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800	Convergence of Stochastic Iterative Dynamic Programming Algorithms Tommi Jaakkola'" Michael I. Jordan Satinder P. Singh Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Increasing attention has recently been paid to algorithms based on dynamic programming (DP) due to the suitability of DP for learning problems involving control. In stochastic environments where the system being controlled is only incompletely known, however, a unifying theoretical account of these methods has been missing. In this paper we relate DP-based learning algorithms to the powerful techniques of stochastic approximation via a new convergence theorem, enabling us to establish a class of convergent algorithms to which both TD("\) and Q-Iearning belong. 1 INTRODUCTION Learning to predict the future and to find an optimal way of controlling it are the basic goals of learning systems that interact with their environment. A variety of algorithms are currently being studied for the purposes of prediction and control in incompletely specified, stochastic environments. Here we consider learning algorithms defined in Markov environments. There are actions or controls (u) available for the learner that affect both the state transition probabilities, and the probability distribution for the immediate, state dependent costs (Ci( u)) incurred by the learner. Let Pij (u) denote the probability of a transition to state j when control u is executed in state i. The learning problem is to predict the expected cost of a ... E-mail: tommi@psyche.mit.edu 703 704 Jaakkola, Jordan, and Singh fixed policy p (a function from states to actions), or to obtain the optimal policy (p) that minimizes the expected cost of interacting with the environment. If the learner were allowed to know the transition probabilities as well as the immediate costs the control problem could be solved directly by Dynamic Programming (see e.g., Bertsekas, 1987). However, when the underlying system is only incompletely known, algorithms such as Q-Iearning (Watkins, 1989) for prediction and control, and TD(>.) (Sutton, 1988) for prediction, are needed. One of the central problems in developing a theoretical understanding of these algorithms is to characterize their convergence; that is, to establish under what conditions they are ultimately able to obtain correct predictions or optimal control policies. The stochastic nature of these algorithms immediately suggests the use of stochastic approximation theory to obtain the convergence results. However, there exists no directly available stochastic approximation techniques for problems involving the maximum norm that plays a crucial role in learning algorithms based on DP. In this paper, we extend Dvoretzky's (1956) formulation of the classical RobbinsMunro (1951) stochastic approximation theory to obtain a class of converging processes involving the maximum norm. In addition, we show that Q-Iearning and both the on-line and batch versions of TD(>.) are realizations of this new class. This approach keeps the convergence proofs simple and does not rely on constructions specific to particular algorithms. Several other authors have recently presented results that are similar to those presented here: Dayan and Sejnowski (1993) for TD(A), Peng and Williams (1993) for TD(A), and Tsitsiklis (1993) for Q-Iearning. Our results appear to be closest to those of Tsitsiklis (1993). 2 Q-LEARNING The Q-Iearning algorithm produces values-"Q-values"-by which an optimal action can be determined at any state. The algorithm is based on DP by rewriting Bellman's equation such that there is a value assigned to every state-action pair instead of only to a state. Thus the Q-values satisfy Q(s,u) = cs(u) +, ~pssl(u)maxQ(sl,ul) L....J 1.).1 (1) 8 ' where c denotes the mean of c. The solution to this equation can be obtained by updating the Q-values iteratively; an approach known as the vaz'ue iteration method. In the learning problem the values for the mean of c and for the transition probabilities are unknown. However, the observable quantity CSt (Ut) +, maxQ(St+l, u) (2) 1.). where St and Ut are the state of the system and the action taken at time t, respectively, is an unbiased estimate of the update used in value iteration. The Q-Iearning algorithm is a relaxation method that uses this estimate iteratively to update the current Q-values (see below). The Q-Iearning algorithm converges mainly due to the contraction property of the value iteration operator. Convergence of Stochastic Iterative Dynamic Programming Algorithms 705 2.1 CONVERGENCE OF Q-LEARNING Our proof is based on the observation that the Q-Iearning algorithm can be viewed as a stochastic process to which techniques of stochastic approximation are generally applicable. Due to the lack of a formulation of stochastic approximation for the maximum norm, however, we need to slightly extend the standard results. This is accomplished by the following theorem the proof of which can be found in Jaakkola et al. (1993). Theorem 1 A random iterative process ~n+I(X) = (l-ll:n(X))~n(x)+lin(x)Fn(x) converges to zero w.p.l under the following assumptions: 1) The state space is finite. 2) Ln ll:n(x) = 00, Ln ll:~(x) < 00, Ln lin(x) = 00, Ln Ii~(x) < 00, and E{lin(x)IPn} ~ E{ll:n(x)IPn} uniformly w.p.1. 3) II E{Fn(x)IPn} Ilw~ 'Y II ~n IlwI where'Y E (0,1). 4) Var{Fn(x)IPn} ~ C(1+ II ~n Ilw)2, where C is some constant. Here Pn = {~n, ~n-I, .. ·' Fn- I, ... , ll:n-I,· .. , lin-I, ... } stands for the past at step n. Fn(x), ll:n(x) and lin(x) are allowed to depend on the past insofar as the above conditions remain valid. The notation II . Ilw refers to some weighted maximum norm. In applying the theorem, the ~n process will generally represent the difference between a stochastic process of interest and some optimal value (e.g., the optimal value function). The formulation of the theorem therefore requires knowledge to be available about the optimal solution to the learning problem before it can be applied to any algorithm whose convergence is to be verified. In the case of Q-Iearning the required knowledge is available through the theory of DP and Bellman's equation in particular. The convergence of the Q-Iearning algorithm now follows easily by relating the algorithm to the converging stochastic process defined by Theorem 1.1 Theorem 2 The Q-learning algorithm given by Qt+I(St, Ut) = (1 - ll:t(St, Ut))Qt(St, ut) + ll:t(St, ut}[CSt(ut) + 'Yvt(St+dJ converges to the optimal Q(s, u) values if 1) The state and action spaces are finite. 2) Lt ll:t(s, u) = 00 and Lt ll:;(s, u) < 00 uniformly w.p.1. 3) Var{cs(u)} is bounded. 1 We note that the theorem is more powerful than is needed to prove the convergence of Q-learning. Its generality, however, allows it to be applied to other algorithms as well (see the following section on TD(>.)). 706 Jaakkola, Jordan, and Singh 3) If, = 1, all policies lead to a cost free terminal state w.p.1. Proof. By subtracting Q(s, u) from both sides of the learning rule and by defining Llt(s, u) = Qt(s, u) - Q(s, u) together with (3) the Q-learning algorithm can be seen to have the form of the process in Theorem 1 with !3t(s, u) = at(s, u). To verify that Ft(s, u) has the required properties we begin by showing that it is a contraction mapping with respect to some maximum norm. This is done by relating Ft to the DP value iteration operator for the same Markov chain. More specifically, maxIE{Ft(i, u)}1 u j < ,max ~Pij(u)maxIQt(j,v) - Q(j,v)1 u 6 v j ,muax LPij(U)Va(j) = T(Va)(i) j where we have used the notation Va(j) = maXv IQt(j, v)-Q(j, v)1 and T is the DP value iteration operator for the case where the costs associated with each state are zero. If, < 1 the contraction property of E{ Ft (i, u)} can be obtained by bounding I:j Pij(U)Va(j) by maxj Va(j) and then including the, factor. When the future costs are not discounted (, = 1) but the chain is absorbing and all policies lead to the terminal state w.p.1 there still exists a weighted maximum norm with respect to which T is a contraction mapping (see e.g. Bertsekas & Tsitsiklis, 1989) thereby forcing the contraction of E{Ft(i, u)}. The variance of Ft(s, u) given the past is within the bounds of Theorem 1 as it depends on Qt(s, u) at most linearly and the variance of cs(u) is bounded. Note that the proof covers both the on-line and batch versions. o 3 THE TD(-\) ALGORITHM The TD(A) (Sutton, 1988) is also a DP-based learning algorithm that is naturally defined in a Markov environment. Unlike Q-learning, however, TD does not involve decision-making tasks but rather predictions about the future costs of an evolving system. TD(A) converges to the same predictions as a version ofQ-learning in which there is only one action available at each state, but the algorithms are derived from slightly different grounds and their behavioral differences are not well understood. The algorithm is based on the estimates 00 V/\(i) = (1 - A) L An-l~(n)(i) (4) n=l where ~(n)(i) are n step look-ahead predictions. The expected values of the ~>"(i) are strictly better estimates of the correct predictions than the lit (i)s are (see Convergence of Stochastic Iterative Dynamic Programming Algorithms 707 Jaakkola et al., 1993) and the update equation of the algorithm Vt+l(it) = vt(it) + adV/(it) - Vt(it)J (5) can be written in a practical recursive form as is seen below. The convergence of the algorithm is mainly due to the statistical properties of the V? (i) estimates. 3.1 CONVERGENCE OF TDP) As we are interested in strong forms of convergence we need to impose some new constraints, but due to the generality of the approach we can dispense with some others. Specifically, the learning rate parameters an are replaced by a n( i) which satisfy Ln an(i) = 00 and Ln a~(i) < 00 uniformly w.p.1. These parameters allow asynchronous updating and they can, in general, be random variables. The convergence of the algorithm is guaranteed by the following theorem which is an application of Theorem 1. Theorem 3 For any finite absorbing Markov chain, for any distribution of starting states with no inaccessible states, and for any distributions of the costs with finite variances the TD(A) algorithm given by 1) 2) m t Vn+1(i) = Vn(i) + an(i) L)Ci t + ,Vn(it+d - Vn(it)] LbA)t-kXi(k) t=l k=l Ln an(i) = 00 and Ln a~(i) < 00 uniformly w.p.i. t Vt+l(i) = Vt(i) + at(i)[ci t + ,Vt(it+d - Vt(id] LbA)t-kXi(k) k=l Lt at(i) = 00 and Ln a;(i) < 00 uniformly w.p.i and within sequences at(i)/maXtESat(i) ----;. 1 uniformly w.p.i. converges to the optimal predictions w.p.i provided" A E [0,1] with ,A < 1. Proof for (1): We use here a slightly different form for the learning rule (cf. the previous section). Vn(i) + an (i)[Gn(i) - E~~~)} Vn(i)] 1 m(i) E{m(i)} {; Vn"(i; k) where Vn"( i; k) is an estimate calculated at the ph occurrence of state i in a sequence and for mathematical convenience we have made the transformation an(i) ----;. E{m(i)}an(i), where m(i) is the number of times state i was visited during the sequence. 708 Jaakkola, Jordan, and Singh To apply Theorem 1 we subtract V* (i), the optimal predictions, from both sides of the learning equation. By identifying an(i) := an(i)m(i)/E{m(i)}, f3n(i) := an(i), and Fn(i) := Gn(i) - V(i)m(i)/E{m(i)} we need to show that these satisfy the conditions of Theorem 1. For an(i) and f3n(i) this is obvious. We begin here by showing that Fn(i) indeed is a contraction mapping. To this end, m?xIE{Fn(i) 1 Vn}1 = I miaxIE{~(i)} E{(VnA(i; 1) - V(i» + (VnA(i;2) - V(i» +···1 Vn}1 which can be bounded above by using the relation IE{VnA(i; k) - V(i) 1 Vn}1 < E { IE{VnA(i; k) - V(i) 1 m(i) ~ k, Vn}IO(m(i) - k) 1 Vn} < P{m(i) ~ k}IE{VnA(i) - V(i) 1 Vn}1 < I P {m( i) > k} m~x 1 Vn (i) - V* (i) 1 I where O(x) = 0 if x < 0 and 1 otherwise. Here we have also used the fact that VnA(i) is a contraction mapping independent of possible discounting. As Lk P {m( i) ~ k} = E{ m( i)} we finally get m~x IE{ Fn( i) 1 Vn} 1 ::; I m?x IVn(i) - V(i)1 I I The variance of Fn (i) can be seen to be bounded by E{ m4} m~xIVn(i)12 I For any absorbing Markov chain the convergence to the terminal state is geometric and thus for every finite k, E{mk}::; C(k), implying that the variance of Fn(i) is within the bounds of Theorem 1. As Theorem 1 is now applicable we can conclude that the batch version of TD(>.) converges to the optimal predictions w.p.l. 0 Proof for (2) The proof for the on-line version is achieved by showing that the effect of the on-line updating vanishes in the limit thereby forcing the two versions to be equal asymptotically. We view the on-line version as a batch algorithm in which the updates are made after each complete sequence but are made in such a manner so as to be equal to those made on-line. Define G~ (i) = G n (i) + G~ (i) to be a new batch estimate taking into account the on-line updating within sequences. Here Gn (i) is the batch estimate with the desired properties (see the proof for (1» and G~ (i) is the difference between the two. We take the new batch learning parameters to be the maxima over a sequence, that is an(i) = maxtES at(i). As all the at(i) satisfy the required conditions uniformly w.p.1 these new learning parameters satisfy them as well. To analyze the new batch algorithm we divide it into three parallel processes: the batch TD( >.) with an (i) as learning rate parameters, the difference between this and the new batch estimate, and the change in the value function due to the updates made on-line. Under the conditions of the TD(>.) convergence theorem rigorous Convergence of Stochastic Iterative Dynamic Programming Algorithms 709 upper bounds can be derived for the latter two processes (see Jaakkola, et al., 1993). These results enable us to write II E{G~ - V} II < II E{Gn - V} II + II G~ II < (-y' + C~) II Vn - V II +C~ where C~ and C~ go to zero with w.p.I. This implies that for any c > 0 and II Vn - V* II~ c there exists I < 1 such that I II E{Gn - V} II::; I II Vn - V II for n large enough. This is the required contraction property of Theorem 1. In addition, it can readily be checked that the variance of the new estimate falls under the conditions of Theorem 1. Theorem 1 now guarantees that for any c the value function in the on-line algorithm converges w.p.1 into some t-bounded region of V* and therefore the algorithm itself converges to V* w.p.I. 0 4 CONCLUSIONS In this paper we have extended results from stochastic approximation theory to cover asynchronous relaxation processes which have a contraction property with respect to some maximum norm (Theorem 1). This new class of converging iterative processes is shown to include both the Q-Iearning and TD(A) algorithms in either their on-line or batch versions. We note that the convergence of the on-line version of TD(A) has not been shown previously. We also wish to emphasize the simplicity of our results. The convergence proofs for Q-Iearning and TD(A) utilize only highlevel statistical properties of the estimates used in these algorithms and do not rely on constructions specific to the algorithms. Our approach also sheds additional light on the similarities between Q-Iearning and TD(A). Although Theorem 1 is readily applicable to DP-based learning schemes, the theory of Dynamic Programming is important only for its characterization of the optimal solution and for a contraction property needed in applying the theorem. The theorem can be applied to iterative algorithms of different types as well. Finally we note that Theorem 1 can be extended to cover processes that do not show the usual contraction property thereby increasing its applicability to algorithms of possibly more practical importance. References Bertsekas, D. P. (1987). Dynamic Programming: Deterministic and Stochastic Models. Englewood Cliffs, NJ: Prentice-Hall. Bertsekas, D. P., & Tsitsiklis, J. N. (1989). Parallel and Distributed Computation: Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall. Dayan, P. (1992). The convergence of TD(A) for general A. Machine Learning, 8, 341-362. 710 Jaakkola, Jordan, and Singh Dayan, P., & Sejnowski, T. J. (1993). TD(>.) converges with probability 1. CNL, The Salk Institute, San Diego, CA. Dvoretzky, A. (1956). On stochastic approximation. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. University of California Press. Jaakkola, T., Jordan, M. I., & Singh, S. P. (1993). On the convergence of stochastic iterative dynamic programming algorithms. Submitted to Neural Computation. Peng J., & Williams R. J. (1993). TD(>.) converges with probability 1. Department of Computer Science preprint, Northeastern University. Robbins, H., & Monro, S. (1951). A stochastic approximation model. Annals of Mathematical Statistics, 22, 400-407. Sutton, R. S. (1988). Learning to predict by the methods of temporal differences. Machine Learning, 3, 9-44. Tsitsiklis J. N. (1993). Asynchronous stochastic approximation and Q-learning. Submitted to: Machine Learning. Watkins, C.J .C.H. (1989). Learning from delayed rewards. PhD Thesis, University of Cambridge, England. Watkins, C.J .C.H, & Dayan, P. (1992). Q-learning. Machine Learning, 8, 279-292.	1993	48
801	High Performance Neural Net Simulation on a Multiprocessor System with "Intelligent" Communication Urs A. Miiller, Michael Kocheisen, and Anton Gunzinger Electronics Laboratory, Swiss Federal Institute of Technology CH-B092 Zurich, Switzerland Abstract The performance requirements in experimental research on artificial neural nets often exceed the capability of workstations and PCs by a great amount. But speed is not the only requirement. Flexibility and implementation time for new algorithms are usually of equal importance. This paper describes the simulation of neural nets on the MUSIC parallel supercomputer, a system that shows a good balance between the three issues and therefore made many research projects possible that were unthinkable before. (MUSIC stands for Multiprocessor System with Intelligent Communication) 1 Overview of the MUSIC System The goal of the MUSIC project was to build a fast parallel system and to use it in real-world applications like neural net simulations, image processing or simulations in chemistry and physics [1, 2]. The system should be flexible, simple to program and the realization time should be short enough to not have an obsolete system by the time it is finished. Therefore, the fastest available standard components were used. The key idea of the architecture is to support the collection and redistribution of complete data blocks by a simple, efficient and autonomously working communication network realized in hardware. Instead of considering where to send data and where from to receive data, each processing element determines which part of a (virtual) data block it has produced and which other part of the same data block it wants to receive for the continuation of the algorithm. 888 Parallel Neural Net Simulation 889 Host computer (Sun, PC, Macintosh) - user terminal - mass storage SCSI r··-.. ····_·· .... _··· __ ·· __ ···_·····_·····_····_··_-···_·1 _ ••• __ •• _ ... _ •• _ •• _- ._ ••••• _ ••• _ •• _ ••• _ •• ! MUSIC board I .:.! Board MUSIC board ! Bo~ I i manager I II' manager I : I : I .. I . - .... ------.1 Transputer links PE PE 32+8 bit, 5 MHz Figure 1: Overview of the MUSIC hardware 110 board vo Outside world Figure 1 shows an overview of the MUSIC architecture. For the realization of the communication paradigm a ring architecture has been chosen. Each processing element has a communication interface realized with a XILINX 3090 programmable gate array. During communication the data is shifted through a 40-bit wide bus (32 bit data and 8 bit token) operated at a 5-MHz clock rate. On each clock cycle, the processing elements shift a data value to their right neighbors and receive a new value from their left neighbors. By counting the clock cycles each communication interface knows when to copy data from the stream passing by into the local memory of its processing element and, likewise, when to insert data from the local memory into the ring. The tokens are used to label invalid data and to determine when a data value has circulated through the complete ring. Three processing elements are placed on a 9 x 8.5-inch board, each of them consisting of a Motorola 96002 floating-point processor, 2 Mbyte video (dynamic) RAM, 1 Mbyte static RAM and the above mentioned communication controller. The video RAM has a parallel port which is connected to the processor and a serial port which is connected to the communication interface. Therefore, data processing is almost not affected by the communication network's activity and communication and processing can overlap in time. This allows to use the available communication bandwidth more efficiently. The processors run at 40 MHz with a peak performance of 60 MFlops. Each board further contains an Inmos T425 transputer as a board 890 Milller, Kocheisen, and Gunzinger N umber of processing elments: Peak performance: Floating-point format: Memory: Programming language: Cabinet: Cooling: Total power consumption: Host computer: 60 3.6 Gflops 44 bit IEEE single extended precision 180 Mbyte C, Assembler 19-inch rack forced air cooling less than 800 Watt Sun workstation, PC or Macintosh Table 1: MUSIC system technical data manager, responsible for performance measurements and data communication with the host (a Sun workstation, PC or Macintosh). In order to provide the fast data throughput required by many applications, special I/O modules (for instance for real-time video processing applications) can be added which have direct access to the fast ring bus. An SCSI interface module for four parallel SCSI-2 disks, which is currently being developed, will allow the storage of huge amount of training data for neural nets. Up to 20 boards (60 processing elements) fit into a standard 19-inch rack resulting in a 3.6-Gflops system. MUSIC's technical data is summarized in Table 1. For programming the communication network just three library functions are necessary: Init_commO to specify the data block dimensions and data partitioning, Data.IeadyO to label a certain amount of data as ready for communication and Wait...ciataO to wait for the arrival of the expected data (synchronization). Other functions allow the exchange and automatic distribution of data blocks between the host computer and MUSIC and the calling of individual user functions. The activity of the transputers is embedded in these functions and remains invisible for the user. Each processing element has its own local program memory which makes MUSIC a MIMD machine (multiple instructions multiple data). However, there is usually only one program running on all processing elements (SPMD = single program multiple data) which makes programming as simple or even simpler as programming a SIMD computer (single instruction multiple data). The difference to SIMD machines is that each processor can take different program pathes on conditional branches without the performance degradation that occurs on SIMD computers in such a case. This is especially important for the simulation of neural nets with nonregular local structures. 2 Parallelization of Neural Net Algorithms The first implemented learning algorithm on MUSIC was the well-known backpropagation applied to fully connected multilayer perceptrons [3]. The motivation was to gain experience in programming the system and to demonstrate its performance on a real-world application. All processing elements work on the same layer a time, each of them producing an individual part of the output vector (or error vector in the backward path) [1]. The weights are distributed to the processing elements accordingly. Since a processing element needs different weight subsets in Parallel Neural Net Simulation 891 200.-----.-----~----._----._----~----_n 50 900-600-30 ----:,.../-:--.. v' 300-200-10 ~~ .......... .;..-.. ~.-.~ .... : .... :;.-.; ... : .... + .... ~ .... ~ .... ~ . ................ + + 203-80-26 .....• ...... I!JI!JI!IDI!JI!JIII!JIiIIiII!JIDIiI •••• ~ II O~ ____ L-____ L-____ ~ ____ ~ ____ ~ ____ -U o 10 20 30 40 50 60 Number of processing elements Figure 2: Estimated (lines) and measured (points) back-propagation performance for different neural net sizes. the forward and in the backward path, two subsets are stored and updated on each processing element. Each weight is therefore stored and updated twice on different locations on the MUSIC system [1]. This is done to avoid the communication of the weights during learning what would cause a saturation of the communication network. The estimated and experimentally measured speedup for different sizes of neural nets is illustrated in Figure 2. Another frequently reported parallelization scheme is to replicate the complete network on all processing elments and to let each of them work on an individual subset of the training patterns [4, 5, 6]. The implementation is simpler and the communication is reduced. However, it does not allow continuous weight update, which is known to converge significantly faster than batch learning in many cases. A comparison of MUSIC with other back-propagation implementations reported in the literature is shown in Table 2. Another category of neural nets that have been implemented on MUSIC are cellular neural nets (CNNs) [10]. A CNN is a two-dimensional array of nonlinear dynamic cells, where each cell is only connected to a local neighborhood [11, 12]. In the MUSIC implementation every processing elment computes a different part of the array. Between iteration steps only the overlapping parts of the neighborhoods need to be communicated. Thus, the computation to communication ratio is very high resulting in an almost linear speedup up to the maximum system size. CNNs are used in image processing and for the modeling of biological structures. 3 A Neural Net Simulation Environment After programming all necessary functions for a certain algorithm (e.g. forward propagate, backward propagate, weight update, etc.) they need to be combined 892 Muller, Kocheisen, and Gunzinger System PC (80486, 50 MHz)_* Sun (Sparcstation 10)* Alpha Station (150 MHz)* Hypercluster [7] Warp [4] CM-2 [6] Cray Y-MP C90* RAP [8] NEC SX-3*** MUSIC* Sandy /8** [9] GFll [5] Own measurements Estimated numbers No. of PEs 1 1 1 64 10 64K 1 40 1 60 256 356 **No published reference available. Performance Cont. forward Learmng Peak weight [MCPS] (McuPS] (%) update 1.1 0.47 38.0 Yes 3.0 1.1 43_0 Yes 8.3 3.2 8.6 Yes 27.0 9.9 17.0 No 180.0 40.0 No 220.3 65.6 Yes 574.0 106.0 50.0 Yes 130.0 9.6 Yes 504.0 247.0 28.0 Yes 583.0 31.0 Yes 901.0 54.0 No Table 2: Comparison of floating-point back-propagation implementations. "PEs" means processing elements, "MCPS" stands for millions of connections per second in the forward path and "MCUPS" is the number of connection updates per second in the learning mode, including both forward and backward path. Note that not all implementations allow continuous weight update. in order to construct and train a specific neural net or to carry out a series of experiments. This can be done using the same programming language that was used to program the neural functions (in case of MUSIC this would be C). In this case the programmer has maximum flexibility but he also needs a good knowledge of the system and programming language and after each change in the experimental setup a recompilation of the program is necessary. Because a set of neural functions is usually used by many different researchers who, in many cases, don't want to be involved in a low-level (parallel) programming of the system, it is desirable to have a simpler front-end for the simulator. Such a front-end can be a shell program which allows to specify various parameters of the algorithm (e.g. number of layers, number of neurons per layer, etc.). The usage of such a shell can be very easy and changes in the experimental setup don't require recompilation of the code. However, the flexibility for experimental research is usually too much limited with a simple shell program. We have chosen a way in between: a command language to combine the neural functions which is interactive and much simpler to learn and to use than an ordinary programming language like C or Fortran. The command language should have the following properties: - interactive - easy to learn and to use - flexible - loops and conditional branches - variables - transparent interface to neural functions. Parallel Neural Net Simulation 893 Instead of defining a new special purpose command language we decided to consider an existing one. The choice was Basic which seems to meet the above requirements best. It is easy to learn and to use, it is widely spread, flexible and interactive. For this purpose a Basic interpreter, named Neuro-Basic, was written that allows the calling of neural (or other) functions running parallel on MUSIC. From the Basic level itself the parallelism is completely invisible. To allocate a new layer with 300 neurons, for instance, one can type a = new_layer(300) The variable a afterwards holds a pointer to the created layer which later can be used in other functions to reference that layer. The following command propagates layer a to layer b using the weight set w propagate (a, b, w) Other functions allow the randomization of weights, the loading of patterns and weight sets, the computation of mean squared errors and so on. Each instruction can be assigned to a program line and can then be run as a program. The sequence 10 a = new_layer(300) 20 b = new_layer(10) 30 w = new_weights(a, b) for instance defines a two-layer perceptron with 300 input and 10 output neurons being connected with the weights w. Larger programs, loops and conditional branches can be used to construct and train complete neural nets or to automatically run complete series of experiments where experimental setups depend on the result of previous experiments. The Basic environment thus allows all kinds of gradations in experimental research, from the interactive programming of small experiments till large off-line learning jobs. Extending the simulator with new learning algorithms means that the programmer just has to write the parallel code of the actual algorithm. It can then be controlled by a Basic program and it can be combined with already existing algorithms. The Basic interpreter runs on the host computer allowing easy access to the input/output devices of the host. However, the time needed for interpreting the commands on the host can easily be in the same order of magnitude as the runtime of the actual functions on the attached parallel processor array. The interpretation of a Basic program furthermore is a sequential part of the system (it doesn't run faster if the system size is increased) which is known to be a fundamental limit in speedup (Amdahls law [13]). Therefore the Basic code is not directly interpreted on the host but first is compiled to a simpler stack oriented meta-code, named b-code, which is afterwards copied and run on all processing elements at optimum speed. The compilation phase is not really noticeable to the user since compiling 1000 source lines takes less than a second on a workstation. Note that Basic is not the programming language for the MUSIC system, it is a high level command language for the easy control of parallel algorithms. The actual programming language for MUSIC is C or Assembler. 894 Muller, Kocheisen, and Gunzinger Of course, Neuro-Basic is not restricted to the MUSIC system. The same principle can be used for neural net simulation on conventional workstations, vector computers or other parallel systems. Furthermore, the parallel algorithms of MUSIC also run on sequential computers. Simulations in Neuro-Basic can therefore be executed locally on a workstation or PC as well. 4 Conclusions Neuro-Basic running on MUSIC proved to be an important tool to support experimental research on neural nets. It made possible to run many experiments which could not have been carried out otherwise. An important question, however, is, how much more programming effort is needed to implement a new algorithm in the Neuro-Basic environment compared to an implementation on a conventional workstation and how much faster does it run. Algorithm additional speedup programming Back-propagation ~ C) x 2 60 Back-propagation (Assembler) x 8 240 Cellular neural nets (CNN) x 3 60 Table 3: Implementation time and performance ratio of a 60-processor MUSIC system compared to a Sun Sparcstation-10 Table 3 contains these numbers for back-propagation and cellular neural nets. It shows that if an additional programming effort of a factor two to three is invested to program the MUSIC system in C, the return of investment is a speedup of approximately 60 compared to a Sun Sparcstation-10. This means one year of CPU time on a workstation corresponds to less than a week on the MUSIC system. Acknowledgements We would like to express our gratitude to the many persons who made valuable contributions to the project, especially to Peter Kohler and Bernhard Baumle for their support of the MUSIC system, Jose Osuna for the CNN implementation and the students Ivo Hasler, Bjorn Tiemann, Rene Hauck, Rolf Krahenbiihl who worked for the project during their graduate work. This work was funded by the Swiss Federal Institute of Technology, the Swiss N ational Science Foundation and the Swiss Commission for Support of Scientific Research (KWF). References [1] Urs A. Miiller, Bernhard Baumle, Peter Kohler, Anton Gunzinger, and Walter Guggenbiihl. Achieving supercomputer performance for neural net simulation with an array of digital signal processors. IEEE Micro Magazine, 12(5):55-65, October 1992. Parallel Neural Net Simulation 895 [2] Anton Gunzinger, Urs A. Miiller, Walter Scott, Bernhard Bliumle, Peter Kohler, Hansruedi Vonder Miihll, Florian Miiller-Plathe, Wilfried F. van Gunsteren, and Walter Guggenbiihl. Achieving super computer performance with a DSP array processor. In Robert Werner, editor, Supercomputing '92, pages 543-550. IEEEj ACM, IEEE Computer Society Press, November 16-20, 1992, Minneapolis, Minnesota 1992. [3] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representation by error propagation. In David E. Rumelhart and James L. McClelland, editors, Parallel Distributet Processing: Explorations in the Microstructure of Cognition, volume 1, pages 318-362. Bradford Books, Cambridge MA, 1986. [4] Dean A. Pomerleau, George L. Gusclora, David S. Touretzky, and H. T. Kung. Neural network simulation at Warp speed: How we got 17 million connections per second. In IEEE International Conference on Neural Networks, pages 11.143-150, July 24-27, San Diego, California 1988. [5] Michael Witbrock and Marco Zagha. An implementation of backpropagation learning on GF11, a large SIMD parallel computer. Parallel Computing, 14(3):329-346, 1990. [6] Xiru Zhang, Michael Mckenna, Jill P. Mesirov, and David L. Waltz. An efficient implementation of the back-propagation algorithm on the Connection Machine CM-2. In David S. Touretzky, editor, Advances in Neural Information Processing Systems (NIPS-89), pages 801-809,2929 Campus Drive, Suite 260, San Mateo, CA 94403, 1990. Morgan Kaufmann Publishers. [7] Heinz Miihlbein and Klaus Wolf. Neural network simulation on parallel computers. In David J. Evans, Gerhard R. Joubert, and Frans J. Peters, editors, Parallel Computing-89, pages 365-374, Amsterdam, 1990. North Holland. [8] Phil Kohn, Jeff Bilmes, Nelson Morgan, and James Beck. Software for ANN training on a Ring Array Processor. In John E. Moody, Steven J. Hanson, and Richard P. Lippmann, editors, Advances in Neural Information Processing Systems 4 (NIPS-91), 2929 Campus Drive, Suite 260, San Mateo, California 94403, 1992. Morgan kaufmann. [9] Hideki Yoshizawa, Hideki Kato Hiroki Ichiki, and Kazuo Asakawa. A highly parallel architecture for back-propagation using a ring-register data path. In 2nd International Conference on Microe/ectrnics for Neural Networks (ICMNN-91), pages 325-332, October 16-18, Munich 1991. [10] J. A. Osuna, G. S. Moschytz, and T. Roska. A framework for the classification of auditory signals with cellular neural networks. In H. Dedieux, editor, Procedings of 11. European Conference on Circuit Theory and Design, pages 51-56 (part 1). Elsevier, August 20 - Sept. 3 Davos 1993. [11] Leon O. Chua and Lin Yang. Cellular neural networks: Theory. IEEE Transactions on Circuits and Systems, 35(10):1257-1272, October 1988. [12] Leon O. Chua and Lin Yang. Cellular neural networks: Applications. IEEE Transactions on Circuits and Systems, 35(10):1273-1290, October 1988. [13] Gene M. Amdahl. Validity of the single processor approach to achieving large scale computing capabilities. In AFIPS Spring Computer Conference Atlantic City, NJ, pages 483-485, April 1967.	1993	49
802	The Power of Amnesia Dana Ron Yoram Singer Naftali Tishby Institute of Computer Science and Center for Neural Computation Hebrew University, Jerusalem 91904, Israel Abstract We propose a learning algorithm for a variable memory length Markov process. Human communication, whether given as text, handwriting, or speech, has multi characteristic time scales. On short scales it is characterized mostly by the dynamics that generate the process, whereas on large scales, more syntactic and semantic information is carried. For that reason the conventionally used fixed memory Markov models cannot capture effectively the complexity of such structures. On the other hand using long memory models uniformly is not practical even for as short memory as four. The algorithm we propose is based on minimizing the statistical prediction error by extending the memory, or state length, adaptively, until the total prediction error is sufficiently small. We demonstrate the algorithm by learning the structure of natural English text and applying the learned model to the correction of corrupted text. Using less than 3000 states the model's performance is far superior to that of fixed memory models with similar number of states. We also show how the algorithm can be applied to intergenic E. coli DNA base prediction with results comparable to HMM based methods. 1 Introduction Methods for automatically acquiring the structure of the human language are attracting increasing attention. One of the main difficulties in modeling the natural language is its multiple temporal scales. As has been known for many years the language is far more complex than any finite memory Markov source. Yet Markov 176 The Power of Amnesia 177 models are powerful tools that capture the short scale statistical behavior of language, whereas long memory models are generally impossible to estimate. The obvious desired solution is a Markov sOUrce with a 'deep' memory just where it is really needed. Variable memory length Markov models have been in use for language modeling in speech recognition for some time [3, 4], yet no systematic derivation, nor rigorous analysis of such learning mechanism has been proposed. Markov models are a natural candidate for language modeling and temporal pattern recognition, mostly due to their mathematical simplicity. It is nevertheless obvious that finite memory Markov models can not in any way capture the recursive nature of the language, nor can they be trained effectively with long enough memory. The notion of a variable length memory seems to appear naturally also in the context of universal coding [6]. This information theoretic notion is now known to be closely related to efficient modeling [7]. The natural measure that appears in information theory is the description length, as measured by the statistical predictability via the Kullback- Liebler (KL) divergence. The algorithm we propose here is based on optimizing the statistical prediction of a Markov model, measured by the instantaneous KL divergence of the following symbols, or by the current statistical surprise of the model. The memory is extended precisely when such a surprise is significant, until the overall statistical prediction of the stochastic model is sufficiently good. We apply this algorithm successfully for statistical language modeling. Here we demonstrate its ability for spelling correction of corrupted English text. We also show how the algorithm can be applied to intergenic E. coli DNA base prediction with results comparable to HMM based methods. 2 Prediction Suffix Trees and Finite State Automata Definitions and Notations Let ~ be a finite alphabet. Denote by ~* the set of all strings over~ . A string s, over ~* of length n, is denoted by s = Sl S2 ... Sn. We denote by e the empty string. The length of a string s is denoted by lsi and the size of an alphabet ~ is denoted by I~I. Let, Prefix(s) = SlS2 .. . Sn-1, denote the longest prefix ofa string s, and let Prefix(s) denote the set of all prefixes of s, including the empty string. Similarly, 5uffix(s) = S2S3 . . . Sn and 5uffix(s) is the set of all suffixes of s. A set of strings is called a prefix free set if, V Sl, S2 E 5: {Sl} nPrefix(s2) = 0. We call a probability measure P, over the strings in ~ proper if P( e) = 1, and for every string s, I:aEl:P(sa) = P(s). Hence, for every prefix free set 5, I:sEsP(s):S 1, and specifically for every integer n 2: 0, I:sEl:n P(s) = 1. Prediction Suffix Trees A prediction suffix tree T over ~, is a tree of degree I~I. The edges of the tree are labeled by symbols from ~, such that from every internal node there is at most one outgoing edge labeled by each symbol. The nodes of the tree are labeled by pairs (s, / s) where s is the string associated with the walk starting from that node and ending in the root of the tree, and /s : ~ --t [0,1] is the output probability function related with s satisfying I:aEE /s(O") = 1. A prediction suffix tree induces 178 Ron, Singer, and Tishby probabilities on arbitrary long strings in the following manner. The probability that T generates a string w = W1W2 .. 'Wn in ~n, denoted by PT(W), is IIi=1/s.-1(Wi), where SO = e, and for 1 :S i :S n 1, sj is the string labeling the deepest node reached by taking the walk corresponding to W1 ... Wi starting at the root of T. By definition, a prediction suffix tree induces a proper measure over ~, and hence for every prefix free set of strings {wl, ... , wm}, L~l PT(Wi) :S 1, and specifically for n 2: 1, then L3 En PT(S) = 1. An example of a prediction suffix tree is depicted in Fig. 1 on the left, where the nodes of the tree are labeled by the corresponding suffix they present. 1'0=0.6 1'1=0.4 0.4 ~,---(~~, ... ... .... ···· ...... ~~.:.6 .... Figure 1: Right: A prediction suffix tree over ~ = {a, I}. The strings written in the nodes are the suffixes the nodes present. For each node there is a probability vector over the next possible symbols. For example, the probability of observing a '1' after observing the string '010' is 0.3. Left: The equivalent probabilistic finite automaton. Bold edges denote transitions with the symbol '1' and dashed edges denote transitions with '0'. The states of the automaton are the leaves of the tree except for the leaf denoted by the string 1, which was replaced by the prefixes of the strings 010 and 110: 01 and 11. Finite State Automata and Markov Processes A Probabilistic Finite Automaton (PFA) A is a 5-tuple (Q, 1:, T, I, 7r), where Q is a finite set of n states, 1: is an alphabet of size k, T : Q x ~ -;. Q is the transition junction, I : Q x ~ -;. [0, 1 J is the output probability junction, and 7r : Q -;. [0, 1 J is the probability distribution over the starting states. The functions I and 7r must satisfy the following requirements: for every q E Q, LUEE I(q, 0') = 1, and LqEQ 7r( q) = 1. The probability that A generates a string s = S1 S2 ... Sn E 1:n is PA(S) = LqOEQ 7r(qO) TI7=l l(qi-1, Si), where qi+l = T(qi, Si). We are interested in learning a sub-class of finite state machines which have the following property. Each state in a machine M belonging to this sub-class is labeled by a string of length at most L over ~, for some L 2: O. The set of strings labeling the states is suffix free. We require that for every two states ql ,q2 E Q and for every symbol 0' E ~, if T(q1, 0') = q2 and ql is labeled by a string s1, then q2 is labeled The Power of Amnesia 179 by a string s2 which is a suffix of s1 . a. Since the set of strings labeling the states is suffix free, if there exists a string having this property then it is unique. Thus, in order that r be well defined on a given set of string S, not only must the set be suffix free, but it must also have the property, that for every string s in the set and every symbol a, there exists a string which is a suffix of sa. For our convenience, from this point on, if q is a state in Q then q will also denote the string labeling that state. A special case of these automata is the case in which Q includes all 2L strings of length L. These automata are known as Markov processes of order L. We are interested in learning automata for which the number of states, n, is actually much smaller than 2£, which means that few states have "long memory" and most states have a short one. We refer to these automata as Markov processes with bounded memory L. In the case of Markov processes of order L, the "identity" of the states (i.e. the strings labeling the states) is known and learning such a process reduces to approximating the output probability function. When learning Markov processes with bounded memory, the task of a learning algorithm is much more involved since it must reveal the identity of the states as well. It can be shown that under a slightly more complicated definition of prediction suffix trees, and assuming that the initial distribution on the states is the stationary distribution, these two models are equivalent up to a grow up in size which is at most linear in L. The proof of this equi valence is beyond the scope of this paper, yet the transformation from a prediction suffix tree to a finite state automaton is rather simple. Roughly speaking, in order to implement a prediction suffix tree by a finite state automaton we define the leaves of the tree to be the states of the automaton. If the transition function of the automaton, r(-, .), can not be well defined on this set of strings, we might need to slightly expand the tree and use the leaves of the expanded tree. The output probability function of the automaton, ,(-, .), is defined based on the prediction values of the leaves of the tree. i.e., for every state (leaf) s, and every symbol a, ,( s, a) = ,s (a). The outgoing edges from the states are defined as follows: r(q1, a) = q2 where q2 E Suffix(q1a). An example of a finite state automaton which corresponds to the prediction tree depicted in Fig. 1 on the left, is depicted on the right part of the figure. 3 Learning Prediction Suffix Trees Given a sample consisting of one sequence of length I or m sequences of lengths 11 ,/2 , ... ,1m we would like to find a prediction suffix tree that will have the same statistical properties of the sample and thus can be used to predict the next outcome for sequences generated by the same source. At each stage we can transform the tree into a Markov process with bounded memory. Hence, if the sequence was created by a Markov process, the algorithm will find the structure and estimate the probabilities of the process. The key idea is to iteratively build a prediction tree whose probability measure equals the empirical probability measure calculated from the sample. We start with a tree consisting of a single node (labeled by the empty string e) and add nodes which we have reason to believe should be in the tree. A node as, must be added to the tree if it statistically differs from its parent node s. A natural measure 180 Ron, Singer, and Tishby to check the statistical difference is the relative entropy (also known as the KullbackLiebler (KL) divergence) [5], between the conditional probabilities PCI s) and PCIO"s). Let X be an observation space and Pl , P2 be probability measures over X then the KL divergence between Pl and P'2 is, DKL(Pl IIP2) = 2:XEx Pl(X) log ;~~:~. Note that this distance is not symmetric and Pl should be absolutely continuous with respect to P2 . In our problem, the KL divergence measures how much additional information is gained by using the suffix crs for prediction instead of predicting using the shorter suffix s. There are cases where the statistical difference is large yet the probability of observing the suffix crs itself is so small that we can neglect those cases. Hence we weigh the the statistical error by the prior probability of observing crs. The statistical error measure in our case is, E1'1'(o"s, s) P(O"s) DKL (P(-IO"s)IIPCls)) P( ) ~ P( 'I ) I P(a'las) O"s L....-a'EE 0" O"S og P(a'ls) ~ P( ') 1 P(a3a') L....