted as a fuzzy membership of data point Xi in cluster Q. The case of supervised clustering can be treated in an analogous fashion (Buhmann, Kuhne1, 1993a) which gives rise to the third layer in the neural network implementation (see Fig. 1). The global minimum of the free energy (4) with respect to Pa, Ya determines the maximum entropy solution of the cost function (1). Note that the optimization problem (1) of a KN state space has been reduced to a K( d + 1) dimensional minimization of the free energy F K (4). To find the optimal parameters Pa . Ya and the number of clusters K which minimize the free energy, we start with one cluster located at the centroid of the data distribution, split that cluster and reestimate Pa, Ya using equation (5,6). The new configuration is accepted as an improved solution if the free energy (4) has been decreased. This splitting and reestimation loop is continued until we fail to find a new configuration with lower free energy. The temperature determines the fuzziness of a clustering solution, whereas the complexity term penalizes excessively many clusters. 3 Meanfield Approximation for Pairwise Clustering The maximum entropy estimation for pairwise clustering constitutes a much harder problem than the calculation of the free energy for central clustering. Analytical expression for the Gibbs distributions are not known except for the quadratic distance measure Dik = (Xi Xk)2. Therefore, we approximate the free energy by a variational principle commonly refered to as meanfield approximation. Given the costfunction (2) we derive a lower bound to the free energy by a system of noninteracting assignment variables. The approximative costfunction with the variational parameters Eiv is K N E~ = L L MivEiv, (8) v=) i=) The original costfunction for pairwise clustering can be written as Ek = E9.: + V with a (small) perturbation term V = Ek - E9.: due to cluster interactions. The partition function L exp (-/3E9.:) exp (-,(3V) Z L exp ( - f1Ek) {Miv} L exp (-f1E~) ...;,..{M_i......:V} ______ _ {M. v } L exp (-/3E9.:) {Mi v } Zo(exp( - /3V)o > Zo exp( - /3(V)o) (9) is bound from below if terms of the order O( ((V - (V)O)3)0) and higher are negligible compared to the quadratic term. The angular brackets denote averages over all configurations of the costfunction without interactions. The averaged perturbation term (V)o amounts to 1 (V}o = LL(MiV}(MkV )2 NDik+ALL(MiV)Cv- LL(Miv)Eiv. (10) · k ~ . . V 1. ' V ., V 1. (Mia) being the averaged assignment variables (Mia) = exp( -/3Eia) L exp( -/3Eiv) v (11) 108 Buhmann and Hofmann The meanfield approximation with the cost function (8) yields a lower bound to the partition function Z of the original pairwise clustering problem. Therefore, we vary the parameters Cia to maximize the quantity In Zo - ,8(V)o which produces the best lower bound of Z based on an interaction free costfunction. Variation of Cia leads to the conditions ViE {1 , ... ,N},a E {l. ... ,K}, (12) ctv being defined as For a given distance matrix V ik the transcendental equations (11,12) have to be solved simultaneously. So far the Cia have been treated as independent variation parameters. An important problem, which is usually discussed in the context of Multidimensional Scaling, is to find an embedding for the data set in an Euclidian space and to cluster the embedded data. The variational framework can be applied to this problem, if we consider the parameters Cia as functions of data coordinates and prototype coordinates, Cia = Via(Xi, Ya), e.g. with a quadratic distortion measure Via (Xi. Ya) = IIXi - Y a11 2 . The variables Xi, Ya E ~d are the variational parameters which have to be determined by maximizing In Zo - {1(V)0. Without imposing the restriction for the prototypes to be the cluster centroids, this leads to the following conditions for the data coordinates After further algebraic manipulations we receive the explicit expression for the data points KiXi = ~ L (Miv) (iIYvIl 2 - ci*v) (Yv - L (MiJ.t)Y J.t), (15) v J.t with the covariance matrix Ki = ((yyT)i - (Y)i(Y);), (Y)i = L.v(Miv)Yv. Let us assume that the matrix Ki is non-singular which imposes the condition K > d and the cluster centers {y al a = 1, ... , K} being in general position. For K < d the equations Cia = cta + Ci are exactly solvable and embedding in dimensions larger than K produces non-unique solutions without improving the lower bound in (9). Varying In Zo - ,8(V)o with respect to Ya yields a second set of stationarity conditions L(Mja ) (1- (Mja)) (Cja -cja) (Xj - Ya) = 0, Va E {I, ... ,K}. (16) j The weighting factors in (16), however, decay exponentially fast with the inverse temperature, i.e., (Mja)(1 - (Mja)) rv 0(,8 exp[-,8c]), C > O. This implies that the optimal solution for the data coordinates displays only a very weak dependence on the special choice of the prototypes in the low temperature regime. Fixing the parameters Ya and solving the transcendental equations (14,15) for Xi, the solution will be very close to the optimal approximation. It is thus possible to choose the prototypes as the cluster centroids Ya = 1/(PaN ) L.i(Mia)Xi and, thereby, to solve Eq. (15) in a self-consistent fashion. Central and Pairwise Data Clustering by Competitive Neural Networks 109 a b * * * * * * * * * * * * * * * * * ir. * * c * Figure 2: A data distribution (4000 data points) (a), generated by four normally distributed sources is clustered with the complexity measure Ca. = -logpa. and.A = 0.4 (b). The plus signs (+) denote the centers ofthe Gaussians and stars (*) denote cluster centers. Figure (c) shows a topology preserving clustering solution with complexity Ca. = 1/ Pa. and external noise (ry = 0.05). If the prototype variables depend on the data coordinates, the derivatives oY a./ OXi will not vanish in general and the condition (14) becomes more complicated. Regardless of this complication the resulting algorithm to estimate data coordinates Xi interleaves the clustering process and the optimization of the embedding in a Euclidian space. The artificial separation of multidimensional scaling from data clustering has been avoided. Data points are embedded and clustered simultaneously. Furthermore, we have derived a maximum entropy approximation which is most robust with respect to changes in the average costs (EK). 4 Clustering Results Non-topological (Ta.'}' = on,},) clustering results at zero temperature for the logarithmic complexity measure (Ca. = 10gpa.) are shown in Fig. 2b. In the limit of very small complexity costs the best clustering solution densely covers the data distribution. The specific choice of logarithmic complexity costs causes an almost homogeneous density of cluster centers, a phenomenon which is known from studies of asymptotic codebook densities and which is explained by the vanishing average complexity costs (Ca.) = -Pa.logpa. of very sparsely occupied clusters (for references see (Buhmann, Kuhnel, 1993b». Figure 2c shows a clustering configuration assuming a one-dimensional topology in index space with nearest neighbor transitions. The short links between neighboring nodes of the neural chain indicate that the distortions due to cluster index transitions have also been optimized. Note, that complexity optimized clustering determines the length of the chain or, for a more general noise distribution, an optimal size of the cluster set. This stopping criterion for adding new cluster nodes generalizes self-organizing feature maps (Kohonen, 1984) and removes arbitrariness in the design of topological mappings. Furthermore, our algorithm is derived from an energy minimization principle in contrast to self-organizing feature maps which "cannot be derived as a stochastic gradient on any energy function" (Erwin et aI., 1992). The complexity optimized clustering scheme has been tested on the real world task of 110 Buhmann and Hofmann a b d ; .... " I ' . ~ . ,'. , '. .~., • ~ . ~ ~ c e , ,I. tI • .... ;'! .... ' .~.~ .,' .'\.~ .. ~ ' . . ""r', I • ~\... .. ' . '11\ll i' •. . . ' 1L .. ~ ._ .. ' \-: #"':. . ' • ':.' ~. . '- ' .. -.. -.. :. 'I, . ' ..... - , ~: .... u:-. .~ ~ "1 '. lfl" . tJ\ \ : .-:~ Figure 3: Quantization of a 128x 128, 8bit, gray-level image. (a) Original picture. (b) Image reconstruction from wavelet coefficients quantized with entropic complexity. (c) Reconstruction from wavelet coefficients quantized by K -means clustering. (d,e) Absolute values of reconstruction errors in the images (b,c). Black is normalized in (d,e) to a deviation of 92 gray values. image compression (Buhmann, Kuhnel, 1993b). Entropy optimized clustering of wavelet decomposed images has reduced the reconstruction error of the compressed images up to 30 percent. Images of a compression and reconstruction experiment are shown in Fig. 3. The compression ratio is 24.5 for a 128 x 128 image. According to our efficiency criterion entropy optimized compression is 36.8% more efficient than K -means clustering for that compression factor. The peak SNR values for (b,c) are 30.1 and 27.1, respectively. The considerable higher error near edges in the reconstruction based on K -means clustering (e) demonstrates that entropy optimized clustering of wavelet coefficients not only results in higher compression ratios but, even more important it preserves psychophysically important image features like edges more faithfully than conventional compression schemes. 5 Conclusion Complexity optimized clustering is a maximum entropy approach to central and pairwise data clustering which determines the optimal number of clusters as a compromise between distortion errors and the complexity of a cluster set. The complexity term turns out to be as important for the design of a cluster set as the distortion measure. Complexity optimized clustering maps onto a winner-take-all network which suggests hardware implementations in analog VLSI (Andreou et al., 1991). Topology preserving clustering provides us with a Central and Pairwise Data Clustering by Competitive Neural Networks 111 cost function based approach to limit the size of self-organizing maps. The maximum entropy estimation for pairwise clustering cannot be solved analytically but has to be approximated by a meanfield approach. This mean field approximation of the pairwise clustering costs with quadratic Euclidian distances establishes a connection between multidimensional scaling and clustering. Contrary to the usual strategy which embeds data according to their dissimilarities in a Euclidian space and, in a separate second step, clusters the embedded data, our approach finds the Euclidian embedding and the data clusters simultaneously and in a selfconsistent fashion. The proposed framework for data clustering unifies traditional clustering techniques like K -means clustering, entropy constraint clustering or fuzzy clustering with neural network approaches such as topological vector quantizers. The network size and the cluster parameters are determined by a problem adapted complexity function which removes considerable arbitrariness present in other non-parametric clustering methods. Acknowledgement: JB thanks H. Kuhnel for insightful discussions. This work was supported by the Ministry of Science and Research of the state Nordrhein-Westfalen. References Amit, D. (1989). Modelling Brain Function. Cambridge: Cambridge University Press. Andreou, A. G., Boahen, K. A., Pouliquen, P.O., Pavasovic, A., Jenkins, R. E., Strohbehn, K. (1991). Current Mode Subthreshold MOS Circuits for Analog VLSI Neural Systems. IEEE Transactions on Neural Networks, 2,205-213. Buhmann, J., Kuhne1, H. (1993a). Complexity Optimized Data Clustering by Competitive Neural Networks. Neural Computation, 5, 75-88. Buhmann, J., Kuhnel, H. (1993b). Vector Quantization with Complexity Costs. IEEE Transactions on Information Theory, 39(4),1133-1145. Chou, P. A., Lookabaugh, T., Gray, R. M. (1989). Entropy-Constrained Vector Quantization. IEEE Transactions on Acoustics, Speech and Signal Processing, 37, 31-42. Erwin, W., Obermayer, K., Schulten, K. (1992). Self-organizing Maps: Ordering, Convergence Properties, and Energy Functions. Biological Cybernetics, 67, 47-55. Gray, R. M. (1984). Vector Quantization. IEEE Acoustics, Speech and Signal Processing Magazine, April, 4-29. Kohonen, T. (1984). Self-organization and Associative Memory. Berlin: Springer. Linde, Y., Buzo, A., Gray, R. M. (1980). An algorithm for vector quantizer design. IEEE Transactions on Communications COM, 28, 84-95. Ritter, H., Martinetz, T., Schulten, K. (1992). Neural Computation and Self-organizing Maps. New York: Addison Wesley. Rose, K., Gurewitz, E., Fox, G. (1990). Statistical Mechanics and Phase Transitions in Clustering. Physical Review Letters, 65(8), 945-948. Tikochinsky, Y., Tishby, N.Z., Levine, R. D. (1984). Alternative Approach to MaximumEntropy Inference. Physical Review A, 30, 2638-2644.
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Processing of Visual and Auditory Space and Its Modification by Experience Josef P. Rauschecker Laboratory of Neurophysiology National Institute of Mental Health Poolesville, MD 20837 Terrence J. Sejnowski Computational Neurobiology Lab The Salk: Institute San Diego, CA 92138 Visual spatial information is projected from the retina to the brain in a highly topographic fashion, so that 2-D visual space is represented in a simple retinotopic map. Auditory spatial information, by contrast, has to be computed from binaural time and intensity differences as well as from monaural spectral cues produced by the head and ears. Evaluation of these cues in the central nervous system leads to the generation of neurons that are sensitive to the location of a sound source in space ("spatial tuning") and, in some animal species, to auditory space maps where spatial location is encoded as a 2-D map just like in the visual system. The brain structures thought to be involved in the multimodal integration of visual and auditory spatial integration are the superior colliculus in the midbrain and the inferior parietal lobe in the cerebral cortex. It has been suggested for the owl that the visual system participates in setting up the auditory space map in the superior. Rearing owls with displacing prisms, for example, shifts the map by a fixed amount. These behavioral and neurobiological findings have been successfully incorporated into a connectionist model of the owl's sound localization system (Rosen, Rumelhart, and Knudsen, 1994). On the other hand, cats that are reared with both eyes sutured shut develop completely normal auditory spatial mechanisms: Precision of sound localization is even improved above normal (Rauschecker and Kniepert, 1994), and a higher number of auditory neurons with sharper spatial tuning is found in parietal cortex of such cats (Rauschecker and Korte, 1993). Non-visual sensory signals and/or motor feedback must be capable, therefore, to calibrate the auditory spatial mechanisms. Activity-dependent Hebbian learning and synaptic competition between inputs to the parietal region from different sensory modalities are sufficient to explain these results. 1186 Processing of Visual and Auditory Space and Its Modification by Experience 1187 The question remains how visual and auditory information are kept in spatial register with each other when the animal moves its eyes or head. Experiments in awake behaving monkeys help to solve this problem. Neurons in the lateral intraparietal area of cortex (LIP) respond to visual and auditory stimuli which call for a movement to the same location in space. Neuronal responses in both modalities are modulated by eye position leading to "gain fields", in which the location of a target in headcentered coordinates is encoded via the response strength in a population of neurons (Andersen, Snyder, Li, and Stricanne, 1993). The neurobiological data from owls, cats and monkeys were used to develop a neural network model of multisensory integration (Pouget and Sejnowski, 1993). A set of basis functions was introduced which replace the conventional allocentric representations and produce gain fields similar to monkey parietal cortex. An extension of the model also incorporates the plasticity of this system. Predictive Hebbian learning is used to bring the visual and auditory maps into register. In the network a Hebb rule is gated by a reinforcement term, which is the difference between actual reinforcement and how much reinforcement is expected by the system. It utilizes the activity of diffuse transmitter projection systems, such as noradrenaline (NA), acetylcholine (ACh), and dopamine (DA) , which are known to play an important role for plasticity in the brain of higher mammals. In summary, it appears extremely fruitful to bring together neuroscientists and neural network modelers, because both groups can profit from each other. Neurobiological data are the flesh for realistic network models, and models are helpful to formalize a biological hypothesis and guide the way for further testing. Andersen RA, Snyder LH, Li C-S, Stricanne B (1993) Coordinate transformations in the representation of spatial information. Curr Opinion Neurobiol3: 171-176. Pouget A, Fisher SA, Sejnowski TJ (1993) Egocentric spatial representation in early vision. J Cog Neurosci 5:150-161. Rauschecker JP and Korte M (1993) Auditory compensation for early blindness in cat cerebral cortex. J Neurosci 13:4538-4548. Rauschecker JP and Kniepert U (1994) Enhanced precision of auditory localization behavior in visually deprived cats. Eur J Neurosci 6 (in press). Rosen D, Rumelhart D, Knudsen E (1994) A connectionist model of the owl's sound localization system. In: Advances in Neural Information Processing Systems 6, Cowan J, Tesauro G, Alspector J (eds), San Mateo, CA: Morgan Kaufmann (in press)
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A Connectionist Model of the Owl's Sound Localization System D alliel J. Rosen· Department of Psychology Stanford University Stanford, CA 94305 David E. Rumelhart Department of Psychology Stanford University Stanford, CA 94305 Eric. I. Knudsen Department of Neurobiology Stanford University Stanford, CA 94305 Abstract ,,"'e do not have a good understanding of how theoretical principles of learning are realized in neural systems. To address this problem we built a computational model of development in the owl's sound localization system. The structure of the model is drawn from known experimental data while the learning principles come from recent work in the field of brain style computation. The model accounts for numerous properties of the owl's sound localization system, makes specific and testable predictions for future experiments, and provides a theory of the developmental process. 1 INTRODUCTION The barn owl, Tyto Alba, has a remarkable ability to localize sounds in space. In complete darkness it catches mice with nearly flawless precision. The owl depends upon this skill for survival, for it is a nocturnal hunter who uses audition to guide ·Current address: Keck Center for Integrative Neuroscience, UCSF, 513 Parnassus Ave., San Francisco, CA 94143-0444. 606 A Connectionist Model of the Owl's Sound Localization System 607 its search for prey (Payne, 1970; Knudsen, Blasdel and Konishi, 1979). Central to the owl's localization system are the precise auditory maps of space found in the owl's optic tectum and in the external nucleus of the inferior colliculus (lex). The development of these sensory maps poses a difficult problem for the nervous system, for their accuracy depends upon changing relationships between the animal and its environment. The owl encodes information about the location of a sound source by the phase and amplitude differences with which the sound reaches the owl's two ears. Yet these differences change dramatically as the animal matures and its head grows. The genome cannot "know" in advance precisely how the animal's head will develop - many environmental factors affect this process - so it cannot encode the precise development of the auditory system. Rather, the genome must design the auditory system to adapt to its environment, letting it learn the precise interpretation of auditory cues appropriate for its head and ears. In order to understand the nature of this developmental process, we built a connectionist model of the owl's sound localization system, using both theoretical principles of learning and knowledge of owl neurophysiology and neuroanatomy. 2 THE ESSENTIAL SYSTEM TO BE MODELED The owl calculates the horizontal component of a sound source location by measuring the interaural time difference (lTD) of a sound as it reaches the two ears (Knudsen and Konishi, 1979). It computes the vertical component of the signal by determining the interaurallevel difference (ILD) of that same sound (Knudsen and Konishi, 1979). The animal processes these signals through numerous sub-cortical nuclei to form ordered auditory maps of space in both the ICx and the optic tectum. Figure 1 shows a diagram of this neural circuit. Neurons in both the ICx and the optic tectum are spatially tuned to auditory stimuli. Cells in these nuclei respond to sound signals originating from a restricted region of space in relation to the owl (Knudsen, 1984). Neurons in the ICx respond exclusively to auditory signals. Cells in the optic tectum, on the other hand, encode both audito!y and visual sensory maps, and drive the motor system to orient to the location of an auditory or visual signal. Researchers study the owl's development by systematically altering the animal's sensory experience, usually in one of two ways. They may fit the animal with a sound attenuating earplug, altering its auditory experience, or they may fit the owl with displacing prisms, altering its visual experience. Disturbance of either auditory or visual cues, during a period when the owl is developing to maturity, causes neural and behavioral changes that bring the auditory map of space back into alignment with the visua.l map, and/or tune the auditory system to be sensitive to the appropriate range of binaural sound signals. The earplug induced changes take place at the level of the VLVp, where ILD is first computed (Mogdans and Knudsen, 1992). The visually induced adjustment of the auditory maps of space seems to take place at the level of the ICx (Brainard and Knudsen, 1993b). The ability of the owl to adjust to altered sensory signals diminishes over time, and is greatly restricted in adulthood (Knudsen and Knudsen, 1990). 608 Rosen, Rumelhart, and Knudsen OVERVIEW of the BARN OWL'. SOUND LOCALIZATION SYSTEM ( NUCLBJS MAGNOCEWJLAAIS T"'*'a ( ~~dIC L"". NUCLBJS MAGNOCB.LULAAIS TIn*'II Figure 1: A chart describing the flow of auditory information in the owl's sound localization system. For simplicity, only the connections leading to the one of the bilateral optic tecta are shown. Nuclei labeled with an asterisk (*) are included in the model. Nuclei that process ILD and/or lTD information are so labeled. 3 THE NETWORK MODEL The model has two major components: a network architecture based on the neurobiology of the owl's localization system, as shown in Figure 1, and a learning rule derived from computational learning theory. The elements of the model are standard connectionist units whose output activations are sigmoidal functions of their weighted inputs. The learning rule we use to train the model is not standard. In the following section we describe how and why we derived this rule. 3.1 DEFINING THE GOAL OF THE NETWORK The goal of the network, and presumably the owl, is to accurately map sound signals to sound source locations. The network must discover a model of the world which best captures the relationship between sound signals and sound source locations. Recent work in connectionist learning theory has shown us ways to design networks that search for the model that best fits the data at hand (Buntine and Weigend, 1991; MacKay, 1992; Rumelhart, Durbin, Golden and Chauvin, in press). In this section we apply such an analysis to the localization network. A Connectionist Model of the Owl's Sound Localization System 609 Table 1: A table showing the mathematical terms used in the analysis. I TERM I MEANING M The Model 1J The Data P(MI1J) Probability of the Model given the Data < X,Y>i The set of i input/target training pairs xi The input vector for training trial i Yi The target vector for training trial i Yi The output vector for training trial i Yij The value of output unit j on training trial i Wij The weight from unit j to unit i 7Jj The netinput to unit j :F(7Jj) The activation function of unit j evaluated at its netinput C The term to be maximized by the network 3.2 DERIVING THE FUNCTION TO BE MAXIMIZED The network should maximize the probability of the model given the data. Using Bayes' rule we write this probability as: P(MI1J) = P(1JIM)P(M) P(1J) . Here M represents the model (the units, weights and associated biases) and D represents the data. We define the data as a set of ordered pairs, [< soundsignal, location - signal >d, which represent the cues and targets normally used to train a connectionist network. In the owl's case the cues are the auditory signals, and the target information is provided by the visual system. (Table 1 lists the mathematical terms we use in this section.) We simplify this equation by taking the natural logarithm of each side giving: In P(MI1J) = In P(1JIM) + InP(M) -In P(1J). Since the natural logarithm is a monotonic transformation, if the network maximizes the second equation it will also maximize the first. The final term in the equation, In P(1J), represents the probability of the ordered pairs the network observes. Regardless of which model the network settles upon, this term remains the same - the data are a constant during training. Therefore we can ignore it when choosing a model. The second term in the equation, In P(M), represents the probability of the model. This is the prior term in Bayesian analysis and is our estimation of how likely it is that a particular model is true, regardless of the data. 'Ve will discuss it below. For now we will concentrate on maximizing In P(1JIM). 610 Rosen, Rumelhart, and Knudsen 3.3 ASSUMPTIONS ABOUT THE NETWORK'S ENVIRONMENT We assume that the training data - pairs of stylized auditory and visual signals are independent of one another and re-write the previous term as: InP(VIM) = L:lnP« i,Y>i 1M), i The i subscript denotes the particular data, or training, pair. We further expand this term to: In P(VIM) = Lin P(ih Iii 1\ M) + L: In P(Xi). i i We ignore the last term, since the sound signals are not dependent on the model. vVe are left, then, with the task of maximizing Li In P(Ui Iii 1\ M). It is important to note that Yi represents a visual signal, not a localization decision. The network attempts to predict its visual experience given its auditory experience. It does not predict the probability of making an accurate localization decision. If we assume that visual signals provide the target values for the network, then this analysis shows that the auditory map will always follow the visual map, regardless of whether this leads to accurate localization behavior or not. Our assumption is supported by experiments showing that, in the owl, vision does guide the formation of auditory spatial maps (Knudsen and Knudsen, 1985; Knudsen, 1988). Next, we must clarify the relationship between the inputs, Xi and the targets, ih. \Ve know that the real world is probabilistic - that for a given input there exists some distribution of possible target values. We need to estimate the shape of this distribution. In this case we assume that the target values are binomially distributed - that given a particular sound signal, the visual system did or did not detect a sound source at each point in owl-centered space. Having made this assumption, we can clarify our interpretation of the network output array, Y~. Each element, Yij, of this vector represents the activity of output unit j on training trial i. We assume that the output activation of each of these units represents the expected value of its corresponding target, Yij. In this case the expected value is the mean of a binomial distribution. So the value of each output unit Yij represents the probability that a sound signal originated from that particular location. vVe now write the probability of the data given the model as: P(yilxi 1\ M) = II yft (1 - Yij )l-Yij . j Taking the natural log of the probability and summing over all data pairs we get: C = L L: Yij In Yij + (1 - Yij) In( 1 - Yij) i j where C is the term we want to maximize. This is the standard cross-entropy term. 3.4 DERIVING THE LEARNING RULE Having defined our goal we derive a learning rule appropriate to achieving that goal. To determine this rule we compute :~ where 7}j is the net input to a unit. (In these A Connectionist Model of the Owl's Sound Localization System 611 equations we have dropped the i subscript, which denotes the particular training trial, since this analysis is identical for all trials.) We write this as: where aF( '1]j) is the derivative of a unit's activation function evaluated at its net input. Next we choose an appropriate activation function for the output units. The logistic function, F('1]j) = ( 1_,,"), is a good choice for two reasons. First, it is bounded by l+e , zero and one. This makes sense since we assume that the probability that a sound signal originated at anyone point in space is bounded by zero and one. Second, when we compute the derivative of the logistic function we get the following result: aF('1]j) = F('1]j)(I- F('1]j)) = 1/j(1- 1/j). This term is the variance of a binomial distribution and when we return to the derivative of our cost function, we see that this variance term is canceled by the denominator. The final derivative we use to compute the weight changes at the output units is therefore: ac ( ~ ) ~ <X Yj Yj . u'1]j The weights to other units in the network are updated according to the standard backpropagation learning algorithm. 3.5 SPECIFYING MODEL PRIORS There are two types of priors in this model. First is the architectural one. We design a fixed network architecture, described in the previous section, based upon our knowledge of the nuclei involved in the owl's localization system. This is equivalent to setting the prior probability of this architecture to 1, and all others to O. We also use a weight elimination prior. This and similar priors may be interpreted as ways to reduce the complexity of a network (\Veigend, Huberman and Rumelhart, 1990). The network, therefore, maximizes an expression which is a function of both its error and complexity. 3.6 TRAINING We train the model by presenting it with input to the core of the inferior colli cui us (ICc), which encodes interaural phase and time differences (IPD/ITD), and the angular nuclei, which encode sound level. The outputs of the network are then compared to target values, presumed to come from the visual system. The weights are adjusted in order to minimize this difference. \Ve mimic plug training by varying the average difference between the two angular input values. We mimic prism training by systematically changing the target values associated with an input. 612 Rosen, Rumelhart, and Knudsen Figure 2: The activity level of lex units in response to a particular auditory input immediately after simulated prism training was begun (left), midway through training (middle) and after training was completed (right). 4 RESULTS and DISCUSSION The trained network localizes accurately, shows appropriate auditory tuning curves in each of the modeled nuclei, and responds appropriately to manipulations that mimic experiments such as blocking inhibition at the level of the lex. The network also shows appropriate responses to changing average binaural intensity at the level of the VLVp, the lateral shell and the lex. Furthermore, the network exhibits many properties found in the developing owl.. The model appropriately adjusts its auditory localization behavior in simulated earplug experiments and this plasticity takes place at the level of the VLVp. As earplug simulations are begun progressively later in training, the network's ability to adapt to plug training gradually diminishes, following a time course of plasticity qualitatively similar to the sensitive and critical periods described in the owl. The network adapts appropriately in simulated prism studies and the changes in response to these simulations primarily take place along the lateral shell to lex connections. As with the plug studies, the network's ability to adapt to prisms diminishes over time. However, unlike the mature owl, a highly trained network retains the ability to adapt in a simulated prism experiment. We also discovered that the principally derived learning rule better models intermediate stages of prism adjustment than does a standard back-propagation network. Brainard and Knudsen (1993a) report observing two peaks of activity across the tectum in response to an auditory stimulus during prism training - one corresponding to the pre-training response and one corresponding to the newly learned response. Over time the pre-trained response diminishes while the newly learned one grows. As shown in Figure 2, the network exhibits this same pattern of learning. Networks we trained under a standard back-propagation learning algorithm do not. Such a A Connectionist Model of the Owl's Sound Localization System 613 result lends support to the idea that the owl's localization system is computing a function similar to the one the network was designed to learn. In addition to accounting for known data, the network predicts results of experiments it was not designed to mimic. Specifically, the network accurately predicted that removal of the animal's facial ruff, which causes ILD to vary with azimuth instead of elevation, would have no effect on the animal's response to varying ILD. The network accomplishes the goals for which it was designed. It accounts for much, though not all, of the developmental data, it makes testable predictions for future experiments, and since we derived the learning rule in a principled fashion, the network provides us with a specific theory of the owl's sound localization system. References Brainard, M. S., & Knudsen, E. 1. (1993a). Dynamics of the visual calibration of the map of interaural time difference in the barn owl's optic tectum. Society Jor Neuroscience Abstracts, 19, 369.8. Brainard, M. S., & Knudsen, E. 1. (1993b). Experience-dependent plasticity in the inferior colliculus: a site for visual calibration of the neural representation of auditory space in the barn owl. The Journal of Neuroscience, 13, 4589-4608. Buntine, W. L., & Weigend, A. S. (1991). Bayesian back-propagation. Complex Systems, 5, 603-612. Knudsen, E. (1984). Auditory properties of space-tuned units in owl's optic tectum. Journal of Neurophysiology, 52(4), 709-723. Knudsen, E. (1988). Early blindness results in a degraded auditory map of space in the optic tectum of the barn owl. Proceedings of the National Academy of Science, U.S.A., 85, 6211-6214. Kuudsen, E., Blasdel, G., & Konishi, M. (1979). Sound localization by the barn owl (tyto alba) measured with the search coil technique. The Journal of Comparative Physiology A, 133, 1-11. Knudsen, E., & Knudsen, P. (1985). Vision guides the adjustment of auditory localization in young barn owls. Science, 230, 545-548. Knudsen, E., & Knudsen, P. (1990). Sensitive and critical periods for visual calibration of sound localization by barn owls. The Journal of Neuroscience, 10(1), 222-232. MacKay, D. J. (1992). Bayesian Methods for Adaptive :Models. Unpublished doctoral dissertation, California Institute of Technology, Pasadena, California. Mogdans, J., & Knudsen, E. 1. (1992). Adaptive adjustment of unit tuning to sound localization cues in response to monaural occlusion in developing owl optic tectum. The Journal of Neuroscience, 12, 3473-3484. Payne, R. S. (1970). Acoustic location of prey by barn owls (tyto alba). The Journal of Experimental Biology, 54, 535-573. Rumelhart, D. E., Durbin, R., Golden, R., & Chauvin, Y. (in press). Backpropagation: The theory. In Y. Chauvin & D. E. Rumelhart (Eds.), Backpropagation: Theory, Architectures and Applications. Hillsdale, N.J.: Lawrence Earlbaum Associates. Weigend, A. S., Huberman, B. A., & Rumelhart, D. E. (1990). Predicting the future: A connectionist approach. International Journal of Neural Systems, 1, 193-209.
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Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study N. Karunanithi Room 2E-378, Bellcore 435 South Street Morristown, NJ 07960 E-mail: karun@faline.bellcore.com Abstract Functional complexity of a software module can be measured in terms of static complexity metrics of the program text. Classifying software modules, based on their static complexity measures, into different fault-prone categories is a difficult problem in software engineering. This research investigates the applicability of neural network classifiers for identifying fault-prone software modules using a data set from a commercial software system. A preliminary empirical comparison is performed between a minimum distance based Gaussian classifier, a perceptron classifier and a multilayer layer feed-forward network classifier constructed using a modified Cascade-Correlation algorithm. The modified version of the Cascade-Correlation algorithm constrains the growth of the network size by incorporating a cross-validation check during the output layer training phase. Our preliminary results suggest that a multilayer feed-forward network can be used as a tool for identifying fault-prone software modules early during the development cycle. Other issues such as representation of software metrics and selection of a proper training samples are also discussed. 793 794 Karunanithi 1 Problem Statement Developing reliable software at a low cost is an important issue in the area of software engineering (Karunanithi, Whitley and Malaiya, 1992). Both the reliability of a software system and the development cost can be reduced by identifying troublesome software modules early during the development cycle. Many measurable program attributes have been identified and studied to characterize the intrinsic complexity and the fault proneness of software systems. The intuition behind software complexity metrics is that complex program modules tend to be more error prone than simple modules. By controlling the complexity of software modules during development, one can produce software systems that are easy to maintain and enhance (because simple program modules are easy to understand). Static complexity metrics are measured from the passive program texts early during the development cycle and can be used as a valuable feedback for allocating resources in future development efforts (future releases or new projects). Two approachs can be applied to relate static complexity measures with faults found or program changes made during testing. In the estimative approach regressions models are used to predict the actual number of faults that will be disclosed during testing (Lipow, 1982; Gaffney, 1984; Shen et al., 1985; Crawford et al., 1985; Munson and Khoshgoftaar, 1992). Regression models assume that the metrics that constitute independent variables are independent and normally distributed. However, most practical measures often violate the normality assumptions and exhibit high correlation with other metrics (i.e., multicollinearity). The resulting fit of the regression models often tend to produce inconsistent predictions. Under the classification approach software modules are categorized into two or more fault-prone classes (Rodriguez and Tsai, 1987; Munson and Khoshgoftaar, 1992; Karunanithi, 1993; Khoshgoftaar et al., 1993). A special case of the classification approach is to classify software modules into either low-fault (non-complex) or high-fault (complex) categories. The main rationale behind this approach is that the software managers are often interested in getting some approximate feedback from this type of models rather than accurate predictions of the number of faults that will be disclosed. Existing two-class categorization models are based on linear discriminant principle (Rodriguez and Tsai, 1987; Munson and Khoshgoftaar, 1992). Linear discriminant models assume that the metrics are orthogonal and that they follow a normal distribution. To reduce multicollinearity, researchers often use principle component analysis or some other dimensionality reduction techniques. However, the reduced metrics may not explain all the variability if the original metrics have nonlinear relationship. In this paper, the applicability of neural network classifiers for identifying fault proneness of software modules is examined. The motivation behind this research is to evaluate whether classifiers can be developed without usual assumptions about the input metrics. In order to study the usefulness of neural network classifiers, a preliminary comparison is made between a simple minimum distance based Gaussian classifier, a single layer perceptron and a multilayer feed-forward network developed using a modified version of Fahlman's Cascade Correlation algorithm (Fahlman and Lebiere, 1990). The modified algorithm incorporates a cross-validation for constraining the growth of the size of the network. In this investigation, other issues Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study 795 such as selection of proper training samples and representation of metrics are also considered. 2 Data Set Used The metrics data used in this study were obtained from a research conducted by Lind and Vairavan (Lind and Vairavan, 1989) for a Medical Imaging System software. The complete system consisted of approximately 4500 modules amounting to about 400,000 lines of code written in Pascal, FORTRAN, PL/M and assembly level. From this set, a random sample of 390 high level language routines was selected for the analysis. For each module in the sample, program changes were recorded as an indication of software fault. The number of changes in the program modules varied from zero to 98. In addition to changes, 11 software complexity metrics were extracted from each module. These metrics range from total lines of code to Belady's bandwidth metric. (Readers curious about these metrics may refer to Table I of Lind and Vairavan, 1989.) For the purpose of our classification study, these metrics represent 11 input (both real and integer) variables of the classifier. A software module is considered as a low fault-prone module (Category I) if there are 0 or 1 changes and as a high fault-prone module (Category II) if there are 10 or more changes. The remaining modules are considered as medium fault category. For the purpose of this study we consider only the low and high fault-prone modules. Our extreme categorization and deliberate discarding of program modules is similar to the approach used in other studies (Rodriguez and Tsai, 1987; Munson and Khoshgoftaar, 1992). After discarding medium fault-prone modules, there are 203 modules left in the data set. Of 203 modules, 114 modules belong to the low fault-prone category while the remaining 89 modules belong to the high fault-prone category. The output layer of the neural nets had two units corresponding to two fault categories. 3 Training Data Selection We had two objectives in selecting training data: 1) to evaluate how well a neural network classifier will perform across different sized training sets and 2) to select the training data as much unbiased as possible. The first objective was motivated by the need to evaluate whether a neural network classifier can be used early in the software development cycle. Thus the classification experiments were conducted using training samples of size S = ~, ~, }, ~, ~, 190 fraction of 203 samples belonging to Categories I and II. The remaining ~ 1-S) fraction of the samples were used for testing the classifiers. In order to avoid bias in the training data, we randomly selected 10 different training samples for each fraction S. This resulted in 6 X 10 (=60) different training and test sets. 796 Karunanithi 4 Classifiers Compared 4.1 A Minimum Distance Classifier In order to compare neural network classifiers and linear discriminant classifiers we implemented a simple minimum distance based two-class Gaussian classifier of the form (Nilsson, 1990): IX - Gil = ((X - Gi)(X - Gi)t)1/2 where Gi, i = 1, 2 represent the prototype points for the Categories I and II, X is a 11 dimensional metrics vector, and t is the transpose operator. The prototype points G1 and G2 are calculated from the training set based on the normality assumption. In this approach a given arbitrary input vector X is placed in Category I if IX - G11 < IX - G21 and in Category II otherwise. All raw component metrics had distributions that are asymmetric with a positive skew (i.e., long tail to the right) and they had different numerical ranges. Note that asymmetric distributions do not conform to the normality assumption of a typical Gaussian classifier. First, to remove the extreme asymmetry of the original distribution of the individual metric we transformed each metric using a natural logarithmic base. Second, to mask the influence of individual component metric on the distance score, we divided each metric by its standard deviation of the training set. These transformations considerably improved the performance of the Gaussian classifier. To be consistent in our comparison we used the log transformed inputs for other classifiers also. 4.2 A Perceptron Classifier A perceptron with a hard-limiting threshold can be considered as a realization of a non-parametric linear discriminant classifier. If we use a sigmoidal unit, then the continuous valued output of the perceptron can be interpreted as a likelihood or probability with which inputs are assigned to different classes. In our experiment we implemented a perceptron with two sigmoidal units (outputs 1 and 2) corresponding to two categories. A given arbitrary vector X is assigned to Category I if the value of the output unit 1 is greater than the output of the unit 2 and to Category II otherwise. The weights of the network are determined iteratively using least square error minimization procedure. In almost all our experiments, the perceptron learned about 75 to 80 percentages of the training set. This implies that the rest of the training samples are not linearly separable. 4.3 A Multilayer Network Classifier To evaluate whether a multilayer network can perform better than the other two classifiers, we repeated the same set of experiments using feed-forward networks constructed by Fahlman's Cascade-Correlation algorithm. The Cascade-Correlation algorithm is a constructive training algorithm which constructs a suitable network architecture by adding one hidden (layer) unit at a time. (Refer to Fahlman and Lebiere, 1990 for more details on the Cascade-Correlation algorithm.) Our initial results suggested that the multilayer layer networks constructed by the CascadeCorrelation algorithm are not capable of producing a better classification accuracy Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study 797 than the other two classifiers. An analysis of the network suggested that the resulting networks had too many free variables (i.e., due to too many hidden units). A further analysis of the rate of decrease of the residual error versus the number of hidden units added to the networks revealed that the Cascade-Correlation algorithm is capable of adding more hidden units to learn individual training patterns at the later stages of the training phase than in the earlier stages. This happens if the training set contains patterns that are interspersed across different decision regions or what might be called "border patterns" (Ahmed, S. and Tesauro, 1989). In an effort to constrain the growth of the size of the network, we modified the Cascade-Correlation algorithm to incorporate a cross-validation check during the output layer training phase. For each training set of size S, one third was used for cross-validation and the remaining two third was used to train the network. The network .construction was stopped as soon as the residual error of the crossvalidation set stopped decreasing from the residual error at the end of the previous output layer training phase. The resulting network learned about 95% of the training patterns. However, the cross-validated construction considerably improved the classification performance of the networks on the test set. Table 1 presented in the next section provides a comparison between the networks developed with and without cross-validation. Training Hidden Unit Error Statistics Set Size Statistics Type I Error Type II Error Sin% Mean I Std Mean I Std Mean I Std Without Cross-Validation 25 5.1 1.5 24.64 7.2 16.38 6.4 33 6.2 1.8 20.24 8.4 17.27 5.5 50 7.4 1.8 18.30 7.4 18.65 6.4 67 9.7 1.7 15.78 6.5 18.05 7.1 75 10.4 1.8 14.54 7.6 16.85 7.3 90 11.2 1.6 10.33 7.2 17.73 8.3 With Cross-Validation 25 1.9 1.3 20.19 5.4 12.11 4.7 33 2.2 1.0 18.24 5.5 12.40 4.1 50 2.0 0.9 17.41 5.6 15.04 5.2 67 2.7 1.1 14.32 5.8 14.08 5.5 75 2.7 1.3 13.27 7.0 13.84 5.4 90 2.9 1.2 9.77 9.4 15.47 5.1 Table 1: A Comparison of Nets With and Without Cross-Validation. 5 Results In this section we present some preliminary results from our classification experiments. First, we provide a comparison between the multilayer networks developed with and without cross-validation. Next, we compare different classifiers in terms of their classification accuracy. Since a neural network's performance can be affected by the weight vector used to initialize the network, we repeated the training experiment 25 times with different initial weight vectors for each training set. This 798 Karunanithi resulted in a total of 250 training trials for each value of S. The results reported here for the neural network classifiers represent a summary statistics for 250 experiments. The performance of the classifiers are reported in terms of classification errors. There are two type of classification errors that a classifier can make: a Type I error occurs when the classifier identifies a low fault-prone (Category I) module as a high fault-prone (Category II) module; a Type II error is produced when a high faultprone module is identified as a low fault-prone module. From a software manager's point of view, these classification errors will have different implications. Type I misclassification will result in waste of test resources (because modules that are less fault-prone may be tested longer than what is normally required). On the other hand, Type II misclassification will result in releasing products that are of inferior quality. From reliability point of view, a Type II error is a serious error than a Type I error. No. of Patterns Error Statistics S I Training I Test % Set Set Gaussian I Perceptron I Multilayer Nets Mean 1 Std Mean 'I Std Mean r Std Type I Error Statistics 25 50 86 13.16 4.7 16.17 5.5 20.19 5.4 33 66 77 11.44 4.0 11.74 3.9 18.24 5.5 50 101 57 12.45 3.2 11.58 3.2 17.41 5.6 67 136 37 9.46 4.1 10.14 3.9 14.32 5.8 75 152 28 8.57 5.4 9.15 5.8 13.27 7.0 90 182 12 14.17 7.9 4.03 4.3 9.77 9.4 Type 11 Error Statistics 25 50 67 15.61 4.2 15.98 7.8 12.11 4.7 33 66 60 15.46 4.6 15.78 6.6 12.40 4.1 50 101 45 16.01 5.1 16.97 6.8 15.04 5.2 67 136 30 16.00 5.4 16.11 7.6 14.08 5.5 75 152 23 17.39 5.8 18.39 6.3 13.84 5.4 90 182 9 21.11 6.3 19.11 5.6 15.47 5.1 Table 2: A Summary of Type I and Type II Error Statistics. Table 1 compares the complexity and the performance of the multilayer networks developed with and without cross-validation. Columns 2 through 7 represent the size and the performance of the networks developed by the Cascade-Correlation without cross-validation. The remaining six columns correspond to the networks constructed with cross-validation. Hidden unit statistics for the networks suggest that the growth of the network can be constrained by adding a cross-validation during the output layer training. The corresponding error statistics for both the Type I and Type II errors suggest that an improvement classification accuracy can be achieved by cross-validating the size of the networks. Table 2 illustrates the preliminary results for different classifiers. The first two columns in Table 2 represent the size of the training set in terms of S as a percentage of all patterns and the number of patterns respectively. The third column represents the number oft est patterns in Categories I (1st half) and the II (2nd half). The remaining six columns represent the error statistics for the three classifiers in Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study 799 terms of percentage mean errors and standard deviations. The percentages errors were obtained by dividing the number of misclassifications by the total number of test patterns in that Category. The Type I error statistics in the first half of the table suggest that the Gaussian and the Perceptron classifiers may be better than multilayer networks at early stages of the software development cycle. However, the difference in performance of the Gaussian classifier is not consistent across all values of S. The neural network classifiers seem to improve their performance with an increase in the size of the training set. Among neural networks, the perceptron classifier seems to perform classification than a multilayer net. However, the Type II error statistics in the second half of the table suggest that a multilayer network classifier may provide a better classification of Category II modules than the other two classifiers. This is an important results from the reliability perspective. 6 Conclusion and Work in Progress We demonstrated the applicability of neural network classifiers for identifying faultprone software modules. We compared the classification efficacy of three different pattern classifiers using a data set from a commercial software system. Our preliminary empirical results are encouraging in that there is a role for multilayer feed-forward networks either during the software development cycle of a subsequent release or for a similar product. The cross-validation implemented in our study is a simple heuristics for constraining the size of the networks constructed by the Cascade-Correlation algorithm. Though this improved the performance of the resulting networks, it should be cautioned that cross-validation may be needed only if the training patterns exhibit certain characteristics. In other circumstances, the networks may have to be constructed using the entire training set. At this stage we have not performed complete analysis on what characteristics of the training samples would require cross-validation for constraining the network growth. Also we have not used other sophisticated structure reduction techniques. We are currently exploring different loss functions and structure reduction techniques. The Cascade-Correlation algorithm always constructs a deep network. Each additional hidden unit develops an internal representation that is a higher order sigmoidal computation than those of previously added hidden units. Such a complex internal representation may not be appropriate in a classification application such as the one studied here. We are currently exploring alternatives to construct shallow networks within the Cascade-Correlation frame work. At this stage, we have not performed any analysis on how the internal representations of a multilayer network correlate with the input metrics. This is currently being studied. References Ahmed, S. and G. Tesauro (1989). "Scaling and Generalization in Neural Networks: A Case Study", Advances in Neural Information Processing Systems 1, pp 160-168, D. Touretzky, ed. Morgan Kaufmann. 800 Karunanithi Crawford, S. G., McIntosh, A. A. and D. Pregibon (1985). "An Analysis of Static Metrics and Faults in C Software", The Journal of Systems and Software, Vol. 5, pp. 37-48. Fahlman, S. E. and C. Lebiere (1990). "The Cascaded-Correlation Learning Architecture," Advances in Neural Information Processing Systems 2, pp 524-532, D. Touretzky, ed. Morgan Kaufmann. Gaffney Jr., J. E. (1984). "Estimating the Number of Faults in Code", IEEE Trans. on Software Eng., Vol. SE-lO, No.4, pp. 459-464. Karunanithi, N, Whitley, D. and Y. K. Malaiya (1992). "Prediction of Software Reliability Using Connectionist Models" , IEEE Trans. on Software Eng., Vol. 18, No.7, pp. 563-574. Karunanithi, N. (1993). "Identifying Fault-Prone Software Modules Using Connectionist Networks", Proc. of the 1st Int'l Workshop on Applications of Neural Networks to Telecommunications, (IWANNT'93), pp. 266-272, J. Alspector et al., ed., Lawrence Erlbaum, Publisher. Khoshgoftaar, T. M., Lanning, D. L. and A. S. Pandya (1993). "A Neural Network Modeling Methodology for the Detection of High-Risk Programs" , Proc. of the 4th Int'l Symp. on Software Reliability Eng. pp. 302-309. Lind, R. K. and K. Vairavan (1989). "An Experimental Investigation of Software Metrics and Their Relationship to Software Development Effort", IEEE Trans. on Software Eng., Vol. 15, No.5, pp. 649-653. Lipow, M. (1982). "Number of Faults Per Line of Code", IEEE Trans. on Software Eng., Vol. SE-8, No.4, pp. 437-439. Munson, J. C. and T. M. Khoshgoftaar (1992). "The Detection of Fault-Prone Programs", IEEE Trans. on Software Eng., Vol. 18, No.5, pp. 423-433. Nilsson, J. Nils (1990). The Mathematical Foundations of Learning Machines, Morgan Kaufmann, Chapters 2 and 3. Rodriguez, V. and W. T. Tsai (1987). "A Tool for Discriminant Analysis and Classification of Software Metrics", Information and Software Technology, Vol. 29, No.3, pp. 137-149. Shen, V. Y., Yu, T., Thebaut, S. M. and T. R. Paulsen (1985). "Identifying ErrorProne Software: An Empirical Study", IEEE Trans. on Software Eng., Vol. SE-ll, No.4, pp. 317-323.