-a'E~ O"SO" og P(a'ls)P(as) Therefore, a node crs is added to the tree if the statistical difference (defined by E1'1'( crs, s)) between the node and its parrent s is larger than a predetermined accuracy c The tree is grown level by level, adding a son of a given leaf in the tree whenever the statistical surprise is large. The problem is that the requirement that a node statistically differs from it's parent node is a necessary condition for belonging to the tree, but is not sufficient. The leaves of a prediction suffix tree must differ from their parents (or they are redundant) but internal nodes might not have this property. Therefore, we must continue testing further potential descendants of the leaves in the tree up to depth L. In order to avoid exponential grow in the number of strings tested, we do not test strings which belong to branches which are reached with small probability. The set of strings, tested at each step, is denoted by 5, and can be viewed as a kind of potential 'frontier' of the growing tree T. At each stage or when the construction is completed we can produce the equivalent Markov process with bounded memory. The learning algorithm of the prediction suffix tree is depicted in Fig. 2. The algorithm gets two parameters: an accuracy parameter t and the maximal order of the process (which is also the maximal depth of the tree) L. The true source probabilities are not known, hence they should be estimated from the empirical counts of their appearances in the observation sequences. Denote by #s the number of time the string s appeared in the observation sequences and by #crls the number of time the symbol cr appeared after the string s. Then, usmg Laplace's rule of succession, the empirical estimation of the probabilities is, ~ #s + 1 ~ #crls + 1 P(s) ~ P(s) = 2: #' I~I P(crls) ~ P(O"ls) = 2: # 'I I I 3'EEIsi S + a'E~ 0" S + ~ 4 A Toy Learning Example The algorithm was applied to a 1000 symbols long sequence produced by the automaton depicted top left in Fig. 3. The alphabet was binary. Bold lines in the figure represent transition with the symbol '0' and dashed lines represent the symbol '1'. The prediction suffix tree is plotted at each stage of the algorithm. At the The Power of Amnesia 181 • Initialize the tree T and the candidate strings S: T consists of a single root node, and S {O" I 0" E ~ /\ p( 0") 2: t} . • While S =I 0, do the following: 1. Pick any s E S and remove it from S. 2. If Err{s, Suffix(s)) 2: E then add to T the node corresponding to s and all the nodes on the path from the deepest node in T (the deepest ancestor of s) until s. 3. If lsi < L then for every 0" E ~ if P(O"s) 2: E add O"S to S. Figure 2: The algorithm for learning a prediction suffix tree. end of the run the correponding automat.on is plotted as well (bottom right.). Note that the original automaton and the learned automaton are the same except for small diffrences in the transition probabilities. 0.7 0.3 0.32. 0.68 o. o. 0.14 0.86 0.69 0.31 Figure 3: The original automaton (top left), the instantaneous automata built along the run of the algorithm (left to right and top to bottom), and the final automaton (bottom left). 5 Applications We applied the algorithm to the Bible with L = 30 and E = 0.001 which resulted in an automaton having less than 3000 states. The alphabet was the english letters and the blank character. The final automaton constitutes of states that are of length 2, like r qu' and r xe', and on the other hand 8 and 9 symbols long states, like r shall be' and r there was'. This indicates that the algorithm really captmes 182 Ron, Singer, and Tishby the notion of variable context length prediction which resulted in a compact yet accurate model. Building a full Markov model in this case is impossible since it requires II:IL = 279 states. Here we demonstrate our algorithm for cleaning corrupted text. A test text (which was taken out of the training sequence) was modified in two different ways. First by a stationary noise that altered each letter with probability 0.2, and then the text was further modified by changing each blank to a random letter. The most probable state sequence was found via dynamic programming. The 'cleaned' observation sequence is the most probable outcome given the knowledge of the error rate. An example of such decoding for these two types of noise is shown in Fig. 4. We also applied the algorithm to intergenic Original Text: and god called the dry land earth and the gathering together of the waters called he seas and god saw that it was good and god said let the earth bring forth grass the herb yielding seed and the fruit tree yielding fruit after his kind Noisy text (1): and god cavsed the drxjland earth ibd shg gathervng together oj the waters dlled re seas aed god saw thctpit was good ann god said let tae earth bring forth gjasb tse hemb yielpinl peed and thesfruit tree sielxing fzuitnafter his kind Decoded text (1): and god caused the dry land earth and she gathering together of the waters called he sees and god saw that it was good and god said let the earth bring forth grass the memb yielding peed and the fruit tree fielding fruit after his kind Noisy text (2): andhgodpcilledjthesdryjlandbeasthcandmthelgatceringhlogetherjfytrezaatersoczlled xherseasaknddgodbsawwthathitqwasoqoohanwzgodcsaidhletdtheuejrthriringmforth bgrasstthexherbyieidingzseedmazdctcybfruitttreeayieidinglfruztbafberihiskind Decoded text (2): and god called the dry land earth and the gathering together of the altars called he seasaked god saw that it was took and god said let the earthriring forth grass the herb yielding seed and thy fruit treescielding fruit after his kind Figure 4: Cleaning corrupted text using a Markov process with bounded memory. regions of E. coli DNA, with L = 20 and f. = 0.0001. The alphabet is: A. C. T. G. The result of the algorithm is an automaton having 80 states. The names of the states of the final automaton are depicted in Fig. 5. The performance of the model can be compared to other models, such as the HMM based model [8], by calculating the normalized log-likelihood (NLL) over unseen data. The NLL is an empirical measure of the the entropy of the source as induced by the model. The NLL of bounded memory Markov model is about the same as the one obtained by the HMM .based model. Yet, the Markov model does not contain length distribution of the intergenic segments hence the overall perform ace of the HMM based model is slightly better. On the other hand, the HMM based model is more complicated and requires manual tuning of its architecture. The Power of Amnesia 183 ACT G AA AC AT CA CC CT CG TA TC TT TG GA GC GT GG AAC AAT AAG ACA ATT CAA CAC CAT CAG CCA CCT CCG CTA CTC CTT CGA CGC CGT TAT TAG TCA TCT TTA TTG TGC GAA GAC GAT GAG GCA GTA GTC GTT GTG GGA GGC GGT AACT CAGC CCAG CCTG CTCA TCAG TCTC TTAA TTGC TTGG TGCC GACC GATA GAGC GGAC GGCA GGCG GGTA GGTT GGTG CAGCC TTGCA GGCGC GGTTA Figure 5: The states that constitute the automaton for predicting the next base of intergenic regions in E. coli DNA. 6 Conclusions and Future Research In this paper we present a new efficient algorithm for estimating the structure and the transition probabilities of a Markov processes with bounded yet variable memory. The algorithm when applied to natural language modeling result in a compact and accurate model which captures the short term correlations. The theoretical properties of the algorithm will be described elsewhere. In fact, we can prove that a slightly different algorithm constructs a bounded memory markov process, which with arbitrary high probability, induces distributions (over I:n for n > 0) which are very close to those induced by the 'true' Markovian source, in the sense of the KL divergence. This algorithm uses a polynomial size sample and runs in polynomial time in the relevent parameters of the problem. We are also investigating hierarchical models based on these automata which are able to capture multi-scale correlations, thus can be used to model more of the large scale structure of the natural language. Acknowledgment We would like to thank Lee Giles for providing us with the software for plotting finite state machines, and Anders Krogh and David Haussler for letting us use their E. coli DN A data and for many helpful discussions. Y.S. would like to thank the Clore foundation for its support. References [1] J.G Kemeny and J.L. Snell, Finite Markov Chains, Springer-Verlag 1982. [2] Y. Freund, M. Kearns, D. Ron, R. Rubinfeld, R.E. Schapire, and L. Sellie, Efficient Learning of Typical Finite Automata from Random Walks, STOC-93. [3] F. Jelinek, Self-Organized Language Modeling for Speech Recognition, 1985. [4] A. N adas, Estimation of Probabilities in the Language Model of the IBM Speech Recognition System, IEEE Trans. on ASSP Vol. 32 No.4, pp. 859-861, 1984. [5] S. Kullback, Information Theory and Statistics, New York: Wiley, 1959. [6] J. Rissanen and G. G. Langdon, Universal modeling and coding, IEEE Trans. on Info. Theory, IT-27 (3), pp. 12-23, 1981. [7] J. Rissanen, Stochastic complexity and modeling, The Ann. of Stat., 14(3),1986. [8] A. Krogh, S.1. Mian, and D. Haussler, A Hidden Markov Model that finds genes in E. coli DNA, UCSC Tech. Rep. UCSC-CRL-93-16.	1993	5
803	An Analog VLSI Model of Central Pattern Generation in the Leech Micah S. Siegel* Department of Electrical Engineering Yale University New Haven, CT 06520 Abstract I detail the design and construction of an analog VLSI model of the neural system responsible for swimming behaviors of the leech. Why the leech? The biological network is small and relatively well understood, and the silicon model can therefore span three levels of organization in the leech nervous system (neuron, ganglion, system); it represents one of the first comprehensive models of leech swimming operating in real-time. The circuit employs biophysically motivated analog neurons networked to form multiple biologically inspired silicon ganglia. These ganglia are coupled using known interganglionic connections. Thus the model retains the flavor of its biological counterpart, and though simplified, the output of the silicon circuit is similar to the output of the leech swim central pattern generator. The model operates on the same time- and spatial-scale as the leech nervous system and will provide an excellent platform with which to explore real-time adaptive locomotion in the leech and other "simple" invertebrate nervous systems. 1. INTRODUCTION A Central Pattern Generator (CPG) is a network of neurons that generates rhythmic output in the absence of sensory input (Rowat and Selverston, 1991). It has been * Present address: Micah Siegel, Computation and Neural Systems, Mail Stop 139-74 California Institute of Technology, Pasadena, CA 91125. 622 An Analog VLSI Model of Central Pattern Generation in the Leech 623 I 'oult ~v ~ "iuhib I lre=ov I Figure l. Silicon neuromime. The circuit includes tonic excitation, inhibitory synapses and an inhibitory recovery time. Note that there are two inhibitory synapses per device. Iionic sets the level of tonic excitatory input; V inhib sets the synaptic strength; Irecov determines the inhibitor recover time. suggested that invertebrate central pattern generation may represent an excellent theatre within which to explore silicon implementations of adaptive neural systems: invertebrate CPG networks are orders of magnitude smaller than their vertebrate counterparts, much detailed information is available about them, and they guide behaviors that may be of technological interest (Ryckebusch et al., 1989). Furthermore, CPG networks are typically embedded in larger neural circuits and are integral to the neural correlates of adaptive behavior in many natural organisms (Friesen, 1989). On strategy for modeling "simple" adaptive behaviors is first to evolve a biologically plausible framework within which to include increasingly more sophisticated and verisimilar adaptive mechanisms; because the model of leech swimming presented in this paper encompasses three levels of organization in the leech central nervous system, it may provide an ideal such structure with which to explore potentially useful adaptive mechanisms in the leech behavioral repertoire. Among others, these mechanisms include: habituation of the swim response (Debski and Friesen, 1985), the local bending reflex (Lockery and Kristan, 1990), and conditioned learning of the stepping and shortening behaviors (Sahley and Ready, 1988). 624 Siegel A Co. Cel ~l ~phase 113 -1O· 11& 10' a", 1O' 0;-, "0' 01-1(12 -..•. ~ .. . ........ 11:1' .1 I , ,ao· e l= 1= .0 120' IJ 150· n leo' C~;;Ip. f1' '10' )I.IY' /0' ,..,. Je(l"1C" p::66e B ~,II ~JllU\ ~~U, 123 L--J ___ I l. ~~I} 'IIL~' 2S J L_-' I ,i~ 27-..J ~ I Figure 2. The individual ganglion. (A) Cycle phases of the oscillator neurons in the biological ganglion (from Friesen, 1989). (B) Somatic potential of the simplified silicon ganglion. (C) Circuit diagram of silicon ganglion using cells and s na tic connections identified in the leech an lion. 2. LOCOMOTORY CPG IN THE LEECH As a first step toward modeling a full repertoire of adaptive behavior in the medicinal leech (Hirundo medicinalis), I have designed, fabricated, and successfully tested an analog silicon model of one critical neural subsystem the coupled oscillatory central pattern generation network responsible for swimming. A leech swims by undulating its segmented body to form a rearward-progressing body wave. This wave is analogous to the locomotory undulations of most elongated aquatic animals (e.g. fish), and some terrestrial amphibians and reptiles (including salamanders and snakes) (Friesen, 1989). The moving crests and troughs in the body wave are produced by phase-delayed contractile rhythms of the dorsal and ventral body wall along successive segments (Stent and Kristan, 1981). The interganglionic neural subsystem that subserves this behavior constitutes an important modeling platform because it guides locomotion in the leech over a wide range of frequencies and adapts to varying intrinsic and extrinsic conditions (Debski and Friesen, 1985). In the medicinal leech, interneurons that coordinate the rearward-progressing swimming contractions undergo oscillations in membrane potential and fire impulses in bursts. It appears that the oscillatory activity of these intemeurons arises from a network rhythm that depends on synaptic interaction between neurons rather than from an endogenous polarization rhythm arising from inherently oscillatory membrane potentials in individual An Analog VLSI Model of Central Pattern Generation in the Leech 625 A ganglion: 9 10 11 .... ~II----- head tail ----i~~ 8 28 __ ----' 9 { 27 -------+--123 { 28 __ ---I ~-------------~~~~----10 27 -,...-____ +-__ 123~ ~---r---------~ { 28 ____ __' 11 27 ______ +-.--I lOOms Figure 3. The complete silicon model. (A) Coupled oscillatory ganglia. As in the leech nervous system, interganglionic connections employ conduction delays. (B) Somatic recording of cells (28, 27, 123) from three midbody ganglia (9,10,11) in the silicon model. Notice the phase-delay in homologous cells of successive ganglia. (The apparent "beat" frequencies riding on the spike bursts are an aliasing artifact of the digital oscilloscope measurement and the time-scale; all spikes are approximately the same hei ht. neurons (Friesen, 1989). The phases of the oscillatory intemeurons fonn groups clustered about three phase points spaced equally around the activity cycle. To first approximation, all midbody ganglia of the leech nerve cord express an identical activity rhythm. However, activity in each ganglion is phase-delayed with respect to more anterior ganglia (Friesen, 1989); presumably this is responsible for the undulatory body wave characteristic of leech swimming. 626 Siegel 3. THE SILICON MODEL The silicon analog model employs biophysically realistic neural elements (neuromimes), connected into biologically realistic ganglion circuits. These ganglion circuits are coupled together using known interganglionic connections. This silicon model thus spans three levels of organization in the nervous system of the leech (neuron, ganglion, system), and represents one of the first comprehensive models of leech swimming (see also Friesen and Stent, 1977). The hope is that this model will provide a framework for the implementation of adaptive mechanisms related to undulatory locomotion in the leech and other invertebrates. The building block of the model CPO is the analog neuromime (see figure I); it exhibits many essential similarities to its biological counterpart. Like CPO interneurons in the leech swim system, the silicon neuromime integrates current across a somatic "capacitance" and uses positive feedback to generate action potentials whose frequency is determined by the magnitude of excitatory current input (Mead, 1989). In the leech swim system. nearly tonic excitatory input is transformed by a system of inhibition to produce the swim pattern (Friesen. 1989); adjustable tonic excitation is therefore included in the individual silicon neuromime. Inhibitory synapses with adjustable weights are also implemented. Like its biological counterpart, the silicon neuromime includes a characteristic recovery time from inhibition. From theoretical and experimental studies. such inhibition recovery time is thought to play an important functional role in the interneurons that constitute the leech swim system (Friesen and Stent, 1977). Axonal delays have been demonstrated in the intersegmental interaction between ganglia in the leech. Similar axonal delays have been implemented in the silicon model using Shifting delay lines. The building block of the distributed model for the leech swim system is the ganglion. These biologically motivated silicon ganglia are constructed using only (though not all) identified cells and synaptic connections between cells in the biological system. Cells 27, 28, and 123 constitute a central inhibitory loop within each ganglion. Figure 2 exhibits the simplified diagram and the cycle phases of oscillatory interneurons in both the biological and the silicon ganglion. As in the leech ganglion, the phase relationships in the model ganglion fall into three groups, with cells 27. 28. and 123 participating each in the appropriate group of the oscillatory cycle. It is interesting that, though the silicon model captures the spirit of the tri-phasic output, the model is imprecise with respect to the exact phase locations of cells 27. 28. and 123 within their respective groups. This discrepancy between the silicon model and the biological system may point to the significance of other swim interneurons for swim pattern generation in the leech. Undoubtedly. the additional oscillatory interneurons sculpt this tri-phasic output significantly. The silicon model of coupled successive segments in the leech is implemented using these silicon neurons and biologically motivated ganglia. The model employs interganglionic connections known to exist in the biological system and generates qualitatively similar output at the same time-scale as the leech system. It appears in the leech that synchronization between ganglia is governed by the interganglionic synaptic interaction of interneurons involved in the oscillatory pattern rather than by autonomous An Analog VLSI Model of Central Pattern Generation in the Leech 627 coordinating neurons (Friesen. 1989). In the silicon model. interganglionic interaction is represented by a projection from more anterior cell 123 to more posterior cell 28; this A B ----.1UJV'lJNil Wf;tU12i'l .. lOOms Figure 4. Phase lag between more anterior and more posterior segments in both systems. (A) Intersegmental phase lag in the leech swim system (from Friesen. 1989). (B) Intersegmental phase lag in the silicon model. Though not shown in the figure, this cycle repeats at the same frequency as the c cle in A. ote chan e of time scale. projection is also observed between cells 123 and 28 of successive ganglia in the leech (Friesen, 1989). however it is by no means the only such interganglionic connection. In addition, the biological system utilizes conduction delays in its interganglionic projections; each of these is modeled in the silicon system by a delay line (Friesen and Stent. 1977) analogous to an active cable with adjustable propagation speed. Figure 3 demonstrates the silicon model of three coupled ganglia with transmission delays. Notice that neuromimes in each successive ganglion are phase-delayed from homologous neuromimes in more anterior ganglia. Figure 4 shows this phase delay more explicitly. 4. DISCUSSION The analog silicon model of central pauern generation in the leech successfully captures design principles from three levels of organization in the leech nervous system and has been tested over a wide range of network parameter values. It operates on the same timescale as its biological counterpart and gives rise to ganglionic activity that is qualitatively similar to activity in the leech ganglion. Furthermore. it maintains biologically plausible phase relationship between homologous elements of successive ganglia. The design of the silicon model is intentionally compatible with analog Very Large Scale Integration (VLSI) technology. making its integrated spatial-scale close to that of the leech nervous system. It is interesting that this highly simplified model captures qualitatively the output both within and between ganglia of the leech; it may be illuminating to explore the functional significance of other swim interneurons by their inclusion in similar silicon networks. The current model provides an important platform for future implementations of invertebrate adaptive behaviors, especially those behaviors related to swim and other locomotory pattern generation. The hope is that such behaviors 628 Siegel can be evolved incrementally using neuromime models of identified adaptive interneurons to modulate the swim central pattern generating network. Acknowledgments I would like to thank the department of Electrical Engineering at Yale University for encouraging and generously supporting independent undergraduate research. References Rowat, P.P. and Selverston, A.I. (1991). Network, 2, 17-41. Ryckebusch, S., Bower, J.M., Mead, C., (1989). In D.Touretzky (ed.), Advances in Neural Information Processing Systems, 384-393. San Mateo, CA: Morgan Kaufmann. Friesen, W.O. (1989). In J. Jacklet (ed), Neuronal and Cellular Oscillators, 269-316. New York: Marcel Dekker. Debski, E.A. and Friesen, W.O. (1985). Journal of Experimental Biology, 116, 169188. Lockery, S.R. and Kristan, W.B. (1990). Journal of Neuroscience, 10(6), 1811-1815. Sahley, C.L. and Ready, D.P. (1988). Journal of Neuroscience, 8(12), 4612-4620. Stent, G.S. and Kristan, W.B. (1981). In K.Muller, J Nicholls, and G. Stent (eds) , Neurobiology of the Leech, 113-146. Cold Spring Harbor: Cold Spring Harbor Laboratory . Mead, C.A. (1989). Analog VLSl and Neural Systems, Reading, MA: Addison-Wesley. Friesen, W.O. and Stent, G.S. (1977). Biological Cybernetics, 28,27-40.	1993	50
804	Non-linear Statistical Analysis and Self-Organizing Hebbian Networks Jonathan L. Shapiro and Adam Priigel-Bennett Department of Computer Science The University, Manchester Manchester, UK M139PL Abstract Neurons learning under an unsupervised Hebbian learning rule can perform a nonlinear generalization of principal component analysis. This relationship between nonlinear PCA and nonlinear neurons is reviewed. The stable fixed points of the neuron learning dynamics correspond to the maxima of the statist,ic optimized under nonlinear PCA. However, in order to predict. what the neuron learns, knowledge of the basins of attractions of the neuron dynamics is required. Here the correspondence between nonlinear PCA and neural networks breaks down. This is shown for a simple model. Methods of statistical mechanics can be used to find the optima of the objective function of non-linear PCA. This determines what the neurons can learn. In order to find how the solutions are partitioned amoung the neurons, however, one must solve the dynamics. 1 INTRODUCTION Linear neurons learning under an unsupervised Hebbian rule can learn to perform a linear statistical analysis ofthe input data. This was first shown by Oja (1982), who proposed a learning rule which finds the first principal component of the variance matrix of the input data. Based on this model, Oja (1989), Sanger (1989), and many others have devised numerous neural networks which find many components of this matrix. These networks perform principal component analysis (PCA), a well-known method of statistical analysis. 407 408 Shapiro and Priigel-Bennett Since PCA is a form of linear analysis, and the neurons used in the PCA networks are linear - the output of these neurons is equal to the weighted sum of inputs; there is no squashing function of sigmoid - it is obvious to ask whether non-linear Hebbian neurons compute some form of non-linear PCA? Is this a useful way to understand the performance of the networks? Do these networks learn to extract features of the input data which are different from those learned by linear neurons? Currently in the literature, the phrase "non-linear PCA" is used to describe what is learned by any non-linear generalization of Oja neurons or other PCA networks (see for example, Oja, 1993 and Taylor, 1993). In this paper, we discuss the relationship between a particular form of non-linear Hebbian neurons (Priigel-Bennett and Shapiro, 1992) and a particular generalization of non-linear PCA (Softky and Kammen 1991). It is clear that non-linear neurons can perform very differently from linear ones. This has been shown through analysis (Priigel-Bennett and Shapiro, 1993) and in application (Karhuenen and Joutsensalo, 1992). It can also be very useful way of understanding what the neurons learn. This is because non-linear PCA is equivalent to maximizing some objective function. The features that this extracts from a data set can be studied using techniques of statistical mechanics. However, non-linear PCA is ambiguous because there are multiple solutions. What the neuron can learn is given by non-linear PCA. The likelihood of learning the different solutions is governed by the dyanamics chosen to implement non-linear PCA, and may differ in different implementations of the dynamics. 2 NON-LINEAR HEBBIAN NEURONS Neurons with non-linear activation functions can learn to perform very different tasks from those learned by linear neurons. Nonlinear Hebbian neurons have been analyzed for general non-linearities by Oja (1991), and was applied to sinusoidal signal detection by Karhuenen and Joutsensalo (1992). Previously, we analysed a simple non-linear generalization of Oja's rule (PriigelBennett and Shapiro, 1993). We showed how the shape of the neuron activation function can control what a neuron learns. Whereas linear neurons learn to a statistic mixture of all of the input patterns, non-linear neurons can learn to become tuned to individual patterns, or to small clusters of closely correlated patterns. In this model, each neuron has weights, Wi is the weight from the ith input, and responds to the usual sum of input times weights through an activation function A(y). This is assumed a simple power-law above a threshold and zero below it. I.e. (1) Here ¢ is the threshold, b controls the power of the power-law, xf is the ith component of the pth pattern, and VP = Li xf Wi. Curves of these functions are shown in figure laj if b = 1 the neurons are threshold-linear. For b > 1 the curves can be thought of as low activation approximations to a sigmoid which is shown in figure 1 b. The generalization of Oja's learning rule is that the change in the weights 8Wi Non-Linear Statistical Analysis and Self-Organizing Hebbian Networks 409 Neuron Activation Function b>1 b<1 • psp A Sigmoid Activation Function Figure 1: a) The form of the neuron activation function. Control by two parameters band <p. When b > 1, this activation function approximates a sigmoid, which is shown in b). is given by 6Wi = LA(VP) [xf - VP Wi ] . (2) P If b < 1, the neuron learns to average a set of patterns. If b = 1, the neuron finds the principal component of the pattern set. When b > 1, the neuron learns to distinguish one of the patterns in the presence of the others, if those others are not too correlated with the pattern. There is a critical correlation which is determined by b; the neuron learns to individual patterns which are less correlated than the critical value, but learns to something like the center of the cluster if the patterns are more correlated. The threshold controls the size of the subset of patterns which the neuron can respond to. For these neurons, the relationship between non-PCA and the activation function was not previously discussed. That is done in the next section. 3 NON-LINEAR peA A non-linear generalization of PCA was proposed by Softky and Kammen (1991). In this section, the relationship between non-linear PCA and unsupervised Hebbian learning is reviewed. 410 Shapiro and Priigel-Bennett 3.1 WHAT IS NON-LINEAR PCA The principal component of a set of data is the direction which maximises the variance. I.e. to find the principal component of the data set, find the vector tV of unit length which maximises (3) Here, Xi denotes the ith component of an input pattern and < .. . > denotes the average over the patterns. Sofky and Kammen suggested that an appropriate generalization is to find the vector tV which maximizes the d-dimensional correlation, (4) They argued this would give interesting results if higher order correlations are important, or ifthe shape ofthe data cloud is not second order. This can be generalized further, of course, maximizing the average of any non-linear function of the input U(y), (5) The equations for the principal components are easily found using Lagrange multipliers. The extremal points are given by < U' (1: WkXk )Xi >= AWi. k These points will be (local) maxima if the Hessian 1lij, 1lij =< U"(I: WkXk)XiXj > -ADij, k Here, A is a Lagrange multiplier chosen to make Iwl2 = 1. 3.2 NEURONS WHICH LEARN PCA (6) (7) A neuron learning via unsupervised Hebbian learning rule can perform this optimization. This is done by associating Wi with the weight from the ith input to the neuron, and the data average < . > as the sum over input patterns xf. The nonlinear function which is optimized is determined by the integral of the activation function of the neuron A(y) = U'(y). In their paper, Softky and Kammen propose a learning rule which does not perform this optimization in general. The correct learning rule is a generalization of Oja's rule (equation (2) above), in this notation, (8) Non-Linear Statistical Analysis and Self-Organizing Hebbian Networks 411 This fixed points of this dynamical equation will be solutions to the extremal equation of nonlinear peA, equation (6), when the a.'3sociations A = (A(V)V) , and A(y) = U'(y) are made. Here (.) is interpreted as sum over patterns; this is batch learning. The rule can also be used incrementally, but then the dynamics are stochastic and the optimization might be performed only on average, and then maybe only for small enough learning rates. These fixed points will be stable when the Hessian llij is negative definite at the fixed point. This is now, which is the same as the previous, equation (7),in directions perpendicular to the fixed point, but contains additional terms in direction of the fixed point which normalize it. The neurons described in section 2 would perform precisely what Softky and Kammen proposed if the activation function was pure power-law and not thresholded; as it is they maximize a more complicated objective function. Since there is a one to one correspondence between the stable fixed points of the dynamics and the local maxima of the non-linear correlation measure, one says that these non-linear neurons compute non-linear peA. 3.3 THEORETICAL STUDIES OF NONLINEAR PCA In order to understand what these neurons learn, we have studied the networks learning on model data drawn from statistical distributions. For very dense clusters p ~ 00, N fixed, the stable fixed point equations are algebraic. In a few simple cases they can be solved. For example, if the data is Gaussian or if the data cloud is a quadratic cloud (a function of a quadratic form), the neuron learns the principal component, like the linear neuron. Likewise, if the patterns are not random, the fixed point equations can be solved in some cases. For large number of patterns in high dimensions fluctuations in the data are important (N and P goes to 00 together in some way). In this case, methods of statistical mechanics can be used to average over the data. The objective function of the non-linear peA acts as (minus) the energy in statistical mechanics. The free energy is formally, F =< IOg(D. J Of, 6 (t wl- I) exp (3U(V) > . (10) In the limit that f3 is large, this calculation finds the local maxima of U. In this form of analysis, the fact that the neuron optimizes an objective function is very important. This technique was used to produce the results outlined in section 2. 412 Shapiro and Priigel-Bennett 3.4 WHAT NON-LINEAR peA FAILS TO REVEAL In the linear peA, there is one unique solution, or if there are many solutions it is because the solutions are degenerate. However, for the non-linear situation, there are many stable fixed points of the dynamics and many local maxima of the non-linear correlation measure. This has two effects. First, it means that you cannot predict what the neuron will learn simply by studying fixed point equations. This tells you what the neuron might learn, but the probability that this solution will be can only be ascertained if the dynamics are understood. This also breaks the relationship between non-linear peA and the neurons, because, in principle, there could be other dynamics which have the same fixed point structure, but do not have the same basins of attraction. Simple fixed point analysis would be incapable of predicting what these neurons would learn. 4 PARTITIONING An important question which the fixed-point analysis, or corresponding statistical mechanics cannot address is: what is the likelihood of learning the different solutions? This is the essential ambiguity of non-linear peA - there are many solutions and the size of the basin of attractions of each is determined by the dynamics, not by local maxima of the nonlinear correlation measure. As an example, we consider the partitioning of the neurons described in section 2. These neurons act much like neurons in competitive networks, they become tuned to individual patterns or highly correlated clusters. Given that the density of patterns in the input set is p(i), what is the probability p(i) that a neuron will become tuned to this pattern. It is often said that the desired result should be p(i) = p(i), although for Kohonen I-d feature maps ha.~ been shown to be p(i) = p(i)2/3 (see for example, Hertz, Krogh, and Palmer 1991). We have found that he partitioning cannot be calculated by finding the optima of the objective function . For example, in the case of weakly correlated patterns, the global maxima is the most likely pattern, whereas all of the patterns are local maxima. To determine the partitioning, the basin of attraction of each pattern must be computed. This could be different for different dynamics with the same fixed point structure. In order to determine the partitioning, the dynamics must be understood. The details will be described elsewhere (Priigel-Bennett and Shapiro, 1994). For the case of weakly correlated patterns, a neuron will learn a pattern for which p(xp)(Vcr/- 1 > p(xq)(Voq)b-l Vq f- p. Here Vcr is the initial overlap (before learning) of the neuron's weights with the pth pattern. This defines the basin of attraction for each pattern. In the large P limit and for random patterns p(i) ~ p(iYx (11) where a ~ 210g(P)/(b -1), P is the number of patterns, and where b is a parameter that controls the non-linearity of the neuron's response. If b is chosen so that a = 1, Non-Linear Statistical Analysis and Self-Organizing Hebbian Networks 413 then the probability of a neuron learning a pattern will be proportional to the frequency with which the pattern is presented. 5 CONCLUSIONS The relationship between a non-linear generalization of Oja's rule and a non-linear generalization of PCA was reviewed. Non-linear PCA is equivalent to maximizing a objective function which is a statistical measure of the data set. The objective function optimized is determined by the form of the activation function of the neuron. Viewing the neuron in this way is useful, because rather than solving the dynamics, one can use methods of statistical mechanics or other methods to find the maxima of the objective function. Since this function has many local maxima, however, these techniques cannot determine how the solutions are partitioned amoung the neurons. To determine this, the dynamics must be solved. Acknowledgements This work was supported by SERC grant GRG20912. References J. Hertz, A. Krogh, and R.G. Palmer. (1991). Introduction to the Theory of Neural Computation. Addison-Wesley. J. Karhunen and J. J outsensalo. (1992) Nonlinear Heb bian algorithms for sinusoidal frequency estimation, in Artificial Neural Networks, 2, I. Akeksander and J . Taylor, editors, North-Holland. Erkki Oja. (1982) A simplified neuron model as a principal component analyzer. em J. Math. Bio., 15:267-273. Erkki Oja. (1989) Neural networks, principal components, and subspaces. Int. J. of Neural Systems, 1(1):61-68. E. Oja, H. Ogawa, and J. Wangviwattan. (1992) Principal Component Analysis by homogeneous neural networks: Part II: analysis and extension of the learning algorithms IEICE Trans. on Information and Systems, E75-D, 3, pp 376-382. E. Oja. (1993) Nonlinear PCA: algorithms and applications, in Proceedings of World Congress on Neural Networks, Portland, Or. 1993. A. Prugel-Bennett and Jonathan 1. Shapiro. (1993) Statistical Mechanics of Unsupervised Hebbian Learning. J. Phys. A: 26, 2343. A. Prugel-Bennett and Jonathan L. Shapiro. (1994) The Partitioning Problem for Unsupervised Learning for Non-linear Neurons. J. Phys. A to appear. T. D. Sanger. (1989) Optimal Unsupervised Learning in a Single-Layer Linear Feedforward Neural Network. Neural Networks 2,459-473. Jonathan L. Shapiro and A. Prugel-Bennett (1992), Unsupervised Hebbian Learning and the Shape of the Neuron Activation Function, in Artificial Neural Networks, 2, I. Akeksander and J. Taylor, editors, North-Holland. 414 Shapiro and Prugel-Bennett W . Softky and D. Kammen (1991). Correlations in High Dimensional or Asymmetric Data Sets: Hebbian Neuronal Processing. Neural Networks 4, pp 337-347. J . Taylor, (1993) Forms of Memory, in Proceedings of World Congress on Neural Networks, Portland, Or. 1993.	1993	51
805	Locally Adaptive Nearest Neighbor Algorithms Dietrich Wettschereck Thomas G. Dietterich Department of Computer Science Oregon State University Corvallis, OR 97331-3202 wettscdGcs.orst.edu Abstract Four versions of a k-nearest neighbor algorithm with locally adaptive k are introduced and compared to the basic k-nearest neighbor algorithm (kNN). Locally adaptive kNN algorithms choose the value of k that should be used to classify a query by consulting the results of cross-validation computations in the local neighborhood of the query. Local kNN methods are shown to perform similar to kNN in experiments with twelve commonly used data sets. Encouraging results in three constructed tasks show that local methods can significantly outperform kNN in specific applications. Local methods can be recommended for on-line learning and for applications where different regions of the input space are covered by patterns solving different sub-tasks. 1 Introduction The k-nearest neighbor algorithm (kNN, Dasarathy, 1991) is one of the most venerable algorithms in machine learning. The entire training set is stored in memory. A new example is classified with the class of the majority of the k nearest neighbors among all stored training examples. The (global) value of k is generally determined via cross-validation. For certain applications, it might be desirable to vary the value of k locally within 184 Locally Adaptive Nearest Neighbor Algorithms 185 different parts of the input space to account for varying characteristics of the data such as noise or irrelevant features. However, for lack of an algorithm, researchers have assumed a global value for k in all work concerning nearest neighbor algorithms to date (see, for example, Bottou, 1992, p. 895, last two paragraphs of Section 4.1). In this paper, we propose and evaluate four new algorithms that determine different values for k in different parts of the input space and apply these varying values to classify novel examples. These four algorithms use different methods to compute the k-values that are used for classification. We determined two basic approaches to compute locally varying values for k. One could compute a single k or a set of k values for each training pattern, or training patterns could be combined into groups and k value(s) computed for these groups. A procedure to determine the k to be used at classification time must be given in both approaches. Representatives of these two approaches are evaluated in this paper and compared to the global kNN algorithm. While it was possible to construct data sets where local algorithms outperformed kNN, experiments with commonly used data sets showed, in most cases, no significant differences in performance. A possible explanation for this behavior is that data sets which are commonly used to evaluate machine learning algorithms may all be similar in that attributes such as distribution of noise or irrelevant features are uniformly distributed across all patterns. In other words, patterns from data sets describing a certain task generally exhibit similar properties. Local nearest neighbor methods are comparable in computational complexity and accuracy to the (global) k-nearest neighbor algorithm and are easy to implement. In specific applications they can significantly outperform kNN. These applications may be combinations of significantly different subsets of data or may be obtained from physical measurements where the accuracy of measurements depends on the value measured. Furthermore, local kNN classifiers can be constructed at classification time (on-line learning) thereby eliminating the need for a global cross-validation run to determine the proper value of k. 1.1 Methods compared The following nearest neighbor methods were chosen as representatives of the possible nearest neighbor methods discussed above and compared in the subsequent experiments: • k-nearest neighbor (kNN) This algorithm stores all of the training examples. A single value for k is determined from the training data. Queries are classified according to the class of the majority of their k nearest neighbors in the training data. • localKNN 1:11 unrelltricted This is the basic local kNN algorithm. The three subsequent algorithms are modifications of this method. This algorithm also stores all of the training examples. Along with each training example, it stores a list of those values of k that correctly classify that example under leave-one-out cross-validation. To classify a query q, the M nearest neighbors of the query are computed, and that k which classifies correctly most of these M 186 Wettschereck and Dietterich neighbors is determined. Call this value kM,q. The query q is then classified with the class of the majority of its kM,q nearest neighbors. Note that kM,q can be larger or smaller than M. The parameter M is the only parameter of the algorithm, and it can be determined by cross-validation. • localKNN kI pruned The list of k values for each training example generally contains many values. A global histogram of k values is computed, and k values that appear fewer than L times are pruned from all lists (at least one k value must, however, remain in each list). The parameter L can be estimated via crossvalidation. Classification of queries is identical to localKNN kI unrestricted. • localKNN one 1: per clau For each output class, the value of k that would result in the correct (leaveone-out) classification of the maximum number of training patterns from that class is determined. A query q is classified as follows: Assume there are two output classes, C1 and C2 • Let kl and k2 be the k value computed for classes Cl and C2, respectively. The query is assigned to class C1 if the percentage of the kl nearest neighbors of q that belong to class C1 is larger than the percentage of the k2 nearest neighbors of q that belong to class C2. Otherwise, q is assigned to class C2. Generalization of that procedure to any number of output classes is straightforward. • localKNN one 1: per cluster An unsupervised cluster algorithm (RPCL, l Xu et al., 1993) is used to determine clusters of input data. A single k value is determined for each cluster. Each query is classified according to the k value of the cluster it is assigned to. 2 Experimental Methods and Data sets used To measure the performance of the different nearest neighbor algorithms, we employed the training set/test set methodology. Each data set was randomly partitioned into a training set containing approximately 70% of the patterns and a test set containing the remaining patterns. After training on the training set, the percentage of correct classifications on the test set was measured. The procedure was repeated a total of 25 times to reduce statistical variation. In each experiment, the algorithms being compared were trained (and tested) on identical data sets to ensure that differences in performance were due entirely to the algorithms. Leave-one-out cross-validation (Weiss & Kulikowski, 1991) was employed in all experiments to estimate optimal settings for free parameters such as k in kNN and M in localKNN. 1 Rival Penalized Competitive Learning is a straightforward modification of the well known k-means clustering algorithm. RPCL's main advantage over k-means clustering is that one can simply initialize it with a sufficiently large number of clusters. Cluster centers are initialized outside of the input range covered by the training examples. The algorithm then moves only those cluster centers which are needed into the range of input values and therefore effectively eliminates the need for cross-validation on the number of clusters in k-means. This paper employed a simple version with the number of initial clusters always set to 25, O'c set to 0.05 and O'r to 0.002. Locally Adaptive Nearest Neighbor Algorithms 187 We report the average percentage of correct classifications and its standard error. Two-tailed paired t-tests were conducted to determine at what level of significance one algorithm outperforms the other. We state that one algorithm significantly outperforms another when the p-value is smaller than 0.05. 3 Results 3.1 Experiments with Constructed Data Sets Three experiments with constructed data sets were conducted to determine the ability of local nearest neighbor methods to determine proper values of k. The data sets were constructed such that it was known before experimentation that varying k values should lead to superior performance. Two data sets which were presumed to require significantly different values of k were combined into a single data set for each of the first two experiments. For the third experiment, a data set was constructed to display some characteristics of data sets for which we assume local kNN methods would work best. The data set was constructed such that patterns from two classes were stretched out along two parallel lines in one part of the input space. The parallel lines were spaced such that the nearest neighbor for most patterns belongs to the same class as the pattern itself, while two out of the three nearest neighbors belong to the other class. In other parts of the input space, classes were well separated, but class labels were flipped such that the nearest neighbor of a query may indicate the wrong pattern while the majority of the k nearest neighbors (k > 3) would indicate the correct class (see also Figure 4). Figure 1 shows that in selected applications, local nearest neighbor methods can lead to significant improvements over kNN in predictive accuracy. Letter Experiment 2 Sine-21 Wave-21 Combined Experiment 3 Constructed 70.0±O.6 -4~~~~~~~~~~~~~~~~~ I. ks pruned • ks unrestricted Q one k per class 0 one k per cluster 1 Figure 1: Percent accuracy of local kNN methods relative to kNN on separate test sets. These differences () were statistically significant (p < 0.05). Results are based on 25 repetitions. Shown at the bottom of each graph are sizes of training sets/sizes of test sets/number of input features. The percentage at top of each graph indicates average accuracy of kN N ± standard error. The best performing lqcal methods are locaIKNNl;, pruned, localKNNl;8 unre,tricted, 188 Wettschereck and Dietterich and 10calKNNone k per cluster. These methods were outperformed by kNN in two of the original data sets. However, the performance of these methods was clearly superior to kNN in all domains where data were collections of significantly distinct subsets. 3.2 Experiments with Commonly Used Data Sets Twelve domains of varying sizes and complexities were used to compare the performance of the various nearest neighbor algorithms. Data sets for these domains were obtained from the UC-Irvine repository of machine learning databases (Murphy & Aha, 1991, Aha, 1990, Detrano et al., 1989). Results displayed in Figure 2 indicate that in most data sets which are commonly used to evaluate machine learning algorithms, local nearest neighbor methods have only minor impact on the performance of kNN. The best local methods are either indistinguishable in performance from kNN (localKNN one k per cluster) or inferior in only one domain (localKNN k, pruned). 105150/4 150/64/9 16 ~"""T'"-f&C:NN -2 I. ks pruned • ks unrestricted Iilll one k per class 0 one k per cluster 1 Figure 2: Percent accuracy of local kNN methods relative to kNN on separate test sets. These differences (*) were statistically significant (p < 0.05). Results are based on 25 repetitions. Shown at the bottom of each graph are sizes of training sets/sizes of test sets/number of input features. The percentage at top of each graph indicates average accuracy of kNN ± standard error. The number of actual k values used varies significantly for the different local methods (Table 1). Not surprisingly, 10calKNNks unrestricted uses the largest number of distinct k values in all domains. Pruning of ks significantly reduced the number of values used in all domains. However, the method using the fewest distinct k values is 10calKNN one k per cluster, which also explains the similar performance of kNN and 10calKNNone k per cluster in most domains. Note that several clusters computed by 10calKNN one k per cluster may use the same k. Locally Adaptive Nearest Neighbor Algorithms 189 Table 1: Average number of distinct values for k used by local kNN methods. Task kNN local kNN methods k! k! one k per one k per J2runed unredricted cia!! clu!ter Letter recos. 1 7.6±1.1 10.8±1.5 6 .4.±O.3 1.8±O.2 Led-16 1 I6.4.±2.5 4.3.3±O.9 9.2±O.1 9.2±O.5 CombinedLL 1 52.0±3.8 71.4.±1.2 H .7±O.4. 3.0±O.2 Sine-21 1 6.6±l.O 27.5±1.1 2.0±O.O l.O±O.O Waveform-21 1 9.1±1.4. 28.0±1.5 2.9±O.1 4..2±O.2 Combined SW 1 13.5±1.5 30.8±1.6 3.0±O.O 4..8±O.2 Constructed 1 1l .8±O.9 15.7±O.5 2.0±O.O 5.4.±O.2 Iris 1 1.6±O.2 2.0±O.2 2.4.±O.1 2.3±O.1 Glasd 1 7.7±O.8 1l.2±O.7 3.3±O.2 1.9±O.2 Wine 1 2.2±O.4. 3.8±O.4. 2.0±O.1 2.6±O.1 Hunsarian 1 4..I±O.6 12.6±O.6 2.0±O.O l.O±O.O Cleveland 1 8.0±l.O 17.2±1.1 1.8±O.1 4. .6±O.2 Votins 1 4..I±O.4. 6.4.±O.3 2.0±O.O 1.3±O.1 Led-7 Display 1 5.6±O.4. 7.6±O.4. 6.1±O.2 1.0±O.O Led-24. Display 1 16.0±2.9 37.4.±1.6 9 .0±O.2 1.6±O.2 Waveform-2I 1 9.7±1.3 27.8±1.2 3.0±O.O 4..3±O.1 Waveform-4.0 1 8.4.±2.0 29.9±1.5 3.0±O.O 4..8±O.1 Iaolet Letter 1 1l.5±2.1 4.3.9±O.6 16.5±O.5 7.1±O.3 Letter reco6' 1 9.4.±1.9 I7.0±2.3 6.0±O.3 2.4.±O.2 Figure 3 shows, for one single run of Experiment 2 (data sets were combined as described in Figure 1), which k values were actually used by the different local methods. Three clusters of k values can be seen in this graph, one cluster at k = 1, one at k = 7,9,11,12 and the third at k = 19,20,21. It is interesting to note that the second and the third cluster correspond to the k values used by kNN in the separate experiments. Furthermore, kNN did not use k = 1 in any of the separate runs. This gives insight into why kNN's performance was inferior to that of the local methods in this experiment: Patterns in the combined data set belong to one of three categories as indicated by the k values used to classify them (k = 1, k ~ 10, k ~ 20). Hence, the performance difference is due to the fact that kNN must estimate at training time which single category will give the best performance while the local methods make that decision at classification time for each query depending on its local neighborhood. • 13 kvalues (bars) • 30 k values (bars) El 3 k values (bars) o S k values (bars) one k per class 0 one k per cluster I Figure 3: Bars show number of times local kNN methods used certain k values to classify test examples in Experiment 2 (Figure 1 (Combined), numbers based on single run). KNN used k = 1 in this experiment. 190 Wettschereck and Dietterich 4 Discussion Four versions of the k-nearest neighbor algorithm which use different values of k for patterns which belong to different regions of the input space were presented and evaluated in this paper. Experiments with constructed and commonly used data sets indicate that local nearest neighbor methods may have superior classification accuracy than kNN in specific domains. Two methods can be recommended for domains where attributes such as noise or relevance of attributes vary significantly within different parts of the input space. The first method, called localKNN 1:" pruned, computes a list of "good" k values for each training pattern, prunes less frequent values from these lists and classifies a query according to the list of k values of a pre-specified number of neighbors of the query. Leave-one-out cross-validation is used to estimate the proper amount of pruning and the size of the neighborhood that should be used. The other method, localKNNone k per du,ter, uses a cluster algorithm to determine clusters of input patterns. One k is then computed for each cluster and used to classify queries which fall into this cluster. LocalKNNone k per du,ter performs indistinguishable from kNN in all commonly used data sets and outperforms kNN on the constructed data sets. This method compared with all other local methods discussed in this paper introduces a lower computational overhead at classification time and is the only method which could be modified to eliminate the need for leave-one-ou t cross-validation. The only purely local method, localKNN k. unre,tricted, performs well on constructed data sets and is comparable to kNN on non-constructed data sets. Sensitivity studies (results not shown) showed that a constant value of 25 for the parameter M gave results comparable to those where cross-validation was used to determine the value of M. The advantage of localKNNk, unrestricted over the other local methods and kNN is that this method does not require any global information whatsoever (if a constant value for M is used). It is therefore possible to construct a localKNN k6 unre,tricted classifier for each query which makes this method an attractive alternative for on-line learning or extremely large data sets. If the researcher has reason to believe that the data set used is a collection of subsets with significantly varying attributes such as noise or number of irrelevant features, we recommend the construction of a classifier from the training data using localKNN on e k per du,ter and comparison of its performance to kNN. If the classifier must be constructed on-line then localKNNk, unre,tricted should be used instead of kNN. We conclude that there is considerable evidence that local nearest neighbor methods may significantly outperform the k-nearest neighbor method on specific data sets. We hypothesize that local methods will become relevant in the future when classifiers are constructed that simultaneously solve a variety of tasks. Acknowledgements This research was supported in part by NSF Grant IRI-8657316, NASA Ames Grant NAG 2-630, and gifts from Sun Microsystems and Hewlett-Packard. Many thanks Locally Adaptive Nearest Neighbor Algorithms 191 to Kathy Astrahantseff and Bill Langford for helpful comments during the revision of this manuscript. References Aha, D.W. (1990). A Study of Instance-Based Algorithms for Supervised Learning Tasks. Technical Report, University of California, Irvine. Bottou, L., Vapnik, V. (1992). Local Learning Algorithms. Neural Computation, 4(6), 888-900. Dasarathy, B.V. (1991). Nearest Neighbor(NN) Norms: NN Pattern Classification Techniques. IEEE Computer Society Press. Detrano, R., Janosi, A., Steinbrunn, W., Pfisterer, M., Schmid, K., Sandhu, S., Guppy, K., Lee, S. & Froelicher, V. (1989). Rapid searches for complex patterns in biological molecules. American Journal of Cardiology, 64, 304-310. Murphy, P.M. & Aha, D.W. (1991). UCI Repository of machine learning databases {Machine-readable data repository}. Technical Report, University of California, Irvine. Weiss, S.M., & Kulikowski, C.A. (1991). Computer Systems that learn. San Mateo California: Morgan Kaufmann Publishers, INC. Xu, L., Krzyzak, A., & Oja, E. (1993). Rival Penalized Competitive Learning for Clustering Analysis, RBF Net, and Curve Detection IEEE Transactions on Neural Networks, 4(4),636-649. kNN correct: local kNN correct: Total correct: 69.3% 66.9% I i I ! : : ! .50 da .. point. ! •• ------ Nol..,(.-da .. -----_ i •• ------ Nol..,rr-da .. -----i .50d."polnta i ! : : ! ~ ! i I kNN:70.0% .51.0'11> 84.6% local kNN: 74.8% 77 . .5% 78.3% Size o( ttalnlnll ..,t: 480 leat act: 120 Figure 4: Data points for the Constructed data set were drawn from either of the two displayed curves (i.e. all data points lie on either of the two curves). Class labels were flipped with increasing probabilities to a maximum noise level of approximately 45% at the respective ends of the two lines. Listed at the bottom is performance of kNN and 10calKNN unre.stricted within different regions of the input space and for the entire input space.	1993	52
806	Bayesian Backprop in Action: Pruning, Committees, Error Bars and an Application to Spectroscopy Hans Henrik Thodberg Danish Meat Research Institute Maglegaardsvej 2, DK-4000 Roskilde thodberg~nn.meatre.dk Abstract MacKay's Bayesian framework for backpropagation is conceptually appealing as well as practical. It automatically adjusts the weight decay parameters during training, and computes the evidence for each trained network. The evidence is proportional to our belief in the model. The networks with highest evidence turn out to generalise well. In this paper, the framework is extended to pruned nets, leading to an Ockham Factor for "tuning the architecture to the data". A committee of networks, selected by their high evidence, is a natural Bayesian construction. The evidence of a committee is computed. The framework is illustrated on real-world data from a near infrared spectrometer used to determine the fat content in minced meat. Error bars are computed, including the contribution from the dissent of the committee members. 1 THE OCKHAM FACTOR William of Ockham's (1285-1349) principle of economy in explanations, can be formulated as follows: 208 If several theories account for a phenomenon we should prefer the simplest which describes the data sufficiently well. Bayesian Backprop in Action 209 The principle states that a model has two virtues: simplicity and goodness of fit. But what is the meaning of "sufficiently well" - i.e. what is the optimal trade-off between the two virtues? With Bayesian model comparison we can deduce this trade-off. We express our belief in a model as its probability given the data, and use Bayes' formula: P(H I D) = P(D IH)P(H) P(D) (1) We assume that the prior belief P(H) is the same for all models, so we can compare models by comparing P(D IH) which is called the evidence for H, and acts as a quality measure in model comparison. Assume that the model has a single tunable parameter w with a prior range ~ Wprior so that pew IH) = 1/ ~Wprior. The most probable (or maximum posterior) value WMP of the parameter w is given by the maximum of P( ID H)= P(Dlw,H)P(wl1i) w, P(DIH) (2) The width of this distribution is denoted ~Wpo8terior. The evidence P(D 11i) is obtained by integrating over the posterior w distribution and approximating the integral: P(DIH) Evidence J P(Dlw,H)P(wIH)dw P(D I WMP, H) ~wpo8terior ~Wprior Likelihood x OckhamFactor The evidence for the model is the product of two factors: (3) (4) (5) • The best fit likelihood, i.e. the probability of the data given the model and the tuned parameters. It measures how well the tuned model fits the data . • The integrated probability of the tuned model parameters with their uncertainties, i.e. the collapse of the available parameter space when the data is taken into account. This factor is small when the model has many parameters or when some parameters must be tuned very accurately to fit the data. It is called the Ockham Factor since it is large when the model is simple. By optimizing the modelling through the evidence framework we can avoid the overfitting problem as well as the equally important "underfitting" problem. 2 THE FOUR LEVELS OF INFERENCE In 1991-92 MacKay presented a comprehensive and detailed framework for combining backpropagation neural networks with Bayesian statistics (MacKay, 1992). He outlined four levels of inference which applies for instance to a regression problem where we have a training set and want to make predictions for new data: 210 Thodberg Level 1 Make predictions including error bars for new input data. Level 2 Estimate the weight parameters and their uncertainties. Level 3 Estimate the scale parameters (the weight decay parameters and the noise scale parameter) and their uncertainties. Level 4 Select the network architecture and for that architecture select one of the w-minima. Optionally select a committee to reflect the uncertainty on this level. Level 1 is the typical goal in an application. But to make predictions we have to do some modelling, so at level 2 we pick a net and some weight decay parameters and train the net for a while. But the weight decay parameters were picked rather arbitrarily, so on level 3 we set them to their inferred maximum posterior (MP) value. We alternate between level 2 and 3 until the network has converged. This is still not the end, because also the network architecture was picked rather arbitrarily. Hence level 2 and 3 are repeated for other architectures and the evidences of these are computed on level 4. (Pruning makes level 4 more complicated, see section 6). When we make inference on each of these levels, there are uncertainties which are described by the posterior distributions of the parameters which are inferred. The uncertainty on level 2 is described by the Hessian (the second derivative of the net cost function with respect to the weights). The uncertainty on level 3 is negligible if the number of weight decays parameters is small compared to the number of weights. The uncertainty on level 4 is described by the committee of networks with highest evidence within some margin (discussed below). The uncertainties are used for two purposes. Firstly they give rise to error bars on the predictions on level 1. And secondly the posterior uncertainty divided by the prior uncertainty (the Ockham Factor) enters the evidence. MacKay's approach differs in two respects from other Bayesian approaches to neural nets: • It assumes the Gaussian approximation to the posterior weight distribution. In contrast, the Monte Carlo approach of (Neal, 1992) does not suffer from this limitation . • It determines maximum posterior values of the weight decay parameters, rather than integrating them out as done in (Buntine and Weigend, 1991). It is difficult to justify these choices in general. The Gaussian approximation is believed to be good when there are at least 3 training examples per weight (MacKay, 1992). The use of MP weight decay parameters is the superior method when there are ill-defined parameters, as there usually is in neural networks, where some weights are typically poorly defined by the data (MacKay, 1993). 3 BAYESIAN NEURAL NETWORKS The training set D consists of N cases of the form (x, t). We model t as a function of x, t = y(x) + II, where II is Gaussian noise and y(x) is computed by a neural Bayesian Backprop in Action 211 network 11. with weights w. The noise scale is a free parameter {3 = 1/(1';. The probability of the data (the likelihood) is P(Dlw,{3,11.) ex exp(-{3En) En ~ L:(y - t)2 where the sum extends over the N cases. (6) (7) In Bayesian modelling we must specify the prior distribution of the model parameters. The model contains k adjustable parameters w, called weights, which are in general split into several groups, for instance one per layer of the net. Here we consider the case with all weights in one group. The general case is described in (MacKay, 1992) and in more details in (Thodberg, 1993). The prior of the weights w 1S p(w\{3,e,11.) ex exp(-{3eEw) Ew _ ~L:w2 (8) (9) {3 and e are called the scales of the model and are free parameters determined by the data. The most probable values of the weights given the data, some values of the scales (to be determined later) and the model, is given by the maximum of P(w\D,{3,e,11.) P(D/w,{3,e, 11.)p(w/{3,e, 11.) ( (3C) p(D\{3,e,11.) ex exp (10) (11) So the maximum posterior weights according to the probabilistic interpretation are identical to the weights obtained by minimising the familiar cost function C with weight decay parameter e. This is the well-known Bayesian account for weight decay. 4 MACKAY'S FORMULAE The single most useful result of MacKay's analysis is a simple formula for the MP value of the weight decay parameter En , eMP =-E N w -, (12) where , is the number of well-determined parameters which can be approximated by the actual number of parameters k, or computed more accurately from the eigenvalues Ai of the Hessian \T\T En: k A' -L: ' ,;=} Ai + eMP (13) The MP value of the noise scale is {3MP = N /(2C). 212 Thodberg The evidence for a neural network 'Jf. is, as in section 1, obtained by integration over the posterior distribution of the inferred parameters, which gives raise to the Ockham Factors: Ev('Jf.) = log P( D 1'Jf.) logOck(w) Ock«(3) N 'Y _ N log 411"C 2 2 N + log Ock(w) + log Ock«(3) + log Ock(e) k "" eMP 'Y ~ L.J log - - + log h! + h log 2 i=l eMP + Ai 2 J411"/(N - 'Y) Ock(e) = .f4iFt logO logO (14) (15) (16) The first line in (14) is the log likelihood. The Ockham Factor for the weights Ock(w) is small when the eigenvalues Ai of the Hessian are large, corresponding to well-determined weights. 0 is the prior range of the scales and is set (subjectively) to 103 . The expression (15) (valid for a network with a single hidden layer) contains a symmetry factor h!2h. This is because the posterior volume must include all w configurations which are equivalent to the particular one. The hidden units can be permuted, giving a factor h! more posterior volume. And the sign of the weights to and from every hidden unit can be changed giving 2h times more posterior volume. 5 COMMITTEES For a given data set we usually train several networks with different numbers of hidden units and different initial weights. Several of these networks have evidence near or at the maximal value, but the networks differ in their predictions. The different solutions are interpreted as components of the posterior distribution and the correct Bayesian answer is obtained by averaging the predictions over the solutions, weighted by their posterior probabilities, i.e. their evidences. However, the evidence is not accurately determined, primarily due to the Gaussian approximation. This means that instead of weighting with Ev('Jf.) we should use the weight exp{log Ev / ~(log Ev», where ~(log Ev) is the total uncertainty in the evaluation of log Ev. As an approximation to this, we define the committee as the models with evidence larger than log Evrnax - ~ log Ev, where Evrnax is the largest evidence obtained, and all members enter with the same weight. To compute the evidence Ev(C) of the committee, we assume for simplicity that all networks in the committee C share the same architecture. Let Nc be the number of truly different solutions in the committee. Of course, we count symmetric realisations only once. The posterior volume i.e. the Ockham Factor for the weights is now Nc times larger. This renders the committee more probable - it has a larger evidence: log Ev(C) = log Nc + log Ev('Jf.) (17) where log Ev('Jf.) denotes the average log evidence of the members. Since the evidence is correlated with the generalisation error, we expect the committee to generalise better than the committee members. Bayesian Backprop in Action 213 6 PRUNING We now extend the Bayesian framework to networks which are pruned to adjust the architecture to the particular problem. This extends the fourth level of inference. At first sight, the factor h! in the Ockham Factor for the weights in a sparsely connected network appears to be lost, since the network is (in general) not symmetric with respect to permutations of the hidden units. However, the symmetry reappears because for every sparsely connected network with tuned weights there are h! other equivalent network architectures obtained by permuting the hidden units. So the factor h! remains. If this argument is not found compelling, it can be viewed as an assumption. If the data are used to select the architecture, which is the case in pruning designed to minimise the cost function, an additional Ockham Factor must be included. With one output unit, only the input-to-hidden layer is sparsely connected, so consider only these connections. Attach a binary pruning parameter to each of the m potential connections. A sparsely connected architecture is described by the values of the pruning parameters. The prior probability of a connection to be present is described by a hyperparameter cP which is determined from the data i.e. it is set to the fraction of connections remaining after pruning (notice the analogy between cP and a weight decay parameter). A non-pruned connection gives an Ockham Factor cP and a pruned 1 cP, assuming the data to be certain about the architecture. The Ockham Factors for the pruning parameters is therefore log Ock(pruning) = m(cPMP log cPMP + (1 - cPMP) 10g(1 - cPMP» (18) The tuning of the meta-parameter to the data gives an Ockham factor Ock( cP) :::::; J2jm, which is rather negligible. From a minimum description length perspective (18) reflects the extra information needed to describe the topology of a pruned net relative to a fully connected net. It acts like a barrier towards pruning. Pruning is favoured only if the negative contribution log Ock(pruning) is compensated by an increase in for instance log Ock(w). 7 APPLICATION TO SPECTROSCOPY Bayesian Backprop is used in a real-life application from the meat industry. The data were recorded by a Tecator near-infrared spectrometer which measures the spectrum of light transmitted through samples of minced pork meat. The absorbance spectrum has 100 channels in the region 850-1050 nm. We want to calibrate the spectrometer to determine the fat content. The first 10 principal components of the spectra are used as input to a neural network. Three weight decay parameters are used: one for the weights and biases of the hidden layer, one for the connections from the hidden to the output layer, and one for the direct connections from the inputs to the output as well as the output bias. The relation between test error and log evidence is shown in figure 1. The test error is given as standard error of prediction (SEP), i.e. the root mean square error. The 12 networks with 3 hidden units and evidence larger than -270 are selected for a 214 Thodberg c: 0 ."" .!Z! "Q I!! c.. 0 ~ e ~ LU "E '" j en 0 C\f ... • 1 hidden unit 0 ¢ 2 hidden units • 3 hidden units X 4 hidden units • 6 hidden units 0 8 hidden units 0 X C! .,... X • co • 0 • X ci • 0 • X 0 • •• • X 0 .. • ~OoO •• X ~ 0 • 0 0 C X • X • d(D X • ~ X -. °0 C • ~mwlDl • • C • . X_. ~ •• X IJ 0 c Ix X -320 -300 -280 -260 log Evidence Figure 1: The test error as a function of the log evidence for networks trained on the spectroscopic data. High evidence implies low test error. committee. The committee average gives 6% lower SEP than the members do on average, and 21% lower SEP than a non-Bayesian analysis using early stopping (see Thodberg, 1993). Pruning is applied to the networks with 6 hidden units. The evidence decreases slightly, i.e. Ock(pruning) dominates. Also the SEP is slightly worse. So the evidence correctly suggests that pruning is not useful for this problem. 1 The Bayesian error bars are illustrated for the spectroscopic data in figure 2. We study the model predictions on the line through input space defined by the second principal component axis, i.e. the second input is varied while all other inputs are zero. The total prediction variance for a new datum x is (19) where Uwu comes from the weight uncertainties (level 2) and Ucu from the committee dissent (level 4). 1 For artificial data generated by a sparsely connected network the evidence correctly points to pruned nets as better models (see Thodberg, 1993). l c: ~ 8 as u.. ~ yo) .0 .Bayesian Backprop in Action 215 '. " . • . . Total U,..rllinty .••• • ,/ CommiUM Prediction ...... ..... : ................... '\ ......... ::::::::::::::: .......... . \ Total Unc.rtlinly ••• •..••• • .•• •• \ ...... " . I \ \ \\ .... P* Unc.rllinly \ ". .I ...... :::::::::.... .! ".,"'. !/ , \~"""~ ' • ConllTitlM Unc.rllinly ". , '. . <'>" .... /j .... .... ,'.!; ,'. ,II ... .... ;. ,I ". \.... 10· Weight Unc«tainty '. \~ "~ '" " \ .... " \ ". .... -.-;.' ,I ........ if .:~" . "./ ,'. I. ... ". \ '. 10· Randcm Noi.. '.<, ij. .... '. .' t'/" ... " . ................................................ ./ ... "./ / , // / ",..... .,... / '....:..-.:::----...::-_"'::-_.-._.-. ~/ ---..----~ o ~------,_----------._----------r_---------.r---------~~--~~ -4 -2 o 2 4 Second Principal Component Figure 2: Prediction of the fat content as a function of the second principal component P2 of the NIR spectrum. 95% of the training data have Ip21 < 2. The total error bars are indicated by a "1 sigma" band with the dotted lines. The total standard errors O'total(X) and the standard errors of its contributions (O'v, O'wu(x) and O'cu(x)) are shown separately, multiplied by a factor of 10. References W.L.Buntine and A.S.Weigend, "Bayesian Back-Propagation", Complex Systems 5, (1991) 603-643. R.M.Neal, "Bayesian Learning via Stochastic Dynamics", Neural Information Processing Systems, Vol.5, ed. C.L.Giles, S.J . Hanson and J .D.Cowan (Morgan Kaufmann, San Mateo, 1993) D.J .C.MacKay, "A Practical Bayesian Framework for Backpropagation Networks" Neural Compo 4 (1992) 448-472. D.J .C.MacKay, paper on Bayesian hyperparameters, in preparation 1993. H.H.Thodberg, "A Review of Bayesian Backprop with an Application to Near Infrared Spectroscopy" and "A Bayesian Approach to Pruning of Neural Networks", submitted to IEEE Transactions of Neural Networks 1993 (in /pub/neuroprose/thodberg.ace-of-bayes*.ps.Z on archive.cis.ohio-state.edu).	1993	53
807	Solvable Models of Artificial Neural Networks Sumio Watanabe Information and Communication R&D Center Ricoh Co., Ltd. 3-2-3, Shin-Yokohama, Kohoku-ku, Yokohama, 222 Japan sumio@ipe.rdc.ricoh.co.jp Abstract Solvable models of nonlinear learning machines are proposed, and learning in artificial neural networks is studied based on the theory of ordinary differential equations. A learning algorithm is constructed, by which the optimal parameter can be found without any recursive procedure. The solvable models enable us to analyze the reason why experimental results by the error backpropagation often contradict the statistical learning theory. 1 INTRODUCTION Recent studies have shown that learning in artificial neural networks can be understood as statistical parametric estimation using t.he maximum likelihood method [1], and that their generalization abilities can be estimated using the statistical asymptotic theory [2]. However, as is often reported, even when the number of parameters is too large, the error for the test.ing sample is not so large as the theory predicts. The reason for such inconsistency has not yet been clarified, because it is difficult for the artificial neural network t.o find the global optimal parameter. On the other hand, in order to analyze the nonlinear phenomena, exactly solvable models have been playing a central role in mathematical physics, for example, the K-dV equation, the Toda lattice, and some statistical models that satisfy the Yang423 424 Watanabe Baxter equation[3]. This paper proposes the first solvable models in the nonlinear learning problem. We consider simple three-layered neural networks, and show that the parameters from the inputs to the hidden units determine the function space that is characterized by a differential equation. This fact means that optimization of the parameters is equivalent to optimization of the differential equation. Based on this property, we construct a learning algorithm by which the optimal parameters can be found without any recursive procedure. Experimental result using the proposed algorithm shows that the maximum likelihood estimator is not always obtained by the error backpropagation, and that the conventional statistical learning theory leaves much to be improved. 2 The Basic Structure of Solvable Models Let us consider a function fc,w( x) given by a simple neural network with 1 input unit, H hidden units, and 1 output unit, H fc,w(x) = L CiIPw;{X), (I) i=1 where both C = {Ci} and w = {Wi} are parameters to be optimized, IPw;{x) is the output of the i-th hidden unit. We assume that {IPi(X) = IPw, (x)} is a set of independent functions in CH -class. The following theorem is the start point of this paper. Theorem 1 The H -th order differential equation whose fundamental system of solution is {IPi( x)} and whose H -th order coefficient is 1 is uniquely given by (Dwg)(x) = (_l)H H!H+l(g,1P1,1P2, .. ·,IPH) = 0, (2) lVH(IP1, IP2, .. ·,IPH) where ltV H is the H -th order Wronskian, (H-l) (H-l) 'PI 'P2 IPH ( 1) IPH (2) 'PH (H -1) IPH For proof, see [4]. From this theorem, we have the following corollary. Corollary 1 Let g(x) be a C H -class function. Then the following conditions for g(x) and w = {wd are equivalent. (1) There exists a set C = {cd such that g{x) = E~l CjIPw;(x). (2) (Dwg)(x) = O. Solvable Models of Artificial Neural Networks 425 Example 1 Let us consider a case, !Pw;(x) = exp(WiX). H g(x) = L Ci exp(WiX) i=l is equivalent to {DH + P1D H- 1 + P2DH-2 + ... + PH }g(x) = 0, where D = (d/dx) and a set {Pi} is determined from {Wi} by the relation, H zH + PlzH- 1 + P2zH-2 + ... + PIl = II(z - Wi) ('Vz E C). i=l Example 2 (RBF) A function g(x) is given by radial basis functions, 11 g(x) = L Ci exp{ -(x - Wi)2}, i=l if and only if e- z2 {DIl + P1DIl-l + P2DIl-2 + ... + PIl }(eZ2 g(x)) = 0, where a set {Pi} is determined from {Wi} by the relation, 11 zll + Plzll- 1 + P2zll-2 + ... + PII = II(z - 2Wi) ('Vz E C). i=l Figure 1 shows a learning algorithm for the solvable models. When a target function g( x) is given, let us consider the following function approximation problem. 11 g(x) = L Ci!Pw;(X) + E(X). (3) i=l Learning in the neural network is optimizing both {cd and {wd such that E( x) is minimized for some error function. From the definition of D w , eq. (3) is equivalent to (Dwg)(x) = (Dw€)(x), where the term (Dwg)(x) is independent of Cj. Therefore, if we adopt IIDwEIl as the error function to be minimized, {wd is optimized by minimizing IIDwgll, independently of {Cj}, where 111112 = J II(x)12dx. After IIDwgll is minimized, we have (Dw.g)(x) ~ 0, where w* is the optimized parameter. From the corollary 1, there exists a set {cn such that g(x) ~ L:ci!Pw~(x), where {en can be found using the ordinary least square method. 3 Solvable Models For a general function !Pw, the differential operator Dw does not always have such a simple form as the above examples. In this section, we consider a linear operator L such that the differential equation of L!pw has a simple form. Definition A neural network L: Cj!PWi (x) is called solvable ifthere exist functions a, b, and a linear operator L such that (L!pwJ(x) = exp{a{wj)x + b(wi)). The following theorem shows that the optimal parameter of the solvable models can be found using the same algorithm as Figure 1. 426 Watanabe H g(X) = L Ci ~ (x) +E(X) i=l i It is difficult to optimize wi independently ?f ci t There exits C i s.t. H g(x) = L Ci <P .(x) i=l wi I equiv. D g(x) = D E(X) w w II D wg II : minimited -- W: optimized ..... -.-----1 q,* g(x) 0 I eqmv. Least Square Method ~ ci : optimized H g(x) = L < <P .(x) i=l wi Figure 1: St.ructure of Solvable Models Theorem 2 For a solvable model of a neuml network, the following conditions are equivalent when Wi "# Wj (i "# j). (1) There exist both {cd and {wd such that g(x) = E:!:l Ci<t'w;(X). (2) There exists {Pi} such that {DH + P1D H- 1 + P2DH-2 + ... + PH }(Lg)(x) = O. (3) For arbitmry Q > 0, we define a sequence {Yn} by Yn = (Lg)(nQ). Then, there exists {qd such that Yn + qlYn-l + q2Yn-2 + ... + qHYn-H = o. Note that IIDwLgl12 is a quadratic form for {pd, which is easily minimized by the least square method. En IYn + qlYn-l + ... + QHYn_HI2 is also a quadratic form for {Qd· Theorem 3 The sequences { wd, {pd, and {qd in the theorem 2 have the following relations. H H+ H-l+ H-2+ + z PIZ P2 Z ... PH IT(z - a(wi)) ('Vz E C), i=l H zH + qlzH-l + q2zH-2 + ... + qH = IT(z - exp(a(Wi)Q)) ('Vz E C). i=l For proofs of the above theorems, see [5]. These theorems show that, if {Pi} or Solvable Models of Artificial Neural Networks 427 {qd is optimized for a given function g( x), then {a( wd} can be found as a set of solutions of the algebraic equation. Suppose that a target function g( x) is given. Then, from the above theorems, the globally optimal parameter w* = {wi} can be found by minimizing IIDwLgll independently of {cd. Moreover, if the function a(w) is a one-to-one mapping, then there exists w* uniquely without permutation of {wi}, if and only if the quadratic form II{DH + P1DH-1 + ... + PH }g1l2 is not degenerate[4]. (Remark that, if it is degenerate, we can use another neural network with the smaller number of hidden units.) Example 3 A neural network without scaling H fb,c(X) = L CiU(X + bi), i=1 (4) is solvable when (F u)( x) I- 0 (a.e.), where F denotes the Fourier transform. Define a linear operator L by (Lg)(x) = (Fg)(x)/(Fu)(x), then, it follows that H (Lfb,c)(X) = L Ci exp( -vCi bi x). (5) i=l By the Theorem 2, the optimal {bd can be obtained by using the differential 01' the sequential equation. Example 4 (MLP) A three-layered perceptron H ~ -1 X + bi fb,c(X) = L Ci tan ( a. ), i=1 z (6) is solvable. Define a linear operator L by (Lg)( x) = x . (F g)( x), then, it follows that H (Lfb,c)(X) = L Ci exp( -(a.i + yCi bdx + Q(ai, bd) (x ~ 0). (7) i=1 where Q( ai, bi ) is some function of ai and bj. Since the function tan -1 (x) is monotone increasing and bounded, we can expect that a neural network given by eq. (6) has the same ability in the function approximation problem as the ordinary three-layered perceptron using the sigmoid function, tanh{x). Example 5 (Finite Wavelet Decomposition) A finite wavelet decomposition H x + bj fb,c(X) = L Cju( ), (8) a.j i=l is solvable when u(x) = (d/dx)n(1/(l + x 2 » (n ~ 1). Define a lineal' operator L by (Lg)(x) = x- n . (Fg)(x) then, it follows that H (Lfb,c)(X) = L Ci exp( -(a.j + vCi bi)x + P(a.j, bi» (x ~ 0). (9) i=1 428 Watanabe where f3(ai, bi) is some function of ai and bi. Note that O"(x) is an analyzing wavelet, and that this example shows a method how to optimize parameters for the finite wavelet decomposition. 4 Learning Algorithm We construct a learning algorithm for solvable models, as shown in Figure 1< <Learning Algorithm> > (0) A target function g(x) is given. (1) {Ym} is calculated by Ym = (Lg)(mQ). (2) {qi} is optimized by minimizing L:m IYm + Q1Ym-l + Q2Ym-2 + ... + QHYm_HI2. (3) {Zi} is calculated by solving zH + q1zH-1 + Q2zH-2 + ... + QH = 0. (4) {wd is determined by a( wd = (l/Q) log Zi. (5) {cd is optimized by minimizing L:j(g(Xj) - L:i Cj<;?w;(Xj»2. Strictly speaking, g(x) should be given for arbitrary x. However, in the practical applicat.ion, if the number of training samples is sufficiently large so that (Lg)( x) can be almost precisely approximated, this algorithm is available. In the third procedure, to solve the algebraic equation, t.he DKA method is applied, for example. 5 Experimental Results and Discussion 5.1 The backpropagation and the proposed method For experiments, we used a probabilit.y density fUllction and a regression function given by Q(Ylx) 1 ((y - h(X»2) exp J27r0"2 20"2 h(x) 1 -1 X - 1/3 1 -1 X - 2/3 -3" tan ( 0.04 ) + 6" tan ( 0.02 ) where 0" = 0.2. One hundred input samples were set at the same interval in [0,1), and output samples were taken from the above condit.ional distribution. Table 1 shows the relation between the number of hidden units, training errors, and regression errors. In the table, the t.raining errol' in the back propagation shows the square error obtained after 100,000 training cycles. The traiuing error in the proposed method shows the square errol' by the above algorithm. And the regression error shows the square error between the true regression curve h( x) and the estimated curve. Figure 2 shows the true and estimated regression lines: (0) the true regression line and sanlple points, (1) the estimated regression line with 2 hidden units, by the BP (the error backpropagation) after 100,000 training cycles, (2) the estimated regression line with 12 hidden units, by the BP after 100,000 training cycles, (3) the Solvable Models of Artificial Neural Networks 429 Table 1: Training errors and regression errors Hidden Backpropagation Proposed Method Units Training Regression Training Regression 2 4.1652 0.7698 4.0889 0.3301 4 3.3464 0.4152 3.8755 0.2653 6 3.3343 0.4227 3.5368 0.3730 8 3.3267 0.4189 3.2237 0.4297 10 3.3284 0.4260 3.2547 0.4413 12 3.3170 0.4312 3.1988 0.5810 estimated line with 2 hidden units by the proposed method, and (4) the estimated line with 12 hidden units by the proposed method. 5.2 Discussion When the number of hidden units was small, the training errors by the BP were smaller, but the regression errors were larger. Vlhen the number of hidden units was taken to be larger, the training error by the BP didn't decrease so much as the proposed method, and the regression error didn't increase so mnch as the proposed method. By the error back propagation , parameters dichl 't reach the maximum likelihood estimator, or they fell into local minima. However, when t.he number of hidden units was large, the neural network wit.hout. t.he maximum likelihood estimator attained the bett.er generalization. It seems that paramet.ers in the local minima were closer to the true parameter than the maximum likelihood estimator. Theoretically, in the case of the layered neural networks, the maximum likelihood estimator may not be subject to asymptotically normal distribution because the Fisher informat.ion matrix may be degenerate. This can be one reason why the experimental results contradict the ordinary st.atistical theory. Adding such a problem, the above experimental results show that the local minimum causes a strange problem. In order to construct the more precise learning t.heory for the backpropagation neural network, and to choose the better parameter for generalization, we maybe need a method to analyze lea1'1ling and inference with a local minimum. 6 Conclusion We have proposed solvable models of artificial neural networks, and studied their learning structure. It has been shown by the experimental results that the proposed method is useful in analysis of the neural network generalizat.ion problem. 430 Watanabe .. ~--------. '. . ..' .. ..... ' : ... .'" .... ".' .. "0 ' .. (0) True Curve and Samples. Sample error sum = 3.6874 "0 e" : .. : .... " ... ". . . . . ' . ' .. (1) BP after 100,000 cycles H = 2, Etrain = 4.1652, E"eg = 0.7698 . . . ..... " : ... ,'. .. (3) Proposed Method . .. . ..... ,". '.' .. ' H = 2, Etrain = 4.0889, Ereg = 0.3301 H : the number of hidden units Etrain : t.he t.raining error E"eg : the regression error . . ~ ...... , . . "0, e" e"' .. .. ....... ~ .. : ........... : ...... :::: .. . '. . ' • • ' 0" • (2) TIP aft.er 100,000 cycles H = 12, E Ir•a;" = 3.3170, E"eg = 0.4312 .. ...... . .:'{: ' .. (4) Proposed Met.hod H = 12, E'm;" = 3.1988, Ereg = 0.5810 Figure 2: Experimental Results References [I] H. White. (1989) Learning in artificial neural networks: a statistical perspective. Neural Computation, 1, 425-464. [2] N.Murata, S.Yoshizawa, and S.-I.Amari.(1992) Learning Curves, Model Selection and Complexity of Neural Networks. Adlla:nces in Neural Information Processing Systems 5, San Mateo, Morgan Kaufman, pp.607-614. [3] R. J. Baxter. (1982) Exactly Solved Models in Statistical Mechanics, Academic Press. [4] E. A. Coddington. (1955) Th.eory of ordinary differential equations, the McGrawHill Book Company, New York. [5] S. Watanabe. (1993) Function approximation by neural networks and solution spaces of differential equations. Submitted to Neural Networks.	1993	54
808	A Comparative Study Of A Modified Bumptree Neural Network With Radial Basis Function Networks and the Standard MultiLayer Perceptron. Richard T .J. Bostock and Alan J. Harget Department of Computer Science & Applied Mathematics Aston University Binningham England Abstract Bumptrees are geometric data structures introduced by Omohundro (1991) to provide efficient access to a collection of functions on a Euclidean space of interest. We describe a modified bumptree structure that has been employed as a neural network classifier, and compare its performance on several classification tasks against that of radial basis function networks and the standard mutIi-Iayer perceptron. 1 INTRODUCTION A number of neural network studies have demonstrated the utility of the multi-layer perceptron (MLP) and shown it to be a highly effective paradigm. Studies have also shown, however, that the MLP is not without its problems, in particular it requires an extensive training time, is susceptible to local minima problems and its perfonnance is dependent upon its internal network architecture. In an attempt to improve upon the generalisation performance and computational efficiency a number of studies have been undertaken principally concerned with investigating the parametrisation of the MLP. It is well known, for example, that the generalisation performance of the MLP is affected by the number of hidden units in the network, which have to be determined empirically since theory provides no guidance. A number of investigations have been conducted into the possibility of automatically determining the number of hidden units during the training phase (BostOCk, 1992). The results show that architectures can be attained which give satisfactory, although generally sub-optimal, perfonnance. Alternative network architectures such as the Radial Basis Function (RBF) network have also been studied in an attempt to improve upon the performance of the MLP network. The RBF network uses basis functions in which the weights are effective over only a small portion of the input space. This is in contrast to the MLP network where the weights are used in a more global fashion, thereby encoding the characteristics of the training set in a more compact form. RBF networks can be rapidly trained thus making 240 Modified Bumptree Neural Network and Standard Multi-Layer Perceptron 241 them particularly suitable for situations where on-line incremental learning is required. The RBF network has been successfully applied in a number of areas such as speech recognition (Renals, 1992) and financial forecasting (Lowe, 1991). Studies indicate that the RBF network provides a viable alternative to the MLP approach and thus offers encouragement that networks employing local solutions are worthy of further investigation. In the past few years there has been an increasing interest in neural network architectures based on tree structures. Important work in this area has been carried out by Omohundro (1991) and Gentric and Withagen (1993). These studies seem to suggest that neural networks employing a tree based structure should offer the same benefits of reduced training time as that offered by the RBF network. The particular tree based architecture examined in this study is the bumptree which provides efficient access to collections of functions on a Euclidean space of interest. A bumptree can be viewed as a natural generalisation of several other geometric data structures including oct-trees, k-d trees, balltrees (Omohundro, 1987) and boxtrees (Omohundro, 1989). In this paper we present the results of a comparative study of the performance of the three types of neural networks described above over a wide range of classification problems. The performance of the networks was assessed in terms of the percentage of correct classifications on a test, or generalisation data set, and the time taken to train the network. Before discussing the results obtained we shall give an outline of the implementation of our bumptree neural network since this is more novel than the other two networks. 2 THE BUMPTREE NEURAL NETWORK Bumptree neural networks share many of the underlying principles of decision trees but differ from them in the manner in which patterns are classified. Decision trees partition the problem space into increasingly small areas. Classification is then achieved by determining the lowest branch of the tree which contains a reference to the specified point. The bumptree neural network described in this paper also employs a tree based structure to partition the problem space, with each branch of the tree being based on multiple dimensions. Once the problem space has been partitioned then each branch can be viewed as an individual neural network modelling its own local area of the problem space, and being able to deal with patterns from multiple output classes. Bumptrees model the problem space by subdividing the space allowing each division to be described by a separate function. Initial partitioning of the problem space is achieved by randomly assigning values to the root level functions. A learning algorithm is applied to determine the area of influence of each function and an associated error calculated. If the error exceeds some threshold of acceptability then the area in question is further subdivided by the addition of two functions; this process continues until satisfactory performance is achieved. The bumptree employed in this study is essentially a binary tree in which each leaf of the tree corresponds to a function of interest although the possibility exists that one of the functions could effectively be redundant if it fails to attract any of the patterns from its parent function. A number of problems had to be resolved in the design and implementation of the bumptree. Firstly, an appropriate procedure had to be adopted for partitioning the 242 Bostock and Harget problem space. Secondly, consideration had to be given to the type of learning algorithm to be employed. And finally, the mechanism for calculating the output of the network had to be determined. A detailed discussion of these issues and the solutions adopted now follows. 2.1 PARTITIONING THE PROBLEM SPACE The bumptree used in this study employed gaussian functions to partition the problem space, with two functions being added each time the space was partitioned. Patterns were assigned to whichever of the functions had the higher activation level with the restriction that the functions below the root level could only be active on patterns that activated their parents. To calculate the activation of the gaussian function the following expression was used: (1) where Afp is the activation of function f on pattern p over all the input dimensions, afi is the radius of function f in input dimension i, Cfi is the centre of function f in input dimension i, and Inpi is the ith dimension of the pth input vector. It was found that the locations and radii of the functions had an important impact on the performance of the network. In the original bumptree introduced by Omohundro every function below the root level was required to be wholly enclosed by its parent function. This restriction was found to degrade the performance of the bumptree particularly if a function had a very small radius since this would produce very low levels of acti vation for most patterns. In our studies we relaxed this constraint by assigning the radius of each function to one, since the data presented to the bumptree was always normalised between zero and one. This modification led to an improved performance. A number of different techniques were examined in order to effectively position the functions in the problem space. The first approach considered, and the simplest, involved selecting two initial sets of centres for the root function with the centre in each dimension being allocated a value between zero and one. The functions at the lower levels of the tree were assigned in a similar manner with the requirement that their centres fell within the area of the problem space for which their parent function was active. The use of nonhierarchical clustering techniques such as the Forgy method or the K-means clustering technique developed by MacQueen provided other alternatives for positioning the functions. The approach finally adopted for this study was the multiple-initial function (MIF) technique. In the MIF procedure ten sets of functions centres were initially defined by random assignment and each pattern in the training set assigned to the function with the highest activation level. A "goodness" measure was then determined for each function over all patterns for which the function was active. The goodness measure was defined as the square of the error between the calculated and observed values divided by the number of active patterns. The function with the best value was retained and the remaining functions that were active on one or more patterns had their centres averaged in each dimension to provide a second function. The functions were then added to the network structure and the patterns assigned to the function which gave the greater activation. Modified Bumptree Neural Network and Standard Multi-Layer Perceptron 243 2.2 THE LEARNING ALGORITHM A bumptree neural network comprises a number of functions each function having its own individual weight and bias parameters and each function being responsive to different characteristics in the training set. The bumptree employed a weighted value for every input to output connection and a single bias value for each output unit. Several different learning algorithms for determining the weight and bias values were considered together with a genetic algorithm approach (Williams, 1993). A one-shot learning algorithm was finally adopted since this gave good results and was computationally efficient. The algorithm used a pseudo-matrix inversion technique to determine the weight and bias parameters of each function after a single presentation of the relevant patterns in the training set had been made. The output of any function for a given pattern p was determined from jmax = "" a * (p) + f.l. GO ipz £..J ijz X j Piz (2) j=l where aoipz is the output of the zth output unit of the ith function on the pth pattern, j is the input unit, jmax is the total number of input units, aijz is the weight that connects the jth input unit to the zth output unit for the ith function, Xj(p) is the element of the pth pattern concerned with the jth input dimension, and ~iz is the bias value for the zth output unit. The weight and bias parameters were determined by minimising the squared error given in (3), where Ei is the error of the ith function across all output dimensions (zmax), for all patterns upon which the function is active (pmax). The desired output for the zth output dimension is tvpz" and aoipz is the actual output of the ith function on the zth dimension of the pth pattern. The weight values are again represented by Ooijz and the bias by ~iz' (3) After the derivatives of aijz and ~iz were determined it was a simple task to arrive at the three matrices used to calculate the weight and bias values for the individual functions. Problems were encountered in the matrix inversion when dealing with functions which were only active on a few patterns and which were far removed from the root level of the tree; this led to difficulties with singular matrices. It was found that the problem could be overcome by using the Gauss-Jordan singular decomposition technique for the pseudoinversion of the matrices. 