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Event-Driven Simulation of Networks of Spiking l'Ieurons Lloyd Watts Synaptics Inc. 2698 Orchard Parkway San Jose CA 95134 11oydGsynaptics.com Abstract A fast event-driven software simulator has been developed for simulating large networks of spiking neurons and synapses. The primitive network elements are designed to exhibit biologically realistic behaviors, such as spiking, refractoriness, adaptation, axonal delays, summation of post-synaptic current pulses, and tonic current inputs. The efficient event-driven representation allows large networks to be simulated in a fraction of the time that would be required for a full compartmental-model simulation. Corresponding analog CMOS VLSI circuit primitives have been designed and characterized, so that large-scale circuits may be simulated prior to fabrication. 1 Introduction Artificial neural networks typically use an abstraction of real neuron behaviour, in which the continuously varying mean firing rate of the neuron is presumed to carry the information about the neuron's time-varying state of excitation [1]. This useful simplification allows the neuron's state to be represented as a time-varying continuous-amplitude quantity. However, spike timing is known to be important in many biological systems. For example, in nearly all vertebrate auditory systems, spiral ganglion cells from the cochlea are known to phase lock to pure-tone stimuli for all but the highest perceptible frequencies [2]. The barn owl uses axonal delays to compute azimuthal spatial localization [3]. Studies in the cat [4] have shown that 927 928 Watts relative timing of spikes is preserved even at the highest cortical levels. Studies in the visual system of the blowfly [5] have shown that the information contained in just three spikes is enough for the fly to make a decision to turn, if the visual input IS sparse. Thus, it is apparent that biological neural systems exploit the spiking and timedependent behavior of the neurons and synapses to perform system-level computation. To investigate this type of computation, we need a simulator that includes detailed neural behavior, yet uses a signal representation efficient enough to allow simulation of large networks in a reasonable time. 2 Spike: Event-Driven Simulation Spike is a fast event-driven simulator optimized for simulating networks of spiking neurons and synapses. The key simplifying assumption in Spike is that all currents injected into a cell are composed of piecewise-constant pulses (i.e., boxcar pulses), and therefore all integrated membrane voltage trajectories are piecewise linear in time. This very simple representation is capable of surprisingly complex and realistic behaviors, and is well suited for investigating system-level questions that rely on detailed spiking behavior. The simulator operates by maintaining a queue of scheduled events. The occurrence of one event (i.e., a neuron spike) usually causes later events to be scheduled in the queue (Le., end of refractory period, end of post-synaptic current pulse). The total current injected into a cell is integrated into the future to predict the time of firing, at which time a neuron spike event is scheduled. If any of the current components being injected into the cell subsequently change, the spike event is rescheduled. The simulator runs until the queue is empty or until the desired run-time has elapsed. A similar event-driven neural simulator was developed by Pratt [6]. The simulator output may be plotted by a number of commercially available plotting programs, including Gnuplot, Mathematica, Xvgr, and Cview. 3 N euraLOG: Neural Schematic Capture NeuraLOG is a schematic entry tool, which allows the convenient entry of "neural" circuit diagrams, consisting of neurons, synapses, test inputs, and custom symbols. NeuraLOG is a customization of the program AnaLOG, by John Lazzaro and Dave Gillespie. The parameters of the neural elements are entered directly on the schematic diagram; these parameters include the neuron refractory period, duration and intensity of the post-synaptic current pulse following an action potential, saturation value of summating post-synaptic currents, tonic input currents, axonal delays, etc. Custom symbols can be defined, so that arbitrarily complex hierarchical designs may be made. It is common to create a complex "neuron" containing many neuron and synapse primitive elements. Spiking inputs may be supplied as external stimuli for the circuit in a number of different formats, including single spikes, periodic spike trains, periodic bursts, poisson random spike trains, and gaussian-jittered periodic Event-Driven Simulation of Networks of Spiking Neurons 929 spike trains. Textual input to Spike is also supported, to allow simulation of circuit topologies that would be too time-consuming to enter graphically. 4 A Simple Example A simple example of a neural circuit is shown in Figure 1. This circuit consists of two neurons (the large disks), several synapses (the large triangles), and two tonic inputs (the small arrows). The text strings associated with each symbol define that symbol's parameters: neuron parameters are identifier labels (Le., ni) and refractory period in milliseconds (ms); synapse parameters are the value of the postsynaptic current in nA, and the duration of the current pulse in ms, and an optional saturation parameter, which indicates how many post-synaptic current pulses may be superposed before saturation; the tonic input parameter is the injected current in nA. Filled symbols (tonic inputs and synapses) indicate inhibitory behavior. -.8108> -815> .132.5> 6.4> -.001 > Figure 1: Graphical input representation of a simple neural circuit, as entered in NeuraLOG. The simulated behavior of the circuit is shown in Figure 2. The neuron ni exhibits an adapting bursting behavior, as seen in the top trace of the plot. The excitatory tonic current input to neuron ni causes ni to fire repeatedly. The weakly excitatory synapse from ni to neuron n2 causes n2 to fire after many spikes from n1. The synaptic current in the synapse from ni to n2 is plotted in the trace labeled snin2. The strongly inhibitory synapse from n2 to ni causes ni to stop firing after n2 fires a spike. The synaptic current in the synapse from n2 to ni is plotted in the trace labeled sn2ni. The combination of the excitatory tonic input to ni and the inhibitory feedback from n2 to ni causes the bursting behavior. The adapting behavior is caused by the self-inhibitory accumulating feedback from neuron ni to itself, via the summating inhibitory synapse in the top left of the diagram. Each spike on ni causes a slightly increased inhibitory current into ni, which gradually slows the rate of firing with each successive pulse. The synaptic current in this inhibitory synapse is plotted in the trace snini; it is similar to the 930 Watts n1 _mJJJD"'----___ JillJJU"'----___ OOJJU~ n2 J J J~ sn1n1 sn2n1 o 10 20 30 40 50 60 70 Time (ms) Figure 2: Simulation results for the circuit of Figure 1, showing adapting bursting behavior. current that would be generated by a calcium-dependent potassium channel. This simple example demonstrates that the summating synapse primitive can be used to model a behavior that is not strictly synaptic in origin; it can be thought of as a general time-dependent state variable. This example also illustrates the principle that proper network topology (summating synapse in a negative feedback loop) can lead to realistic system-level behavior (gradual adaptation), even though the basic circuit elements may be rather primitive (boxcar current pulses). 5 Applications of the Simulation Tools NeuraLOG and Spike have been used by the author to model spiking associative memories, adaptive structures that learn to predict a time delay, and chaotic spiking circuits. Researchers at Caltech [7, 8] and the Salk Institute have used the tools in their studies of locust central pattern generators (CPGs) and cortical oscillations. The cortical oscillation circuits contain a few hundred neurons and a few thousand synapses. A CPG circuit, developed by Sylvie Ryckebusch, is shown in Figure 3; the corresponding simulation output is shown in Figure 4. NeuraLOG and Spike are distributed at no charge under the GNU licence. They are currently supported on HP and Sun workstations. The tools are supplied with a user's manual and working tutorial examples. Event-Driven Simulation of Networks of Spiking Neurons 931 in)>--~. ~)>--~. pill IwI)>--i11 chI)>---..... i11) ~ levi) ~ pill) levi) <l ~ pill) <l dll I-I) ItIVr) ~ I_r) <l ItIVr) in) ~in ItIVr) pirr ItIVr) i1r dar) i1r) ItIVr ) ~ pirr) ItIVr) I_r pirr) <l dar ItIVr ) levi) levi) <l IwI) Figure 3: Sylvie Ryckebusch's locust CPG circuit. For clarity, the synapse parameters have been omitted. 932 Watts In I dfl dsl i11 levi pirl dfr dsr j j J J J J j j J 11r " t -L t -I 1 -I L -L t -I levr j j J j .J j j j J pirr • 0 20 40 60 80 100 Time (ms) Figure 4: Simulation results for Sylvie Ryckebusch's locust CPG circuit. 6 The Link to Analog VLSI Analog VLSI circuit primitives that can be modelled by Spike have been designed and tested. The circuits are shown in Figure 5, and have been described previously [9, 10]. These circuits have been used by workers at Caltech to implement VLSI models of central pattern generators. The software simulation tools allow simulation of complex neural circuits prior to fabrication, to improve the likelihood of success on first silicon, and to allow optimization of shared parameters (bias wires). 7 Conclusion NeuraLOG and Spike fill a need for a fast neural simulator that can model large networks of biologically realistic spiking neurons. The simple computational primitives within Spike can be used to create complex and realistic neural behaviors in arbitrarily complex hierarchical designs. The tools are publicly available at no charge. NeuraLOG and Spike have been used by a number of research labs for detailed modeling of biological systems. Acknowledgements NeuraLOG is a customization of the program AnaLOG, which was written by John Lazzaro and David Gillespie. Lloyd Watts gratefully acknowledges helpful discussions with Carver Mead, Sylvie Ryckebusch, Misha Mahowald, John Lazzaro, Event-Driven Simulation of Networks of Spiking Neurons 933 .......................•.......... ..•••....... ~ ....................•.....•....................•...................... I emI IKJ I . ~ ............ ~y.f!~.P.s.~ .............. I~~.i~ .. L ............................... ~~':Ir~!" ............................ . Figure 5: CMOS Analog VLSI circuit primitives. The neuron circuit models a voltage-gated sodium channel and a delayed rectifier potassium channel to produce a spiking mechanism. The tonic circuit allows constant currents to be injected onto the membrane capacitance em. The synapse circuit creates a boxcar current pulse in response to a spike input. David Gillespie, Mike Vanier, Brad Minch, Rahul Sarpeshkar, Kwabena Boahen, John Platt, and Steve Nowlan. Thanks to Sylvie Ryckebusch for permission to use her CPG circuit example. References [1] J. Hertz, A. Krogh and R. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley, 1991. [2] N. Y-S. Kiang, T. Watanabe, E. C. Thomas, L. F. Clark, "Discharge Patterns of Single Fibers in the Cat's Auditory Nerve", MIT Res. Monograph No. 35, (MIT, Cambridge, MA). [3] M. Konishi, T.T. Takahashi, H. Wagner, W.E. Sullivan, C.E. Carr, "Neurophysiological and Anatomical Substrates of Sound Localization in the Owl", In Auditory Function, G.M. Edelman, W.E. Gall, and W.M. Cowan, eds., pp. 721-745, Wiley, New York. [4] D. P. Phillips and S. E. Hall, "Response Timing Constraints on the Cortical Representation of Sound Time Structure" , Journal of the Acoustical Society of America, 88 (3), pp. 1403-1411, 1990. [5] R.R. de Ruyter van Steveninck and W. Bialek, "Real-time Performance of a movement-sensitive neuron in the blowfly visual system: Coding and infor934 Watts mation transfer in short spike sequences", Proceedings of the Royal Society of London, Series B, 234, 379-414. [6] G. A. Pratt, Pulse Computation, Ph.D. Thesis, Massachusetts Institute of Technology, 1989. [7] M. Wehr, S. Ryckebusch and G. Laurent, Western Nerve Net Conference, Seattle, Washington, 1993. [8] S. Ryckebusch, M. Wehr, and G. Laurent, "Distinct rhythmic locomotor patterns can be generated by a simple adaptive neural circuit: biology, simulation, and VLSI implementation", in review, Journal of Computational Neuroscience. [9] R. Sarpeshkar, L. Watts, C.A. Mead, "Refractory Neuron Circuits", Internal Memorandum, Physics of Computation Laboratory, California Institute of Technology, 1992. [10] L. Watts, "Designing Networks of Spiking Silicon Neurons and Synapses", Proceedings of Computation and Neural Systems Meeting CNS*92, San Francisco, CA,1992. PART VIII VISUAL PROCESSING
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Grammatical Inference by Attentional Control of Synchronization in an Oscillating Elman Network Bill Baird Dept Mathematics, U.C.Berkeley, Berkeley, Ca. 94720, baird@math.berkeley.edu Todd Troyer Dept of Phys., U.C.San Francisco, 513 Parnassus Ave. San Francisco, Ca. 94143, todd@phy.ucsf.edu Abstract Frank Eeckman Lawrence Livermore National Laboratory, P.O. Box 808 (L-270), Livermore, Ca. 94550, eeckman@.llnl.gov We show how an "Elman" network architecture, constructed from recurrently connected oscillatory associative memory network modules, can employ selective "attentional" control of synchronization to direct the flow of communication and computation within the architecture to solve a grammatical inference problem. Previously we have shown how the discrete time "Elman" network algorithm can be implemented in a network completely described by continuous ordinary differential equations. The time steps (machine cycles) of the system are implemented by rhythmic variation (clocking) of a bifurcation parameter. In this architecture, oscillation amplitude codes the information content or activity of a module (unit), whereas phase and frequency are used to "softwire" the network. Only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules contributes incoherent crosstalk noise. Attentional control is modeled as a special subset of the hidden modules with ouputs which affect the resonant frequencies of other hidden modules. They control synchrony among the other modules and direct the flow of computation (attention) to effect transitions between two subgraphs of a thirteen state automaton which the system emulates to generate a Reber grammar. The internal crosstalk noise is used to drive the required random transitions of the automaton. 67 68 Baird, Troyer, and Eeckman 1 Introduction Recordings of local field potentials have revealed 40 to 80 Hz oscillation in vertebrate cortex [Freeman and Baird, 1987, Gray and Singer, 1987]. The amplitude patterns of such oscillations have been shown to predict the olfactory and visual pattern recognition responses of a trained animal. There is further evidence that although the oscillatory activity appears to be roughly periodic, it is actually chaotic when examined in detail. This preliminary evidence suggests that oscillatory or chaotic network modules may form the cortical substrate for many of the sensory, motor, and cognitive functions now studied in static networks. It remains be shown how networks with more complex dynamics can performs these operations and what possible advantages are to be gained by such complexity. 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 views cortex as a set of coupled oscillatory associative memories, and is guided by the principle that attractors must be used by macroscopic systems for reliable computation in the presence of noise. This system must function reliably in the midst of noise generated by crosstalk from it's own activity. 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". In a similar manner, the network we have constructed is a symbol processing system, but with analog input and oscillatory subsymbolic representations. The architecture operates as a thirteen state finite automaton that generates the symbol strings of a Reber grammar. It is designed to demonstrate and study the following issues and principles of neural computation: (1) Sequential computation with coupled associative memories. (2) Computation with attractors for reliable operation in the presence of noise. (3) Discrete time and state symbol processing arising from continuum dynamics by bifurcations of attractors. (4) Attention as selective synchronization controling communication and temporal program flow. (5) chaotic dynamics in some network modules driving randomn choice of attractors in other network modules. The first three issues have been fully addressed in a previous paper [Baird et. al., 1993], and are only briefly reviewed. ".le focus here on the last two. 1.1 Attentional Processing 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 irrelevant to the present computation are contributing crosstalk noise. This is smilar to the problem of coding messages in a computer architecture like the Connection Machine so that they can be picked up from the common communication buss line by the proper receiving module. Periodic or nearly periodic (chaotic) variation of a signal introduces additional degrees of freedom that can be exploited in a computational architecture. We investigate the principle that selective control of synchronization, which we hypopthesize to be a model of "attention", can be used to solve this coding problem and control communication and program flow in an architecture with dynamic attractors. The architecture illust.rates the notion that synchronization not only "binds" senGrammatical Inference by Attentional Control of Synchronization 69 sory inputs into "objects" [Gray and Singer, 1987], but binds the activity of selected cortical areas into a functional whole that directs behavior. It is a model of "attended activity" as that subset which has been included in the processing of the moment by synchronization. This is both a spatial and temporal binding. Only the inputs which are synchronized to the internal oscillatory activity of a module can effect previously learned transitions of at tractors within it. For example, consider two objects in the visual field separately bound in primary visual cortex by synchronization of their components at different phases or frequencies. One object may be selectively attended to by its entrainment to oscillatory processing at higher levels such as V4 or IT. These in turn are in synchrony with oscillatory activity in motor areas to select the attractors there which are directing motor output. In the architecture presented here, we have constrained the network dynamics so that there exist well defined notions of amplitude, phase, and frequency. The network has been designed so that amplitude codes the information content or activity of a module, whereas phase and frequency are used to "softwire" the network. An oscillatory network module has a passband outside of which it will not synchronize with an oscillatory input. Modules can therefore easily be de synchronized by perturbing their resonant frequencies. Furthermore, only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules contributes incoherant crosstalk or noise. The flow of communication between modules can thus be controled by controlling synchrony. By changing the intrinsic frequency of modules in a patterned way, the effective connectivity of the network is changed. The same hardware and connection matrix can thus subserve many different computations and patterns of interaction between modules without crosstalk problems. The crosstalk noise is actually essential to the function of the system. It serves as the noise source for making random choices of output symbols and automaton state transitions in this architecture, as we discuss later. In cortex there is an issue as to what may constitute a source of randomness of sufficient magnitude to perturb the large ensemble behavior of neural activity at the cortical network level. It does not seem likely that the well known molecular fluctuations which are easily averaged within one or a few neurons can do the job. The architecture here models the hypothesis that deterministic chaos in the macroscopic dynamics of a network of neurons, which is the same order of magnitude as the coherant activity, can serve this purpose. In a set of modules which is desynchronized by perturbing the resonant frequencies of the group, coherance is lost and "random" phase relations result. The character of the model time traces is irregular as seen in real neural ensemble activity. The behavior of the time traces in different modules of the architecture is similar to the temporary appearance and switching of synchronization between cortical areas seen in observations of cortical processing during sensory/motor tasks in monkeys and humans [Bressler and Nakamura, 1993]. The structure of this apparently chaotic signal and its use in network learning and operation are currently under investigation. 2 Normal Form Associative Memory Modules The mathematical foundation for the construction of network modules is contained in the normal form projection algorithm [Baird and Eeckman, 1993]. This is a learning algorithm for recurrent analog neural networks which allows associative memory storage of analog patterns, continuous periodic sequences, and chaotic 70 Baird, Troyer, and Eeckman attractors in the same network. An N node module can be shown to function as an associative memory for up to N /2 oscillatory, or N /3 chaotic memory attractors [Baird and Eeckman, 1993]. 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]. The network modules of this architecture were developed previously as models of olfactory cortex with distributed patterns of activity like those observed experimentally [Baird, 1990, Freeman and Baird, 1987]. Such a biological network is dynamically equivalent to a network in normal form and may easily be designed, simulated, and theoretically evaluated in these coordinates. When the intramodule competition is high, they are "memory" or winner-take-all cordinates where attractors have one oscillator at maximum amplitude, with the other amplitudes near zero. In figure two, the input and output modules are demonstrating a distributed amplitude pattern ( the symbol "T"), and the hidden and context modules are two-attractor modules in normal form coordinates showing either a right or left side active. In this paper all networks are discussed in normal form coordinates. By analyzing the network in these coordinates, the amplitude and phase dynamics have a particularly simple interaction. When the input to a module is synchronized with its intrinsic oscillation, the amplitude of the periodic activity may be considered separately from the phase rotation. The module may then be viewed as a static network with these amplitudes as its activity. To illustrate the behavior of individ ualnetwork modules, we examine a binary (twoattractor) module; the behavior of modules with more than two attractors is similar. Such a unit is defined in polar normal form coordinates by the following equations of the Hopf normal form: rli 1l.irli - Cdi + (d - bsin(wclockt))rlir5i + L wtlj cos(Oj - Oli) j rOi 1l.jr Oi - crgi + (d - bsin(wclockt))roirii + L wijlj cos(Oj - OOi) j Oli Wi + L wt(Ij /1·li) sin(Oj - Oli) j OOi Wi + L wij(Ij/rOi) sin(Oj - OOi) j The clocked parameter bsin(wclockt) is used to implement the discrete time machine cycle of the Elman architecture as discussed later. It has lower frequency (1/10) than the intrinsic frequency of the unit Wi. Examination of the phase equations shows that a unit has a strong tendency to synchronize with an input of similar frequency. Define the phase difference cp = 00 OJ = 00 - wJt between a unit 00 and it's input OJ. For either side of a unit driven by an input of the same frequency, WJ = Wo, There is an attractor at zero phase difference cp = 00 OJ = ° and a repellor at cp = 180 degrees. In simulations, the interconnected network of these units described below synchronizes robustly within a few cycles following a perturbation. If the frequencies of some modules of the architecture are randomly dispersed by a significant amount, WJ - Wo #- 0, phase-lags appear first, then synchronization is lost in those units. An oscillating module therefore acts as a band pass filter for oscillatory inputs. Grammatical Inference by Attentional Control of Synchronization 71 When the oscillators are sychronized with the input, OJ - Oli = 0, the phase terms cos(Oj - Oli) = cos(O) = 1 dissappear. This leaves the amplitude equations rli and rOi with static inputs Ej wt;Ij and E j wijlj. Thus we have network modules which emulate static network units in their amplitude activity when fully phaselocked to their input. Amplitude information is transmitted between modules, with an oscillatory carrier. For fixed values of the competition, in a completely synchronized system, the internal amplitude dynamics define a gradient dynamical system for a fourth order energy fUllction. External inputs that are phase-locked to the module's intrinsic oscillation simply add a linear tilt to the landscape. 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 along one axis over the other. Thinking of Tli as the "acitivity" of the unit, this acitivity becomes a monotonically increasing function of input. The module behaves as an analog connectionist unit whose transfer function can be approximated by a sigmoid. We refer to this as the "analog" mode of operation of the module. With high levels of competition, the unit will behave as a binary (bistable) digital flip-flop element. There are two deep potential wells, one on each axis. Hence the module performs a winner-take-all choice on the coordinates of its initial state and maintains that choice "clamped" and independent of external input. This is the "digital" or "quantized" mode of operation of a module. We think of one attractor within the unit as representing "1" (the right side in figure two) and the other as representing "0" . 3 Elman Network of Oscillating Associative Memories 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. Previously we cons structed a system which functions as the six Figure 1. state finite automaton that perfectly recognizes or generates the set of strings defined by the Reber grammar described in Cleeremans et. al. [Cleeremans et al., 1989]. We found the connections for this network by using the backpropagation algorithm in a static network that approximates the behavior of the amplitudes of oscillation in a fully synchronized dynamic network [Baird et al., 1993]. Here we construct a system that emulates the larger 13 state automata similar (less one state) to the one studied by Cleermans, et al in the second part of their paper. The graph of this automaton consists of two subgraph branches each of which has the graph structure of the automaton learned as above, but with different assignments of transition output symbols (see fig. 1). T 72 Baird, Troyer, and Beckman We use two types of modules in implementing the Elman network architecture shown in figure two below. The input and output layer each consist of a single associative memory module with six oscillatory attractors (six competing oscillatory modes), one for each of the six symbols in the grammar. The hidden and context layers consist of the binary "units" above composed of a two oscillatory attractors. The architecture consists of 14 binary modules ill the hidden and context layers - three of which are special frequency control modules. The hidden and context layers are divided into four groups: the first three correspond to each of the two subgraphs plus the start state, and the fourth group consists of three special control modules, each of which has only a special control output that perturbs the resonant frequencies of the modules (by changing their values in the program) of a particular state coding group when it is at the zero attractor, as illustrated by the dotted control lines in figure two. This figure shows control unit two is at the one attractor (right side of the square active) and the hidden units coding for states of subgraph two are in synchrony with the input and output modules. Activity levels oscillate up and down through the plane of the paper. Here in midcycle, competition is high in all modules. Figure 2. OSCILLATING ELMAN NETWORK OUTPUT INPUT The discrete machine cycle of the Elman algorithm is implemented by the sinusoidal variation (clocking) of the bifurcation parameter in the normal form equations that determines the level of intramodule competition [Baird et al., 1993]. At the beginning of a machine cycle, when a network is generating strings, the input and context layers are at high competition and their activity is clamped at the bottom of deep basins of attraction. The hidden and output modules are at low competition and therefore behave as a traditional feedforward network free to take on analog values. In this analog mode, a real valued error can be defined for the hidden and output units and standard learning algorithms like backpropagation can be used to train the connections. Then the situation reverses. For a Reber grammar there are always two equally possible next symbols being activated in the output layer, and we let the crosstalk noise Grammatical Inference by Attentional Control of Synchronization 73 break this symmetry so that the winner-take-all dynamics of the output module can chose one. High competition has now also "quantized" and clamped the activity in the hidden layer to a fixed binary vector. Meanwhile, competition is lowered in the input and context layers, freeing these modules from their attractors. An identity mapping from hidden to context loads the binarized activity of the hidden layer into the context layer for the next cycle, and an additional identity mapping from the output to input module places the chosen output symbol into the input layer to begin the next cycle. 4 Attentional control of Synchrony We introduce a model of attention as control of program flow by selective synchronization. The attentional controler itself is modeled in this architecture as a special set of three hidden modules with ouputs that affect the resonant frequencies of the other corresponding three subsets of hidden modules. Varying levels of intramodule competition control the large scale direction of information flow between layers of the architecture. To direct information flow on a finer scale, the attention mechanism selects a subset of modules within each layer whose output is effective in driving the state transition behavior of the system. By controling the patterns of synchronization within the network we are able to generate the grammar obtained from an automaton consisting of two subgraphs connected by a single transition state (figure 1). During training we enforce a segregation of the hidden layer code for the states of the separate subgraph branches of the automaton so that different sets of synchronized modules learn to code for each subgraph of the automaton. Then the entire automaton is hand constructed with an additional hidden module for the start state between the branches. Transitions in the system from states in one subgraph of the automaton to the other are made by "attending" to the corresponding set of nodes in the hidden and context layers. This switching of the focus of attention is accomplished by changing the patterns of synchronization within the network which changes the flow of communication between modules. Each control module modulates the intrinsic frequency of the units coding for the states a single su bgraph or the unit representing the start state. The control modules respond to a particular input symbol and context to set the intrinsic frequency of the proper subset of hidden units to be equal to the input layer frequency. As described earlier, modules can easily be desynchronized by perturbing their resonant frequencies. By perturbing the frequencies of the remaining modules away from the input frequency, these modules are no longer communicating with the rest of the network. Thus coherent information flows from input to output only through one of three channels. Viewing the automata as a behavioral program, the control of synchrony constitutes a control of the program flow into its subprograms (the subgraphs of the automaton). When either exit state of a subgraph is reached, the "B" (begin) symbol is then emitted and fed back to the input where it is connected through the first to second layer weight matrix to the attention control modules. It turns off the synchrony of the hidden states of the subgraph and allows entrainment of the start state to begin a new string of symbols. This state in turn activates both a "T" and a "P' in the output module. The symbol selected by the crosstalk noise and fed back to the input module is now connected to the control modules through the weight matrix. It desynchronizes the start state module, synchronizes in the subset of hidden units 74 Baird. Troyer. and Eeckman coding for the states of the appropriate subgraph, and establishes there the start state pattern for that subgraph. Future work will investigate the possibilities for self-organization of the patterns of synchrony and spatially segregated coding in the hidden layer during learning. The weights for entire automata, including the special attention control hidden units, should be learned at once. 4.1 Acknowledgments 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, 1990] Baird, B. (1990). 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 and Eeckman, 1993] Baird, B. and Eeckman, F. H. (1993). A normal form projection algorithm for associative memory. In Hassoun, M. H., editor, Associative Neural Memories: Theory and Implementation, New York, NY. Oxford University Press. [Baird et al., 1993] Baird, B., Troyer, T., and Eeckman, F. H. (1993). Synchronization and gramatical inference in an oscillating elman network. In Hanson, S., Cowan, J., and Giles, C., editors, Advances in Neural Information Processing Systems S, pages 236-244. Morgan Kaufman. [Bressler and Nakamura, 1993] Bressler, S. and Nakamura. (1993). Interarea synchronization in Macaque neocortex during a visual discrimination task. In Eeckman,F. H., and Bower, J., editors, Computation and Neural Systems, page 515. Kluwer. [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, 1987] Freeman, W. and Baird, B. (1987). Relation of olfactory EEG to behavior: Spatial analysis. Behavioral Neuroscience, 101:393-408. [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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Bayesian Self-Organization Alan L. Yuille Division of Applied Sciences Harvard University Cambridge, MA 02138 Stelios M. Smirnakis Lyman Laboratory of Physics Harvard University Cambridge, MA 02138 Lei Xu * Dept. of Computer Science HSH ENG BLDG, Room 1006 The Chinese University of Hong Kong Shatin, NT Hong Kong Abstract Recent work by Becker and Hinton (Becker and Hinton, 1992) shows a promising mechanism, based on maximizing mutual information assuming spatial coherence, by which a system can selforganize itself to learn visual abilities such as binocular stereo. We introduce a more general criterion, based on Bayesian probability theory, and thereby demonstrate a connection to Bayesian theories of visual perception and to other organization principles for early vision (Atick and Redlich, 1990). Methods for implementation using variants of stochastic learning are described and, for the special case of linear filtering, we derive an analytic expression for the output. 1 Introduction The input intensity patterns received by the human visual system are typically complicated functions of the object surfaces and light sources in the world. It *Lei Xu was a research scholar in the Division of Applied Sciences at Harvard University while this work was performed. 1001 1002 Yuille, Smimakis, and Xu seems probable, however, that humans perceive the world in terms of surfaces and objects (Nakayama and Shimojo, 1992). Thus the visual system must be able to extract information from the input intensities that is relatively independent of the actual intensity values. Such abilities may not be present at birth and hence must be learned. It seems, for example, that binocular stereo develops at about the age of two to three months (Held, 1981). Becker and Hinton (Becker and Hinton, 1992) describe an interesting mechanism for self-organizing a system to achieve this. The basic idea is to assume spatial coherence of the structure to be extracted and to train a neural network by maximizing the mutual information between neurons with disjoint receptive fields. For binocular stereo, for example, the surface being viewed is assumed flat (see (Becker and Hinton, 1992) for generalizations of this assumption) and hence has spatially constant disparity. The intensity patterns, however, do not have any simple spatial behaviour. Adjusting the synaptic strengths of the network to maximize the mutual information between neurons with non-overlapping receptive fields, for an ensemble of images, causes the neurons to extract features that are spatially coherent thereby obtaining the disparity [fig. 1]. maximize I (a;b) ( : I : I ~ I ~ ) Figure 1: In Hinton and Becker's initial scheme (Becker and Hinton, 1992), maximization of mutual information between neurons with spatially disjoint receptive fields leads to disparity tuning, provided they train on spatially coherent patterns (i.e. those for which disparity changes slowly with spatial position) Workers in computer vision face a similar problem of estimating the properties of objects in the world from intensity images. It is commonly stated that vision is illposed (Poggio et al, 1985) and that prior assumptions about the world are needed to obtain a unique perception. It is convenient to formulate such assumptions by the use of Bayes' theorem P(SID) = P(DIS)P(S)/ P(D). This relates the probaBayesian Self-Organization 1003 bility P(SID) of the scene S given the data D to the prior probability of the scene P(S) and the imaging model P(DIS) (P(D) can be interpreted as a normalization constant). Thus a vision theorist (see (Clark and Yuille, 1990), for example) determines an imaging model P(DIS), picks a set of plausible prior assumptions about the world P(S) (such as natural constraints (Marr, 1982)), applies Bayes' theorem, and then picks an interpretation S* from some statistical estimator of P(SID) (for example, the maximum a posteriori (MAP) estimator S* = ARG{M AXsP(SID)}.) An advantage of the Bayesian approach is that, by nature of its probabilistic formulation, it can be readily related to learning with a teacher (Kersten et aI, 1987). It is unclear, however, whether such a teacher will always be available. Moreover, from Becker and Hinton's work on self-organization, it seems that a teacher is not always necessary. This paper proposes a way for generalizing the self-organization approach, by starting from a Bayesian perspective, and thereby relating it to Bayesian theories of vision. The key idea is to force the activity distribution of the outputs to be close to a pre-specified prior distribution Pp(S). We argue that this approach is in the same spirit as (Becker and Hinton, 1992), because we can choose the prior distribution to enforce spatial coherence, but it is also more general since many other choices of the prior are possible. It also has some relation to the work performed by Atick and Redlich (Atick and Redlich, 1990) for modelling the early visual system. We will take the viewpoint that the prior Pp(S) is assumed known in advance by the visual system (perhaps by being specified genetically) and will act as a selforganizing principle. Later we will discuss ways that this might be relaxed. 2 Theory We assume that the input D is a function of a signal L that the system wants to determine and a distractor N [fig.2]. For example L might correspond to the disparities of a pair of binocular stereo images and N to the intensity patterns. The distribution of the inputs is PD(D) and the system assumes that the signal L has distribution Pp(L). Let the output of the system be S = G(D, ,) where G is a function of a set of parameters, to be determined. For example, the function G(D, ,) could be represented by a multi-layer perceptron with the , 's being the synaptic weights. By approximation theory, it can be shown that a large varidy of neural networks can approximate any input-output function arbitrarily well given enough hidden nodes (Hornik et aI, 1989). The aim of self-organizing the network is to ensure that the parameters, are chosen so that the outputs S are as close to the L as possible. We claim that this can be achieved by adjusting the parameters, so as to make the derived distribution of the outputs PDD(S : ,) = f 8(S - G(D, ,))PD (D)[dD] as close as possible to Pp(S). This can be seen to be a consistency condition for a Bayesian theory as from Bayes formula we obtain the equation: J P(SID)PD(D)[dD] = J P(DIS)Pp(S)[dD] = Pp(S). (1) 1004 Yuille, Smimakis, and Xu which is equivalent to our condition, provided we choose to identify P(SID) with 6(S - C(D, -y». To make this more precise we must define a measure of similarity between the two distributions Pp(S) and PDD(S : -y). An attractive measure is the Kullback-Leibler distance (the entropy of PDD relative to Pp): D= F(~,N) ~(~) J PDD(S:-y) I( L(-y) = PDD(S : -y) log Pp(S) [dS]. S=G(D,r) (2) Figure 2: The parameters -yare adjusted to minihu~e the Kullback-Leibler distance between the prior (Pp) distribution of the true signal (E) and the derived distribution (PDD) of the network output (8). This measure can be divided into two parts: (i) - I PDD(S : -y) log Pp(S)[dS] and (ii) I PDD(S : -y) log PDD(S : -y)[dS). The second term encourages variability of the output while the first term forces similarity to the prior distribution. Suppose that Pp(S) can be expressed as a Markov random field (i.e. the spatial distribution of Pp(S) has a local neighbourhood structure, as is commonly assumed in Bayesian models of vision). Then, by the Hammersely-Clifford theorem, we can write Pp(S) = e-fJEp(S) /Z where Ep(S) is an energy function with local connections (for example, Ep(S) = Li(S, - Si+1)2), {3 is an inverse temperature and Z is a normalization constant. Then the first term can be written (Yuille et ai, 1992) as -J PDD(S : -y) log Pp(S)[d8) = {3{Ep(G(D, -Y»)D + log Z. (3) Bayesian Self-Organization 1005 We can ignore the log Z term since it is a constant (independent of ,). Minimizing the first term with respect to , will therefore try to minimize the energy of the outputs averaged over the inputs - (Ep(G(D,')))D - which is highly desirable (since it has a close connection to the minimal energy principles in (Poggio et aI, 1985, Clark and Yuille, 1990)). It is also important, however, to avoid the trivial solution G(D,,) = 0 as well as solutions for which G(D,,) is very small for most inputs. Fortunately these solutions are discouraged by the second term: J PDD(D,,) log PDD(D, ,)[dD], which corresponds to the negative entropy of the derived distribution of the network output. Thus, its minimization with respect to , is a maximum entropy principle which will encourage variability in the outputs G( D,,) and hence prevent the trivial solutions. 3 Reformulating for Implementation. Our theory requires us to minimize the Kullback-Leibler distance, equation 2, with respect to ,. We now describe two ways in which this could be implemented using variants of stochastic learning. First observe that by substituting the form of the derived distribution into equation 2 and integrating out the 5 variable we obtain: " J PDD(G(D,,) : ,) J\L({) = PD(D) log Pp(G(D,,)) [dD]. (4) Assuming a representative sample {DJ.t : JJ fA} of inputs we can approximate K L(,) by LJ.ttA log[PDD(G(DJ.t,,) : ,)/ Pp(G(DJ.t, ,))]. We can now, in principle, perform stochastic learning using backpropagation: pick inputs DJ.t at random and update the weights, using log[PDD(G(DJ.t,,): ,)/Pp(G(DJ.t,,))] as the error function. To do this, however, we need expressions for PDD(G(DJ.t,,) : ,) and its derivative with repect to,. If the function G(D,,) can be restricted to being 1-1 (increasing the dimensionality of the output space if necessary) then we can obtain (Yuille et aI, 1992) analytic expressions PDD(G(D,,) :,) = PD(D)/I det(oG/oD)1 and (ologPDD(G(D,,) : ,)/0,) = -(oG/OD)-1(02G/oDo,), where [-1] denotes the matrix inverse. Alternatively we can perform additional sampling to estimate PDD(G(D,,):,) and (ologPDD(G(D,,): ,)/0,) directly from their integral representations. (This second approach is similar to (Becker and Hinton, 1992) though they are only concerned with estimating the first and second moments of these distributions. ) 4 Connection to Becker and Hinton. The Becker and Hinton method (Becker and Hinton, 1992) involves maximizing the mutual information between the output of two neuronal units 5 1,52 [fig.l]. This is given by : where the first two terms correspond to maximizing the entropies of 51 and 52 while the last term forces 51 :::::: 52. 1006 Yuille, Smirnakis, and Xu By contrast, our version tries to minimize the quantity: If we then ensure that Pp (S 1, S2) = 6 (S 1 - S2) our second term will force S 1 ~ S2 and our first term will maximize the entropy of the joint distribution of Sl, S2. We argue that this is effectively the same as (Becker and Hinton, 1992) since maximizing the joint entropy of Sl, S2 with Sl constrained to equal S2 is equivalent to maximizing the individual entropies of SI and S2 with the same constraint. To be more concrete, we consider Becker and Hinton's implementation of the mutual information maximization principle in the case of units with continuous outputs. They assume that the outputs of units 1, 2 are Gaussian 1 and perform steepest descent to maximize the symmetrized form of the mutual information between SI and S2: where VO stands for variance over the set of inputs. They assume that the difference between the two outputs can be expressed as un correlated additive noise, SI = S2 + N. We reformalize their criterion as maximizing EBH(V(S2), V(N)) where EBH(V(S2), V(N)) = log{V(S2) + V(N)} + log V(S2) - 210g V(N). (6) For our scheme we make similar assumptions about the distributions of SI and S2. We see that < logPDD(SI,S2) >= -log{< si >< S~ > < S1S2 >2} = -log{V(S2)V(N)} (since < S1S2 >=< (S2 + N)S2 >= V(S2) and < Sf >= V(S2) + V(N)). Using the prior distribution PP(Sl' S2) ~ e- r (Sl-S2)2 our criterion corresponds to minimizing EYSX(V(S2), V(N)) where: Ey SX(V(S2), V(N)) = -log V(S2) - log V(N) + rV(N). (7) It is easy to see that maximizing EBH (V(S2), V(N)) will try to make V(S2) as large as possible and force V(N) to zero (recall that, by definition, V(N) ~ 0). Minimizing our energy will try to make V(S2) as large as possible and will force V(N) to 1/r (recall that r appears as the inverse of the variance of a Gaussian prior distribution for SI - S2 so making r large will force the prior distribution to approach 6(Sl - S2).) Thus, provided r is very large, our method will have the same effect as Becker and Hinton's. 5 Application to Linear Filtering. We now describe an analysis of these ideas for the case of linear filtering. Our approach will be contrasted with the traditional Wiener filter approach. 1 We assume for simplicity that these Gaussians have zero mean. Bayesian Self-Organization 1007 Consider a process ofthe form D(i) = ~(i)+N(i) where D(i) denotes the input to the system, ~(i) is the true signal which we would like to predict, and N(i) is the n?ise corrupting the signal. The resulting Wiener filter Aw (i) has fourier transform Aw = ~~ , ~/«h:: , ~ + ~N,N) where ~~,~ and ~N,N are the power spectrum of the signal and the noise respectively. By contrast, let us extract a linear filter Ab by applying our criterion. In the case that the noise and signal are independent zero mean Gaussian distributions this filter can be calculated explicitly (Yuille et aI, 1992). It has fourier transform with squared magnitude given by IAbl2 = ~!:,~/(~~,~ + ~N,N) . Thus our filter can be thought of as the square root of the Wiener filter. It is important to realize that although our derivation assumed additive Gaussian noise our system would not need to make any assumptions about the noise distribution. Instead our system would merely need to assume that the filter was linear and then would automatically obtain the "correct" result for the additive Gaussian noise case. We conjecture that the system might detect non-Gauusian noise by finding it impossible to get zero Kullback-Liebler distance with the linear ansatz. 6 Conclusion The goal of this paper was to introduce a Bayesian approach to self-organization using prior assumptions about the signal as an organizing principle. We argued that it was a natural generalization of the criterion of maximizing mutual information assuming spatial coherence (Becker and Hinton, 1992). Using our principle it should be possible to self-organize Bayesian theories of vision, assuming that the priors are known, the network is capable of representing the appropriate functions and the learning algorithm converges. There will also be problems if the probability distributions of the true signal and the distractor are too similar. If the prior is not correct then it may be possible to detect this by evaluating the goodness of the Kullback-Leibler fit after learning 2. This suggests a strategy whereby the system increases the complexity of the priors until the Kullback-Leibler fit is sufficiently good (this is somewhat similar to an idea proposed by Mumford (Mumford, 1992)). This is related to the idea of competitive priors in vision (Clark and Yuille, 1990). One way to implement this would be for the prior probability itself to have a set of adjustable parameters that would enable it to adapt to different classes of scenes. We are currently (Yuille et aI, 1992) investigating this idea and exploring its relationships to Hidden Markov Models. Ways to implement the theory, using variants of stochastic learning, were described. We sketched the relation to Becker and Hinton. As an illustration of our approach we derived the filter that our criterion would give for filtering out additive Gaussian noise (possibly the only analytically tractable case). This had a very interesting relation to the standard Wiener filter. 2This is reminiscent of Barlow's suspicious coincidence detectors (Barlow, 1993), where we might hope to determine if two variables x & yare independent or not by calculating the Kullback-Leibler distance between the joint distribution P(x, y) and the product of the individual distributions P( x) P(y). 1008 Yuille, Smirnakis, and Xu Acknowledgements We would like to thank DARPA for an Air Force contract F49620-92-J-0466. Conversations with Dan Kersten and David Mumford were highly appreciated. References J.J. Atick and A.N. Redlich. "Towards a Theory of Early Visual Processing". Neural Computation. Vol. 2, No.3, pp 308-320. Fall. 1990. H.B. Barlow. "What is the Computational Goal of the Neocortex?" To appear in: Large scale neuronal theories of the brain. Ed. C. Koch. MIT Press. 1993. S. Becker and G.E. Hinton. "Self-organizing neural network that discovers surfaces in random-dot stereograms". Nature, Vol 355. pp 161-163. Jan. 1992. J .J. Clark and A.L. Yuille. Data Fusion for Sensory Information Processing Systems. Kluwer Academic Press. Boston/Dordrecht/London. 1990. R. Held. "Visual development in infants". In The encyclopedia of neuroscience, vol. 2. Boston: Birkhauser. 1987. K. Hornik, S. Stinchocombe and H. White. "Multilayer feed-forward networks are universal approximators". Neural Networks 4, pp 251-257. 1991. D. Kersten, A.J. O'Toole, M.E. Sereno, D.C. Knill and J .A. Anderson. "Associative learning of scene parameters from images". Optical Society of America, Vol. 26, No. 23, pp 4999-5006. 1 December, 1987. D. Marr. Vision. W.H. Freeman and Company. San Francisco. 1982. D. Mumford. "Pattern Theory: a unifying perspective". Dept. Mathematics Preprint. Harvard University. 1992. K. Nakayama and S. Shimojo. "Experiencing and Perceiving Visual Surfaces". Science. Vol. 257, pp 1357-1363. 4 September. 1992. T. Poggio, V. Torre and C. Koch. "Computational vision and regularization theory" . Nature, 317, pp 314-319. 1985. A.L. Yuille, S.M. Smirnakis and L. Xu. "Bayesian Self-Organization". Harvard Robotics Laboratory Technical Report. 1992. PART IX SPEECH AND SIGNAL PROCESSING
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The "Softmax" Nonlinearity: Derivation Using Statistical Mechanics and Useful Properties as a Multiterminal Analog Circuit Element I. M. Elfadel J. L. Wyatt, Jr. Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139 Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139 Abstract We use mean-field theory methods from Statistical Mechanics to derive the "softmax" nonlinearity from the discontinuous winnertake-all (WTA) mapping. We give two simple ways of implementing "soft max" as a multiterminal network element. One of these has a number of important network-theoretic properties. It is a reciprocal, passive, incrementally passive, nonlinear, resistive multiterminal element with a content function having the form of informationtheoretic entropy. These properties should enable one to use this element in nonlinear RC networks with such other reciprocal elements as resistive fuses and constraint boxes to implement very high speed analog optimization algorithms using a minimum of hardware. 1 Introduction In order to efficiently implement nonlinear optimization algorithms in analog VLSI hardware, maximum use should be made of the natural properties of the silicon medium. Reciprocal circuit elements facilitate such an implementation since they 882 The "Softmax" Nonlinearity 883 can be combined with other reciprocal elements to form an analog network having Lyapunov-like functions: the network content or co-content. In this paper, we show a reciprocal implementation of the "softmax" nonlinearity that is usually used to enforce local competition between neurons [Peterson, 1989]. We show that the circuit is passive and incrementally passive, and we explicitly compute its content and co-content functions. This circuit adds a new element to the library of the analog circuit designer that can be combined with reciprocal constraint boxes [Harris, 1988] and nonlinear resistive fuses [Harris, 1989] to form fast, analog VLSI optimization networks. 2 Derivation of the Softmax Nonlinearity To a vector y E ~n of distinct real numbers, the discrete winner-take-all (WTA) mapping W assigns a vector of binary numbers by giving the value 1 to the component of y corresponding to maxl<i<n Yi and the value 0 to the remaining components. Formally, W is defined as - W(y) = (Wl(y), ... I Wn(y»T where for every 1 ~ j ~ n, { 1 if YJ' > Yi, V 1 ~ i ~ n Wj (y) = 0 otherwise Following [Geiger, 1991], we assign to the vector y the "energy" function n Ey(z) = - L ZkYk = _zT y, z E Vnl k=l (1) (2) where Vn is the set of vertices of the unit simplex Sn = {x E ~n, Zi > 0, 1 < i < n and E~=l Zk = 1}. Every vertex in the simplex encodes one possible winner. It is then easy to show that W(y) is the solution to the linear programming problem n max LZkYk. ZE'V .. k=l Moreover, we can assign to the energy Ey(z) the Gibbs distribution e-Ey(Z)/T Py(z) = Py(Zl' ... , zn) = ZT where T is the temperature of the heat bath and ZT is a normalizing constant. Then one can show that the mean of Zj considered as a random variable is given by [Geiger, 1991] A _ ey;/T ey;/T Fj(y/T) = Zj = -z = En/T· T i=l eY' The mapping F : ~n -+ ~n whose components are the Fj's, 1 ~ j < n, is the generalized sigmoid mapping [Peterson, 1989] or "soft max" . It plays, in WTA networks, a role similar to that of the sigmoidal function in Hopfield and backpropagation 884 Elfadel and Wyatt v Figure 1: A circuit implementation of softmax with 5 inputs and 5 outputs. This circuit is operated in subthreshold mode, takes the gates voltages as inputs and gives the drain currents as outputs. This circuit is not a reciprocal multiterminal element. networks [Hopfield, 1984, Rumelhart, 1986] and is usually used for enforcing competitive behavior among the neurons of a single cluster in networks of interacting clusters [Peterson, 1989, Waugh, 1993]. A For y E ~n, we denote by FT(y) = F(y IT). The softmax mapping satisfies the following properties: 1. The mapping FT converges pointwise to W over ~n as T -+ 0 and to the center of mass of Sn, *e = *(1,1, . . . , 1)T E ~n, as T -+ +00. 2. The Jacobian DF of the softmax mapping is a symmetric n x n matrix that satisfies DF(y) = diag (F,,(y» - F(y)F(y)T. (3) It is always singular with the vector e being the only eigenvector corresponding to the zero eigenvalue. Moreover, all its eigenvalues are upper-bounded by maxlS"Sn F,,(y) < 1. 3. The soft max mapping is a gradient map, i.e, there exists a "potential" function 'P : ~n -+ ~ such that F = V'P. Moreover 'P is convex. The symbol 'P was chosen to indicate that it is a potential function. It should be noted that if F is the gradient map of 'P then FT is the gradient map of T'PT where 'PT(Y) = 'P(yIT). In a related paper [Elfadel, 1993], we have found that the convexity of'P is essential in the study of the global dynamics of analog WTA networks. Another instance where the convexity of 'P was found important is the one reported in [Kosowsky, 1991] where a mean-field algorithm was proposed to solve the linear assignment problem. The "Softmax" Nonlinearity 885 v, Vz v" Va Ta Figure 2: Modified circuit implementation of softmax. In this circuit all the transistors are diode-connected, and all the drain currents are well in saturation region. Note that for every transistor, both the voltage input and the current output are on the same wire - the drain. This circuit is a reciprocal multiterminal element. 3 Circuit Implementations and Properties Now we propose two simple CMOS circuit implementations of the generalized sigmoid mapping. See Figures 1 and 2. When the transistors are operated in the subthreshold region the drain currents i l , .. . ,in are the outputs of a softmax mapping whose inputs are the gate voltages Vl, •.. , Vn . The explicit v - i characteristics are given by (4) where K, is a process-dependent parameter and Vo is the thermal voltage ([Mead, 1989],p. 36). These circuits have the interesting properties of being unclocked and parallel. Moreover, the competition constraint is imposed naturally through the KCL equation and the control current source. From a complexity point of view, this circuit is most striking since it computes n exponentials, n ratios, and n - 1 sums in one time constant! A derivation similar to the above was independently carried out in [Waugh, 1993] for the circuit of Figure 1. Although the first circuit implements softmax, it has two shortcomings. The first is practical: the separation between inputs and outputs implies additional wiring. The second is theoretical: this circuit is not a reciprocal multiterminal element, and therefore it can't be combined with other reciprocal elements like resistive fuses or constraint boxes to design analog, reciprocal optimization networks. Therefore, we only consider the circuit of Figure 2 and let v and i be the ndimensional vectors representing the input voltages and the output currents, respectively. 1 The softmax mapping i = F(v) represents a voltage-controlled, nonlinear, lCompare with Lazarro et. al.'s WTA circuit [Lazzaro, 1989] whose inputs are currents and outputs are voltages. 886 Elfadel and Wyatt resistive multiterminal element. The main result of our paper is the following: 2 Theorem 1 The softmax multiterminal element F is reciprocal, passive, locally passive and has a co-content function given by 1 n ~(v) = K, Ie Vo In L exp(K,vm/Vo) (5) m=l and a content function given by ..w..*(O) _ IeVo ~ im 1 im 'If I - -- L.J n-. K, m=l Ie Ie (6) Thus, with this reciprocal, locally passive implementation of the softmax mapping, we have added a new circuit element to the library of the circuit designer. Note that this circuit element implements in an analog way the constraint L:~=1 y" = 1 defining the unit simplex Sn. Therefore, it can be considered a nonlinear constraint box [Harris, 1988] that can be used in reciprocal networks to implement analog optimization algorithms. The expression of ~* is a strong reminder of the information-theoretic definition of entropy. We suggest the name "entropic resistor" for the circuit of Figure 2. 4 Conclusions In this paper, we have discussed another instance of convergence between the statistical physics paradigm of Gibbs distributions and analog circuit implementation in the context of the winner-take-all function. The problem of using the simple, reciprocal circuit implementation of softmax to design analog networks for finding near optimal solutions of the linear assignment problem [Kosowsky, 1991] or the quadratic assignment problem [Simic, 1991] is still open and should prove a challenging task for analog circuit designers. Acknowledgements I. M. Elfadel would like to thank Alan Yuille for many helpful discussions and Fred Waugh for helpful discussions and for communicating the preprint of [Waugh, 1993]. This work was supported by the National Science Foundation under Grant No. MIP-91-17724. References [Peterson, 1989] C. Peterson and B. Soderberg. A new method for mapping optimization problems onto neural networks. International Journal of Neural Systems, 1(1):3 - 22, 1989. 2The concepts of reciprocity, passivity, content, and co-content are fundamental to nonlinear circuit theory. They are carefully developed in [Wyatt, 1992]. The "Softmax" Nonlinearity 887 [Harris, 1988] J. G. Harris. Solving early vision problems with VLSI constraint networks. In Neural Architectures for Computer Vision Workshop, AAAI-88, Minneapolis, MN, 1988. [Harris,1989] J. G. Harris, C. Koch, J. Luo, and J. Wyatt. Resistive fuses: Analog hardware for detecting discontinuities in early vision. In C. Mead and M. Ismail, editors, Analog VLSIImplemenation of Neural Systems. Kluwer Academic Publishers, 1989. [Geiger, 1991] D. Geiger and A. Yuille. A common framework for image segmentation. Int. J. Computer Vision, 6:227 - 253, 1991. [Hopfield, 1984] J. J. Hopfield. Neurons with graded responses have collective computational properties like those of two-state neurons. Proc. Nat'l Acad. Sci., USA, 81:3088-3092, 1984. [Rumelhart, 1986] D. E. Rumelhart et. al. Parallel Distributed Processing, volume 1. MIT Press, 1986. [Waugh, 1993] F. R. Waugh and R. M. Westervelt. Analog neural networks with local competition. I. dynamics and stability. Physical Review E, 1993. in press. [Elfadel, 1993] I. M. Elfadel. Global dynamics of winner-take-all networks. In SPIE Proceedings, Stochastic and Neural Methods in Image Processing, volume 2032, pages 127 - 137, San Diego, CA, 1993. [Kosowsky, 1991] J. J. Kosowsky and A. L. Yuille. The invisible hand algorithm: Solving the assignment problem with statistical physics. TR # 91-1, Harvard Robotics Laboratory, 1991. [Mead, 1989] Carver Mead. Analog VLSI and Neural Systems. Addison-Wesley, 1989. [Lazzaro,1989] J. Lazarro, S. Ryckebush, M. Mahowald, and C. Mead. Winnertake-all circuits of O(n) complexity. In D. S. Touretsky, editor, Advances in Neural Information Processing Systems I, pages 703 - 711. Morgan Kaufman, 1989. [Wyatt, 1992] J. L. Wyatt. Lectures on Nonlinear Circuit Theory. MIT VLSI memo # 92-685,1992. [Simic, 1991] P. D. Simic. Constrained nets for graph matching and other quadratic assignment problems. Neural Computation, 3:169 - 281, 1991.