2.3 CALCULATION OF THE NETWORK OUTPUT The difficulty in determining the output of the bumptree was that there were usually functions at different levels of the tree that gave slightly different outputs for each active pattern. Several different approaches were studied in order to resolve the difficulty including using the normalised output of all the active functions in the tree irrespective of their level in the structure. A technique which gave good results and was used in this 244 Bostock and Harget study calculated the output for a pattern solely on the output of the lowest level active function in the tree. The final output class of a pattern being given by the output unit with the highest level of activation. 3 NETWORK PERFORMANCES The perfonnance of the bumptree neural network was compared against that of the standard MLP and RBF networks on a number of different problems. The bumptree used the MIF placing technique in which the radius of each function was set to one. This particular implementation of the bumptree will now be referred to as the MIF bumptree. The MLP used the standard backpropagation algorithm (Rumelhart, 1986) with a learning rate of 0.25 and a momentum value of 0.9. The initial weights and bias values of the network were set to random values between -2 and +2. The number of hidden units assigned to the network was determined empirically over several runs by varying the number of hidden units until the best generalisation perfonnance was attained. The RBF network used four different types of function, they were gaussian, multi-quadratic, inverse multi-quadratic and thin plate splines. The RBF network placed the functions using sample points within the problem space covered by the training set 3.1 INITIAL STUDIES In the initial studies. a set of classical non-linear problems was used to compare the perfonnance of the three types of networks. The set consisted of the XOR, Parity(6) and Encoder(8) problems. The average results obtained over 10 runs for each of the data sets are shown in Table 1 - the figures presented are the percentage of patterns correctly classified in the training set together with the standard deviation. Table 1. Percentage of Patterns Correctly Classified for the three Data Sets for each Network type. DATA SET MLP RBF MIF XOR Parity(6) Encoder(8) 100 100 100 100 92.1 ± 4.7 82.5 ± 16.8 100 98.3 ± 4.2 100 For the XOR problem the MLP network required an average of 222 iterations with an architecture of 4 hidden units, for the parity problem an architecture of 10 hidden units and an average of 1133 iterations. and finally for the encoder problem the network required an average of 1900 iterations for an architecture consisting of three hidden units. The RBF network correctly classified all the patterns of the XOR data set when four multi-quadratic. inverse multi-quadratic or gaussian functions were used. For the parity(6) problem the best result was achieved with a network employing between 60 and 64 inverse multi-quadratic functions. In the case of the encoder problem the best performance was obtained using a network of 8 multi-quadratic functions. The MIF bumptree required two functions to achieve perfect classification for the XOR and encoder problems and an average of 40 functions in order to achieve the best perfonnance on the parity problem. Thus in the case of the XOR and encoder problems no further functions were required additional to the root functions. Modif1ed Bumptree Neural Network and Standard Multi-Layer Perceptron 245 A comparison of the training times taken by each of the networks revealed considerable differences. The MLP required the most extensive training time since it used the backpropagation training algorithm which is an iterative procedure. The RBF network required less training time than the MLP, but suffered from the fact that for all the patterns in the training set the activity of all the functions had to be calculated in order to arrive at the optimal weights. The bumptree proved to have the quickest training time for the parity and encoder problems and a training time comparable to that taken by the RBF network for the XOR problem. This superiority arose because the bumptree used a noniterative training procedure, and a function was only trained on those members of the training set for which the function was active. In considering the sensitivity of the different networks to the parameters chosen some interesting results emerge. The performance of the MLP was found to be dependent on the number of hidden units assigned to the network. When insufficient hidden units were allocated the performance of the MLP degraded. The performance of the RBF network was also found to be highly influenced by the values taken for various parameters, in particular the number and type of functions employed by the network. The bumptree on the other hand was assigned the same set of parameters for all the problems studied and was found to be less sensitive than the other two networks to the parameter settings. 3.2 COMPARISON OF GENERALISATION PERFORMANCE The performance of the three different networks was also measured for a set of four 'realworld' problems which allowed the generalisation performance of each network to be determined. A summary of the results taken over 10 runs is given in Table 2. Table 2 Performance of the Networks on the Training and Generalisation Data Sets of the Test Problems. DATA NETWORK FUNCTIONS TRAINING TEST HIDDEN UNITS Iris MLP 4 100 95.7 ± 0.6 RBF 75 gaussians 100 96.0 ± 0.0 MIF 8 100 97.5 ± 0.4 Skin Cancer MLP 6 88.7 ± 4.3 79.2 ± 1.7 RBF 10 multi-quad 84.4 ± 3.2 80.3 ± 4.4 MIF 4 79.8 ± 5.2 80.8 ± 1.9 Vowel Data MLP 20 82.4 ± 5.3 77.1 ± 6.6 RBF 50 Thin plate spl. 82.1 ± 1.5 77.8 ± 1.4 MIF 104 86.5 ± 5.6 73.6 ± 4.6 Diabetes MLP 16 82.5 ± 2.7 78.9 ± 1.2 RBF 25 Thin plate spl. 76.0 ± 0.8 78.9 ± 0.9 MIF 3 76.5 ± 1.2 80.0 ± 1.1 All three networks produce a comparable performance on the test problems, but in the case of the bumptree this was achieved with a training time substantially less than that required by the other networks. Inspection of the results also shows that the bumptree required fewer functions in general than the RBF network. 246 Bostock and Harget The results shown above for the bumptree were obtained with the same set of parameters used in the initial study which further confirms its lack of sensitivity to parameter settings. 4. CONCLUSION A comparative study of the performance of three different types of networks, one of which is novel, has been conducted on a wide range of problems. The results show that the performance of the bumptree compared very favourably, both in terms of generalisation and training times, with the more traditional MLP and RBF networks. In addition, the performance of the bumptree proved to be less sensitive to the parameters settings than the other networks. These results encourage us to continue further investigation of the bumptree neural network and lead us to conclude that it has a valid place in the list of current neural networks. Acknowledgement We gratefully acknowledge the assistance given by Richard Rohwer. References Bostock R.T 1. & Harget Al. (1992) Towards a Neural Network Based System for Skin Cancer Diagnosis: lEE Third International Conference on Artificial Neural Networks: P21S-220. Broomhead D.S. & Lowe D. (1988) Radial Basis Functions, Multi-Variable Functional Interpolation and Adaptive Networks: RSRE Memorandum No. 4148, Royal Signals and Radar Establishment, Malvern, England. Gentric P. & Withagen H.C.A.M. (1993) Constructive Methods for a New Classifier Based on a Radial Basis Function Network Accelerated by a Tree: Report, Eindhoven Technical University, Eindhoven, Holland. Lowe D. & Webb A.R. (1991) Time Series Prediction by Adaptive Networks: A Dynamical Systems Perspective: lEE Proceedings-F, vol. 128(1), Feb." P17-24. Moody J. & Darken C. (1988) Learning With Localized Receptive Fields: Research Report YALE UID CSIRR-649. Omohundro S.M. (1987) Efficient Algorithms With Neural Network Behaviour; in Complex Systems 1 (1987): P273-347. Omohundro S.M. (1989) Five Balltree Construction Algorithms: International Computer Science Institute Technical Report TR-89-063. Omohundro S.M. (1991) Bumptrees for Efficient Function, Constraint, and Classification Learning: Advances in Neural Information Processing Systems 3, P693699. Renals S. & Rohwer R.J. (1989) Phoneme Classification Experiments Using Radial Basis Functions: Proceedings of the IJCNN, P461-467. Rumelhart D.E., Hinton G.E. & Williams Rl. (1986) Learning Internal Representations by Error Propagation: in Parallel Distributed Processing, vol. 1 P318-362. Cambridge, MA : MIT Press. Williams B.V., Bostock R.TJ., Bounds D.G. & Harget A.J. (1993) The Genetic Bumptree Classifier: Proceedings of the BNSS Symposium on Artificial Neural Networks: to be published.	1993	55
809	Generalization Error and The Expected Network Complexity Chuanyi Ji Dept. of Elec., Compt. and Syst Engl'. Rensselaer Polytechnic Inst.itu( e Troy, NY 12180-3590 chuanyi@ecse.rpi.edu Abstract For two layer networks with n sigmoidal hidden units, the generalization error is shown to be bounded by O(E~) O( (EK)d l N) K + N og , where d and N are the input dimension and the number of training samples, respectively. E represents the expectation on random number K of hidden units (1 :::; I\ :::; n). The probability Pr(I{ = k) (1 :::; k :::; n) is (kt.erl11ined by a prior distribution of weights, which corresponds to a Gibbs distribtt! ion of a regularizeI'. This relationship makes it possible to characterize explicitly how a regularization term affects bias/variance of networks. The bound can be obtained analytically for a large class of commonly used priors. It can also be applied to estimate the expected net.work complexity Ef{ in practice. The result provides a quantitative explanation on how large networks can generalize well . 1 Introduction Regularization (or weight-decay) methods are widely used in supervised learning by adding a regularization term t.o an energy function. Although it is well known that such a regularization term effectively reduces network complexity by introducing more bias and less variance[4] to the networks, it is not clear whether and how the information given by a regularization term can be used alone to characterize the effective network complexity and how the estimated effective network complexity relates to the generaliza.tion error. This research attempts to provide answers to t.hese questions for two layer feedforward networks with sigmoidal hidden units. 367 368 Ji Specifically) the effective network complexity is ch(lJ'act.erized by the expected nUI11bel' of hidden units determined by a Gibbs dist.ribution corresponding to a regula L'ization tenl1. The generalization error can then be bounded by the expected network complexity) and thus be tighter than the original bound given by Barron[2]. The new bound shows explicitly) through a bigger approximation error and a smaller estimation error I how a regularization term introduces more bias and less varia nce to the networks. It therefore provides a quantitative explanation on how a network larger than necessary can also generalize well under certain conditions) which can not be explained by the existing learning theory[9]. For a class of commonly-used regularizers) the expecced network complexity can be obtained in a closed form. It is then used to estimate the expected network complexity for Gaussion mixture model[6]. 2 Background and Previous Results A relationship has been developed by Barron[2] between generalization error and network complexity for two layer net.works used for function approximation. "Ve will briefly describe this result in this section and give our extension subsequently. Consider a class of two layer networks of fixed architecture with n sigmoidal hidden units a.nd one (linear) output unit. Let fn(x; w) = twF)91(wP)T x) be a neiW01'k 1=1 function) where wEen is the network weight vcctor comprising both Wf2) and wP) for 1 ::; l ::; n. w}l) and W}2) are the incoming weights to the l-th hidden unit and the weight from the l-th hidden unit to the output) respectively. en ~ Rn(d+1) is t.he weight space for n hidden unit.s (and input dimension d) . Each sigmoid unit !JI(Z) is assumed to be of tanh type: !J/(z) --+ ±1 as z --+ ±oo for 1 ::; I :S n 1. The input is xED ~ Rd. ''''ithout loss of generality) D is assumed to be a unit hypercube in R d ) i.e.) all the components of x are in [·-1) 1]. Let f( x) be a target function defined in the sa.me domain D and satisfy some smoot.hness conditions [2]. Consider N training samples independently drawn from some distribution p(:/.:): (x1)f(:I:1)), ... ) (xN)f(;t.v)). Define an energy function e) where e = f1 + A LTI.~~(1U) . Ln ,N(W) is a regularization term as a function of tv for a. fixed II . A is a const.ant.. C1 is a quadratic error function on N training lV ') samples: e1 = J: 'L,(fn(Xi;W) - f(Xi)t· Let fll,l'.,r(x;-t'iJ) be t.he (optimal) network i=l . function such t.hat 'ttl minimizes t.he energy function e: tV = arg min e. The genwEen eralization error Eg is defined to be the squared L'2 norm E9 = Ell f - fn,N 112 = EJU(x) - fn,N(X; w))2dp(x)) where E is the expectation over all training sets of D size N drawn from the same distributioll. Thus) the generalization error measnres the mean squared distance between the unknown function an' I the best network function that can be obtained for training sets of size N . The generalization error 1 In the previous ,\'ork by Barron) t.he sigmoillal hidden units atC' '1,( ~)+1. It is easy t.o show that his results are applica.ble to the class of .t!1(Z))S we consider h;re. Generalization Error and the Expected Network Complexity 369 Eg is shown[2] to be bounded as Eg ::; O(Rn,N), (1) where Rn ,N, called the index of resol vability [2], can be expressed as Rn ,N = min {II .f _ in 112 + Ln,~( tv)}, wEen (2) where III is the clipped fn(x; tv) (see [2]). The index of resolvability can be further bounded as Rn,N :::; O(~) + O(',~~logN). Therefore, the generalization error IS bounded as 1 nd E!! :::; 0(;;) + O( N logN), (3) where O(~) and 0(';1 logN) are t.he bounds for approxima.tion error (bia.s) and esti ;:l.lnt.ion error (varia.nce), respectively. In addition, t.he hOllnd for E9 can be minimized if all additional regularization term LN (71) is used in the energy function to minimize the number of hidden units, i.e., r=N Eg :::; O( V dlogN ). 3 Open Questions and Motivations Two open questions, which can not be answered by the previous result, are of the primary interest of this work. I) How do large networks generalize? The largc networks refer to those wit.h a rat.io ~~ to he somewhat big, where TV and N are the t.ot.al number of independent.ly modifiable weights (lV ~ nel, for 11 lcugc) and the number of training samples, respectively. Networks tra.ined with reglll<Hization t.erms may fall int.o this category. Such large networks are found (0 Jw abk to generalize well sometimes. JImH'H'J', when '~~{ is big, the bonnel in Eqll ahon (~:l) is t.oo loose t.o bOllnd the actual generaliza t.ion error meaningfully. Therefme. for the large networks, the tot.al number of hidden ullits n ma.y no longer be a. good est.imate for network complexity. Efforts have been made to develop measures on effective net.work complexity both analytically and cmpirically[1][5][10] . These measures depend on training data as well as a regularization term in an implicit way which make it difficult to see direct. effects of a regulariza.tion term on generaliza.tion error. This naturally leads t.o our second question. 2) Is it possible to characterize network complexit.y for a cLI~~ of networks using only the information given by a regularizat.ion term:!? How t.o relat.e the estimated network complexity rigorously with generalization error? In practice, when a regularization term (L I1 .N(W)) is used to penalize the m;l~llitude of weights, it effectively minimizes the number of hidden units as ,,,,'ell even til' '1lgb a.n additional regularization term LN(n) is not used. This is dne to the fact tbll. some of the hidden units may only operate in the lineal' region of a sigmoid when their 2This was posed as an open problem hy Solia. ei..al. [8] 370 Ji incoming weights are small and inputs are bounded. Therefore, a Ln,N(W) term can effectively act like a LN(n) term that reduces the effective number of hidden units, and thus result in a degenerate parameter space whose degrees of freedom is fewer than nd. This fact was not taken into consideration in the previous work, and as shown later in this work, will lead to a tighter bound on Rn,N. In what follows, we will first define the expected network complexity, then use it to bound the generalization error. 4 The Expected Network C0111plexity For reasons that will hecome apparent, we choose to define the effective complexity of a feedforward two layer network as the expected number of hidden unit.s EE (1 :::; J{ :::; 11) ,vhich are effectively nonlinear, i.e. operating outside t.he central linear regions of their sigmoid response function g(.::). '''''e define the linear region as an interval 1 z 1< b with b a positive constant. Consider the presynaptic input:: = wiT x to a hidden unit g(z), where Wi is the incoming weight vector for the unit. Then the unit is considered to be effectively linear if 1 z 1< b for all xED. This will happen if 1 Zl 1< b, where z' = wiT x' with x' being any vertex of the unit hypercube D. This is b~cause 1 z I:::; wiT X, where x is the vertex of D whose elements are t.he 8gn functions of the elements of Wi. Next, consider network weights as random variaJ)lcs wit.h a distribution p(w) = Aex1J( - Ln,N (tv)), ,,,hich corresponds t.o a. Gibbs distribution of a regularization term wit.h a normalizing constant. A. Consider the vector ;'1;' to be a random vector also wit.h eqnally probable l~s ,Hld -l's. Then I::' 1< b will be a random event. The probability for this hidden unit to be effectively nonlin0.ill' equals to 1- Pr(1 z 1< b), which can be completely determined by the distributions of weights p( 'W) and x' (equally probable). Let. f{ be the number of hidden units which are effectively nonlinear. Then t.he probability, Pr(K = k) (1 :::; k :::; n), can be determined through a joint probabilit.y of k hidden units that are operating beyond the central linear region of sigmoid fUllctions. The expected network complexity, EI<, can then be obtained through Pr(I< = k), which is determined by the Gibbs distribution of LN,n (w). The motivation on utilizing such a Gibbs distribution comes from the fact that Rk,N is independent of training samples but dependent. of a regularization term which corresponds to a prior distribution of weights. Using sHch a formulation, as will be shown later, the effect of a regularization term on bias and va riance ca.n be characterized explicitly. 5 A New Bound for The Generalization Error To develop a t.ightcr houucl for the generalizat.ion error, we consider subspa.ces of t.he weights indexed by different number of effectively nonlinc(lr hidden units: 8 1 ~ 8 2 . .. ~ 8 n . For ead, 8 j , there are j out of 11 hidden unit.s which are effectively nonlinear fo], 1 :; j :::; n. '1'11e1'e1'ore, the index ofl'esolvability T?71,N ca.n be expressed as (4) Generalization Error and the Expected Network Complexity 371 where each Rk,N = min {II f - in 112 + Ln.~(w)}. Next let us consider the number wEe" of effectively nonlinear units to be random. Since the minimum is no bigger than the average, we have (5) where the expectation is taken over the random variable J{ utilizing the probability Pr(I{ = k). For each K , however, the t,yO terms in Rf(,N can be bounded as by the t.rian.gle ine4uality, where fn-l":,n is the actuallletwork function with n J{ hidden units operating in the region bounded by the constant b, and ff( is the correspondillg network funct.ion which t.rea ts the 11 J{ units as linear units. In addition, we have . ') I{d Ln,N(W) ::; O(II.fn-K,n - jI{ W) + O( N logN), (7) \vhere the f-irst term also results from the triangle inequality, and the second term is obtained by cliscretizing the degenerate parameter space e J{ using similar techl1lques as in [2]3. Applying Taylor expansion on the t.erm \\ fn-K,n - ff( \\2, \\'e have \\ fn-K,n - ff{ \\2 ::; O(b13(n - K)). (8) Putting Equations (5) (6) (7) and (8) into Equation (1), \\'(' have 1 (EK)d 6 () Eg ::; O(E !{) + O( N logN) + O(b (11 - EX)) + o(b)), (9) where EX is the expected number of hidden units which are effectively nonlinear. If b ::; O( -\-), we have n3 1 (EI{)d Eg ::; O(E J() + O( N logN) . (10) 6 A Closed Fornl Expression For a Class of Regularization Ternls For commonly used regularization terms, how can \"e actually find the probability distribution of the number of (nonlinear) hidden units Pr(I{ = k)? And how shall we evaluate EK and E J( ? As a simple example, we consider a special class of prior distrihutions for iid weights, i.e, p( w) = TIiP( Wi), where W.i are the "i<'ments of wEen. This corresponds to a large class of regularization terms ,,'hicIt minimize the magnitudes of individual weights indepcndently[7]. Consider each weight as a random variable with zero mean and a common variance (J. Then for large input dimension el, v7zZ' is approximately normal with zero-mean 3 Deta.ils \Yill be given ill iL longer version of the pa.per in prepa.ra.tion. 372 Ji and varia.nce (J by the Central Limit Theorem[3]. Let q denote the probability that a. unit is effectively nonlinear. We have b q = 2Q(r,)' (Jyd (11 ) -x :;l where Q( -;1.:) = );- J e- T ely. Next consider the probability that J( out of n -co hidden units are nonlinear. Based 011 our (independence) assumptions on w' a.nd x', I( has a binomial distribution (71.) k n /.; Pr(I{ = I.:) = k q (1 - q) , where 1 < k < n. Then n-1 . EX = nq. 1 1 E}, = +~, \ n (12) (1:3) (14) where ~ = L HI - qr-~ + (1 - qt· Then the generalization error Eo satisfies i=1 1 nqd Eg :::; 0(- +~) + O(-N logN) . n (15) 7 Application As an example for applica.t.ions of t.he tJleoretical results, the expected network complexity EJ{ is estimat.ed for G<:tussian mixture model used for time-series prediction (details can he found in [6]) 4. In genera.l, llsillg only a prior dist.ribut.ion of ,,,eights to est.ima.te the network COlllplexit.y EJ{ may lead to a less accurate measure on the effective net.work complexiLy than incorporat.ing informat.ion on training data also. However, if parameters of a regularization term also get optimized during training, as shown in this example, the resulting Gibbs prior distribution of weights may lead to a good estimate of the effective number of hidden units. Specifically, the corresponding Gibbs distribution p( 'W) of the weights from the Gaussion mixture is iicl, which consists of a linear combination of eight Gaussia.n distributions. This function results in a skewed distribntion with a sharp peak around the zero (see [6]). The mean and variance of the presynaptic inputs z t.o the hidden units can thus be estimated as 0.02 and 0.04, respectively. The other parameters used are n = 8, d = 12. b = 0.6 is chosen. Then q ~ 004 is obtained through Equation (11). The effective network complexity is EJ{ ~ 3 (or 4). The empirical result(10], which estima.tes the effective number of hidden units using the dominated eigenvalues at the hidden layer, results in about ;3 effective hidden units. 4 Strictly speaking, the theoretical resnlts deal with l'egulariza tion terms with discrete weight.s. It. can a.nd ha.s been extended to continuous weight.s by D.F. McCaffrey and A.R. Galla.nt. Details are beyond the content of this paper. Generalization Error and the Expected Network Complexity 373 5r---------.----------r---------.----------r-------~ 4.5 4 variance 0.5 increase in bias 0.2 0.4 0.6 0.8 q Figure 1: Illustration of an increase .6.. in bias and variance Bqn as a function of q. A sca.ling fadar J3 = 0.25 is used for t.he convenience of the plot. 11 = 20 is chosen. 8 Discussions Is this new bound for the generalization tighter than the old one which takes no account of l1etwork-weight.-dependent information? If so . what does it tell us? Compared wit.h the bOllnd in Equation (3), the new bound results in an increase .6.. in approximation error (bias), and qn instea.d of n as ~sLimatjon errol' (variallce). These two terms are plotted as functions of q in Figure (1). Since q is a. function of (J which characterizes how strongly the magnitude of the weights is penalized, the larger the (J, the less the weights get penalized, the larger the q, the more hidden uni ts are likely to be effectively nonlinear, thus the smaller the bias and larger the variance. ,\Vhen q = 1, all the hidden units are effectively nonlinear and the new bound reduces to the old one. This shows ho",- a regulariza.tion t.erm directly affects bias / variance. '\i\Then the estimation error dominates, the bound for the generalization error will be proportional to nq inst.ead of n. The value of 1'/,I}, however, depends on the choice of a. For small (J, the new bound can be much tighter than the old one, especially for large netwOl'ks with n large but nq small. This will provide a practical method to cstilllate gCltcrnlizn.tion errol' for large nctworks as well as an explanation of when rllld why hn~e networks can generalize ,,-ell. How tight the bound really is depends on how well Ln,l\ (lL!) is chosen. Let no denote t.he optimallll1ll1ber of (nonlinear) hidden units needeJ to approximate I(x). If Ln,N(W) is chosen so that. the corresponding 1J(W) is almost a delta. function a.t no, t.hen ERK,i\' ~ Rno,N, which gives a. very tight bound. Otherwise, if, for insta.nce, 374 Ii Ln,N(W) penalizes network complexity so little that ERJ(,N :=:::: Rn,N, the bound will be as loose as the original one. It should also be noted that an exact value for the bound cannot be obtained unless some information on the unknown function f itself is available. For commonly used regularization terms, the expected network complexity can be estimated through a close form expression. Such expected network complexity is shown to be a good estimate for the actual network complexity if a Gibbs prior distribution of weights also gets optimized through training, and is also sharply peaked. More research will be done to evaluate the applica.bility of the theoretical results. A cknow ledgeluent The support of National Science Foundation is gratefully acknowledged. References [1] S. Amari and N. Murata, "Statistical Theory of Learning Curves under Entropic Loss Criterion," Neural Computation, 5, 140-153, 1993. [2] A. Barron, "Approximation a.nd Estimation Bounds for Artificial Neural Networks," Proc. of The 4th Workshop on Computational Learning Theory, 243249, 1991. [3] Vv. Feller, An Introduction to Probability Theory and Its Applications, New York: John \Viley and Sons, 1968. [4] S. Geman, E. Bienenstock, and R. Doursat, "Neural Networks and the Bias/Variance Dilemma," Neural Comp1tiation, 4, 1-58, 1992. [5] J. Moody, "Generalization, vVeight Decay, and Architecture Selection for N onlinear Learning Systems," Proc. of Neural Information Processing Systems, 1991. [6] S.J. Nowlan, and G.E. Hinton, "Simplifying Neural Networks by Soft \Veight Sha.ring," Neural computation, 4,473-493(1992). [7] R. Reed, "Pruning Algorithms-A Survey," IEEE Trans. Neural Networks Vol. 4, 740-'i'47, (1993). [8] S. Solla, "The Emergence of Generalization Ability in Learning," Presented at NIPS92. [9] V. Vapnik, "Estimation of Dependences Based on Empirical Data," SpringerVerlag, New York, 1982. [10] A.S . V\'eigend and D.E . Rumelhart, "The Effective Dimension of the Space of Hidden Units," Proc. of International Joint Conference on Ne1tral Networks, 1992.	1993	56
810	COMBINED NEURAL NETWORKS FOR TIME SERIES ANALYSIS Iris Ginzburg and David Horn School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Science Tel-Aviv University Tel-A viv 96678, Israel Abstract We propose a method for improving the performance of any network designed to predict the next value of a time series. Vve advocate analyzing the deviations of the network's predictions from the data in the training set. This can be carried out by a secondary network trained on the time series of these residuals. The combined system of the two networks is viewed as the new predictor. We demonstrate the simplicity and success of this method, by applying it to the sunspots data. The small corrections of the secondary network can be regarded as resulting from a Taylor expansion of a complex network which includes the combined system. \\Te find that the complex network is more difficult to train and performs worse than the two-step procedure of the combined system. 1 INTRODUCTION The use of neural networks for computational tasks is based on the idea that the efficient way in which the nervous system handles memory and cognition is worth immitating. Artificial implementations are often based on a single network of mathematical neurons. We note, however, that in biological systems one can find collections of consecutive networks, performing a complicated task in several stages, with later stages refining the performance of earlier ones. Here we propose to follow this strategy in artificial applications. 224 Combined Neural Networks for Time Series Analysis 225 We study the analysis of time series, where the problem is to predict the next element on the basis of previous elements of the series. One looks then for a functional relation Yn = f (Yn -1 , Yn - 2, ... , Yn - m) . (1 ) This type of representation is particularly useful for the study of dynamical systems. These are characterized by a common continuous variable, time, and many correlated degrees of freedom which combine into a set of differential equations. Nonetheless, each variable can in principle be described by a lag-space representation of the type 1 . This is valid even if the Y = y(t) solution is unpredictable as in chaotic phenomena. Weigend Huberman and Rumelhart (1990) have studied the experimental series of yearly averages of sunspots activity using this approach. They have realized the lag-space representation on an (m, d, 1) network, where the notation implies a hidden layer of d sigmoidal neurons and one linear output. Using m = 12 and a weight-elimination method which led to d = 3, they obtained results which compare favorably with the leading statistical model (Tong and Lim, 1980). Both models do well in predicting the next element of the sunspots series. Recently, Nowlan and Hinton (1992) have shown that a significantly better network can be obtained if the training procedure includes a complexity penalty term in which the distribution of weights is modelled as a mixture of multiple gaussians whose parameters vary in an adaptive manner as the system is being trained. We propose an alternative method which is capable of improving the performance of neural networks: train another network to predict the errors of the first one, to uncover and remove systematic correlations that may be found in the solution given by the trained network, thus correcting the original predictions. This is in agreement with the general philosophy mentioned at the beginning, where we take from Nature the idea that the task does not have to be performed by one complicated network; it is advantageous to break it into stages of consecutive analysis steps. Starting with a network which is trained on the sunspots data with back-propagation, we show that the processed results improve considerably and we find solutions which match the performance of Weigend et. al. 2 CONSTRUCTION OF THE PRIMARY NETWORK Let us start with a simple application of back-propagation to the construction of a neural network describing the sunspots data which are normalized to lie between o and 1. The network is assumed to have one hidden layer of sigmoidal neurons, hi i = 1" . " d, which receives the input of the nth vector: m hi = 0'(2: WijYn-j - Oi) j=l The output of the network, Pn, is constructed linearly, d Pn = 2: Wi hi O. i=l (2) (3) 226 Ginzburg and Hom The error-function which we minimize is defined by 1 N E = 2 L (Pn - Yn)2 n=m+l (4) where we try to equate Pn, the prediction or output of the network, with Yn, the nth value of the series. This is the appropriate formulation for a training set of N data points which are viewed as N - m strings of length m used to predict the point following each string. We will work with two sets of data points. One will be labelled T and be used for training the network, and the other P will be used for testing its predictive power. Let us define the average error by 1 {s = jjSfj 2:(Pn - Yn)2 nES (5) where the set S is either Tor P. An alternative parameter was used by Weigend et. al. ,in which the error is normalized by the standard deviation of the data. This leads to an average relative variance (arv) which is related to the average error through (6) Following Weigend et. al. we choose m = 12 neurons in the first layer and IITII = 220 data points for the training set. The following IIPII = 35 years are used for testing the predictions of our network. We use three sigmoidal units in the hidden layer and run with a slow convergence rate for 7000 periods. This is roughly where cross-validation would indicate that a minimum is reached. The starting parameters of our networks are chosen randomly. Five examples of such networks are presented in Table 1. 3 THE SECONDARY NETWORK Given the networks constructed above, we investigate their deviations from the desired values qn = Yn - Pn· (7) A standard statistical test for the quality of any predictor is the analysis of the correlations between consecutive errors. If such correlations are found, the predictor must be improved. The correlations reflect a systematic deviation of the primary network from the true solution. We propose not to improve the primary network by modifying its architecture but to add to it a secondary network which uses the residuals qn as its new data. The latter is being trained only after the training session of the primary network has been completed. Clearly one may expect some general relation of the type (8) to exist. Looking for a structure of this kind enlarges considerably the original space in which we searched for a solution to 1 . We wish the secondary network Combined Neural Networks for Time Series Analysis 227 to do a modest task, therefore we assume that much can be gained by looking at the interdependence of the residuals qn on themselves. This reduces the problem to finding the best values of Tn = b(qn-l, qn-2,"', qn-I) which would minimize the new error function 1 N E2='2 L (Tn-qn)2. n=I+1 (9) (10) Alternatively, one may try to express the residual in terms of the functional values Tn = !2(Yn-1, Yn-2,"', Yn-I) (11) minimizing again the expression 10 . When the secondary network completes its training, we propose to view tn = Pn + Tn (12) as the new prediction of the combined system. We will demonstrate that a major improvement can be obtained already with a linear perceptron. This means that the linear regression or 1 Tn = L aIqn-i + /31 i=l 1 (13) Tn = L a;Yn-i + /32 (14) i=l is sufficient to account for a large fraction of the systematic deviations of the primary networks from the true function that they were trained to represent. 4 NUMERICAL RESULTS We present in Table 1 five examples of results of (12,5,1) networks, i.e. m = 12 inputs, a hidden layer of three sigmoidal neurons and a linear output neuron. These five examples were chosen from 100 runs of simple back-propagation networks with random initial conditions by selecting the networks with the smallest R values (Ginzburg and Horn, 1992). This is a weak constraint which is based on letting the network generate a large sequence of data by iterating its own predictions, and selecting the networks whose distribution of function values is the closest to the corresponding distribution of the training set. The errors of the primary networks, in particular those of the prediction set €p, are quite higher than those quoted by Weigend et. al. who started out from a (12,8,1) network and brought it down through a weight elimination technique to a (12,5,1) structure. They have obtained the values €T = 0.059 €p = 0.06. We can reduce our errors and reach the same range by activating a secondary network with I = 11 to perform the linear regression (3.6) on the residuals of the predictions of the primary network. The results are the primed errors quoted in the table. Characteristically we observe a reduction of €T by 3 - 4% and a reduction of €p by more than 10%. 228 Ginzburg and Hom # fT f' T {p {' P 1 0.0614 0.0587 0.0716 0.0620 2 0.0600 0.0585 0.0721 0.0663 3 0.0611 0.0580 0.0715 0.0621 4 0.0621 0.0594 0.0698 0.0614 5 0.0616 0.0589 0.0681 0.0604 Table 1 Error parameters of five networks. The unprimed errors are those of the primary networks. The primed errors correspond to the combined system which includes correction of the residuals by a linear perceptron with I = 11 , which is an autoregressions of the residuals. Slightly better results for the short term predictions are achieved by corrections based on regression of the residuals on the original input vectors, when the regression length is 13 (Table 2). # {T fT fp f' p 1 0.061 0.059 0.072 0.062 2 0.060 0.059 0.072 0.065 3 0.061 0.058 0.072 0.062 4 0.062 0.060 0.070 0.061 5 0.062 0.059 0.068 0.059 Table 2 Error parameters for the same five networks. The primed errors correspond to the combined system which includes correction of the residuals by a linear perceptron based on original input vectors with I = 13. 5 LONG TERM PREDICTIONS When short term prediction is performed, the output of the original network is corrected by the error predicted by the secondary network. This can be easily generalized to perform long term predictions by feeding the corrected output produced by the combined system of both networks back as input to the primary network. The corrected residuals predicted by the secondary network are viewed as the residuals needed as further inputs if the secondary network is the one performing autoregression of residuals. We run both systems based on regression on residuals and regression on functional values to produce long term predictions. In table 3 we present the results of this procedure for the case of a secondary network performing regression on residuals. The errors of the long term predictions are averaged over the test set P of the next 35 years. We see that the errors of the primary networks are reduced by about 20%. The quality of these long term predictions is within the range of results presented by Weigend et. al. Using the regression on (predicted) functional values, as in Eq. 14 , the results are improved by up to 15% as shown in Table 4. Combined Neural Networks for Time Series Analysis # f2 fj f5 f~ fll , f11 1 0.118 0.098 0.162 0.109 0.150 0.116 2 0.118 0.106 0.164 0.125 0.131 0.101 3 0.117 0.099 0.164 0.112 0.136 0.099 4 0.116 0.099 0.152 0.107 0.146 0.120 5 0.113 0.097 0.159 0.112 0.147 0.123 Table 3 Long term predictions into the future. fn denotes the average error of n time steps predictions over the P set. The unprimed errors are those of the primary networks. The primed errors correspond to the combined system which includes correction of the residuals by a linear perceptron. # f2 f' f' f11 , 2 f5 5 f11 1 0.118 0.098 0.162 0.107 0.150 0.101 2 0.118 0.104 0.164 0.117 0.131 0.089 3 0.117 0.098 0.164 0.108 0.136 0.086 4 0.117 0.098 0.152 0.105 0.146 0.105 5 0.113 0.096 0.159 0.110 0.147 0.109 Table 4 Long term predictions into the future. The primed errors correspond to the combined system which includes correction of the residuals by a linear perceptron based on the original inputs. 6 THE COMPLEX NETWORK Since the corrections of the secondary network are much smaller than the characteristic weights of the primary network, the corrections can be regarded as resulting from a Taylor expansion of a complex network which include's the combined system. This can be simply implemented in the case of Eq. 14 which can be incorporated in the complex network as direct linear connections from the input layer to the output neuron, in addition to the non-linear hidden layer, i.e., d m tn = L:: Wihi + L viYn-i () . (15) i=l i=l We train such a complex network on the same problem to see how it compares with the two-step approach of the combined networks described in the previous chapters. The results depend strongly on the training rates of the direct connections, as compared with the training rates of the primary connections (i.e. those of the primary network). When the direct connections are trained faster than the primary ones, the result is a network that resembles a linear perceptron, with non-linear 229 230 Ginzburg and Hom corrections. In this case, the assumption of the direct connections being small corrections to the primary ones no longer holds. The training error and prediction capability of such a network are worse than those of the primary network. On the other hand, when the primary connections are trained using a faster training rate, we expect the final network to be similar in nature to the combined system. Still, the quality of training and prediction of these solutions is not as good as the quality of the combined system, unless a big effort is made to find the correct rates. Typical results of the various systems are presented in Table 5. type of network primary network learning rate of linear weights = 0.1 learning rate of linear weights = 0.02 combined system Table 5 0.061 0.062 0.061 0.058 0.072 0.095 0.068 0.062 Short term predictions of various networks. The learning rate of primary weights is 0.04. The performance of the complex network can be better than that of the primary network by itself, but it is surpassed by the achievements of the combined system. 7 DISCUSSION It is well known that increasing the complexity of a network is not the guaranteed solution to better performance (Geman et. al. 1992). In this paper we propose an alternative which increases very little the number of free parameters, and focuses on the residual errors one wants to eliminate. Still one may raise the question whether this cannot be achieved in one complex network. It can, provided we are allowed to use different updating rates for different connections. In the extreme limit in which one rate supersedes by far the other one, this is equivalent to a disjoint architecture of a combined two-step system. This emphasizes the point that a solution of a feedforward network to any given task depends on the architecture of the network as well as on its training procedure. The secondary network which we have used was linear, hence it defined a simple regression of the residual on a series of residuals or a series of function values. In both cases the minimum which the network looks for is unique. In the case in which the residual is expressed as a regression on function values, the problem can be recast in a complex architecture. However, the combined procedure guarantees that the linear weights will be small, i.e. we look for a small linear correction to the prediction of the primary network. If one trains all weights of the complex network at the same rate this condition is not met, hence the worse results. We advocate therefore the use of the two-step procedure of the combined set of networks. We note that combined set of networks. We note that the secondary networks perform well on all possible tests: they reduce the training errors, they Combined Neural Networks for Time Series Analysis 231 improve short term predictions and they do better on long term predictions as well. Since this approach is quite general and can be applied to any time-series forecasting problem, we believe it should be always tried as a correction procedure. REFERENCES Geman, S., Bienenstock, E., & Doursat, R., 1992. Neural networks and the bias/variance dilemma. Neural Compo 4, 1-58. Ginzburg, I. & Horn, D. 1992. Learning the rule of a time series. Int. Journal of Neural Systems 3, 167-177. Nowlan, S. J. & Hinton, G. E. 1992. Simplifying neural networks by soft weightsharing. Neural Compo 4, 473-493. Tong, H., & Lim, K. S., 1980. Threshold autoregression, limit cycles and cyclical data. J. R. Stat. Soc. B 42, 245. Weigend, A. S., Huberman, B. A. & Rumelhart, D. E., 1990. Predicting the Future: A Connectionist Approach, Int. Journal of Neural Systems 1, 193-209.	1993	57
811	Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina Joachim M. Buhmann Rheinische Friedrich-Wilhelms-U niversitiit Institut fUr Informatik II, RomerstraBe 164 0-53117 Bonn, Germany Martin Lades Ruhr-Universitiit Bochum Institut fiir Neuroinformatik 0-44780 Bochum, Germany Frank Eeckman Lawrence Livermore National Laboratory ISCR, P.D.Box 808, L-426 Livermore, CA 94551 Abstract Changes in lighting conditions strongly effect the performance and reliability of computer vision systems. We report face recognition results under drastically changing lighting conditions for a computer vision system which concurrently uses a contrast sensitive silicon retina and a conventional, gain controlled CCO camera. For both input devices the face recognition system employs an elastic matching algorithm with wavelet based features to classify unknown faces. To assess the effect of analog on -chip preprocessing by the silicon retina the CCO images have been "digitally preprocessed" with a bandpass filter to adjust the power spectrum. The silicon retina with its ability to adjust sensitivity increases the recognition rate up to 50 percent. These comparative experiments demonstrate that preprocessing with an analog VLSI silicon retina generates image data enriched with object-constant features. 1 Introdnction Neural computation as an information processing paradigm promises to enhance artificial pattern recognition systems with the learning capabilities of the cerebral cortex and with the 769 770 Buhmann, Lades, and Eeckman adaptivity of biological sensors. Rebuilding sensory organs in silicon seems to be particularly promising since their neurophysiology and neuroanatomy, including the connections to cortex, are known in great detail. This knowledge might serve as a blueprint for the design of artificial sensors which mimic biological perception. Analog VLSI retinas and cochleas, as designed by Carver Mead (Mead, 1989; Mahowald, Mead, 1991) and his collaborators in a seminal research program, will ultimately be integrated in vision and communication systems for autonomous robots and other intelligent information processing systems. The study reported here explores the influence of analog retinal preprocessing on the recognition performance of a face recognition system. Face recognition is a challenging classification task where object inherent distortions, like facial expressions and perspective changes, have to be separated from other image variations like changing lighting conditions. Preprocessing with a silicon retina is expected to yield an increased recognition rate since the first layers of the retina adjust their local contrast sensitivity and thereby achieve invariance to variations in lighting conditions. Our face recognizer is equipped with a silicon retina as an adaptive camera. For comparison purposes all images are registered simultaneously by a conventional CCD camera with automatic gain control. Galleries with images of 109 different test persons each are taken under three different lighting conditions and two different viewing directions (see Fig. 1). These different galleries provide separate statistics to measure the sensitivity of the system to variations in light levels or contrast and image changes due to perspective distortions. Naturally, the performance of an object recognition system depends critically on the classification strategy pursued to identify unknown objects in an image with the models stored in a database. The matching algorithm selected to measure the performance enhancing effect of retinal preprocessing deforms prototype faces in an elastic fashion (Buhmann et aI., 1989; Buhmann et al., 1990; Lades et al., 1993). Elastic matching has been shown to perform well on the face classification task recognizing up to 80 different faces reliably (Lades et al., 1993) and in a translation, size and rotation invariant fashion (Buhmann et aI., 1990). The face recognition algorithm was initially suggested as a simplified version of the Dynamic Link A rchitecture (von der Malsburg, 1981), an innovative neural classification strategy with fast changes in the neural connectivity during recognition stage. Our recognition results and conclusions are expected to be qualitatively typical for a whole range of face/object recognition systems (Turk, Pentland, 1991; Yuille, 1991; Brunelli, Poggio, 1993), since any image preprocessing with emphasis on object constant features facilitates the search for the correct prototype. 2 The Silicon Retina The silicon retina used in the recognition experiments models the interactions between receptors and horizontal cells taking place in the outer plexiform layer of the vertebrate retina. All cells and their interconnections are explicitly represented in the chip so that the following description simultaneously refers to both biological wetware and silicon hardware. Receptors and horizontal cells are electrically coupled to their neighbors. The weak electrical coupling between the receptors smoothes the image and reduces the influence of voltage offsets between adjacent receptors. The horizontal cells have a strong lateral electrical coupling and compute a local background average. There are reciprocal excitatory-inhibitory synapses between the receptors and the horizontal cells. The horizontal cells use shunting inhibition to adjust the membrane conductance of the receptors and Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina 771 thereby adjust their sensitivity locally. This feedback interaction produces an antagonistic center/surround organization of receptive fields at the output The center is represented by the weakly coupled excitatory receptors and the surround by the more strongly coupled inhibitory horizontal cells. The center/surround organization removes the average intensity and expands the dynamic range without response compression. Furthennore, it enhances edges. In contrast to this architecture, a conventional CCD camera can be viewed as a very primitive retina with only one layer of non-interacting detectors. There is no DC background removal, causing potential over- and underexposure in parts of the image which reduces the useful dynamic range. A mechanical iris has to be provided to adjust the mean luminance level to the appropriate setting. Since cameras are designed for faithful image registration rather than vision, on-chip pixel processing, if provided at all, is used to improve the camera resolution and signal-to-noise ratio. Three adjustable parameters allow us to fine tune the retina chip for an object recognition experiment: (i) the diffusivity of the cones (ii) the diffusivity ofthe horizontal cells (iii) the leak in the horizontal cell membrane. Changes in the diffusivities affect the shape of the receptive fields, e.g., a large diffusivity between cones smoothes out edges and produces a blurred image. The other extreme of large diffusivity between horizontal cells pronounces edges and enhances the contrast gain. The retina chip has a resolution of 90 x 92 pixels, it was designed by (Boahen, Andreou, 1992) and fabricated in 2flm n-well technology by MOSIS. 3 Elastic Matching Algorithm for Face Recognition Elastic matching is a pattern classification strategy which explicitly accounts for local distortions. A prototype template is elastically deformed to measure local deviations from a new, unknown pattern. The amount of deformation and the similarity oflocal image features provide us with a decision criterion for pattern classification. The rubbersheet-like behavior of the prototype transformation makes elastic matching a particularly attractive method for face recognition where ubiquitous local distortions are caused for example by perspective changes and different facial expressions. Originally, the technique was developed for handwritten character recognition (Burr, 1981). The version of elastic matching employed for our face recognition experiments is based on attributed graph matching. A detailed description with a plausible interpretation in neural networks terms is published in (Lades et al., (993). Each prototype face is encoded as a planar graph with feature vectors attached to the vertices of the graph and metric information attached to the edges. The feature vectors extract local image information at pixel Xi in a multiscale fashion, i.e., they are functions of wavelet coefficients. Each feature vector establishes a correspondence between a vertex i of a prototype graph and a pixel Xi in the image. The components of a feature vector are defined as the magnitudes of the convolution of an image with a set of two-dimensional, DC free Gaussian kernels centered at pixel Xi. The kernels with the form fl (flx2) [ ( -) 1 1/!'k (X) = (72 exp 2(72 exp ikX - exp (-(72/2) (I) are parameterized by the wave vector k defining their orientations and their sizes. To construct a self-similar set of filter functions we select eight different orientations and five 772 Buhmann, Lades, and Eeckman different scales according to k(v, tt) = ~ Tv/2 (cos( itt), sin( itt)) (2) with v E {O, ... ,4};tt E {O, ... , 7}. The multi-resolution data format represents local distortions in a robust way, i.e., only feature vectors in the vicinity x of an image distortion are altered by the changes. The edge labels encode metric information, in particular we choose the difference vectors AXij == Xi - Xj as edge labels. To generate a new prototype graph for the database, the center of a new face is determined by matching a generic face template to it. A 7 x 10 rectangular grid with 10 pixel spacing between vertices and edges between adjacent vertices is then centered at that point. The saliency of image points is taken into account by deforming that generic grid so that each vertex is moved to the nearest pixel with a local maximum in feature vector length. The classification of an unknown face as one of the models in the database or its rejection as an unclassified object is achieved by computing matching costs and distortion costs. The matching costs are designed to maximize the similarity between feature vector J;M of vertex i in the model graph (M) and feature vector Jl (Xi) associated with pixel Xi in the new image (I). The cosine of the angle between both feature vectors -[...., -M S(JI(x) jM) = J (Xi) . Ji (3) '" Ilf1(Xi)IIIIJ;M II is suited as a similarity function for elastic matching since global contrast changes in images only scale feature vectors but do not rotate them. Besides maximizing the similarity between feature vectors the elastic matching algorithm penalizes large distortions. The distortion cost term is weighted by a factor ,\ which can be interpreted as a prior for expected distortions. The combined matching cost function which is used in the face recognition system compromises between feature similarity and distortion, i.e, it minimizes the cost function (4) for the model M in the face database with respectto the correspondence points {xf}. (i, j) in Eq. (4) denotes that index j runs over the neighborhood of vertex i and index i runs over all vertices. By minimizing Eq. (4) the algorithm assigns pixel x; in the new image I to vertex i in the prototype graph M. Numerous classification experiments revealed that a steepest descent algorithm is sufficient to minimize cost function (4) although it is nonconvex and local minima may cause non-optimal correspondences with reduced recognition rates. During a recognition experiment all prototype graphs in the database are matched to the new image. A new face is classified as prototype A if H A is minimal and if the significance criterion (5) is fulfilled. The average costs (Ji) and their standard deviation LH are calculated excluding match A. This heuristic is based on the assumption that a new face image strongly Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina n3 ~ .... ~ l> gr.tl rr.m.1 l>gr.rr~ l>~"~ l>~.tr .. llo'" ~1Ib.ka-:2to,.. > Workstation Datacube Figure I: Laboratory setup of the face recognition experiments. correlates with the correct prototype but the matching costs to all the other prototype faces is approximately Gaussian distributed with mean (1l) and standard deviation I.H. The threshold parameter 0 is used to limit the rate of false positive matches, i.e., to exclude significant matches to wrong prototypes. 4 Face Recognition Results To measure the recognition rate of the face recognition system using a silicon retina or a CCD camera as input devices, pictures of 109 different persons are taken under 3 different lighting conditions and 2 different viewing directions. This setup allows us to quantify the influence of changes in lighting conditions on the recognition performance separate from the influence of perspective distortions. Figure 2 shows face images of one person taken under two different lighting setups. The images in Figs. 2a,c with both lights on are used as the prototype images for the respective input devices. To test the influence of changing lighting conditions the left light is switched off. The faces are now strongly illuminated from the right side. The CCD camera images (Figs. 2a,b) document the drastic changes of the light settings. The corresponding responses of the silicon retina shown in Figs. 2c,d clearly demonstrate that the local adaptivity of the silicon retina enables the recognition system to extract object structure from the bright and the dark side of the face. For control purposes all recognition experiments have been repeated with filtered CCD camera images. The filter was adjusted such that the power spectra of the retina chip images and the filtered CCD images are identical. The images (e,f) are filtered versions of the images (a,b). It is evident that information in the dark part of image (b) has been erased due to saturation effects of the CCD camera and cannot be recovered by any local filtering procedure. We first measure the performance of the silicon retina under uniform lighting conditions, b ~ ... '. - ,. '\. .... • .. ... C .. ~ ~ • . .... .. ~ ... ~ It .~ • . 1 ..... • Figure 2: (a) Conventional CCD camera images (a,b) and silicon retina image (c,d) under different lighting conditions. The images (e,O are filtered CCD camera images with a power spectrum adjusted to the images in (c,d). The images (a,c) are used to generate the Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina 775 Table 1: (a) Face recognition results in a well illuminated environment and (b) in an environment with drastic changes in lighting conditions. f. p. rate silicon retina cony. CCD filt. CCD a 100% 83.5 86.2 85.3 100/0 81.7 83.5 84.4 50/0 76.2 82.6 80.7 10/0 71.6 79.8 75.2 b 1000/0 96.3 80.7 78.0 10% 96.3 76.2 75.2 50/0 96.3 72.5 72.5 10/0 93.6 64.2 62.4 i.e., both lamps are on and the person looks 20-30 degrees to the right. The recognition system has to deal with perspective distortions only. A gallery of 109 faces is matched to a face database of the same 109 persons. Table la shows that the recognition rate reaches values between 80 and 90 percent if we accept the best match without checking its significance. Such a decision criterion is unpractical for many applications since it corresponds to a false positive rate (f. p. rate) of 100 percent. If we increase the threshold E> to limit false positive matches to less than 1 percent the face recognizer is able to identify three out of four unknown faces. Filtering the CCD imagery does not hurt the recognition performance as the third column in Table 1a demonstrates. All necessary information for recognition is preserved in the filtered CCD images. The situation changes dramatically when we switch off the lamp on the left side of the test person. We compare a test gallery of persons looking straight ahead, but illuminated only from the right side, to our model gallery. Table 1 b summarizes the recognition results for different false positive rates. The advantage of using a silicon retina are 20 to 45 percent higher recognition rates than for a system with a CCD camera. For a false positive rate below one percent a silicon retina based recognition system identifies two third more persons than a conventional system. Filtering does not improve the recognition rate of a system that uses a CCD camera as can be seen in the third column. Our comparative face recognition experiment clearly demonstrates that a face recognizer with a retina chip is performing substantially better than conventional CCD camera based systems in environments with uncontrolled, substantially changing lighting conditions. Retina-like preprocessing yields increased recognition rates and increased significance levels. We expect even larger discrepancies in recognition rates if object without a bilateral symmetry have to be classified. In this sense the face recognition task does not optimally explore the potential of adaptive preprocessing by a silicon retina. Imagine an object recognition task where the most significant features for discrimination are hardly visible or highly ambiguous due to poor illumination. High error rates and very low significance levels are an inevitable consequence of such lighting conditions. The limited resolution and poor signal-to-noise ratio of silicon retina chips are expected to be improved by a new generation of chips fabricated in 0.7 /lm CMOS technology with a 776 Buhmann, Lades, and Eeckman potential resolution of256 x 256 pixels. Lighting conditions as simulated in ourrecognition experiment are ubiquitous in natural environments. Autonomous robots and vehicles or surveillance systems are expected to benefit from the silicon retina technology by gaining robustness and reliability. Silicon retinas and more elaborate analog VLSI chips for low level vision are expected to be an important component of an Adaptive Vision System. Acknowledgement: It is a pleasure to thank K. A. Boahen for providing us with the retina chips. We acknowledge stimulating discussions with C. von der Malsburg and C. Mead. This work was supported by the German Ministry of Science and Technology (lTR-8800-H 1) and by the Lawrence Livermore National Laboratory (W-7405-Eng-48). References Boahen, K., Andreou, A. 1992. A Contrast Sensitive Silicon Retina with Reciprocal Synapses. Pages 764-772 of: NIPS91 Proceedings. IEEE. Brunelli, R., Poggio, T. (1993). Face Recognition: Features versus Templates. IEEE Trans. on Pattern Analysis Machine Intelligence, 15, 1042-1052. Buhmann, J., Lange, J., von der Malsburg, C. 1989. Distortion Invariant Object Recognition by Matching Hierarchically Labeled Graphs. Pages I 155-159 of' Proc. llCNN, Washington. IEEE. Buhmann, J., Lades, M., von der Malsburg, C. 1990. Size and Distortion Invariant Object Recognition by Hierarchical Graph Matching. Pages II 411-416 of' Proc. llCNN, SanDiego. IEEE. Burr, D. J. (1981). Elastic Matching of Line Drawings. IEEE Trans. on Pat. An. Mach. Intel., 3, 708-713. Lades, M., Vorbriiggen, J.C., Buhmann, J., Lange, J., von der Malsburg, C., Wurtz, R.P., Konen, W. (1993). Distortion Invariant Object Recognition in the Dynamic Link Architecture. IEEE Transactions on Computers, 42, 300-311. Mahowald, M., Mead, C. (1991). The Silicon Retina. Scientific American, 264(5), 76. Mead, C. (1989). Analog VLSI and Neural Systems. New York: Addison Wesley. Turk, M., Pentland, A. (1991). Eigenfaces for Recognition. J. Cog. Sci., 3, 71-86. von der Malsburg, Christoph. 1981. The Correlation Theory of Brain Function. Internal Report. Max-Planck-Institut, Biophys. Chern., Gottingen, Germany. Yuille, A. (1991). Deformable Templates for Face Recognition. J. Cog. Sci., 3, 60-70.	1993	58
812	Address Block Location with a Neural Net System Hans Peter Graf AT&T Bell Laboratories Crawfords Corner Road Holmdel, NJ 07733, USA Abstract Eric Cosatto We developed a system for finding address blocks on mail pieces that can process four images per second. Besides locating the address block, our system also determines the writing style, handwritten or machine printed, and moreover, it measures the skew angle of the text lines and cleans noisy images. A layout analysis of all the elements present in the image is performed in order to distinguish drawings and dirt from text and to separate text of advertisement from that of the destination address. A speed of more than four images per second is obtained on a modular hardware platform, containing a board with two of the NET32K neural net chips, a SP ARC2 processor board, and a board with 2 digital signal processors. The system has been tested with more than 100,000 images. Its performance depends on the quality of the images, and lies between 85% correct location in very noisy images to over 98% in cleaner images. 1 INTRODUCTION The system described here has been integrated into an address reading machine developed for the 'Remote Computer Reader' project of the United States Postal Service. While the actual reading of the text is done by other modules, this system solves one of the major problems, namely, finding reliably the location of the destination address. There are only a few constraints on how and where an address has to be written, hence they may appear in a wide variety of styles and layouts. Often an envelope contains advertising that includes images as well as text. 785 786 Graf and Cosatto Sometimes. dirt covers part of the envelope image. including the destination address. Moreover. the image captured by the camera is thresholded and the reader is given a binary image. This binarization process introduces additional distortions; in particular. often the destination address is surrounded by a heavy texture. The high complexity of the images and their poor quality make it difficult to find the location of the destination address. requiring an analysis of all the elements present in the image. Such an analysis is compute-intensive and in our system it turned out to be the major bottleneck for a fast throughput. In fact. finding the address requires much more computation than reading it. Special-purpose hardware in the form of the NET32K neural net chips (Graf. Henderson. 90) is used to solve the address location problem. Finding address blocks has been the focus of intensive research recently. as several companies are developing address reading machines (United States Postal Service 92). The wide variety of images that have to be handled has led other researchers to apply several different analysis techniques to each image and then try to combine the results at the end. see e.g. (palumbo et a1. 92). In order to achieve the throughput required in an industrial application. special purpose processors for finding connected components and/or for executing Hough transforms have been applied. In our system we use the NET32K processor to extract geometrical features from an image. The high compute power of this chip allows the extraction of a large number of features simultaneously. From this feature representation. an interpretation of the image's content can then be achieved with a standard processor. Compared to an analysis of the original image. the analysis of the feature maps requires several orders of magnitude less computation. Moreover. the feature representation introduces a high level of robustness against noise. This paper gives a brief overview of the hardware platfOlm in section 2 and then describes the algorithms to find the address blocks in section 3. 2 THE HARDWARE The NET32K system has been designed to serve as a high-speed image processing platform. where neural nets as well as conventional algorithms can be executed. Three boards form the whole system. Two NET32K neural net chips are integrated with a sequencer and data formatting circuits on one board. The second board contains two digital signal processors (DSPs). together with 6 Mbytes of memOly. Control of the whole system is provided by a board containing a SP ARC2 processor plus 64 Mbytes of memory. A schematic of this system is shown in Figure 1. Image buffering and communication with other modules in the address reader are handled by the board with the SP ARC2 processor. When an image is received. it is sent to the DSP board and from there over to the NET32K processor. The feature maps produced by the NET32K processor are stored on the DSP board. while the SP ARC2 starts with the analysis of the feature maps. The DSP's main task is formatting of the data. while the NET32K processor extracts all the features. Its speed of computation is more than 100 billion multiply-accumulates per second with operands that have one or two bits of resolution. Images with a size of Sl2xS 12 pixels are processed at a rate of more than 10 frames per second. and 64 convolution kernels. each with a size of 16x 16 pixels. can be scanned simultaneously over the image. Each such kernel IS tuned to detect the presence of a feature. such as a line, an edge or a comer. Address Block Location with a Neural Net System 787 .................................................................................... ! NET32K MODULE I N~K I N~ r1~·~·~·:·A·T .. r ........ -.... -.-....... -..... ::::.fr=:::::::.~~~::=::::.~~=.=:::.=n:::=:: ..... _._ ...... _ ...... ! i v. . : I I • ~ It ~ "> ~~ " ). Afr ~lt ~ ~r ~ " ~ ~" ~U 'I ~ '1 ~ .... '1 SRAM DSP32C DRAM DSP32C SRAM 1 MEG 4 MEG 1 MEG L. .... ~~~ .. ~.~.~.~~., .... -.-.... ---.-.. -.. l-........ --... _ .... + .... _ ...... :;:~ ........... ..l ~ ________________________ ~. SPARC VME BUS Figure 1: Schematic of the whole NET32K system. Each of the dashed boxes represents one 6U VME board. The aITOWS show the conununication paths. 3. SEQUENCE OF ALGORITHMS The final result of the address block location system is a box describing a tight bmmd around the destination address, if the address is machine printed. Of handwritten addresses, only the zip code is read, and hence, one has to find a tight boundary around the zip code. This information is then passed along to reader modules of the address reading machine. There is no a priori knowledge about the writing style. Therefore the system first has to discriminate between handwritten and machine Plinted text. At the end of the address block location process, additional algorithms are executed to improve the accuracy of the reader. An overview of the sequence of algorithms used to solve these tasks is shown in Figure 2. The whole process is divided into three major steps: Preprocessing, feature extraction. and high-level analysis based on the feature information. 3.1. Preprocessing To quickly get an idea about the complexity of the image, a coarse evaluation of its layout is done. By sampling the density of the black pixels in various places of the image, one can see already whether the image is clean or noisy and whether the text is lightly printed or is dark. 788 Oraf and Cosatto The images are divided into four categories, depending on their darkness and the level of noise. 'This infonnation is used in the subsequent processing to guide the choice of the features. Only about one percent of the pixels are taken into account for this analysis, therefore, it can be executed quickly on the SP ARC2 processor. Preprocessing Extract features NET32K clean. light clean. dark =-.P IF. ~ = .... -.... ~ 16 Feature maps 8 Feature maps ---. ~.= : :~, ' ,,' -I'" ..... Extract text lines Cluster lines into groups --- Classify groups of lines MACHINE PRINT Analyse group of lines Determine level of noise Clean with NET32K; HANDWRITIEN Cluster text segments into lines Analyse group of lines Segment lines to find ZIP Determine slanVskew angle; Figure 2: Schematic of the sequence of algorithms for finding the position of the address blocks. 3.2. Feature Extraction After the preprocessing, the image is sent to the NET32K board where simple geometrical features, such as edges, corners and lines are extracted. Up to 16 different feature maps are generated, where a pixel in one of the maps indicates the presence of a feature in this location. Some of these feature maps are used by the host processor, for example, to decide whether text is handwritten or machine printed. Other feature maps are combined and sent once more through the NET32K processor in order to search for combinations of features representing more complex features. Typically, the feature maps are thresholded, so that only one bit per pixel is kept. More resolution of the computation results is available from the neural net chips. but in this way the amount of data that has to be analyzed is minimal. and one bit of resolution turned out to be sufficient. Examples of kernels used for the detection of strokes and text lines are shown in Figure 3. In the chip, usually four line detectors of increasing height plus eight stroke detectors of different orientations are stored. Other detectors are tuned to edges and strokes of machine printed text. The line detectors respond to any black line of the proper height. Due to the large width of 16 Address Block Location with a Neural Net System 789 pixels. a kernel stretches over one or even several characters. Hence a text line gives a response similar to that produced by a continuous black line. When the threshold is set properly. a text line in the original image produces a continuous line in the feature map. even across the gaps between characters and across small empty spaces between words. For an interpretation of a line feature map only the left and right end points of each connected component are stored. In this way one obtains a compact representation of the lines' positions that are well suited for the high-level analysis of the layout. Kernel: Line detector Image • t the NET32K syste IC::GUla Feature Kernel: Stroke detector Feature map Figure 3:Examples of convolution kernels and their results. The kernels' sizes are 16x 16 pixels, and their pixels' values are + 1, O. -1 . The upper part illustrates the response of a line detector on a machine printed text line. The lower kernel extracts strokes of a celtain orientation from handwritten text. Handwritten lines are detected by a second technique, because they are more irregular in height and the characters may be spaced apm1 widely. Detectors for strokes, of the type shown in the lower half ofFigw-e 3. are well suited for sensing the presence of handwritten text. The feature maps resulting form handwritten text tend to exhibit blobs of pixels along the text line. By smearing such feature maps in horizontal direction the responses of individual strokes are merged into lines that can then be used in the same way as described for the machine printed lines. Horizontal smearing of text lines. combined with connected component analysis is a well-known 790 Graf and Cosatto technique, often applied in layout analysis, to find words and whole lines of text. But when applied to the pixels of an image, such an approach works well only in clean images. As soon as there is noise present, this technique produces ilTegular responses. The key to success in a real world environment is robustness against noise. By extracting features first and then analyzing the feature maps, we drastically reduce the influence of noise. Each of the convolution kernels covers a range of 256 pixels and its response depends on several dozens of pixels inside this area. If pixels in the image are corrupted by noise, this has only a minor effect on the result of the convolution and, hence, the appearance of the feature map. When the analysis is started, it is unknown, whether the address is machine printed or hand written. In order to distinguish between the two writing styles, a simple one-layer classifier looks at the results of four stroke detectors and of four line detectors. It can determine reliably whether text is handwritten or machine printed. Additional useful information that can be extracted easily from the feature maps, is the skew angle of handwritten text. People tend to write with a skew anywhere from -45 degrees to almost +90 degrees. In order to improve the accuracy of a reader, the text is first deskewed. The most time consuming part of this operation is to determine the skew angle of the writing. The stroke detector with the maximum response over a line is a good indicator of the skew angle of the text. We compared this simple technique with several alternatives and found it to be as reliable as the best other algorithm and much faster to compute. 3.3. High-level Analysis The results of the feature extraction process are line segments, each one marked as handwritten or machine printed. Only the left and right end points of such lines are stored. At this point, there may still be line segments in this group that do not correspond to text, but rather to solid black lines or to line drawings. Therefore each line segment is checked, to determine whether the ratio of black and white pixels is that found typically in text. Blocks of lines are identified by clustering the line segments into groups. Then each block is analyzed, to see whether it can represent the destination address. For this purpose such features as the number of lines in the block, its size, position, etc. are used. These features are entered into a classifier that ranks each of the blocks. Certain conditions, such as a size that is too large, or if there are too many text lines in the block, will lead to an attempt to split blocks. If no good result is obtained, clustering is tried again with a changed distance metric, where the horizontal and the vertical distances between lines are weighted differently. If an address is machine printed, the whole address block is passed on to the reader, since not only the zip code, but the whole address, including the city name, the street name and the name of the recipient have to be read. A big problem for the reader present images of poor quality, particularly those with background noise and texture. State-of-the-art readers handle machine printed text reliably if the image quality is good, but they may fail totally if the text is buried in noise. For that reason, an address block is cleaned before sending it to the reader. Feature extraction with the NET32K board is used once more for this task, this time with detectors tuned to find all the strokes of the machine printed text. Applying stroke detectors with the proper width allows a good discrimination between the text and any noise. Even texture that consists of lines can be rejected reliably, if the line thickness of the texture is not the same as that of the text. . : . ..... Address Block Location with a Neural Net System 791 "3"" /"ksiQ \i~.\. Cal! [~ ~"S'~e".I • . ~ .. ~ ~ ===t ,o;;;r;;;a.e;2 . t1r .: " . ',' ····ee-5AT'fO··t;~a.····· "'~;Au'j'':f;,:.)i'''\i·bl,..~~···~t ....... "·~·S\;.·\·.cs.",~·A'···"" -~.W" .. -,\e"'..4*!~ .. _Q33.~2..:Figure 4: Example of an envelope image at various stages of the processing. Top: The result of the clustering process to find the bounding box of the address. Bottom right: The text lines within the address block are marked. Bottom left: Cuts in the text line with the zip code and below that the result of the reader. (The zip code is actually the second segment sent to the reader; the first one is the string 'USA'). If the address is handwritten, only the zip code is sent to the reader. In order to find the zip code, an analysis of the internal stmcture of the address block has to be done, which starts with finding the true text lines. Handwritten lines are often not straight, may be heavily skewed, and may contain large gaps. Hence simple techniques, such as connected component analysis, do not provide proper results. ClusteJing of the line segments obtained from the feature maps, provides a reliable solution of this problem. Once the lines are found, each one is segmented into words and some of them are selected as candidates for the zip code and are sent to the reader. Figure 4 shows an example of an envelope image as it progresses through the various processing steps. The system has been tested extensively on overall more than 100,000 images. Most of these tests were done in the assembled address reader, but during development of the system, large 792 Graf and Cosatto tests were also done with the address location module alone. One of the problems for evaluating the peIformance is the lack of an objective quality measure. When has an address been located correctly? Cutting off a small part of the address may not be detrimental to the final interpretation, while a bounding box that includes some additional text may slow the reader down too much. or it may throw off the interpretation. Therefore, it is not always clear when a bounding box, describing the address' location, is tight enough. Another important factor affecting the accw-acy numbers is, how many candidate blocks one actually considers. For all these reasons, accw-acy numbers given for address block location have to be taken with some caution. The results mentioned here were obtained by judging the images by eye. If images are clean and the address is surrounded by a white space larger than two line heights, the location is found correctly in more than 98% of the cases. Often more than one text block is found and of these the destination address is the first choice in 90% of the images, for a typical layout. If the image is very noisy, which actually happens surprisingly often, a tight bound around the address is found in 85% of the cases. These results were obtained with 5,000 images, chosen from more than 100,000 images to represent as much variety as possible. Of these 5,000 images more than 1,200 have a texture around the address, and often this texture is so dark that a human has difficulties to make out each character. 4. CONCLUSION Most of our algorithms described here consist of two parts: feature extraction implemented with a convolution and interpretation, typically implemented with a small classifier. Surprisingly many algorithms can be cast into such a fOimat. This common framework for algorithms has the advantage of facilitating the implementation, in particular when algorithms are mapped into hardware. Moreover, the feature extraction with large convolution kernels makes the system robust against noise. This robustness is probably the biggest advantage of our approach. Most existing automatic reading systems are very good as long as the images are clean, but they deteriorate rapidly with decreasing image quality. 'The biggest drawback of convolutions is that they require a lot of computation. In fact, without special purpose hardware, convolutions are often too slow. Our system relies on the NET32K new-al net chips to obtain the necessary throughput. The NET32K system is, we believe, at the moment the fastest board system for this type of computation. This speed is obtained by systematically exploiting the fact that only a low resolution of the computation is required. This allows to use analog computation inside the chip and hence much smaller circuits than would be the case in an all-digital circuit. References United States Postal Service, (1992), Proc. Advanced Technology Conf., Vol. 3, Section on address block location: pp. 1221 - 1310. P.W. Palumbo, S.N. Srihari, J. Soh, R. Sridhar, V. Demjanenko, (1992), !'Postal Address Block Location in Real Time", IEEE COMPUTER, Vol. 25n, pp. 34 - 42. H.P. Oraf and D. Henderson, (1990), "A Reconfigurable CMOS Neural Network", Digest IEEE Int. Solid State Circuits Conf. p. 144.	1993	59
813	On the Non-Existence of a Universal Learning Algorithm for Recurrent Neural Networks Herbert Wiklicky Centrum voor Wiskunde en Informatica P.O.Box 4079, NL-1009 AB Amsterdam, The Netherlands· e-mail: herbert@cwi.nl Abstract We prove that the so called "loading problem" for (recurrent) neural networks is unsolvable. This extends several results which already demonstrated that training and related design problems for neural networks are (at least) NP-complete. Our result also implies that it is impossible to find or to formulate a universal training algorithm, which for any neural network architecture could determine a correct set of weights. For the simple proof of this, we will just show that the loading problem is equivalent to "Hilbert's tenth problem" which is known to be unsolvable. 1 THE NEURAL NETWORK MODEL It seems that there are relatively few commonly accepted general formal definitions of the notion of a "neural network". Although our results also hold if based on other formal definitions we will try to stay here very close to the original setting in which Judd's NP completeness result was given [Judd, 1990]. But in contrast to [Judd, 1990] we will deal here with simple recurrent networks instead of feed forward architectures. Our networks are constructed from three different types of units: .E-units compute just the sum of all incoming signals; for II -units the activation (node) function is given by the product of the incoming signals; and with E)-units - depending if the input signal is smaller or larger than a certain threshold parameter fl - the output is zero or one. Our units are connected or linked by real weighted connections and operate synchronously. Note that we could base our construction also just on one general type of units, namely what usually is called .E II -units. Furthermore, one could replace the II -units in the below 431 432 Wiklicky construction by (recurrent) modules of simple linear threshold units which had to perform unary integer multiplication. Thus, no higher order elements are actually needed. As we deal with recurrent networks, the behavior of a network now is not just given by a simple mapping from input space to output space (as with feed forward architectures). In geneml, an input pattern now is mapped to an (infinite) output sequence. But note, that if we consider as the output of a recurrent network a certain final, stable output pattern, we could return to a more static setting. 2 THE MAIN RESULT The question we will look at is how difficult it is to construct or train a neural network of the described type so that it actually exhibits a certain desired behavior, i.e. solves a given learning task. We will investigate this by the following decision problem: Decision 1 Loading Problem INSTANCE: A neural network architecture N and a learning task T . QUESTION: Is there a configuration C for N such that T is realized by C? By a network configuration we just think of a certain setting of the weights in a neural network. Our main result concerning this problem now just states that it is undecidable or unsolvable. Theorem 1 There exists no algorithm which could decide for any learning task T and any (recurrent) neural network (consisting of"£.., TI-, and 8-units) if the given architecture can peiformT. The decision problem (as usual) gives a "lower bound" on the hardness of the related constructive problem [Garey and Johnson, 1979]. If we could construct a correct configuration for all instances, it would be trivial to decide instantly if a correct configuration exists at all. Thus we have: Corollary 2 There exists no universal learning algorithm for (recurrent) neural networks. 3 THE PROOF The proof of the above theorem is by constructing a class of neural networks for which it is impossible to decide (for all instance) if a certain learning task can be satisfied. We will refer for this to "Hilbert's tenth problem" and show that for each of its instances we can construct a neuml network, so that solutions to the loading problem would lead to solutions to the original problem (and vice versa). But as we know that Hilbert's tenth problem is unsolvable we also have to conclude that the loading problem we consider is unsolvable. 3.1 fiLBERT'S TENTH PROBLEM Our reference problem - of which we know it is unsolvable - is closely related to several famous and classical mathematical problems including for example Fermat's last theorem. On the Non-Existence of a Universal Learning Algorithm for Recurrent Neural Networks 433 Definition 1 A diophantine equation is a polynomial D in n variables with integer coefficients. that is D(.1:J, :J:2, ... ,.1",,) = L di(3:1, .T2, ... ,.r n ) t with each term d i of the form di( 3:1, .1:2, ... , .1:rt) = r.i . J: i • . J: iz .... . J: im, where the indices {i I, £2, ... , ; rrt} are taken from {I, 2, ... , 11 } and the coefficient r.i E Z. The concrete problem, first formulated in [Hilbert, 1900] is to develop a universal algorithm how to find the integer solutions for all D, i.e. a vector (3: J, .1:2, ... ,3:,1) with .1: i E Z (or IN), such that D( 3: 1,3:2, ... , .1: rt) = O. The corresponding decision problem therefore is the following: Decision 2 Hilbert's Tenth Problem INSTANCE: Given a diophantine equation D. QUESTION: Is there an integer solutionfor D? Although this problem might seem to be quite simple - it formulation is actually the shortest among D. Hilbert's famous 23 problems - it was not until 1970 when Y. Matijasevich could prove that it is unsolvable or undecidable [Matijasevich, 1970]. There is no recursive computable predicate for diophantine equations which holds if a solution in Z (or N) exists and fails otherwise [Davis, 1973, Theorem 7.4]. 3.2 THE NETWORK ARCIDTECTURE The construction of a neural network IV for each diophantine D is now straight forward (see FigJ). It is just a three step construction. First, each variable .1: i of D is represented in IV by a small sub-network. The structure of these modules is quite simple (left side of Fig.1). Note that only the self-recurrent connection for the unit at the bottom of these modules is "weighted" by 0.0 < 'II! < 1.0. All other connection transmit their signals unaltered (i.e. w = 1.0). Second, the terms di in D are represent by Il-units in IV (as show in Fig.1). Therefore, the connections to these units from the sub-modules representing the variables .1: i of D correspond to the occurrences of these variables in each term d i. Finally, the output signals of all these Il-units is multiplied by the corresponding coefficients C:i and summed up by the ~-unit at the top. 3.3 THE SUB.MODULES The fundamental property of the networks constructed in the above way is given by the simple fact that the behavior of such a neural network IV corresponds uniquely to the evaluation of the original diophantine D. First, note that the behavior of N only depends on the weights Wi in each of the variable modules. Therefore, we will take a closer look at the behavior of these sub-modules. Suppose, that at some initial moment a signal of value 1.0 is received by each variable module. After that the signal is reset again to 0.0. 434 Wiklicky The "seed" signal starts circling via Wi. With each update circle this signal becomes a little bit smaller. On the other hand, the same signal is also sent to the central 8-unit, which sends a signal 1.0 to the top accumulator unit as long as the "circling" activation of the bottom unit is larger then the (preset) threshold 0,. The top unit (which also keeps track of its former activiations via a recurrent connection) therefore just counts how many updates it takes before the activiation of the bottom unit drops below 0,. The final, maximum, value which is emitted by the accumulator unit is some integer .1:, for which we have: We thus have a correspondence between Wi and the integer .1: i = l ~ I~/i J ' where L-T J the largest integer which is smaller or equal to .1:. Given .1: i we also can construct an appropriate weight Wi by choosing it from the interval (exp (~~) ,exp (:r.1~!1))' 3.4 THE EQUIVALENCE To conclude the proof, we now have to demonstrate the equivalence of Hilbert's tenth problem and the loading problem for the discussed class of recurrent networks and some learning task. The learning task we will consider is the following: Map an input pattern with all signals equal to 1.0 (presented only once) to an output sequence which after afinite number of steps On the Non-Existence of a Universal Learning Algorithm for Recurrent Neural Networks 435 is constant equal to 0.0. Note that - as discussed above - we could also consider a more static learing task where a final state, which detennines the (single) output of the network, was detennined by the condition that the outgoing signals of all 8-units had to be zero. Considering this learing task and with what we said about the behavior of the sub-modules it is now trivial to see that the constructed network just evaluates the diophantine polynomial for a set of variables ;r i corresponding to the (final) output signals of the sub-modules (which are detennined uniquely by the weight values !lii) if the input to the network is a pattern of all 1.0s. If we had a solution .1.' i of the original diophantine equation D, and if we take the corresponding values Wi (according to the above relation) as weights in the sub-modules of N, then this would also solve the loading problem for this architecture. On the other hand, if we knew the correct weights Wi for any such network N, then the corresponding integers 3: i would also solve the corresponding diophantine equation D. In particular, if it would be possible to decide if a correct set of weights Wi for N exists (for the above learning task), we could also decide if the corresponding diophantine D had a solution 3: i E :IN (and vice versa). As the whole construction was trivial, we have shown that both problems are equivalent. 4 CONCLUSIONS We demonstrated that the loading problem not only is NP-complete - as shown for simple feed fOIward architectures in [Judd, 1990], [Lin and Vitter, 1991], [Blum and Rivest, 1992], etc. - but actually unSOlvable, i.e. that the training of (recurrent) neural networks is among those problems which "indeed are intractable in an especially strong sense" [Garey and Johnson, 1979, P 12]. A related non-existence result concerning the training of higher order neural networks with integer weights was shown in [Wiklicky, 1992, WIklicky, 1994]. One should stress once again that the fact that no general algorithm exists for higher order or recurrent networks, which could solve the loading problem (for all its instances), does not imply that all instances of this problem are unsolvable or that no solutions exist. One could hope, that in most relevant cases - whatever that could mean - or, when we restrict the problem, a sub-class of problems things might become tractable. But the difference between solvable and unsolvable problems often can be very small. In particular, it is known that the problem of solving linear diophantine equations (instead of general ones) is polynomially computable, while if we go to quadratic diophantine equations the problem already becomes;V P complete [Johnson, 1990]. And for general diophantine the problem is even unsolvable. Moreover, it is also known that this problem is unsolvable if we consider only diophantine equations of maximum degree 4, and there exists a universal diophantine with only 13 variables which is unsolvable [Davis et al., 1976]. But we think, that one should interpret the "negative" results on NP-complexity as well as on undecidability of the loading problem not as restrictions for neural networks, but as related to their computational power. As it was shown that concrete neural networks can be constructed, so that they simulate a universal Turing machine [Siegelmann and Sontag, 1992, Cosnard et al., 1993]. It is mere the complexity of the problem one attempts to solve which simply cannot disappear and not some intrinsic intractability of the neural network approach. 436 Wiklicky Acknowledgement This work was started during the author's affiliation with the "Austrian Research Institute for Artificial Intelligence", Schottengasse 3, A-101O Wien, Austria. Further work was supported by a grant from the Austrian "Fonds zur Forderung der wissenschaftlichen Forschung" as Projekt J0828-PHY. References [Blum and Rivest, 1992] Avrim L. Blum and Ronald L. Rivest. Training a 3-node neural network is NP-complete. Neural Networks, 5:117-127,1992. [Cosnard et al., 1993] Michael Cosnard, Max Garzon, and Pascal Koiran. Computability properties of low-dimensional dynamical systems. In Symposium on Theoretical Aspects of Computer Science (STACS '93), pages 365-373, Springer-Verlag, BerlinNew York, 1993. [Davis, 1973] Martin Davis. Hilbert's tenth problem is unsolvable. Amer. Math. Monthly, 80:233-269, March 1973. [Davis et aI., 1976] Martin Davis, Yuri Matijasevich, and Julia Robinson. Hilbert's tenth problem - diophantine equations: Positive aspects of a negative solution. In Felix E. Browder, editor, Mathematical developments arising from Hilbert, pages 323-378, American Mathematical Society, 1976. [Garey and Johnson, 1979] Michael R. Garey and David S. Johnson. Computers and Intractability -A Guide to the Theory of NP-Complete ness. W. H. Freeman, New York, 1979. [Hilbert, 1900] David Hilbert. Mathematische Probleme. Nachr. Ges. Wiss. G6ttingen, math.-phys.Kl., :253-297, 1900. [Johnson, 1990] David S. Johnson. A catalog of complexity classes. In Handbook of Theoretical Computer Science (Volume A: Algorithms and Complexity), chapter 2, pages 67-161, Elsevier - MIT Press, Amsterdam - Cambridge, Massachusetts, 1990. [Judd, 1990] J. Stephen Judd. Neural Network Design and the Complexity of Learning. MIT Press, Cambridge, Massachusetts - London, England, 1990. [Lin and Vitter, 1991] Jyh-Han Lin and Jeffrey Scott Vitter. Complexity results on learning by neural networks. Machine Learning, 6:211-230,1991. [Matijasevich, 1970] Yuri Matijasevich. Enumerable sets are diophantine. Dokl. Acad. Nauk., 191:279-282, 1970. [Siegelmann and Sontag, 1992] Hava T. Siegelmann and Eduardo D. Sontag. On the computational power of neural nets. In Fifth Workshop on Computational Learning Theory (COLT 92), pages 440-449, 1992. [Wiklicky, 1992] Herbert Wiklicky. SyntheSis and Analysis of Neural Networks On a Framework for Artificial Neural Networks. PhD thesis, University of Vienna Technical University of Vienna, September 1992. [WIklicky, 1994] Herbert Wiklicky. The neural network loading problem is undecidable. In Euro-COLT '93 - Conference on Computational Learning Theory, page (to appear), Oxford University Press, Oxford, 1994. PART III THEORETICAL ANALYSIS: DYNAMICS AND STATISTICS	1993	6