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Robust Parameter Estimation And Model Selection For Neural Network Regression Yong Liu Department of Physics Institute for Brain and Neural Systems Box 1843, Brown University Providence, RI 02912 yong~cns.brown.edu Abstract In this paper, it is shown that the conventional back-propagation (BPP) algorithm for neural network regression is robust to leverages (data with :n corrupted), but not to outliers (data with y corrupted). A robust model is to model the error as a mixture of normal distribution. The influence function for this mixture model is calculated and the condition for the model to be robust to outliers is given. EM algorithm [5] is used to estimate the parameter. The usefulness of model selection criteria is also discussed. Illustrative simulations are performed. 1 Introduction In neural network research, the back-propagation (BPP) algorithm is the most popular algorithm. In the regression problem y = 7](:n, w) + £, in which 7](:n, 8) denote a neural network with weight 8, the algorithm is equivalent to modeling the error as identically independently normally distributed (i.i.d.), and using the maximum likelihood method to estimate the parameter [13]. Howerer, the training data set may contain surprising data points either due to errors in y space (outliers) when the response vectors ys of these data points are far away from the underlying function surface, or due to errors in :n space (leverages), when the the feature vectors 192 Robust Parameter Estimation and Model Selection for Neural Network Regression 193 xs of these data points are far away from the mass of the feature vectors of the rest of the data points. These abnormal data points may be able to cause the parameter estimation biased towards them. A robust algorithm or robust model is the one that overcome the influence of the abnormal data points. A lot of work has been done in linear robust regression [8, 6, 3]. In neural network. it is generally believed that the role of sigmoidal function of the basic computing unit in the neural net has some significance in the robustness of the neural net to outliers and leverages. In this article, we investigate this more thoroughly. It turns out the conventional normal model (BPP algorithm) is robust to leverages due to sigmoidal property of the neurons, but not to outliers (section 2). From the Bayesian point of view [2], modeling the error as a mixture of normal distributions with different variances, with flat prior distribution on the variances, is more robust. The influence function for this mixture model is calculated and condition for the model to be robust to outliers is given (section 3.1). An efficient algorithm for parameter estimation in this situation is the EM algorithm [5] (section 3.2). In section 3.3, we discuss a choice of prior and its properties. In order to choose among different probability models or different forms of priors, and neural nets with different architecture, we discuss the model selection criteria in section 4. Illustrative simulations on the choice of prior, or the t distribution model, and the normal distribution model are given. Model selection statistics, is used to choose the degree of freedom oft distribution, different neural network, and choose between a t model and a normal model (section 4 and 5). 2 Issue Of Robustness In Normal Model For Neural Net Regression One way to think of the outliers and leverages is to regard them as a data perturbation on the data distribution of the good data. Remember that a estimated parameter T = T(F) is an implicit function of the underlying data distribution F. To evaluate the influence of T by this distribution perturbation, we use the influence function [6] of estimator T at point z = (x, y) with data distribution F, which is defined as IF(T ) -1· T((1 - t)F + t~z) - T(F) , z, F Imt -+ 0+ ----.:'-'------'------'-----'----'t (1 ) in which ~:r: has mass 1 at x. 1 This definition is equivalent to a definition of derivative with respect to F except what we are dealing now is the derivative of functional. This definition gives the amount of change in the estimator T with respect to a distribution perturbation t~z at point z = (x, y). For a robust estimation of the parameter, we expect the estimated parameter does not change significantly with respect to a data perturbation. In another word, the influence function is bounded for a robust estimation. Denote the conditional probability model of y given x as i.i.d. f(ylx,8) with parameter 8. If the error function is the negative log-likelihood plus or not plus a penalty term, then a general property of the influence function of the estimated parameter B, is IF(B, (Xi, Yi), F) ex \71l1ogf(ydxi, B) (for proof, see [11]). Denote the neural lThe probability density of the distribution D.", is 6(y - 2:). 194 Liu net, with h hidden units and the dimension of the output being one (dy = 1), as h 17(:z:,8) = L akO"( Wk:Z: + tk) (2) k=l in which O"(:z:) is the sigmoidal function or 1/(1 + exp(:z:)) and 8 = {ak, Wk, td. For a normal model, f(yl:z:, 8, 0") = JV(Yj 17(:Z:, 8), 0") in which .N'(y; c, 0") denotes dyvariate normal distribution with mean c and covariance matrix 0"2 I. Straightforward calculation yield (dy = 1) ( (O"(Wi:z:+ti))hXl ) IF(8, (:Z:i' Yi), F) <X (y - 17(X, 8)) (O,:O"',(w:x + t~)x ) (3) aiO" (WiX + td hx 1 Since y with a large value makes the influence function unbounded, thus the normal model or the back-propagation algorithm for regression is not robust to outliers. Since 0"' (wx +t) tends to be zero for x that is far away from the projection wx +i = 0, the influence function is bounded for a abnormal x, or the normal model for regression is robust to leverages. This analysis can be easily extented to a neural net with multiple hidden layers and multiple outputs. Since the neural net model is robust to leverages, we shall concentrate on the discussion of robustness with respect to outliers afterwards. 3 Robust Probability Model And Parameter Estimation 3.1 Mixture Model One method for the robust estimation is by the Bayesian analysis [2J. Since our goal is to overcome the influence of outliers in the data set, we model the error as a mixture of normal distributions, or, f(yl:z:,8,0") = J f(ylx,8,q,0")7r(q)dq (4) with f(ylx, 8, q, 0") = N'(y; 17(x, 8), 0"2 /q) and the prior distribution on q is denoted as 7r(q). Intuitively, a mixture of different normal distributions with different qs, or different variances, somehow conveys the idea that a data point is generated from a normal distribution with large variance, which can be considered to be outliers, or from that with small variance, which can be considered to be good data. This requires 7r(q) to be flat to accommodate the abnormal data points. A case of extreme non-flat prior is to choose 7r(q) = 6(q - 1), which will make f(ylx, 8, 0") to be a normal distribution model. This model has been discussed in previous section and it is not robust to outliers. Calculation yields (dy = 1) the influence function as ~ ( (O"(ti\X+ti))hXl ) IF(8, (x, y), F) <X (y - 17(x, 8)) w ( a:a',( wix + t~)x ) (5) ai a (Wi X + td h x 1 Robust Parameter Estimation and Model Selection for Neural Network Regression 195 in which (6) where expectation is taken with respect to the posterior distribution of q, or 7r(qly x 0- 8) = f(ylx,9,q,u)'1r(q) For the influence function to be bounded for a y , , , f(ylx,9,u) with large value, (y - 7](x, 8))w must be bounded. This is the condition on 7r(q) when the distribution f(ylx, 8, 0-) is robust to outliers. It can be noticed that the mixture model is robust to leverages for the same reason as in the case of the normal distribution model. 3.2 Algorithm For Parameter Estimation An efficient parameter estimation method for the model in equation 4 is the EM algorithm [5]. In EM algorithm, a portion ofthe parameter is regarded as the missing observations. During the estimation, these missing observations are estimated through previous estimated parameter of the model. Afterwards, these estimated missing observations are combined with the real observations to estimate the parameter of the model. In our mixture model, we shall regard {qi, i = 1, ... n} as the missing observations. Denote w = {Xi, Yi, i = 1, ... n} as the training data set. It is a straight forward calculation for the EM algorithm (see Liu, 1993b) once one w~it~ ~o~n the full probability f( {Yi, qdl{xd, 0-, 8). The algorithm is equivalent to mmimIzmg n L w~S-l)(Yi - 7](Xi' 8))2 (7) i=l and estimating 0- at the s step by (0-2)(S) = ~ l:~l W~!-l)(Yi - 7](Xi' 8(5»))2. If we use f(ylx, 8, cr) oc exp( -p(IY-7]( x, 8) 110-)) and denote 1/J(z) = p' (z), calculation yield, w = E [qly, x, 0-, 8] = ""~z) Iz=IY-71(X,9)I/u' This has exact the same choice of weight wr S - 1) as in the iterative reweighted regression algorithm [7]. What we have here, different from the work of Holland et al., is that we link the EM algorithm with the iterative reweighted algorithm, and also extend the algorithm to a much more general situation. The weighting Wi provides a measure of the goodness of a data point. Equation 7 estimates the parameters based on the portion of the data that are good. A penalization term on 8 can also be included in equation 7. 2 3.3 Choice Of Prior There are a lot choices of prior distribution 7r(q) (for discussion, see [11]). We only discuss the choice IIq '" X~, i.e., a chi distribution with II degree of freedom. By intergrating equation 4 f(Ylx 8 0-) = r'( v+dl/)/2) (1 + (Y-71(x,9»2)-(Il+dl/)/2. , " r(V/2)(q2V'1r)ctl//2 vu 2 It is a dy variate t distribution with II degree of freedom, mean 0 and covariance matrix cr2 I. Calculation yields, E [qly, x, 0-, 8] = v + (Y-~tx~9»)27u2 The t distribution 2 A prior on 8 can be 1r(8) ex: e- a (A ,9)/(2cr 2 ), which yields a additional penalization term 0:( A, 8) in equation 7, in which A denotes a tunning parameters of the penalization. 196 Liu becomes a normal distribution as 1.1 goes to infinity. For finite 1.1, it has heavier tail than the normal distribution and thus is appropriate for regression with to outliers. Actually the condition for robustness, (y - 1J(x,8))w being bounded for a y with large value, is satisfied. The weighting w ex: 1/{1 + [Y -1J(x,8)f /a 2 } balances the influence of the ys with large values, achieving robustness with respect to outliers for the t distribution. 4 Model Selection Criteria The meaning of model is in a broad sense. It can be the degree of penalization, or a probability model, or a neural net architecture, or the combination of the above. A lot of work has been done in model selection [1, 17, 15, 4, 13, 14, 10, 12]. The choice of a model is based on its prediction ability. A natural choice is the expected negative log-likelihood. This is equivalent to using the Kullback-Leibler measure [9] for model selection, or -E [logf(ylx, model)] + E [log f(ylx, true model)]. This has problem if the model can not be normalized as in the case of a improper prior. Equation 7 implies that we can use n 1 ,,"",* A 2 Tm(w) = Lt Wi (Yi - 1J(Xi' (Ld) neff i=l (8) as the cross-validation [16] assessment of model m, in which neff = Ei wi, wi is the convergence limit of w~s), or equation 6, and 8_i is the estimator of 8 with ith data deleted from the full data set. The successfulness of the cross-validation method depends on a robust parameter estimation. The cross-validation method is to calculate the average prediction error on a data based on the rest of the data in the training data set. In the presence of outliers, predicting an outlier based on the rest of the data, is simply not meaningful in the evaluation of the model. Equation 8 takes consideration of the outliers. Using result from [10], we can show [11] with penalization term 0:(>',8), 1 ~ * A 2 Tm(w) ~ -Ltwi (Yi-1J(Xi,8)) neff i=1 (9) + ~ t wirigJ [2: wi (gigJ - ri(i) + 'VI) 'VI)O:(>', 8)]-1 rigi (10) eff i=1 i in which gi = 'V1)1J(xi,8), (i = 'V1)'V~1J(xi,8) and ri = Yi -1J(xi,8). Thus if the models in comparison contains a improper prior, the above model selection statistics can be used. If the models in comparison have close forms of f(ylx, 8, u), the average negative log-likelihood can be used as the model selection criteria. In Liu's work [10], an approximation form for the unbiased estimation of expected negative log-likelihood was provided. If we use the negative log-likelihood plus a penalty term 0:(>.,8) as the parameter optimization criteria, the model selection statistics is 1 ~ A 1 ~ A 1 -1 Tm(w) = -- Ltlogf(Yilxi,8_i) ~ -- Ltlogf(Yilxi,8) + -Tr(C D) n . n n ~=1 i=1 (11) Robust Parameter Estimation and Model Selection for Neural Network Regression 197 10 which C E~=l V' (1 log f(Yi lXi, B)V'~ log f(Yi lXi, B) and D = - E~=l V'(1V'pogf(Yilxi,B) + V'(1\7~a().,8). The optimal model is the one that minimizes this statistics. If the true underlying distribution is the normal distribution model and there is no penalization terms, it is easy to prove C -+ D as n goes to infinite. Then the statistics becomes AIC [1]. o S o o o o o 00 -1.5 ~--__ ~ ______ ~~ __ ~ ______ ~ ____ ~ ______ ~ ____ ~ o 1 2 3 4 5 6 7 Figure 1: BPP fit to data set with leverages, and comparison with BPP fit to the data set without the leverages. An one hidden layer neural net with 4 hidden units, is fitted to a data set with 10 leverage, which are on the right side of X = 3.5, by using the conventional BPP method. The main body of the data (90 data points) was generated from Y = sin(x) + € with € '"V .V(€j 0, a = 0.2). It can be noticed that the fit on the part of good data points was not dramatically influenced by the leverages. This verified our theoretical result about the robustness of a neural net with respect to leverages 5 Illustrative Simulations For the results shown in figure 2 and 3, the training data set contains 93 data point from Y == sin( x) + € and seven Y values (outliers) randomly generated from region [1, 2), in which € '" .:\:'( €j 0, a = 0.2). The neural net we use is of the form in equation 2. Denote h as the number of hidden units in the neural net. The caption of each figure (1, 2, 3) explains the usefulness of the parameter estimation algorithm and the model selection. Acknowledgements The author thanks Leon N Cooper, M. P. Perrone. The author also thanks his wife Congo This research was supported by grants from NSF, ONR and ARO. References [1] H. Akaike. Information theory and an extension of the maximum likelihood 198 Liu 1.1 .--.--~--~~--~--~--~~--~--~--~~--~~ 1 0.9 Tm statistics 0.8 MSE on the test set (x 10- 1 ) 0.7 0.6 0.5 OA'=-----'-__ -'--__ ..L----l __ ---'-__ ---L--_L-----..l __ ---L-__ ..l.--_L-..---'-__ -'---..d (3,3)(2,3)( 4,3)(2,4)(3,4)(5,3)(3,5)( 1,3)(3,7)( n,3X n,4X n,5X n, 7) models (/.I, h), n stands for normal distribution model (BPP fit) Figure 2: Model selection statistics Tm for fits to data set with outliers, tests on a independent data set with 1000 data points from y = sin(:z:) + €, where € '" JV(f.; 0, U = 0.2). it can be seen that Tm statistics is in consistent with the error on the test data set. The Tm statistics favors t model with small /.I than for the normal distribution models. 2 0 0 1.5 0 00 0 0 1 0.5 Y 0 0 -0.5 t3 model with outliers -1 BPP fit with outliers PP fit without outliers -1.5 0 1 2 3 4 5 6 7 :z: Figure 3: Fits to data set with outliers, and comparison with BPP fit to the data set without the outliers. The best fit in the four BPP fits (h = 3), according to Tm statistics, was influenced by the outliers, tending to shift upwards. Although the distribution is not a t distribution at all, the best fit by the EM algorithm under the t model (/.I = 3, h = 3), also according to Tm statistics, gives better result than the BPP fit, actually is almost the same as the BPP fit (h = 3) to the training data set without the outliers. This is due to the fact that a t distribution has a heavy tail to accommodate the outliers Robust Parameter Estimation and Model Selection for Neural Network Regression 199 principle. In Petrov and Czaki, editors, Proceedings of the 2nd International Symposium on Information Theory, pages 267-281, 1973. [2] J. O. Berger. Statistical Decision Theory and Bayesian Analysis. SpringerVerlag, 1985. [3] R. D. Cook and S. Weisberg. Characterization of an empirical influence function for detecting influential cases in regression. Technometrics, 22:495-508, 1980. [4] P. Craven and G. Wahba. Smoothing noisy data with spline functions:estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math., 31:377-403, 1979. [5] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. (with discussion). J. Roy. Stat. Soc. Ser. B, 39:1-38, 1977. [6] F.R. Hampel, E.M. Rouchetti, P.l. Rousseeuw, and W.A. Stahel. Robust Statistics: The approach based on influence functions. Wiley, 1986. [7] P.J. Holland and R.E. Welsch. Robust regression using iteratively reweighted least-squares. Commun. Stat. A, 6:813-88, 1977. [8] P.l. Huber. Robust Statistics. New York: Wiley, 1981. [9] S. Kullback and R.A. Leibler. On information and sufficiency. Ann. Stat., 22:79-86, 1951. [10] Y. Liu. Neural Network Model Selection Using Asymptotic Jackknife Estimator and Cross-Validation Method. In C.L. Giles, S.l.and Hanson, and J.D. Cowan, editors, Advances in neural information processing system 5. Morgan Kaufmann Publication, 1993. [11] Y. Liu. Robust neural network parameter estimation and model selection for regression. Submitted., 1993. [12] Y. Liu. Unbiased estimate of generalization error and model selection criterion in neural network. Submitted to Neural Network, 1993. [13] D. MacKay. Bayesian methods for adaptive models. PhD thesis, California Institute of Technology, 1991. [14] J. E. Moody. The effective number of parameters, an analysis of generalization and regularization in nonlinear learning system. In l. E. Moody, S. l. Hanson, and R. P. Lippmann, editors, Advances in neural information processing system 4, pages 847-854. Morgan Kaufmann Publication, 1992. [15] G. Schwartz. Estimating the dimension of a model. Ann. Stat, 6:461-464, 1978. [16] M. Stone. Cross-validatory choice and assessment of statistical predictions (with discussion). J. Roy. Stat. Soc. Ser. B, 36:111-147, 1974. [17] M. Stone. An asymptotic equivalence of choice of model by cross-validation and Akaike's criterion. J. Roy. Stat. Soc., Ser. B, 39(1):44-47, 1977.
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Pulling It All Together: Methods for Combining Neural Networks Michael P. Perrone Institute for Brain and Neural Systems Brown University Providence, RI mpp@cns. brown. edu The past several years have seen a tremendous growth in the complexity of the recognition, estimation and control tasks expected of neural networks. In solving these tasks, one is faced with a large variety of learning algorithms and a vast selection of possible network architectures. After all the training, how does one know which is the best network? This decision is further complicated by the fact that standard techniques can be severely limited by problems such as over-fitting, data sparsity and local optima. The usual solution to these problems is a winner-take-all cross-validatory model selection. However, recent experimental and theoretical work indicates that we can improve performance by considering methods for combining neural networks. This workshop examined current neural network optimization methods based on combining estimates and task decomposition, including Boosting, Competing Experts, Ensemble Averaging, Metropolis algorithms, Stacked Generalization and Stacked Regression. The issues covered included Bayesian considerations, the role of complexity, the role of cross-validation, incorporation of a priori knowledge, error orthogonality, task decomposition, network selection techniques, overfitting, data sparsity and local optima. Highlights of each talk are given below. To obtain the workshop proceedings, please contact the author or Norma Caccia (norma_caccia@brown.edu) and ask for IBNS ONR technical report #69. M. Perrone (Brown University, "Averaging Methods: Theoretical Issues and Real World Examples") presented weighted averaging schemes [7], discussed their theoretical foundation [6], and showed that averaging can improve performance whenever the cost function is (positive or negative) convex which includes Mean Square Error, a general class of Lp-norm cost functions, Maximum Likelihood Estimation, Maximum Entropy, Maximum Mutual Information, the Kullback-Leibler Information (Cross Entropy), Penalized Maximum Likelihood Estimation and Smoothing Splines [6]. Averaging was shown to improve performance on the NIST OCR data, a human face recognition task and a time series prediction task [5]. J. Friedman (Stanford, "A New Approach to Multiple Outputs Using Stacking") presented a detailed analysis of a method for averaging estimators and noted simulations showed that averaging with a positivity constraint was better than cross1188 Pulling It All Together: Methods for Combining Neural Networks 1189 validation estimator selection [1]. S. Nowlan (Synaptics, "Competing Experts") emphasized the distinctions between static and dynamic algorithms and between averaged and stacked algorithms; and presented results of the mixture of experts algorithm [3] on a vowel recognition task and a hand tracking task. H. Drucker (AT&T, "Boosting Compared to Other Ensemble Methods") reviewed the boosting algorithm [2] and showed how it can improve performance for OCR data. J. Moody (OGI, "Predicting the U.S. Index ofIndustrial Production") showed that neural networks make better predictions for the US IP index than standard models [4] and that averaging these estimates improves prediction performance further. W. Buntine (NASA Ames Research Cent.er, "Averaging and Probabilistic Networks: Automating the Process") discussed placing combination techniques within the Bayesian framework. D. Wolpert (Santa Fe Institute, "Infen ing a Function vs. Inferring an Inference Algorithm") argued that theory can not, in general, identify the optimal network; so one must make assumptions in order to improve performance. H. Thodberg (Danish Meat Research Institute, "Error Bars on Predictions from Deviations among Committee Member~ (within Bayesian Backprop)") raised the provocative (and contentious) point that Bayesian arguments support averaging while Occam's Razor (seemingly?) does not. S. Hashem (Purdue University, "Merits of Combining Neural Networks: Potential Benefits and Risks") emphasized the importance of dealing with collinearity when using averaging methods. References [1] Leo Breiman. Stacked regression. Technical Report TR-367, Department of Statistics, University of California, Berkeley, August 1992. [2] Harris Drucker, Robert Schapire, and Patrice Simard. Boosting performance in neural networks. International Journal of Pattern Recognition and Artificial Intelligence, [To appear]. [3] R. A. Jacobs, M. 1. Jordan, S. J. Nowlan, and G. E. Hinton. Adaptive mixtures of local experts. Neural Computation, 3(2), 1991. [4] U. Levin, T. Leen, and J. Moody. Fa.st pruning using principal components. In Steven J. Hanson, Jack D. Cowan, and C. Lee Giles, editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmann, 1994. [5] M. P. Perrone. Improving Regression Estimation: A veraging ~Methods for Variance Reduction with Eztensions to General Convez Measure Optimization. PhD thesis, Brown University, Institute for Brain and Neural Systems; Dr. Leon N Cooper, Thesis Supervisor, May 1993. [6] M. P. Perrone. General averaging results for convex optimization. In Proceedings of the 1993 Connectionist Models Su,mmer School, pages 364-371, Hillsdale, N.T, 1994. Erlbaum Associates. [7] M. P. Perrone and L. N Cooper. '!\Then networks disagree: Ensemble method for neural networks. In Artificial Neuml Networks for Speech and l!ision. ChapmanHall, 1993. Chapter 10.
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Computational Elements of the Adaptive Controller of the Human Arm Reza Shadmehr and Ferdinando A. Mussa-Ivaldi Dept. of Brain and Cognitive Sciences M. I. T ., Cambridge, MA 02139 Email: reza@ai.mit.edu, sandro@ai.mit.edu Abstract We consider the problem of how the CNS learns to control dynamics of a mechanical system. By using a paradigm where a subject's hand interacts with a virtual mechanical environment, we show that learning control is via composition of a model of the imposed dynamics. Some properties of the computational elements with which the CNS composes this model are inferred through the generalization capabilities of the subject outside the training data. 1 Introduction At about the age of three months, children become interested in tactile exploration of objects around them. They attempt to reach for an object, but often fail to properly control their arm and end up missing their target. In the ensuing weeks, they rapidly improve and soon they can not only reach accurately, they can also pick up the object and place it. Intriguingly, during this period of learning they tend to perform rapid, flailing-like movements of their arm, as if trying to "excite" the plant that they wish to control in order to build a model of its dynamics. From a control perspective, having a model of the arm's skeletal dynamics seems necessary because of the relatively low gain of the fast acting feedback system in the spinal neuro-muscular controllers (Crago et al. 1976), and the long delays in transmission of sensory information to the supra-spinal centers. Such a model could be used by the CNS to predict the muscular forces that need to be produced in order to move the arm along a desired trajectory. Yet, this model by itself is not sufficient 1077 1078 Shadmehr and Mussa-Ivaldi for performing a contact task because most objects which our hand interacts with change the arm's dynamics significantly. We are left with a situation in which we need to be able to quickly acquire a model of an object's dynamics so that we can incorporate it in the control system for the arm. How we learn to construct a model of a dynamical system and how our brains represent the composed model are the subjects of this research. 2 Learning Dynamics of a Mechanical System To make the idea behind learning dynamics evident, consider the example of controlling a robotic arm. The arm may be seen as an inertially dominated mechanical admitance, accepting force as input and producing a change in state as its output: q = H(q)-l (F - C(q, q)) (1) where q is the configuration of the robot, H is the inertia tensor, F is the input force from some controllable source (e.g., motors), and C is the coriolis/centripetal forces. In learning to control the arm, i.e., having it follow a certain state trajectory or reach a final state, we form a model which has as its input the desired change in the state of the arm and receive from its output a quantity representing the force that should be produced by the actuators. Therefore, what needs to be learned is a map from state and desired changes in state to force: iJ(q, q, iid) = if(q)qd + C(q, q) Combine the above model with a simple PD feedback system, F = iJ + if K(q - qd) + if B(q - qd) (2) and the dynamics of the system in Eq. (1) can now be written in terms of a new variable s = q - qd, i.e., the error in the trajectory. It is easy to see that if we have if ~ Hand C ~ C, and if J( and B are positive definite, then s will be a decreasing function of time, i.e., the system will be globally stable. Learning dynamics means forming the map in Eq. (2). The computational elements which we might use to do this may vary from simple memory cells that each have an address in the state space (e.g., Albus 1975, Raibert & Wimberly 1984, Miller et al. 1987), to locally linear functions restricted to regions where we have data (Moore & Atkeson 1994), to sigmoids (Gomi & Kawato 1990) and radial basis functions which can broadly encode the state space (Botros & Atkeson 1991). Clearly, the choice that we make in our computational elements will affect how the learned map will generalize its behavior to regions of the state space outside of the training data. Furthermore, since the task is to learn dynamics of a mechanical system (as opposed to, for example, dynamics of a financial market), certain properties of mechanical systems can be used to guide us in our choice for the computational elements. For example, the map from states to forces for any mechanical system can be linearly parameterized in terms of its mass properties (Slotine and Li, 1991). In an inertially dominated system (like a multi-joint arm) these masses may be unknown, but the fact that the dynamics can be linearized in terms of the unknowns makes the task of learning control much simpler and orders of magnitude faster than using, for example, an unstructured memory based approach. . ,~ .. -., ... " . ,' ..... . c Computational Elements of the Adaptive Controller of the Human Arm 1079 ~ "C !.'! Lfl o 1.sec J ji /i f t B 2.5 I~ --.J Lfl 1. sec ci Figure 1: Dynamics of a real 2 DOF robot was learned so to produce a desired trajectory. A: Schematic of the robot. The desired trajectory is the quarter circle. Performance of a PD controller is shown by the gray line, as well as in B, where joint trajectories are drawn: the upper trace is the shoulder joint and the lower trace is the elbow joint. Desired joint trajectory is solid line, actual trajectory is the gray line. C: Performance when the PD controller is coupled with an adaptive model. D: Error in trajectory. Solid line is PD, Gray line is PD+adaptation. To illustrate this point, consider the task of learning to control a real robot arm. Starting with the assumption that the plant has 2 degrees of freedom with rotational joints, inertial dynamics of Eq. (2) can be written as a product of a known matrix-function of state-dependent geometric transformations Y, and an unknown (but constant) vector a, representing the masses, center of masses, and link lengths: D( q , q, qd) = Y (q, q, qd) a . The matrix Y serves the function of referring the unknown masses to their center of rotation and is a geometric transformation which can be derived from our assumption regarding the structure of the robot. It is these geometric transformations that can guide us in choosing the computational elements for encoding the sensory data (q and q). We used this approach to learn to control a real robot. The adaptation law was derived from a Lyapunov criterion, as shown by Slotine and Li (1991): ~ = _yT (q, q, qd) (q - qd(t) + q - qd(t)) The system converged to a very low trajectory tracking error within only three periods of the movement (Fig. 1). This performance is achieved despite the fact that our model of dynamics ignores frictional forces, noise and delay in the sensors, and dynamics of the actuators. In contrast, using a sigmoid function as the basic com1080 Shadmehr and Mussa-Ivaldi putational element of the map and training via back-propagation led to comparable levels of performance in over 4000 repetitions of the training data (Shadmehr 1990). The difference in performance of these two approaches was strictly due to the choice of the computational elements with which the map of Eq. (2) was formed. Now consider the task of a child learning dynamics of his arm, or that of an adult picking up a hammer and pounding a nail. We can scarcely afford thousands of practice trials before we have built an adequate model of dynamics. Our proposal is that because dynamics of mechanical systems are distinctly structured, perhaps our brains also use computational elements that are particularly suited for learning dynamics of a motor task (as we did in learning to control the robot in Fig. 1). How to determine the structure of these elements is the subject of the following sections. 3 A Virtual Mechanical Environment To understand how humans represent learned dynamics of a motor task, we designed a paradigm where subjects reached to a target while their hand interacted with a virtual mechanical environment. This environment was a force field produced by a manipulandum whose end-effector was grasped by the subject. The field of forces depended only on the velocity of the hand, e.g., F = Bx, as shown in Fig. 2A, and significantly changed the dynamics of the limb: When the robot's motors were turned off (null field condition), movements were smooth, straight line trajectories to the target (Fig. 2B). When coupled with the field however, the hand's trajectory was now significantly skewed from the straight line path (Fig. 2C). It has been suggested that in making a reaching movement, the brain formulates a kinematic plan describing a straight hand path along a smooth trajectory to the target (Morasso 1981). Initially we asked whether this plan was independent of the dynamics of the moving limb. If so, as the subject practiced in the environment, the hand path should converge to the straight line, smooth trajectory observed in the null field. Indeed , with practice, trajectories in the force field did converge to those in the null field. This was quantified by a measure of correlation which for all eight subjects increased monotonically with practice time. If the CNS adapted to the force field by composing a model of its dynamics, then removal of the field at the onset of movement (un-be-known to the subject) should lead to discrepancies between the actual field and the one predicted by the subject's model, resulting in distorted trajectories which we call after-effects. The expected dynamics of these after-effects can be predicted by a simple model of the upper arm (Shadmehr and Mussa-Ivaldi 1994). Since the after-effects are a by-product of the learning process, we expected that as subjects adapted to the field, their performance in the null field would gradually degrade. We observed this gradual growth of the after-effects, leading to grossly distorted trajectories in the null field after subjects had adapted to the force field (Fig. 2D). This evidence suggested that the CNS composed a model of the field and used this model to compensate for the forces which it predicted the hand would encounter during a movement. The information contained in the learned model is a map whose input is the state and the desired change in state of the limb, and whose output is force (Eq. 2). How is this map implemented by the CNS? Let us assume that the approximation is via I 0.5 ~ ~ 0 ~ >-g .. r -0.5 A -1 Computational Elements of the Adaptive Controller of the Human Arm 1081 15o,------------. -0.5 0.5 B -150 '--------,-1~00--=:-50-~-------:5::-:-0 -1:-:"':OO--,J150 Hand x-velocrty (rrV$) Displacement (mm) Figure 2: A: The virtual mechanical environment as a force field. B: Trajectory of reaching movements (center-out) to 8 targets in a null field. C: Average±standard-deviation of reaches to same targets when the field was on, before adaptation. D: After-affects of adaptation, i.e., when moving in a null field but expecting the field. a distributed set of computational elements (Poggio 1990). What are the properties of these elements? An important property may be the spatial bandwidth, i.e_, the size of the receptive field in the input space (the portion of the input space where the element generates a significant output). This property greatly influences how the eNS might interpolate between states which it has visited during training, and whether it can generalize to regions beyond the boundary of the training data. For example, in eye movements, it has been suggested that a model of dynamics of the eye is stored in the cerebellum (Shidara et al. 1992). Cells which encode this model (Purkinje cells) vary their firing rate as a linear function of the state of the eye, and the sum of their outputs (firing rates) correlates well with the force that the muscles need to produce to move the eye. Therefore, the model of eye's dynamics is encoded via cells with very large receptive fields. On the other hand, cells which take part in learning a visual hyperacuity task may have very small receptive fields (Poggio et al. 1992), resulting in a situation where training in a localized region does not lead to generalization. In learning control of our limbs, one possibility for the computational elements is the neural control circuits in the spinal cord (Mussa-Ivaldi 1992). Upon activation of 1082 Shadmehr and Mussa-Ivaldi Test workspace Trained Workspace A -fo:::;;...-+---:>10 em X 150 100 50 I 0 -50 -100 B -150 -100 -50 50 100 150 c -1 -0.5 o 0.5 Displacement (mm) Hand .-velocity (""s) Figure 3: A: Schematic of subject's arm and the trained region of the workspace where the force field was presented and the "test" region where the transferred effects were measured. B: After-effects at the test region. C: A joint-based translation of the force field shown in Fig. 2A to the novel workspace. This is the field that the subject expected at the test region. one such circuit, muscles produce a time varying force field, i.e., forces which depend on the state of the limb (position and velocity) and time (Mussa-Ivaldi et al. 1990). Let us call the force function produced by one such motor element h(q, q, t) . It turns out that as one changes the amount of activation to a motor element, the output forces essentially scale. When two such motor elements are activated, the resulting force field is a linear combination of the two individual fields (Bizzi et al. 1991): f = 2::;=1 Cdi(q, q, t). N ow consider the task of learning to move in the field shown in Fig. 2A. The model that the eNS builds is a map from state of the limb to forces imposed by the environment. Following the above scenario, the task is to find coefficients Ci for each element such that the output field is a good approximation of the environmental field. Unlike the computational elements of a visual task however, we may postulate that the motor elements are characterized by their broad receptive fields. This is because muscular force changes gradually as a function of the state of the limb and therefore its output force is non zero for wide region of the state space. It follows that if learning dynamics was accomplished through formation of a map whose computational elements were these motor functions, then because of the large spatial bandwidth of the elements the composed model should be able to generalize to well beyond the region of the training data. Computational Elements of the Adaptive Controller of the Human Arm 1083 To test this, we limited the region of the input space for which training data was provided and quantified the subject's ability to generalize to a region outside the training set. Specifically, we limited the workspace where practice movements in the force field took place and asked whether local exposure to the field led to after-effects in other regions (Fig. 3A). We found that local training resulted in after-effects in parts ofthe workspace where no exposure to the field had taken place (Fig. 3B). This indicated that the model composed by the CNS predicted specific forces well outside the region in which it had been trained. The existence of this generalization showed that the computational elements with which the internal model was implemented had broad receptive fields. The transferred after-effects (Fig. 3B) show that at the novel region of the workspace, the subject's model of the environment predicted very different forces than the one on which the subject had been trained on (compare with Fig. 2D). This rejected the hypothesis that the composed model was a simple mapping (i.e., translation in variant) in a hand-based coordinate system, i.e., from states of the arm to forces on the hand. The alternate hypothesis was that the composed model related observed states of the arm to forces that needed to be produced by the muscles and was translation invariant in a coordinate system based on the joints and muscles. This would be the case, for example, if the computational elements encoded the state of the arm linearly (analogous to Purkinje cells for the case of eye movements) in joint space. To test this idea, we translated the field in which the subject had practiced to the novel region in a coordinate system defined based on the joint space of the subject's arm, resulting in the field shown in Fig. 3C. We recorded the performance of the subjects in this new field at the novel region of the workspace (after they had been trained on field of Fig. 2A) and found that performance was near optimum at the first exposure. This indicated that the geometric structure of the composed model supported transfer of information in an intrinsic, e.g., joint based, coordinate system. This result is consistent with the hypothesis that the computational elements involved in this learning task broadly encode the state space and represent their input in a joint-based coordinate system and not a hand-based one. 4 Conclusions In learning control of an inertially dominated mechanical system, knowledge of the system's geometric constraints can direct us to choose our computational elements such that learning is significantly faciliated. This was illustrated by an example of a real robot arm: starting with no knowledge of its dynamics, a reasonable model was learned within 3 periods of a movements (as opposed to thousands of movements when the computational elements were chosen without regard to the geometric properties). We argued that in learning to control the human arm, the CNS might also make assumption regarding geometric properties of its links and use specialized computational elements which facilitate learning of dynamics. One possibility for these elements are the discrete neuronal circuits found in the spinal cord. The function of these circuits can be mathematically formulated such that a map representing inverse dynamics of the arm is formed via a combination of the elements. Because these computational elements encode their input space 1084 Shadmehr and Mussa-Ivaldi broadly, i.e., has significant output for a wide region of the input space, we expected that if subjects learned a dynamical process from localized training data, then the formed model should generalize to novel regions of the state space. Indeed we found that the subjects transferred the training information to novel regions of the state space, and this transfer took place in a coordinate system similar to that of the joints and muscles. We therefore suggest that the eNS learns control of the arm through formation of a model whose computational elements broadly encode the state space, and that these elements may be neuronal circuits of the spinal cord. Acknowledgments: Financial support was provided in part by the NIH (AR26710) and the ONR (N00014/90/J/1946). R. S. was supported by the McDonnell-Pew Center for Cognitive Neurosciences and the Center for Biological and Computational Learning. References Albus JS (1975) A new approach to manipulator control: The cerebellar model articulation controller (CMAC). Trans ASME J Dyn Syst Meas Contr 97:220-227. Bizzi E, Mussa-Ivaldi FA, Giszter SF (1991) Computations underlying the execution of movement: a novel biological perspective. Science 253:287-291. Botros SM, Atkeson CG (1991) Generalization properties of radial basis functions. In: Lippmann et al., Adv. in Neural Informational Processing Systems 3:707-713. Crago, Houk JC, Hasan Z (1976) Regulatory actions of human stretch reflex. J NeurophysioI39:5-19. Gomi H, Kawato M (1990) Learning control for a closed loop system using feedback error learning. Proc IEEE Conf Decision Contr. Miller WT, Glanz FH, Kraft LG (1987) Application of a general learning algorithm to the control of robotic manipulators. Int J Robotics Res 6(2):84-98. Moore AW, Atkeson CG (1994) An investigation of memory-based function approximators for learning control. Machine Learning, submitted. Mussa-Ivaldi FA, Giszter SF (1992) Vector field approximation: a computational paradigm for motor control and learning. BioI Cybern 67:491- 500. Mussa-Ivaldi FA, Giszter SF, Bizzi E (1990) Motor-space coding in the central nervous system. Cold Spring Harbor Symp Quant BioI 55:827-835. Poggio T (1990) A theory of how the brain might work. Cold Spring Harbor Symp Quant Bioi 55:899-910. Poggio T, Fahle M, Edelman S (1992) Fast perceptual learning in visual hyperacuity. Science 256:1018-1021. Raibert MH, Wimberly Fe (1984) Tabular control of balance in a dynamic legged system. IEEE Trans Systems, Man, Cybernetics SMC-14(2):334-339. Shadmehr R (1990) Learning virtual equilibrium trajectories for control of a robot arm. Neural Computation 2:436-446. Shadmehr R, Mussa-Ivaldi FA (1994) Adaptive representation of dynamics during learning of a motor task. J Neuroscience, in press. Shidara M, Kawano K, Gomi H, Kawato M (1993) Inverse-dynamics model eye movement control by Purkinje cells in the cerebellum. Nature 365:50-52. Slotine JJE, Li W (1991) Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey.
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Transition Point Dynamic Programming Kenneth M. Buckland'" Dept. of Electrical Engineering University of British Columbia Vancouver, B.C, Canada V6T 1Z4 buckland@pmc-sierra.bc.ca Peter D. Lawrence Dept. of Electrical Engineering University of British Columbia Vancouver, B.C, Canada V6T 1Z4 peterl@ee.ubc.ca Abstract Transition point dynamic programming (TPDP) is a memorybased, reinforcement learning, direct dynamic programming approach to adaptive optimal control that can reduce the learning time and memory usage required for the control of continuous stochastic dynamic systems. TPDP does so by determining an ideal set of transition points (TPs) which specify only the control action changes necessary for optimal control. TPDP converges to an ideal TP set by using a variation of Q-Iearning to assess the merits of adding, swapping and removing TPs from states throughout the state space. When applied to a race track problem, TPDP learned the optimal control policy much sooner than conventional Q-Iearning, and was able to do so using less memory. 1 INTRODUCTION Dynamic programming (DP) approaches can be utilized to determine optimal control policies for continuous stochastic dynamic systems when the state spaces of those systems have been quantized with a resolution suitable for control (Barto et al., 1991). DP controllers, in lheir simplest form, are memory-based controllers that operate by repeatedly updating cost values associated with every state in the discretized state space (Barto et al., 1991). In a slate space of any size the required quantization can lead to an excessive memory requirement, and a related increase in learning time (Moore, 1991). This is the "curse of dimensionality". ·Nowat: PMC-Sierra Inc., 8501 Commerce Court, Burnaby, B.C., Canada V5A 4N3. 639 640 Buckland and Lawrence Q-Iearning (Watkins, 1989, Watkins et al., 1992) is a direct form of DP that avoids explicit system modeling - thereby reducing the memory required for DP control. Further reductions are possible if Q-Ieal'l1ing is modified so that its DP cost values (Q-values) are associated only with states where control action changes need to be specified. Transition point dynamic programming (TPDP), the control approach described in this paper, is designed to take advantage of this DP memory reduction possibility by determining the states where control action changes must be specified for optimal control, and what those optimal changes are. 2 GENERAL DESCRIPTION OF TPDP 2.1 TAKING ADVANTAGE OF INERTIA TPDP is suited to the control of continuous stochastic dynamic systems that have inertia. In such systems "uniform regions" are likely to exist in the state space where all of the (discretized) states have the same optimal control action (or the same set of optimal actionsl ). Considering one such uniform region, if the optimal action for that region is specified at the "boundary states" of the region and then maintained throughout the region until it is left and another uniform region is entered (where another set of boundary states specify the next action), none of the "dormant states" in the middle of the region need to specify any actions themselves. Thus dormant states do not have to be represented in memory. This is the basic premise of TPDP. The association of optimal actions with boundary states is done by "transition points" (TPs) at those states. Boundary states include all of the states that can be reached from outside a uniform region when that region is entered as a result of stochastic state transitions. The boundary states of anyone uniform region form a hyper-surface of variable thickness which mayor may not be closed. The TPs at boundary states must be represented in memory, but if they are small in number compared to the dormant states the memory savings can be significant. 2.2 ILLUSTRATING THE TPDP CONCEPT Figure 1 illustrates the TPDP concept when movement control of a "car" on a one dimensional track is desired. The car, with some initial positive velocity to the right, must pass Position A and return to the left. The TPs in Figure 1 (represented by boxes) are located at boundary states. The shaded regions indicate all of the states that the system can possibly move through given the actions specified at the boundary states and the stochastic response of the car. Shaded states without TPs are therefore dormant states. Uniform regiolls consist of adjacent boundary states where the same action is specified, as well as the shaded region through which that action is maintained before another boundary is encountered. Boundary states that do not seem to be on the main sta.te transition routes (the one identified in Figure 1 for example) ensure that any stochastic deviations from those routes are realigned. Unshaded states are "external states" the system does not reach. IThe simplifying assumption t.hat t.here is ouly oue optimal action in each uniform region will be made throughout this paper. TPDP operates the same regardless. Transition Point Dynamic Programming 641 + ~ '0 00 Q) > A Position Each 13 is a transition point (TP), niform Region Boundary State Figure 1: Application of TPDP to a One Dimension Movement Control Task 2.3 MINIMAL TP OPTIMAL CONTROL The main benefit of the TPDP approach is that, where uniform regions exist, they can be represented by a relatively small number of DP elements (TPs) - depending on the shape of the boundaries and the size of the uniform regions they encompass. This reduction in memory usage results in an accompanying reduction in the learning time required to learn optimal control policies (Chapman et al., 1991). TPDP operates by learning optimal points of transition in the control action specification, where those points can be accurately located in highly resolved state spaces. To do this TPDP must determine which states are boundary states that should have TPs, and what actions those TPs should specify. In other words, TPDP must find the right TPs for the right states. When it has done so, "minimal TP optimal control" has been achieved. That is, optimal control with a minimal set of TPs. 3 ACHIEVING MINIMAL TP OPTIMAL CONTROL 3.1 MODIFYING A SET OF TPs Given an arbitrary initial set of TPs, TPDP must modify that set so that it is transformed into a minimal TP optimal control set. Modifications can include the "addition" and "removal" of TPs throughout the state space, and the "swapping" of one TP for another (each specifying a different action) at the same state. These 642 Buckland and Lawrence modifications are performed one at a time in arbitl'ary order, and can continue indefinitely. TPDP operates so that each TP modification results in an incremental movement towards minimal TP optimal control (Buckland, 1994). 3.2 Q-LEARNING TPDP makes use of Q-Iearning (Watkins, 1989, Watkins et ai., 1992) to modify the TP set. Normally Q-Iearning is used to determine the optimal control policy J-t for a stochastic dynamic system subjected to immediate costs c(i, u) when action u is applied in each state i (Barto et ai., 1991). Q-learning makes use of "Q-values" Q( i, u), which indicate the expected total infini te-horizon discounted cost if action u is applied in state i, and actions defined by the existing policy J-t are applied in all future states. Q-values are learned by using the following updating equation: Qt+l(St, Ut) = (1 - Ctt)Qt(St, ud + at [c(St, ud + 'YVt(St+l)] (1) Where at is the update rate, l' is the discount factor, and St and Ut are respectively the state at time step t and the action taken at that time step (all other Q-values remain the same at time step t). The evaluation function value lit ( i) is set to the lowest Q-value action of all those possible U(i) in each state i: Vt(i) = min Qt(i, u) (2) UEU(i) If Equations 1 and 2 are employed during exploratory movement of the system, it has been proven that convergence to optimal Q-values Q* (i, u) and optimal evaluation function values VI-'. (i) will result (given that the proper constraints are followed, Watkins, 1989, Watkins et ai., 1992, Jaakkola et ai., 1994). From these values the optimal action in each state can be determined (the action that fulfills Equation 2). 3.3 ASSESSING TPs WITH Q-LEARNING TPDP uses Q-Iearning to determiue how an existing set of TPs should be modified to achieve minimal TP optimal control. Q-values can be associated with TPs, and the Q-values of two TPs at the same "TP state", each specifying different actions, can be compared to determine which should be maintained at that state - that is, which has the lower Q-value. This is how TPs are swapped (Buckland, 1994). States which do not have TPs, "non-TP states", have no Q-values from which evaluation function values vt(i) can be determined (using Equation 2). As a result, to learn TP Q-values, Equation 1 must be modified to facilitate Q-value updating when the system makes d state transitions from one TP state through a number of non-TP states to another TP state: Qt+.( St, Ut) = (1 - a,jQt (5t, Ut) + "t [ (~'Yn c( St+n, Ut)) + 'Y.v,( St+.)] (3) When d = 1, Equation 3 takes the form of Equation 1. When d > 1, the intervening non-TP states are effectively ignored and treated as inherent parts of the stochastic dynamic behavior of the system (Buckla.nd, 1994). If Equation 3 is used to determine the costs incurred when no action is specified at a state (when the action specified at some previous state is maintained), an "Rvalue" R( i) is the result. R-values can be used to expediently add and remove TPs Transition Point Dynamic Programming 643 from each state. If the Q-value of a TP is less than the R-value of the state it is associated with, then it is worthwhile having that TP at that state; otherwise it is not (Buckland, 1994). 3.4 CONVERGENCE TO MINIMAL TP OPTIMAL CONTROL It has been proven that a random sequence of TP additions, swaps and removals attempted at states throughout the state space will result in convergence to minimal TP optimal control (Buckland, 1994). This proof depends mainly on all TP modifications "locking-in" any potential cost reductions which are discovered as the result of learning exploration. The problem with this proof of convergence, and the theoretical form of TPDP described up to this point, is that each modification to the existing set of TPs (each addition, swap and removal) requires the determination of Q-values and R-values which are negligibly close to being exact. This means that a complete session of Q-Iearning must occur for every TP modification. 2 The result is excessive learning times - a problem circumvented by the practical form of TPDP described next. 4 PRACTICAL TPDP 4.1 CONCURRENT TP ASSESSMENT To solve the problem of the protracted learning time required by the theoretical form of TPDP, many TP modifications can be assessed concurrently. That is, Q-Iearning can be employed not just to determine the Q-values and R-values for a single TP modification, but instead to learn these values for a number of concurrent modifications. Further, the modification attempts, and the learning of the values required for them, need not be initiated simultaneously. The determination of each value can be made part of the Q-Iearning process whenever new modifications are randomly attempted. This approa.ch is called "Pra.ctical TPDP". Practical TPDP consists of a continually running Q-Ieal'l1ing process (based on Equations 2 and 3), where the Q-values and R-values of a constantly changing set of TPs are learned. 4.2 USING WEIGHTS FOR CONCURRENT TP ASSESSMENT The main difficulty that arises when TPs are assessed concurrently is that of determining when an assessment is complete. That is, when the Q-values and R-values associated with each TP ha.ve been learned well enough for a TP modification to be made based on them. The technique employed to address this problem is to associate a "weight" wei, u) with ea.ch TP that indicates the general merit of that TP. The basic idea of weights is to facilita.te the random addition of trial TPs to a TP "assessment group" with a low initial weight Winitial. The Q-values and Rvalues of the TPs in the assessment group are learned in an ongoing Q-Iearning process, and the weights of the TPs are adjusted heuristically using those values. Of those TPs at any state i whose weights wei, u) have been increased above Wthr 2The TPDP proof allows for more than one TP swap to be assessed simultaneously, but this does little to reduce the overall problem being described (Buckland, 1994). 644 Buckland and Lawrence ..c C> C Q) .....J ..c CU a... Q) C> ~ Q) ~ 100 50 o o Practical TPDP Conventional Q-Iearning Epoch Number 2500 Figure 2: Performance of Practical TPDP on a Race Track Problem (Winitial < Wthr < wmax ), the one with the lowest Q-value Q(i, u) is swapped into the "policy TP" role for that state. The heuristic weight adjustment rules are: 1. New, trial TPs are given an initial weight of Wjnitial (0 < Winitial < Wthr). 2. Each time the Q-value of a TP is updated, the weight w(i, u) of that TP is incremented if Q(i, u) < R(i) and decremented otherwise. 3. Each TP weight w( i, u) is limited to a maximum value of wmax . This prevents anyone weight from becoming so large that it cannot readily be reduced again. 4. If a TP weight w(i, u) is decremented to 0 the TP is removed. An algorithm for Practical TPDP implementation is described in Buckland (1994). 4.3 PERFORMANCE OF PRACTICAL TPDP Practical TPDP was applied to a continuous version of a control task described by Barto et al. (1991) - that of controlling the acceleration of a car down a race track (specifically the track shown in Figures 3 and 4) when that car randomly experiences control action non-responsiveness. As shown in Figure 2 (each epoch in this Figure consisted of 20 training trials and 500 testing trials), Practical TPD P learned the optimal control policy much sooner than conventional Q-Iearning, and it was able to do so when limited to only 15% of the possible number of TPs (Buckland, 1994). The possible number of TPs is the full set of Q-values required by conventional Q-Iearning (one for each possible state and action combination). The main advantage of Practical TPDP is that it facilitates rapid learning of preliminary control policies. Figure 3 shows typical routes followed by the car early Starting Positions Starting Positions Transition Point Dynamic Programming 645 Figure 3: Typical Race Track Routes After 300 Epochs Figure 4: Typical Race Track Routes After 1300 Epochs Finishing Positions Finishing Positions in the learning process. With the addition of relatively few TPs, the policy of accelerating wildly down the track, smashing into the wall and continuing on to the finishing positions was learned. Further learning centered around this preliminary policy led to the optimal policy of sweeping around the left turn. Figure 4 shows typical routes followed by the car during this shift in the learned policy - a shift indicated by a slight drop in the learning curve shown in Figure 2 (around 1300 epochs). After this shift, learning progressed rapidly until roughly optimal policies were consistently followed. A problem which occurs in Practical TPDP is that of the addition of superfluous TPs after the optimal policy has bac;ically been learned. The reasons this occurs are described in Buckland (1994), ac; well as a number of solutions to the problem. 5 CONCLUSION The practical form of TPDP performs very well when compared to conventional Q-Iearning. When applied to a race track problem it was able to learn optimal policies more quickly while using less memory. Like Q-learning, TPDP has all the 646 Buckland and Lawrence advantages and disadvantages that result from it being a direct control approach that develops no explicit system model (Watkins, 1989, Buckland, 1994). In order to take advantage of the sparse memory usage that occurs in TPDP, TPs are best represented by ACAMs (associative content addressable memories, Atkeson, 1989). A localized neural network design which operates as an ACAM and which facilitates Practical TPDP control is described in Buckland et al. (1993) and Buckland (1994). The main idea of TPDP is to, "try this for a while and see what happens". This is a potentially powerful approach, and the use of TPs associated with abstracted control actions could be found to have substantial utility in hierarchical control systems. Acknowledgements Thanks to John Ip for his help on this work. This work was supported by an NSERC Postgraduate Scholarship, and NSERC Operating Grant A4922. References Atkeson, C. G. (1989), "Learning arm kinematics and dynamics", Annual Review of Neuroscience, vol. 12, 1989, pp. 157-183. Barto, A. G., S. J. Bradtke and S. P. Singh (1991), "Real-time learning and control using asynchronous dynamic programming", COINS Technical Report 91-57, University of Massachusetts, Aug. 1991. Buckland, K. M. and P. D. Lawrence (1993), "A connectionist approach to direct dynamic programming control" , Proc. of the IEEE Pacific Rim Conf. on Communications, Computers and Signal Processing, Victoria, 1993, vol. 1, pp. 284-287. Buckland, K. M. (1994), Optimal Control of Dynamic Systems Through the Reinforcement Learning of Transition Points, Ph.D. Thesis, Dept. of Electrical Engineering, University of British Columbia, 1994. Chapman, D. and L. P. Kaelbling (1991), "Input generalization in delayed reinforcement-learning: an algorithm a.nd performance comparisons", Proc. of the 12th Int. Joint Con/. on Artificial Intelligence, Sydney, Aug. 1991, pp. 726-731. Jaakkola, T., M. I. Jordan and S. P. Singh (1994), "Stocha'ltic convergence of iterative DP algorithms", A dvances in N eM'al Information Processing Systems 6, eds.: J. D. Cowen, G. Tesauro and J. Alspector, San Francisco, CA: Morgan Kaufmann Publishers, 1994. Moore, A. W. (1991), "Variable resolution dynamic programming: efficiently learning action maps in multivariate real-valued state-spaces", Machine Learning: Proc. of the 8th Int. Workshop, San Mateo, CA: Morgan Kaufmann Publishers, 1991. Watkins, C. J. C. H. (1989), Learning from Delayed Rewards, Ph.D. Thesis, Cambridge University, Cambridge, England, 1989. Watkins, C. J. C. H. and P. Dayan (1992), "Q-Iearning", Machine Learning, vol. 8, 1992, pp. 279-292.