814	Assessing the Quality of Learned Local Models Stefan Schaal Christopher G. Atkeson Department of Brain and Cognitive Sciences & The Artifical Intelligence Laboratory Massachusetts Institute of Technology 545 Technology Square, Cambridge, MA 02139 email: sschaal@ai.mit.edu, cga@ai.mit.edu Abstract An approach is presented to learning high dimensional functions in the case where the learning algorithm can affect the generation of new data. A local modeling algorithm, locally weighted regression, is used to represent the learned function. Architectural parameters of the approach, such as distance metrics, are also localized and become a function of the query point instead of being global. Statistical tests are given for when a local model is good enough and sampling should be moved to a new area. Our methods explicitly deal with the case where prediction accuracy requirements exist during exploration: By gradually shifting a "center of exploration" and controlling the speed of the shift with local prediction accuracy, a goal-directed exploration of state space takes place along the fringes of the current data support until the task goal is achieved. We illustrate this approach with simulation results and results from a real robot learning a complex juggling task. 1 INTRODUCTION Every learning algorithm faces the problem of sparse data if the task to be learned is sufficiently nonlinear and high dimensional. Generalization from a limited number of data points in such spaces will usually be strongly biased. If, however, the learning algorithm has the ability to affect the creation of new experiences, the need for such bias can be reduced. This raises the questions of (1) how to sample data the most efficient, and (2) how to assess the quality of the sampled data with respect to the task to be learned. To address these questions, we represent the task to be learned with local linear models. Instead of constraining the number of linear models as in other approaches, infinitely many local models are permitted. This corresponds to modeling the task with the help of (hyper-) tangent planes at every query point instead of representing it in a piecewise linear fashion. The algorithm applied for this purpose, locally weighted regression (L WR), stems from nonparametric regression analysis (Cleveland, 1979, Muller, 1988, Hardie 1990, Hastie&Tibshirani, 1991). In Section 2, we will briefly outline LWR. Section 3 discusses 160 Assessing the Quality of Learned Local Models 161 several statistical tools for assessing the quality of a learned linear L WR model, how to optimize the architectural parameters of L WR, and also how to detect outliers in the data. In contrast to previous work, all of these statistical methods are local, i.e., they depend on the data in the proximity of the current query point and not on all the sampled data. A simple exploration algorithm, the shifting setpoint algorithm (SSA), is used in Section 4 to demonstrate how the properties of L WR can be exploited for learning control. The SSA explicitly controls prediction accuracy during learning and samples data with the help of optimal control techniques. Simulation results illustrate that this method work well in high dimensional spaces. As a final example, the methods are applied to a real robot learning a complex juggling task in Section 5. 2 LOCALLY WEIGHTED REGRESSION Locally linear models constitute a good compromise between locally constant models such as nearest neighbors or moving average and locally higher order models; the former tend to introduce too much bias while the latter require fitting many parameters which is computationally expensive and needs a lot of data. The algorithm which we explore here, locally weighted regression (LWR) (Atkeson, 1992, Moore, 1991, Schaal&Atkeson, 1994), is closely related to versions suggested by Cleveland et al. (1979, 1988) and Farmer&Siderowich (1987). A LWR model is trained by simply storing every experience as an input/output pair in memory. If an output Y, is to be generated from a given input x" the it is computed by fitting a (hyper-) tangent plane at x by means of weightd . , e regressIOn: (1) where X is an mx(n+ 1) matrix of inputs to the regression, y the vector of corresponding outputs, P(x,) the vector of regression parameters, and W the diagonal mxm matrix of weights. The requested Y,results from evaluating the tangent plane at x ,i.e., Y = x~p. The elements of W give points which are close to the current query poi~t x, a l~ger influence than those which are far away. They are determined by a Gaussian kernel: w;(x,) = exp( (x; - x,lD(x,)(x; - x,) / 2k(x,)2) (2) w; is the weight'for the i rh data point (xj,Yj) in memory given query point x . The matrix D(x,) weights the contribution of the individual input dimensions, and the factor k(x,) determines how local the regression will be. D and k are architectural parameters of L WR and can be adjusted to optimize the fit of the local model. In the following we will just focus on optimizing k, assuming that D normalizes the inputs and needs no further adjustment; note that, with some additional complexity, our methods would also hold for locally tuning D. 3 ASSESSING THE LOCAL FIT In order to measure the goodness of the local model, several tests have been suggested. The most widely accepted one is leave-one-out cross validation (CV) which calculates the prediction error of every point in memory after recalculating (1) without this point (Wahba&Wold 1975, Maron&Moore 1994). As an alternative measure, Cleveland et al. (1988) suggested Mallow's Cp-test, originally developed as a way to select covariates in linear regression analysis (Mallow, 1966). Hastie&Tibshirani (1991) showed that CV and the Cp-test are closely related for certain classes of analyses. Hastie&Tibshirani (1991) 162 Schaal and Atkeson also presented pointwise standard-error bands to assess the confidence in a fitted value which correspond to confidence bands in the case of an unbiased fit All these tests are essentially global by requiring statistical analysis over the entire range of data in memory. Such a global analysis is computationally costly, and it may also not give an adequate measure at the current query site Xq: the behavior of the function to be approximated may differ significantly in different places, and an averaging over all these behaviors is unlikely to be representative for all query sites (Fan&Gijbels, 1992). It is possible to convert some of the above measures to be local. Global cross validation has a relative in linear regression analysis, the PRESS residual error (e.g., Myers, 1990), here formulated as a mean squared local cross validation error: n is the number of data points in memory contributing with a weight Wj greater than some small constant (e.g., Wi> 0.01) to the regression, and p is the dimensionality of ~. The PRESS statistic performs leave-one-out cross validation computationally very efficient by not requiring the recalculation of ~ (Eq.(1)) for every excluded point. Analogously, prediction intervals from linear regression analysis (e.g., Myers, 1990) can be transformed to be a local measure too: 1'1 = x;~ ± (a/2,11'-p' S~1 + x: (XTWTWXfl Xq where S2 is an estimate of the variance at x'I: S2(X ) = (X~ - ytWTW(X~ - y) q n' - p' (4) (5) and (a/2,,.'-' isStudent'st-valueof n'-p' degrees of freedom fora l00(I-a)% prediction bound. The direct interpretation of (4) as prediction bounds is only possible if y is an unbiased estimate, which is usually hard to determine. 'I Finally, the PRESS statistic can also be used for local outlier detection. For this PUIJJOse it is reformulated as a standardized individual PRESS residual: eiC,..,.. .. (x q )= ~ , T T T -1 S 1- w·x. (X W wx) X.W. I I I I (6) This measure has zero mean and unit variance. If it exceeds a certain threshold for a point Xi' the point can be called an outlier. An important ingredient to forming the measures (3)-(6) lies in the definition of n' and p' as given in (3). Imagine that the weighting function (2) is not Gaussian but rather a function that clips data points whose distance from the current query point exceeds a certain threshold and that the remaining r data points all contribute with unit weight. This reduced data regression coincides correctly with a r -data regression since n' = r . In the case of the soft-weighting (2). the definition of n' ensures the proper definition of the moments of the data. However, the definition of p', i.e., the degrees of freedom of the regression, is somewhat arbitrary since it is unclear how many degrees of freedom have acAssessing the Quality of Learned Local Models 163 tually been used. Defining p' as in (3) guarantees that p' < n' and renders all results more pessimistic when only a small number of data points contribute to the regression. A , ,.. A , ,.. 2 The statistical tests (3) and (4) can not only be 1.5 0.5 (a) : used as a diagnostic tool, but they can also .1 \. serve to optimize the architectural parameters J; ., :, ofLWR. This results in a function fitting technique which is called supersmoothing in statistics (Hastie&Tibshirani, 1991). Fan&Gijbels (1992) investigated a method for this purpose that required estimation of the second deriva.0.S.o+.2~+-+-0 ~"0.""2 c..,.....,....,..O' • ..;..;........~0T-.8~"0.~e ~-i--r'~1.2 tive of the function to be approximated and the 1.5 0.5 (b) data density distribution. These two measures are not trivially obtained in high dimensions and we would like to avoid using them. Figure 1 shows fits of noisy data from the function y = x- sin\2n:x3 ) COS(2n:x3) exp(x4) with 95% prediction intervals around the fitted values. In Figure la, global one-leave-out cross validation was applied to optimize k (cf. .o .5.+0.2~""""~"0.-'2 ~""'0 .• """"""~0f-.8~"0.8~""""""".....-'r-...-112 Eq.(2». In the left part of the graph the fit 1.5 0.5 o (c) -_. _. predcton int.rv. " nai., data x ··> starts to follow noise. Such behavior is to be expected since the global optimization of k also took into account the quickly changing regions on the right side of the graph and thus chose a rather small k. In Figure 1b minimization of the local one-leave-out cross validation error was applied to fit the data, and in Figure 1c prediction intervals were mini.0 . 5.+0.2..,......,.-.,.....,~...,0.2-.-,......,.,..0 .• ...,.....,....~0r-.8,....,.....,r-r0.8~...,...,-..,......,.-.,....,1.2 mized. These two fits cope nicely with both J(--> Figure 1: Optimizing the L WR fit using: (a) global cross validation; (b) local cross validation; (c) local prediction intervals. the high frequency and the low frequency regions of the data and recover the true function rather well. The extrapolation properties of local cross validation are the most appropriate given that the we know the true function. Interestingly, at the right end of Figure 1c, the minimization of the prediction intervals suddenly detects that global regression has a lower prediction interval than local regression and jumps into the global mode by making k rather large. In both local methods there is always a competition between local and global regression. But sudden jumps take place only when the prediction interval is so large that the data is not trustworthy anyway. To some extend, the statistical tests (3)-(6) implicitly measure the data density at the current query point and are thus sensitive towards little data support, characterized by a small n'. This property is desirable as a diagnostic tool, particularly if the data sampling process can be directed towards such regions. However, if a fixed data set is to be analyzed which has rather sparse and noisy data in several regions, a fit of the data with local optimization methods may result in too jagged an approximation since the local fitting mistakes the noise in such regions as high frequency portion of the data. Global methods avoid this effect by biasing the function fitting in such unfavorable areas with knowledge from other data regions and will produce better results if this bias is appropriate. 164 Schaal and Atkeson 4 THE SHIFTING SETPOINT EXPLORATION ALGORITHM In this section we want to give an example of how LWR and its statistical tools can be used for goal directed data sampling in learning control. If the task to be learned is high dimensional it is not possible to leave data collection to random exploration; on the one hand this would take too much time. and on the other hand it may cause the system to enter unsafe or costly regions of operation. We want to develop an exploration algorithm which explicitly avoids with such problems. The shifting setpoint algorithm (SSA) attempts to decompose the control problem into two separate control tasks on different time scales. At the fast time scale. it acts as a nonlinear regulator by trying to keep the controlled system at some chosen setpoints in order to increase the data density at these setpoints. On a slower time scale. the setpoints are shifted by controlling local prediction accuracy to accomplish a desired goal. In this way the SSA builds a narrow tube of data support in which it knows the world. This data can be used by more sophisticated control algorithms for planning or further exploration. The algorithm is graphically illustrated in the example of a mountain car in Figure 2. The task of the car is to drive at a given constant horizontal speed xdesired from the left to the right of Figure 2a. xduired need not be met precisely; the car should also minimize its fuel consumption. Initially. the car knows nothing about the world and cannot look ahead. but it has noisy feedback of its position and velocity. Commands. which correspond to the thrust F of the motor. can be generated at 5Hz. The mountain car starts at its start point with one arbitrary initial action for the first time step; then it brakes and starts all over again. assuming the system can be reset somehow. The discrete one step dynamics of the car are modeled by an L WR forward model: x...,xt = f(Xc..,.,.elll. F). where x = (x.xl (7) After a few trials~ the SSA searches the data in memory for the point (x;u"elll.F,x~«xt)resl whose outcome x lI«xt can be predicted with the smallest local prediction interval. This best point is declared the setpoint of this stage: ( T F T )T (T FAT)T XS,ill' S ,XS,OIl' = XC~IIl' 'Xllm bltSl (8) and its local linear model results from a corresponding LWR lookup: A XS,OIll = f(xS,u.,Fs):::: AxS;1I + BFs + C (9) Based on this liDear model. an optimal LQ controller (e.g., Dyer&McReynolds. 1970) can be constructed. This results in a control law of the form: (10) After these calculations. the mountain car learned one controlled action for the first time step, However. since the initial action was chosen arbitrarily, XS,OIII will be significantly away from the desired speed Xdesir«d. A reduction of this error is achieved as follows, First, the SSA repeats one step actions with the LQ controller until suffjcient data is collected to reduce the prediction intervals ofLWR lookups for (x~,ill,Fs) (Eq.(9)) below a certain threshold. Then it shifts the setpoint towards the goal according to the procedure: 1) calculate the error of the predicted output state: err S o,d = xde . d Xs III 2) take the derivfltive of the error with respect to the comm'and Fs sr;om a LWR lookup for (XIill.FS) (cf. (9)): Assessing the Quality of Learned Local Models 165 aerr S,OI" = aerr S,Old aXS,OMI = _ aXS,Old = _ B aFs aXSpld aFs aFs and calculate a correction Ms from solving: -BMs = a errs old ; a E [0,1] determines how much of the error should be compensated for in one step. 3) update Fs: Fs = Fs - Ms and calculate the new X SOM1 with LWR (Eq.(9». 4) assess the fit for the updated setpoint with prediction intervals. If the quality is above a certain threshold, continue with I), otherwise terminate shifting. Figure 2: The mountain car: (a) landscape across which the car has to drive at constant velocity of 0.8 mIs, (b) contour plot of data density in phase space as generated by using multistage SSA, (c) contour plot of data density in position-action space, (d) 2-dimensional mountain car 0.1 In this way, the output state of the setpoint shifts towards the goal until the data support falls below a threshold. Now the mountain car perfonns several new trials with the new setpoint and the correspondingly updated LQ controller. After the quality of fit statis10 2D 30 40 10 tics rise above a threshold, the setpoint can • Polltlon E" ... [III) [J Ylloclty EITOf ["'") be shifted again. As soon as the first stage's Figure 3: Mean prediction error of local models setpoint reduces the error Xdesj~d Xs old sufficiently, a new stage is created and the mountain car tries to move one step further in its world. The entire procedure is repeated for each new stage until the car knows how to move across the landscape. Figure 2b and Figure 2c show the thin band of data which the algorithm collected in state space and position-action space, These two pictures together form a narrow tube of knowledge in the input space of the forward model. 166 Schaal and Atkeson The example of the mountain car can easily be scaled up to arbitrarily high dimensions by making the mountain a multivariate function. We tried versions up to a 5-dimensional mountain corresponding to a 9\15 ~ 9\10 forward model; Figure 2d shows the 2-dimensional version. The results of learning had the same quality as in the ID example. Figure 3 shows the prediction errors of the local models after learning for the ID. 2D •...• and 5D mountain car. To obtain these errors. the car was started at random positions within its data support from where it drove along the desired trajectory. The difference between the predicted next state and the actual outcome at each time step was averaged. Position errors stayed within 2-4 cm on the 10m long landscape. and velocity errors within 0.020.05 m/s. The dimensionality of the problem did not affect the outcome significantly. (a) (b) (;j ~,~~--------------------~ ~1OIIJ " 21 3, 4' 51 Trial Number (C) 5 ROBOT JUGGLING To test our algorithms in a real world experiment. we implemented them on a juggling robot. The juggling task to be performed. devil sticking. is illustrated in Figure 4a. For the robot. devil sticking was slightly simplified by attaching the devil stick to a boom. as illustrated in Figure 4b. The task state was encoded as a 5-dimensional state vector. taken at the moment when the devilstick hit one of the hand sticks; the throw action was parameterized as 5-dimensional action vector. This resulted in a 9\10 ~ 9\5 discrete forward model of the task. Initially the robot was given default actions for the left-hand and right-hand throws; the quality of these throws. however. was far away from achieving steady juggling. The robot started with no initial experiences and tried to build controllers to perform continuous juggling. The goal states for the SSA developed automatically from the requirement that the left hand had to learn to throw the devilstick to a place where the right hand had sufficient data support to control the devilstick. and vice versa. Figure 4c shows a typical learning curve for this task. It took about 40 trials before the left and the right hand learned to throw the devilstick such that both hands were able to Figure 4: (a) illustration of devilsticking, (b) a cooperate. Then. performance quickly went devils ticking robot, (c) learning curve of robot up to long runs up to 1200 consecutive hits. Humans usually need about one week of one hour practicing per day before they achieve decent juggling performance. In comparison to this. the learning algorithm performed very well. However. it has to be pointed out that the learned controllers were only local and could not cope with larger perturbations. A detailed description of this experiment can be found in Schaal&Atkeson (1994). Assessing the Quality of Learned Local Models 167 CONCLUSIONS One of the advantages of memory-based nonparametric learning methods lies in the least commitment strategy which is associated with them. Since all data is kept in memory, a lookup can be optimized with respect to the architectural parameters. Parametric approaches do not have this ability if they discard their training data; if they retain it, they essentially become memory-based. The origin of nonparametric modeling in traditional statistics provides many established statistical methods to inspect the quality of what has been learned by the system. Such statistics formed the backbone of the SSA exploration algorithm. So far we have only examined some of the most obvious statistical tools which directly relate to regression analysis. Many other methods from other statistical frameworks may be suitable as well and will be explored by our future work. Acknowledgements Support was provided by the Air Force Office of Scientific Research, by the Siemens Corporation, the German Scholarship Foundation and the Alexander von Humboldt Foundation to Stefan Schaal, and a National Science Foundation Presidential Young Investigator Award to Christopher G. Atkeson. We thank Gideon Stein for implementing the first version of L WR on a DSP board, and Gerrie van Zyl for building the devil sticking robot and implementing the first version of learning of devil sticking. References Atkeson, C.G. (1992), "Memory-Based Approaches to Approximating Continuous Functions", in: Casdagli, M.; Eubank, S. (eds.): Nonlinear Modeling and Forecasting. Redwood City, CA: Addison Wesley (1992). Cleveland, W.S., Devlin, S.l, Grosse, E. (1988), "Regression by Local Fitting: Methods, Properties, and Computational Algorithms". Journal of &onometrics 37,87 -114, North-Holland (1988). Cleveland, W.S. (1979), "Robust Locally-Weighted Regression and Smoothing Scatterplots". Journal of the American Statistical Association ,no.74, pp.829-836 (1979). Dyer, P., McReynolds, S.R. (1970), The Computation and Theory of Optima I Comrol, New York: Academic Press (1970). Fan, J., Gijbels, I. (1992), "Variable Bandwidth And Local Linear Regression Smoothers", The Annals of Statistics, vol.20, no.4, pp.2008-2036 (1992). Farmer, J.D., Sidorowich, J.I (1987), "Predicting Chaotic Dynamics", Kelso, IA.S., Mandell, AJ., Shies inger, M.F., (eds.):Dynamic Patterns in Complex Systems, World Scientific Press (1987). HardIe, W. (1991), Smoothing Techniques with Implementation in S, New York, NY: Springer. Hastie, T.l; Tibshirani, R.J. (1991), Generalized Additive Models, Chapman and Hall. Mallows, C.L. (1966), "Choosing a Subset Regression", unpublished paper presented at the annual meeting of the American Statistical Association, Los Angles (1966). Maron, 0., Moore, A.W. (1994), "Hoeffding Races: Accelerating Model Selection Search for Classification and Function Approximation", in: Cowan, J. , Tesauro, G., and Alspector, 1. (eds.) Advances in Neural Information Processing Systems 6, Morgan Kaufmann (1994). Muller, H.-G. (1988), Nonparametric Regression Analysis of Longitudinal Data, Lecture Notes in Statistics Series, vo1.46, Berlin: Springer (1988). Myers, R.H. (1990), Classical And Modern Regression With Applications, PWS-KENT (1990). Schaal, S., Atkeson, C.G. (1994), "Robot Juggling: An Implementation of Memory-based Learning", to appear in: Control Systems Magazine, Feb. (1994). Wahba, G., Wold, S. (1975), "A Completely Automatic French Curve: Fitting Spline Functions By Cross-Validation", Communications in Statistics, 4(1) (1975).	1993	60
815	Synchronization, oscillations, and 1/ f noise in networks of spiking neurons Martin Stemmler, Marius Usher, and Christof Koch Computation and Neural Systems, 139-74 California Institute of Technology Pasadena, CA 91125 Zeev Olami Dept. of Chemical Physics Weizmann Institute of Science Rehovot 76100, Israel Abstract We investigate a model for neural activity that generates long range temporal correlations, 1/ f noise, and oscillations in global activity. The model consists of a two-dimensional sheet of leaky integrateand-fire neurons with feedback connectivity consisting of local excitation and surround inhibition. Each neuron is independently driven by homogeneous external noise. Spontaneous symmetry breaking occurs, resulting in the formation of "hotspots" of activity in the network. These localized patterns of excitation appear as clusters that coalesce, disintegrate, or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters. The emergent cross-correlation functions have a dual structure, with a sharp peak around zero on top of a much broader hill. The power spectrum associated with single units shows a 1/ f decay for small frequencies and is flat at higher frequencies, while the power spectrum of the spiking activity averaged over many cells-equivalent to the local field potential-shows no 1/ f decay but a prominent peak around 40 Hz. 629 630 Stemmler, Usher, Koch, and Olami 1 The model The model consists of a 100-by-l00 lattice of integrate-and-fire units with cyclic lattice boundary conditions. Each unit represents the nerve cell membrane as a simple RC circuit (r = 20 msec) with the addition of a reset mechanism; the refractory period TreJ is equal to one iteration step (1 msec). Units are connected to each other within the layer by local excitatory and inhibitory connections in a center-surround pattern. Each unit is excitatorily connected to N = 50 units chosen from a Gaussian probability distribution of u = 2.5 (in terms of the lattice constant), centered at the unit's position N inhibitory connections per unit are chosen from a uniform probability distribution on a ring eight to nine lattice constants away. Once a unit reaches the threshold voltage, it emits a pulse that is transmitted in one iteration (1 msec) to connected neighboring units, and the potential is reset by subtracting the threshold from resting potential. \Ii(t + 1) = (exp( -l/r)\Ii(t) + h (t)) O[vth - V(t)]. (1) Ii is the input current, which is the sum of lateral currents from presynaptic units and external current. The lateral current leads to an increase (decrease) in the membrane potential of excitatory (inhibitorily ) connected cells. The weight of the excitation and inhibition, in units of voltage threshold, is ~ and J3 ~. The values a = 1.275 and J3 = 0.67 were used for simulations. The external input is modeled independently for each cell as a Poisson process of excitatory pulses of magnitude 1/ N, arriving at a mean rate "ext. Such a simple cellular model mimics reasonably well the discharge patterns of cortical neurons [Bernander et al., 1994, Softky and Koch, 1993]. 2 Dynamics and Pattern Formation In the mean-field approximation, the firing rate of an integrate-and-fire unit is a function of the input current [Amit and Tsodyks, 1991] given by f(I) = (TreJ - r In[l - 1/(1 r)])-l, (2) where Tref is the refractory period and r the membrane time constant. In this approximation, the dynamics associated with eq. 1 simplify to ~i = -Ii + L j Wijf(Ij) + Itxt , (3) where Wij represents the connection strength matrix from unit j to unit i. Homogeneous firing activity throughout the network will result as long as the connectivity pattern satisfies W(k)-l < 0 for all k, where W(k) is the Fourier transform of Wij . As one increases the total strength of lateral connectivity, clusters of high firing activity develop. These clusters form a hexagonal grid across the network; for even stronger lateral currents, the clusters merge to form stripes. The transition from a homogeneous state to hexagonal clusters to stripes is generic to many nonequilibrium systems in fluid mechanics, nonlinear optics, reactiondiffusion systems, and biology. (The classic theory for fluid mechanics was first Synchronization, Oscillations, and IlfNoise in Networks of Spiking Neurons 631 developed by [Newell and Whitehead, 1969], see [Cross and Hohenberg, 1993] for an extensive review. Cowan (1982) was the first to suggest applying the techniques of fluid mechanics to neural systems.) The richly varied dynamics of the model, however, can not be captured by a meanfield description. Clusters in the quasi-hexagonal state coalesce, disintegrate, or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters. 16 14 E 12 ... " " 10 t'.: B 8 ; 6 R~ndom Walk of Clusters o~~--~~--~~--~~--~~ o 2 6 8 10 12 14 16 18 x (latt~ce un~t~) Figure 1: On the left, the summed firing activity for the network over 50 msec of simulation is shown. Lighter shades denote higher firing rates (maximum firing rate 120 Hz). Note the nearly hexagonal pattern of clusters or "hotspots" of activity. On the right, we illustrate the motion of a typical cluster. Each vertex in the graph represents a tracked cluster's position averaged over 50 msec. Repulsive interactions with surrounding clusters generally constrain the motion to remain within a certain radius. This vibratory motion of a cluster is occasionally punctuated by longerrange diffusion. Statistical fluctuations, diffusion and synchronization of clusters, and noise in the external input driving the system lead to 1/ I-noise dynamics, long-range correlations, and oscillations in the local field potential. These issues shall be explored next. 3 1/ f Noise The power spectra of spike trains from individual units are similar to those published in the literature for nonbursting cells in area MT in the behaving monkey [Bair et al., 1994]. Power spectra were generally flat for all frequencies above 100 Hz. The effective refractory period present in an integrate-and-fire model introduces a dip at low frequencies (also seen in real data). Most noteworthy is the l/lo.s component in the power spectrum at low frequencies. Notice that in order to see such a decay for very low frequencies in the spectrum, single units must be recorded for on the order of 10-100 sec, longer than the recording time for a typical trial in neurophysiology. We traced a cluster of neuronal activity as it diffused through the system, and 632 Stemmler. Usher. Koch. and Olami Spike Tra~n Power Spectrum 3r-----~----~------~----_r----~ 2.5 2 1.5 1 0.5 20 40 60 80 100 Hz ... 0.7 0.5 0.3 0.2 0.15 0.1 30. lSI distribution 50. 70. 100. 150. 200. msec Figure 2: Typical power spectrum and lSI distribution of single units over 400 sec of simulation. At low frequencies, the power spectrum behaves as f- O.S±O.017 up to a cut-off frequency of ~ 8 Hz. The lSI distribution on the right is shown on a log-log scale. The lSI histogram decays as a power law pet) ex t-1.70±O.02 between 25 and 300 msec. In contrast, a system with randomized network connections will have a Poisson-distributed lSI histogram which decays exponentially. measured the lSI distribution at a fixed point relative to the cluster center. In the cluster frame of reference, activity should remain fairly constant, so we expect and do find an interspike interval (lSI) distribution with a single characteristic relaxation time: Pr(t) = A(r)exp(-tA(r)) , where the firing rate A(r) is now only a function of the distance r in cluster coordinates. Thus Pr(t) is always Poisson for fixed r. If a cluster diffuses slowly compared to the mean interspike interval, a unit at a fixed position samples many lSI distributions of varying A(r) as the cluster moves. The lSI distribution in the fixed frame reference is thus pet) = j A(r)2 exp( -t A(r»)dr. (4) Depending on the functional form of A(r), pet) (the lSI distribution for a unit at a fixed position) will decay as a power law, and not as an exponential. Empirically, the distribution of firing rates as a function of r can be approximated (roughly) by a Gaussian. A Gaussian A(r) in eq. 4 leads to pet) f'oi t- 2 for t at long times. In turn, a power-law (fractal) pet) generates 1/ f noise (see Table 1). 4 Long-Range Cross-Correlations Excitatory cross-correlation functions for units separated by small distances consist of a sharp peak at zero mean time delay followed by a slower decay characterized by a power law with exponent -0.21 until the function reaches an asymptotic level. Nelson et al. (1992) found this type of cross-correlation between neurons-a "castle on a hill" -to be the most common form of correlation in cat visual cortex. Inhibitory Synchronization, Oscillations, and lifNoise in Networks of Spiking Neurons 633 cross-correlations show a slight dip that is much less pronounced than the sharp excitatory peak at short time-scales. 1000 750 500 250 1000 750 500 250 -300 -300 Cross-Correlation at d 1 -200 -100 o msec 100 200 Cross-Correlation at d 9 -200 -100 o 100 200 msec 300 300 Figure 3: Cross-correlation functions between cells separated by d units of the lattice. Given the center-surround geometry of connections, the upper curve corresponds to mutually excitatory coupling and the lower to mutually inhibitory coupling. Correlations decay as l/tO.21 , consistent with a power spectrum of single spike trains that behaves as 1/ fo .8. Since correlations decay slowly in time due to the small exponent of the power, long temporal fluctuations in the firing rate result, as the 1/ f-type power spectra of single spike trains demonstrate. These fluctuations in turn lead to high variability in the number of events over a fixed time period. In fact, the decay in the auto-correlation and power spectrum, as well as the rise in the variability in the number of events, can be related back to the slow decay in the interspike interval (lSI) distribution. If the lSI distribution decays as a power law pet) ,...., t- II , then the point process giving rise to it is fractal with a dimension D = v-I [Mandelbrot, 1983]. Assuming that the simulation model can be described as a fully ergodic renewal process, all these quantities will scale together [Cox and Lewis, 1966, Teich, 1989, Lowen and Teich, 1993, Usher et al., 1994]: 634 Stemmler, Usher, Koch, and Olami 10-4 (lJ :J: 0 P... Table 1: Scaling Relations and Empirical Results Var(N) Auto-correlation Power Spectrum lSI Distribution Var(N) "-J Nil A(t) "-J t ll - 2 S(I) "-J /-11+1 pet) ""' t- II Var(N) '" N1.54 A(t) "-J t- 0.21 S(I) ""' /-0.81 pet) "-J c1.7O These relations will be only approximate if the process is nonrenewal or nonergodic, or if power-laws hold over a limited range. The process in the model is clearly nonrenewal, since the presence of a cluster makes consecutive short interspike intervals for units within that cluster more likely than in a renewal process. Hence, we expect some (slight) deviations from the scaling relations outlined above. 5 Cluster Oscillations and the Local Field Potential The interplay between the recurrent excitation that leads to nucleation of clusters and the "firewall" of inhibition that restrains activity causes clusters of high activity to oscillate in size. Fig 4 is the power spectrum of ensemble activity over the size of a typical cluster. Power Spectrum of Cluster ActlVlty withln radlus d=9 25 20 15 10 5 0 0 20 40 60 80 100 Hz Figure 4: Power spectrum of the summed spiking activity over a circular area the size of a single cluster (with a radius of 9 lattice constants) recorded from a fixed point on the lattice for 400 seconds. Note the prominent peak centered at 43 Hz and the loss of the 1// component seen in the single unit power spectra (Fig. 2). These oscillations can be understood by examining the cross-correlations between cells. By the Wiener-Khinchin theorem, the power spectrum of cluster activity is the Fourier transform of the signal's auto-correlation. Since the cluster activity is the sum of all single-unit spiking activity within a cluster of N cells, the autocorrelation of the cluster spiking activity will be the sum of N auto-correlations functions of the Synchronization, Oscillations, and lifNoise in Networks of Spiking Neurons 635 individual cells and N x (N - 1) cross-correlation functions among individual cells within the cluster. The ensemble activity is thus dominated by cross-correlations. In general, the excitatory "castles" are sharp relative to the broad dip in the crosscorrelation due to inhibition (see Fig. 3). In Fourier space, these relationships are reversed: broader Fourier transforms of excitatory cross-correlations are paired with narrower Fourier transforms of inhibitory cross-correlations. Superposition of such transforms leads to a peak in the 30-70 Hz range and cancellation of the 1/ f component which was present the single unit power spectrum. Interestingly, the power spectra of spike trains of individual cells within the network (Fig. 2) show no evidence of a peak in this frequency band. Diffusion of clusters disrupts any phase relationship between single unit firing and ensemble activity. The ensemble activity corresponds to the local field potential in neurophysiological recordings. While oscillations between 30 and 90 Hz have often been seen in the local field potential (or sometimes even in the EEG) measured in cortical areas in the anesthetized or awake cat and monkey, these oscillations are frequently not or only weakly visible in multi- or single-unit data (e.g., [Eeckman and Freeman, 1990, Kreiter and Singer, 1992, Gray et al., 1990, Eckhorn et al., 1993]). We here offer a general explanation for this phenomenon. Acknowledgments: We are indebted to William Softky, Wyeth Bair, Terry Sejnowski, Michael Cross, John Hopfield, and Ernst Niebur, for insightful discussions. Our research was supported by a Myron A. Bantrell Research Fellowship, the Howard Hughes Medical Institute, the National Science Foundation, the Office of Naval Research and the Air Force Office of Scientific Research. References [Amit and Tsodyks, 1991] Amit, D. J. and Tsodyks, M. V. (1991). Quantitative study of attractor neural network retrieving at low rates: 1. substrate spikes, rates and neuronal gain. Network Com., 2(3):259-273. [Bair et al., 1994] Bair, W., Koch, C., Newsome, W., and Britten, K. (1994). Power spectrum analysis of MT neurons in the behaving monkey. J. Neurosci., in press. [Bernander et al., 1994] Bernander, 0., Koch, C., and Usher, M. (1994). The effect of synchronized inputs at the single neuron level. Neural Computation, in press. [Cowan, 1982] Cowan, J. D. (1982). Spontaneous symmetry breaking in large scale nervous activity. Int. J. Quantum Chemistry, 22:1059-1082. [Cox and Lewis, 1966] Cox, D. and Lewis, P. A. W. (1966). The Statistical Analysis of Series of Events. Chapman and Hall, London. [Cross and Hohenberg, 1993] Cross, M. C. and Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Rev. Mod. Phys., 65(3):851-1112. [Eckhorn et al., 1993] Eckhorn, R., Frien, A., Bauer, R., Woelbern, T., and Harald, K. (1993). High frequency (60-90 hz) oscillations in primary visual cortex of awake monkey. Neuroreport, 4:243-246. 636 Stemmler, Usher, Koch, and Olami [Eeckman and Freeman, 1990] Eeckman, F . and Freeman, W. (1990). Correlations between unit firing and EEG in the rat olfactory system. Brain Res., 528(2):238244. [Grayet al., 1990] Gray, C. M., Engel, A. K., Konig, P., and Singer, W. (1990). Stimulus dependent neuronal oscillations in cat visual cortex: receptive field properties and feature dependence. Europ. J. Neurosci., 2:607-619. [Kreiter and Singer, 1992] Kreiter, A. K. and Singer, W. (1992). Oscillatory neuronal responses in the visual cortex of the awake macaque monkey. Europ. J. Neurosci., 4:369-375. [Lowen and Teich, 1993] Lowen, S. B. and Teich, M. C. (1993). Fractal renewal processes generate Iff noise. Phys. Rev. E, 47(2):992-1001. [Mandelbrot, 1983] Mandelbrot, B. B. (1983). The fractal geometry of nature. W. H. Freeman, New York. [Nelson et al., 1992] Nelson, J. I., Salin, P. A., Munk, M. H.-J., Arzi, M., and Bullier, J. (1992). Spatial and temporal coherence in cortico-cortical connections: A cross-correlation study in areas 17 and 18 in the cat. Visual Neuroscience, 9:21-38. [Newell and Whitehead, 1969] Newell, A. C. and Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection. J. Fluid Mech., 38:279-303. [Softky and Koch, 1993] Softky, W. R. and Koch, C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci., 13(1):334-350. [Teich, 1989] Teich, M. C. (1989). Fractal character of the auditory neural spike train. IEEE Trans. Biomed. Eng., 36(1):150-160. [Usher et al., 1994] Usher, M., Stemmler, M., Koch, C., and Olami, Z. (1994). Network amplification of local fluctuations causes high spike rate variability, fractal firing patterns, and oscillatory local field potentials. Neural Computation, in press. PART V CONTROL, NAVIGATION, AND PLANNING	1993	61
816	What Does the Hippocampus Compute?: A Precis of the 1993 NIPS Workshop Mark A. Gluck Center for Molecular and Behavioral Neuroscience Rutgers University Newark, NJ 07102 gluck@pavlov.rutgers.edu Computational models of the hippocampal-region provide an important method for understanding the functional role of this brain system in learning and memory. The presentations in this workshop focused on how modeling can lead to a unified understanding of the interplay among hippocampal physiology, anatomy, and behavior. Several approaches were presented. One approach can be characterized as "top-down" analyses of the neuropsychology of memory, drawing upon brain-lesion studies in animals and humans. Other models take a "bottom-up" approach, seeking to infer emergent computational and functional properties from detailed analyses of circuit connectivity and physiology (see Gluck & Granger, 1993, for a review). Among the issues discussed were: (1) integration of physiological and behavioral theories of hippocampal function, (2) similarities and differences between animal and human studies, (3) representational vs. temporal properties of hippocampaldependent behaviors, (4) rapid vs. incremental learning, (5) mUltiple vs. unitary memory systems, (5) spatial navigation and memory, and (6) hippocampal interaction with other brain systems. Jay McClelland, of Carnegie-Mellon University, presented one example of a topdown approach to theory development in his talk, "Complementary roles of neocortex and hippocampus in learning and memory" McClelland reviewed findings indicating that the hippocampus appears necessary for the initial acquisition of some forms of memory, but that ultimately all forms of memory are stored independently of the hippocampal system. Consolidation in the neocortex appears to occur over an extended period -- in humans the process appears to extend over several years. McClelland suggested that the cortex uses interleaved learning to extract the structure of events and experiences while the hippocampus provides a special system for the rapid initial storage of traces of specific events and experiences in a form that minimizes mutual interference between memory traces. According to this view, the hippocampus is necessary to avoid the catastrophic 1173 1174 Gluck interference that would result if memories were stored directly in the neocortex. Consolidation is slow to allow the gradual integration of new knowledge via continuing interleaved learning (McClelland, 1994/in press). In another example of top-down modeling, Mark Gluck of Rutgers University discussed "Stimulus representation and hippocampal function in animal and human learning." He described a computational account of hippocampal-region function in classical conditioning (Gluck & Myers, 1993; Myers & Gluck, 1994). In this model, the hippocampal region constructs new stimulus representations biased by two opponent constraints: first, to differentiate representations of stimuli which predict different future events, and second, to compress together representations of cooccurring or redundant stimuli. This theory accurately describe the role of the hippocampal region in a wide range of conditioning paradigms. Gluck also presented an extension of this theory which suggests that stimulus compression may arise from the operation of circuitry in the superficial layers of entorhinal cortex, whereas stimulus differentiation may arise from the operation of constituent circuits of the hippocampal formation. Discussing more physiologically-motivated "bottom-up" research, Michael Hasselmo, of Harvard University, talked about "The septohippocampal system: Feedback regulation of cholinergic modulation." Hasselmo presented a model in which feedback regulation sets appropriate dynamics for learning of novel input or recall of familiar input. This model extends previous work on cholinergic modulation of the piriform cortex (Hasselmo, 1993; Hasselmo, 1994). This model depends upon a comparison in region CAl between self-organized input from entorhinal cortex and recall of patterns of activity associated with CA3 input. When novel afferent input is presented, the inputs to CA 1 do not match, and cholinergic modulation remains high, allowing storage of a new association. For familiar input, the match between input patterns suppresses modulation, allowing recall dynamics dominated by input from CA3. Michael Recce and Neil Burgess, from England, presented their work on "Using phase coding and wave packets to represent places." They are attempting to model the spatial behavior of rats in terms of the firing of single cells in the hippocampus. A reinforcement signal enables a set of "goal cells" to learn a population vector encoding the direction of the rat from the goal. This is achieved by exploiting the apparent phase-coding of place cell firing, and the presence of head-direction cells. The model shows rapid latent-learning and robust navigation to previously encountered goal locations (Burgess, O'Keefe, & Recce, 1993; Burgess, Recce, & 0' Keefe, 1994). Spatial trajectories and cell firing characteristics compare well with experimental data. Richard Granger, of U .C. Irvine, was originally scheduled to talk on "Distinct biology and computation of entorhinal, dentate, CA3 and CAl." Granger and colleagues have noted that synaptic changes in each component of the hippocampus (i.e., DG, CA3 and CAl) exhibit different time courses, specificities, and reversibility. As such, they suggest that subtypes of memory operate serially, in an What Does the Hippocampus Compute?: A Precis of the 1993 NIPS Workshop 1175 "assembly line" of specialized functions, each of which adds a unique aspect to the processing of memories (Granger et al, 1994). In other talks, Bruce McNaughton of the University of Arizona discussed models of spatial navigation (McNaughton et aI, 1991) and William Levy from the University of Virginia presented a theory of how sparse recurrence of CA3 and several other, less direct feedback systems, leads to an ability to learn and compress sequences (Levy, 1989). Mathew Shapiro, of McGill University, had been scheduled to talk on computing locations and trajectories with simulated hippocampal place fields. References Burgess N, O'Keefe 1 & Recce M (1993) Using hippocampal "place cells" for navigation, exploiting phase coding, in: Hanson S 1, Giles C L & Cowan 1 D (eds.) Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann. Burgess N, Recce M and O'Keefe 1 (1994) A model of hippocampal function, Neural Networks, Special Issue on Neurodynamics and Behavior, to be published. Gluck, M. and Granger, R. (1993). Computational models of the neural bases of learning and memory. Annual Review of Neuroscience. 16, 667-706. Gluck, M., & Myers, C. (1993). Hippocampal mediation of stimulus representation: A computational theory. Hippocampus, 3., 491-516. Granger, R., Whitson, 1., Larson, 1. and Lynch, G. (1994). Non-Hebbian properties of L TP enable high-capacity encoding of temporal sequences. Proc. Nat'l. Acad. Sci., (in press). Hasselmo, M.E. (1993) Acetylcholine and learning in a cortical associative memory. Neural Computation 5,32-44. Hasselmo, M.E. (1994) Runaway synaptic modification in models of cortex: Implications for Alzheimer's disease. Neural Networks, in press. Levy, W. B (1989) A computational approach to hippocampal function. In: Computational Models of Learning in Simple Neural Systems. (R.D. Hawkins and G.H. Bower, Eds.), New York: Academic Press, pp. 243-305. McClelland, 1. L. (1994/in press). The organization of memory: A parallel distributed processing perspective. Revue Neurologique, Masson, Paris McNaughton, B., Chen, L., & Markus, E. (1991). "Dead reckoning", landmark learning, and the sense of direction: A neurophysiological and computational hypothesis. 10urnal of Cognitive Neuroscience, 3.(2), 190-202.	1993	62