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Dopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina Eric Boussard Jean-Fran~ois Vibert B3E, INSERM U263 Faculte de medecine Saint-Antoine 27 rue Chaligny 75571 Paris cedex 12 Abstract The fovea of a mammal retina was simulated with its detailed biological properties to study the local preprocessing of images. The direct visual pathway (photoreceptors, bipolar and ganglion cells) and the horizontal units, as well as the D-amacrine cells were simulated. The computer program simulated the analog non-spiking transmission between photoreceptor and bipolar cells, and between bipolar and ganglion cells, as well as the gap-junctions between horizontal cells, and the release of dopamine by D-amacrine cells and its diffusion in the extra-cellular space. A 64 x 64 photoreceptors retina, containing 16,448 units, was carried out. This retina displayed contour extraction with a Mach effect, and adaptation to brightness. The simulation showed that the dopaminergic amacrine cells were necessary to ensure adaptation to local brightness. 1 INTRODUCTION The retina is the first stage in visual information processing. One of its functions is to compress the information received from the environment by removing spatial and temporal redundancies that occur in the light input signal. Modelling and computer simulations present an efficient means to investigate and characterize the physiological mechanisms that underlie such a complex process. In fact, filtering depends on the quality of the input image (van Hateren, 1992): 559 560 Boussard and Vibert l.High mean light intensity (high signal to noise ratio). A high-pass filter enhances the edges (contour extraction) and the temporal changes of the input. 2.Low mean light intensity (low signal to noise ratio). The sensitivity of highpass filters to noise makes them inefficient in this case. A low-pass filter, averaging the signal over several receptors, is required to extract the relevant information. There are three aspects in the filtering adaptivity displayed by the retina: adaptivity to i) the global spatial changes in the image, ii) the local spatial changes in the image, iii) the temporal changes in the image. We will focus on the second feature. A biologically plausible mammalian retina was modelled and simulated to explore the local preprocessing of the images. A first model (Bedfer & Vibert, 1992), that did not take into account the dopamine neuromodulation, reproduced some of the behaviors found in the living retina, like a progressive decrease of ganglion cells' firing rate in response to a constant image presented to photoreceptors, reversed post-image, and optic illusion (Hermann grid). The model, however, displayed a poor local adaptivity. It could not give both a good contrast rendering and a Mach effect. The Mach effect is a psychophysical law that is characterized by an edge enhancement (Ratliff, 1965). The retina network produces a double lighter and darker contour from the frontier line between two areas of different brightness in the stimulus. This phenomenon is indispensable for contour extraction. This paper will first present the conditions in which high-pass filtering and low-pass filtering occur exclusively in the retina model. These results are then compared to those obtained with a model that includes dopamine neuromodulation, thus illustrating the role played by dopamine in local adaptivity (Besharse & Iuvone, 1992). 2 METHODS The retina is an unusual neural structure: i) the photoreceptors respond to light by an hyperpolarization, ii) signal transmission from photoreceptors to bipolar units does not involve spikes, neurotransmitter release at these synapses is a continuous function of the membrane potential (Buser & Imbert, 1987). Only ganglion cells generate spikes. Furthermore, horizontal cells are connected by dopamine dependent gap-junctions. Dopamine is an ubiquitous neurotransmitter and neuromodulator in the central nervous system. In the visual pathway, dopamine affects several types of retinal neurons (Witkovsky & Dearry, 1992). Dopamine is released by stimulated D-amacrine and interplexiform cells. It diffuses in the extra-cellular space, and produces: cone shortening and rod elongation, reduced permeability of gap-junctions, increased conductance of glutamate-induced current among horizontal cells, increased conductance of the cone-to- horizontal cell synapse, and retroinhibition on D-amacrine cells (Djamgoz & Wagner, 1992). Our model focused on the adaptive filtering mechanism in the fovea that enables the retina to simultaneously perform both high-pass and low-pass filtering. Therefore, dopamine action on gap-junction between horizontal cells and the retro-inhibition on D-amacrine cells was the only dopamine effect implemented (fig. 1). Our model included the three neuron types of the direct pathway - photoreceptors, bipolar and ganglion units - as well as two types of the indirect pathway - the horizontal and dopaminergic amacrine cells. Only the On pathway of a mammal fovea was studied here. Each neuron type has been modelled with its own anatomical and electrophysiologDopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina 561 ~ Excitation .. lnhibition .11 II' Gap-junction ~ ~fea:!ne Figure 1: The dopaminergic amacrine units in the modelled retina. The connections of an On center pathway in the simulated retina. Photo: Photoreceptors. Horiz: Horizontal units. Bip: Bipolar units. Gang: Ganglion Units. DA: Dopaminergic Amacrine unit. DA units are stimulated by many bipolar units. With an enough excitation, they can release dopamine in the extracellular space. This released dopamine goes to modulates the conductance value of horizontal gapjunctions. 562 Boussard and Vibert ical properties (Wiissle & Boycott, 1991)(Lewick & Dvorak, 1986). The temporal evolution of the membrane potential of each unit can be recorded. 3 RESULTS A 64x64 photoreceptors retina was constructed as a noisy hexagonal frame where photoreceptors, bipolar and ganglion units were connected to their nearest neighbours. Horizontal units were connected to their 18 nearest photoreceptors and bipolar units, with a number of synaptic boutons decreasing as a function of distance. They did not retroact on the nearest photoreceptor. This horizontal layer architecture produces lateral inhibition. Each modelled D-amacrine unit was connected to about fifty bipolar units. The diffusion of released dopamine in the extra-cellular space was simulated. The modelled retina consisted of 16,448 units and 862,720 synapses. At each simulation, the photoreceptors layer was stimulated by an input image. Stimulations were given as a 256x256 pixel image presented to the simulated 64x64 photoreceptor retina. Since the localization of photoreceptors was not regular, each receptor received the input from 16 pixels on the average. The output image was reconstructed using the ganglion units response. For each of the 4096 ganglion units the spike frequency was measured during a given time (according to the experiment) and coded in a grey level for the given unit retinotopic position. Thus, each simulation produced an image of the retina output. This output image was compared to the input image. The input image (stimulus) consisted here of one white disk on a dark background. The results presented, in fig. 2, were obtained after 750 ms of stationary stimulations. The stimuli were here a white disk on a black background. The inputs were stationary to avoid temporal effects owing to evolving inputs. Output images of stationary inputs, however, vanished after 1000 ms. The time was limited to 750 ms to optimize the quality of the output image. Biological datas available on the conductance value suggest that in the mammalian retina the conductance does not remain constant and undergoes a dynamical tuning depending on the local brightness [?]. This provides a range of possible values for the conductance. The behavior of the model was tested for values within this range. Different values lead to different network behaviors. Three types of results were obtained from the simulations : l.Without dopamine action, the conductance values were fixed for all gap-junctions to 1O-6S (fig. 2-A). The output image rendered well the contrast in the input image, but did not display the Mach effect (low-pass filtering). 2.Without dopamine action, the conductance values were fixed for all gap-junctions to 1O- 10S (fig. 2-B). The low conductance value allowed a pronounced Mach effect, but the contrast in the output image was strongly diminished (high-pass filtering). This contrast appears like an average of the two brightness. Only the contour delimited by Mach effect allows the disk to be distinguished. 3.With dopamine, the conductance values were initially set to 1O- 7S (fig. 2-C). The output image displayed both the contrast rendering and the Mach effect (locally Dopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina 563 22 23 24 26 27 28 29 30 32 ' 33 34 40 32 24 16 8 A °0~---1-1--~22~-3~4---4~5--~56 B 40 32 ..-----,. 24 16 8 0~~1~2--~24~~3~6--~4=8--~60 65 52 39 26 13 c LJV'\....I'-_I 0~~12~~2~4--~3~7--4~9~~61 Figure 2: ~ontour extraction (Mach effect) according to gap-junctions conductances. On the left, results obtained after 750 ms of stimulation for an zmage of a white disk on a black background. On the right, sections through the corresponding image. A bscissa: spike count; Ordinates: geographic position of the unit, from the left side to the middle of the left panel. A: without dopamine (fixed Ggap = 10-6 S). B: without dopamine (fixed Ggap = lO-lOS). C: with dopamine release (starting Ggap = 1O-7S). A gives a good contrast rendering, but no Mach effect. B gives a Mach effect, but there is an averaging between darker and lighter areas. C, with dopa minergic neuromodulation, gives both a Mach effect and a good contrast rendering. 564 Boussard and Vibert adaptive filtering). 4 DISCUSSION These results show that the conductance cannot be fixed at a single value for all the gap-junctions. If the conductance value is high (fig. 2-A), the model acts like a low-pass filter. A good contrast rendering was obtained, but there was no Mach effect. If the conductance value is low (fig. 2-B), the model becomes a high-pass filter. A Mach effect was obtained, but the contrast in the post-retinal image was dramatically deteriorated: an undesirable averaging of the brightness between the darker and the more illuminated areas appeared. Therefore in this model the Mach effect was only obtained at the expense of the contrast. A mammalian retina is able to perform both contrast rendering and contour extraction functions together. It works like an adaptive filter. To obtain a similar result, it is necessary to have a variable communication between horizontal units. The simulated retina needs low gap-junctions conductance in the high light intensity areas and high conductance in the low light intensity areas. The conductance of each gap-junction must be tuned according to the local stimulation. The model used to obtain the fig. 2-C takes into account the dopamine release by the D-amacrine cells. Here, the network performs the two antagonist functions of filtering. Dopamine provides our model with the capacity to have a biological behaviour. What is the action of dopamine on network? Dopamine is released by D-amacrine units. Then, it diffuses from its release point into the extra-cellular space among the neurons, reaches gap-junctions and decreases their conductance value. Thus the conductance modulation depends in time and in intensity on the distance between gap-junction and D-amacrine unit. In addition, this action is transient. 5 CONCLUSION Thanks to dopamine neuromodulation, the network is able to subdivise itself into several subnetworks, each having the appropriate gap-junction conductance. Each subnetwork is thus adapted for a better processing of the external stimulus. Dopamine neuromodulation is a chemically addressed system, it acts more diffusely and more slowly than transmission through the axo-synaptic connection system. Therefore neuromodulation adds a dynamical plasticity to the network. References G. Bedfer & J .-F. Vibert. (1992) Image preprocessing in simulated biologicalretina. Proc. 14th Ann. Conf. IEEE EMBS 1570-157l. J. Besharse & P. Iuvone. (1992). Is dopamine a light-adaptive or a dark-adaptive modulator in retina? NeuroChemistry International 20:193-199. P. Buser & M. Imbert. (1987) Vision. Paris: Hermann. M. Djamgoz & H.-J. Wagner. (1992) Localization and function of dopamine in the adult vertebrate retina. NeuroChemistry InternationaI20:139-19l. L. Dowling. (1986) Dopamine: a retinal neuromodulator? Trends In NeuroSciences Dopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina 565 9:236-240. W . Levick & D. Dvorak. (1986) The retina - from molecule to network. Trends In NeuroSciences 9:181-185. F. Ratliff. (1965) Mach bands: quantitative studies on neural network in the retina. Holden-Day. J . H. van Hateren. (1992) Real and optimal images in early vision. Nature 360:6870. H. Wassle & B. B. Boycott. (1991) Functional architecture of the mammalian retina. Physiological Reviews 71(2):447-479. P. Witkovsky & A. Dearry. (1992) Functional roles of dopamine in the vertebrate retina. Retinal Research 11:247-292.
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Monte Carlo Matrix Inversion and Reinforcement Learning Andrew Barto and Michael Duff Computer Science Department University of Massachusetts Amherst, MA 01003 Abstract We describe the relationship between certain reinforcement learning (RL) methods based on dynamic programming (DP) and a class of unorthodox Monte Carlo methods for solving systems of linear equations proposed in the 1950's. These methods recast the solution of the linear system as the expected value of a statistic suitably defined over sample paths of a Markov chain. The significance of our observations lies in arguments (Curtiss, 1954) that these Monte Carlo methods scale better with respect to state-space size than do standard, iterative techniques for solving systems of linear equations. This analysis also establishes convergence rate estimates. Because methods used in RL systems for approximating the evaluation function of a fixed control policy also approximate solutions to systems of linear equations, the connection to these Monte Carlo methods establishes that algorithms very similar to TD algorithms (Sutton, 1988) are asymptotically more efficient in a precise sense than other methods for evaluating policies. Further, all DP-based RL methods have some of the properties of these Monte Carlo algorithms, which suggests that although RL is often perceived to be slow, for sufficiently large problems, it may in fact be more efficient than other known classes of methods capable of producing the same results. 687 688 Barto and Duff 1 Introduction Consider a system whose dynamics are described by a finite state Markov chain with transition matrix P, and suppose that at each time step, in addition to making a transition from state Xt = i to XHI = j with probability Pij, the system produces a randomly determined reward, rt+1! whose expected value is R;. The evaluation junction, V, maps states to their expected, infinite-horizon discounted returns: It is well known that V uniquely satifies a linear system of equations describing local consistency: V = R + -yPV, or (I - -yP)V = R. ( 1) The problem of computing or estimating V is interesting and important in its own right, but perhaps more significantly, it arises as a (rather computationallyburdensome) step in certain techniques for solving Markov Decision Problems. In each iteration of Policy-Iteration (Howard, 1960), for example, one must determine the evaluation function associated with some fixed control policy, a policy that improves with each iteration. Methods for solving (1) include standard iterative techniques and their variantssuccessive approximation (Jacobi or Gauss-Seidel versions), successive overrelaxation, etc. They also include some of the algorithms used in reinforcement learning (RL) systems, such as the family of TD algorithms (Sutton, 1988). Here we describe the relationship between the latter methods and a class of unorthodox Monte Carlo methods for solving systems of linear equations proposed in the 1950's. These methods recast the solution of the linear system as the expected value of a statistic suitably defined over sample paths of a Markov chain. The significance of our observations lies in arguments (Curtiss, 1954) that these Monte Carlo methods scale better with respect to state-space size than do standard, iterative techniques for solving systems of linear equations. This analysis also establishes convergence rate estimates. Applying this analysis to particular members of the family of TD algorithms (Sutton, 1988) provides insight into the scaling properties of the TD family as a whole and the reasons that TD methods can be effective for problems with very large state sets, such as in the backgammon player of Tesauro (Tesauro, 1992). Further, all DP-based RL methods have some of the properties of these Monte Carlo algorithms, which suggests that although RL is often slow, for large problems (Markov Decision Problems with large numbers of states) it is in fact far more practical than other known methods capable of producing the same results. First, like many RL methods, the Monte Carlo algorithms do not require explicit knowledge of the transition matrix, P. Second, unlike standard methods for solving systems of linear equations, the Monte Carlo algorithms can approximate the solution for some variables without expending the computational effort required to approximate Monte Carlo Matrix Inversion and Reinforcement Learning 689 the solution for all of the variables. In this respect, they are similar to DP-based RL algorithms that approximate solutions to Markovian decision processes through repeated trials of simulated or actual control, thus tending to focus computation onto regions of the state space that are likely to be relevant in actual control (Barto et. al., 1991). This paper begins with a condensed summary of Monte Carlo algorithms for solving systems of linear equations. We show that for the problem of determining an evaluation function, they reduce to simple, practical implementations. Next, we recall arguments (Curtiss, 1954) regarding the scaling properties of Monte Carlo methods compared to iterative methods. Finally, we conclude with a discussion of the implications of the Monte Carlo technique for certain algorithms useful in RL systems. 2 Monte Carlo Methods for Solving Systems of Linear Equations The Monte Carlo approach may be motivated by considering the statistical evaluation of a simple sum, I:k ak. If {Pk} denotes a set of values for a probability mass function that is arbitrary (save for the requirement that ak =P 0 imply Pk =P 0), then I:k ak = I:k (~) Pk, which may be interpreted as the expected value of a random variable Z defined by Pr { Z = ~ } = Pk. From equation (1) and the Neumann series representation of the inverse it is is clear that V = (1 - -yp)-l R = R + -yP R + -y2 p2 R + ... whose ith component is Vi = R; + -y L P"l R;l + -y2 L P"lP'1'2 R;2 + ... . . . + -yk L Pii1 ... P,/o-li/oR;/o + ... (2) and it is this series that we wish to evaluate by statistical means. A technique originated by Ulam and von-Neumann (Forsythe & Leibler, 1950) utilizes an arbitrarily defined Markov chain with transition matrix P and state set {I, 2, "., n} (V is assumed to have n components). The chain begins in state i and is allowed to make k transitions, where k is drawn from a geometric distribution with parameter Pdep; i.e., Pr{k state transitions} = P~tep(1 - P,tep)' The Markov chain, governed by P and the geometrically-distributed stopping criterion, defines a mass function assigning probability to every trajectory of every length starting in state i, Xo = io = i --+ Zl = i l --+ ... --+ Zk = ik, and to each such trajectory there corresponds a unique term in the sum (2). For the cas/~ of value estimation, "Z" is defined by 690 Barto and Duff which for j> = P and P,tep = 'Y becomes k Pr {z = 1~" } = 'Yk(1 - 'Y) IT Pij_li;'Y ;=1 The sample average of sampled values of Z is guaranteed to converge (as the number of samples grows large) to state i's expected, infinite-horizon discounted return. In Wasow's method (Wasow, 1952), the truncated Neumann series ~ = R; + 'Y LPiilR;l + 'Y2 LPiilPili2R;2 + ... + 'YN L Piil ·· ·PiN_liNR;N is expressed as R; plus the expected value of the sum of N random variables ZlI Z2, ... , ZN, the intention being that E(Zk) = 'Yk L PihPili2" ·pi"_d,,R;,,· i 1 ···i" Let trajectories of length N be generated by the Markov chain governed by P. A given term 'Y"Pii1Pili2·· 'Pi"_li"R;" is associated with all trajectories i -+ i1 -+ i2 -+ ... -+ ik -+ ik+1 -+ ... -+ iN whose first k + 1 states are i, ill ... , ik. The measure of this set of trajectories is just Pii1Pili2 ... Pi"_li". Thus, the random variables Zk, k = 1, N are defined by If P = P, then the estimate becomes an average of sample truncated, discounted returns: ~ = R; + 'YR;1 + 'Y2 R;.2 + ... + 'YN R;N. The Ulam/von Neumann approach may be reconciled with that of Wasow by processing a given trajectory a posteriori, converting it into a set of terminated paths consistent with any choice of stopping-state transition probabilities. For example, for a stopping state transition probability of 1 'Y, a path of length k has probability 'Yk(1 - 'Y). Each "prefix" of the observed path x(O) -+ x(1) -+ z(2) -+ ... can be weighted by the probability of a path of corresponding length, resulting in an estimate, V, that is the sampled, discounted return: 00 V = L -rk RZ(k). k=O 3 Complexity In (Curtiss, 1954) Curtiss establishes a theoretical comparison of the complexity (number of multiplications) required by the Ulam/von Neumann method and a stationary linear iterative process for computing a single component of the solution to a system of linear equations. Curtiss develops an analytic formula for bounds on the conditional mean and variance of the Monte-Carlo sample estimate, V, and mean and variance of a sample path's time to absorption, then appeals to the Monte Carlo Matrix Inversion and Reinforcement Learning 691 n 1000 900 800 700 600 500 400 300 200 100 )"=.5 ),,=.7 )"=.9 O~----------~----~--~--~--~ a 100 200 300 400 500 600 700 800 900 1000 1/~ Figure 1: Break-even size of state space versus accuracy. Central Limit Theorem to establish a 95%-confidence interval for the complexity of his method to reduce the initial error by a given factor, e. 1 For the case of. value-estimation, Curtiss' formula for the Monte-Carlo complexity may be written as WORKMonte-Carlo = 1 ~ "'; (1 + e 22 ) . (3) This is compared to the complexity of the iterative method, which for the valueestimation problem takes the form of the classical dynamic programming recursion, v(n+l) = R + ",;pv(n): ( lOge) 2 WORKiterati'lle = 1 + log",; n + n. The iterative methodts complexity has the form an2 + n, with a > It while the Monte-Carlo complexity is independent of n-it is most sensitive to the amount of error reduction desired, signified bye. Thus, given a fixed amount of computation, for large enough n, the Monte-Carlo method is likely (with 95% confidence level) to produce better estimates. The theoretical "break-even" points are plotted in Figure It and Figure 2 plots work versus state-space size for example values of",; and e. IThat is, for the iterative method, e is defined via IIV(oo) - yen) II < eIlV(oo) - yeO) II, while for the Monte Carlo method, e is defined via IV(OD)(i) - VMI < eIlV(OD) - V(O)II, where VM is the average over M sample V's. 692 Barto and Duff .::&.50000 .... o ~45000 I I I I 40000~------------~/------~-------I 35000 I 30000 25000 20000 15000 10000 5000 , I I I Iterative Monte Carlo Gauss O~~--~~~~--~~--~--~~--~ o 10 20 30 40 50 60 70 80 90 100 n Figure 2: Work versus number of states for"Y = .5 and e = .01. 4 Discussion It was noted that the analytic complexity Curtiss develops is for the work required to compute one component of a solution vector. In the worst case, all components could be estimated by constructing n separate, independent estimators. This would multiply the Monte-Carlo complexity by a factor of n, and its scaling supremacy would be only marginally preserved. A more efficient approach would utilize data obtained in the course of estimating one component to estimate other components as well; Rubinstein (Rubinstein, 1981) decribes one way of doing this, using the notion of "covering paths." Also, it should be mentioned that substituting more sophisticated iterative methods, such as Gauss-Seidel, in place of the simple successive approximation scheme considered here, serves only to improve the condition number of the underlying iterative operator-the amount of computation required by iterative methods remains an2 + n, for some a> 1. An attractive feature of the the analysis provided by Curtiss is that, in effect, it yields information regarding the convergence rate of the method; that is, Equation 4 can be re-arranged in terms of e. Figure 3 plots e versus work for example values of"Y and n. The simple Monte Carlo scheme considered here is practically identical to the limiting case of TD-A with A equal to one (TD-l differs in that its averaging of sampled, discounted returns is weighted with recency). Ongoing work (Duff) explores the connection between TD-A (Sutton, 1988), for general values of A, and Monte Carlo methods augmented by certain variance reduction techniques. Also, Barnard (Barnard) has noted that TD-O may be viewed as a stochastic approxima~ 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 Monte Carlo Matrix Inversion and Reinforcement Learning 693 ... Iterative Monte Carlo 10000 20000 30000 40000 50000 Work Figure 3: Error reduction versus work for "y = .9 and n = 100. tion method for solving (1). On-line RL methods for solving Markov Decision Problems, such as Real-Time Dynamic Programming (RTDP)(Barto et. al., 1991), share key features with the Monte Carlo method. As with many algorithms, RTDP does not require explicit knowledge of the transition matrix, P, and neither, of course, do the Monte Carlo algorithms. RTDP approximates solutions to Markov Decision Problems through repeated trials of simulated or actual control, focusing computation upon regions of the state space likely to be relevant in actual control. This computational "focusing" is also a feature of the Monte Carlo algorithms. While it is true that a focusing of sorts is exhibited by Monte Carlo algorithms in an obvious way by virtue of the fact that they can compute approximate solutions for single components of solution vectors without exerting the computational labor required to compute all solution components, a more subtle form of computational focusing also occurs. Some of the terms in the Neumann series (2) may be very unimportant and need not be represented in the statistical estimator at all. The Monte Carlo method's stochastic estimation process achieves this automatically by, in effect, making the appearance of the representative of a non-essential term a very rare event. These correspondences-between TD-O and stochastic approximation, between TD). and Monte Carlo methods with variance reduction, between DP-based RL algorithms for solving Markov Decision Problems and Monte Carlo algorithms together with the comparatively favorable scaling and convergence properties enjoyed by the simple Monte Carlo method discussed in this paper, suggest that DPbased RL methods like TD/stochastic-approximation or RTDP, though perceived to be slow, may actually be advantageous for problems having a sufficiently large 694 Barto and Duff number of states. Acknowledgement This material is based upon work supported by the National Science Foundation under Grant ECS-9214866. References E. Barnard. Temporal-Difference Methods and Markov Models. Submitted for publication. A. Barto, S. Bradtke, & S. Singh. (1991) Real-Time Learning and Control Using Asynchronous Dynamic Programming. Computer Science Department, University of Massachusetts, Tech. Rept. 91-57. 1. Curtiss. (1954) A Theoretical Comparison of the Efficiencies of Two Classical Methods and a Monte Carlo Method for Computing One Component of the Solution of a Set of Linear Algebraic Equations. In H. A. Mayer (ed.), Symposium on Monte Carlo Methods, 191-233. New york, NY: Wiley. M. Duff. A Control Variate Perspective for the Optimal Weighting of Truncated, Corrected Returns. In Preparation. S. Forsythe & R. Leibler. (1950) Matrix Inversion by a Monte Carlo Method. Math. Tables Other Aids Comput., 4:127-129. R. Howard. (1960) Dynamic Programming and Markov Proceses. Cambridge, MA: MIT Press. R. Rubinstein. (1981) Simulation and the Monte Carlo Method. New York, NY: Wiley. R. Sutton. (1988) Learning to Predict by the Method of Temporal Differences. Machine Learning 3:9-44. G. Tesauro. (1992) Practical Issues in Temporal Difference Learning. Machine Learning 8:257-277. W. Wasow. (1952) A Note on the Inversion of Matrices by Random Walks. Math. Tables Other Aids Comput., 6:78-81.
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Globally Trained Handwritten Word Recognizer using Spatial Representation, Convolutional Neural Networks and Hidden Markov Models Yoshua Bengio ... Dept. Informatique et Recherche Operationnelle Universite de Montreal Montreal, Qc H3C-3J7 Donnie Henderson AT&T Bell Labs Holmdel NJ 07733 Abstract Yann Le Cun AT&T Bell Labs Holmdel NJ 07733 We introduce a new approach for on-line recognition of handwritten words written in unconstrained mixed style. The preprocessor performs a word-level normalization by fitting a model of the word structure using the EM algorithm. Words are then coded into low resolution "annotated images" where each pixel contains information about trajectory direction and curvature. The recognizer is a convolution network which can be spatially replicated. From the network output, a hidden Markov model produces word scores. The entire system is globally trained to minimize word-level errors. 1 Introduction Natural handwriting is often a mixture of different "styles", lower case printed, upper case, and cursive. A reliable recognizer for such handwriting would greatly improve interaction with pen-based devices, but its implementation presents new *also, AT&T Bell Labs, Holmdel NJ 07733 937 938 Bengio, Le Cun, and Henderson technical challenges. Characters taken in isolation can be very ambiguous, but considerable information is available from the context of the whole word. We propose a word recognition system for pen-based devices based on four main modules: a preprocessor that normalizes a word, or word group, by fitting a geometrical model to the word structure using the EM algorithm; a module that produces an "annotated image" from the normalized pen trajectory; a replicated convolutional neural network that spots and recognizes characters; and a Hidden Markov Model (HMM) that interprets the networks output by taking word-level constraints into account. The network and the HMM are jointly trained to minimize an error measure defined at the word level. Many on-line handwriting recognizers exploit the sequential nature of pen trajectories by representing the input in the time domain. While these representations are compact and computationally advantageous, they tend to be sensitive to stroke order, writing speed, and other irrelevant parameters. In addition, global geometric features, such as whether a stroke crosses another stroke drawn at a different time, are not readily available in temporal representations. To avoid this problem we designed a representation, called AMAP, that preserves the pictorial nature of the handwriting. In addition to recognizing characters, the system must also correctly segment the characters within the words. One approach, that we call INSEG, is to recognize a large number of heuristically segmented candidate characters and combine them optimally with a postprocessor (Burges et al 92, Schenkel et al 93). Another approach, that we call OUTSEG, is to delay all segmentation decisions until after the recognition, as is often done in speech recognition. An OUTSEG recognizer must accept entire words as input and produce a sequence of scores for each character at each location on the input. Since the word normalization cannot be done perfectly, the recognizer must be robust with respect to relatively large distortions, size variations, and translations. An elastic word model -e.g., an HMM- can extract word candidates from the network output. The HMM models the long-range sequential structure while the neural network spots and classifies characters, using local spatial structure. 2 Word Normalization Input normalization reduces intra-character variability, simplifying character recognition. This is particularly important when recognizing entire words. We propose a new word normalization scheme, based on fitting a geometrical model of the word structure. Our model has four "flexible" lines representing respectively the ascenders line, the core line, the base line and the descenders line (see Figure 1). Points on the lines are parameterized as follows: y = fk(X) = k(x XO)2 + s(x - xo) + YOk (1) where k controls curvature, s is the skew, and (xo,Yo) is a translation vector. The parameters k, s, and Xo are shared among all four curves, whereas each curve has its own vertical translation parameter YOk. First the set of local maxima U and minima L of the vertical displacement are found. Xo is determined by taking the average abscissa of extrema points. The lines of the model are then fitted to the extrema: the upper two lines to the maxima, and the lower two to the minima. The fit is performed using a probabilistic model for the extrema points given the lines. The idea is to find the line parameters 8* that maximize the probability of Globally Trained Handwritten Word Recognizer 939 --' -----Figure 1: Word Normalization Model: Ascenders and core curves fit y-maxima whereas descenders and baseline curves fit y-minima. There are 6 parameters: a (ascenders curve height relative to baseline), b (baseline absolute vertical position), c (core line position), d (descenders curve position), k (curvature), s (angle). generating the observed points. 0* = argmax log P(X I 0) + log P(O) (J (2) The above conditional distribution is chosen to be a mixture of Gaussians (one per curve) whose means are the y-positions obtained from the actual x-positions through equation 1: 3 P(Xi, Yi 1 0) = log L WkN(Yi; fk(xd, (J'y) (3) k=O where N(x; J1, (J') is a univariate Normal distribution of mean J1 and standard deviation (J'. The Wk are the mixture parameters, some of which are set to 0 in order to constrain the upper (lower) points to be fitted to the upper (lower) curves. They are computed a-priori using measured frequencies of associations of extrema to curves on a large set of words. The priors P(O) on the parameters are required to prevent the collapse of the curves. They can be used to incorporate a-priori information about the word geometry, such as the expected position of the baseline, or the height of the word. These priors for each parameter are chosen to be independent normal distributions whose standard deviations control the strength of the prior. The variables that associate each point with one of the curves are taken as hidden variables of the EM algorithm. One can thus derive an auxiliary function which can be analytically (and cheaply) solved for the 6 free parameters O. Convergence of the EM algorithm was typically obtained within 2 to 4 iterations (of maximization of the auxiliary function). 3 AMAP The recognition of handwritten characters from a pen trajectory on a digitizing surface is often done in the time domain. Trajectories are normalized, and local 940 Bengio, Le Cun, and Henderson geometrical or dynamical features are sometimes extracted. The recognition is performed using curve matching (Tappert 90), or other classification techniques such as Neural Networks (Guyon et al 91). While, as stated earlier, these representations have several advantages, their dependence on stroke ordering and individual writing styles makes them difficult to use in high accuracy, writer independent systems that integrate the segmentation with the recognition. Since the intent of the writer is to produce a legible image, it seems natural to preserve as much of the pictorial nature of the signal as possible, while at the same time exploit the sequential information in the trajectory. We propose a representation scheme, called AMAP, where pen trajectories are represented by low-resolution images in which each picture element contains information about the local properties of the trajectory. More generally, an AMAP can be viewed as a function in a multidimensional space where each dimension is associated with a local property of the trajectory, say the direction of motion e, the X position, and the Y position of the pen. The value of the function at a particular location (e, X, Y) in the space represents a smooth version of the "density" of features in the trajectory that have values (e, X, Y) (in the spirit of the generalized Hough transform). An AMAP is a multidimensional array (say 4x10x10) obtained by discretizing the feature density space into "boxes". Each array element is assigned a value equal to the integral of the feature density function over the corresponding box. In practice, an AMAP is computed as follows. At each sample on the trajectory, one computes the position of the pen (X, Y) and orientation of the motion () (and possibly other features, such as the local curvature c). Each element in the AMAP is then incremented by the amount of the integral over the corresponding box of a predetermined point-spread function centered on the coordinates of the feature vector. The use of a smooth point-spread function (say a Gaussian) ensures that smooth deformations of the trajectory will correspond to smooth transformations of the AMAP. An AMAP can be viewed as an "annotated image" in which each pixel is a feature vector. A particularly useful feature of the AMAP representation is that it makes very few assumptions about the nature of the input trajectory. It does not depend on stroke ordering or writing speed, and it can be used with all types of handwriting (capital, lower case, cursive, punctuation, symbols). Unlike many other representations (such as global features), AMAPs can be computed for complete words without requiring segmentation. 4 Convolutional Neural Networks Image-like representations such as AMAPs are particularly well suited for use in combination with Multi-Layer Convolutional Neural Networks (MLCNN) (Le Cun 89, Le Cun et al 90). MLCNNs are feed-forward neural networks whose architectures are tailored for minimizing the sensitivity to translations, rotations, or distortions of the input image. They are trained with a variation of the Back-Propagation algorithm (Rumelhart et al 86, Le Cun 86). The units in MCLNNs are only connected to a local neighborhood in the previous layer. Each unit can be seen as a local feature detector whose function is determined by the learning procedure. Insensitivity to local transformations is built into the network architecture by constraining sets of units located at different places to use identical weight vectors, thereby forcing them to detect the same feature on different parts of the input. The outputs of the units at identical locations in different feature maps can be collectively thought of as a local feature vector. Features of increasing Globally Trained Handwritten Word Recognizer 941 complexity and globality are extracted by the neurons in the successive layers. This weight-sharing technique has two interesting side effects. First, the number of free parameters in the system is greatly reduced since a large number of units share the same weights. Classically, MLCNNs are shown a single character at the input, a.nd have a single set of outputs. However, an essential feature of MLCNNs is that they can be scanned (replicated) over large input fields containing multiple unsegmented characters (whole words) very economically by simply performing the convolutions on larger inputs. Instead of producing a single output vector, SDNNs produce a series of output vectors. The outputs detects and recognize characters at different (and overlapping) locations on the input. These multiple-input, multipleoutput MLCNN are called Space Displacement Neural Networks (SDNN) (Matan et al 92). One of the best networks we found for character recognition has 5 layers arranged as follows: layer 1: convolution with 8 kernels of size 3x3, layer 2: 2x2 subsampling, layer 3: convolution with 25 kernels of size 5x5, layer 4 convolution with 84 kernels of size 4x4, layer 5: 2x2 subsampling. The subsampling layers are essential to the network's robustness to distortions. The output layer is one (single MLCNN) or a series of (SDNN) 84-dimensional vectors. The target output configuration for each character class was chosen to be a bitmap of the corresponding character in a standard 7x12 (=84) pixel font. Such a code facilitates the correction of confusable characters by the postprocessor. 5 Post-Processing The convolutional neural network can be used to give scores associated to characters when the network (or a piece of it corresponding to a single character output) has an input field, called a segment, that covers a connected subset of the whole word input. A segmentation is a sequence of such segments that covers the whole word input. Because there are in general many possible segmentations, sophisticated tools such as hidden Markov models and dynamic programming are used to search for the best segmentation. In this paper, we consider two approaches to the segmentation problem called INSEG (for input segmentation) and OUTSEG (for output segmentation). The postprocessor can be generally decomposed into two levels: 1) character level scores and constraints obtained from the observations, 2) word level constraints (grammar, dictionary). The INSEG and OUTSEG systems share the second level. In an INSEG system, the network is applied to a large number of heuristically segmented candidate characters. A cutter generates candidate cuts, which can potentially represent the boundary between two character segments. It also generates definite cuts, which we assume that no segment can cross. Using these, a number of candidate segments are constructed and the network is applied to each of them separately. Finally, for each high enough character score in each of the segment, a character hypothesis is generated, corresponding to a node in an observation graph. The connectivity and transition probabilities on the arcs of the observation graph represent segmentation and geometrical constraints (e.g., segments must not overlap and must cover the whole word, some transitions between characters are more or less likely given the geometrical relations between their images). In an OUTSEG system, all segmentation decisions are delayed until after the recog942 Bengio, Le Cun, and Henderson nition, as is often done in speech recognition. The AMAP of the entire word is shown to an SDNN, which produces a sequence of output vectors equivalent to (but obtained much more cheaply than) scanning the single-character network over all possible pixel locations on the input. The Euclidean distances between each output vector and the targets are interpreted as log-likelihoods of the output given a class. To construct an observation graph, we use a set of character models (HMMs) . Each character HMM models the sequence of network outputs observed for that character. We used three-state HMMs for each character, with a left and right state to model transitions and a center state for the character itself. The observation graph is obtained by connecting these character models, allowing any character to follow any character. On top of the constraints given in the observation graph, additional constraints that are independent of the observations are given by what we call a gram mar graph, which can embody lexical constraints. These constraints can be given in the form of a dictionary or of a character-level grammar (with transition probabilities), such as a trigram (in which we use the probability of observing a character in the context of the two previous ones). The recognition finds the best path in the observation graph that is compatible with the grammar graph. The INSEG and OUTSEG architecture are depicted in Figure 2. OUTSEG ARCHITECTURE FOR WORD RECOGNITION raw word word normalization normalized word ~--~'''''---"'''''1i s~f Mi.pf AMAP computation ':':":: :",:, ~~'::: .. , .. ~: ': ::::. ::.:. :~:~: t AMAP SDNN graph ofchar~a~c~e~r--'~----~ Character candi-'r-__ f-_ _ """'II dates ~~~} t} INSEG ARCHITECTURE FOR WORD RECOGNITION raw w0i"'r_d_""",,, ___ ~ Sec; p t ~~~r~~~ ~,= .. . """'_-_ <5~t'>r,ff: . Cut hypotheses I generation segme~n~""""_",,.. __ """"'1 graph \r"!:~'1":"IWPII"""'~W AMAP Convolutional Neural Network HMMs h graph ~~_",-__ -d! S .... c ..... r ...... i .... p .... t ~~~arliooa-cte-r----~ of character s .... e ..... n ..... e.j ... o.T candl cdaantedsi 5 ...... a ... i ... u ...... p .... .f dates""--+ --"""",!! Lexical Lexical constraints constraints wo r""'d--""""",---J " Script " "Script" word Figure 2: INSEG and OUTSEG architectures for word recognition. A crucial contribution of our system is the joint training of the neural network and the post-processor with respect to a single criterion that approximates word-level errors. We used the following discriminant criterion: minimize the total cost (sum of negative log-likelihoods) along the "correct" paths (the ones that yield the correct interpretations) , while minimizing the costs of all the paths (correct or not). The discriminant nature of this criterion can be shown with the following example. If Globally Trained Handwritten Word Recognizer 943 the cost of a path associated to the correct interpretation is much smaller than all other paths, then the criterion is very close to 0 and no gradient is back-propagated. On the other hand, if the lowest cost path yields an incorrect interpretation but differs from a path of correct interpretation on a sub-path, then very strong gradients will be propagated along that sub-path, whereas the other parts of the sequence will generate almost no gradient. \Vithin a probabilistic framework, this criterion corresponds to the maximizing the mutual information (MMI) between the observations and the correct interpretation. During global training, it is optimized using (enhanced) stochastic gradient descent with respect to all the parameters in the system, most notably the network weights. Experiments described in the next section have shown important reductions in error rates when training with this word-level criterion instead of just training the network separately for each character. Similar combinations of neural networks with HMMs or dynamic programming have been proposed in the past, for speech recognition problems (Bengio et al 92). 6 Experimental Results In a first set of experiments, we evaluated the generalization ability of the neural network classifier coupled with the word normalization preprocessing and AMAP input representation. All results are in writer independent mode (different writers in training and testing). Tests on a da tabase of isolated characters were performed separately on four types of characters: upper case (2.99% error on 9122 patterns), lower case (4.15% error on 8201 patterns), digits (1.4% error on 2938 patterns), and punctuation (4.3% error on 881 patterns). Experiments were performed with the network architecture described above. The second and third set of experiments concerned the recognition of lower case words (writer independent). The tests were performed on a database of 881 words. First we evaluated the improvements brought by the word normalization to the INSEG system. For the OUTSEG system we have to use a word normalization since the network sees a whole word at a time. With the INSEG system, and before doing any word-level training, we obtained without word normalization 7.3% and 3.5% word and character errors (adding insertions, deletions and substitutions) when the search was constrained within a 25461-word dictionary. When using the word normalization preprocessing instead of a character level normalization, error rates dropped to 4.6% and 2.0% for word and character errors respectively, i.e., a relative drop of 37% and 43% in word and character error respectively. In the third set of experiments, we measured the improvements obtained with the joint training of the neural network and the post-processor with the word-level criterion, in comparison to training based only on the errors performed at the character level. Training was performed with a database of 3500 lower case words. For the OUTSEG system, without any dictionary constraints, the error rates dropped from 38% and 12.4% word and character error to 26% and 8.2% respectively after word-level training, i.e., a relative drop of 32% and 34%. For the INSEG system and a slightly improved architecture, without any dictionary constraints, the error rates dropped from 22.5% and 8.5% word and character error to 17% and 6.3% respectively, i.e., a relative drop of 24.4% and 25.6%. With a 25461-word dictionary, errors dropped from 4.6% and 2.0% word and character errors to 3.2% and 1.4% respectively after word-level training, i.e., a relative drop of 30.4% and 30.0%. Finally, some further improvements can be obtained by drastically reducing the size of the dictionary to 350 words, yielding 1.6% and 0.94% word and character errors. 944 Bengio, Le Cun, and Henderson 7 Conclusion We have demonstrated a new approach to on-line handwritten word recognition that uses word or sentence-level preprocessing and normalization, image-like representations, Convolutional neural networks, word models, and global training using a highly discriminant word-level criterion. Excellent accuracy on various writer independent tasks were obtained with this combination. References Bengio, Y., R. De Mori and G. Flammia and R. Kompe. 1992. Global Optimization of a Neural Network-Hidden Markov Model Hybrid. IEEE Transactions on Neural Networks v.3, nb.2, pp.252-259. Burges, C., O. Matan, Y. Le Cun, J. Denker, L. Jackel, C. Stenard, C. Nohl and J. Ben. 1992. Shortest Path Segmentation: A Method for Training a Neural Network to Recognize character Strings. Proc. IJCNN'92 (Baltimore), pp. 165-172, v.3. Guyon, 1., Albrecht, P., Le Cun, Y., Denker, J. S., and Weissman, H. 1991 design of a neural network character recognizer for a touch terminal. Pattern Recognition, 24(2):105-119. Le Cun, Y. 1986. Learning Processes in an Asymmetric Threshold Network. In Bienenstock, E., Fogelman-Soulie, F., and Weisbuch, G., editors, Disordered systems and biological organization, pages 233-240, Les Houches, France. Springer-Verlag. Le Cun, Y. 1989. Generalization and Network Design Strategies. In Pfeifer, R., Schreter, Z., Fogelman, F., and Steels, L., editors, Connectionism in Perspective, Zurich, Switzerland. Elsevier. an extended version was published as a technical report of the University of Toronto. Le Cun, Y., Matan, 0., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W., Jackel, L. D., and Baird, H. S. 1990. Handwritten Zip Code Recognition with Multilayer Networks. In IAPR, editor, Proc. of the International Conference on Pattern Recognition, Atlantic City. IEEE. Matan, 0., Burges, C. J. C., LeCun, Y., and Denker, J. S. 1992. Multi-Digit Recognition Using a Space Displacement Neural Network. In Moody, J. M., Hanson, S. J., and Lippman, R. P., editors, Neural Information Processing Systems, volume 4. Morgan Kaufmann Publishers, San Mateo, CA. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. 1986. Learning internal representations by error propagation. In Parallel distributed processing: Explorations in the microstructure of cognition, volume I, pages 318-362. Bradford Books, Cambridge, MA. Schenkel, M., Guyon, I., Weissman, H., and Nohl, C. 1993. TDNN Solutions for Recognizing On-Line Natural Handwriting. In Advances in Neural Information Processing Systems 5. Morgan Kaufman. Tappert, C., Suen, C., and Wakahara, T. 1990. The state of the art in on-line handwriting recognition. IEEE Trans. PAM!, 12(8).
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Robust Reinforcement Learning Motion Planning Satinder P. Singh'" Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 singh@psyche.mit.edu • In Andrew G. Barto, Roderic Grupen, and Christopher Connolly Department of Computer Science University of Massachusetts Amherst, MA 01003 Abstract While exploring to find better solutions, an agent performing online reinforcement learning (RL) can perform worse than is acceptable. In some cases, exploration might have unsafe, or even catastrophic, results, often modeled in terms of reaching 'failure' states of the agent's environment. This paper presents a method that uses domain knowledge to reduce the number of failures during exploration. This method formulates the set of actions from which the RL agent composes a control policy to ensure that exploration is conducted in a policy space that excludes most of the unacceptable policies. The resulting action set has a more abstract relationship to the task being solved than is common in many applications of RL. Although the cost of this added safety is that learning may result in a suboptimal solution, we argue that this is an appropriate tradeoff in many problems. We illustrate this method in the domain of motion planning. "'This work was done while the first author was finishing his Ph.D in computer science at the University of Massachusetts, Amherst. 655 656 Singh, Barto, Grupen, and Connolly An agent using reinforcement learning (Sutton et al., 1991; Barto et al., to appear) (RL) to approximate solutions to optimal control problems has to search, or explore, to improve its policy for selecting actions. Although exploration does not directly affect performance (Moore & Atkeson, 1993) in off-line learning with a model of the environment, exploration in on-line learning can lead the agent to perform worse than is acceptable. In some cases, exploration might have unsafe, or even catastrophic, results, often modeled in terms of reaching 'failure' states of the agent's environment. To make on-line RL more practical, especially if it involves expensive hardware, task-specific minimal levels of performance should be ensured during learning, a topic not addressed by prior RL research. Although the need for exploration cannot be entirely removed, domain knowledge can sometimes be used to define the set of actions from which the RL agent composes a control policy so that exploration is conducted in a space that excludes most of the unacceptable policies. We illustrate this approach using a simulated dynamic mobile robot in two different environments. 1 Closed-loop policies as actions RL agents search for optimal policies in a solution space determined in part by the set of actions available to the agent. With a few exceptions (e.g., Mahadevan & Connell, 1990; Singh, 1992), researchers have formulated RL tasks with actions that are primitive in the sense that they are low-level, are available in very state, are executed open-loop, and last a single time-step. We propose that this is an arbitrary, and self-imposed, restriction, and that in general the set of actions can have a much more abstract relationship to the problem being solved. Specifically, what are considered 'actions' by the RL algorithm can themselves be closed-loop control policies that meet important subgoals of the task being solved. In this paper, the following general advantages afforded by using closed-loop policies as actions are demonstrated in the domain of motion planning: 1. It is possible to design actions to meet certain hard constraints so that RL maintains acceptable performance while simultaneously improving performance over time. 2. It is possible to design actions so that the action space for the learning problem has fewer dimensions than the actual dimension of the physical action space. The robustness and greatly accelerated learning resulting from the above factors can more than offset the cost of designing the actions. However, care has to be taken in defining the action space to ensure that the resulting policy space contains at least one policy that is close to optimal. 2 RL with Dirichlet and Neumann control policies The motion planning problem arises from the need to give an autonomous robot the ability to plan its own motion, i.e., to decide what actions to execute in order to achieve a task specified by initial and desired spatial arrangements of objects. Robust Reinforcement Learning in Motion Planning 657 First consider geometric path planning, i.e., the problem of finding safe paths for a robot with no dynamical constraints in an environment with stationary obstacles. A safe path in our context is one that avoids all obstacles and terminates in a desired configuration. Connolly (1992) has developed a method that generates safe paths by solving Laplace's equation in configuration space with boundary conditions determined by obstacle and goal configurations (also see, Connolly & Grupen, 1993). Laplace's equation is the partial differential equation n {j2ljJ V2ljJ L {)x~ = 0, (1) i=l I whose solution is a harmonic function, ljJ, with no interior local minima. In practice, a finite difference approximation to Equation 1 is solved numerically via Gauss Sidel relaxation on a mesh over configuration space. Safe paths are generated by gradient descent on the resulting approximate harmonic function. In the general motion planning problem, we are interested in finding control policies that not only keep the robot safe but also extremize some performance criterion, e.g., minimum time, minimum jerk, etc. Two types of boundary conditions, called Dirichlet and Neumann boundary conditions, give rise to two different harmonic functions, <I> D and <I> N, that generate different types of safe paths. Robots following paths generated from <I> D tend to be repelled perpendicularly from obstacles. In contrast, robots following paths generated from <I>N tend to skirt obstacles by moving parallel to their boundaries. Although the state space in the motion planning problem for a dynamic robot in a planar environment is {x, x, y, if}, harmonic functions are derived in two-dimensional position space. These functions are inexpensive to compute (relative to the expense of solving the optimal control problem) because they are independent of the robot dynamics and criterion of optimal control. The closed-loop control policies that follow the gradient of the Dirichlet or Neumann solutions, respectively denoted 1rD and 1rN, are defined approximately as follows: 1rD(S) = V<I>D(§), and 1rN(S) = V<I>N(§), where § is the projection of the state s onto position space.1 Instead of formulating the motion planning problem as a RL task in which a control policy maps states into primitive control actions, consider the formulation in which a policy maps each state s to a mixing parameter k( s) so that the actual action is : [1- k(S)]1rD(S) + [k(S)]1rN(S) , where 0 ~ k(s) ~ 1. Figure 1B shows the structure of this kind of policy. In Appendix B, we present conditions guaranteeing that for a robot with no dynamical constraints, this policy space contains only acceptable policies, i.e., policies that cause the robot to reach the goal configuration without contacting an obstacle. Although this guarantee does not strictly hold when the robot has dynamical constraints, e.g., when there are bounds on acceleration, this formulation still reduces the risk of unacceptable behavior. 3 Simulation Results In this paper we present a brief summary of simulation results for the two environments shown in Figures 2A and 3A. See Singh (1993) for details. The first 1 In practice, the gradients of the harmonic functions act as reference signals to a PDcontroller. 658 Singh, Barto, Grupen, and Connolly environment consists of two rooms connected by a corridor. The second environment is a horseshoe-shaped corridor. The mobile robot is simulated as a unit-mass that can accelerate in any direction. The only dynamical constraint is a bound on the maximum acceleration. A Q(state. action) • • X X Y Y state (s) k mixing coefficient B State Policy (s) (s) Policy 1 k(s) 1 - k(s) PoliCy 2 Neumann (s) Figure 1: Q-Iearning Network and Policy Structure. Panel A: 2-layer Connectionist Network Used to Store Q-values. Network inversion was used to find the maximum Q-value (Equation 2) at any state and the associated greedy action. The hidden layer consists of radial-basis units. Panel B: Policy Structure. The agent has to learn a mapping from state s to a mixing coefficient 0 < k( s) < 1 that determines the proportion in which to mix the actions specifies by the pure Dirichlet and Neumann policies. The learning task is to approximate minimum time paths from every point inside the environment to the goal region without contacting the boundary wall. A reinforcement learning algorithm called Q-Iearning (Watkins, 1989) (see Appendix A) was used to learn the mixing function, k. Figure lA shows the 2-layer neural network architecture used to store the Q-values. The robot was trained in a series of trials, each trial starting with the robot placed at a randomly chosen state and ending when the robot enters the goal region. The points marked by stars in Figures 2A and 3A were the starting locations for which statistics were collected to produce learning curves. Figures 2B, 2C, 3A and 3B show three robot trajectories from two randomly chosen start states; the black-filled circles mark the Dirichlet trajectory (labeled D), the white-filled circles mark the Neumann trajectory (labeled N), and the grey-filled circles mark the trajectory after learning (labeled Q). Trajectories are shown by taking snapshots of the robot at every time step; the velocity of the robot can be judged by the spacing between successive circles on the trajectory. Figure 2D shows the mixing function for zero-velocity states in the two-room environment, while Figure 3C shows the mixing function for zero velocity states in the horseshoe environment. The darker the region, the higher the proportion of the Neumann Robust Reinforcement Learning in Motion Planning 659 policy in the mixture. In the two-room environment, t.he agent learns to follow the Neumann policy in the left-hand room and to follow the Dirichlet policy in the right-hand room. Figure 2E shows the average time to reach the goal region as a function of the number of trials in the two-room environment. The solid-line curve shows the performance of the Q-Iearning algorithm. The horizontal lines show the average time to reach the goal region for the designated pure policies. Figure 3D similarly presents the results for the horseshoe environment. Note that in both cases the RL agent learns a policy that is better than either the pure Dirichlet or the pure Neumann policies. The relative advantage of the learned policy is greater in the two-room environment than in the horseshoe environment. On the two-room environment we also compared Q-Iearning using harmonic functions, as described above, with Q-Iearning using primitive accelerations of the mobile robot as actions. The results are summarized along three dimensions: a) speed of learning: the latter system took more than 20,000 trials to achieve the same level of performance achieved by the former in 100 trials, b) safety: the simulated robot using the latter system crashed into the walls more than 200 times, and c) asymptotic performance: the final solution found by the latter system was 6% better than the one found by the former. 4 Conclusion Our simulations show how an RL system is capable of maintaining acceptable performance while simultaneously improving performance over time. A secondary motivation for this work was to correct the erroneous impression that the proper, if not the only, way to formulate RL problems is with low-level actions. Experience on large problems formulated in this fashion has contributed to the perception that RL algorithms are hopelessly slow for real-world applications. We suggest that a more appropriate way to apply RL is as a "component technology" that uses experience to improve on partial solutions that have already been found through either analytical techniques or the cumulative experience and intuitions of the researchers themselves. The RL framework is more abstract, and hence more flexible, than most current applications of RL would lead one to believe. Future applications of RL should more fully exploit the flexibility of the RL framework. A Q-learning On executing action a in state St at time t, the following update on the Q-value function is performed: where R( St, a) is the payoff, 0 ::; I ::; 1 is the discount factor, and a is a learning rate parameter. See Watkins (1989) for further details. 660 Singh, Barto, Grupen, and Connolly A * * * * * * * ------------------. GOAl . B c * * * ~ CJ .&:. CJ m ex: 0 Q) E i= Q) C) ca .... Q) ~ 7000 E 8000 5000 _0 300 0 2000 00 0 Q-Iearning Neumann Dirichlet 9000 18000 27000 3eOOO Number of Trials 45000 Figure 2: Results for the Two-Room Environment. Panel A: Two-Room Environment. The stars mark the starting locations for which statistics were computed. Panel B: Sample Trajectories from one Starting Location. The black-filled circles labeled D show a pure Dirichlet trajectory, the white-filled circles labeled N show a pure Neumann trajectory, and the grey-filled circles labeled Q show the trajectory after learning. The trajectories are shown by taking snapshots at every time step; the velocity of the robot can be judged by the distance between successive points on the trajectory. Panel C: Three Sample Trajectories from a Different Starting Location. Panel D: Mixing Function Learned by the Q-Iearning Network for Zero Velocity States. The darker the region the higher the proportion of the Neumann policy in the resulting mixture. Panel E: Learning Curve. The curve plots the time taken by the robot to reach the goal region, averaged over the locations marked with stars in Panel A, as a function of the number of Q-Iearning trials. The dashed line shows the average time using the pure Neumann policy; the dotted line for the pure Dirichlet policy; and the solid line for Q-Iearning. The mixed policy formed by Q-Iearning rapidly outperforms both pure harmonic function policies. A * * * * * B * • • .......... 1... .. G • N : ) • D 8 o · : ____ ••• __ A. __ • A.._ GOAL Robust Reinforcement Learning in Motion Planning 661 20000 ~ 1&000 <!J s= m a::: '0000 $2 Q) E ~ Q) ~ Q) ~ • • ••• • M •• • • M •••• _ •••• M . .. . ....... . ..... _ ..... . ........ , ••• • • • ••••••• _ .................... . . _ . . ..................... . .... _ • • •• M D Q-Iearning Neumann Dirichlet ------------------------_. ooo~--------~,oooo~------~~~-------~~------~ Number of_Trials Figure 3: Results for the Horseshoe Environment. Panel A: Horseshoe-Shaped Environment. Locations marked with stars are the starting locations for which statistics were computed. It also shows sample trajectories from one starting location; the black-filled circles marked D show a Dirichlet trajectory, the white-filled circles marked N show a Neumann trajectory, and the grey-filled circles marked Q show the trajectory after learning. The trajectories are shown by taking snapshots at every time step; the velocity of the robot can be judged by the distance between successive points on the trajectory. Panel B: Three Sample Trajectories from a Different Starting Location. Panel C: Mixing Function Learned by the Q-Iearning Network for Zero Velocity States. The darker the region the higher the proportion of the Neumann policy in the resulting mixture. Panel D: Learning Curve. The curve plots the time taken by the robot to reach the goal region, averaged over the locations marked with stars in Panel A, as a function of the number of Q-Iearning trials. The dashed line shows the average time for the pure Neumann policy; the dotted line for the pure Dirichlet policy; and the solid line for Q-Iearning. Q-Iearning rapidly outperforms both pure harmonic function policies. 662 Singh, Barto, Grupen, and Connolly B Safety Let L denote the surface whose gradients at any point are given by the closed-loop policy under consideration. Then for there to be no minima in L, the gradient of L should not vanish in the workspace, i.e., (1- k(S»\7<1>D(S) + k(S)\7<1>N(S) ;/; O. The only way it can vanish is if 'Vi k(s) 1- k(s) (3) where [·Ji is the ith component of vector [.J. The algorithm can explicitly check for that possibility and prevent it. Alternatively, note that due to the finite precision in any practical implementation, it is extremely unlikely that Equation 3 will hold in any state. Also note that 7r( s) for any point s on the boundary will always point away from the boundary because it is the convex sum of two vectors, one of which is normal to the boundary, and the other of which is parallel to the boundary. Acknowledgements This work was supported by a grant ECS-9214866 to Prof. A. G. Barto from the National Science Foundation, and by grants IRI-9116297, IRI-9208920, and CDA8922572 to Prof. R. Grupen from the National Science Foundation. References Barto, A.G., Bradtke, S.J., & Singh, S.P. (to appear). Learning to act using realtime dynamic programming. Artificial Intelligence. Connolly, C. (1992). Applications of harmonic functions to robotics. In The 1992 International Symposium on Intelligent Control. IEEE. Connolly, C. & Grupen, R. (1993). On the applications of harmonic functions to robotics. Journal of Robotic Systems, 10(7), 931-946. Mahadevan, S. & Connell, J. (1990). Automatic programming of behavior-based robots using reinforcement learning. Technical report, IBM Research Division, T.J.Watson Research Center, Yorktown Heights, NY. Moore, A.W. & Atkeson, C.G. (1993). Prioritized sweeping: Reinforcement learning with less data and less real time. Machine Learning, 13(1). Singh, S.P. (1992). Transfer of learning by composing solutions for elemental sequential tasks. Machine Learning, 8(3/4), 323-339. Singh, S.P. (1993). Learning to Solve Markovian Decision Processes. PhD thesis, Department of Computer Science, University of Massachusetts. also, CMPSCI Technical Report 93-77. Sutton, R.S., Barto, A.G., & Williams, R.J. (1991). Reinforcement learning is direct adaptive optimal control. In Proceedings of the American Control Conference, pages 2143-2146, Boston, MA. Watkins, C.J.C.H. (1989). Learning from Delayed Rewards. PhD thesis, Cambridge Univ., Cambridge, England.
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Optimal Stopping and Effective Machine Complexity in Learning Changfeng Wang Department of SystE'IIlS Sci. (Iud Ell/!,. Salltosh S. Venkatesh Dp»artn}(,llt (If Elf'drical EugiJlPprinJ!, U IIi v('rsi ty (If Ppnllsyl va nia Philadelphia, PA, U.S.A. 19104 UJliversity of PPIIIlsylv1I.Ili(l Philadelphin, PA, U.S.A. I!HlJ4 J. Stephen Judd Siemens Corporate Research 755 College Rd. East, Princeton, NJ, U.S.A. 08540 Abstract We study tltt' problem of when to stop If'arning a class of feedforward networks - networks with linear outputs I1PUrOIl and fixed input weights - when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there a.re in general three distinct phases in the generalization performance in the learning process, and in particular, the network has hetter gt'neralization pPTformance when learning is stopped at a certain time before til(' global miniIl111lu of the empirical error is reachert. A notion of effective size of a machine is rtefil1e<i and used to explain the trade-off betwf'en the complexity of the marhine and the training error ill the learning process. The study leads nat.urally to a network size selection critt'rion, which turns Ol1t to be a generalization of Akaike's Information Criterioll for the It'arning process. It if; shown that stopping Iparning before tiJt' global minimum of the empirical error has the effect of network size splectioll. 1 INTRODUCTION The primary goal of learning in neural nets is to find a network that gives valid generalization. In achieving this goal, a central issue is the trade-off between the training error and network complexity. This usually reduces to a problem of network size selection, which has drawn much research effort in recent years. Various principles, theories, and intuitions, including Occam's razor, statistical model selection criteria such as Akaike's Information Criterion (AIC) [11 and many others [5, 1, 10,3,111 all quantitatively support the following PAC prescription: between two machines which have the same empirical error, the machine with smaller VC-dimf'nsion generalizes better. However, it is noted that these methods or criteria do not npcpssarily If'ad to optimal (or llearly optimal) generalization performance. Furthermore, all of these m<.'thods are valid only at th~ global minimum of thf' empirical error function (e.g, the likelihood function for AIC), and it is not clear by these methods how the generalization error is f'ffected by network complexity or, more generally, how a network generalizes during the learning process. This papPI acldrf'f;sPs these issues. 303 304 Wang, Venkatesh, and Judd Recently, it has often been observed that when a network is 'trained by a gradient descent algorithm, there exists a critical region in the training epochs where the trained network generalizes best, and after that region the generalization error will increase (frequently called over-training). Our numerical experiments with gradient-type algorithms in training feedforward networks also indicate that in this critical region, as long as the network is large enougb to learn the examples, the size of the network plays little role in the (hest) generalization performance of the network. Does this mean we must revise Occam's principle? How should one define the complexity of a network and go about tuning it to optimize geIlNalization performance? When should one stop learning? Although relevant learning processes wen' treatccJ by nUlll<'TOIIS authors [2, 6, 7, 4], the formal theoretical studies of these problems are abeyant. Under rather general regularity conditions (Section 1), we give in Section 2 a theorem which relates the generalization error at each epoch of learning to that at the global minimum of the training error. Its consequence is that for any linear machine whose VC-dimension is finite but large enough to learn the target concept, the number of iterations needed for the best generalization to occur is at the order of the logarithm of the sample size, rather than at the global minimum of the training error; it also provides bounds on the improvement expected. Section 3 deals with the relation between the size of the machine and generalization error by appealing to the concept of effective size. Section 4 concerns the application of these results to the problem of network size selection, where the AIC is generalized to cover the time evolution of the learning process. Finally, we conclude the paper with comments on practical implementation and further research in this direction. 2 THE LEARNING MACHINE The machine we considf'f acc.epts input v('ctors X from an arbitrary input space and produc('s scalar outputs d Y = 2: 1./;,(X)n', + € = 1/J(X)'o:* + €. (1) .=1 Here, 0:* = (0:*1, . . . ,0:' d)' is a fixed vect.or of real weights, for eac.h i, 1./;,(X) is a fixed real fUBction of the inpnts, with 1/J(X) = (1/JI (X), . . . ,t/Jd(X)), the corresponding vedor of functions, and ~ is a random noise term. The machine (1) can be thought of as a feedforward nenral network with a fixed front end and variable weights at the output. In particular, the functions 1/J; can represent fixed polynomials (higher-order or sigma-pi neural networks), radial basis functions with fixed centers, a fixed hidden-layer of sigmoidal neurons, or simply a linear map. In this context, N. J. Nilsson [8) has called similar structures cI>-machines. We consider the problem of learning from examples a relationship between a random variable Y and an n-dimensional random vector X. We assume that this function is given by (1) for some fixed integer d, the random vector X and random variable ~ are defined on the same probability space, that E [~IX) = 0, and (12(X) = Var{€lX) = constant < 00 almost surely. The smallest eigenvalue of the matrix 1fJ(x)1fJ(x ) is assumed to be bounded from below by the inverse of some square integrable function. Note that it can be shown that the VC-dirnension of the class of cI>-machines with d neurons is d under the last assumption. The learning-theoretic properties of the system will be determined largely by the eigen structure of cI>. Accordingly, let >'1 ~ >'2 ~ ... ~ >'d denote the eigenvalues of cI>. The goal of the learning is to finei the true concept 0: given independently drawn examples (X, y) from (1). Given any hypothesis (vector) W = (WI, ... ,Wd)' for consideration as an approximation to the true concept 0:, the performance measure we use is the mean-square prediction (or ensemble) error £(W) = E (Y -1/J(X)'w( (2) Note that the true concept 0:* is the mean-square solution 0:* = argmin£(w) = cI>-IE (1/J(X)y), tv (3) Optimal Stopping and Effective Machine Complexity in Learning 305 and the minimum predict.ion error is given by £(0) = lllinw E.(w) = u'l. Let 11. be the nUmbE'f of samples of (X,} -). WE' assume that an independent, ic\entkally distributed sample (X(1),y(J), ... , (x(n),y(n), generated according to the joint distribution of (X, Y) induced by (1), is provided to thE:' IE:'arner. To simplify notation, define thE' matrix 'It == [",,(X(l) . . . ""(X(",) ) and the corresponding vector of outputR y = (y(l), . . . , y(n))'. In analogy with (2) define the empirical error 011 the sample by Let a denote the hypothesis vector for which t.he empirical error 011 the sample is minimized: 'Vw£(o) = O. Analogously with (3) we can thell show that (4) where cj, = t-'It 'It' is the empirical covariallre matrix, whirh is almost surely nonsingular for large n. The terms in (4) are the empirical counterparts of the ensemble averages in (3). The gradient descent algorithm is givf'n by: (5) where 0 = (01,02, . . . ,03 )" t is the number of iterations, and € is the rate of learning. From this we can get a, = (I ~(t»o + ~(t.)oo, (6) where ~(t) = (I - €ci»t, clIld 00 is the initial weight vector. The limit of Ot is n when t. goes to infinity, provided ci> is positive definite and the learning rate € is small enough (Le., smaller than the smallest eigenvalue of ci». This implies that the gradient descent algorithm converges to the least squarE'S solution, starting from any point in Rn. 3 GENERALIZATION DYNAMICS AND STOPPING TIME 3.1 MAIN THEOREM OF GENERALIZATION DYNAMICS Even if the true concept (i.e., the precise relation between Y and X in the current problem) is in the class of models we consider, it is usually hopeless to find it using only a finite number of examples, except in some trivial cases. Our goal is hf'nce less ambitious; we seek to find the best approximation of the true concept, the approach entailing a minimization of the training or empirical error, and then taking the global minimum of the empirical error a as the approximation. As we have seen the procedure is unbiased and consistent. Does this then imply that training should always be carried out to the limit? Surprisingly, the answer is 110. This assertion follows from the next theorem. Theorem 3.1 Let Mn > 0 be an arbitrary f'eal constant (possibly depending on 11.), and suppose a.~s1tm.ptions Al to A.'I af'C satisfied; then the rwnrralizatioll dynamics in the training process are gOllerned "y the follolllinq rquatioll.: uniformly for all initial weight ver.iors, no in the d-dim.ensional ball {n* + 8 : 11811 ~ M n , 8 E R d }, and for all t > O. 0 306 Wang, Venkatesh, and Judd 3.2 THREE PHASES IN GENERALIZATION By Theorem 3.1, the mean gelleralil':ation <'nor at each epoch of the t.raining proc£'ss is characterized by the following function: ¢J(t) == t ['\,8~(1 - f'\,)2' 202 (1 - €,\;)t[1- ~(1 - f,\;)'I] . . n 2 ,=1 The analysis of the evolution of generalization with training is facilitated by treating ¢J(.) as a function of a continuotls tinlf' parameter f. \Ve will show that. there are three distinct phases in generalization dynamics. These results are givpn in the following in form by several corollaries of Theorem 3.1. Without loss of geIH'rality, w<' assnllJ(' th<' init.ial w<'ight ve-ctm is pickerl 11)1 in a region with 11811 ~ Mn = 0(11°), and in particular, 181 = O(I/ n ). L<'t 1", = 11I(l/1:.~·.x·r1)f, the-II for all 0 S; t < f, _ 2111(1;'t'1~ • .x.r1)' we have 0::; 7', < ~,and thus d :.I tI L :.I 2' 1 1 20- L ' '\i"; (1 - f,\r/) = O( --2 »> 0(-13 -. ) = (1 - f'\;) . "T, n'" n .=-1 ;= 1 The quantity 8; (1 - €A;) 2t in the fi rst term of t he above inequalities is related to the elimination of initial error, and can be defined as the approximation error (or fitting error); the last term is related to the effective complexity of the network at t (in fact, an order O( ~) shift of the complexity (,rror). The definition and observations here will be discussed in more detail ill the next section. We call the learning process during the time interval 0 ~ t S; tl the first phase of learning. Siuce ill this interval ¢J(t) = 0(,,-2,·,) is n lIlollutollkally df'Cfeasillg function of i, the g('neralil':atiou error decreases monotonically ill the first phase of It'arning. At the end of first phase of learning ¢J(tl) = O( ~), therefore the generalization error is £(nt,) = [(nco) + O( ~). As a snmmary of these statements we have the following mroJlary. Corollary 3.2 In the first phase of learning, the complexity error is dominated by the approximation error, and within an order of O( ~) I the generalization eTTor decrea.5es monotonically in the lrarnin.q process to £ (noo) + O( ~) at the end of first pha.~e. 0 For f > t 1 , we can show by Thp.orem 3.1 t ha t t It(' g<'llcralizatioll dynamics is given by thp. following equation, where 8" == n(tl) n~, 2 cI [ ] 20, 1 2 , _1 £a(at,+t) = £a(ao) - L(l - f'\i) 1 - - (1 + Pi) (1 - f'\i) + O(n 2), n 2 ,=1 when~ p~ == ,\j8;(tl )n./0-2 , which is, with probahility a.pproaching one, of ordPf O(nO). Without causing confusion, WP. stillnse ¢J(-) for the new time-varying part of the gf'neralization error. The function ¢J(.) has much more complex behavior after tl than in the first phase of learninr;. As we will see, it decreases for some time, and finally begins to increase again. In particular, we found the best generalization at tha.t t where ¢J( t) is minimized. (It is noted that 8tl is a random variable now, and the following statements of the generalization dynamics are ill the sense of with probability approaching one as n -+ 00.) Define the optimal stopping tim<': f",ill == argmin{£(a,) : t E [D,oo!}, i.e., the epoch corresponding to the smallest gPllPralization Pfror. Then we can prove the following corollaries: Corollary 3.3 The optimal stoppin.q time t",ill = O(ln 71.), p1'Ovided 0-2 > D. In particular, the following inequalities hold: 2 2 ., . In(l+p,) d In(I+Pj) b h 1. tt S; tmin ~ tl/, 111 tere tf = t\ + nlIn, In(I/[1 -,.x,I) an ttL = tl + max, 111(1/[1-,,\.)) are ot finite real numbers. Th at is, the smallest generalization occurs before the global minimum of the empirical err07' is 1·eachcd. Optimal Stopping and Effective Machine Complexity in Learning 307 2. ¢C) (tmcking the gencmlizntio7! eTTor) decreases monotonically for t < tf and increases monotonically tn zero for t > tu; fuf'thermore, tmin is unique if tt + In 2 :x > tu. In(I/[I-< I» 3. _",2 "d 1 --L,., < ¢(tmin) < _",2 -2!L[...h.~d ]1' where'" = 11l(1-<~I) and (12 _ "d p2 ,. 0t= H.pi n 1+1' 1'+1 +p' 'In(I-( d)' 0t=1 i' In accordance with our earlier definitions, We' call the learning proeess during the time intl'rval between tl and t" til(' s('cond pitas(' of l('aruinl1;; and the rest of timl' til(' third phasf' of learning. According to Corollary 3.3, for t > tlL sufficiently large, the gell('ralization error is uniformly better than at the globalminimuIn, a, of the empirieal error, although minimum generalization error is achieved betwel'n t f and tu. The generalization error is redllced hy at least. ",2 -2!L [...h.+ AJ1' . ,. 1+1' l' I n+p over that for a if we stop training at. a prop<'f time. For a fixed nUlnlwr of it'aming examples, the larger is the ratio d/lI, the larger is til(' improvement in generalization error if the algorithm is stopped before the glohal minimum n° is reariwd. 4 THE EFFECTIVE SIZE OF THE MACHINE Our concentration on dynamics and our seeming disregard for complexity do not conflict with the learning-theoretic focus on VC-dimension; in fact, the two attitudes fit nicely together. This section explains the generalization dynamics by introducing the the concept of effective complexity of the machine. It is argued that early stopping in drect sets the l'ffective size of the network to a value smaller than its VC-dimension. The effective size of the machine at time t is defined to be d(t) == L~=1 [1 - (1 - d.,)fJ2, which increases monotonically to d, the VC-dimensioll of the network, as t -+ 00. This definition is justified after the following theorem: Theorem 4.1 Under the a.5sumptions of Them'em 3.1, the following equation holds uniformly for nil no such that 1151 ~ 111 n, In the limit of learning, we have by letting t -+ 00 in the above equation, 2 £(a) =£(a*)+ ~d+O(n-~) n (7) o (8) Hence, to an order of O(n-1.5), the generalization error at the limit of training breaks into two parts: the approximation error £(0 0 ), and the complexity error ~0'2 . Clearly, the latter is proportional to d, the VC-dimension of the network. For all d's larger than necessary, £(a*) remains a constant, and the generalization error is determined solely by ~. The term £(a.,t) differs from £(0*) only in terms of initial error, and is identified to be the approximation error at t. Comparison of the above 2 two equations thus shows that it is reasonable to define ':. d(t) as the complexity error at t, and justifies the definition of d(t) as the effective size of the machine at the same time. The quantity d(t) captures the notion of the degree to which the capacity of the machine is used at t. It depends on the machine parameters, the a.lgorithm being IIsed, and the marginal distribution of X. Thus, we see from (7) that the generalization error at epoch t falls into the same two parts as it does at the limit: the approximation error (fitting error) and the complexity error (determined by the effective size of the machine). As we have show in the last section, during the first phase of learning, the complexity error is of higher order in n compared to the fitting error during the first phase of learning, if the initial error is of order O(nO) or largN. Thus derrpase of til(' fitting error (which is proportional to the training error, as we will see in the next section) illl plies the decrpase of the generalization error. However, 308 Wang, Venkatesh, and Judd when the fitting error is brought down to the order O( ~), thE' decreas~ of fitting error will no longer imply th~ decreasE' of the' genc>rali?:ation error. In fact, by the ahoVf' t.heorem , the generali?:ation error at t + tl can be written as The fitting error and the complexity error compete at order O( ~) during the second phase oflearning. After the second the phase of icarning, th(' complexity error dominates the fitting error, still at tilE' order of O( ~) . Furthermore, if we define K == 1 ~. [#I d~lp2 J', then by the above equation and (3.3), we have Corollary 4.2 At the optimal 8topping time, flip following u1J11er bound (m the generalization error holds, Since K is a quantity of order 0(71,°), (1 - K)d is strictly smaller than d. Thus stopping training at tmin has the same effed as using a smaller machine of size less than (1 - K)d and carrying training out to the limit! A more detailed analysis reveals how the effective size of the machine is affected by each neuron in thE' learning process (omitted dne to the space limit). REMARK: The concept of effE'ctive size of the machine can be defined similarly for an arbitrary starting point. However, to compare the degree to which the capacity of the machine has been used at t, one must specify at what distance between the hypothesis a and the truth o' is such comparison started. While each point in the d-diuwnsional Euclidean space can be rega.rded as a hypothesis (machine) about 0*, it is intuitively dear that earh of these machines has a different capacity to approximate. it. But it is r('asonable to think that all of the machines that a.re 011 the same sphere {a : 10 - 0*1 = r}, for each ,. > 0, haW' the same capacity in approximating 0*. Thus, to compare the capacity being llsed at t, we mllst specify a sl)('cifk sphere as the starting point; defining the effective size of the marhillc at t withont spedfying the starting sphere is clearly meallingless. As we have seen, r ~ 7; is found to be a good choice for our purposes. 5 NETWORK SIZE SELECTION The next theorem relates the generalization error and training error at E'ach epoch of learning, and forms the basis for choosing the optimal stopping time as well as the best size of the machine during the learning process. In the limit of the learning process, the criterion reduces to the well-known Akaike Information Criterion (AIC) for statistical model selection. Comparison of the two criteria reveals that our criterion will result in better generalization than AIC, since it incorporates the information of each individual neuron rather than just the total number of neurons as in the Ale. Theorem 5.1 A.9suming the learning algorithm converges, and the conditions of Theorem 3.1 are satisfied; then the following equation holds: £ ( (t,) = (1 + () ( ] » E £ 1I ( 0, ) + r( d, t) + 0 ( ~ ) IIIhr.rr r(d, t) = 2~~_ 2:7-1 [J -- (1 -- rAj)'1 (9) o A(~('ording to this th('orl'lIl, We' find an M;.YIIlJllotically unbiased estimate of £(u,) to ht' £,,(0,) + C(d, t) when (J"2 is known. This results in the following criterion for finding the optimal stopping time and network size: min{£n(at) + C(d, t) : d, t = 1,2, .. . } (10) Optimal Stopping and Effective Machine Complexity in Learning 309 When t goes to infinity, the above criterion becomes: min{£,,(&) + 2(12d : d = 1,2, . . . } n (11) which is the AIC for choosing the b!:'st siz!:' of networks. Therefore, (10) can be viewed as an extension of the AIC to the learning process. To understand the differences, consider the case when ~ has standard normal distribution N(O, (12). Under this assumption, the Maximum Likelihood (ML) estimation of the weiglJt vectors is the saine as the Mean Square estimation. The AIC was obtained by minimizing E !::~:: i~l, the K ullback-Leibler distance of the density function f 0 M L (X) with aML being the ML estimation of n and that of the true density 10' This is equivalent. to minimizing Iimt--+ooE(Y - lo,(X))2 = E(Y - fOML(X))2 (assuming the limit and the expectation are interchangeable). Now it is dear that while AIC chooses networks only at the limit of learning, (10) does this in the whole learning procef1s. Observe that the matrix 4' is now exactly the Fisher Information Matrix of the density function f.,(X), and Ai is a measure of the capacity of 'ljJi in fitting the relation b!:'tween X and Y. Therefore Hllr criterion incorporates the information about each specific neuron provided by the Fisher Information Matrix, which is a measure of how well the data fit the model. This implies that there are two aspects in finding the trade-off between the model complexity and the empirical error in order to minimize the generalization error: one is to have the smallest number of neurons and the other is to minimize the utilization of each neuron. The AIC (and in fact most statistical model selection criteria) are aimed at the former, while our criterion incorporates the two aspects at the same time. We have seen in the earlier discussions that for a given number of neurons, this is done by using the capacit.y of each neuron in fitting the data only to the degree 1 - (1 - fA,)t",;" rather than to its limit. 6 CONCLUDING REMARKS To the best of our knowledge, the results described in this paper provide for the first time a precise language to describe overtraining phenomena in [('arning machin!:'s such as neural networks. We have studied formally the generalization process of a linear machine when it is trained with a gradient descent algorithm. The concept of effective size of a machine was introduced to break the generalization error into two parts: t.he approximation error and the error caused by a complexity term which is proportional to effective size; the former decreases monotonically and the later increases monotonically in the learning proress. When the machine is trained on a finite number of examples, there are in general three distinct phases of l!:'arning according to the relative magnitude of the fitting and complexity errors. In particular, there exists an optimal stopping time tmin = O(lnn) for minimizing generalization error which occurs before the global minimum of the empirical error is reached. These results lead to a generalization of the AIC in which the effect of certain network parameters and time of learning are together taken into account in the network size selection process. For practical application of neural networks, these results demonstrate that training a network to its limits is not desirable. From the learning-theoretic- point of view, the concept of effective dimension of a network t!:'Us us that we need more than thp VC-dimension of a machine to describe the generalization properties of a machine, excppt in the limit of learning. The generalization of the AIC reveals some unknown factf1 ill statistical model selection theory: namely, the generalization error of a network is affeded not only by the number of parameters but also by the degree to which each parametf'r is act.ually used in the learning process. Occam's principle therefore stands in a subtler form: Make minimal ILse of the ca.pacity of a network for encoding the information provided by learning samples. Our results hold for weaker assumptions than were made herein about the distributions of X and~. The case of machines that have vector (rather than scalar) outputs is a simple generalization. Also, our theorems have recently been generalized to the case of general nonlinear machines and are not restricted to the squared error loss function. While the problem of inferring a rule from the observational data has been studied for a long time in learning theory as well as in other context sHch (IS in Linear and Nonlinear Regression, the 310 Wang, Venkatesh, and Judd study of the problem as a dynamical process seems to open a new ave~ue for looking at the problem. Many problems are open. For example, it is interesting to know what could be learned from a finite number of examples in a finite number of itf'rations in the case where the size of the machine is not small compared to the sample size. Acknowledgments C. Wang thanks Siemens Corporate Research for slIpport during the summer of 1992 whE'n t.his research was initiated. Thp work of C. Wang aud S. Venkatesh has bf'en supported in part by thp Air Force Office of Srif'lIt.ific Rpsparrh unrler grant. F49620-93-1-0120. References [1) Akaike, H. (1974) Informat.ion theory and an extension of the maximum likelihood principle. Second International Sym.lJosimn on Information Theory, Ed. B.N. Krishnaiah, North Holland, Amsterdam, 27-4l. [2) Baldi, P. and Y. Chauvin (1991) Temporal evolution of generalization during learning in linear networks. Neural Communication. 3,589-603. [3] Chow, G. C. (1981) A comparison of the information and posterior probability criteria for model selection. Journal of Econometrics 16, 21-34. (4) Hansen, Lars Ka.i (1993) Stochastic linear learning: E'xact test and training error averages. Neural Network.~, 4, 393-396. [5] Haussler, D. (1989) Decision theoretical gE'neralization of the PAC model for neural networks and other learning applications. Preprillt. (6) Heskes, Tom M. and Bert Kappen (1991) Learning processes in neural networks. Physical Review A, Vol 44, No.4, 2718- 2726. [7) Kroght, Anders and John A. Herts Generalization in a linear percept ron in the presence of noise. Preprint. [8) Nilsson, N. J. Learning Machine.5. New York: McGraw Hill. (9) Pinelis, I., and S. Utev (1984) Estimates of moments of SUms of independent random variables. Theory of Probability and It.5 Application.5. 29 (1984) 574-577. [10] Rissanen, J. (1987) Stochastic complE'xity. J. Royal Stati.5tical Society. Series B, Vol. 49, No. 3, 223-265. [l1J Schwartz, G. (1978) Estimating till' dimellsion of a model. Annals of Stati.stic.9 6, 461-464. [12] Sazonov, V. (1982). On the accuracy of normal approximation. Journal of multivariate analysis. 12, 371-384. [13] Senatov, V. (1980) Uniform estimates of the rate of convergence in the multi-dimensional central limit theorem. Theory of Probability and Its Applications. 25 (1980) 745-758. [14] Vapnik, V. (1992) Measuring the capacity of learning machines (I). Preprint. [15) Weigend, S.A. and Rllmelhart (1991). Generalization through minimal networks with application to forcasting. INTERFACE'91-23rd Symposium on the Interface: Computing Science and Statistics, ed. E. M., Keramidas, pp362-370. Interface Foundation of North America.
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Exploiting Chaos to Control the Future Gary W. Flake* Guo-Zhen Sunt Yee-Chun Leet Hsing-Hen Chent Institute for Advance Computer Studies University of Maryland College Park, MD 20742 Abstract Recently, Ott, Grebogi and Yorke (OGY) [6] found an effective method to control chaotic systems to unstable fixed points by using only small control forces; however, OGY's method is based on and limited to a linear theory and requires considerable knowledge of the dynamics of the system to be controlled. In this paper we use two radial basis function networks: one as a model of an unknown plant and the other as the controller. The controller is trained with a recurrent learning algorithm to minimize a novel objective function such that the controller can locate an unstable fixed point and drive the system into the fixed point with no a priori knowledge of the system dynamics. Our results indicate that the neural controller offers many advantages over OGY's technique. 1 Introduction Recently, Ott, Grebogi and Yorke (OGY) [6] proposed a simple but very good idea. Since any small perturbation can cause a large change in a chaotic trajectory, it is possible to use a very small control force to achieve a large trajectory modification. Moreover, due to the ergodicity of chaotic motion, any state in a chaotic *Department of Computer Science, peyote@umiacs.umd.edu tLaboratory for Plasma Research 647 648 Flake, Sun, Lee, and Chen attractor can be reached by a small control force. Since OGY published their work, several experiments and simulations have proven the usefulness of OGY's method. One prominent application of OGY's method is the prospect of controlling cardiac chaos [1]. We note that there are several unfavorable constraints on OGY's method. First, it requires a priori knowledge of the system dynamics, that is, the location of fixed points. Second, due to the limitation of linear theory, it will not work in the presence of large noise or when the control force is as large as beyond the linear region from which the control law was constructed. Third, although the ergodicity theory guarantees that any state after moving away from the desired fixed point will eventually return to its linear vicinity, it may take a very long time for this to happen, especially for a high dimensional chaotic attractor. In this paper we will demonstrate how a neural network (NN) can control a chaotic system with only a small control force and be trained with only examples from the state-space. To solve this problem, we introduced a novel objective function which measures the distance between the current state and its previous average. By minimizing this objective function, the NN can automatically locate the fixed point. As a preliminary step, a training set is used to train a forward model for the chaotic dynamics. The work of Jordan and Rumelhart [4] has shown that control problems can be mapped into supervised learning problems by coupling the outputs of a controller NN (the control signals) to the inputs of a forward model of a plant to form a multilayer network that is indirectly recurrent. A recurrent learning a.lgorithm is used to train the controller NN. To facilitate learning we use an extended radial basis function (RBF) network for both the forward model and the controller. To benchmark with OGY's result, the Himon map is used as a numerical example. The numerical results have shown the preliminary success of the proposed scheme. Details will be given in the following sections. In the next section we give our methodology and describe the general form of the recurrent learning algorithm used in our experiments. In Section 3, we discuss RBF networks and reintroduce a more powerful version. In Section 4, the numerical results are presented in detail. Finally, in Section 5, we give our conclusions. 2 Recurrent Learning for Control Let kC) denote a NN whose output, tit, is composed through a plant, l(·), with unknown dynamics. The output of the unknown plant (the state), it+l' forms part of the input for the NN a.t the next time step, hence the recurrency. At each time step the state is also passed to an output function , gC), which computes the sensation, Yt+l. The time evolution of this system is more accurately described by fit k(it,!h+l'W) it+l {(it, fit) Yt+l g(Xt+I), where ii7+1 is the desired sensation for time t + 1 and W represents the trainable weights for the network. Additionally, we define the temporally local and global Exploiting Chaos to Control the Future 649 error functionals Jt = ~11Y7 - Ytl1 2 and E = L~I Ji, where N is the final time step for the system. The real-time recurrent learning (RTRL) algorithm [9] for training the network weights to minimize E is based on the fair assumption that minimizing the local error functionals with a small learning rate at each time step will correspond to minimizing the global error. To derive the learning algorithm, we can imagine the system consisting of the plant, controller, and error functionals as being unfolded in time. From this perspective we can view each instance of the controller NN as a separate NN and thus differentiate the error functionals with respect to the network weights at different times. Hence, we now add a time index to Wt to represent this fact. However, when we use W without the time index, the term should be understood to be time invariant. We can now define the matrix f t = ~ a~t = ~it aiIt-1 (axt aiIt-1 ait ) L.J £) £) a + £)a.... + a.... ft-I, i=O UWi UUt-l Wt-I UUt-l Xt-I Xt-l which further allows us to define aJi aw aE aw (1) (2) (3) Equation 2 is the gradient equation for the RTRL algorithm while Equation 3 is for the backpropagation through time (BPTT) learning algorithm [7]. The gradients defined by these equations are usually used with gradient descent on a multilayer perceptron (MLP). We will use them on RBF networks. 3 The CNLS Network The Connectionist Normalized Local Spline (eNLS) network [3] is an extension of the more familiar radial basis function network of Moody and Darken [5]. The forward operation of the network is defined by (4) where (5) All of the equations in this section assume a single output. Generalizing them for multiple outputs merely adds another index to the terms. For all of our simulations, we choose to distribute the centers, iii, based on a sample of the input space. 650 Flake, Sun, Lee, and Chen Additionally, the basis widths, f3i' are set to an experimentally determined constant. Because the output, <p, is linear in the terms Ii and d~, training them is very fast. To train the CNLS network on a prediction problem we, can use a quadratic error function of the form E = ~(y(i) - qj(i»2, where y(i) is the target function that we wish to approximate. We use a one-dimensional Newton-like method [8] which yields the update equations If + 7J (y(i) - <P(i»L'~~~i)' ~ + 7J (y(x) <p(x»~=---=-!J...£...!..lo..::....L-The right-most update rules form the learning algorithm when using the CNLS network for prediction, where 7J is a learning rate that should be set below 1.0. The left-most update rules describe a more general learning algorithm that can be used when a target output is unknown. When using the CNLS network architecture as part of a recurrent learning algorithm we must be able to differentiate the network outputs with respect to the inputs. Note that in Equations 1 and 2 each of the terms aXt/aUt-l, aUt-daxt-l, ait/Bit- 1 , and Biii/ aii can either be exactly solved or approximated by differentiating a CNLS network. Since the CNLS output is highly nonlinear in its inputs, computing these partial derivatives is not quite as elegant as it would be in a MLP. Nevertheless, it can be done. We skip the details and just show the end result: ann a: = ~ d~Pi(X) + 2 ~(pj (x) qj f3j (aj - i)) - 2<p(x)::;, l=l J=l (6) 4 Adaptive Control By combining the equations from the last two sections, we can construct a recurrent learning scheme for RBF networks in a similar fashion to what has been done with MLP networks. To demonstrate the utility of our technique, we have chosen a wellstudied nonlinear plant that has been successfully modeled and controlled by using non-neural techniques. Specifically, we will use the Henon map as a plant, which has been the focus of much of the research of OGY [6]. We also adopt some of their notation and experimental constraints. 4.1 The Himon Map The Henon map [2] is described by the equations (7) (8) Exploiting Chaos to Control the Future 651 where A = Ao + p and p is a control parameter that may be modified at each time step to coerce the plant into a desirable state. For all simulations we set Ao = 1.29 and B = 0.3 which gives the above equations a chaotic attracter that also contains an unstable fixed point. Our goal is to train a CNLS network that can locate and drive the map into the unstable fixed point and keep it there with only a minimal amount of information about the plant and by using only small values of p. The unstable fixed point (XF, YF) in Equations 7 and 8 can be easily calculated as XF = YF ~ 0.838486. Forcing the Henon map to the fixed point is trivial if the controller is given unlimited control of the parameter. To make the problem more realistic we define p* as the maximum magnitude that p can take and use the rule below on the left if Ipi < p* if p > p* if p < -p* _ {p if Ipl < p* Pn 0 if Ipl > p* while OGY use the rule on the right. The reason we avoid the second rule is that it cannot be modeled by a CNLS network with any precision since it is step-like. The next task is to define what it means to "control" the Henon map. Having analytical knowledge of the fixed point in the attracter would make the job of the controller much easier, but this is unrealistic in the case where the dynamics of the plant to control are unknown. Instead, we use an error function that simply compares the current state of the plant with an average of previous states: 1 [ 2 2] et=2 (Xt-(x)r) +(Yt-(Y)r) , (9) where (.)r is the average of the last T values of its argument. This function approaches zero when the map is in a fixed point for time length greater than T. This function requires no special knowledge about the dynamics of the plant, yet it still enforces our constraint of driving the map into a fixed point. The learning algorithm also requires the partial derivatives of the error function with respect to the plant state variables, which are oet!f)xt = Xt - (x}r and oet!oYt = Yt (Y)r. These two equations and the objective function are the only special purpose equations used for this problem. All other equations generalize from the derivation of the algorithm. Additionally, since the "output" representation (as discussed earlier) is identical to the state representation, training on a distinct output function is not strictly necessary in this case. Thus, we simplify the problem by only using a single additional model for the unknown next-state function of the Henon map. 4.2 Simulation To facilitate comparison between alternate control techniques, we now introduce the term f6t where 6t is a random variable and f is a small constan~ which specifies the intensity of the noise. We use a Gaussian distribution for bt such that the distribution has a zero mean, is independent, and has a variance of one. In keeping with [6], we discard any values of 6t which are greater in magnitude than 10. For training we set f = 0.038. However, for tests on the real controller, we will show results for several values of f. 652 Flake, Sun, Lee, and Chen (a) • • • (b) • • • (c) • • • """ 'r (d) • • - (e) . • . (f) • Figure 1: Experimental results from training a neural controller to drive the Himon map into a fixed point. From (a) to (f), the values of fare 0.035, 0.036, 0.038, 0.04,0.05, and 0.06, respectively. The top row corresponds to identical experiments performed in [6]. We add the noise in two places. First, when training the model, we add noise to the target output of the model (the next state). Second, when testing the controller on the real Henon map, we add the noise to the input of the plant (the previous state). In the second case, we consider the noise to be an artifact of our fictional measurements; that is, the plant evolves from the previous noise free state. Training the controller is done in two stages: an off-line portion to tune the model and an on-line stage to tune the controller. To train the model we randomly pick a starting state within a region (-1.5, 1.5) for the two state variables. We then iterate the map for one hundred cycles with p = 0 so that the points will converge onto the chaotic attractor. Next, we randomly pick a value for p in the range of (-p*, p*). The last state from the iteration is combined with this control parameter to compute a target state. We then add the noise to the new state values. Thus, the model input consists of a clean previous state and a control parameter and the target values consist of the noisy next state. We compute 100 training patterns in this manner. Using the prediction learning algorithm for the CNLS network we train the model network on each of the 100 patterns (in random order) for 30 epochs. The model quickly converges to a low average error. In the next stage, we use the model network to train the controller network in two ways. First, the model acts as the plant for the purposes of computing a next state. Additionally, we differentiate the model for values needed for the RTRL algorithm. We train the controller for 30 epochs, where each epoch consists of 50 cycles. At the beginning of each epoch we initialize the plant state to some random values (not necessarily on the chaotic attracter ,) and set the recurrent history matrix, Exploiting Chaos to Control the Future 653 ... - .... .._ ..... _ ... _--.. _.... . .. _. -.-_ ..... _ .. ... -. . -.- -_ ... _---.-- . _ ... ...... __ ..... -. - ..... . .-.-.. _. . ... _ ... _-.. (a) (b) (c) Figure 2: Experimental results from [6]. From left to right, the values of f. are 0.035, 0.036, and 0.038, respectively. r t, to zero. Then, for each cycle, we feed the previous state into the controller as input. This produces a control parameter which is fed along with the previous state as input into the model network, which in turn produces the next state. This next state is fed into the error function to produce the error signal. At this point we compute all of the necessary values to train the controller for that cycle while maintaining the history matrix. In this way, we train both the model and control networks with only 100 data points, since the controller never sees any of the real values from the Henon map but only estimates from the model. For this experiment both the control and model RBF networks consist of 40 basis functions. 4.3 Summary Our results are summarized by Figure 1. As can be seen, the controller is able to drive the Henon Map into the fixed point very rapidly and it is capable of keeping it there for an extended period of time without transients. As the level of noise is increased, it can be seen that the plant maintains control for quite some time. The first visible spike can be observed when f. = 0.04. These results are an improvement over the results generated from the best nonneural technique available for two reasons: First, the neural controller that we have trained is capable of driving the Henon map into a fixed point with far fewer transients then other techniques. Specifically, alternate techniques, as illustrated in Figure 2, experience numerous spikes in the map for values of f. for which our controller is spike-free (0.035 - 0.038). Second, our training technique has smaller data requirements and uses less special purpose information. For example, the RBF controller was trained with only 100 data points compared to 500 for the nonneural. Additionally, non-neural techniques will typically estimate the location of the fixed point with an initial data set. In the case of [6] it was assumed that the fixed point could be easily discovered by some technique, and as a result all of their experiments rely on the true (hard-coded) fixed point. This, of course, could be discovered by searching the input space on the RBF model, but we have instead allowed the controller to discover this feature on its own. 654 Flake, Sun, Lee, and Chen 5 Conclusion and Future Directions A crucial component of the success of our approach is the objective function that measures the distance between the current state and the nearest time average. The reason why this objective function works is that during the control stage the learning algorithm is minimizing only a small distance between the current point and the "moving target." This is in contrast to minimizing the large distance between the current point and the target point, which usually causes unstable long time correlation in chaotic systems and ruins the learning. The carefully designed recurrent learning algorithm and the extended RBF network also contribute to the success of this approach. Our results seem to indicate that RBF networks hold great promise in recurrent systems. However, further study must be done to understand why and how NNs could provide more useful schemes to control real world chaos. Acknowledgements We gratefully acknowledge helpful comments from and discussions with Chris Barnes, Lee Giles, Roger Jones, Ed Ott, and James Reggia. This research was supported in part by AFOSR grant number F49620-92-J-0519. References [1] A. Garfinkel, M.L. Spano, and W.L. Ditto. Controlling cardiac chaos. Science, 257(5074):1230, August 1992. [2] M. HEmon. A two-dimensional mapping with a strange attractor. Communications in Mathematical Physics, 50:69-77, 1976. [3] R.D. Jones, Y.C. Lee, C.W. Barnes, G.W. Flake, K. Lee, P.S. Lewis, and S. Qian. Function approximation and time series prediction with neural network. In Proceedings of the International Joint Conference on Neural Networks, 1990. [4] M.1. Jordan and D.E. Rumelhart. Forward models: Supervised learning with a distal teacher. Technical Report Occasional Paper #40, MIT Center for Cognitive Science, 1990. [5] J. Moody and C. Darken. Fast learning in networks of locally-tuned processing units. Neural Computation, 1:281-294, 1989. [6] E. Ott, C. Grebogi, and J .A. Yorke. Controlling chaotic dynamical systems. In D.K. Campbell, editor, CHAOS: Soviet-American Perspectives on Nonlinear Science, pages 153-172. American Institute of Physics, New York, 1990. [7] F.J. Pineda. Generalization of back-propagation to recurrent neural networks. Physical Review Letters, 59:2229-2232, 1987. [8] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes. Cambridge University Press, Cambridge, 1986. [9] R.J. Williams and D. Zipser. Experimental analysis of the real-time recurrent learning algorithm. Connection Science, 1:87-111, 1989.