817	Unsupervised Learning of Mixtures of Multiple Causes in Binary Data Eric Saund Xerox Palo Alto Research Center 3333 Coyote Hill Rd., Palo Alto, CA, 94304 Abstract This paper presents a formulation for unsupervised learning of clusters reflecting multiple causal structure in binary data. Unlike the standard mixture model, a multiple cause model accounts for observed data by combining assertions from many hidden causes, each of which can pertain to varying degree to any subset of the observable dimensions. A crucial issue is the mixing-function for combining beliefs from different cluster-centers in order to generate data reconstructions whose errors are minimized both during recognition and learning. We demonstrate a weakness inherent to the popular weighted sum followed by sigmoid squashing, and offer an alternative form of the nonlinearity. Results are presented demonstrating the algorithm's ability successfully to discover coherent multiple causal representat.ions of noisy test data and in images of printed characters. 1 Introduction The objective of unsupervised learning is to identify patterns or features reflecting underlying regularities in data. Single-cause techniques, including the k-means algorithm and the standard mixture-model (Duda and Hart, 1973), represent clusters of data points sharing similar patterns of Is and Os under the assumption that each data point belongs to, or was generated by, one and only one cluster-center; output activity is constrained to sum to 1. In contrast, a multiple-cause model permits more than one cluster-center to become fully active in accounting for an observed data vector. The advantage of a multiple cause model is that a relatively small number 27 28 Saund of hidden variables can be applied combinatorially to generate a large data set. Figure 1 illustrates with a test set of nine 121-dimensional data vectors. This data set reflects two independent processes, one of which controls the position of the black square on the left hand side, the other controlling the right. While a single cause model requires nine cluster-centers to account for this data, a perspicuous multiple cause formulation requires only six hidden units as shown in figure 4b. Grey levels indicate dimensions for which a cluster-center adopts a "don't-know /don't-care" assertion . ••••••••• Figure 1: Nine 121-dimensional test data samples exhibiting multiple cause structure. Independent processes control the position of the black rectangle on the left and right hand sides. While principal components analysis and its neural-network variants (Bourlard and Kamp, 1988; Sanger, 1989) as well as the Harmonium Boltzmann Machine (Freund and Haussler, 1992) are inherently multiple cause models, the hidden representations they arrive at are for many purposes intuitively unsatisfactory. Figure 2 illustrates the principal components representation for the test data set presented in figure 1. Principal components is able to reconstruct the data without error using only four hidden units (plus fixed centroid), but these vectors obscure the compositional structure of the data in that they reveal nothing about the statistical independence of the left and right hand processes. Similar results obtain for multiple cause unsupervised learning using a Harmonium network and for a feedforward network using the sigmoid nonlinearity. We seek instead a multiple cause formulation which will deliver coherent representations exploiting "don't-know/don't-care" weights to make explicit the statistical dependencies and independencies present when clusters occur in lower-dimensional subspaces of the full J -dimensional data space. Data domains differ in ways that underlying causal processes interact. The present discussion focuses on data obeying a WRITE-WHITE-AND-BLACK model, under which hidden causes are responsible for both turning "on" and turning "off" the observed variables. a b Figure 2: Principal components representation for the test data from figure 1. (a) centroid (white: -1, black: 1). (b) four component vectors sufficient to encode the nine data points. (lighter shadings: Cj,k < 0; grey: Cj,k = 0; darker shading: Cj,/.: > 0). Unsupervised Learning of Mixtures of Multiple Causes in Binary Data 29 2 Mixing Functions A large class of unsupervised learning models share the architecture shown in figure 3. A binary vector Di = (di ,l,di ,2, ... di,j, ... di,J) is presented at the data layer, and a measurement, or response vector mi = (mi,l, mi,2, ... mi ,k, ... mi ,K) is computed at the encoding layer using "weights" Cj,k associating activity at data dimension j with activity at hidden cluster-center k. Any activity pattern at the encoding layer can be turned around to compute a prediction vector ri = (ri,l" ri,2, ... ri,j, ... ri,J) at the data layer. Different models employ different functions for performing the measurement and prediction mappings, and give different interpretations to the weights. Common to most models is a learning procedure which attempts to optimize an objective function on errors between data vectors in a training set, and predictions of these data vectors under their respective responses at the encoding layer. encoding layer ( cluster-centers) data layer d j (observed data) r. (predicted) J pMietion Figure 3: Architecture underlying a large class of unsupervised learning models. The key issue is the mixing function which specifies how sometimes conflicting predictions from individual hidden units combine to predict values on the data dimensions. Most neural-network formulations, including principal components variants and the Boltzmann Machine, employ linearly weighted sum of hidden unit activity followed by a squashing, bump, or other nonlinearity. This form of mixing function permits an error in prediction by one cluster center to be cancelled out by correct predictions from others without consequence in terms of error in the net prediction. As a result, there is little global pressure for cluster-centers to adopt don't-know values when they are not quite confident in their predictions. Instead, a mult.iple cause formulation delivering coherent cluster-centers requires a form of nonlinearit.y in which active disagreement must result in a net "uncertain" or neutral prediction that results in nonzero error. 30 Saund 3 Multiple Cause Mixture Model Our formulation employs a zero-based representation at the data layer to simplify the mathematical expression for a suitable mixing function. Data values are either 1 or -1; the sign of a weight Cj ,k indicates whether activity in cluster-center k predicts a 1 or -1 at data dimension j, and its magnitude (ICj,kl ~ 1) indicates strength of belief; Cj ,k = 0 corresponds to "don't-know /don't-care" (grey in figure 4b). The mixing function takes the form, L mi ,k(-c),k) II (1 + m"kCj,k) - 1 + L mi,kc) ,k I- II (1 m"kCj,k) r.,) = k <".<0 k <". <0 k <".>0 k <".>0 This formula is a computationally tractable approximation to an idealized mixing function created by linearly interpolating boundary values on the extremes of mi,k E {O, I} and Cj,k E {-I, 0, I} rationally designed to meet the criteria outlined above. Both learning and measurement operate in the context of an objective function on predictions equivalent to log-likelihood. The weights Cj,k are found through gradient ascent in this objective function, and at each training step the encoding mi of an observed data vector is found by gradient ascent as well. 4 Experimental Results Figure 4 shows that the model converges to the coherent multiple cause representation for the test data of figure 1 starting with random initial weights. The model is robust with respect to noisy training data as indicated in figure 5. In figure 6 the model was trained on data consisting of 21 x 21 pixel images of registered lower case characters. Results for J( = 14 are shown indicating that the model has discovered statistical regularities associated with ascenders, descenders, circles, etc. a b ...----.-Figure 4: Multiple Cause Mixture Model representation for the test data from figure 1. (a) Initial random cluster-centers. (b) Cluster-centers after seven training iterations (white: Cj,k = -1; grey: Cj,k = 0; black: Cj,k = 1). Unsupervised Learning of Mixtures of Multiple Causes in Binary Data 31 5 Conclusion Ability to compress data, and statistical independence of response activities (Barlow, 1989), are not the only criteria by which to judge the success of an encoder network paradigm for unsupervised learning. For many purposes, it is equally important that hidden units make explicit statistically salient structure arising from causally distinct processes. The difficulty lies in getting the internal knowledge-bearing entities sensibly to divvy up responsibility for training data not just pointwise, but dimensionwise. Mixing functions based on linear weighted sum of activities (possibly followed by a nonlinearity) fail to achieve this because they fail to pressure the hidden units into giving up responsibility (adopting "don't know" values) for data dimensions on which they are prone to be incorrect. We have outlined criteria, and offered a specific functional form, for nonlinearly combining beliefs in a predictive mixing function such that statistically coherent hidden representations of multiple causal structure can indeed be discovered in binary data. References Barlow, H.; [1989], "Unsupervised Learning," Neural Computation, 1: 295-31l. Bourlard, H., and Kamp, Y.; [1988], Auto-Association by Multilayer Perceptrons and Singular Value Decomposition," Biological Cybernetics, 59:4-5, 291-294. Duda, R., and Hart, P.; [1973], Pattern Classification and Scene Analysis, Wiley, New York. Foldiak, P.; [1990], "Forming sparse representations by local anti-Hebbian learning," Biological Cybernetics, 64:2, 165-170. Freund, Y., and Haussler, D.; [1992]' "Unsupervised learning of distributions on binary vectors using two-layer networks," in Moody, J., Hanson, S., and Lippman, R., eds, Advances in Neural Information Processing Systems 4, Morgan Kauffman, San Mateo, 912-919. Nowlan, S.; [1990], "Maximum Likelihood Competitive Learning," in Touretzky, D., ed., Advances in Neural Information Processing Systems 2, Morgan Kauffman, San Mateo, 574-582. Sanger, T.; [1989], "An Optimality Principle for Unsupervised Learning," in Touretzky, D., ed., Advances in Neural Information Processing Systems, Morgan Kauffman, San Mateo, 11-19. 32 Saund c a b observpd data d, • • • • • nlf'a~ l1I cme nt s 1n" k predictions r, • '., . .. , .;.' .~ x ::.:,;,; • .. , .... , .. , ,';, . '.' .:: ' .;;., • • • Figure 5: Multiple Cause Mixture Model results for noisy training data. (a) Five test data sample suites with 10% bit-flip noise. Twenty suites were used to train from random initial cluster-centers, resulting in the representation shown in (b) . (c) Left: Five test data samples di ; Middle: Numerical activities mi,k for the most active cluster-centers (the corresponding cluster-center is displayed above each mi,k value); Right: reconstructions (predictions) ri based on the activities. N ot.e how these "clean up" the noisy samples from which they were computed. b Unsupervised Learning of Mixtures of Multiple Causes in Binary Data 33 a Figure 6: (a) Training set of twenty-six 441-dimensional binary vectors. (b) Multiple Cause Mixt.ure Model representation at J{ = 14. (c) Left: Five test data samples di ; Middle: Numerical activities mi,k for the most active cluster-centers (the corresponding cluster-center is displayed above each mi,k value); Right: reconstructions (predictions) ri based on the activities. 34 Saund observed data d; measurements m;,k predictions ri c	1993	63
818	A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation Kenji Doya* Allen I. Selverston Department of Biology University of California, San Diego La Jolla, CA 92093-0357, USA Abstract Peter F. Rowat A gradient descent algorithm for parameter estimation which is similar to those used for continuous-time recurrent neural networks was derived for Hodgkin-Huxley type neuron models. Using membrane potential trajectories as targets, the parameters (maximal conductances, thresholds and slopes of activation curves, time constants) were successfully estimated. The algorithm was applied to modeling slow non-spike oscillation of an identified neuron in the lobster stomatogastric ganglion. A model with three ionic currents was trained with experimental data. It revealed a novel role of A-current for slow oscillation below -50 mY. 1 INTRODUCTION Conductance-based neuron models, first formulated by Hodgkin and Huxley [10], are commonly used for describing biophysical mechanisms underlying neuronal behavior. Since the days of Hodgkin and Huxley, tens of new ionic channels have been identified [9]. Accordingly, recent H-H type models have tens of variables and hundreds of parameters [1, 2]. Ideally, parameters of H-H type models are determined by voltage-clamp experiments on individual ionic currents. However, these experiments are often very difficult or impossible to carry out. Consequently, many parameters must be hand-tuned in computer simulations so that the model behavior resembles that of the real neuron. However, a manual search in a high dimensional current address: The Salk Institute, CNL, P.O. Box 85800, San Diego, CA 92186-5800. 566 A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 567 I Figure 1: A connectionist's view of the H-H neuron model. parameter space is very unreliable. Moreover, even if a good match is found between the model and the real neuron, the validity of the parameters is questionable because there are, in general, many possible settings that lead to apparently the same behavior. We propose an automatic parameter tuning algorithm for H-H type neuron models [5]. Since a H-H type model is a network of sigmoid functions, multipliers, and leaky integrators (Figure 1), we can tune its parameters in a manner similar to the tuning of connection weights in continuous-time neural network models [6, 12]. By training a model from many initial parameter points to match the experimental data, we can systematically estimate a region in the parameter space, instead of a single point. We first test if the parameters of a spiking neuron model can be identified from the membrane potential trajectories. Then we apply the learning algorithm to a model of slow non-spike oscillation of an identified neuron in the lobster stomatogastric ganglion [7]. The resulting model suggests a new role of A-current [3] for slow oscillation in the membrane potential range below -50 m V. 2 STANDARD FORM OF IONIC CURRENTS Historically, different forms of voltage dependency curves have been used to represent the kinetics of different ionic channels. However, in order to derive a simple, efficient learning algorithm, we chose a unified form of voltage dependency curves which is based on statistical physics of ionic channels [11] for all the ionic currents in the model. The dynamics of the membrane potential v is given by Gil = I - LIj, j (1) where G is the membrane capacitance and I is externally injected current. The j-th ionic current Ij is the product of the maximum conductance 9j, activation variable 568 Doya, Selverston, and Rowat aj, inactivation variable bj , and the difference of the membrane potential v from the reversal potential Vrj. The exponents Pi and qj represent multiplicity of gating elements in the ionic channels and are usually an integer between 0 and 4. Variables aj and bj are assumed to obey the first order differential equation (2) Their steady states ajoo and bjoo are sigmoid functions of the membrane potential 1 xoo(v) = ()' (x=aj,bj ), (3) 1 + e-~'" v-v", where Vx and Sx represent the threshold and slope of the steady state curve, respectively. The rate coefficients ka · (v) and kb · (v) have the voltage dependence [11] ]] k ( ) 1 h sx( v - vx) x v cos , tx 2 where tx is the time constant. 3 ERROR GRADIENT CALCULUS Our goal is to minimize the average error over a cycle with period T: E = ~ iT ~(v(t) - v(t»2dt, where v(t) is the target membrane potential trajectory. (4) (5) We first derive the gradient of E with respect to the model parameters ( ... , Oi, ... ) = ( ... , 9j, va], Saj' taj' ... ). In studies of recurrent neural networks, it has been shown that teacher forcing is very important in training autonomous oscillation patterns [4, 6, 12, 13]. In H-H type models, teacher forcing drives the activation and inactivation variables by the target membrane potential v(t) instead of vet) as follows. x = kx(v(t»· (-x +xoo(v(t») (x = aj,bj ). (6) We use (6) in place of (2) during training. The effect of a small change in a parameter Oi of a dynamical system x = F(X; ... , Oi, ... ), is evaluated by the variation equation . of of y = oX y + OOi' (7) (8) which is an n-dimensional linear system with time-varying coefficients [6, 12]. In general, this variation calculus requires O(n2 ) arithmetics for each parameter. However, in the case of H-H model with teacher forcing, (8) reduces to a first or second order linear system. For example, the effect of a small change in the maximum conductance 9j on the membrane potential v is estimated by (9) A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 569 where GCt) = l:k 9kak(t)Pkbk(t)Qk is the total membrane conductance. Similarly, the effect of the activation threshold va] is estimated by the equations GiJ = -G(t)y - 9jpjaj(t)pj- 1bj(t)Qj(v(t) - Vrj) Z, Z = -kaj(t) [z + 8;j {aj(t) + ajoo(t) - 2aj(t)ajoo(t)}] . (10) The solution yet) represents the perturbation in v at time t, namely 8;b~). The error gradient is then given by aE 1 fT .. av(t) OBi = T Jo (v(t) - v (t)) OBi dt. (11) 4 PARAMETER UPDATE Basically, we can use arbitrary gradient-based optimization algorithms, for example, simple gradient descent or conjugate gradient descent. The particular algorithm we used was a continuous-time version of gradient descent on normalized parameters. Because the parameters of a H-H type model have different physical dimensions and magnitudes, it is not appropriate to perform simple gradient descent on them. We represent each parameter by the default value Oi and the deviation Bi as below. (12) Then we perform gradient descent on the normalized parameters Bi . Instead of updating the parameters in batches, i.e. after running the model for T and integrating the error gradient by (11), we updated the parameters on-line using the running average of the gradient as follows. . 1.. av(t) OBi Ta.D.o; = -.D.o, + T(v(t) - v (t)) OBi oBi' Bi = -€.D.o, , (13) where Ta is the averaging time and € is the learning rate. This on-line scheme was less susceptible to 2T-periodic parameter oscillation than batch update scheme and therefore we could use larger learning rates. 5 PARAMETER ESTIMATION OF A SPIKING MODEL First, we tested if a model with random initial parameters can estimate the parameters of another moqel by training with its membrane potential trajectories. The default parameters Bi of the model was set to match the original H-H model [10] (Table 1). Its membrane potential trajectories at five different levels of current injection (I = 0,15,30,45, and 60J..lA/cm2 ) were used alternately as the target v(t). We ran 100 trials after initializing Bi randomly in [-0.5,+0.5]. In 83 cases, the error became less than 1.3 m V rms after 100 cycles of training. Figure 2a is an example of the oscillation patterns of the trained model. The mean of the normalized 570 Doya, Selverston, and Rowat Table 1: Parameters of the spiking neuron model. Subscripts L, Na and K specifies leak, sodium and potassium currents, respectively. Constants: C=1J.lF/cm2 , vNa=55mV, vK=-72mV, vL=-50mV, PNa=3, QNa=l, PK=4, QK=PL=qL=O, Llv=20mV, (=0.1, Ta = 5T. v[ a_No [ b_Na[ ________ a_K [-------.....o 10 gL gNa VaNa SaNa taNa VbNa SbNa tbNa gK VaK SaK taK 20 time (ms) (a) ()i after learning default value iii mean s.d. 0.3 mS/cm -0.017 0.252 120.0 mS/cm2 -0.002 0.248 -36.0 mV 0.006 0.033 0.1 l/mV -0.052 0.073 0.5 msec -0.103 0.154 -62.0 mV 0.012 0.202 -0.09 l/mV -0.010 0.140 12.0 msec 0.093 0.330 40.0 mS/cm2 0.050 0.264 -50.0 mV -0.021 0.136 0.06 l/mV -0.061 0.114 5.0 msec -0.073 0.168 taX saK vaK gK IbNa sbNa vbNa taNa saNa vaNa gNa gL 30 gL gNa vaNa saNa taNa vbNasbNa IbNa gK vaK saK taK (b) Figure 2: (a) The trajectory of the spiking neuron model at I = 30J.lA/cm2 • v: membrane potential (-80 to +40 mY). a and b: activation and inactivation variables (0 to 1). The dotted line in v shows the target trajectory v(t). (b) Covariance matrix of the normalized parameters Oi after learning. The black and white squares represent negative and positive covariances, respectively. A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 571 Table 2: Parameters of the DG cell model. Constants: C=1J.lF/cm2, vA=-80mV, VH= -lOmV, vL=-50mV, PA=3, qA=l, PH=l, QH=PL=qL=O, ~v=20mV, (=0.1, Ta = 2T. iJ· tuned (}i v[ t gL 0.01 0.025 mS cm gA 50 41.0 mS/cm2 a~[ VaA -12 -11.1 mV --SaA 0.04 0.022 1/mV b~ [ taA 7.0 7.0 msec VbA -62 -76 mV '_H[ SbA -0.16 -0.19 1/mV tllA 300 292 msec gH 0.1 0.039 mS/cm2 VaH -70 -75.1 mV I_L SaH -0.14 -0.11 1/mV ~ ~ """'" taH 3000 4400 msec I_A 10000 20000 30000 40000 50000 tlme(msl Figure 3: Oscillation pattern of the DG cell model. v: membrane potential (-70 to -50 mY). a and b: activation and inactivation variables (0 to 1). I: ionic currents (-1 to +1 pAlcm2 ). parameters iii were nearly zero (Table 1), which implies that the original parameter values were successfully estimated by learning. The standard deviation of each parameter indicates how critical its setting is to replicate the given oscillation patterns. From the covariance matrix of the parameters (Figure 2b), we can estimate the distribution of the solution points in the parameter space. 6 MODELING SLOW NON-SPIKE OSCILLATION Next we applied the algorithm to experimental data from the "DG cell" of the lobster stomatogastric ganglion [7]. An isolated DG cell oscillates endogenously with the acetylcholine agonist pilocarpine and the sodium channel blocker TTX. The oscillation period is 5 to 20 seconds and the membrane potential is approximately between -70 and -50 m V. From voltage-clamp data from other stomatogastric neurons [8], we assumed that A-current (potassium current with inactivation) [3] and H-current (hyperpolarization-activated slow inward current) are the principal active currents in this voltage range. The default parameters for these currents were taken from [2] (Table 2). 572 Doya, Selverston, and Rowat 2 1 o -2 ,r .. , .. ~' W If " ionic currents ./ ~ ~ ../ V .... .. .... ~ ~ -60 -40 -20 0 20 40 v (mV) Figure 4: Current-voltage curves of the DG cell model. Outward current is positive. Figure 3 is an example of the model behavior after learning for 700 cycles. The actual output v of the model, which is shown in the solid curve, was very close to the target output v*(t), which is shown in the dotted curve. The bottom three traces show the ionic currents underlying this slow oscillation. Figure 4 shows the steady state I-V curves of three currents. A-current has negative conductance in the range from -70 to -40 m V. The resulting positive feedback on the membrane potential destabilizes a quiescent state. If we rotate the I-V diagram 180 degrees, it looks similar to the I-V diagram for the H-H model; the faster outward A-current in our model takes the role of the fast inward sodium current in the H-H model and the slower inward H-current takes the role of the outward potassium current. 7 DISCUSSION The results indicate that the gradient descent algorithm is effective for estimating the parameters of H-H type neuron models from membrane potential trajectories. Recently, an automatic parameter search algorithm was proposed by Bhalla and Bower [1]. They chose only the maximal conductances as free parameters and used conjugate gradient descent. The error gradient was estimated by slightly changing each of the parameters. In our approach, the error gradient was more efficiently derived by utilizing the variation equations. The use of teacher forcing and parameter normalization was essential for the gradient descent to work. In order for a neuron to be an endogenous oscillator, it is required that a fast positive feedback mechanism is balanced with a slower negative feedback mechanism. The most popular example is the positive feedback by the sodium current and the negative feedback by the potassium current in the H-H model. Another common example is the inward calcium current counteracted by the calcium dependent outward potassium current. We found another possible combination of positive and negative feedback with the help of the algorithm: the inactivation of the outward A-current and the activation of the slow inward H-current. A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 573 Acknowledgements The authors thank Rob Elson and Thom Cleland for providing physiological data from stomatogastric cells. This study was supported in part by ONR grant N0001491-J-1720. References [1] U. S. Bhalla and J. M. Bower. Exploring parameter space in detailed single neuron models: Simulations of the mitral and granule cells of the olfactory bulb. Journal of Neurophysiology, 69:1948-1965, 1993. [2] F. Buchholtz, J. Golowasch, I. R. Epstein, and E. Marder. Mathematical model of an identified stomatogastric ganglion neuron. Journal of Neurophysiology, 67:332-340, 1992. [3] J. A. Connor, D. Walter, and R. McKown. Neural repetitive firing, modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons. Biophysical Journal, 18:81-102, 1977. [4] K. Doya. Bifurcations in the learning of recurrent neural networks. In Proceedings of 1992 IEEE International Symposium on Circuits and Systems, pages 6:2777-2780, San Diego, 1992. [5] K. Doya and A. I. Selverston. A learning algorithm for Hodgkin-Huxley type neuron models. In Proceedings of IJCNN'93, pages 1108-1111, Nagoya, Japan, 1993. [6] K. Doya and S. Yoshizawa. Adaptive neural oscillator using continuous-time back-propagation learning. Neural Networks, 2:375-386, 1989. [7] R. C. Elson and A. I. Selverston. Mechanisms of gastric rhythm generation in the isolated stomatogastric ganglion of spiny lobsters: Bursting pacemaker potential, synaptic interactions, and muscarinic modulation. Journal of Neurophysiology, 68:890-907, 1992. [8] J. Golowasch and E. Marder. Ionic currents of the lateral pyloric neuron of stomatogastric ganglion of the crab. Journal of Neurophysiology, 67:318-331, 1992. [9] B. Hille. Ionic Channels of Excitable Membranes. Sinauer, 1992. [10] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane currents and its application to conduction and excitation in nerve. Journal of Physiology, 117:500-544, 1952. [11] H. Lecar, G. Ehrenstein, and R. Latorre. Mechanism for channel gating in excitable bilayers. Annals of the New York Academy of Sciences, 264:304-313, 1975. [12] P. F. Rowat and A.I. Selverston. Learning algorithms for oscillatory networks with gap junctions and membrane currents. Network, 2:17-41, 1991. [13] R. J. Williams and D. Zipser. Gradient based learning algorithms for recurrent connectionist networks. Technical Report NU-CCS-90-9, College of Computer Science, Northeastern University, 1990.	1993	64
819	Memory-Based Methods for Regression and Classification Thomas G. Dietterich and Dietrich Wettschereck Department of Computer Science Oregon State University Corvallis, OR 97331-3202 Chris G. Atkeson MIT AI Lab 545 Technology Square Cambridge, MA 02139 Andrew W. Moore School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Memory-based learning methods operate by storing all (or most) of the training data and deferring analysis of that data until "run time" (i.e., when a query is presented and a decision or prediction must be made). When a query is received, these methods generally answer the query by retrieving and analyzing a small subset of the training data-namely, data in the immediate neighborhood of the query point. In short, memory-based methods are "lazy" (they wait until the query) and "local" (they use only a local neighborhood). The purpose of this workshop was to review the state-of-the-art in memory-based methods and to understand their relationship to "eager" and "global" learning algorithms such as batch backpropagation. There are two essential components to any memory-based algorithm: the method for defining the "local neighborhood" and the learning method that is applied to the training examples in the local neighborhood. We heard several talks on issues related to defining the "local neighborhood". Federico Girosi and Trevor Hastie reviewed "kernel" methods in classification and regression. A kernel function K(d) maps the distance d from the query point to a training example into a real value. In the well-known Parzen window approach, the kernel is a fixed-width gaussian, and a new example is classified by taking a weighted vote of the classes of all training examples, where the weights are determined by the gaussian kernel. Because of the "local" shape of the gaussian, distant training examples have essentially no influence on the classification decision. In regression problems, a common approach is to construct a linear regression fit to the data, where the squared error from each data point is weighted by the kernel. Hastie described the kernel used in the LOESS method: K(d) = (1_d3)3 (0::; d::; 1 and K(d) = 0 otherwise). To adapt to the local density of training examples, this kernel is scaled to cover the kth nearest neighbor. Many other kernels have been explored, with particular attention to bias and variance at the extremes of the 1165 1166 Dietterich, Wettschereck, Atkeson, and Moore training data. Methods have been developed for computing the effective number of parameters used by these kernel methods. Girosi pointed out that some "global" learning algorithms (e.g., splines) are equivalent to kernel methods. The kernels often have informative shapes. If a kernel places most weight near the query point, then we can say that the learning algorithm is local, even if the algorithm performs a global analysis of the training data at learning time. An open problem is to determine whether multi-layer sigmoidal networks have equivalent kernels and, if so, what their shapes are. David Lowe described a classification algorithm based on gaussian kernels. The kernel is scaled by the mean distance to the k nearest neighbors. His Variablekernel Similarity Metric (VSM) algorithm learns the weights of a weighted Euclidean distance in order to maximize the leave-one-out accuracy of the algorithm. Excellent results have been obtained on benchmark tasks (e.g., NETtalk). Patrice Simard described the tangent distance method. In optical character recognition, the features describing a character change as that character is rotated, translated, or scaled. Hence, each character actually corresponds to a manifold of points in feature space. The tangent distance is a planar approximation to the distance between two manifolds (for two characters). Using tangent distance with the nearest neighbor rule gives excellent results in a zip code recognition task. Leon Bottou also employed a sophisticated distance metric by using the Euclidean distance between the hidden unit activations of the final hidden layer in the Bell Labs "LeNet" character recognizer. A simple linear classifier (with weight decay) was constructed to classify each query. Bottou also showed that there is a tradeoff between the quality of the distance metric and the locality of the learning algorithm. The tangent distance is a near-perfect metric, and it can use the highly local firstnearest-neighbor rule. The hidden layer of the LeNet gives a somewhat better metric, but it requires approximately 200 "local" examples. With the raw features, LeNet itself requires all of the training examples. We heard several talks on methods that are local but not lazy. John Platt described his RAN (Resource Allocating Network) that learns a linear combination of radial basis functions by iterative training on the data. Bernd Fritzke described his improvements to RAN. Stephen Omohundro explained model merging, which initially learns local patches and, when the data justifies, combines primitive patches into larger high-order patches. Dietrich Wettschereck presented BNGE, which learns a set of local axis-parallel rectangular patches. Finally, Andrew Moore, Chris Atkeson, and Stefan Schaal described integrated memory-based learning systems for control applications. Moore's system applies huge amounts of cross-validation to select distance metrics, kernels, kernel widths, and so on. Atkeson advocated radical localism-all algorithm parameters should be determined by lazy, local methods. He described algorithms for obtaining confidence intervals on the outputs of local regression as well as techniques for outlier removal. One method seeks to minimize the width of the confidence intervals. Some of the questions left unanswered by the workshop include these: Are there inherent computational penalties that lazy methods must pay (but eager methods can avoid)? How about the reverse? For what problems are local methods appropriate?	1993	65
820	Neural Network Methods for Optimization Problems Arun Jagota Department of Mathematical Sciences Memphis State University Memphis, TN 38152 E-mail: jagota~nextl.msci.memst.edu In a talk entitled "Trajectory Control of Convergent Networks with applications to TSP", Natan Peterfreund (Computer Science, Technion) dealt with the problem of controlling the trajectories of continuous convergent neural networks models for solving optimization problems, without affecting their equilibria set and their convergence properties. Natan presented a class of feedback control functions which achieve this objective, while also improving the convergence rates. A modified Hopfield and Tank neural network model, developed through the proposed feedback approach, was found to substantially improve the results of the original model in solving the Traveling Salesman Problem. The proposed feedback overcame the 2n symmetric property of the TSP problem. In a talk entitled "Training Feedforward Neural Networks quickly and accurately using Very Fast Simulated Reannealing Methods", Bruce Rosen (Asst. Professor, Computer Science, UT San Antonio) presented the Very Fast Simulated Reannealing (VFSR) algorithm for training feedforward neural networks [2]. VFSR Trained networks avoid getting stuck in local minima and statistically guarantee the finding of an optimal weights set. The method can be used when network activation functions are nondifferentiable, and although often slower than gradient descent, it is faster than other Simulated Annealing methods. The performances of conjugate gradient descent and VFSR trained networks were demonstrated on a set of difficult logic problems. In a talk entitled "A General Method for Finding Solutions of Covering problems by Neural Computation", Tal Grossman (Complex Systems, Los Alamos) presented a neural network algorithm for finding small minimal covers of hypergraphs. The network has two sets of units, the first representing the hyperedges to be covered and the second representing the vertices. The connections between the units are determined by the edges of the incidence graph. The dynamics of these two types of units are different. When the parameters of the units are correctly tuned, the stable states of the system correspond to the possible covers. As an example, he found new large square free subgraphs of the hypercube. In a talk entitled "Algebraic and Grammatical Design of Relaxation Nets", Eric 1184 Neural Network Methods for Optimization Problems 1185 Mjolsness (Professor, Computer Science, Yale University) presented useful algebraic notation and computer-algebraic syntax for general "programming" with optimization ideas; and also some optimization methods that can be succinctly stated in the proposed notation. He addressed global versus local optimization, time and space cost, learning, expressiveness and scope, and validation on applications. He discussed the methods of algebraic expression (optimization syntax and transformations, grammar models), quantitative methods (statistics and statistical mechanics, multiscale algorithms, optimization methods), and the systematic design approach. In a talk entitled "Algorithms for Touring Knights", Ian Parberry (Associate Professor, Computer Sciences, University of North Texas) compared Takefuji and Lee's neural network for knight's tours with a random walk and a divide-and-conquer algorithm. The experimental and theoretical evidence indicated that the neural network is the slowest approach, both on a sequential computer and in parallel, and for the problems of generating a single tour, and generating as many tours as possible. In a talk entitled "Report on the DIMACS Combinatorial Optimization Challenge" , Arun Jagota (Asst. Professor, Math Sciences, Memphis State University) presented his work, towards the said challenge, on neural network methods for the fast approximate solution of the Maximum Clique problem. The Mean Field Annealing algorithm was implemented on the Connection Machine CM-5. A fast (twotemperature) annealing schedule was experimentally evaluated on random graphs and on the challenge benchmark graphs, and was shown to work well. Several other algorithms, of the randomized local search kind, including one employing reinforcement learning ideas, were also evaluated on the same graphs. It was concluded that the neural network algorithms were in the middle in the solution quality versus running time trade-off, in comparison with a variety of conventional methods. In a talk entitled "Optimality in Biological and Artificial Networks" , Daniel Levine (Professor, Mathematics, UT Arlington) previewed a book to appear in 1995 [1]. Then he expanded his own view, that human cognitive functioning is sometimes, but not always or even most of the time, optimal. There is a continuum from the most "disintegrated" behavior, associated with frontal lobe damage, to stereotyped or obsessive-compulsive behavior, to entrenched neurotic and bureaucratic habits, to rational maximization of some measurable criteria, and finally to the most "integrated" , self-actualization (Abraham Maslow's term) which includes both reason and intuition. He outlined an alternative to simulated annealing, whereby a network that has reached an energy minimum in some but not all of its variables can move out of it through a "negative affect" signal that responds to a comparison of energy functions between the current state and imagined alternative states. References [1] D.S. Levine & W. Elsberry, editors. Optimality in Biological and Artificial Networks? Lawrence Erlbaum Associates, 1995. [2] B. E. Rosen & J. M. Goodwin. Training hard to learn networks using advanced simulated annealing methods. In Proc. of A CM Symp. on Applied Comp ..	1993	66
821	A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction Thomas G. Dietterich Arris Pharmaceutical Corporation and Oregon State University Corvallis, OR 97331-3202 Ajay N. Jain Arris Pharmaceutical Corporation 385 Oyster Point Blvd., Suite 3 South San Francisco, CA 94080 Richard H. Lathrop and Tomas Lozano-Perez Arris Pharmaceutical Corporation and MIT Artificial Intelligence Laboratory 545 Technology Square Cambridge, MA 02139 Abstract In drug activity prediction (as in handwritten character recognition), the features extracted to describe a training example depend on the pose (location, orientation, etc.) of the example. In handwritten character recognition, one of the best techniques for addressing this problem is the tangent distance method of Simard, LeCun and Denker (1993). Jain, et al. (1993a; 1993b) introduce a new technique-dynamic reposing-that also addresses this problem. Dynamic reposing iteratively learns a neural network and then reposes the examples in an effort to maximize the predicted output values. New models are trained and new poses computed until models and poses converge. This paper compares dynamic reposing to the tangent distance method on the task of predicting the biological activity of musk compounds. In a 20-fold cross-validation, 216 A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction 217 dynamic reposing attains 91 % correct compared to 79% for the tangent distance method, 75% for a neural network with standard poses, and 75% for the nearest neighbor method. 1 INTRODUCTION The task of drug activity prediction is to predict the activity of proposed drug compounds by learning from the observed activity of previously-synthesized drug compounds. Accurate drug activity prediction can save substantial time and money by focusing the efforts of chemists and biologists on the synthesis and testing of compounds whose predicted activity is high. If the requirements for highly active binding can be displayed in three dimensions, chemists can work from such displays to design new compounds having high predicted activity. Drug molecules usually act by binding to localized sites on large receptor molecules or large enyzme molecules. One reasonable way to represent drug molecules is to capture the location of their surface in the (fixed) frame of reference of the (hypothesized) binding site. By learning constraints on the allowed location of the molecular surface (and important charged regions on the surface), a learning algorithm can form a model of the binding site that can yield accurate predictions and support drug design. The training data for drug activity prediction consists of molecules (described by their structures, i.e., bond graphs) and measured binding activities. There are two complications that make it difficult to learn binding site models from such data. First, the bond graph does not uniquely determine the shape of the molecule. The bond graph can be viewed as specifying a (possibly cyclic) kinematic chain which may have several internal degrees of freedom (i.e., rotatable bonds). The conformations that the graph can adopt, when it is embedded in 3-space, can be assigned energies that depend on such intramolecular interactions as the Coulomb attraction, the van der Waal's force, internal hydrogen bonds, and hydrophobic interactions. Algorithms exist for searching through the space of conformations to find local minima having low energy (these are called "conformers"). Even relatively rigid molecules may have tens or even hundreds of low energy conformers. The training data does not indicate which of these conformers is the "bioactive" one-that is, the conformer that binds to the binding site and produces the observed binding activity. Second, even if the bioactive conformer were known, the features describing the molecular surface-because they are measured in the frame of reference of the binding site-change as the molecule rotates and translates (rigidly) in space. Hence, if we consider feature space, each training example (bond graph) induces a family of 6-dimensional manifolds. Each manifold corresponds to one conformer as it rotates and translates (6 degrees of freedom) in space. For a classification task, a positive decision region for "active" molecules would be a region that intersects at least one manifold of each active molecule and no manifolds of any inactive molecules. Finding such a decision region is quite difficult, because the manifolds are difficult to compute. 218 Dietterich, Jain, Lathrop, and Lozano-Perez A similar "feature manifold problem" arises in handwritten character recognition. There, the training examples are labelled handwritten digits, the features are extracted by taking a digitized gray-scale picture, and the feature values depend on the rotation, translation, and zoom of the camera with respect to the character. We can formalize this situation as follows. Let Xi, i = 1, ... , N be training examples (i.e., bond graphs or physical handwritten digits), and let I(Xi) be the label associated with Xi (i.e., the measured activity of the molecule or the identity of the handwritten digit). Suppose we extract n real-valued features V( Xi) to describe object Xi and then employ, for example, a multilayer sigmoid network to approximate I(x) by j(x) = g(V(x». This is the ordinary supervised learning task. However, the feature manifold problem arises when the extracted features depend on the "pose" of the example. We will define the pose to be a vector P of parameters that describe, for example, the rotation, translation, and conformation of a molecule or the rotation, translation, scale, and line thickness of a handwritten digit. In this case, the feature vector V(x,p) depends on both the example and the pose. Within the handwritten character recognition community, several techniques have been developed for dealing with the feature manifold problem. Three existing approaches are standardized poses, the tangent-prop method, and the tangent-distance method. Jain et al. (1993a, 1993b) describe a new method-dynamic reposingthat applies supervised learning simultaneously to discover the "best" pose pi of each training example Xi and also to learn an approximation to the unknown function I(x) as j(Xi) = g(V(Xi'p;». In this paper, we briefly review each of these methods and then compare the performance of standardized poses, tangent distance, and dynamic reposing to the problem of predicting the activity of musk molecules. 2 FOUR APPROACHES TO THE FEATURE MANIFOLD PROBLEM 2.1 STANDARDIZED POSES The simplest approach is to select only one of the feature vectors V( Xi, Pi) for each example by constructing a function, Pi = S(Xi), that computes a standard pose for each object. Once Pi is chosen for each example, we have the usual supervised learning task-each training example has a unique feature vector, and we can approximate 1 by j(x) = g(V(x, S(x»). The difficulty is that S can be very hard to design. In optical character recognition, S typically works by computing some pose-invariant properties (e.g., principal axes of a circumscribing ellipse) of Xi and then choosing Pi to translate, rotate, and scale Xi to give these properties standard values. Errors committed by OCR algorithms can often be traced to errors in the S function, so that characters are incorrectly positioned for recognition. In drug activity prediction, the standardizing function S must guess which conformer is the bioactive conformer. This is exceedingly difficult to do without additional information (e.g., 3-D atom coordinates of the molecule bound in the binding A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction 219 site as determined by x-ray crystallography). In addition, S must determine the orientation of the bioactive conformers within the binding site. This is also quite difficult-the bioactive conformers must be mutually aligned so that shared potential chemical interactions (e.g., hydrogen bond donors) are superimposed. 2.2 TANGENT PROPAGATION The tangent-prop approach (Simard, Victorri, LeCun, & Denker, 1992) also employs a standardizing function S, but it augments the learning procedure with the constraint that the output of the learned function g(V( x, p)) should be invariant with respect to slight changes in the poses of the examples: II\7p g(V(x,p)) Ip=S(x) II = 0, where II . II indicates Euclidean norm. This constraint is incorporated by using the left-hand-side as a regularizer during backpropagation training. Tangent-prop can be viewed as a way of focusing the learning algorithm on those input features and hidden-unit features that are invariant with respect to slight changes in pose. Without the tangent-prop constraint, the learning algorithm may identify features that "accidentally" discriminate between classes. However, tangent-prop still assumes that the standard poses are correct. This is not a safe assumption in drug activity prediction. 2.3 TANGENT DISTANCE The tangent-distance approach (Simard, LeCun & Denker, 1993) is a variant of the nearest-neighbor algorithm that addresses the feature manifold problem. Ideally, the best distance metric to employ for the nearest-neighbor algorithm with feature manifolds is to compute the "manifold distance"-the point of nearest approach between two manifolds: This is very expensive to compute, however, because the manifolds can have highly nonlinear shapes in feature space, so the manifold distance can have many local mInIma. The tangent distance is an approximation to the manifold distance. It is computed by approximating the manifold by a tangent plane in the vicinity of the standard poses. Let Ji be the Jacobian matrix defined by (Jdik = 8V(Xi,Pi)ij8(Pih, which gives the plane tangent to the manifold of molecule Xi at pose Pi. The tangent distance is defined as where PI = S(xI) and P2 = S(X2)' The column vectors a and b give the change in the pose required to minimize the distance between the tangent planes approximating the manifolds. The values of a and b minimizing the right-hand side can be computed fairly quickly via gradient descent (Simard, personal communication). In practice, only poses close to S(xd and S(X2) are considered, but this provides 220 Dietterich, Jain, Lathrop, and Lozano-Perez more opportunity for objects belonging to the same class to adopt poses that make them more similar to each other. In experiments with handwritten digits, Simard, LeCun, and Denker (1993) found that tangent distance gave the best performance of these three methods. 2.4 DYNAMIC REPOSING All of the preceding methods can be viewed as attempts to make the final predicted output j(x) invariant with respect to changes in pose. Standard poses do this by not permitting poses to change. Tangent-prop adds a local invariance constraint. Tangent distance enforces a somewhat less local invariance constraint. In dynamic reposing, we make j invariant by defining it to be the maximum value (taken over all poses p) of an auxiliary function g: j(x) = max g(V(x,p)). p The function 9 will be the function learned by the neural network. Before we consider how 9 is learned, let us first consider how it can be used to predict the activity of a new molecule x'. To compute j(x'), we must find the pose p'. that maximizes g(V(x',p'*». We can do this by performing a gradient ascent starting from the standard pose S(x) and moving in the direction of the gradient of 9 with respect to the pose: \7plg(V(X',p'». This process has an important physical analog in drug activity prediction. If x' is a new molecule and 9 is a learned model of the binding site, then by varying the pose p' we are imitating the process by which the molecule chooses a low-energy conformation and rotates and translates to "dock" with the binding site. In handwritten character recognition, this would be the dual of a deformable template model: the template (g) is held fixed, while the example is deformed (by rotation, translation, and scaling) to find the best fit to the template. The function 9 is learned iteratively from a growing pool of feature vectors. Initially, the pool contains only the feature vectors for the standard poses of the training examples (actually, we start with one standard pose of each low energy conformation of each training example). In iteration j, we apply backpropagation to learn hypothesis gj from selected feature vectors drawn from the pool. For each molecule, one feature vector is selected by performing a forward propagation (i.e., computing 9(V(Xi' Pi»)) of all feature vectors of that molecule and selecting the one giving the highest predicted activity for that molecule. After learning gj, we then compute for each conformer the pose P1+1 that maximizes gj(V(Xi' p»: ·+1 Pi = argmax gj(V(Xi'p». p From the chemical perspective, we permit each of the molecules to "dock" to the current model gj of the binding site. ·+1 The feature vectors V(Xi,Pi ) corresponding to these poses are added to the pool of poses, and a new hypothesis gj+l is learned. This process iterates until the poses A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction 221 cease to change. Note that this algorithm is analogous to the EM procedure (Redner & Walker, 1984) in that we accomplish the simultaneous optimization of 9 and the poses {Pi} by conducting a series of separate optimizations of 9 (holding {Pi} fixed) and {pd (holding 9 fixed). We believe the power of dynamic reposing results from its ability to identify the features that are critical for discriminating active from inactive molecules. In the initial, standard poses, a learning algorithm is likely to find features that "accidentally" discriminate actives from inactives. However, during the reposing process, inactive molecules will be able to reorient themselves to resemble active molecules with respect to these features. In the next iteration, the learning algorithm is therefore forced to choose better features for discrimination. Moreover, during reposing, the active molecules are able to reorient themselves so that they become more similar to each other with respect to the features judged to be important in the previous iteration. In subsequent iterations, the learning algorithm can "tighten" its criteria for recognizing active molecules. In the initial, standard poses, the molecules are posed so that they resemble each other along all features more-or-Iess equally. At convergence, the active molecules have changed pose so that they only resemble each other along the features important for discrimination. 