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Backpropagation Convergence Via Deterministic Nonmonotone Perturbed Minimization o. L. Mangasarian & M. v. Solodov Computer Sciences Department University of Wisconsin Madison, WI 53706 Email: olvi@cs.wisc.edu, solodov@cs.wisc.edu Abstract The fundamental backpropagation (BP) algorithm for training artificial neural networks is cast as a deterministic nonmonotone perturbed gradient method. Under certain natural assumptions, such as the series of learning rates diverging while the series of their squares converging, it is established that every accumulation point of the online BP iterates is a stationary point of the BP error function. The results presented cover serial and parallel online BP, modified BP with a momentum term, and BP with weight decay. 1 INTRODUCTION We regard training artificial neural networks as an unconstrained minimization problem N min f(x) := ~ h(x) xERn ~ j=l (1) where h : ~n --+ ~, j = 1, ... , N are continuously differentiable functions from the n-dimensional real space ~n to the real numbers~. Each function Ii represents the error associated with the j-th training example, and N is the number of examples in the training set. The n-dimensional variable space here is that of the weights associated with the arcs of the neural network and the thresholds of the hidden and 383 384 Mangasarian and Solodov output units. For an explicit description of f(x) see (Mangasarian, 1993). We note that our convergence results are equally applicable to any other form of the error function, provided that it is smooth. BP (Rumelhart,Hinton & Williams, 1986; Khanna, 1989) has long been successfully used by the artificial intelligence community for training artificial neural networks. Curiously, there seems to be no published deterministic convergence results for this method. The primary reason for this is the nonmonotonic nature of the process. Every iteration of online BP is a step in the direction of negative gradient of a partial error function associated with a single training example, e.g. Ii (x) in (1). It is clear that there is no guarantee that such a step will decrease the full objective function f( x), which is the sum of the errors for all the training examples. Therefore a single iteration of BP may, in fact, increase rather than decrease the objective function f( x) we are trying to minimize. This difficulty makes convergence analysis of BP a challenging problem that has currently attracted interest of many researchers (Mangasarian & Solodov, 1994; Gaivoronski, 1994; Grippo, 1994; Luo & Tseng, 1994; White, 1989). By using stochastic approximation ideas (Kashyap,Blaydon & Fu, 1970; Ermoliev & Wets, 1988), White (White, 1989) has shown that, under certain stochastic assumptions, the sequence of weights generated by BP either diverges or converges almost surely to a point that is a stationary point of the error function. More recently, Gaivoronski obtained stronger stochastic results (Gaivoronski, 1994). It is worth noting that even if the data is assumed to be deterministic, the best that stochastic analysis can do is to establish convergence of certain sequences with probability one. This means that convergence is not guaranteed. Indeed, there may exist some noise patterns for which the algorithm diverges, even though this event is claimed to be unlikely. By contrast, our approach is purely deterministic. In particular, we show that online BP can be viewed as an ordinary perturbed nonmonotone gradient-type algorithm for unconstrained optimization (Section 3). We note in the passing, that the term gradient descent which is widely used in the backpropagation and neural networks literature is incorrect. From an optimization point of view, online BP is not a descent method, because there is no guaranteed decrease in the objective function at each step. We thus prefer to refer to it as a nonmonotone perturbed gradient algorithm. We give a convergence result for a serial (Algorithm 2.1), a parallel (Algorithm 2.2) BP, a modified BP with a momentum term, and BP with weight decay. To the best of our knowledge, there is no published convergence analysis, either stochastic or deterministic, for the latter three versions of BP. The proposed parallel algorithm is an attempt to accelerate convergence of BP which is generally known to be relatively slow. 2 CONVERGENCE OF THE BACKPROPAGATION ALGORITHM AND ITS MODIFICATIONS We now turn our attention to the classical BP algorithm for training feedforward artificial neural networks with one layer of hidden units (Rumelhart,Hinton & Backpropagation Convergence via Deterministic Nonmonotone Perturbed Minimization 385 Williams, 1986; Khanna, 1989). Throughout our analysis the number of hidden units is assumed to be fixed. The choice of network topology is a separate issue that is not addressed in this work. For some methods for choosing the number of hidden units see (Courrien, 1993; Arai, 1993). We now summarize our notation. N : Nonnegative integer denoting number of examples in the training set. i = 1,2, ... : Index number of major iterations (epochs) of BP. Each major iteration consists of going through the entire set of error functions !1(x), ... , fN(X). j = 1, ... ,N : Index of minor iterations. Each minor iteration j consists of a step in the direction of the negative gradient - \7 fmU)(zi,j) and a momentum step. Here m(j) is an element of the permuted set {I, ... , N}, and zi,j is defined immediately below. Note that if the training set is randomly permuted after every epoch, the map m(·) depends on the index i. For simplicity, we skip this dependence in our notation. xi : Iterate in ~n of major iteration (epoch) i = 1,2, .... zi,; : Iterate in ~n of minor iteration j = 1, ... , N, within major iteration i 1,2, .... Iterates zi,j can be thought of as elements of a matrix with N columns and infinite number of rows, with row i corresponding to the i-th epoch of BP. By 1}i we shall denote the learning rate (the coefficient multiplying the gradient), and by (ki the momentum rate (the coefficient multiplying the momentum term). For simplicity we shall assume that the learning and momentum rates remain fixed within each major iteration. In a manner similar to that of conjugate gradients (Polyak, 1987) we reset the momentum term to zero periodically. Algorithm 2.1. Serial Online BP with a Momentum Term. Start with any xO E ~n. Having xi, stop if \7 f(x i ) = 0, else compute xi+l as follows: zi,j+l = zi,j TJi \7 fmu)(i,j) + aif1zi,j, j = 1, ... , N xi+l = zi,N+l where if j = 1 otherwise (2) (3) (4) (5) Remark 2.1. Note that the stopping criterion of this algorithm is typically that used in first order optimization methods, and is not explicitly related to the ability of the neural network to generalize. However, since we are concerned with convergence properties of BP as a numerical algorithm, this stopping criterion is 386 Mangasarian and Solodov justified. Another point related to the issue of generalization versus convergence is the following. Our analysis allows the use of a weight decay term in the objective function (Hinton, 1986; Weigend,Huberman & Rumelhart, 1990) which often yields a network with better generalization properties. In the latter case the minimization problem becomes N min I(x) := ~ hex) + Allx l1 2 xElRn Li=l (6) where A is a small positive scaling factor. Remark 2.2. The choice of C¥i = 0 reduces Algorithm 2.1 to the original BP without a momentum term. Remark 2.3. We can easily handle the "mini-batch" methods (M!2l11er, 1992) by merely redefining the meaning of the partial error function Ii to represent the error associated with a subset of training examples. Thus "mini-batch" methods also fall within our framework. We next present a parallel modification of BP. Suppose we have k parallel processors, k 2: 1. We consider a partition of the set {l, ... , N} into the subsets J" 1 = 1, ... ,k, so that each example is assigned to at least one processor. Let N, be the cardinality of the corresponding set J,. In the parallel BP each processor performs one (or more) cycles of serial BP on its set of training examples. Then a synchronization step is performed that consists of averaging the iterates computed by all the k processors. From the mathematical point of view this is equivalent to each processor I E {I, ... , k} handling the partial error function I' (x) associated with the corresponding set of training examples J,. In this setting we have k J'(x)=~Ii(x), f(x)=~f'(x) iEJI 1=1 We note that in training a neural network it might be advantageous to assign some training examples to more than one parallel processor. We thus allow for the possibility of overlapping sets J,. The notation for Algorithm 2.2 is similar to that for Algorithm 2.1, except for the index 1 that is used to label the partial error function and minor iterates associated with the l-th parallel processor. We now state the parallel BP with a momentum term. Algorithm 2.2. Parallel Online BP with a Momentum Term. Start with any xO E ~n. Having xi, stop if xi+l = xi, else compute xi+l as follows (i) Parallelization. For each parallel processor I E {I, ... , k} do i,l i z, = x i,i+l _ i,i '~f' (iIi) + . A i,i . 1 N z, - z, 7], v m(j) z, c¥,uz" J = , ... , I where ~zlili = { 0 z;,i - z;,i- l otherwise if j = 1 (7) (8) (9) Backpropagation Convergence via Deterministic Nonmonotone Perturbed Minimization 387 o < TJi < 1, O:s a i < 1 (ii) Synchronization k Xi+l = ~ L z;,Nr+l 1=1 We give below in Table 1 a flowchart of this algorithm. i 1 . Z ' .- x' 1 .Serial BP on examples in Jl Major iteration i : xi / ..... ~ .~ i 1 . z' '- x' I .~ Serial BP on examples in J, ~ i,Nr+l z, i 1 . z' '- x' k .~ Serial BP on examples in Jk ~ i,N,,+I zk J / M ·· . . 1 ,'+1 1 "k i Nr+ 1 aJor IteratIOn z + : x = k L.....I=1 z,' Table 1. Flowchart of the Parallel BP (10) Remark 2.4. It is well known that ordinary backpropagation is a relatively slow algorithm. One appealing remedy is parallelization (Zhang,Mckenna,Mesirov & Waltz, 1990). The proposed Algorithm 2.2 is a possible step in that direction. Note that in Algorithm 2.2 all processors typically use the same program for their computations. Thus load balancing is easily achieved. Remark 2.5. We wish to point out that synchronization strategies other than (10) are possible. For example, one may choose among the k sets of weights and thresholds the one that best classifies the training data. To the best of our knowledge there are no published deterministic convergence 388 Mangasarian and Solodov proofs for either of Algorithms 2.1,2.2. Using new convergence analysis for a class of nonmonotone optimization methods with perturbations (Mangasarian & Solodov, 1994), we are able to derive deterministic convergence properties for online BP and its modifications. Once again we emphasize the equivalence of either of those methods to a deterministic nonmonotone perturbed gradient-type algorithm. We now state our main convergence theorem. An important result used in the proof is given in the Mathematical Appendix. We refer interested readers to (Mangasarian & Solodov, 1994) for more details. Theorem 2.1. If the learning and momentum rates are chosen such that 00 00 00 L l7i = 00, L 171 < 00, L O:'il7i < 00, (11) i=O i=O i=O then for any sequence {xi} generated by any of the Algorithms 2.1 or 2.2, it follows that {/(xiH converges, {\7 !(xi)} 0, and for each accumulation point x of the sequence {x'}, \7 I( x) = O. Remark 2.6. We note that conditions (11) imply that both the learning and momentum rates asymptotically tend to zero. These conditions are similar to those used in (White, 1989; Luo & Tseng, 1994) and seem to be the inevitable price paid for rigorous convergence. For practical purposes the learning rate can be fixed or adjusted to some small but finite number to obtain an approximate solution to the minimization problem. For state-of-the-art techniques of computing the learning rate see (Ie Cun, Simard & Pearlmutter, 1993). Remark 2.7. We wish to point out that Theorem 2.1 covers BP with momentum and/or decay terms for which there is no published convergence analysis of any kind. Remark 2.8. We note that the approach of perturbed minimization provides theoretical justification to the well known properties of robustness and recovery from damage for neural networks (Sejnowski & Rosenberg, 1987). In particular, this approach shows that the net should recover from any reasonably small perturbation. Remark 2.9. Establishing convergence to a stationary point seems to be the best one can do for a first-order minimization method without any additional restrictive assumptions on the objective function. There have been some attempts to achieve global descent in training, see for example, (Cetin,Burdick & Barhen, 1993). However, convergence to global minima was not proven rigorously in the multidimensional case. 3 MATHEMATICAL APPENDIX: CONVERGENCE OF ALGORITHMS WITH PERTURBATIONS In this section we state a new result that enables us to establish convergence properties of BP. The full proof is nontrivial and is given in (Mangasarian & Solodov, 1994). Backpropagation Convergence via Deterministic Nonmonotone Perturbed Minimization 389 Theorem 3.1. General Nonmonotonic Perturbed Gradient Convergence (subsumes BP convergence). Suppose that f(x) is bou?,!-ded below and that \1 f(x) is bounded and Lipschitz continuous on the sequence {x'} defined below. Consider the following perturbed gradient algorithm. Start with any x O E ~n. Having xi, stop if \1 f(x i ) = 0, else compute (12) where di = -\1f(xi ) + ei (13) for some ei E ~n, TJi E~, TJi > 0 and such that for some I > 0 00 00 00 L TJi = 00, L TJl < 00, L TJdleili < 00, Ileill ~ I Vi (14) ;=0 i=O i=O It follows that {f(xi)} converges, {\1 f(xi)} -+ 0, and for each accumulation point x of the sequence {x'}, V' f(x) = O. If, in addition, the number of stationary points of f(x) is finite, then the sequence {xi} converges to a stationary point of f(x). Remark 3.1. The error function of BP is nonnegative, and thus the boundedness condition on f(x) is satisfied automatically. There are a number of ways to ensure that f(x) has Lipschitz continuous and bounded gradient on {xi} . In (Luo & Tseng, 1994) a simple projection onto a box is introduced which ensures that the iterates remain in the box. In (Grippo, 1994) a regularization term as in (6) is added to the error function so that the modified objective function has bounded level sets. We note that the latter provides a mathematical justification for weight decay (Hinton, 1986; Weigend,Huberman & Rumelhart, 1990). In either case the iterates remain in some compact set, and since f( x) is an infinitely smooth function, its gradient is bounded and Lipschitz continuous on this set as desired. Acknowledgements This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grant CCR9101801. References M. Arai. (1993) Bounds on the number of hidden units in binary-valued three-layer neural networks. Neural Networks, 6:855-860. B. C. Cetin, J. W. Burdick, and J. Barhen. (1993) Global descent replaces gradient descent to avoid local minima problem in learning with artificial neural networks. In IEEE International Conference on Neural Networks, (San Francisco), volume 2, 836-842. P. Courrien.(1993) Convergent generator of neural networks. Neural Networks, 6:835-844. Yu. Ermoliev and R.J.-B. Wets (editors). (1988) Numerical Techniques for Stochastic Optimization Problems. Springer-Verlag, Berlin. 390 Mangasarian and Solodov A.A. Gaivoronski. (1994) Convergence properties of backpropagation for neural networks via theory of stochastic gradient methods. Part 1. Optimization Methods and Software, 1994, to appear. 1. Grippo. (1994) A class of unconstrained minimization methods for neural network training. Optimization Methods and Software, 1994, to appear. G. E. Hinton. (1986) Learning distributed representations of concepts. In Proceedings of the Eighth Annual Conference of the Cognitive Science Society, 1-12, Hillsdale. Erlbaum. R. 1. Kashyap, C. C. Blaydon and K. S. Fu. (1970) Applications of stochastic approximation methods. In J .M.Mendel and K.S. Fu, editors, Adaptive, Learning, and Pattern Recognition Systems. Academic Press. T. Khanna. (1989) Foundations of neural networks. Addison-Wesley, New Jersey. Y. Ie Cun, P.Y. Simard, and B. Pearlmutter. (1993) Automatic learning rate maximization by on-line estimation of the Hessian's eigenvectors. In C.1.Giles S.J .Hanson, J .D.Cowan, editor, Advances in Neural Information Processing Systems 5, 156-163, San Mateo, California, Morgan Kaufmann. Z.-Q. Luo and P. Tseng. (1994) Analysis of an approximate gradient projection method with applications to the backpropagation algorithm. Optimization Methods and Software, 1994, to appear. 0.1. Mangasarian. (1993) Mathematical programming in neural networks. ORSA Journal on Computing, 5(4), 349-360. 0.1. Mangasarian and M.V. Solodov. (1994) Serial and parallel backpropagation convergence via nonmonotone perturbed minimization. Optimization Methods and Software, 1994, to appear. Proceedings of Symposium on Parallel Optimization 3, Madison July 7-9, 1993. M.F. M!2Sller. (1992) Supervised learning on large redundant training sets. In Neural Networks for Signal Processing 2. IEEE Press. B.T. Polyak. (1987) Introduction to Optimization. Optimization Software, Inc., Publications Division, New York. D.E. Rumelhart, G.E. Hinton, and R.J. Williams. (1986) Learning internal representations by error propagation. In D.E. Rumelhart and J.1. McClelland, editors, Parallel Distributed Processing, 318-362, Cambridge, Massachusetts. MIT Press. T.J. Sejnowski and C.R. Rosenberg. (1987) Parallel networks that learn to pronounce english text. Complex Systems, 1:145-168. A.S. Weigend, B.A. Huberman, and D.E. Rumelhart. (1990) Predicting the future:a connectionist approach. International Journal of Neural Systems, 1 :193-209. H. White. (1989) Some asymptotic results for learning in single hidden-layer feedforward network models. Journal of the American Statistical Association, 84( 408): 1003-1013. X. Zhang, M. Mckenna, J. P. Mesirov, and D. 1. Waltz. (1990) The backpropagation algorithm on grid and hypercube architectures. Parallel Computing, 14:317-327.
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Fool.s Gold: Extracting Finite State Machines From Recurrent Network Dynamics John F. Kolen Laboratory for Artificial Intelligence Research Department of Computer and Information Science The Ohio State University Columbus,OH 43210 kolen-j @cis.ohio-state.edu Abstract Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize a formal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the network. Some researchers have begun to extracting finite state machines from the internal state trajectories of their recurrent networks. This paper describes how sensitivity to initial conditions and discrete measurements can trick these extraction methods to return illusory finite state descriptions. INTRODUCTION Formal language learning (Gold, 1969) has been a topic of concern for cognitive science and artificial intelligence. It is the task of inducing a computational description of a formal language from a sequence of positive and negative examples of strings in the target language. Neural information processing approaches to this problem involve the use of recurrent networks that embody the internal state mechanisms underlying automata models (Cleeremans et aI., 1989; Elman, 1990; Pollack, 1991; Giles et aI, 1992; Watrous & Kuhn, 1992). Unlike traditional automata-based approaches, learning systems relying on recurrent networks have an additional burden: we are still unsure as to what these networks are doing.Some researchers have assumed that the networks are learning to simulate finite state 501 502 Kolen machines (FSMs) in their state dynamics and have begun to extract FSMs from the networks' state transition dynamics (Cleeremans et al., 1989; Giles et al., 1992; Watrous & Kuhn, 1992). These extraction methods employ various clustering techniques to partition the internal state space of the recurrent network into a finite number of regions corresponding to the states of a finite state automaton. This assumption of finite state behavior is dangerous on two accounts. First, these extraction techniques are based on a discretization of the state space which ignores the basic definition of information processing state. Second, discretization can give rise to incomplete computational explanations of systems operating over a continuous state space. SENSITIVITY TO INITIAL CONDITIONS In this section, I will demonstrate how sensitivity to initial conditions can confuse an FSM extraction system. The basis of this claim rests upon the definition of information processing state. Information processing (lP) state is the foundation underlying automata theory. Two IP states are the same if and only if they generate the same output responses for all possible future inputs (Hopcroft & Ullman, 1979). This definition is the fulcrum for many proofs and techniques, including finite state machine minimization. Any FSM extraction technique should embrace this definition, in fact it grounds the standard FSM minimization methods and the physical system modelling of Crutchfield and Young (Crutchfield & Young, 1989). Some dynamical systems exhibit exponential divergence for nearby state vectors, yet remain confined within an attractor. This is known as sensitivity to initial conditions. If this divergent behavior is quantized, it appears as nondeterministic symbol sequences (Crutchfield & Young, 1989) even though the underlying dynamical system is completely deterministic (Figure 1). Consider a recurrent network with one output and three recurrent state units. The output unit performs a threshold at zero activation for state unit one. That is, when the activation of the first state unit of the current state is less than zero then the output is A. Otherwise, the output is B. Equation 1 presents a mathematical description. Set) is the current state of the system 0 (t) is the current output. S (t + 1) = tanh ( 0 0 2 1 . S(t)) r 2 -2 0 -J [ ~ o 0 2 -1 1 (1) Figure 2 illustrates what happens when you run this network for many iterations. The point in the upper left hand state space is actually a thousand individual points all within a ball of radius 0.01. In one iteration these points migrate down to the lower corner of the state space. Notice that the ball has elongated along one dimension. After ten iterations the original ball shape is no longer visible. After seventeen, the points are beginning to spread along a two dimensional sheet within state space. And by fifty iterations, we see the network reaching the its full extent of in state space. This behavior is known as sensitivity to initial conditions and is one of three conditions which have been used to characterize chaotic dynamical systems (Devaney, 1989). In short, sensitivity to initial conditions implies Fool's Gold: Extracting Finite State Machines from Recurrent Network Dynamics 503 x~4x(l-x) { : x<O.5 @A O(x) = x>O.5 x~2x mod 1 A 1 x<-3 O(x) = B 1 2 - <x<3 3 C 2 -<x 3 C A x ~ 3.68x(l-x) C x<O.5 O(x) = x>O.5 Figure 1: Examples of deterministic dynamical systems whose discretize trajectories appear nondeterministic. that any epsilon ball on the attractor of the dynamical will exponentially diverge, yet still be contained within the locus of the attractor. The rate of this divergence is illustrated in Figure 3 where the maximum distance between two points is plotted with respect to the number of iterations. Note the exponential growth before saturation. Saturation occurs as the point cloud envelops the attractor. No matter how small one partitions the state space, sensitivity to initial conditions will eventually force the extracted state to split into multiple trajectories independent of the future input sequence. This is characteristic of a nondeterministic state transition. Unfortunately, it is very difficult, and probably intractable, to differentiate between a nondeterministic system with a small number of states or a deterministic with large number of states. In certain cases, however, it is possible to analytically ascertain this distinction (Crutchfield & Young, 1989). THE OBSERVERS' PARADOX One response to this problem is to evoke more computationally complex models such as push-down or linear-bounded automata. Unfortunately, the act of quantization can actually introduce both complexion and complexity in the resulting symbol sequence. Pollack and I have focused on a well-hidden problems with the symbol system approach to understanding the computational powers of physical systems. This work (Kolen & Pollack, 1993; S04 Kolen 1 I output=A 1 Start (e<O.Ol) 1 I output=A,B 1 17 iterations 1 I output=B 1 1 iteration 1 output=A,B 1 25 iterations 1 I output=A 1 10 iterations I 1 1 50 iterations Figure 2: The state space of a recurrent network whose next state transitions are sensitive to initial conditions. The initial epsilon ball contains 1000 points. These points first straddle the output decision boundary at iteration seven. Kolen & Pollack, In press) demonstrated that computational complexity, in terms of Chomsky's hierarchy of formal languages (Chomsky, 1957; Chomsky, 1965) and Newell and Simon's physical symbol systems (Newell & Simon, 1976), is not intrinsic to physical systems. The demonstration below shows how apparently trivial changes in the partitioning of state space can produce symbol sequences from varying complexity classes. Consider a point moving in a circular orbit with a fixed rotational velocity, such as the end of a rotating rod spinning around a fixed center, or imagine watching a white dot on a spinning bicycle wheel. We measure the location of the dot by periodically sampling the location with a single decision boundary (Figure 4, left side). If the point is to the left of boundary at the time of the sample, we write down an "1". Likewise, we write down an "r" when the point is on the other side. (The probability of the point landing on the boundary is zero and can arbitrarily be assigned to either category without affecting the results below.) In the limit, we will have recorded an infinite sequence of symbols containing long sequences of r's and l's. The specific ordering of symbols observed in a long sequence of multiple rotations is Fool's Gold: Extracting Finite State Machines from Recurrent Network Dynamics 505 ••••• • • ell 2.5 ...... c:: .... • 0 0.. • • c:: 2 • 0 0 ~ ...... 0 • • .0 1.5 0 u c:: • ~ ...... ell 1 .... "1:;) 8 ::s • S 0.5 • • .... ~ • ::E • •• • • • • • 10 20 30 40 50 Iteration number Figure 3: Spread of initial points across the attractor as measured by maximum distance. 1 r 1 r c Figure 4: On the left, two decision regions which induce a context free language. 9 is the current angle of rotation. At the time of sampling, if the point is to the left (right) of the dividing line, an 1 (r) is generated. On the right, three decision regions which induce a context sensitive language. dependent upon the initial rotational angle of the system. However, the sequence does possess a number of recurring structural regularities, which we call sentences: a run of r's followed by a run of l's. For a fixed rotational velocity (rotations per time unit) and sampling rate, the observed system will generate sentences of the form r n1 m (n, m > 0). (The notation rn indicates a sequence of n r's.) For a fixed sampling rate, each rotational velocity specifies up to three sentences whose number of r's and l's differ by at most one. These sentences repeat in an arbitrary manner. Thus, a typical subsequence of a rotator which produces sentences r n1 n, r n1 n+l ,rn+11 n would look like 506 Kolen rnln+lrnlnrnln+lrn+l1nrnlnrnln+l. A language of sentences may be constructed by examining the families of sentences generated by a large collection of individuals, much like a natural language is induced from the abilities of its individual speakers. In this context, a language could be induced from a population of rotators with different rotational velocities where individuals generate sentences of the form {r"l n, r"l "+1 ,r"+ll"}, n > O. The reSUlting language can be described by a context free grammar and has unbounded dependencies; the number of 1 's is a function of the number of preceding r's. These two constraints on the language imply that the induced language is context free. To show that this complexity class assignment is an artifact of the observational mechanism, consider the mechanism which reports three disjoint regions: 1, c, and r (Figure 4, right side). Now the same rotating point will generate sequences ofthe form ... rr ... rrcc ... ccll. .. llrr ... rrcc ... ccll ... ll .... For a fixed sampling rate, each rotational velocity specifies up to seven sentences, r nc ffil k, when n, m, and k can differ no by no more than one. Again, a language of sentences may be constructed containing all sentences in which the number ofr's, c's, and l's differs by no more than one. The resulting language is context sensitive since it can be described by a context sensitive grammar and cannot be context free as it is the finite union of several context sensitive languages related to r"c"l n. CONCLUSION Using recurrent neural networks as the representation underlying the language learning task has revealed some inherent problems with the concept of this task. While formal languages have mathematical validity, looking for language induction in physical systems is questionable, especially if that system operates with continuous internal states. As I have shown, there are two major problems with the extraction of a learned automata from our models. First, sensitivity to initial conditions produces nondeterministic machines whose trajectories are specified by both the initial state of the network and the dynamics of the state transformation. The dynamics provide the shape of the eventual attractor. The initial conditions specify the allowable trajectories toward that attractor. While clustering methods work in the analysis of feed-forward networks because of neighborhood preservation (as each layer is a homeomorphism), they may fail when applied to recurrent network state space transformations. FSM construction methods which look for single transitions between regions will not help in this case because the network eventually separates initially nearby states across several FSM state regions. The second problem with the extraction of a learned automata from recurrent network is that trivial changes in observation strategies can cause one to induce behavioral descriptions from a wide range of computational complexity classes for a single system. It is the researcher's bias which determines that a dynamical system is equivalent to a finite state automata. Fool's Gold: Extracting Finite State Machines from Recurrent Network Dynamics 507 One response to the first problem described above has been to remove and eliminate the sources of nondeterminism from the mechanisms. Zeng et. a1 (1993) corrected the secondorder recurrent network model by replacing the continuous internal state transformation with a discrete step function. (The continuous activation remained for training purposes.) This move was justified by their focus on regular language learning, as these languages can be recognized by finite state machines. This work is questionable on two points, however. First, tractable algorithms already exist for solving this problem (e.g. Angluin, 1987). Second, they claim that the network is self-clustering the internal states. Self-clustering occurs only at the comers of the state space hypercube because of the discrete activation function, in the same manner as a digital sequential circuit "clusters" its states. Das and Mozer (1994), on the other hand, have relocated the clustering algorithm. Their work focused on recurrent networks that perform internal clustering during training. These networks operate much like competitive learning in feed-forward networks (e.g. Rumelhart and Zipser, 1986) as the dynamics of the learning rules constrain the state representations such that stable clusters emerge. The shortcomings of finite state machine extraction must be understood with respect to the task at hand. The actual dynamics of the network may be inconsequential to the final product if one is using the recurrent network as a pathway for designing a finite state machine. In this engineering situation, the network is thrown away once the FSM is extracted. Neural network training can be viewed as an "interior" method to finding discrete solutions. It is interior in the same sense as linear programming algorithms can be classified as either edge or interior methods. The former follows the edges of the simplex, much like traditional FSM learning algorithms search the space of FSMs. Internal methods, on the other hand, explore search spaces which can embed the target spaces. Linear programming algorithms employing internal methods move through the interior of the defined simplex. Likewise, recurrent neural network learning methods swim through mechanisms with mUltiple finite state interpretations. Some researchers, specifically those discussed above, have begun to bias recurrent network learning to walk the edges (Zeng et al, 1993) or to internally cluster states (Das & Mozer, 1994). In order to understand the behavior of recurrent networks, these devices should be regarded as dynamical systems (Kolen, 1994). In particular, most common recurrent networks are actually iterated mappings, nonlinear versions of Barnsley's iterated function systems (Barnsley, 1988). While automata also fall into this class, they are a specialization of dynamical systems, namely discrete time and state systems. Unfortunately, information processing abstractions are only applicable within this domain and do not make any sense in the broader domains of continuous time or continuous space dynamical systems. Acknowledgments The research reported in this paper has been supported by Office of Naval Research grant number NOOOI4-92-J-1195. I thank all those who have made comments and suggestions for improvement of this paper, especially Greg Saunders and Lee Giles. References Angluin, D. (1987). Learning Regular Sets from Queries and Counterexamples. Information 508 Kolen and Computation, 75,87-106. Barnsley, M. (1988). Fractals Everywhere. Academic Press: San Diego, CA. Chomsky, N. (1957). Syntactic Structures. The Hague: Mounton & Co. Chomsky, N. (1965). Aspects of the Theory of Syntax. Cambridge, Mass.: MIT Press. Cleeremans, A, Servan-Schreiber, D. & McClelland, J. L. (1989). Finite state automata and simple recurrent networks. Neural Computation, 1,372-381. Crutchfield, J. & Young, K. (1989). Computation at the Onset of Chaos. In W. Zurek, (Ed.), Entropy, Complexity, and the Physics of Information. Reading: Addison-Wesely. Das, R. & Mozer, M. (1994) A Hybrid Gradient-Descent/Clustering Technique for Finite State Machine Induction. In Jack D. Cowan, Gerald Tesauro, and Joshua Alspector, (Eds.), Advances in Neural Information Processing Systems 6. Morgan Kaufman: San Francisco. Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems. Addison-Wesley. Elman, J. (1990). Finding structure in time. Cognitive Science, 14, 179-211. Giles, C. L., Miller, C. B., Chen, D., Sun, G. Z., Chen, H. H. & C.Lee, Y. (1992). Extracting and Learning an Unknown Grammar with Recurrent Neural Networks. In John E. Moody, Steven J. Hanson & Richard P. Lippman, (Eds.), Advances in Neural Information Processing Systems 4. Morgan Kaufman. Gold, E. M. (1969). Language identification in the limit. Information and Control, 10,372381. Hopcroft, J. E. & Ullman, J. D. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesely. Kolen, J. F. (1994) Recurrent Networks: State Machines or Iterated Function Systems? In M. C. Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman, & AS. Weigend (Eds.), Proceedings of the 1993 Connectionist Models Summer School. (pp. 203-210) Hillsdale, NJ: Erlbaum Associates. Kolen, J. F. & Pollack, J. B. (1993). The Apparent Computational Complexity of Physical Systems. In Proceedings of the Fifteenth Annual Conference of the Cognitive Science Society. Laurence Earlbaum. Kolen, J. F. & Pollack, J. B. (In press) The Observers' Paradox: The Apparent Computational Complexity of Physical Systems. Journal of Experimental and Theoretical Artificial Intelligence. Pollack, J. B. (1991). The Induction Of Dynamical Recognizers. Machine Learning, 7.227252. Newell, A. & Simon, H. A (1976). Computer science as empirical inquiry: symbols and search. Communications of the Associationfor Computing Machinery, 19, 113-126. Rumelhart, D. E., and Zipser, D. (1986). Feature Discovery by Competitive Learning. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, (Eds.), Parallel Distributed Processing. Volume 1. 151-193. MIT Press: Cambridge, MA Watrous, R. L. & Kuhn, G. M. (1992). Induction of Finite-State Automata Using SecondOrder Recurrent Networks. In John E. Moody, Steven J. Hanson & Richard P. Lippman, (Eds.), Advances in Neural Information Processing Systems 4. Morgan Kaufman. Zeng, Z., Goodman, R. M., Smyth, P. (1993). Learning Finite State Machines With Self-Clustering Recurrent Networks. Neural Computation, 5, 976-990 PART IV NEUROSCIENCE
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Cross-Validation Estimates IMSE Mark Plutowski t* Shinichi Sakata t Halbert White t* t Department of Computer Science and Engineering t Department of Economics * Institute for Neural Computation University of California, San Diego Abstract Integrated Mean Squared Error (IMSE) is a version of the usual mean squared error criterion, averaged over all possible training sets of a given size. If it could be observed, it could be used to determine optimal network complexity or optimal data subsets for efficient training. We show that two common methods of cross-validating average squared error deliver unbiased estimates of IMSE, converging to IMSE with probability one. These estimates thus make possible approximate IMSE-based choice of network complexity. We also show that two variants of cross validation measure provide unbiased IMSE-based estimates potentially useful for selecting optimal data subsets. 1 Summary To begin, assume we are given a fixed network architecture. (We dispense with this assumption later.) Let zN denote a given set of N training examples. Let QN(zN) denote the expected squared error (the expectation taken over all possible examples) of the network after being trained on zN. This measures the quality of fit afforded by training on a given set of N examples. Let IMSEN denote the Integrated Mean Squared Error for training sets of size N. Given reasonable assumptions, it is straightforward to show that IMSEN = E[Q N(ZN)] 0"2, where the expectation is now over all training sets of size N, ZN is a random training set of size N, and 0"2 is the noise variance. Let CN = CN(zN) denote the "delete-one cross-validation" squared error measure for a network trained on zN. CN is obtained by training networks on each of the N training sets of size N -1 obtained by deleting a single example; the measure follows 391 392 Plutowski, Sakata, and White by computing squared error for the corresponding deleted example and averaging the results. Let G N,M = G N,M (zN , zM) denote the "generalization" measure obtained by separating the available data of size N + M into a training set zN of size N, and a validation ("test") set zM of size M; the measure follows by training on zN and computing averaged squared error over zM. We show that eN is an unbiased estimator of E[QN_l(ZN)], and hence, estimates 1M SEN-l up to noise variance. Similarly, GN,M is an unbiased estimator of E[QN(ZN, ZM] . Given reasonable conditions on the estimator and on the data generating process we demonstrate convergence with probability 1 of GN,M and eN to E[QN(ZN)] as Nand M grow large. A direct consequence of these results is that when choice is restricted to a set of network architectures whose complexity is bounded above a priori, then choosing the architecture for which either eN (or G N,M) is minimized leads to choice of the network for which 1MSEN is nearly minimized for all N (respectively, N, M) sufficiently large. We also provide results for training sets sampled at particular inputs. Conditional 1M S E is an appealing criterion for evaluating a particular choice of training set in the presence of noise. These results demonstrate that delete-one cross-validation estimates average MSE (the average taken over the given set of inputs,) and that holdout set cross-validation gives an unbiased estimate of E[QN(ZN)IZN = (xN , yN)], given a set of N input values xN for which corresponding (random) output values yN are obtained. Either cross-validation measure can therefore be used to select a representative subset of the entire dataset that can be used for data compaction, or for more efficient training (as training can be faster on smaller datasets) [4]. 2 Definitions 2.1 Learning Task We consider the learning task of determining the relationship between a random vector X and a random scalar y, where X takes values in a subset X of ~r, and Y takes values in a subset Y of~. (e.g. X = ~r and Y = ~). We refer to X as the input space. The learning task is thus one of training a neural network with r inputs and one output. It is straightforward to extend the following analysis to networks with multiple targets. We make the following assumption on the observations to be used in the training of the networks. Assumption 1 X is a Borel subset of ~r and Y is a Borel subset of~. Let Z = X x Y and n = Zoo = Xi:1Z. Let (n,.1', P) be a probability space with.1' = 8(n). The observations on Z = (X', Y)' to be used in the training of the network are a realization of an i.i.d. stochastic process {Z, _ (Xi, Yi)' : n ---+ X x V}. When wEn is fixed, we write z, = Z,(w) for each i = 1,2, .... Also write ZN = (Zl, ... ,ZN) and zn = (Zl, .. . ,zn). Assumption 1 allows uncertainty caused by measurement errors of observations as well as a probabilistic relationship between X and Y . It, however, does not prevent a deterministic relation ship between X and Y such that Y = g(X) for some measurable mapping g : ~r ---+ ~. Cross-Validation Estimates IMSE 393 We suppose interest attaches to the conditional expectation of Y given X, written g( x) = E(Y IX). The next assumption guarantees the existence of E(Yi IXi) and E(cdXi), Ci = Yi - E(YiIXi). Next, for convenience, we assume homoscedasticity of the conditional variance of Yi given Xi. Assumption 2 E(y2) < 00. Assumption 3 E(ci IX1) = u2, where u 2 is a strictly positive constant. 2.2 Network Model Let fP(.,.) : X X WP -- Y be a network function with the "weight space" WP, where p denotes the dimension of the "weight space" (the number of weights.) We impose some mild conditions on the network architecture. Assumption 4 For each p E {I, 2, ... ,p}, pEN, WP is a compact subset of ~P, and fP : X x WP -- ~ satisfies the following conditions: 1. fP(., w) : X -- Y is measurable for each wE WP; 2. fP(x,·) : WP -- Y is continuous for all x E X. We further make a joint assumption on the underlying data generating process and the network architecture to assure that the training dataset and the networks behaves appropriately. Assumption 5 There exists a function D : X -- ~+ = [0,00) such that for each x E X and w E WP, IfP(x, w)1 ~ D(x), and E [(D(X»2] < 00. Hence, fP is square integrable for each wP E WP. We will measure network performance using mean squared error, which for weights wP is given by ).(wP;p) = E [(Y - fP(X, wP)2]. The optimal weights are the weights that minimize ).(wP;p). The set of all optimal weights are given by WP· = {w· E WP : ).( w· ; p) ~ ).( w; p) for any w E Wp}. The index of the best network is p. , given by the smallest p minimizing minwl'Ewl' ).(wP;p), p E {I, 2, ... ,pl. 2.3 Least-Squares Estimator When assumptions I and 4 hold, the nonlinear least-squares estimator exists. Formally, we have Lemma 1 Suppose that Assumptions 1 and 4 hold. Then 1. For each N EN, there exists a measurable function INC; p) : ZN -- WP such that IN(ZN; p) solves the following problem with probability one: minwEWI' N- 1 E~l (Yi - J(Xi, w»2 . 2. ).(.; p) : WP __ ~ is continuous on WP, and WP· is not empty. For convenience, we also define ~ : n -- WP by ~(w) = IN(ZN (w);p) for each wEn. Next let i1 , i2 , •.. , iN be distinct natural numbers and let ZN = (Zil' ... , ZiN)" Then IN(ZN) given above solves ;; Ef=l (Yi; - f(Xi;, wP»2 with probability one. In particular, we will consider the estimate using the dataset Z~i made by deleting the ith observation from zN. Let Z~i be a random matrix made 394 Plutowski, Sakata, and White by deleting the ith row from ZN. Thus, IN -1 (Z~i; p) is a measurable least squares estimator and we can consider its probabilistic behavior. 3 Integrated Mean Squared Error Integrated Mean Squared Error (IMSE) has been used to regulate network complexity [9]. Another (conditional) version of IMSE is used as a criterion for evaluating training examples [5, 6, 7, 8]. The first version depends only on the sample size, not the particular sample. The second (conditional) version depends additionally upon the observed location of the examples in the input space. 3.1 Unconditional IMSE The (unconditional) mean squared error (MSE) of the network output at a particular input value x is MN(X;p)=E [{g(x)-!(x,IN(ZN;p))}2]. (1) Integrating MSE over all possible inputs gives the unconditional IMSE: IMSEN(p) J [MN(X, ;p)] J.L(dx) (2) E [MN(XjP)], (3) where J.L is the input distribution. 3.2 Conditional IMSE To evaluate exemplars obtained at inputs x N , we modify Equation (1) by conditioning on x N , giving MN(xlxN ;p) = E [{g(x) - !(x, IN(ZN))PIXN = xN] . The conditional IMSE (given inputs x N ) is then IMSEN(xN;p) J MN(xlxN;p)JL(dx) E [MN(XlxN;p)] . 4 Cross-Validation (4) (5) Cross-validatory measures have been used successfully to assess the performance of a wide range of estimators [10, 11, 12, 13, 14, 15]. Cross-validatory measures have been derived for various performance criteria, including the Kullback-Liebler Information Criterion (KLIC) and the Integrated Squared Error (ISE, asymptotically equivalent to IMSE) [16]. Although provably inappropriate in certain applications [17, 18], optimality and consistency results for the cross-validatory measures have been obtained for several estimators, including linear regression, orthogonal series, splines, histograms, and kernel density estimators [16, 19,20, 21, 22, 23, 24]. The authors are not aware of similar results applicable to neural networks, although two more general, but weaker results do apply [26]. A general result applicable to neural networks shows asymptotic equivalence between cross-validation and Akaike's Criterion for network selection [25,29]' as well as between cross-validation and Moody's Criterion [30, 29]. Cross-Validation Estimates IMSE 395 4.1 Expected Network Error Given our assumptions, we can relate cross-validation to IMSE. For clarity and notational convenience, we first introduce a measure of network error closely related to IMSE. For each weight wP E WP, we have defined the mean squared error A( wP ; p) in Section 2.2. We define QN to map each dataset to the mean squared error of the estimated network QN(ZN;p) = A(lN(zN;p);p). When Assumption 3 holds, we have A(wP;p) = E [(g(X) - f(X, WP))2] + u2 = E [(g(XN+d - f(XN+l, wP))2] + u2 as is easily verified. We therefore have QN(zN; p) = E [(g(XN+d - f(XN+l, IN(ZN ;p)))2IZN = zN] + u2. Thus, by using the law of iterated expectations, we have E[QN(ZN;p)] = IMSEN(p)+u2. Likewise, given x N E X N , E[QN(ZN; p)IXN = xN] = IMSE(xN ;p) + u2. 4.2 Cross-Validatory Estimation of Error (6) In practice we work with observable quantities only. In particular, we must estimate the error of network p over novel data ("generalization") from a finite set of examples. Such an estimate is given by the delete-one cross-validation measure: N CN(zN;p) = ~ L (Yi - f(Xi,IN_l(zl!i;P)))2 (7) i=l ~ere zl!i, denotes the training set obtained by deleting the ith example. Using z_i insteaa of z avoids a downward bias due to testing upon examples used in training, as we show below (Theorem 3.) Another version of cross-validation is commonly used for evaluating "generalization" when an abundant supply of novel data is available for use as a "hold-out" set: M GN,M(zN,zM;p) = ~ L (iii - J(xi,IN(zN;p)))2, i=l (8) where zM = (ZN+l' ... , ZN+M)' 5 Expectation of the Cross-Validation Measures We now consider the relation between cross-validation measure and IMSE. We examine delete-one cross-validation first. Proposition 1 (Unbiasedness of CN) Let Assumptions 1 through 5 hold. Then for given N, CN is an unbiased estimator of 1M SEN-l (p) + u2 : E [CN(ZN;p)] = IMSEN-1(p) +u2. (9) 396 Plutowski, Sakata, and White With hold-out set cross-validation, the validation set ZM gives i.i.d. information regarding points outside of the training set ZN. Proposition 2 (Unbiasedness of GN,M) Let Assumptions 1 through 5 hold. Let ZM = (ZN+l, .. . ,ZM)'. Then for given Nand M, GN,M is an unbiased estimator of IMSEN(p) + u 2 : [ N -M ] 2 E GN,M(Z , Z ;p) = IMSEN(p) + u . (10) The latter result is appealing for large M, N. We expect delete-one cross-validation to be more appealing when training data is not abundant. 6 Expectation of Cross-Validation when Sampling at Selected Inputs We obtain analogous results for training sets obtained by sampling at a given set of inputs x N . We first consider the result for delete-one cross-validation. Proposition 3 (Expectation of CN given xN1 Let Assumptions 1 through 5 hold. Then, given a particular set of inputs, x , CN is an unbiased estimator of average MSEN-l + u 2 , the average taken over xN : N 1 " (I N . 2 N L.J M N - 1 Xi x_i' p) + u , i=l where X~i is a matrix made by deleting the ith row of x N • This essentially gives an estimate of MSEN-l limited to x E x N , losing a degree of freedom while providing no estimate of the M S E off of the training points. For this average to converge to IMSEN-l, it will suffice for the empirical distribution of x N , p,N, to converge to J-lN, i.e., P,N => J-lN. We obtain a stronger result for hold-out set cross-validation. The hold-out set gives independent information on M SEN off of the training points, resulting in an estimate of IMSEN for given xN . Proposition 4 (Expectation of GN,M given x N ) Let Assumptions 1 through 5 hold. Let ZM = (ZN+1, .. . , ZN+M )'. Then, given a particular set of inputs, x N , GN,M is an unbiased estimator of of IMSEN(xN ; p) + u 2 : E [GN,M(ZN,ZM;p)IXN =xN] IMSEN(xN;p) +u2 • 7 Strong Convergence of Hold-Out Set Cross-Validation Our conditions deliver not only unbiasedness, but also convergence of hold-out set cross-validation to IMSEN, with probability 1. Theorem 1 (Convergence of Hold-Out Set w.p. 1) Let Assumptions 1 through 5 hold. Also let ZM = (ZN+l, ... , ZN+M)'. If for some A > 0 a sequence {MN} of natural numbers satisfies MN > AN for any N = 1,2, ... , then Cross-Validation Estimates IMSE 397 8 Strong Convergence of Delete-One Cross-Validation Given an additional condition (uniqueness of optimal weights) we can show strong convergence for delete-one cross-validation. First we establish uniform convergence of the estimators WP(Z~i) to optimal weights (uniformly over 1 < i < N.) Theorem 2 Let Assumptions 1 through 5 hold. Let Z~k be the dataset made by deleting the kth observation from ZN. Then max d (IN-1(Z~i;P), Wp*) --+ 0 a.s.-P as N --+ 00, (11) l~i~N where d(w, Wp*) = infw.Ewp.llw - w*lI. This convergence result leads to the next result that the delete-one cross validation measure does not under-estimate the optimized MSE, namely, infwPEWp ).(wP;p). Theorem 3 Let Assumptions 1 through 5 hold. Then liminfCN(ZN ;p) > min ).(w;p) a.s.-P. N-oo wEWp When the optimum weight is unique, we have a stronger result about convergence of the delete-one cross validation measure. Assumption 6 Wp* is a singleton, i.e., wp* has only one element. Theorem 4 Let Assumptions 1 through 6 hold. Then CN (ZN ;p) - E [QN(ZN ;p)] --+ 0 a.s. as N --+ 00. 9 Conclusion Our results justify the intuition that cross-validation measures unbiasedly and consistently estimate the expected squared error of networks trained on finite training sets, therefore providing means of obtaining 1M S E-approximate methods of selecting appropriate network architectures, or for evaluating particular choice of training set. Use of these cross-validation measures therefore permits us to avoid underfitting the data, asymptotically. Note, however, that although we also thereby avoid overfitting asymptotically, this avoidance is not necessarily accomplished by choosing a minimally complex architecture. The possibility exists that IMSEN-1(p) = 1M S EN -1 (p + 1). Because our cross-validated estimates of these quanti ties are random we may by chance observe CN(ZN;p) > CN(ZN;p+ 1) and therefore select the more complex network, even though the less complex network is equally good. Of course, because the IMSE's are the same, no performance degradation (overfitting) will result in this solution. Acknowledgements This work was supported by NSF grant IRI 92-03532. We thank David Wolpert, J an Larsen, Jeff Racine, Vjachislav Krushkal, and Patrick Fitzsimmons for valuable discussions. 398 Plutowski, Sakata, and White References [1] White, H. 1989. "Learning in Artificial Neural Networks: A Statistical Perspective." Neural Computation, 1 i, pp.i25-i6i. MIT Press, Cambridge, MA. (2] Plutowski, Mark E., Shinichi Sakata, and Halbert White. 1993. "Cross-validation delivers strongly consistent unbiased estimates of Integrated Mean Squared Error." To appear. (3] Plutowski, Mark E., and Halbert White. 1993. "Selecting concise training sets from clean data." IEEE Transactions on Neural Networks. 4, 3, pp.305-318. (i) Plutowski, Mark E., Garrison Cottrell, and Halbert White. 1992. "Learning Mackey-Glass from 25 examples, Plus or Minus 2." (Presented at 1992 Neural Information Processing Systems conference.) Jack D. Cowan, Gerald Tesauro, Joshua Aspector (eds.), Advances in neural information processing systems 6, San Mateo, CA: Morgan Kaufmann Publishers. (5] Fedorov, V.V. 1972. Theory of Optimal Experimenta. Academic Prell, New York. (6] Box,G., and N.Draper. 1987. Empirical ModelBuilding and Reaponae Surfacea. Wiley, New York. (7] Khuri, A.I., and J.A.Cornell. 1987. Reaponae Surfacea (Deaigna and Analyaea). Marcel Dekker, Inc., New York . (8] Faraway, Julian J. 1990. "Sequential design for the nonparametric regression of curves and surfaces." Technical Report #177, Department of Statistics, The University of Michigan. (9] Geman, Stuart, Elie Bienenstock, and Rene Doursat. 1992. "Neural network. and the bias/variance dilemma." 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Two Iterative Algorithms for Computing the Singular Value Decomposition from Input / Output Samples Terence D. Sanger Jet Propulsion Laboratory MS 303-310 4800 Oak Grove Drive Pasadena, CA 91109 Abstract The Singular Value Decomposition (SVD) is an important tool for linear algebra and can be used to invert or approximate matrices. Although many authors use "SVD" synonymously with "Eigenvector Decomposition" or "Principal Components Transform", it is important to realize that these other methods apply only to symmetric matrices, while the SVD can be applied to arbitrary nonsquare matrices. This property is important for applications to signal transmission and control. I propose two new algorithms for iterative computation of the SVD given only sample inputs and outputs from a matrix. Although there currently exist many algorithms for Eigenvector Decomposition (Sanger 1989, for example), these are the first true samplebased SVD algorithms. 1 INTRODUCTION The Singular Value Decomposition (SVD) is a method for writing an arbitrary nons quare matrix as the product of two orthogonal matrices and a diagonal matrix. This technique is an important component of methods for approximating nearsingular matrices and computing pseudo-inverses. Several efficient techniques exist for finding the SVD of a known matrix (Golub and Van Loan 1983, for example). 144 Singular Value Decomposition 145 p r--------------------------U __ --I~ s I 1 _____________________________ __ I Figure 1: Representation of the plant matrix P as a linear system mapping inputs u into outputs y. LT SR is the singular value decomposition of P. However, for certain signal processing or control tasks, we might wish to find the SVD of an unknown matrix for which only input-output samples are available. For example, if we want to model a linear transmission channel with unknown properties, it would be useful to be able to approximate the SVD based on samples of the inputs and outputs of the channel. If the channel is time-varying, an iterative algorithm for approximating the SVD might be able to track slow variations. 2 THE SINGULAR VALUE DECOMPOSITION The SVD of a nonsymmetric matrix P is given by P = LT SR where Land Rare matrices with orthogonal rows containing the left and right "singular vectors" , and S is a diagonal matrix of "singular values". The inverse of P can be computed by inverting S, and approximations to P can be formed by setting the values of the smallest elements of S to zero. For a memoryless linear system with inputs u and outputs y = Pu, we can write y = LT SRu which shows that R gives the "input" transformation from inputs to internal "modes", S gives the gain of the modes, and LT gives the "output" transformation which determines the effect of each mode on the output. Figure 1 shows a representation of this arrangement. The goal of the two algorithms presented below is to train two linear neural networks Nand G to find the SVD of P. In particular, the networks attempt to invert P by finding orthogonal matrices Nand G such that NG ~ p- 1 , or P NG = I. A particular advantage of using the iterative algorithms described below is that it is possible to extract only the singular vectors associated with the largest singular values. Figure 2 depicts this situation, in which the matrix S is shown smaller to indicate a small number of significant singular values. There is a close relationship with algorithms that find the eigenvalues of a symmetric matrix, since any such algorithm can be applied to P pT = LT S2 Land pT P = RT S2 R in order to find the left and right singular vectors. But in a behaving animal or operating robotic system it is generally not possible to compute the product with pT, since the plant is an unknown component of the system. In the following, I will present two new iterative algorithms for finding the singular value decomposition 146 Sanger u p , -----------' I I I I I I __ _ _ _ __ _ _ _ _ _ J N G y Figure 2: Loop structure of the singular value decomposition for control. The plant is P = LT SR, where R determines the mapping from control variables to system modes, and LT determines the outputs produced by each mode. The optimal sensory network is G = L, and the optimal motor network is N = RT S-l. Rand L are shown as trapezoids to indicate that the number of nonzero elements of S (the "modes") may be less than the number of sensory variables y or motor variables u. of a matrix P given only samples of the inputs u and outputs y. 3 THE DOUBLE GENERALIZED HEBBIAN ALGORITHM The first algorithm is the Double Generalized Hebbian Algorithm (DGHA), and it is described by the two coupled difference equations b-.G = l(zyT - LT[zzT]G) (1) b-.NT = l(zuT - LT[zzT]NT) (2) where LT[ ] is an operator that sets the above diagonal elements of its matrix argument to zero, y = Pu, z = Gy, and I is a learning rate constant. Equation 1 is the Generalized Hebbian Algorithm (Sanger 1989) which finds the eigenvectors of the autocorrelation matrix of its inputs y. For random uncorrelated inputs u, the autocorrelation of y is E[yyT] = LT S2 L, so equation 1 will cause G to converge to the matrix of left singular vectors L . Equation 2 is related to the Widrow-Hoff (1960) LMS rule for approximating uT from z, but it enforces orthogonality of the columns of N. It appears similar in form to equation 1, except that the intermediate variables z are computed from y rather than u. A graphical representation of the algorithm is given in figure 3. Equations 1 and 2 together cause N to converge to RT S-l , so that the combination N G = RT S-l L is an approximation to the plant inverse. Theorem 1: (Sanger 1993) If y = Pu, z = Gy, and E[uuT] = I, then equations 1 and 2 converge to the left and right singular vectors of P . Singular Value Decomposition 147 u p y IGHA Figure 3: Graphic representation of the Double Generalized Hebbian Algorithm. G learns according to the usual G HA rule, while N learns using an orthogonalized form of the Widrow-Hoff LMS Rule. Proof: After convergence of equation 1, E[zzT] will be diagonal, so that E[LT[zzT]] = E[zzT]. Consider the Widrow-Hoff LMS rule for approximating uT from z: ~NT = 'Y(zuT - zzT NT). (3) After convergence of G, this will be equivalent to equation 2, and will converge to the same attractor. The stable points of 3 occur when E[uzT - NzzT] = 0, for which N = RT 5- 1 • The convergence behavior of the Double Generalized Hebbian Algorithm is shown in figure 4. Results are measured by computing B = GP N and determining whether B is diagonal using a score " ...... b~. L...I~) I) €= L 2 . b· 1 1 The reduction in € is shown as a function of the number of (u, y) examples given to the network during training, and the curves in the figure represent the average over 100 training runs with different randomly-selected plant matrices P. Note that the Double Generalized Hebbian Algorithm may perform poorly in the presence of noise or uncontrollable modes. The sensory mapping G depends only on the outputs y, and not directly on the plant inputs u. So if the outputs include noise or autonomously varying uncontrollable modes, then the mapping G will respond to these modes. This is not a problem if most of the variance in the output is due the inputs u, since in that case the most significant output components will reflect the input variance transmitted through P. 4 THE ORTHOGONAL ASYMMETRIC ENCODER The second algorithm is the Orthogonal Asymmetric Encoder (OAE) which is described by the equations (4) 148 Sanger 0.7 0.& 0.5 j 0.4 j ! I 0.3 is 0.2 0.1 Double Generalized Hebbian Algorithm ... ... . . . . . . . . . . . . . . . . . ~~~-~~~~~~~~~-~~~~~ Exomple Figure 4: Convergence of the Double Generalized Hebbian Algorithm averaged over 100 random choices of 3x3 or 10xlO matrices P. (5) where z = NT u. This algorithm uses a variant of the Backpropagation learning algorithm (Rumelhart et al. 1986). It is named for the "Encoder" problem in which a three-layer network is trained to approximate the identity mapping but is forced to use a narrow bottleneck layer. I define the "Asymmetric Encoder Problem" as the case in which a mapping other than the identity is to be learned while the data is passed through a bottleneck. The "Orthogonal Asymmetric Encoder" (OAE) is the special case in which the hidden units are forced to be uncorrelated over the data set. Figure 5 gives a graphical depiction of the algorithm. Theorem 2: (Sanger 1993) Equations 4 and 5 converge to the left and right singular vectors of P. Proof: Suppose z has dimension m. If P = LT SR where the elements of S are distinct, and E[uuT ] = I, then a well-known property of the singular value decomposition (Golub and Van Loan 1983, , for example) shows that E[IIPu - CT NT ullJ (6) is minimized when CT = LrnU, NT = V Rm , and U and V are any m x m matrices for which UV = 1mS/;;". (L~ and Rm signify the matrices of only the first m columns of LT or rows of R.) If we want E[zzT] to be diagonal, then U and V must be diagonal. OAE accomplishes this by training the first hidden unit as if m = 1, the second as if m = 2, and so on. For the case m = 1, the error 6 is minimized when C is the first left singular vector of P and N is the first right singular vector. Since this is a linear approximation problem, there is a single global minimum to the error surface 6, and gradient descent using the backpropagation algorithm will converge to this solution. u p Singular Value Decomposition 149 y I , ~as!pr.2Pa~ti~ I Figure 5: The Orthogonal Asymmetric Encoder algorithm computes a forward approximation to the plant P through a bottleneck layer of hidden units. After convergence, the remaining error is E[II(P - GT N T )ull1. If we decompose the plant matrix as i=l where Ii and ri are the rows of Land R, and Si are the diagonal elements of S, then the remaining error is n P2 = LlisirT i=2 which is equivalent to the original plant matrix with the first singular value set to zero. If we train the second hidden unit using P2 instead of P, then minimization of E[IIP2u - GT NT ull1 will yield the second left and right singular vectors. Proceeding in this way we can obtain the first m singular vectors. Combining the update rules for all the singular vectors so that they learn in parallel leads to the governing equations of the OAE algorithm which can be written in matrix form as equations 4 and 5 . • (Bannour and Azimi-Sadjadi 1993) proposed a similar technique for the symmetric encoder problem in which each eigenvector is learned to convergence and then subtracted from the data before learning the succeeding one. The orthogonal asymmetric encoder is different because all the components learn simultaneously. After convergence, we must multiply the learned N by S-2 in order to compute the plant inverse. Figure 6 shows the performance of the algorithm averaged over 100 random choices of matrix P. Consider the case in which there may be noise in the measured outputs y. Since the Orthogonal Asymmetric Encoder algorithm learns to approximate the forward plant transformation from u to y, it will only be able to predict the components of y which are related to the inputs u. In other words, the best approximation to y based on u is if ~ Pu, and this ignores the noise term. Figure 7 shows the results of additive noise with an SNR of 1.0. 150 Sanger 0.7 0.8 0.5 I!! c§ 0.4 c j 0.3 ii c 0 GO .l!I 0 0.2 0.1 0 0 " 50 Orthogonal Asymmetric Encoder , .. . . . ..... . . . . . ..... 100 150 200 250 300 350 400 450 500 550 800 850 700 750 800 850 900 950 Example Figure 6: Convergence of the Orthogonal Asymmetric Encoder averaged over 100 random choices of 3x3 or 10xlO matrices P. Acknowledgements 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 Bannour S., Azimi-Sadjadi M. R., 1993, Principal component extraction using recursive least squares learning, submitted to IEEE Transactions on Neural Networks. Golub G. H., Van Loan C. F., 1983, Matrix Computations, North Oxford Academic P., Oxford. Rumelhart D. E., Hinton G. E., Williams R. J., 1986, Learning internal representations by error propagation, In Parallel Distributed Processing, chapter 8, pages 318-362, MIT Press, Cambridge, MA. Sanger T. D., 1989, Optimal unsupervised learning in a single-layer linear feedforward neural network, Neural Networks, 2:459-473. Sanger T. D., 1993, Theoretical Elements of Hierarchical Control in Vertebrate Motor Systems, PhD thesis, MIT. Widrow B., Hoff M. E., 1960, Adaptive switching circuits, In IRE WESCON Conv. Record, Part 4, pages 96-104. 2 ! .§ j 1 j I CI Singular Value Decomposition 151 OAE with 50% Added Noise 1- 3x3 • • 101110 . . . ... . . . . . . . . . . . . oL-.---,--,--~~~:;:::::;:::::;~~ o 50 100 150 200 250 300 350 400 450 500 650 &00 860 700 750 &00 860 000 860 ElIIlmple Figure 7: Convergence of the Orthogonal Asymmetric Encoder with 50% additive noise on the outputs, averaged over 100 random choices of 3x3 or 10xlO matrices P.
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Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering Patrice Y. Simard AT&T Bell Laboratories Holmdel, NJ 07733 Abstract By their very nature, memory based algorithms such as KNN or Parzen windows require a computationally expensive search of a large database of prototypes. In this paper we optimize the searching process for tangent distance (Simard, LeCun and Denker, 1993) to improve speed performance. The closest prototypes are found by recursively searching included subset.s of the database using distances of increasing complexit.y. This is done by using a hierarchy of tangent distances (increasing the Humber of tangent. vectors from o to its maximum) and multiresolution (using wavelets). At each stage, a confidence level of the classification is computed. If the confidence is high enough, the c.omputation of more complex distances is avoided. The resulting algorithm applied to character recognition is close to t.hree orders of magnitude faster than computing the full tangent dist.ance on every prot.ot.ypes. 1 INTRODUCTION Memory based algorithms such as KNN or Parzen windows have been extensively used in pattern recognition. (See (Dasal'athy, 1991) for a survey.) Unfortunately, these algorithms often rely 011 simple distances (such a<; Euclidean distance, Hamming distance, etc.). As a result, t.hey suffer from high sensitivity to simple transformations of the input patterns that should leave the classification unchanged (e.g. translation or scaling for 2D images). To make the problem worse, these algorithms 168 Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering 169 are further limited by extensive computational requirements due to the large number of distance computations. (If no optimization technique is used, the computational cost is given in equation 1.) computational cost ~ number of prototypes x dist.ance complexity (1) Recently, the problem of transformation sensitivity has been addressed by the introduction of a locally transformation-invariant metric, the tangent distance (Simard, LeCun and Denker, 1993). The basic idea is that instead of measuring the distance d(A, B) between two patterns A and B, their respective sets of transformations TA and TB are approximated to the first order, and the distance between these two approximated sets is computed. Unfortunately, the tangent distance becomes computationally more expensive as more transformations are taken into consideration, which results in even stronger speed requirements. The good news is that memory based algorithms are well suited for optimization using hierarchies of prototypes, and that this is even more true when the distance complexity is high. In this paper, we applied these ideas to tangent distance in two ways: 1) Finding the closest prototype can be done by recursively searching included subsets of the database using distances of increasing complexity. This is done by using a hierarchy of tangent distances (increasing the number of tangent vectors from 0 to its maximum) and l11ultiresolution (using wavelets). 2) A confidence level can be computed fm each distance. If the confidence in the classification is above a threshold early on, there is no need to compute the more expensive distances. The two methods are described in the next section. Their application on a real world problem will be shown in the result section. 2 FILTERING USING A HIERARCHY OF DISTANCES Our goal is to compute the distance from one unknown pattern to every prototype in a large database in order to determine which one is the closest. It is fairly obvious that some patterns are so different from each other that a very crude approximation of our distance can tell us so. There is a wide range of variation in computation time (and performance) depending on the choice of the distance. For instance, computing the Euclidean distance on n-pixel images is a factor 11/ k of the computation of computing it on k-pixels images. Similarly, at a given resolution, computing the tangent distance with 111 tangent vectors is (m + 1)2 times as expensive as computing the Euclidean distance (m = ° tangent vectors). This observations provided us wit.h a hierarchy of about a dozen different distances ranging in computation time from 4 multiply/adds (Euclidean distance on a 2 x 2 averaged image) to 20,000 multiply /adds (tangent distance, 7 tangent vectors, 16 x 16 pixel images). The resulting filtering algorithm is very straightforward and is exemplified in Figure 1. The general idea is to store the database of prototypes several times at different resolutions and with different tangent. vectors. Each of these resolutions and groups of tangent vectors defines a distance di . These distances are ordered in increasing 170 Simard ( Proto types Euc. Dist ~ 2x2 10~OOO Cost: 4 ~ Confidence Unknown Pattern Euc. Dist ~ 4x4 ~ {soc soo Cost: 16 ~ Confidence Tang.Dist Category 14 vectors t----t~ 16x16 Confidence Figure 1: Pattern recognition using a hierarchy of distance. The filter proceed from left (starting with the whole database) to right (where only a few prototypes remain). At each stage distances between prototypes and the unknown pattern are computed, sorted and the best candidate prototypes are selected for the next stage. As the complexity of the distance increases, the number of prototypes decreases, making computation feasible. At each stage a classification is attempted and a confidence score is computed. If the confidence score is high enough, the remaining stages are skipped. accuracy and complexity. The first distance dl is computed on all (1\0) prototypes of the database. The closest J\ 1 pat.terns are then selected and identified to the next stage. This process is repeated for each of the distances; i.e. at each stage i, the distance di is computed on each J\i-l patterns selected by the previous stage. Of course, the idea is that as the complexity of the distance increases, the number of patterns on which this distance must be computed decreases. At the last stage, the most complex and accurate distance is computed on all remaining patterns to determine the classificat.ion. The only difficult part is to det.ermine the minimum I<i patterns selected at each stage for which the filtering does not decrease t.he overall performance. Note that if the last distance used is the most accurat.e distance, setting all J\j to the number of patterns in the database will give optimal performance (at the most expensive cost). Increasing I<i always improves the performance in the sense that it allows to find patterns that are closer for the next distance measure dj + 1 . The simplest way to determine I<i is by selecting a validation set and plotting t.he performance on this validation set as a function of !\j. The opt.imal !\·i is then determined graphically. An automatic way of computing each 1\; is currently being developed. This method is very useful when the performance is not degraded by choosing small J{j. In this case, the dist.ance evaluation is done using distance metrics which are relatively inexpensive to compute. The computation cost becomes: Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering 171 computational cost ~ L number of prototypes X at stage i distance complexity at stage i (2) Curves showing the performance as a function of the value of !{i will be shown in the result section. 3 PRUNING THE SEARCH USING CONFIDENCE SCORES If a confidence score is computed at each stage of the distance evaluation, it is possible for certain patterns to avoid completely computing the most expensive distances. In the extreme case, if the Euclidean distance between two patterns is 0, there is really no need to compute the tangent distance. A simple (and crude) way to compute a confidence score at a given stage i, is to find the closest prototype (for distance di ) in each of the possible classes. The distance difference between the closest class and the next closest class gives an approximation of a confidence of this classification. A simple algorithm is then to compare at stage i the confidence score Cip of the current unknown patt.ern p to a threshold ()j, and to stop the classification process for this pattern as soon as Cip > ()j. The classification will then be determined by the closest prototype at this stage. The computation time will therefore be different depending on the pattern to be classified. Easy patterns will be recognized very quickly while difficult. patterns will need to be compared to some of the prototypes using the most complex distance. The total computation cost is therefore: computational cost ~ L number of prototypes X at stage i distance complexity X at. stage i probabili ty to reach stage i (3 ) Note that if all ()j are high, the performance is maximized but so is the cost. We therefore wish to find the smallest value of Oi which does not degrade the performance (increasing (Jj a.lways improves the performance). As in the previous section, the simplest way to determine the optimal ()j is graphically with a validation set.. Example of curves representing the perfornlance as a function of ()j will be given in the result section. 4 CHOSING A GOOD HIERARCHY, OPTIMIZATION 4.1 k-d tree Several hierarchies of distance are possible for optimizing the search process. An incremental nearest neighbor search algorithm based on k-d tree (Broder, 1990) was implemented. The k-d tree structure was interesting because it can potentially be used with tangent distance. Indeed, since the separating hyperplanes have n-1 dimension, they can be made parallel to many tangent vectors at the same time. As much as 36 images of 256 pixels ,,,ith each 7 t.angent. vectors can be separat.ed into two group of 18 images by Olle hyperplane which is parallel to all tangent 172 Simard vectors. The searching algorithm is taking some knowledge of the transformation invariance into account when it computes on which side of each hyperplane the unknown pattern is. Of course, when a leaf is reached, the full tangent. distance must be computed. The problem with the k-d tree algorithm however is that in high dimensional space, the distance from a point to a hyperplane is almost always smallel' than the distance between any pair of points. As a result, the unknown pattern must be compared to many prototypes to have a reasonable accuracy. The speed up factor was comparable to our multiresolution approach in the case of Euclidean distance (about 10), but we have not been able to obtain both good performance and high speedup with the k-d tree algorithm applied to tangent distance. This algorithm was not used in our final experiments. 4.2 Wavelets One of the main advantages of the multiresolution approach is that it is easily implemented with wavelet transforms (i\'1allat, 1989), and that in the wavelet space, the tangent distance is conserved (with orthonormal wavelet bases). Furthermore, the multiresolution decomposition is completely orthogonal to the tangent distance decomposition. In our experiment.s, the Haar transform was used. 4.3 Hierarchy of tangent distance Many increasingly accurate approximations can be made for the tangent distance at a given resolution. For instance, the tangent distance can be computed by an iterative process of alternative projections onto the tangent hyperplanes. A hierarchy of distances results, derived from the number of projections performed. This hierarchy is not very good because the initial projection is already fairly expensive. It is more desirable to have a better efficiency in the first stages since only few patterns will be left for the latter stages. Our most successful hierarchy consisted in adding tangent vectors one by one, on both sides. Even though this implies solving a new linear system at each stage, the computational cost is mainly dominated by computing dot products between tangent vectors. These dot-products are then reused in the subsequent stages to create larger linear systems (invol ving more tangent vectors). This hierarchy has the advantage that the first stage is only twice as expensive, yet much more accurate, than the Euclidean distance. Each subsequent stage brings a lot of accuracy at a reasonable cost. (The cost inCl'eases quicker toward the lat.er stages since solving the linear system grows with the cube of the number of tangent vector.) In addition, the last stage is exactly the full tangent distance. As we will see in section 5 the cost in the final stages is negligible. Obviously, the tangent vectors can be added in different order. \Ve did not try to find the optimal order. For character recognition application adding translations first, followed by hyperbolic deformations, the scalings, the thickness deformations and the rotations yielded good performance. Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering 173 z # of T.V. Reso # of prot.o (Ki) # of prod Probab # of mul/add 0 0 4 9709 1 1.00 40,000 1 0 16 3500 1 1.00 56,000 2 0 64 500 1 1.00 32,000 3 1 64 125 2 0.90 14,000 4 2 256 50 5 0.60 40,000 5 4 256 45 7 0.40 32,000 6 6 256 25 9 0.20 11,000 7 8 256 15 11 0.10 4,000 8 10 256 10 13 0.10 3,000 9 12 256 5 15 0.05 1,000 10 14 256 5 17 0.0.5 1,000 Table 1: Summary computation for the classification of 1 pattern: The first column is the distance index, the second column indicates the number of tangent vector (0 for the Euclidean distance), and the third column indicates the resolution in pixels, the fourth is J{j or the number of prototypes on which the distance di must be computed, the fifth column indicat.es the number of additional dot products which must be computed to evaluate distance di, the sixth column indicates the probability to not skip that stage after the confidence score has been used, and the last column indicates the total average number of multiply-adds which must be performed (product of column 3 to 6) at each stage. 4.4 Selecting the k closests out of N prototypes in O(N) In the multiresolution filter, at the early stages we must select the k closest prototypes from a large number of protot.ypes. This is problematic because the prototypes cannot be sorted since O( N ZagN) is expensive compared to computing N distances at very low resolution (like 4 pixels). A simple solution consist.s in using a variation of "quicksort" or "finding the k-t.h element" (Aho, Hopcroft and Ullman, 1983), which can select the k closests out of N prototypes in O(N). The generic idea is to compute the mean of the distances (an approximation is actually sufficient) and then to split the distances into two halves (of different sizes) according to whether they are smaller or larger than the mean distance. If they are more dist.ances smaller than the mean than k, the process is reiterat.ed on the upper half, ot.herwise it is reiterated on the lowel' half. The process is recursively executed until there is only one distance in each half. (k is then reached and all the k prototypes in the lower halves are closer to the unknown pattei'll than all the N ~~ prototypes in the upper halves.) Note that. the elements are not sorted and t.hat only t.he expected t.ime is O(N), but this is sufficient for our problem. 5 RESULTS A simple task of pattern classification was used to test the filtering. The prototype set and the test set consisted l'especti vely of 9709 and 2007 labeled images (16 by 16 pixels) of handwritten digit.s. The prot.otypes were also averaged t.o lower 174 Simard 5 Error in % 4 Resol ution 16 pixels 3 K (in 1000) 8 71 Error in % t 6 5 4 3 / Resolution 16 pIXels ResolutIOn 64 pixels Reso lutlO n 64 pixels 1 tangent veC10r 2~~~~~~ __ ~ __ ~~~~ o 10 20 30 40 50 60 70 80 90 100 % of pat. kept. Figure 2: Left: Raw error performance as a function of Kl and 1\2. The final chosen values were J{ 1 = 3500 and [\'2 = 500. Right: Raw error as a function of the percentage of pattern which have not exceeded the confidence threshold Oi. A 100% means all the pattern were passed to the next stage. resolutions (2 by 2, 4 by 4 and 8 by 8) and copied to separate databases. The 1 by 1 resolution was not useful for anything. Therefore the fastest distance was the Euclidean distance on 2 by 2 images, while the slowest distance was the full tangent distance with 7 tangent vectors for both the prototype and the unknown pattern (Simard, LeCun and Denker, 1993). Table 1 summarizes the results. Several observations can be made. First, simple distance metrics are very useful to eliminate large proportions of Pl"Ototypes at no cost in performances. Indeed the Euclidean distance computed on 2 by 2 images can remove 2 third of the prototypes. Figure 2, left, shows the performance as a function of J{l and 1\2 (2 .5 % raw error was considered optimal performance). It can be noticed that for J{j above a certain value, the performance is optimal alld c.onstant. The most complex distances (6 and 7 tangent vectors on each side) need only be computed for 5% of the prototypes. The second observation is that the use of a confidence score can greatly reduce the number of distance evaluations in later stages. For instance the dominant phases of the computation would be with 2, 4 and 6 tangent vectors at resolution 256 if there were not reduced to 60%, 40% and 20% respectively using the confidence sc.ores. Figure 2, right, shows the raw error performance as a function of the percentage of rejection (confidence lower than OJ) at stage i. It can be noticed that above a certain threshold, the performance are optimal and constant. Less than 10% of the unknown patterns need the most. complex distances (5, 6 and 7 tangent vectors on each side), to be comput.ed. Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering 175 6 DISCUSSION Even though our method is by no way optimal (the order of the tangent vector can be changed, intermediate resolution can be used, etc ... ), the overall speed up we achieved was about 3 orders of magnitude (compared with computing the full tangent distance on all the patterns). There was no significant decrease in performances. This classification speed is comparable with neural network method, but the performance are better with tangent distance (2.5% versus 3%). Furthermore the above methods require no learning period which makes them very attractive for application were the distribution of the patterns to be classified is changing rapidly. The hierarchical filtering can also be combined with learning the prototypes using algorithms such as learning vector quantization (LVQ). References Aho, A. V., Hopcroft, J. E., and Ullman, J. D. (1983). Data Structure and Algorithms. Addison-\V'esley. Broder, A. J. (1990). Strategies for Efficient Incremental Nearest Neighbor Search. Pattern Recognition, 23: 171-178. Dasarathy, B. V. (1991). Nearest Neighbor (NN) Norms: NN Pattern classification Techniques. IEEE Computer Society Press, Los Alamitos, California. Mallat, S. G. (1989). A Theory for I\,Iultiresolution Signal Decomposition: The Wavelet Representation. IEEE Transactions 011 Pattern Analysis and Machine Intelligence, 11, No. 7:674-693. Simard, P. Y., LeCun, Y., and Denker, J. (1993). Efficient Pattern Recognition Using a New Transformation Distance. In Neural Information Processing Systems, volume 4, pages 50-58, Sa.n Mateo, CA.
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Statistics of Natural Images: Scaling in the Woods Daniel L. Ruderman* and William Bialek NEe Research Institute 4 Independence Way Princeton, N.J. 08540 Abstract In order to best understand a visual system one should attempt to characterize the natural images it processes. We gather images from the woods and find that these scenes possess an ensemble scale invariance. Further, they are highly non-Gaussian, and this nonGaussian character cannot be removed through local linear filtering. We find that including a simple "gain control" nonlinearity in the filtering process makes the filter output quite Gaussian, meaning information is maximized at fixed channel variance. Finally, we use the measured power spectrum to place an upper bound on the information conveyed about natural scenes by an array of receptors. 1 Introduction Natural stimuli are playing an increasingly important role in our understanding of sensory processing. This is because a sensory system's ability to perform a task is a statistical quantity which depends on the signal and noise characteristics. Recently several approaches have explored visual processing as it relates to natural images (Atick & Redlich '90, Bialek et al '91, van Hateren '92, Laughlin '81, Srinivasan et al '82). However, a good characterization of natural scenes is sorely lacking. In this paper we analyze images from the woods in an effort to close this gap. We • Current address: The Physiological Laboratory, Downing Street, Cambridge CB2 3EG, England. 551 552 Ruderman and Bialek further attempt to understand how a biological visual system should best encode these images. 2 The Images Our images consist of 256 x 256 pixels 1(x) which are calibrated against luminance (see Appendix). We define the image contrast logarithmically as cf;(x) = In(I(x)/10), where 10 is a reference intensity defined for each image. We choose this constant such that Ex cf;(x) = 0; that is, the average contrast for each image is zero. Our analysis is of the contrast data cf;( x). 3 Scaling Recent measurements (Field '87, Burton & Moorhead '87) suggest that ensembles of natural scenes are scale-invariant. This means that and any quantity defined on a given scale has statistics which are invariant to any change in that scale. This seems sensible in light of the fact that the images are composed of objects at all distances, and so no particular angular scale should stand out. (Note that this does not imply that any particular image is fractal! Rather, the ensemble of scenes has statistics which are invariant to scale.) 3.1 Distribution of Contrasts We can test this scaling hypothesis directly by seeing how the statistics of various quantities change with scale. We define the contrast averaged over a box of size N x N (pixels) to be N cf;N = ~2 L cf;( i, j). i,j=l We now ask: "How does the probability P( cf;N) change with N?" In the left graph of figure 1 we plot log(P( cf;N / cf;~MS)) for N = 1,2,4,8,16,32 along with the parabola corresponding to a Gaussian of the same variance. By dividing out the RMS value we simply plot all the graphs on the same contrast scale. The graphs all lie atop one another, which means the contrast scales-the distribution's shape is invariant to a change in angular scale. Note that the probability is far from Gaussian, as the graphs have linear, and not parabolic, tails. Even after averaging nearly 1000 pixels (in the case of 32x32), it remains non-Gaussian. This breakdown of the central limit theorem implies that the pixels are correlated over very long distances. This is analogous to the physics of a thermodynamic system at a critical point. 3.2 Distribution of Gradients As another example of scaling, we consider the probability distribution of image gradients. We define the magnitude of the gradient by a discrete approximation Statistics of Natural Images: Scaling in the Woods 553 ., · 15 -2.5 ·35 ., '---............ -~--'---~-----'-----' ., -2 ., Figure 1: Left: Semi-log plot of P(</JN/(VJMS ) for N 1,2,4,8,16,32 with a Gaussia~ of the same variance for comparison (solid line). Right: Semi-log plot of P(GN/GN) for same set of N's with a Rayleigh distribution for comparison (solid line) . such that G(x) = IG(x)1 ~ 1 'V</J (x) I· We examine this quantity over different scales by first rescaling the images as above and then evaluating the gradient at the new scale. We plot log( P( G N / G N )) for N = 1,2,4,8,16,32 in the right graph of figure 1, along with the Rayleigh distribution, P ~ G exp( -aG2 ). If the images had Gaussian statistics, local gradients would be Rayleigh distributed. Note once again scaling of the distribution. 3.3 Power Spectrum Scaling can also be demonstrated at the level of the power spectrum. If the ensemble is scale-invariant, then the spectrum should be of the form A S(k) = k2-'7' where k is measured in cycles/degree, and S is the power spectrum averaged over orientations. The spectrum is shown in figure 2 on log-log axes. It displays overlapping data from the two focal lengths, and shows that the spectrum scales over about 2.5 decades in spatial frequency. We determine the parameters as A = (6.47±0.13) x 1O-3deg.(O.19) and 1J = 0.19 ± 0.01. The integrated power spectrum up to 60 cycles/degree (the human resolution limit) gives an RMS contrast of about 30%. 4 Local Filtering The early stages of vision consist of neurons which respond to local patches of images. What do the statistics of these local processing units look like? We convolve images with the filter shown in the left of figure 3, and plot the histogram of its output on a semi-log scale on the right of the figure. 554 Ruderman and Bialek ~ < " • • ~ '" • 'tl ~ ., " ~ ~ ., ~ 0 ~ ~ • ) 0 .': 0 rl '" 0 ..., -1 -2 -3 -4 -5 -6 -1. 5 -1 -0 . 5 0 0.5 1 1.5 LoglO[Spatial Frequency (cycles/degree») • • Figure 2: Power spectrum of the contrast of natural scenes (log-log plot). The distribution is quite exponential over nearly 4 decades in probability. In fact, almost any local linear filter which passes no DC has this property, including centersurround receptive fields. Information theory tells us that it is best to send signals with Gaussian statistics down channels which have power constraints. It is of interest, then, to find some type of filtering which transforms the exponential distributions we find into Gaussian quantities. Music, as it turns out, has some similar properties. An amplitude histogram from 5 minutes of "The Blue Danube" is shown on the left of figure 4. It is almost precisely exponential over 4 decades in probability. We can guess what causes the excesses over a Gaussian distribution at the peak and the tails; it's the dynamics. When a quiet passage is played the amplitudes lie only near zero, and create the excess in the peak. When the music is loud the fluctuations are large, thus creating the ·0. ·1 + .,. .. + ·2' ·2 Figure 3: Left: 2 X 2 local filter. Right: Semi-log plot of histogram of its output when filtering natural scenes. -<15 ., -15 -2 ·25 I I I / i I I I ; I I ! / i -4 Statistics of Natural Images: Scaling in the Woods 555 ".. .. .. .. ".""." .••••• ".,1, •••••••••••• ," •• .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. " " " :::::::1:::::::0:::::::1::::::: ... ... .. .. I I ........ 111,.,., II •• I ••• II. I ••• ,1111,. -2 Figure 4: Left: Semi-log histogram of "The Blue Danube" with a Gaussian for comparison (dashed). Right: 5 x 5 center-surround filter region. tails. Most importantly, these quiet and loud passages extend coherently in time; so to remove the peak and tails, we can simply slowly adjust a "volume knob" to normalize the fluctuations. The images are made of objects which have coherent structure over space, and a similar localized dynamic occurs. To remove it, we need some sort of gain control. To do this, we pass the images through a local filter and then normalize by the local standard deviation of the image (analogous to the volume of a sound passage): ./,( ) = ¢(x) - ¢(x) 'f/ X O'(x)' Here ¢(x) is the mean image contrast in the N x N region surrounding x, and O'(x) is the standard deviation within the same region (see the right of figure 4) . .-",-/ -1 " """" /' , --,\ > ~ - 1 , ! - 1 , ~ I , ~ ! / , \ i - 2 \ I \, ~ -, , ,/' \ 3 -, , I \ I ; \ -J I \ i \ -J , i \ . J , :' \\ i " ."~, "''\'' \'~" -. • c 0 , 1 5 :l 2.5 ) S Contr .... t Gradlent (UrHtl of Me.n1 Figure 5: Left: Semi-log plot of histogram of 1/J, with Gaussian for comparison (dashed). Right: Semi-log plot of histogram of gradients of 1/J, with Rayleigh distribution shown for comparison (dashed). We find that for a value N = 5 (ratio of the negative surround to the positive center), the histograms of 1/J are the closest to Gaussian (see the left of figure 5). Further, the histogram of gradients of 1/J is very nearly Rayleigh (see the right of 556 Ruderman and Bialek figure 5). These are both signatures of a Gaussian distribution. Functionally, this "variance normalization" procedure is similar to contrast gain control found in the retina and LGN (Benardete et ai, '92). Could its role be in "Gaussianizing" the image statistics? 5 Information in the Retina From the measured statistics we can place an upper bound on the amount of information an array of photo receptors conveys about natural images. We make the following assumptions: • Images are Gaussian with the measured power spectrum. This places an upper bound on the entropy of natural scenes, and thus an upper bound on the information represented. • The receptors sample images in a hexagonal array with diffraction-limited optics. There is no aliasing. • Noise is additive, Gaussian, white, and independent of the image. The output of the nth receptor is thus given by Yn = J d2x ¢(x) M(x - xn) + 'f/n, where Xn is the location of the receptor, M(x) is the point-spread function of the optics, and 'f/n is the noise. For diffraction-limited optics, M(k) ~ 1 - Ikl/kc, where kc is the cutoff frequency of 60 cycles/degree. In the limit of an infinite lattice, Fourier components are independent, and the total information is the sum of the information in each component: += Ac fkCdkklog[1+A1 2 IM(k)1 2S(k)]. 47J" Jo cu Here I is the information per receptor, Ac is the area of the unit cell in the lattice, and u 2 is the variance of the noise. We take S(k) = A/k 2- fJ , with A and 'f/ taking their measured values, and express the noise level in terms of the signal-to-noise ratio in the receptor. In figure 6 we plot the information per receptor as a function of SN R along with the information capacity (per receptor) of the photoreceptor lattice at that SN R, which is 1 C = 2 log [1 + S N R] . The information conveyed is less than 2 bits per receptor per image, even at SN R = 1000. The redundancy of this representation is quite high, as seen by the gap between the curves; at least as much of the information capacity is being wasted as is being used. Statistics of Natural Images: Scaling in the Woods 557 I (bits) 5 4 0.5 1 1.5 2 2.5 3 LoglO[SNR) Figure 6: Information per receptor per image (in bits) as a function of 10g(SN R) (lower line). Information capacity per receptor ( upper line). 6 Conclusions We have shown that images from the forest have scale-invariant, highly nonGaussian statistics. This is evidenced by the scaling of the non-Gaussian histograms and the power-law form of the power spectrum. Local linear filtering produces values with quite exponential probability distributions. In order to "Gaussianize," we must use a nonlinear filter which acts as a gain control. This is analogous to contrast gain control, which is seen in the mammalian retina. Finally, an array of receptors which encodes these natural images only conveys at most a few bits per receptor per image of information, even at high SN R. At an image rate of 50 per second, this places an information requirement of less than about 100 bits per second on a foveal ganglion cell. Appendix Snapshots were gathered using a Sony Mavica MVC-5500 still video camera equipped with a 9.5-123.5mm zoom lens. The red, green, and blue signals were combined according to the standard CIE formula Y = 0.59 G + 0.30 R + 0.11 B to produce a grayscale value at each pixel. The quantity Y was calibrated against incident luminance to produce the image intensity I(x). The images were cropped to the central 256 x 256 region. The dataset consists of 45 images taken at a 15mm focal length (images subtend 150 of visual angle) and 25 images at an 80mm focal length (3 0 of visual angle) . All images were of distant objects to avoid problems of focus. Images were chosen by placing the camera at a random point along a path and rotating the field of view until no nearby objects appeared in the frame. The camera was tilted by less than 100 up or down in an effort to avoid sky and ground. The forested environment (woods in New Jersey in springtime) consisted mainly of trees, rocks, hillside, and a stream. 558 Ruderman and Bialek Acknowledgements We thank H. B. Barlow, B. Gianulis, A. J. Libchaber, M. Potters, R. R. de Ruyter van Stevenink, and A. Schweitzer. Work was supported in part by a fellowship from the Fannie and John Hertz Foundation (to D.L.R.). References J .J. Atick and N. Redlich. Towards a theory of early visual processing Neural Computation, 2:308, 1990. E. A. Benardete, E. Kaplan, and B. W. Knight. Contrast gain control in the primate retina: P cells are not X-like, some M-cells are. Vis. Neuosci., 8:483-486, 1992. W. Bialek, D. L. Ruderman, and A. Zee. The optimal sampling of natural images: a design principle for the visual system?, in Advances in Neural Information Processing systems, 3, R. P. Lippman, J. E. Moody and D. S. Touretzky, eds., 1991. G. J. Burton and I. R. Moorhead. Color and spatial structure in natural scenes. Applied Optics, 26:157-170, 1987. D. J. Field. Relations between the statistics of natural images and the response properties of cortical cells. I. Opt. Soc. Am. A, 4:2379, 1987. J. H. van Hateren. Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation. I. Compo Physiol. A, 171:157-170, 1992. S. B. Laughlin. A simple coding procedure enhances a neuron's information capacity. Z. Naturforsh., 36c:910-912, 1981. M. V. Srinivasan, S. B. Laughlin, and A. Dubs. Predictive coding: a fresh view of inhibition in the retina. Proc. R. Soc. Lond. B, 216:427-459, 1982.
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GDS: Gradient Descent Generation of Symbolic Classification Rules Reinhard Blasig Kaiserslautern University, Germany Present address: Siemens AG, ZFE ST SN 41 81730 Miinchen, Germany Abstract Imagine you have designed a neural network that successfully learns a complex classification task. What are the relevant input features the classifier relies on and how are these features combined to produce the classification decisions? There are applications where a deeper insight into the structure of an adaptive system and thus into the underlying classification problem may well be as important as the system's performance characteristics, e.g. in economics or medicine. GDSi is a backpropagation-based training scheme that produces networks transformable into an equivalent and concise set of IF-THEN rules. This is achieved by imposing penalty terms on the network parameters that adapt the network to the expressive power of this class of rules. Thus during training we simultaneously minimize classification and transformation error. Some real-world tasks demonstrate the viability of our approach. 1 Introduction This paper deals with backpropagation networks trained to perform a classification task on Boolean or real-valued data. Given such a classification task in most cases it is not too difficult to devise a network architecture that is capable of learning the input-output relation as represented by a number of training examples. Once training is finished one has a black box which often does a quite good job not 1 Gradient Descent Symbolic Rule Generation 1093 1094 Blasig only on the training patterns but also on some previously unseen test patterns. A good generalization performance indicates that the network has grasped part of the structure inherent in the classification task. The net has figured out which input features are relevant to make a classification decision and which are not. It has also modelled the way the relevant features have to be combined in order to produce the classifying output. In many applications it is important to get an understanding of this information hidden inside the neural network. Not only does this help to create or verify a domain theory, the analysis of this information may also serve human experts to determine, when and in what way the classifier will fail. In order to explicate the network's implicit information, we transform it into a set of rules. This idea is not new, cf. (Saito and Nakano, 1988), (Bochereau and Bourgine, 1990), (Y. Hayashi, 1991) and (Towell and Shavlik, 1992). In contrast to these approaches, which extract rules after BP-training is finished, we apply penalty terms during training to adapt the network's expressive power to that of the rules we want to generate. Consequently the net will be transformable into an equivalent set of rules. Due to their good comprehensibility we restrict the rules to be of the form IF < premise> THEN < conclusion >, where the premise as well as the conclusion are Boolean expressions. To actually make the transformation two problems have to be solved: • Neural nets are well known for their distributed representation of information; so in order to transform a net into a concise and comprehensible rule set one has to find a way of condensing this information without substantially changing it . • In the case of backpropagation networks a continuous activation function determines a node's output depending on its activation. However, the dynamic of this function has no counterpart in the context of rule-based descriptions. We address these problems by introducing a penalty function Ep, which we add to the classification error Ec yielding the total back propagation error ET = ED + A * Ep. (1) 2 The Penalty Term The term Ep is intended to have two effects on the network weights. First, by a weight decay component it aims at reducing network complexity by pushing a (hopefully large) fraction of the weights to O. The smaller the net, the more concise the rules describing its behavior will be. As a positive side effect, this component will tend to act as a form of "Occam's razor": simple networks are more likely to exhibit good generalization than complex ones. Secondly, the penalty term should minimize the error caused by transforming the network into a set of rules. Adopting the common approach that each non-input neuron represents one rule, there would be no transformation error if the neurons' activation function were threshold functions; the Boolean node output would then indicate, whether the conclusion is drawn or not. But since backpropagation neurons use continuous activation functions like GDS: Gradient Descent Generation of Symbolic Classification Rules 1095 y = tanh (x) to transform their activation value x into the output value y, we are left with the difficulty of interpreting the continuous output of a neuron. Thus our penalty term will be designed to produce a high penalty for those neurons of the backpropagation net, whose behavior cannot be well approximated by threshold neurons, because their activation values are likely to fall into the nonsaturated region of the tanh-function2 . 1.00 0.00 -1.00 ,.-------------I I I I I I -3.00 0.00 3.00 Figure 1: We regard Ixl > 3 with Iyl = I tanh(x)I > 0.9 as the regions, where a sigmoidal neuron can be approximated by a threshold neuron. The nonsaturated region is marked by the dashed box. For a better understanding of our penalty term one has to be aware of the fact that IF-THEN rules with a Boolean premise and conclusion are essentially Boolean functions. It can easily be shown that any such function can be calculated by a network of threshold neurons provided there is one (sufficiently large) hidden layer. This is still true if we restrict connection weights to the values {-I, 0, I} and node thresholds to be integers (Hertz, Krogh and Palmer, 1991). In order to transfer this scenario to nets with sigmoidal activation functions and having in mind that the activation values of the sigmoidal neurons should always exceed ±3 (see figure 1), we require the nodes' biases to be odd multiples of ±3 and the weights Wji to obey Wji E {-6,0,6}. (2) We shortly comment on the practical problem that sometimes bias values as large as ±6m, (mi being the fan-in of node i) may be necessary to implement certain Boolean functions. This may slow down or even block the learning process. A simple solution to this problem is to use some additional input units with a constant output of +1. If the connections to these units are also subject to the penalty function Ep, it is sufficient to restrict the bias values to hi E {-3, 3}. (3) 2We have to point out that the conversion of sigmoidal neurons to threshold neurons will reduce the net's computational power: there are Boolean functions which can be computed by a net of sigmoidal neurons, but which exceed the capacity of a threshold net of the same topology (Maass, Schnitger and Sontag, 1991). Note that the objective to use threshold units is a consequence of the decision to search for rules of the type IF < premise > THEN < conclusion >. A failure of the net to simultaneously minimize both parts of the error measure may indicate that other rule types are more adequate to handle the given classification task. 1096 Blasig Now we can define penalty functions that push the biases and weights to the desired values. Obviously Eb (the bias penalty) and Ew (the weight penalty) have to be different: Eb(bi ) = 13-lbill (4) E (w .. ) - { 16 - IWji11 for IWjil ~ e w J' IWjil for IWjil < e (5) The parameter e determines whether a weight should be subject to decay or pushed to attain the value 6 (or -6 respectively). Figure 2 displays the graphs ofthe penalty functions. -3.0 3.0 -6.0 -8 8 6.0 Figure 2: The penalty functions Eb and Ew. The value of e is chosen with the objective that only those weights should exceed this value, which almost certainly have to be nonzero to solve the given classification task. Since we initialize the network with weights uniformly distributed in the interval [-0.5,0.5]' E> = 1.5 works well at the beginning of the training process. The penalty term then has the effect of a pure weight decay. When learning proceeds and the weights converge, we can slowly reduce the value of e, because superfluous weights will already have decayed. So after each sequence of 100 training patterns, say, we decrease e by a factor of 0.995. Observation shows that weights which once exceeded the value of e quickly reach 6 or -6 and that there are relatively few cases where a large weight is reduced again to a value smaller than e. Accordingly, the number of weights in {-6, 6} successively grows in the course of learning, and the criterion to stop training thus influences the number of nonzero weights. The end of training is determined by means of cross validation. However, we do not examine the cross validation performance of the trained net, but that of the corresponding rule set. This is accomplished by calculating the performance of the original net with all weights and biases replaced by their optimal values according to (2) and (3). The weighting factor A of the penalty term (see equation 1) is critical for good learning performance. We pursued the strategy to start learning with A = 0, so that the network parameters first move into a region where the classification error is small. If this error falls below a prespecified tolerance level L, A is incremented by 0.001. The factor A goes down by the same amount, when the error grows larger GDS: Gradient Descent Generation of Symbolic Classification Rules 1097 than L3. By adjusting the weighting factor every 100 training patterns we keep the classification error close to the tolerance level. The choice of L of course depends on the learning task. As a heuristic, L should be slightly larger than the classification error attainable by a non-penalized network. 3 Splice-Junction Recognition The DNA, carrying the genetic information of biological cells, can be thought to be composed of two types of subsequences: exons and introns. The task is to classify each DNA position as either an exon-to-intron transition (EI), an intronto-exon transition (IE) or neither (N). The only information available is a sequence of 30 nucleotides (A, C, G or T) before and 30 nucleotides after the position to be classified. Splice-junction recognition is a classification task that has already been investigated by a number of machine learning researchers using various adaptive models. The pattern reservoir contains about 3200 DNA samples, 30% of which were used for training, 10% for cross-validation and 60% for testing. Since we used a grandmothercell coding for the input DNA sequence, the network has an input layer of 4*60 neurons. With a hidden layer of 20 neurons4 and two output units for the classes EI and IE, this amounts to about 5000 free parameters. The following table compares the classification performance of our penalty term approach and other machine learning algorithms, cf. (Murphy and Aha, 1992). Table 1: Splice-junction recognition: error (in percent) of various machine learning algorithms algorithm N EI IE total KBANN 4.62 7.56 8.47 6.32 GDS 6.71 4.43 9.24 6.75 Backprop 5.29 5.74 10.75 6.77 Perceptron 3.99 16.32 17.41 10.43 ID3 8.84 10.58 13.99 10.56 Nearest Neighbor 31.11 11.65 9.09 20.74 Surprisingly, the GDS network turned out to be very small. The weight decay component of our penalty term managed to push all but 61 weights to zero, making use of only three hidden neurons. Thus in addition to performing very well, the network is transformable into a concise rule set, as follows5: 3Negative A-values are not allowed. 4. A reasonable size, considering the experiments described in (Shavlik et al., 1991) 5We adopt a. notation commonly used in this domain: @n denotes the position of the first nucleotide in the given sequence being left (negative n) or right (positive n) to the point to be classified. Nucleotide 'V' stands for (,C' or 'T'), 'X' is a.ny of {A, C, G, T}. Consequently, e.g. neuron hidden(2) is active iff at least four of the five nucleotides of the sequence 'GTAXG' are identical to the input pattern at positions 1 to 5 right of the possible splice junction. 1098 Blasig hidden(2): at least 4 nucleotides match sequence 11: 'GTAXG' hidden(11): at least 3 nucleotides match sequence 1-3: 'YAG' hidden(17): at least 1 nucleotides matches sequence 1-1: 'GG' class EI: hidden(2) AID hidden(11) class IE: IOT(hidden(2» AID hidden(17) 4 Prediction of Interest Rates This is an application, where the network input is a vector of real numbers. Since our approach can only handle binary input, we supplement the net with a discretization layer that provides a thermometer code representation (Hancock 1988) of the continuous valued input. In contrast to pure Boolean learning algorithms (Goodman, Miller and Smyth, 1989), (Mezard and Nadal, 1989), which can also be endowed with discretization facilities, here the discretization process is fully integrated into the learning scheme, as the discretization intervals will be adapted by the backpropagation algorithm. The data comprises a total of 226 patterns, which we distribute randomly on three sets: training set (60%), cross-validation set (20%) and test set (20%). The input represents the monthly development of 14 economic time series during the last 19 years. The Boolean target indicates, whether the interest rates will go up or down during the six months succeeding the reference month6 • The time series include among others month of the year, income of private households or the amount of German foreign investments. For some time series it is useful not to take the raw feature measurements as input, but the difference between two succeeding measurements; this is advantageous if the underlying time series show only small changes relative to their absolute values. All series were normalized to have values in the range from -1 to +1. We used a network containing a discretization layer of two neurons per input dimension. So there are 28 discretization neurons, which are fully connected to the 10 hidden nodes. The output layer consists of a single neuron. Since our data set is relatively small, the intention to obtain simple rules is not only motivated by the objective of comprehensibility, but also by the notion that we cannot expect a large rule set to be justified by a small amount of training data. In fact, during training 90% of the weights were set to zero and three hidden units proved to be sufficient for this task. Nevertheless the prediction error on the test set could be reduced to 25%. This compares to an error rate of about 20% attainable by a standard backpropagation network with one hidden layer of ten neurons and no input discretization. We thus sacrificed 5% of prediction performance to yield a very compact net, that can be easily transformed into a set of rules. Some of the generated rules are shown below. The first rule e.g. states that interest rates will rise if private income increases AND foreign investments decrease by a certain amount during the reference month. If the rules produce contradicting predictions for a given input, the final decision will be made according to a majority vote. A tie is broken by the bias value of the 61.e. the month where the input data has been measured. GDS: Gradient Descent Generation of Symbolic Classification Rules 1099 output unit, which states that by default interest rates will rise. IF (at least 2 ot { increase ot private income < 0.73%, decrease ot toreign investments < 64 MID DM }) THE! (interest rates will rise) ELSE (interest rates will fall). IF (at least 3 ot { increase of business climate estimate < 1.76%, treasury bonds yields (11 month ago) > 7.36%, treasury bonds yields (12 month ago) > 8.2%, increase ot foreign investments < 60 MID DM }) THE! (interest rates will tall) ELSE (interest rates will rise). 5 Conclusion and Future Work G DS is a learning algorithm that utilizes a penalty term in order to prepare a backpropagation network for rule extraction. The term is designed to have two effects on the network's weights: • By a weight decay component, the number of nonzero weights is reduced: thus we get a net that can hopefully be transformed into a concise and comprehensible rule set . • The penalty term encourages weight constellations that keep the node activations out of the nonsaturated part of the activation function. This is motivated by the fact that rules of the type IF < premise > THEN < conclusion > can only mimic the behavior of threshold units. The important point is that our penalty function adapts the net to the expressive power of the type of rules we wish to obtain. Consequently, we are able to transform the network into an equivalent rule set. The applicability of GDS was demonstrated on two tasks: splice-junction recognition and the prediction of German interest rates. In both cases the generated rules not only showed a generalization performance close to or even superior to what can be attained by other machine learning approaches such as MLPs or ID3. The rules also prove to be very concise and comprehensible. This is even more remarkable, since both applications represent real-world tasks with a large number of inputs. Clearly the applied penalty terms impose severe restrictions on the network parameters: besides minimizing the number of nonzero weights, the weights are restricted to a small set of distinct values. Last but not least, the simplification of sigmoidal to threshold units also affects the net's computational power. There are applications, where such a strong bias may negatively influence the net's learning capabilities. Furthermore our current approach is only applicable to tasks with binary target patterns. These limitations can be overcome by dealing with more general rules than those of the Boolean IF-THEN type. Future work will go into this direction. 1100 Blasig Acknowledgements I wish to thank Hans-Georg Zimmermann and Ferdinand Hergert for many useful discussions and for providing the data on interest rates, and Patrick Murphy and David Aha for providing the UCI Repository of ML databases. This work was supported by a grant of the Siemens AG, Munich. References L. Bochereau, P. Bourgine. (1990) Extraction of Semantic Features and Logical Rules from a Multilayer Neural Network. Proceedings of the 1990 IJCNN - Washington DC, Vol.II 579-582. R.M. Goodman, J .W. Miller, P. Smyth. (1989) An Information Theoretic Approach to Rule-Based Connectionist Expert Systems. Advances in Neural Information Processing Systems 1, 256-263. San Mateo, CA: Morgan Kaufmann. P.J .B. Hancock. (1988) Data Representation in Neural Nets: an Empirical Study. Proc. Connectionist Summer School. Y. Hayashi. (1991) A Neural Expert System with Automated Extraction of Fuzzy If-Then Rules and its Application to Medical Diagnosis. Advances in Neural Information Processing Systems 3, 578-584. San Mateo, CA: Morgan Kaufmann. J. Hertz, A. Krogh, R.G. Palmer. (1991) Introduction to the Theory of Neural Computation. Addison-Wesley. C.M. Higgins, R.M. Goodman. (1991) Incremental Learning with Rule-Based Neural Networks. Proceedings of the 1991 IEEE INNS International Joint Conference on Neural Networks - Seattle, Vol.1 875-880. M. Mezard, J .-P. Nadal. (1989) Learning in Feedforward Layered Networks: The Tiling Algorithm. J. Phys. A: Math. Gen. 22, 2191-2203. W. Maass, G. Schnitger, E.D. Sontag. (1991) On the Computational Power of Sigmoids versus Boolean Threshold Circuits. Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, 767-776. P.M. Murphy, D.W. Aha. (1992). UCI Repository of machine learning databases [ftp-site: ics.uci.edu: pub/machine-Iearning-databases]. Irvine, CA: University of California, Department of Information and Computer Science. J .R. Quinlan. (1986) Induction of Decision Trees. Machine Learning, 1: 81-106. K. Siato, R. Nakano. (1988) Medical diagnostic expert systems based on PDP model. Proc. IEEE International Conference on Neural Networks Vol. I 255-262. V. Tresp, J. Hollatz, S. Ahmad. (1993) Network Structuring and Training Using Rule-Based Knowledge. Advances in Neural Information Processing Systems 5, 871-878. San Mateo, CA: Morgan Kaufman. G.G. Towell, J.W. Shavlik. (1991) Training Knowledge-Based Neural Networks to Recognize Genes in DNA Sequences. In: Lippmann, Moody, Touretzky (eds.), Advances in Neural Information Processing Systems 3, 530-536. San Mateo, CA: Morgan Kaufmann.
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Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach Justin A. Boyan School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Michael L. Littman· Cognitive Science Research Group Bellcore Morristown, NJ 07962 Abstract This paper describes the Q-routing algorithm for packet routing, in which a reinforcement learning module is embedded into each node of a switching network. Only local communication is used by each node to keep accurate statistics on which routing decisions lead to minimal delivery times. In simple experiments involving a 36-node, irregularly connected network, Q-routing proves superior to a nonadaptive algorithm based on precomputed shortest paths and is able to route efficiently even when critical aspects of the simulation, such as the network load, are allowed to vary dynamically. The paper concludes with a discussion of the tradeoff between discovering shortcuts and maintaining stable policies. 1 INTRODUCTION The field of reinforcement learning has grown dramatically over the past several years, but with the exception of backgammon [8, 2], has had few successful applications to large-scale, practical tasks. This paper demonstrates that the practical task of routing packets through a communication network is a natural application for reinforcement learning algorithms. *Now at Brown University, Department of Computer Science 671 672 Boyan and Littman Our "Q-routing" algorithm, related to certain distributed packet routing algorithms [6, 7], learns a routing policy which balances minimizing the number of "hops" a packet will take with the possibility of congestion along popular routes. It does this by experimenting with different routing policies and gathering statistics about which decisions minimize total delivery time. The learning is continual and online, uses only local information, and is robust in the face of irregular and dynamically changing network connection patterns and load. The experiments in this paper were carried out using a discrete event simulator to model the transmission of packets through a local area network and are described in detail in [5]. 2 ROUTING AS A REINFORCEMENT LEARNING TASK A packet routing policy answers the question: to which adjacent node should the current node send its packet to get it as quickly as possible to its eventual destination? Since the policy's performance is measured by the total time taken to deliver a packet, there is no "training signal" for directly evaluating or improving the policy until a packet finally reaches its destination. However, using reinforcement learning, the policy can be updated more quickly and using only local information. Let Qx(d, y) be the time that a node x estimates it takes to deliver a packet P bound for node d by way of x's neighbor node y, including any time that P would have to spend in node x's queue. l Upon sending P to y, x immediately gets back y's estimate for the time remaining in the trip, namely t = . min Q y ( d, z) zEnelghbors of y If the packet spent q units of time in x's queue and s units of time in transmission between nodes x and y, then x can revise its estimate as follows: new estimate old estimate ~~ LlQx(d,Y)=17( q+s+t Qx(d,y)) where 17 is a "learning rate" parameter (usually 0.5 in our experiments). The resulting algorithm can be characterized as a version of the Bellman-Ford shortest paths algorithm [1, 3] that (1) performs its path relaxation steps asynchronously and online; and (2) measures path length not merely by number of hops but rather by total delivery time. We call our algorithm "Q-routing" and represent the Q-function Qx( d, y) by a large table. We also tried approximating Qx with a neural network (as in e.g. [8, 4]), which allowed the learner to incorporate diverse parameters of the system, including local queue size and time of day, into its distance estimates. However, the results of these experiments were inconclusive. 1 We denote the function by Q because it corresponds to the Q function used in the reinforcement learning technique of Q-learning [10]. Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach 673 - .... .... - •• •• Figure 1: The irregular 6 x 6 grid topology 3 RESULTS We tested the Q-routing algorithm on a variety of network topologies, including the 7-hypercube, a 116-node LATA telephone network, and an irregular 6 x 6 grid. Varying the network load, we measured the average delivery time for packets in the system after learning had settled on a routing policy, and compared these delivery times with those given by a static routing scheme based on shortest paths. The result was that in all cases, Q-routing is able to sustain a higher level of network load than could shortest paths. This section presents detailed results for the irregular grid network pictured in Figure l. Under conditions of low load, the network learns fairly quickly to route packets along shortest paths to their destinations. The performance vs. time curve plotted in the left part of Figure 2 demonstrates that the Q-routing algorithm, after an initial period of inefficiency during which it learns the network topology, performs about as well as the shortest path router, which is optimal under low load. As network load increases, however, the shortest path routing scheme ceases to be optimal: it ignores the rising levels of congestion and soon floods the network with packets. The right part of Figure 2 plots performance vs. time for the two routing schemes under high load conditions: while shortest path is unable to tolerate the packet load, Q-routing learns an efficient routing policy. The reason for the learning algorithm's success is apparent in the "policy summary diagrams" in Figure 3. These diagrams indicate, for each node under a given policy, how many of the 36 x 35 point-to-point routes go through that node. In the left part of Figure 3, which summarizes the shortest path routing policy, two nodes in the center of the network (labeled 570 and 573) are on many shortest paths and thus become congested when network load is high. By contrast, the diagram on the right shows that Q-routing, under conditions of high load, has learned a policy which routes 674 Boyan and Littman Q-routing Q-routing 500 Shortest paths ----. 500 ~ Shortest paths ----. I ! !: l I I V ,I II ~ II 1\ , I • ,I ,I, ~ I .,' I J~ .,' ~' I " I " I "' , • I ,ft I I I I :' :1 ~,' I: IVI~" II I 1"1 ~ I~ I " II 'I" I I I ",~ 1 ,,\II : ,'''" I ~ I', II I f I I I I ) "', I 400 400 I, f ' ~'~I I' I I ",' I' l ", ~ ,I \I I I 1 ,I !: ~ ~ ':1 ",1' Q) :: :~ E " I, F I ~ Q) 300 300 , .~ , , Q) , I 0 ~ , Q) ~/~ Cl , ~ I , Q) , > 200 200 !~ « " .,' , , I , I , , , 100 100 " " j " I I , , \ : 0 ------0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Simulator Time Simulator Time Figure 2: Performance under low load and high load 1 ~4--131-+1-7------------1+6--125--1~5 364--392--396-----------396--393--367 , , , , , , , , , , , , , , , , , , , , , , 5~----~2 3T7 54----4;3 1 $9 , , , 375 102---5.9 , , , 207 45----5:4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3~4--2t2--2~8-----------~--2tu--3~3 , • • • I I , " . , ' , .. '" ,. . , ' 2rS445'l'{)- ........ 5fS3$&2'?8 ' . • I • , . . , I 1 tB--1~--219 2$s--1~5--110 2$2--21-8--t1 4 , , , 2?7--2<t>1--2-1 7 , , , , , , , , , , , , , , , , , , 1 rB--1t8---~3 ~---1-t3--146 , , , 1 ~4--1t9--1~8 1 $O--1-+1--H~2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 45----7-6----58 79---105---=15 108--121---=14 Figure 3: Policy summaries: shortest path and Q-routing under high load Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach 675 some traffic over a longer than necessary path (across the top of the network) so as to avoid congestion in the center of the network. The basic result is captured in Figure 4, which compares the performances of the shortest path policy and Q-routing learned policy at various levels of network load. Each point represents the median (over 19 trials) of the mean packet delivery time after learning has settled. When the load is very low, the Q-routing algorithm routes nearly as efficiently as the shortest path policy. As load increases, the shortest path policy leads to exploding levels of network congestion, whereas the learning algorithm continues to route efficiently. Only after a further significant increase in load does the Q-routing algorithm, too, succumb to congestion. 1B Q-routing Shortest paths '' . W 16 u c: Ql u III 14 Ql '5 0-w ~ 12 Ql E i= ~ 10 Ql . ~ Qi 0 Ql B 0> ~ Ql ~ 6 , , _ .. --"-- " 4 0.5 1.5 2 2.5 3 3.5 4 4.5 Network Load Level Figure 4: Delivery time at various loads for Q-routing and shortest paths 3.1 DYNAMICALLY CHANGING NETWORKS One advantage a learning algorithm has over a static routing policy is the potential for adapting to changes in crucial system parameters during network operation. We tested the Q-routing algorithm, unmodified, on networks whose topology, traffic patterns, and load level were changing dynamically: Topology We manually disconnected links from the network during simulation. Qualitatively, Q-routing reacted quickly to such changes and was able to continue routing traffic efficiently. Traffic patterns We caused the simulation to oscillate periodically between two very different request patterns in the irregular grid: one in which all traffic was directed between the upper and lower halves of the network, and one in which all traffic was directed between the left and right halves. Again, 676 Boyan and Littman after only a brief period of inefficient routing each time the request pattern switched, the Q-routing algorithm adapted successfully. Load level When the overall level of network traffic was raised during simulation, Q-routing quickly adapted its policy to route packets around new bottlenecks. However, when network traffic levels were then lowered again, adaptation was much slower, and never converged on the optimal shortest paths. This effect is discussed in the next section. 3.2 EXPLORATION Given the similarity between the Q-routing update equation and the Bellman-Ford recurrence for shortest paths, it seems surprising that there is any difference whatsoever between the performance of Q-routing and shortest paths routing at low load, as is visible in Figure 4. However, a close look at the algorithm reveals that Q-routing cannot fine-tune a policy to discover shortcuts, since only the best neighbor's estimate is ever updated. For instance, if a node learns an overestimate of the delivery time for an optimal route, then it will select a suboptimal route as long as that route's delivery time is less than the erroneous estimate of the optimal route's delivery time. This drawback of greedy Q-Iearning is widely recognized in the reinforcement learning community, and several exploration techniques have been suggested to overcome it [9]. A common one is to have the algorithm select actions with some amount of randomness during the initial learning period[10]. But this approach has two serious drawbacks in the context of distributed routing: (1) the network is continuously changing, thus the initial period of exploration never ends; and more significantly, (2) random traffic has an extremely negative effect on congestion. Packets sent in a suboptimal direction tend to add to queue delays, slowing down all the packets passing through those queues, which adds further to queue delays, etc. Because the nodes make their policy decisions based on only local information, this increased congestion actually changes the problem the learners are trying to solve. Instead of sending actual packets in a random direction, a node using the "full echo" modification of Q-routing sends requests for information to its immediate neighbors every time it needs to make a decision. Each neighbor returns a single number-using a separate channel so as to not contribute to network congestion in our model-giving that node's current estimate of the total time to the destination. These estimates are used to adjust the Qx(d, y) values for each neighbor y. When shortcuts appear, or if there are inefficiencies in the policy, this information propagates very quickly through the network and the policy adjusts accordingly. Figure 5 compares the performance of Q-routing and shortest paths routing with "full echo" Q-routing. At low loads the performance of "full echo" Q-routing is indistinguishable from that of the shortest path policy, as all inefficiencies are purged. Under high load conditions, "full echo" Q-routing outperforms shortest paths but the basic Q-routing algorithm does better still. Our analysis indicates that "full echo" Q-routing constantly changes policy under high load, oscillating between using the upper bottleneck and using the central bottleneck for the majority of crossnetwork traffic. This behavior is unstable and generally leads to worse routing times under high load. Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach 677 18 Q-routing Shortest paths ----Full Echo ----2l 16 c: Q) 0 Ul 14 .!!! '" CT Q; ::: 12 ..!! Q) E i= ~ 10 Q) .~ Qi 0 Q) 8 Cl ~ Q) > « 6 ~,""-,~~ ••. _.".L.~~" •• ~"··----'::···· . 4 0.5 1.5 2 2.5 3 3.5 4 4.5 Network Load Level Figure 5: Delivery time at varIOUS loads for Q-routing, shortest paths and "full echo" Q-routing Ironically, the "drawback" of the basic Q-routing algorithm-that it does no exploration and no fine-tuning after initially learning a viable policy-actually leads to improved performance under high load conditions. We still know of no single algorithm which performs best under all load conditions. 4 CONCLUSION This work considers a straightforward application of Q-Iearning to packet routing. The "Q-routing" algorithm, without having to know in advance the network topology and traffic patterns, and without the need for any centralized routing control system, is able to discover efficient routing policies in a dynamically changing network. Although the simulations described here are not fully realistic from the standpoint of actual telecommunication networks, we believe this paper has shown that adaptive routing is a natural domain for reinforcement learning. Algorithms based on Q-routing but specifically tailored to the packet routing domain will likely perform even better. One of the most interesting directions for future work is to replace the table-based representation of the routing policy with a function approximator. This could allow the algorithm to integrate more system variables into each routing decision and to generalize over network destinations. Potentially, much less routing information would need to be stored at each node, thereby extending the scale at which the algorithm is useful. We plan to explore some of these issues in the context of packet routing or related applications such as auto traffic control and elevator control. 678 Boyan and Littman Acknowledgements The authors would like to thank for their support the Bellcore Cognitive Science Research Group, the National Defense Science and Engineering Graduate fellowship program, and National Science Foundation Grant IRI-9214873. References [1] R. Bellman. On a routing problem. Quarterly of Applied Mathematics, 16(1):87-90, 1958. [2] J. Boyan. Modular neural networks for learning context-dependent game strategies. Master's thesis, Computer Speech and Language Processing, Cambridge University, 1992. [3] L. R. Ford, Jr. Flows in Networks. Princeton University Press, 1962. [4] L.-J . Lin. Reinforcement Learning for Robots Using Neural Networks. PhD thesis, School of Computer Science, Carnegie Mellon University, 1993. [5] M. Littman and J. Boyan. A distributed reinforcement learning scheme for network routing. Technical Report CMU-CS-93-165, School of Computer Science, Carnegie Mellon University, 1993. [6] H. Rudin. On routing and delta routing: A taxonomy and performance comparison of techniques for packet-switched networks. IEEE Transactions on Communications, COM-24(1):43-59, January 1976. [7] A. Tanenbaum. Computer Networks. Prentice-Hall, second edition edition, 1989. [8] G. Tesauro. Practial issues in temporal difference learning. Machine Learning, 8(3/4), May 1992. [9] Sebastian B. Thrun. The role of exploration in learning control. In David A. White and Donald A. Sofge, editors, Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. Van Nostrand Reinhold, New York, 1992. [10] C. Watkins. Learning from Delayed Rewards. PhD thesis, King's College, Cambridge, 1989.
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A Massively-Parallel SIMD Processor for Neural Network and Machine Vision Applications Michael A. Glover Current Technology, Inc. 99 Madbury Road Durham, NH 03824 W. Thomas Miller, III Department of Electrical and Computer Engineering The University of New Hampshire Durham, NH 03824 Abstract This paper describes the MM32k, a massively-parallel SIMD computer which is easy to program, high in performance, low in cost and effective for implementing highly parallel neural network architectures. The MM32k has 32768 bit serial processing elements, each of which has 512 bits of memory, and all of which are interconnected by a switching network. The entire system resides on a single PC-AT compatible card. It is programmed from the host computer using a C++ language class library which abstracts the parallel processor in terms of fast arithmetic operators for vectors of variable precision integers. 1 INTRODUCTION Many well known neural network techniques for adaptive pattern classification and function approximation are inherently highly parallel, and thus have proven difficult to implement for real-time applications at a reasonable cost. This includes 843 844 Glover and Miller a variety of learning systems such as radial basis function networks [Moody 1989], Kohonen self-organizing networks [Kohonen 1982], ART family networks [Carpenter 1988], and nearest-neighbor interpolators [Duda 1973], among others. This paper describes the MM32k, a massively-parallel SIMD computer which is easy to program, high in performance, low in cost and effective for implementing highly parallel neural network architectures. The MM32k acts as a coprocessor to accelerate vector arithmetic operations on PC-AT class computers, and can achieve giga-operation per second performance on suitable problems. It is programmed from the host computer using a C++ language class library, which overloads typical arithmetic operators, and supports variable precision arithmetic. The MM32k has 32768 bit serial PEs, or processing elements, each of which has 512 bits of memory, and all of which are interconnected by a switching network. The PEs are combined with their memory on an single DRAM memory chip giving 2048 processors per chip. The entire 32768 processor system resides on a single ISA bus compatible card. It is much more cost effective than other SIMD processors [Hammerstrom 1990; Hillis 1985; Nickolls 1990; Potter 1985] and more flexible than fixed purpose chips [Holler 1991]. 2 SIMD ARCHITECTURE The SIMD PE array contains 32768 one bit processors, each with 512 bits of memory and a connection to the interconnection network. The PE array design is unique in that 2048 PEs, including their PE memory, are realized on a single chip. The total PE array memory is 2 megabytes and has a peak memory bandwidth is 25 gigabytes per second. The PE array can add 8 bit integers at 2.5 gigaoperations per second. It also dissipates less than 10 watts of power and is shown in Figure 1. Each PE has three one bit registers, a 512 bit memory, and a one bit AL U. It performs bit serial arithmetic and can therefore vary the number of bits of precision to fit the problem at hand, saving SIMD instruction cycles and SIMD memory. There are 17 instructions in the PE instruction set, all of which execute at a 6.25 MIPS rate. The PE instruction set is functionally complete in that it can perform boolean NOT and OR functions and can therefore perform any operation, including arithmetic and conditional operations. A single PE is shown in Figure 2. The interconnection network allows data to be sent from one PE to another. It is implemented by a 64*64 full crossbar switch with 512 PEs connected to each port of the switch. It allows data to be sent from one PE to another PE, an arbitrary distance away, in constant time. The peak switch bandwidth is 280 megabytes per second. The switch also allows the PE array to perform data reduction operations, such as taking the sum or maximum over data elements distributed across all PEs. 3 C++ PROGRAMMING ENVIRONMENT The purpose of the C++ programming environment is to allow a programmer to declare and manipulate vectors on the MM32k as if they were variables in a program running on the host computer. Programming is performed entirely on the host, using standard MS-DOS or Windows compatible C++ compilers. The C++ programming environment for the MM32k is built around a C++ class, named A Massively-Parallel SIMD Processor for Neural Network and Machine Vision Applications 845 Host Computer (PC-AT) Vector Instructions and Data Controller PE Instructions and Data 1-11PE PE PE PE PE PE 0 1 2 3 ... j . .. 3276 '--,.- -,.'--,-Switch Figure 1: A block diagram of the MM32k. Bit 511 Bit 510 Bit 509 9 Bit Address from Controller .. 512 Bit Memory Address Bus Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Bit 0 PE ALU Opcode .. from Controller Data Bus· Data to Data from Switch A Register M Register B Register Switch 1 Bit 1 Bit 1 Bit Figure 2: A block diagram of a single processing element (PE). 846 Glover and Miller Table 1: 8 Bit Operations With 32768 and 262144 Elements 8 bit operation copy vector+vector vector+scalar vector*vector vector*scalar vector>scalar align( vector ,scalar) sum( vector) maximum( vector) Actual MOPS with length of 32768 1796 1455 1864 206 426 1903 186 52 114 Actual MOPS with length of 262144 9429 2074 3457 215 450 6223 213 306 754 MM_ VECTOR, which represents a vector of integers. Most of the standard C arithmetic operators, such as +, -, *, I, =, and> have been overloaded to work with this class. Some basic functions, such as absolute value, square root, minimum, maximum, align, and sum, have also been overloaded or defined to work with the class. The significance of the class MM_ VECTOR is that instances of it look and act like ordinary variables in a C++ program. So a programmer may add, subtract, assign, and manipulate these vector variables from a program running on the host computer, but the storage associated with them is in the SIMD memory and the vector operations are performed in parallel by the SIMD PEs. MM_ VECTORs can be longer than 32768. This is managed (transparent to the host program) by placing two or more vector elements in the SIMD memory of each PE. The class library keeps track of the number of words per PE. MM_ VECTORs can be represented by different numbers of bits. The class library automatically keeps track of the number of bits needed to represent each MM_ VECTOR without overflow. For example, if two 12 bit integers were added together, then 13 bits would be needed to represent the sum without overflow. The resulting MM_VECTOR would have 13 bits. This saves SIMD memory space and SIMD PE instruction cycles. The performance of the MM32k on simple operators running under the class library is listed in Table 1. 4 NEURAL NETWORK EXAMPLES A common operation found in neural network classifiers (Kohonen, ART, etc.) is the multi-dimensional nearest-neighbor match. If the network has a large number of nodes, this operation is particularly inefficient on single processor systems, which must compute the distance metric for each node sequentially. Using the MM32k, the distance metrics for all nodes (up to 32768 nodes) can be computed simultaneously, and the identification of the minimum distance can be made using an efficient tree compare included in the system microcode. A Massively-Parallel SIMD Processor for Neural Network and Machine Vision Applications 847 Table 2: Speedup on Nearest Neighbor Search Processor Time for Time for MM32k MM32k 32768 nodes 65536 nodes speedup for speedup for 32768 nodes 65536 nodes MM32k 2.2 msec 3.1 msec 1:1 1:1 i486 350 msec 700 msec 159:1 226:1 MIPS 970 msec 1860 msec 441:1 600:1 Alpha 81 msec 177 msec 37:1 57:1 SPARC 410 msec 820 msec 186:1 265:1 Figure 3 shows a C++ code example for performing a 16-dimensional nearest neighbor search over 32768 nodes. The global MM_ VECTOR variable state[16] defines the 16-dimensionallocation of each node. Each logical element of state[ ] (state[O], state[l], etc.) is actually a vector with 32768 elements distributed across all processors. The routine find_besLmatchO computes the euclidean distance between each node's state and the current test vector test_input[ ], which resides on the host processor. Note that the equations appear to be scalar in nature, but in fact direct vector operations to be performed by all processors simultaneously. The performance of the nearest neighbor search shown in Figure 3 is listed in Table 2. Performance on the same task is also listed for four comparison processors: a Gateway2000 mode14DX2-66V with 66 MHz 80486 processor (i486), a DECstation 5000 Model 200 with 25 MHz MIPS R3000A processor (MIPS), a DECstation 3000 Model 500AXP with 150 MHz Alpha AXP processor (Alpha), and a Sun SPARCstation 10 Model 30 with 33 MHz SuperSPARC processor (SPARC). There are 16 subtractions, 16 additions, 16 absolute values, one global minimum, and one global first operation performed. The MM32k is tested on problems with 32768 and 65536 exemplars and compared against four popular serial machines performing equivalent searches. The MM32k requires 3.1 milliseconds to search 65536 exemplars which is 265 times faster than a SPARC 10. The flexibility of the MM32k for neural network applications was demonstrated by implementing compl~te fixed-point neural network paradigms on the MM32k and on the four comparison processors (Table 3). Three different neural network examples were evaluated. The first was a radial basis function network with 32,768 basis functions (rational function approximations to gaussian functions). Each basis function had 9 8-bit inputs, 3 16-bit outputs (a vector basis function magnitude), and independent width parameters for each of the nine inputs. The performances listed in the table (RBF) are for feedforward response only. The second example was a Kohonen self-organizing network with a two-dimensional sheet of Kohonen nodes of dimension 200x150 (30,000 nodes). The problem was to map a nonlinear robotics forward kinematics transformation with eight degrees of freedom (8-bit parameters) onto the two-dimensional Kohonen layer. The performances listed in the table (Kohonen) are for self-organizing training. The third example problem was a neocognitron for target localization in a 256x256 8-bit input image. The first hidden layer of the neocognitron had 8 256x256 sheets of linear convolution units 848 Glover and Miller 1* declare 16-D ""32k exemplars *1 ""_VECTOR state[16] = { ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168) , ""_VECTOR(32168) , ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168), ""_VECTOR(32168) , ""_VECTOR(32168) }; 1* return PE number of processor with closest match */ long find_best_match(long test_input[16]) { } int i; ""_VECTOR difference(32168); ""_VECTOR distance(32168); 1* differences *1 1* distances *1 1* compute the 16-D distance scores *1 distance = OJ for (i=O; i<16; ++i) { difference = state[i] - test_input[i]; distance = distance + (difference * difference); } 1* return the PE number for minimum distance *1 return first(distance == minimum(distance»; Figure 3: A C++ code example implementing a nearest neighbor search. A Massively-Parallel SIMD Processor for Neural Network and Machine Vision Applications 849 Table 3: MM32k Speedup for Select Neural Network Paradigms Processor MM32k i486 MIPS Alpha SPARC RBF 1:1 161:1 180:1 31:1 94:1 Kohonen NCGTRN 1:1 1:1 76:1 336:1 69:1 207:1 11:1 35:1 49:1 378:1 with 16x16 receptive fields in the input image. The second hidden layer of the neocognitron had 8 256x256 sheets of sigmoidal units (fixed-point rational function approximations to sigmoid functions) with 3x3x8 receptive fields in the first hidden layer. The output layer of the neocognitron had 256x256 sigmoidal units with 3x3x8 receptive fields in the second hidden layer. The performances listed in the table (NCGTRN) correspond to feedforward response followed by backpropagation training. The absolute computation times for the MM32k were 5.1 msec, 10 msec, and 1.3 sec, for the RBF, Kohonen, and NCGTRN neural networks, respectively. Acknowledgements This work was supported in part by a grant from the Advanced Research Projects Agency (ARPA/ONR Grant #NOOOI4-92-J-1858). References J. 1. Potter. (1985) The Massively Parallel Processor, Cambridge, MA: MIT Press. G. A. Carpenter and S. Grossberg. (1988) The ART of adaptive pattern recognition by a self-organizing neural network. Computer vol. 21, pp. 77-88. R. O. Duda and P. E. Hart. (1973) Pattern Classification and Scene Analysis. New York: Wiley. D. Hammerstrom. (1990) A VLSI architecture for high-performance, low cost, onchip learning, in Proc. IJCNN, San Diego, CA, June 17-21, vol. II, pp. 537-544. W. D. Hillis. (1985) The Connection Machine. Cambridge, MA: MIT Press. M. Holler. (1991) VLSI implementations oflearning and memory systems: A review. In Advances in Neural Information Processing Systems 3, ed. by R. P. Lippman, J. E. Moody, and D. S. Touretzky, San Francisco, CA: Morgan Kaufmann. T. Kohonen. (1982) Self-organized formation of topologically correct feature maps. Biological Cybernetics, vol. 43, pp. 56-69. J. Moody and C. Darken. (1989) Fast learning in networks of locally- tuned processing units. Neural Computation, vol. 1, pp. 281-294. J. R. Nickolls. (1990) The design of the MasPar MP-1: A cost-effective massively parallel computer. In Proc. COMPCON Spring '90, San Francisco, CA, pp. 25-28 ..
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How to Choose an Activation Function H. N. Mhaskar Department of Mathematics California State University Los Angeles, CA 90032 hmhaska@calstatela.edu c. A. Micchelli IBM Watson Research Center P. O. Box 218 Yorktown Heights, NY 10598 cam@watson.ibm.com Abstract We study the complexity problem in artificial feedforward neural networks designed to approximate real valued functions of several real variables; i.e., we estimate the number of neurons in a network required to ensure a given degree of approximation to every function in a given function class. We indicate how to construct networks with the indicated number of neurons evaluating standard activation functions. Our general theorem shows that the smoother the activation function, the better the rate of approximation. 1 INTRODUCTION The approximation capabilities of feedforward neural networks with a single hidden layer has been studied by many authors, e.g., [1, 2, 5]. In [10], we have shown that such a network using practically any nonlinear activation function can approximate any continuous function of any number of real variables on any compact set to any desired degree of accuracy. A central question in this theory is the following. If one needs to approximate a function from a known class of functions to a prescribed accuracy, how many neurons will be necessary to accomplish this approximation for all functions in the class? For example, Barron shows in [1] that it is possible to approximate any function satisfying certain conditions on its Fourier transform within an L2 error of O(1/n) using a feedforward neural network with one hidden layer comprising of n2 neurons, each with a sigmoidal activation function. On the contrary, if one is interested in a class of functions of s variables with a bounded gradient on [-1, I]S , 319 320 Mhaskar and Micchelli then in order to accomplish this order of approximation, it is necessary to use at least 0(11$) number of neurons, regardless of the activation function (cf. [3]). In this paper, our main interest is to consider the problem of approximating a function which is known only to have a certain number of smooth derivatives. We investigate the question of deciding which activation function will require how many neurons to achieve a given order of approximation for all such functions. We will describe a very general theorem and explain how to construct networks with various activation functions, such as the Gaussian and other radial basis functions advocated by Girosi and Poggio [13] as well as the classical squashing function and other sigmoidal functions. In the next section, we develop some notation and briefly review some known facts about approximation order with a sigmoidal type activation function. In Section 3, we discuss our general theorem. This theorem is applied in Section 4 to yield the approximation bounds for various special functions which are commonly in use. In Section 5, we briefly describe certain dimension independent bounds similar to those due to Barron [1], but applicable with a general activation function. Section 6 summarizes our results. 2 SIGMOIDAL-TYPE ACTIVATION FUNCTIONS In this section, we develop some notation and review certain known facts. For the sake of concreteness, we consider only uniform approximation, but our results are valid also for other LP -norms with minor modifications, if any. Let s 2: 1 be the number of input variables. The class of all continuous functions on [-1, IP will be denoted by C$. The class of all 27r- periodic continuous functions will be denoted by C$*. The uniform norm in either case will be denoted by II . II. Let IIn,I,$,u denote the set of all possible outputs of feedforward neural networks consisting of n neurons arranged in I hidden layers and each neuron evaluating an activation function (j where the inputs to the network are from R$. It is customary to assume more a priori knowledge about the target function than the fact that it belongs to C$ or cn. For example, one may assume that it has continuous derivatives of order r 2: 1 and the sum of the norms of all the partial derivatives up to (and including) order r is bounded. Since we are interested mainly in the relative error in approximation, we may assume that the target function is normalized so that this sum of the norms is bounded above by 1. The class of all the functions satisfying this condition will be denoted by W: (or W:'" if the functions are periodic). In this paper, we are interested in the universal approximation of the classes W: (and their periodic versions). Specifically, we are interested in estimating the quantity (2.1) where (2.2) sup En,l,$,u(f) JEW: En,l,$,u(f) := p Anf III - PII· E n,l,s,1T The quantity En,l,s ,u(l) measures the theoretically possible best order of approximation of an individual function I by networks with 11 neurons. We are interested How to Choose an Activation Function 321 in determining the order that such a network can possibly achieve for all functions in the given class. An equivalent dual formulation is to estimate (2.3) En,l,s,O'(W:) := min{m E Z : sup Em,l,s,O'(f) ~ lin}. fEW: This quantity measures the minimum number of neurons required to obtain accuracy of lin for all functions in the class W:. An analogous definition is assumed for W:* in place of W: . Let IH~ denote the class of all s-variable trigonometric polynomials of order at most n and for a continuous function f, 27r-periodic in each of its s variables, (2.4) E~(f):= min Ilf - PII· PEIH~ We observe that IH~ can be thought of as a subclass of all outputs of networks with a single hidden layer comprising of at most (2n + 1)" neurons, each evaluating the activation function sin X. It is then well known that (2.5) Here and in the sequel, c, Cl, ... will denote positive constants independent of the functions and the number of neurons involved, but generally dependent on the other parameters of the problem such as r, sand (j. Moreover, several constructions for the approximating trigonometric polynomials involved in (2.5) are also well known. In the dual formulation, (2.5) states that if (j(x) := sinx then (2.6) It can be proved [3] that any "reasonable" approximation process that aims to approximate all functions in W:'" up to an order of accuracy lin must necessarily depend upon at least O(ns/r) parameters. Thus, the activation function sin x provides optimal convergence rates for the class W:*. The problem of approximating an r times continuously differentiable function f R s --+ R on [-1, I]S can be reduced to that of approximating another function from the corresponding periodic class as follows. We take an infinitely many times differentiable function 1f; which is equal to 1 on [-2,2]S and 0 outside of [-7r, 7rp. The function f1f; can then be extended as a 27r-periodic function. This function is r times continuously differentiable and its derivatives can be bounded by the derivatives of f using the Leibnitz formula. A function that approximates this 27r-periodic function also approximates f on [-I,I]S with the same order of approximation. In contrast, it is not customary to choose the activation function to be periodic. In [10] we introduced the notion of a higher order sigmoidal function as follows. Let k > O. We say that a function (j : R --+ R is sigmoidal of order k if (2.7) lim (j( x) - 1 lim (j(x) - 0 x-+oo xk -, x-+-oo xk , and (2.8) xE R. 322 Mhaskar and Micchelli A sigmoidal function of order 0 is thus the customary bounded sigmoidal function. We proved in [10] that for any integer r ~ 1 and a sigmoidal function (j of order r - 1, we have (2.9) if s = 1, if s > 2. Subsequently, Mhaskar showed in [6] that if (j is a sigmoidal function of order k > 2 and r ~ 1 then, with I = O(log r/ log k)), (2.10) Thus, an optimal network can be constructed using a sigmoidal function of higher order. During the course of the proofs in [10] and [6], we actually constructed the networks explicitly. The various features of these constructions from the connectionist point of view are discussed in [7, 8, 9]. In this paper, we take a different viewpoint. We wish to determine which activation function leads to what approximation order. As remarked above, for the approximation of periodic functions, the periodic activation function sin x provides an optimal network. Therefore, we will investigate the degree of approximation by neural net.works first in terms of a general periodic activation function and then apply these results to the case when the activation function is not periodic. 3 A GENERAL THEOREM In this section, we discuss the degree of approximation of periodic functions using periodic activation functions. It is our objective to include the case of radial basis functions as well as the usual "first. order" neural networks in our discussion. To encompass both of these cases, we discuss the following general formuation. Let s ~ d 2: 1 be integers and ¢J E Cd •. We will consider the approximation of functions in ca. by linear combinat.ions of quantities of the form ¢J(Ax + t) where A is a d x s matrix and t E Rd. (In general, both A and t are parameters ofthe network.) When d = s, A is the identity matrix and ¢J is a radial function, then a linear combination of n such quantities represents the output of a radial basis function network with n neurons. When d = 1 then we have the usual neural network with one hidden layer and periodic activation function ¢J. We define the Fourier coefficients of ¢J by the formula (3.1) , 1 1 . t ¢J(m) := (2 )d ¢J(t)e-zm. dt, 7r [-lI',lI']d Let (3.2) and assume that there is a set J co Itaining d x s matrices with integer entries such that (3.3) How to Choose an Activation Function 323 where AT denotes the transpose of A. If d = 1 and ¢(l) #- 0 (the neural network case) then we may choose S4> = {I} and J to be Z8 (considered as row vectors). If d = sand ¢J is a function with none of its Fourier coefficients equal to zero (the radial basis case) then we may choose S4> = zs and J = {Is x s}. For m E Z8, we let km be the multi-integer with minimum magnitude such that m = ATkm for some A = Am E J. Our estimates will need the quantities (3.4) mn := min{I¢(km)1 : -2n::; m::; 2n} and (3.5) Nn := max{lkml : -2n::; m < 2n} where Ikml is the maximum absolute value of the components of km. In the neural network case, we have mn = 1¢(1)1 and Nn = 1. In the radial basis case, Nn = 2n. Our main theorem can be formulated as follows. THEOREM 3.1. Let s ~ d ~ 1, n ~ 1 and N ~ Nn be integers, f E C n , ¢J E C d*. It is possible to construct a network (3.6) such that (3.7) In (3.6), the sum contains at most O( n S Nd) terms, Aj E J, tj E R d, and dj are linear functionals of f, depending upon n, N, <p. The estimate (3.7) relates the degree of approximation of f by neural networks explicitly in terms of the degree of approximation of f and ¢J by trigonometric polynomials. Well known estimates from approximation theory, such as (2.5), provide close connections between the smoothness of the functions involved and their degree of trigonometric polynomial approximation. In particular, (3.7) indicates that the smoother the function ¢J the better will be the degree of approximation. In [11], we have given explicit constructions of the operator Gn,N,4>. The formulas in [11] show that the network can be trained in a very simple manner, given the Fourier coefficients of the target function. The weights and thresholds (or the centers in the case of the radial basis networks) are determined universally for all functions being approximated. Only the coefficients at the output layer depend upon the function . Even these are given explicitly as linear combinations of the Fourier coefficients of the target function. The explicit formulas in [11] show that in the radial basis case, the operator Gn ,N,4> actually contains only O( n + N)S summands. 4 APPLICATIONS In Section 3, we had assumed that the activation function ¢J is periodic. If the activation function (J is not periodic, but satisfies certain decay conditions near 324 Mhaskar and Micchelli 00, it is still possible to construct a periodic function for which Theorem 3.1 can be applied. Suppose that there exists a function 1j; in the linear span of Au,J := {(T( Ax + t) A E J, t E R d}, which is integrable on R d and satisfies the condition that (4.1) for some T> d. Under this assumption, the function ( 4.2) 1j;0 (x):= L 1j;(x - 27rk) kEZ d is a 27r-periodic function integrable on [-7r, 7r]s. We can then apply Theorem 3.1 with 1j;0 instead of ¢. In Gn,N,tjJo, we next replace 1j;0 by a function obtained by judiciously truncating the infinite sum in (4.2). The error made in this replacement can be estimated using (4.1). Knowing the number of evaluations of (T in the expression for '1/) as a finite linear combination of elements of Au,J, we then have an estimate on the degree of approximation of I in terms of the number of evaluations of (T. This process was applied on a number of functions (T. The results are summarized in Table 1. 5 DIMENSION INDEPENDENT BOUNDS In this section, we describe certain estimates on the L2 degree of approximation that are independent of the dimension of the input space. In this section, II· II denotes th(' L2 norm on [-1, I]S (respectively [-7r, 7r]S) and we approximate functions in the class S Fs defined by (5.1 ) SFtI := {I E C H : II/l1sF,s:= L li(m)l::; I}. mEZ' Analogous to the degree of approximation from IH~, we define the n-th degree of approximation of a function I E CS* by the formula (5.2) En s(f) := inf III - L i(m)eimOxlI , ACZ' ,IAI~n mEA where we require the norm involved to be the L2 norm. In (5.2), there is no need to assume that n is an integer. Let ¢ be a square integrable 27r-periodic function of one variable. We define the L2 degree of approximation by networks with a single hidden layer by the formula (.5.3) E~~~)f) := PEj~~l"'~ III - PII where m is the largest integer not exceeding n. Our main theorem in this connection is the following THEOREM 5.1. Let s 2: 1 be an integer, IE SFs , ¢ E Li and J(1) f:. O. Then, for integers n, N 2: 1, (5.4) How to Choose an Activation Function 325 Table 1: Order of magnitude of En,l,s,o-(W:) for different O"S --Function 0' En IsoRemarks Sigmoidal, order r - 1 n1/ r s=d=I,/=1 Sigmoidal, order r - 1 n lJ / r+(s+2r)/r2 s ~ 2, d = 1, I = 1 xk, if x ~ 0, 0, if x < O. n IJ / r+ (2r+s )/2r k k ~ 2, s ~ 2, d = 1, I = 1 (1 + e-x)-l nlJ/r(log n)2 s~2,d=I,/=1 Sigmoidal, order k n lJ / r k ~ 2, s ~ 1, d = 1, I = o (log r/ log k)) exp( -lxl2 /2) n2s /r s=d>2/=1 , Ixlk(log Ixl)6 n( IJ /r)(2+(3s+2r)/ k) S = d > 2, k > 0, k + seven, 6 = 0 if s odd, 1 if s even, I = 1 where {6n} is a sequence of positive numbers, 0 ::; 6n ::; 2, depending upon f such that 6n --- 0 as n --- 00. Moreover, the coefficients in the network that yields (5.1,) are bounded, independent of nand N. We may apply Theorem 5.1 in the same way as Theorem 3.1. For the squashing activation fUllction, this gives an order of approximation O(n-l/2) with a network consisting of n(lo~ n)2 neurons arranged in one hidden layer. With the truncated power function x + (cf. Table 1, entry 3) as the activation function, the same order of approximation is obtained with a network with a single hidden layer and O(n1+1/(2k») neurons. 6 CONCLUSIONS. We have obtained estimates on the number of neurons necessary for a network with a single hidden layer to provide a gi ven accuracy of all functions under the only a priori assumption that the derivatives of the function up to a certain order should exist. We have proved a general theorem which enables us to estimate this number 326 Mhaskar and Micchelli in terms of the growth and smoothness of the activation function. We have explicitly constructed networks which provide the desired accuracy with the indicated number of neurons. Acknowledgements The research of H. N. Mhaskar was supported in part by AFOSR grant 2-26 113. References 1. BARRON, A. R., Universal approximation bounds for superposition of a sigmoidal function, IEEE Trans. on Information Theory, 39. 2. CYBENKO, G., Approximation by superposition of sigmoidal functions, Mathematics of Control, Signals and Systems, 2, # 4 (1989), 303-314. 3. DEVORE, R., HOWARD , R. AND MICCHELLI, C.A., Optimal nonlinear approximation, Manuscripta Mathematica, 63 (1989), 469-478. 4. HECHT-NILESEN, R., Thoery of the backpropogation neural network, IEEE International Conference on Neural Networks, 1 (1988), 593-605. 5. HORNIK, K., STINCHCOMBE, M. AND WHITE, H ., Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989),359-366. 6. MHASKAR, H. N., Approximation properties of a multilayered feedforward artificial neural network, Advances in Computational Mathematics 1 (1993), 61-80. 7. MHASKAR, H. N., Neural networks for localized approximation of real functions, in "Neural Networks for Signal Processing, III", (Kamm, Huhn, Yoon, Chellappa and Kung Eds.), IEEE New York, 1993, pp. 190-196. 8. MHASKAR, H. N., Approximation of real functions using neural networks, in Proc. of Int. Conf. on Advances in Comput. Math., New Delhi, India, 1993, World Sci. Publ., H. P. Dikshit, C. A. Micchelli eds., 1994. 9. MHASKAR, H. N., Noniterative training algorithms for neural networks, Manuscript, 1993. 10. MHASKAR, H. N. AND MICCHELLI, C. A. , Approximation by superposition of a sigmoidal function and radial basis functions, Advances in Applied Mathematics, 13 (1992),350-373. 11. MHASKAR, H. N. AND MICCHELLI, C. A., Degree of approximation by superpositions of a fixed function, in preparation. 12. MHASKAR, H. N. AND MICCHELLI, C. A., Dimension independent bounds on the degree of approximation by neural networks, Manuscript, 1993. 13. POGGIO, T. AND GIROSI, F., Regularization algorithms for learning that are equivalent to multilayer networks, Science, 247 (1990), 978-982.
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Efficient Simulation of Biological Neural Networks on Massively Parallel Supercomputers with Hypercube Archi tect ure Ernst Niebur Computation and Neural Systems California Institute of Technology Pasadena, CA 91125, USA Dean Brettle Booz, Allen and Hamilton, Inc. 8283 Greensboro Drive McLean, VA 22102-3838, USA Abstract We present a neural network simulation which we implemented on the massively parallel Connection Machine 2. In contrast to previous work, this simulator is based on biologically realistic neurons with nontrivial single-cell dynamics, high connectivity with a structure modelled in agreement with biological data, and preservation of the temporal dynamics of spike interactions. We simulate neural networks of 16,384 neurons coupled by about 1000 synapses per neuron, and estimate the performance for much larger systems. Communication between neurons is identified as the computationally most demanding task and we present a novel method to overcome this bottleneck. The simulator has already been used to study the primary visual system of the cat. 1 INTRODUCTION Neural networks have been implemented previously on massively parallel supercomputers (Fujimoto et al., 1992, Zhang et al., 1990). However, these are implementations of artificial, highly simplified neural networks, while our aim was explicitly to provide a simulator for biologically realistic neural networks. There is also at least one implementation of biologically realistic neuronal systems on a moderately 904 Efficient Simulation of Biological Neural Networks 905 parallel but powerful machine (De Schutter and Bower, 1992), but the complexity of the used neuron model makes simulation of larger numbers of neurons impractical. Our interest here is to provide an efficient simulator of large neural networks of cortex and related subcortical structures. The most important characteristics of the neuronal systems we want to simulate are the following: • Cells are highly interconnected (several thousand connections per cell) but far from fully interconnected. • Connections do not follow simple deterministic rules (like, e.g., nearest neighbor connections). • Cells communicate with each other via delayed spikes which are binary events ("all-or-nothing"). • Such communication events are short (1 ms) and infrequent (1 to 100 per second). • The temporal fine structure of the spike trains may be an important information carrier (Kreiter and Singer, 1992, Richmond and Optican, 1990, Softky and Koch, 1993). 2 IMPLEMENTATION The biological network was modelled as a set of improved integrate-and-fire neurons which communicate with each other via delayed impulses (spikes). The single-cell model and details of the connectivity have been described in refs. (Wehmeier et al., 1989, Worgotter et al., 1991). Despite the rare occurrence of action potentials, their processing accounts for the major workload of the machine. The efficient implementation of inter-neuron communication is therefore the decisive factor which determines the efficacy of the simulator implementation. By "spike propagation" we denote the process by which a neuron communicates the occurrence of an action potential to all its postsynaptic partners. While the most efficient computation of the neuronal equations is obtained by mapping each neuron on one processor, this is very inefficient for spike propagation. This is due to the fact that spikes are rare events and that in the SIMD architecture used, each processor has to wait for the completion of the current tasks of all other processors. Therefore, only very few processors are active at any given time step. A more efficient data representation than provided by this "direct" algorithm is shown in Fig. 1. In this "transposed" scheme, a processor changes its role from simulating one of the neurons to simulating one synapse, which is, in general, not a synapse of the neuron simulated by the processor (see legend of Fig. 1). At any given time step, the addresses of the processors representing spiking neurons are broadcast along binary trees which are implemented efficiently (in time wmplexity log2M for M processors) in a hypercube architecture such as the CM-2. We obtain further computational efficiency by dividing the processor array into "partitions" of size M and by implementing partially parallel I/O scheduling (both not discussed here). 906 Niebur and Brettle 1 2 3 4 5 M-1 M 1 ,1 1,2 1,3 1,4 · · 1 , i · · 1,M 2,1 2,2 2,3 2,4 · · 2, i · · 2,M 3,1 3,2 3,3 3,4 · 3, i · · 3,M J...-............ _'-__ ..IIl.I _ ......... ,.... ... ......... -.. _ .............. _ .. i,1 i,2 i,3 i,4 · ..... "". .. ....... ~ .. _\,a .... ..,. --_ ...... r-- t · i, i ............. 1----- ._-................. i, M ........ -- --_ ..... ......... - r.-. ......... ---~r' -- t--...... .. ..... ..,.- 1----- --- ......... -..--... --+ + + + + + M,1 M,2 M,3 M,4 M,i . M,M Figure 1: Transposed storage method for connections. The storage space for each of the N processors is represented by a vertical column. A small part of this space is used for the time-dependent variables describing each of the N neurons (upper part of each column, "Cell data"). The main part of the storage is used for datasets consisting of the addresses, weights and delays of the synapses ("Synapse data"), represented by the indices i, j in the figure. For instance, "1, I" stands for the first synapse of neuron 1, "1,2" for the second synapse of this neuron and so on. Note that the storage space of processor i does not hold the synapses of neuron i. If neuron i generates a spike, all M processors are used for propagating the spike (black arrows) Efficient Simulation of Biological Neural Networks 907 3 PERFORMANCE ANALYSIS In order to accurately compare the performance of the described spike propagation algorithms, we implemented both the direct algorithm and the transposed algorithm and compared their performances with analytical estimates. Cf) ::s ........ E-t 10 1 0.1 0.01 0.001 0.0001 0.001 0.01 0.1 1 p Figure 2: Execution time for the direct algorithm (diamonds) and the transposed algorithm (crosses) as function of the spiking probability p for each cell. If all cells fire at each time step, there is no advantage for the transposed algorithm; in fact, it is at a disadvantage due to the overhead discussed in the text. Therefore, the two curves cross at a value just below p = 1. As expected, the largest difference between them is found for the smallest values of p. Figure 2 compares the time required for the direct algorithm to the time required for the transposed algorithm as a function of p, the average number of spikes per neuron per time step. Note that while the time required rises much more rapidly for the transposed algorithm than the direct algorithm, it takes significantly less time for p < 0.5. The peak speedup was a factor of 454 which occurred at p = 0.00012 (or 1.2 impulses per second at a timestep of O.lms, corresponding approximately to spontaneous spike rates). The absolutely highest possible speedup, obtained if there is exactly one spike in every partition at every time step, is equal to M (M == 1024 in this simulation). The average speedup is determined by the maximal number of spiking neurons per time step in any partition, since the processors in all partitions have to wait until the last partition has propagated all of its spikes. The average maximal number of spikes in a system of N partitions, each one consisting of M 908 Niebur and Brettle neurons IS M N Nmar(p, M, N) = {; k J; ( ~ ) TI(k)mft(k)N-m (1) where p is the spiking probability of one cell, II(k) is the probability that a given partition has k spikes and k-l IT(k) = L II(i) (2) i=O 1000 100 10 1 0.0001 0.001 0.01 0.1 1 p Figure 3: Speedup of the transposed algorithm over the direct algorithm as a function of p for different VP ratios; M = 1024. The ideal speedup (uppermost curve; diamonds), computed in eq. 3 essentially determines the observed speedup. (lower curves; "+" signs: VP-ratio=1, diamonds: VP-raio=2, crosses: VP-ratio=4.). The difference between the ideal and the effectively obtained speedup is due to communication and other overhead of the transposed algorithm. Note that the difference in speedup for different VP ratios (difference between lower curves) is relatively small, which shows that the penalty for using larger neuron numbers is not large. As expected, the speedup approaches unity for p ~ 1 in all cases. It can be shown that for independent neurons and for low spike rates, II( k) is the Poisson distribution and IT(k) the incomplete r function. The average maximal Efficient Simulation of Biological Neural Networks 909 number of spikes for M = 1024 and different values of P (eq. 1) can be shown to be a mildly growing function of the number of partitions which shows that the performance will not be limited crucially by changing the number of partitions. Therefore, the algorithm scales well with increasing network size and the performance-limiting factor is the activity level in the network and not the size of the network. This is also evident in Fig. 3 which shows the effectively obtained speedup compared to the ideal speedup, which would be obtained if the transposed algorithm were limited only by eq. 1 and would not require any additional communication or other overhead. Using Nmax(P, M, N) from eq. 1 it is clear that this ideal speedup is given by M (3) Nmax(P, M, N) The difference between theory and experiment can be attributed to the time required for the spread operation and other additional overhead associated with the transposed algorithm. At P = 0.0010 (or 10 ips) the obtained speedup is a factor of 106. 4 VERY LARGE SYSTEMS U sing the full local memory of the machine and the "Virtual Processor" capability of the CM-2, the maximal number of neurons that can be simulated without any change of algorithm is as high as 4,194,304 ("4M"). Figure 3 shows that the speedup is reduced only slightly as the number of neurons increases, when the additional neurons are simulated by virtual processors. The performance is essentially limited by the mean network activity, whose effect is expressed by eq. 3, and the additional overhead originating from the higher "VP ratio" is small. This corroborates our earlier conclusion that the algorithm scales well with the size of the simulated system. Although we did not study the scaling of execution time with the size of the simulated system for more than 16,384 real processors, we expect the total execution time to be basically independent of the number of neurons, as long as additional neurons are distributed on additional processors. Acknowlegdements We thank U. Wehmeier and F. Worgotter who provided us with the code for generating the connections, and G. Holt for his retina simulator. Discussions with C. Koch and F. W orgotter were very helpful. We would like to thank C. Koch for his continuing support and for providing a stimulating research atmosphere. We also acknowledge the Advanced Computing Laboratory of Los Alamos National Laboratory, Los Alamos, NM 87545. Some of the numerical work was performed on computing resources located at this facility. This work was supported by the National Science Foundation, the Office of Naval Research, and the Air Force Office of Scientific Research. 910 Niebur and Brettle References De Schutter E. and Bower J .M. (1992). Purkinje cell simulation on the Intel Touchstone Delta with GENESIS. In Mihaly T. and Messina P., editors, Proceedings of the Grand Challenge Computing Fair, pages 268-279. CCSF Publications, Caltech, Pasadena CA. Fujimoto Y., Fukuda N., and Akabane T. (1992). Massively parallel architectures for large scale neural network simulations. IEEE Transactions on Neural Networks, 3(6):876-888. Kreiter A.K. and Singer W. (1992). Oscillatory neuronal responses in the visualcortex of the awake macaque monkey. Europ. 1. Neurosci., 4(4):369-375. 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. Softky W. and Koch C. (1993). The highly irregular firing of cortical-cells is inconsistent with temporal integration of random epsps. 1. Neurosci., 13(1):334-350. Wehmeier U., Dong D., Koch C., and van Essen D. (1989). Modeling the visual system. In Koch C. and Segev I., editors, Methods in Neuronal Modeling, pages 335-359. MIT Press, Cambridge, MA. Worgotter F., Niebur E., and Koch C. (1991). Isotropic connections generate functional asymmetrical behavior in visual cortical cells. J. Neurophysiol., 66(2):444-459. Zhang X., Mckenna M., Mesirov J., and Waltz D. (1990). An efficient implementation of the back-propagation algorithm on the Connection Machine CM-2. In Touretzky D.S., editor, Neural Information Processing Systems 2, pages 801-809. Morgan-Kaufmann, San Mateo, CA.
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