3 AN EXPERIMENTAL COMPARISON 3.1 MUSK ACTIVITY PREDICTION We compared dynamic reposing with the tangent distance and standard pose methods on the task of musk odor prediction. The problem of musk odor prediction has been the focus of many modeling efforts (e.g., Bersuker, et al., 1991; Fehr, et al., 1989; Narvaez, Lavine & Jurs, 1986). Musk odor is a specific and clearly identifiable sensation, although the mechanisms underlying it are poorly understood. Musk odor is determined almost entirely by steric (i.e., "molecular shape") effects (Ohloff, 1986). The addition or deletion of a single methyl group can convert an odorless compound into a strong musk. Musk molecules are similar in size and composition to many kinds of drug molecules. We studied a set of 102 diverse structures that were collected from published studies (Narvaez, Lavine & Jurs, 1986; Bersuker, et al., 1991; Ohloff, 1986; Fehr, et al., 1989). The data set contained 39 aromatic, oxygen-containing molecules with musk odor and 63 homologs that lacked musk odor. Each molecule was conformationally searched to identify low energy conformations. The final data set contained 6,953 conformations of the 102 molecules (for full details of this data set, see Jain, et al., 1993a). Each of these conformations was placed into a starting pose via a hand-written S function. We then applied nearest neighbor with Euclidean distance, nearest neighbor with the tangent distance, a feed-forward network without reposing, and a feed-forward network with the dynamic reposing method. For dynamic reposing, five iterations of reposing were sufficient for convergence. The time required to compute the tangent distances far exceeds the computation times of the other algorithms. To make the tangent distance computations feasible, we only 222 Dietterich, Jain, Lathrop, and Lozano-Perez Table 1: Results of 20-fold cross-validation on 102 musk molecules. Method Percent Correct Nearest neighbor (Euclidean distance) 75 Neural network (standard poses) 75 Nearest neighbor (Tangent distance) 79 Neural network (dynamic reposing) 91 Table 2: Neural network cross-class predictions (percent correct) N Molecular class: Standard poses Dynamic reposing 85 100 76 90 74 85 57 71 computed the tangent distance for the 200 neighbors that were nearest in Euclidean distance. Experiments with a subset of the molecules showed that this heuristic introduced no error on that subset. Table 1 shows the results of a 20-fold cross-validation of all four methods. The tangent distance method does show improvement with respect to a standard neural network approach (and with respect to the standard nearest neighbor method). However, the dynamic reposing method outperforms the other two methods substantially. An important test for drug activity prediction methods is to predict the activity of molecules whose molecular structure (i.e., bond graph) is substantially different from the molecules in the training set. A weakness of many existing methods for drug activity prediction (Hansch & Fujita, 1964; Hansch, 1973) is that they rely on the assumption that all molecules in the training and test data sets share a common structural skeleton. Because our representation for molecules concerns itself only with the surface of the molecule, we should not suffer from this problem. Table 2 shows four structural classes of molecules and the results of "class holdout" experiments in which all molecules of a given class were excluded from the training set and then predicted. Cross-class predictions from standard poses are not particularly good. However, with dynamic reposing, we obtain excellent cross-class predictions. This demonstrates the ability of dynamic reposing to identify the critical discriminating features. Note that the accuracy of the predictions generally is determined by the size of the training set (i.e., as more molecules are withheld, performance drops). The exception to this is the right-most class, where the local geometry of the oxygen atom is substantially different from the other three classes. A Comparison of Dynamic Reposing and Tangent Distance for Drug Activity Prediction 223 4 CONCLUDING REMARKS The "feature manifold problem" arises in many application tasks, including drug activity prediction and handwritten character recognition. A new method, dynamic reposing, exhibits performance superior to the best existing method, tangent distance, and to other standard methods on the problem of musk activity prediction. In addition to producing more accurate predictions, dynamic reposing results in a learned binding site model that can guide the design of new drug molecules. Jain, et al., (1993a) shows a method for visualizing the learned model in the context of a given molecule and demonstrates how the model can be applied to guide drug design. Jain, et al., (1993b) compares the method to other state-of-the-art methods for drug activity prediction and shows that feed-forward networks with dynamic reposing are substantially superior on two steroid binding tasks. The method is currently being applied at Arris Pharmaceutical Corporation to aid the development of new pharmaceutical compounds. Acknowledgements Many people made contributions to this project. The authors thank Barr Bauer, John Burns, David Chapman, Roger Critchlow, Brad Katz, Kimberle Koile, John Park, Mike Ross, Teresa Webster, and George Whitesides for their efforts. References Bersuker, I. B., Dimoglo, A. S., Yu. Gorbachov, M., Vlad, P. F., Pesaro, M. (1991). New Journal of Chemistry, 15, 307. Fehr, C., Galindo, J., Haubrichs, R., Perret, R. (1989). Helv. Chim. Acta, 72, 1537. Hansch, C. (1973). In C. J. Cavallito (Ed.), Structure-Activity Relationships. Oxford: Pergamon. Hansch, C., Fujita, T. (1964). J. Am. Chem. Soc., 86, 1616. Jain, A. N., Dietterich, T. G., Lathrop, R. H., Chapman, D., Critchlow, R. E., Bauer, B. E., Webster, T. A., Lozano-Perez, T. (1993a). A shape-based method for molecular design with adaptive alignment and conformational selection. Submitted. Jain, A., Koile, K., Bauer, B., Chapman, D. (1993b). Compass: A 3D QSAR method. Performance comparisons on a steroid benchmark. Submitted. Narvaez, J. N., Lavine, B. K., Jurs, P. C. (1986). Chemical Senses, 11, 145-156. Ohloff, G. (1986). Chemistry of odor stimuli. Experientia, 42, 271. Redner, R. A., Walker, H. F. (1984). Mixture densities, maximum likelihood, and the EM algorithm. SIAM Review, 26 (2) 195-239. Simard, P. Victorri, B., Le Cun, Y. Denker, J. (1992). Tangent Prop-A formalism for specifying selected invariances in an adaptive network. In Moody, J. E., Hanson, S. J., Lippmann, R. P. (Eds.) Advances in Neural Information Processing Systems 4. San Mateo, CA: Morgan Kaufmann. 895-903. Simard, P. Le Cun, Y., Denker, J. (1993). Efficient pattern recognition using a new transformation distance. In Hanson, S. J., Cowan, J. D., Giles, C. L. (Eds.) Advances in Neural Information Processing Systems 5, San Mateo, CA: Morgan Kaufmann. 50-58.	1993	67
822	Hoo Optimality Criteria for LMS and Backpropagation Babak Hassibi Information Systems Laboratory Stanford University Stanford, CA 94305 Ali H. Sayed Dept. of Elec. and Compo Engr. University of California Santa Barbara Santa Barbara, CA 93106 Thomas Kailath Information Systems Laboratory Stanford University Stanford, CA 94305 Abstract We have recently shown that the widely known LMS algorithm is an H OO optimal estimator. The H OO criterion has been introduced, initially in the control theory literature, as a means to ensure robust performance in the face of model uncertainties and lack of statistical information on the exogenous signals. We extend here our analysis to the nonlinear setting often encountered in neural networks, and show that the backpropagation algorithm is locally H OO optimal. This fact provides a theoretical justification of the widely observed excellent robustness properties of the LMS and backpropagation algorithms. We further discuss some implications of these results. 1 Introduction The LMS algorithm was originally conceived as an approximate recursive procedure that solves the following problem (Widrow and Hoff, 1960): given a sequence of n x 1 input column vectors {hd, and a corresponding sequence of desired scalar responses { di }, find an estimate of an n x 1 column vector of weights w such that the sum of squared errors, L:~o Idi - hi w1 2 , is minimized. The LMS solution recursively 351 352 Hassibi. Sayed. and Kailath updates estimates of the weight vector along the direction of the instantaneous gradient of the squared error. It has long been known that LMS is an approximate minimizing solution to the above least-squares (or H2) minimization problem. Likewise, the celebrated backpropagation algorithm (Rumelhart and McClelland, 1986) is an extension of the gradient-type approach to nonlinear cost functions of the form 2:~o I di - hi ( W ) 12 , where hi ( .) are known nonlinear functions (e. g., sigmoids). It also updates the weight vector estimates along the direction of the instantaneous gradients. We have recently shown (Hassibi, Sayed and Kailath, 1993a) that the LMS algorithm is an H<Xl-optimal filter, where the H<Xl norm has recently been introduced as a robust criterion for problems in estimation and control (Zames, 1981). In general terms, this means that the LMS algorithm, which has long been regarded as an approximate least-mean squares solution, is in fact a minimizer of the H<Xl error norm and not of the JI2 norm. This statement will be made more precise in the next few sections. In this paper, we extend our results to a nonlinear setting that often arises in the study of neural networks, and show that the backpropagation algorithm is a locally H<Xl-optimal filter. These facts readily provide a theoretical justification for the widely observed excellent robustness and tracking properties of the LMS and backpropagation algorithms, as compared to, for example, exact least squares methods such as RLS (Haykin, 1991). In this paper we attempt to introduce the main concepts, motivate the results, and discuss the various implications. \Ve shall, however, omit the proofs for reasons of space. The reader is refered to (Hassibi et al. 1993a), and the expanded version of this paper for the necessary details. 2 Linear HOO Adaptive Filtering \Ve shall begin with the definition of the H<Xl norm of a transfer operator. As will presently become apparent, the motivation for introducing the H<Xl norm is to capture the worst case behaviour of a system. Let h2 denote the vector space of square-summable complex-valued causal sequences {fk, 0 :::; k < oo}, viz., <Xl h2 = {set of sequences {fk} such that L f; fk < oo} k=O with inner product < {Ik}, {gd > = 2:~=o f; gk ,where * denotes complex conjugation. Let T be a transfer operator that maps an input sequence {ud to an output sequence {yd. Then the H<Xl norm of T is equal to IITII<Xl = sup IIyl12 utO,uEh 2 II u l1 2 where the notation 111/.112 denotes the h2-norm of the causal sequence {ttd, viz., 2 ~<Xl * Ilull:? = L...Jk=o ttkUk The H<Xl norm may thus be regarded as the maximum energy gain from the input u to the output y. Hoc Optimality Criteria for LMS and Backpropagation 353 Suppose we observe an output sequence {dd that obeys the following model: di = hT W + Vi (1) where hT = [hi1 hi2 hin ] is a known input vector, W is an unknown weight vector, and {Vi} is an unknown disturbance, which may also include modeling errors. We shall not make any assumptions on the noise sequence {vd (such as whiteness, normally distributed, etc.). Let Wi = F(do, di, ... , di) denote the estimate of the weight vector W given the observations {dj} from time 0 up to and including time i. The objective is to determine the functional F, and consequently the estimate Wi, so as to minimize a certain norm defined in terms of the prediction error ei = hT W - h T Wi-1 which is the difference between the true (uncorrupted) output hT wand the predicted output hT Wi -1. Let T denote the transfer operator that maps the unknowns {w - W_1, {vd} (where W-1 denotes an initial guess of w) to the prediction errors {ed. The HOO estimation problem can now be formulated as follows. Problem 1 (Optimal HOC Adaptive Problem) Find an Hoc -optimal estimation strategy Wi = F(do, d1, ... , di) that minimizes IITlloc' and obtain the resulting !~ = inf IITII!:, = inf sup :F :F w,vEh 2 (2) where Iw - w_11 2 = (w - w-1f (w - W-1), and J1- is a positive constant that reflects apriori knowledge as to how close w is to the initial guess W-1 . Note that the infimum in (2) is taken over all causal estimators F. The above problem formulation shows that HOC optimal estimators guarantee the smallest prediction error energy over all possible disturbances offixed energy. Hoc estimators are thus over conservative, which reflects in a more robust behaviour to disturbance variation. Before stating our first result we shall define the input vectors {hd exciting if, and only if, N lim L hT hi = 00 N-+oc i=O Theoreln 1 (LMS Algorithm) Consider the model (1), and suppose we wish to minimize the Hoc norm of the transfer operator from the unknowns w - W-1 and Vi to the prediction errors ei. If the input vectors hi are exciting and o < J1- < i~f h:h. tit (3) then the minimum H oo norm is !Opt = 1. In this case an optimal Hoo estimator is given by the LA-IS alg01'ithm with learning rate J1-, viz. (4) 354 Hassibi, Sayed, and Kailath In other words, the result states that the LMS algorithm is an H oo -optimal filter. Moreover, Theorem 1 also gives an upper bound on the learning rate J-t that ensures the H oo optimality of LMS. This is in accordance with the well-known fact that LMS behaves poorly if the learning rate is too large. Intuitively it is not hard to convince oneself that "'{opt cannot be less than one. To this end suppose that the estimator has chosen some initial guess W-l. Then one may conceive of a disturbance that yields an observation that coincides with the output expected from W-l, i.e. hT W-l = hT W + Vi = di In this case one expects that the estimator will not change its estimate of w, so that Wi = W-l for all i. Thus the prediction error is ei = hTw - hTwi-l = hTw - hTw-l = -Vi and the ratio in (2) can be made arbitrarily close to one. The surprising fact though is that "'{opt is one and that the LMS algorithm achieves it. What this means is that LMS guarantees that the energy of the prediction error will never exceed the energy of the disturbances. This is not true for other estimators. For example, in the case of the recursive least-squares (RLS) algorithm, one can come up with a disturbance of arbitrarily small energy that will yield a prediction error of large energy. To demonstrate this, we consider a special case of model (1) where hi is now a scalar that randomly takes on the values + 1 or -1. For this model J-t must be less than 1 and we chose the value J-t = .9. We compute the Hoo norm of the transfer operator from the disturbances to the prediction errors for both RLS and LMS. We also compute the worst case RLS disturbance, and show the resulting prediction errors. The results are illustrated in Fig. 1. As can be seen, the H OO norm in the RLS case increases with the number of observations, whereas in the LMS case it remains constant at one. Using the worst case RLS disturbance, the prediction error due to the LMS algorithm goes to zero, whereas the prediction error due to the RLS algorithm does not. The form of the worst case RLS disturbance is also interesting; it competes with the true output early on, and then goes to zero. We should mention that the LMS algorithm is only one of a family of HOO optimal estimators. However, LMS corresponds to what is called the central solution, and has the additional properties of being the maximum entropy solution and the risksensitive optimal solution (Whittle 1990, Glover and Mustafa 1989, Hassibi et al. 1993b). If there is no disturbance in (1) we have the following Corollary 1 If in addition to the assumptions of Theorem 1 there is no disturbance in {1J, then LMS guarantees II e II~:::; J-t-1Iw - w_11 2 , meaning that the prediction error converges to zero. Note that the above Corollary suggests that the larger J-t is (provided (3) is satisfied) the faster the convergence will be. Before closing this section we should mention that if instead of the prediction error one were to consider the filtered error ej,i = hjw - hjwj, then the HOO optimal estimator is the so-called normalized LMS algorithm (Hassibi et al. 1993a). Hoo Optimality Criteria for LMS and Backpropagation 355 a 2.5 .----------''-=--------, 1 0.98 0.96 0.94 0.92 0.5L-------------J o 50 0.9 0 50 (e) (d) 0.5 r------>-=--------, 0.5 \ , o " 1"'-" -0.5 -l~---------~ o 50 -1L-------------------~ o 50 Figure 1: Hoo norm of transfer operator as a function of the number of observations for (a) RLS, and (b) LMS. The true output and the worst case disturbance signal (dotted curve) for RLS are given in (c). The predicted errors for the RLS (dashed) and LMS (dotted) algorithms corresponding to this disturbance are given in (d). The LMS predicted error goes to zero while the RLS predicted error does not. 3 Nonlinear HOO Adaptive Filtering In this section we suppose that the observed sequence {dd obeys the following nonlinear model (5) where hi (.) is a known nonlinear function (with bounded first and second order derivatives), W is an unknown weight vector, and {vd is an unknown disturbance sequence that includes noise and/or modelling errors. In a neural network context the index i in hi (.) will correspond to the nonlinear function that maps the weight vector to the output when the ith input pattern is presented, i.e., hi(W) = h(x(i), w) where x(i) is the ith input pattern. As before we shall denote by Wi = :F(do, ... , di) the estimate of the weight vector using measurements up to and including time i, and the prediction error by I ei = hi(w) - hi(Wi-1) Let T denote the transfer operator that maps the unknowns/disurbances { W W -1 , { vd} to the prediction errors {e;}. Problem 2 (Optimal Nonlinear HOO Adaptive Problem) Find an Hoo-optimal estimation strategy Wi = :F(do, d1, . .. , di) that minimizes IITllooI 356 Hassibi, Sayed, and Kailath and obtain the resulting i'~ = inf IITII~ = inf sup :F :F w,vEh2 (6) Currently there is no general solution to the above problem, and the class of nonlinear functions hi(.) for which the above problem has a solution is not known (Ball and Helton, 1992). To make some headway, though, note that by using the mean value theorem (5) may be rewritten as di = hi(wi-d + ~~ T (wi_d.(w - Wi-I) + Vi (7) where wi-l is a point on the line connecting wand Wi-I. Theorem 1 applied to (7) shows that the recursion (8) will yield i' = 1. The problem with the above algorithm is that the wi's are not known. But it suggests that the i'opt in Problem 2 (if it exists) cannot be less than one. Moreover, it can be seen that the backpropagation algorithm is an approximation to (8) where wi is replaced by Wi. To pursue this point further we use again the mean value theorem to write (5) in the alternative form ohi T ) 1 T 02hi(_ di = hi(wi-d+ ow (wi-d·(w-Wi-l +2(W-Wi-d . ow2 wi-d·(w-Wi-d+Vi (9) where once more Wi-l lies on the line connecting Wi-l and w. Using (9) and Theorem 1 we have the following result. Theorem 2 (Backpropagation Algorithm) Consider backpropagation algorithm the model (5) and the ohi Wi = Wi-l + J.L Ow (wi-d(di - hi(wi-d) (10) then if the ~~i (Wi- d are exciting, and . f 1 o < J.L < In --::T=------i ill!.. ( ) ill!..( ) ow Wi-I· ow wi-l (11) then for all nonzero w, v E h2: II ~~i T (wi-d(w - wi-d II~ -----------~~=-~--~~--~~--------------- < 1 J.L-11w - w_112+ II Vi + !(w - wi_dT ~:::J (wi-d·(w - Wi-I) II~ where Hoo Optimality Criteria for LMS and Backpropagation 357 The above result means that if one considers a new disturbance v; = Vi + ~ (w Wi_I)T ~::J (Wi-I).(W - Wi-I), whose second term indicates how far hi(w) is from a first order approximation at point Wi-I, then backpropagation guarantees that the energy of the linearized prediction error ~~ T (wi-d(w - Wi-I) does not exceed the energy of the new disturbances W W-l and v:. It seems plausible that if W-I is close enough to w then the second term in v~ should be small and the true and linearized prediction errors should be close, so that we should be able to bound the ratio in (6). Thus the following result is expected, where we have defined the vectors {hd persistently exciting if, and only if, for all a E nn Theorem 3 (Local Hoc Optimality) Consider the model (5) and the backpropagation algorithm (10). Suppose that the ~':: (Wi-I) are persistently exciting, and that (11) is satisfied. Then for each ( > 0, there exist cSt, ch > 0 such that for all Iw - w-ti < cSt and all v E h2 with IVil < 82, we have , 12 II ej I 2 < 1 + ( Il-Ilw - w_112+ II v II~ The above Theorem indicates that the backpropagation algorithm is locally HOC optimal. In other words for W-l sufficiently close to w, and for sufficiently small disturbance, the ratio in (6) can be made arbitrarily close to one. Note that the conditions on wand Vi are reasonable, since if for example W is too far from W-l, or if some Vi is too large, then it is well known that backpropagation may get stuck in a local minimum, in which case the ratio in (6) may get arbitrarily large. As before (11) gives an upper bound on the learning rate Il, and indicates why backpropagation behaves poorly if the learning rate is too large. If there is no disturbance in (5) we have the following Corollary 2 If in addition to the assumptions in Theorem 3 there is no disturbance in (5), then for every ( > 0 there exists a 8 > 0 such that for all Iw - w-il < 8, the backpropagation algorithm will yield II e' II~:::; 1l-18(1 + (), meaning that the prediction error converges to zero. Moreover Wi will converge to w. Again provided (11) is satisfied, the larger Il is the faster the convergence will be. 4 Discussion and Conclusion The results presented in this paper give some new insights into the behaviour of instantaneous gradient-based adaptive algorithms. We showed that ifthe underlying observation model is linear then LMS is an HOC optimal estimator, whereas if the underlying observation model is nonlinear then the backpropagation algorithm is locally HOC optimal. The HOC optimality of these algorithms explains their inherent robustness to unknown disturbances and modelling errors, as opposed to other estimation algorithms for which such bounds are not guaranteed. 358 Hassibi, Sayed, and Kailath Note that if one considers the transfer operator from the disturbances to the prediction errors, then LMS (backpropagation) is H OO optimal (locally), over all causal estimators. This indicates that our result is most applicable in situations where one is confronted with real-time data and there is no possiblity of storing the training patterns. Such cases arise when one uses adaptive filters or adaptive neural networks for adaptive noise cancellation, channel equalization, real-time control, and undoubtedly many other situations. This is as opposed to pattern recognition, where one has a set of training patterns and repeatedly retrains the network until a desired performance is reached. Moreover, we also showed that the Hoo optimality result leads to convergence proofs for the LMS and backpropagation algorithms in the absence of disturbances. We can pursue this line of thought further and argue why choosing large learning rates increases the resistance of backpropagation to local minima, but we shall not do so due to lack of space. In conclusion these results give a new interpretation of the LMS and backpropagation algorithms, which we believe should be worthy of further scrutiny. Acknowledgements This work was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command under Contract AFOSR91-0060 and in part by a grant from Rockwell International Inc. References J. A. Ball and J. W. Helton. (1992) Nonlinear H oo control theory for stable plants. Math. Control Signals Systems, 5:233-261. K. Glover and D. Mustafa. (1989) Derivation of the maximum entropy H oo controller and a state space formula for its entropy. Int. 1. Control, 50:899-916. B. Hassibi, A. H. Sayed, and T. Kailath. (1993a) LMS is HOO Optimal. IEEE Conf. on Decision and Control, 74-80, San Antonio, Texas. B. Hassibi, A. H. Sayed, and T. Kailath. (1993b) Recursive linear estimation in Krein spaces - part II: Applications. IEEE Conf. on Decision and Control, 34953501, San Antonio, Texas. S. Haykin. (1991) Adaptive Filter Theory. Prentice Hall, Englewood Cliffs, NJ. D. E. Rumelhart, J. L. McClelland and the PDP Research Group. (1986) Parallel distributed processing: explorations in the microstructure of cognition. Cambridge, Mass. : MIT Press. P. Whittle. (1990) Risk Sensitive Optimal Control. John Wiley and Sons, New York. B. Widrow and M. E. Hoff, Jr. (1960) Adaptive switching circuits. IRE WESCON Conv. Rec., Pt.4:96-104. G. Zames. (1981) Feedback optimal sensitivity: model preference transformation, multiplicative seminorms and approximate inverses. IEEE Trans. on Automatic Control, AC-26:301-320.	1993	68
823	Classifying Hand Gestures with a View-based Distributed Representation Trevor J. Darrell Perceptual Computing Group MIT Media Lab Abstract Alex P. Pentland Perceptual Computing Group MIT Media Lab We present a method for learning, tracking, and recognizing human hand gestures recorded by a conventional CCD camera without any special gloves or other sensors. A view-based representation is used to model aspects of the hand relevant to the trained gestures, and is found using an unsupervised clustering technique. We use normalized correlation networks, with dynamic time warping in the temporal domain, as a distance function for unsupervised clustering. Views are computed separably for space and time dimensions; the distributed response of the combination of these units characterizes the input data with a low dimensional representation. A supervised classification stage uses labeled outputs of the spatio-temporal units as training data. Our system can correctly classify gestures in real time with a low-cost image processing accelerator. 1 INTRODUCTION Gesture recognition is an important aspect of human interaction, either interpersonally or in the context of man-machine interfaces. In general, there are many facets to the "gesture recognition" problem. Gestures can be made by hands, faces, or one's entire body; they can be static or dynamic, person-specific or cross-cultural. Here we focus on a subset of the general task, and develop a method for interpreting dynamic hand gestures generated by a specific user. We pose the problem as one of spotting instances of a set of known (previously trained) gestures. In this context, a gesture can be thought of as a set of hand views observed over time, or simply as a sequence of images of hands over time. These images may occur at different temporal rates, and the hand may have different spatial 945 946 Darrell and Pentland offset or gross illumination condition. We would like to achieve real- or near real-time performance with our system, so that it can be used interactively by users. To achieve this level of performance, we take advantage of the principle of using only as much "representation" as needed to perform the task. Hands are complex, 3D articulated structures, whose kinematics and dynamics are difficult to fully model. Instead of performing explicit model-based reconstruction, and attempting to extract these 3D model parameters (for example see [4, 5, 6]), we use a simpler approach which uses a set of 2D views to represent the object. Using this approach we can perform recognition on objects which are either too difficult to model or for which a model recovery method is not feasible. As we shall see below, the view-based approach affords several advantages, such as the ability to form a sparse representation that only models the poses of the hands that are relevant to the desired recognition tasks, and the ability to learn the relevant model directly from the data using unsupervised clustering. 2 VIEW-BASED REPRESENTATION Our task is to recognize spatio-temporal sequences of hand images. To reduce the dimensionality of the matching involved, we find a set of view images and a matching function such that the set of match scores of a new image with the view images is adequate for recognition. The matching function we use is the normalized correlation between the image and the set of learned spatial views. Each view represents a different pose of the object being tracked or recognized. We construct a set of views that "spans" the set of images seen in the training sequences, in the sense that at least one view matches every frame in the sequence (given a distance metric and threshold value). We can then use the view with the maximum score (minimum distance) to localize the position of the object during gesture performance, and use the ensemble response of the view units (at the location of maximal response) to characterize the actual pose of the object. Each model is based on one or more example images of a view of an object, from which mean and variance statistics about each pixel in the view are computed. The general idea of view-based representation has been advocated by Ullman [12] and Poggio [9] for representing 3-D objects by interpolating between a small set of 2-D views. Recognition using views was analyzed by Breuel, who established bounds on the number of views needed for a given error rate [3]. However the view-based models used in these approaches rely on a feature-based representation of an image, in which a "view" is the list of vertex locations of semantically relevant features. The automatic extraction of these features is not a fully solved problem. (See [2] for a nearly automated system of finding corresponding points and extracting views.) Most similar to our work is that of Murase and Nayar[8] and Turk[11] which use loworder eigenvectors to reduce the dimensionality of the signal and perform recognition. Our work differs from theirs in that we use normalized-correlation model images instead of eigenfunctions and can thus localize the hand position more directly, and we extend into the temporal domain, recognizing image sequences of gestures rather than static poses. A particular view model will have a range of parameter values of a given transformation (e.g., rotation, scale, articulation) for which the correlation score shows a roughly convex "tuning curve". If we have a set of view models which sample the transformation parameter Classifying Hand Gestures with a View-Based Distributed Representation 947 (a) (b) -"-~ _00 <> ... ..0 ... '" :ao ,.... ~ -s=-==== (c) )~!l __ ~ (d) Figure 1: (a) Three views of an eyeball: +30, O. and -30 of gaze angle. (a) Normalized correlation scores of the +30 degree view model when tracking a eyeball rotating from approximately -30 to +30 degrees of gaze angle. (b) Score for 0 degree view model. (c) Score for - 30 degree model. finely enough, it is possible to infer the actual transform parameters for new views by examining the set of model correlation scores. For example, Figure la shows three views of an eyeball that could be used for gaze tracking; one looking 30 degrees left, one looking center-on, and one looking 30 degrees to the right. The three views span a ±30 degree subspace of the gaze direction parameter. Figure I (b,c,d) shows the normalized correlation score for each view model when tracking a rotating eyeball. Since the tuning curves produced by these models are fairly broad with respect to gaze angle, one could interpolate from their responses to obtain a good estimate of the true angle. When objects are non-rigid, either constructed out of flexible materials or an articulated collection of rigid parts (like a hand), then the dimensionality of the space of possible views becomes much larger. Full coverage of the view space in these cases is usually not possible since enumerating it even with very coarse sampling would be prohibitively expensive in terms of storage and search computation required. However, many parts of a high dimensional view space may never be encountered when processing real sequences, due to unforeseen additional constraints. These may be physical (some joints may not be completely independent), or behavioral (some views may never be used in the actual communication between user and machine). A major advantage of our adaptive scheme is that it has no difficulty with sparse view spaces, and derives from the data which regions of the space are full. 948 Darrell and Pentland ( Figure 2: (a) Models automatically acquired from a sequence of images of a rotating box. (b) Normalized correlation scores for each model as a function of image sequence frame number. 3 UNSUPERVISED LEARNING OF VIEW UNITS To derive a set of new view models, we use a simple form of unsupervised clustering in which the first example forms a new view, and subsequent examples that are below a distance threshold are merged into the nearest existing view. A new view is created when an example is below the threshold distance for all views in the current set, but is above a base threshold which establishes that the object is still (roughly) being tracked. Over time, this "follow-the-Ieader" algorithm results in a family of view models that sample the space of object poses in the training data. This method is similar to those commonly used in vector quantization [7]. Variance statistics are updated for each model pixel, and can be used to exclude unreliable points from the correlation computation. For simple objects and transformations, this adaptive scheme can build a model which adequately covers the entire space of possible views. For example, for a convex rigid body undergoing aID rotation with fixed relative illumination, a relatively small number of view models can suffice to track and interpolate the position of the object at any rotation. Figures 2 illustrates this with a simple example of a rotating box. The adaptive tracking scheme was run with a camera viewing a box rotating about a fixed axis. Figure 2a shows the view models in use when the algorithm converged, and all possible rotations were matched with score greater than 0\. To demonstrate the tuning properties of each model under rotation, Figure 2b shows the correlation scores for each model plotted as a function of input frame Classifying Hand Gestures with a View-Based Distributed Representation 949 Figure 3: Four spatial views found by unsupervised clustering method on sequence containing two hand-waving gestures: side-to-side and up-down. I ~ I IT] I I m ... ~ ... c:::::J ~ ~ ~ Yt4 lx spatial temporal views views Figure4: Overview of unsupervised clustering stage to learn spatial and temporal views. An input image sequence is reduced to sequence of feature vectors which record the maximum value in a normalized correlation network corresponding to each spatial view. A similar process using temporal views reduces the spatial feature vectors to a single spatia-temporal feature vector. number of a demonstration sequence. In this sequence the box was held fixed at its initial position for the first 5 frames, and then rotated continuously from 0 to 340 degrees. The responses of each model are broadly tuned as a function of object angle, with a small number of models sufficing to represent/interpolate the object at all rotations (at least about a single axis). We ran our spatial clustering method on images of hands performing two different "waving" gestures. One gesture was a side-to-side wave, with the fingers rigid, and the other was an up-down wave, with the wrist held fixed and the fingers bending towards the camera in synchrony. Running instances of both through our view learning method, with a base threshold of Bo=0.6 and a "new model" threshold of BI = 0.7, the clustering method found 4 four spatial templates to span all of the images in the both sequences Figure 3 shows the pixel values for these four models. 950 Darrell and Pentland Figure 5: Surface plot of temporal templates found by unsupervised clustering method on sequences of two hand-waving gestures. Vertical axis is score, horizontal axis is time, and depth axis is spatial view index. 3.1 TEMPORAL VIEWS The previous sections provide a method for finding spatial views to reduce the dimensionality in a tracking task. The same method can be applied in the temporal domain as well, using a set of "temporal views". Figure 4 shows an overview of these two stages. We construct temporal views using a similar method to that used for spatial views, but with temporal segmentation cues provided by the user. Sequences of spatial-feature vector outputs (the normalized correlation scores of the spatial views) are passed as input to the unsupervised clustering method, yielding a set of temporal views. To find the distance between two sequences, we again use a normalized correlation metric, with Dynamic Time Warping (DlW) method [1, 10]. This allows the time course of a gesture to vary, as long as the same series of spatial poses is present. In this way a set of temporal views acting on spatial views which in turn act on image intensities, is created. The responses of these composi te views yield a single spatio-temporal stimulus vector which describes spatial and temporal properties of the input signal. As an example, for the "hand-waving" example shown above, two temporal views were found by the clustering method. These are shown as surface plots in Figure 5. Empirically we have found that the spatio-temporal units capture the salient aspects of the spatial and temporal variation of the hand gestures in a low-dimensional representation, so efficient classification is possible. The response of these temporal view units on an input sequence containing three instances of each gesture is shown in Figure 6. 4 CLASSIFICATION OF GESTURES The spatio-temporal units obtained by the unsupervised procedure described above are used as inputs to a supervised learning/classification stage (Figure 7(a)). We have implemented two different classification strategies, a traditional Diagonal Gaussian Classifier, and a multi-layer perceptron. Classifying Hand Gestures with a View-Based Distributed Representation 951 (a) (b) Figure 6: (a) surface plot of spatial view responses on input sequence containing three instances of each hand-waving gesture. (b) final spatio-temporal view unit response: the time-warped, normalized correlation score of temporal views on spatial view feature vectors. As an experiment, we collected 42 examples of a "hello" gesture, 26 examples of "goodbye" and 10 examples of other gestures intended to generate false alarms in the classifier. All gestures were performed by a single user under similar imaging conditions. For each trial we randomly selected half of the target gestures to train the classifier, and tested on the remaining half. (All of the conflictor gestures were used in both training and testing sets since they were few in number.) Figure 7(b) summarizes the results for the different classification strategies. The Gaussian classifier (DG) achieved an hit rate of 67%, with zero false alarms. The multi-layer perceptron (MLP) was more powerful but less conservative, with a hit rate of 86% and a false alarm rate of 5%. We found the results of the MLP classifier to be quite variable; on many of the trials the classifier was stuck in a local minima and failed to converge on the test set. Additionally there was considerable dependence on the number of units in the hidden layer; empirically we found 12 gave best performance. Nonetheless, the MLP classifier provided good performance. When we excluded the trials on which the classifier failed to converge on the training set, the performance increased to 91 % hit rate, 2% false alarm rate. 5 CONCLUSION We have demonstrated a system for tracking and recognition of simple hand gestures. Our entire recognition system, including time-warping and classification, runs in real time (over 10Hz). This is made possible through the use of a special purpose normalized correlation search co-processor. Since the dimensionality of the feature space is low, the dynamic time warping and classifications steps can be implemented on conventional workstations and still achieve real-time performance. Because of this real-time performance, our system is 952 Darrell and Pentland II hello II c:::J --... ~ CLASSIFIER .... ~ "bye" ST unit outputs Figure 7: Overview of supervised classification stage and results obtained for different types of classifiers. directly applicable to interactive "glove-free" gestural user interfaces. References [1] Bellman, R E., (1957) Dynamic Programming. Princeton, NJ: Princeton Univ. Press. [2] Beymer, D., Shashua, A., and Poggio, T., (1993) ''Example Based Image Analysis and Synthesis", MIT AI Lab Memo No. 1431 [3] Breuel, T., (1992) "View-based Recognition", IAPR Workshop on Machine Vision Applications. [4] Cipolla, R, Okamotot, Y., and Kuno, Y., (1992) "Qualitative visual interpretation of 3D hand gestures using motion parallax", IAPR Workshop on Machine Vision Applications. [5] Fukumoto, M., Mase, K., and Suenaga, Y., (1992) "Real-Time Detection of Pointing Actions for a Glove-Free Interface", IAPR Workshop on Machine Vision Applications. [6] Ishibuchi, K., Takemura, H., and Kishino, F., "Real-Time Hand Shape Recognition using Pipe-line Image Processor", (1992) IEEE Workshop on Robot and Human Communication, pp. 111-116. [7] Makhoul, J., Roucos, S., and Gish, H., (1985) "Vector Quantization in Speech Coding" Proc. IEEE, Vol. 73, No. 11, pp. 1551-1587. [8] Murase, H.,and Nayar, S. K., (1993) "Learning and Recognition of 3D Objects from Appearance", Proc. IEEE Qualitative Vision Workshop, New York City, pp. 39-49. [9] Poggio, T., and Edelman, S., (1990) "A Network that Learns to Recognize Three Dimensional Objects," Nature, Vol. 343, No. 6255, pp. 263-266. [10] Sakoe, H., and Chiba, S., (1980) "Dynamic Programming optimization for spoken word recognition", IEEE Trans. ASSP, Vol. 26, pp. 623-625. [11] Turk, M., and Pentland, A. P., (1991) "Eigenfaces for Recognition", Journal of Cognitive Neuroscience, vol. 3, pp. 71-89. [12] Ullman, S., and Basri, R, (1991)"Recognition by Linear Combinations of Models," IEEE PAMI, Vol. 13, No. 10, pp. 992-1007.	1993	69
824	Constructive Learning Using Internal Representation Conflicts Laurens R. Leerink and Marwan A. J abri Systems Engineering & Design Automation Laboratory Department of Electrical Engineering The University of Sydney Sydney, NSW 2006, Australia Abstract We present an algorithm for the training of feedforward and recurrent neural networks. It detects internal representation conflicts and uses these conflicts in a constructive manner to add new neurons to the network. The advantages are twofold: (1) starting with a small network neurons are only allocated when required; (2) by detecting and resolving internal conflicts at an early stage learning time is reduced. Empirical results on two real-world problems substantiate the faster learning speed; when applied to the training of a recurrent network on a well researched sequence recognition task (the Reber grammar), training times are significantly less than previously reported. 1 Introduction Selecting the optimal network architecture for a specific application is a nontrivial task, and several algorithms have been proposed to automate this process. The first class of network adaptation algorithms start out with a redundant architecture and proceed by pruning away seemingly unimportant weights (Sietsma and Dow, 1988; Le Cun et aI, 1990). A second class of algorithms starts off with a sparse architecture and grows the network to the complexity required by the problem. Several algorithms have been proposed for growing feedforward networks. The upstart algorithm of Frean (1990) and the cascade-correlation algorithm of Fahlman (1990) are examples of this approach. 279 280 Leerink and Jabri The cascade correlation algorithm has also been extended to recurrent networks (Fahlman, 1991), and has been shown to produce good results. The recurrent cascade-correlation (RCC) algorithm adds a fully connected layer to the network after every step, in the process attempting to correlate the output of the additional layer with the error. In contrast, our proposed algorithm uses the statistical properties of the weight adjustments produced during batch learning to add additional units. The RCC algorithm will be used as a baseline against which the performance of our method will be compared. In a recent paper, Chen et al (1993) presented an algorithm which adds one recurrent neuron with small weights every N epochs. However, no significant improvement in training speed was reported over training the corresponding fixed size network, and the algorithm will not be further analyzed. To the authors knowledge little work besides the two mentioned papers have applied constructive algorithms to recurrent networks. In the majority of our empirical studies we have used partially recurrent neural networks, and in this paper we will focus our attention on such networks. The motivation for the development of this algorithm partly stemmed from the long training times experienced with the problems of phoneme and word recognition from continuous speech. However, the algorithm is directly applicable to feedforward networks. The same criteria and method used to add recurrent neurons to a recurrent network can be used for adding neurons to any hidden layer of a feed-forward network. 2 Architecture In a standard feedforward network, the outputs only depend on the current inputs, the network architecture and the weights in the network. However, because of the temporal nature of several applications, in particular speech recognition, it might be necessary for the network to have a short term memory. Partially recurrent networks, often referred to as Jordan (1989) or Elman (1990) networks, are well suited to these problems. The architecture examined in this paper is based on the work done by Robinson and Fallside (1991) who have applied their recurrent error propagation network to continuous speech recognition. A common feature of all partially recurrent networks is that there is a special set of neurons called context units which receive feedback signals from a previous time step. Let the values of the context units at time t be represented by C(t). During normal operation the input vector at time t are applied to the input nodes I(t), and during the feedforward calculation values are produced at both the output nodes O(t + 1) and the context units C(t + 1). The values of the context units are then copied back to the input layer for use as input in the following time step. Several training algorithms exist for training partially recurrent neural networks, but for tasks with large training sets the back-propagation through time (Werbos, 1990) is often used. This method is computationally efficient and does not use any approximations in following the gradient. For an application where the time information is spread over T. input patterns, the algorithm simply duplicates the network T times - which results in a feedforward network that can be trained by a variation of the standard backpropagation algorithm. Constructive Learning Using Internal Representation Conflicts 281 3 The Algorithm For partially recurrent networks consisting of input, output and context neurons, the following assertions can be made: • The role of the context units in the network is to extract and store all relevant prior information from the sequence pertaining to the classification problem. • For weights entering context units the weight update values accumulated during batch learning will eventually determine what context information is stored in the unit (the sum of the weight update values is larger than the initial random weights). • We assume that initially the number of context units in the network is insufficient to implement this extraction and storage of information (we start training with a small network). Then, at different moments in time during the recognition of long temporal sequences, a context unit could be required to preserve several different contexts. • These conflicts are manifested as distinct peaks in the distribution of the weight update values during the epoch. All but the last fact follows directly from the network architecture and requires no further elaboration. The peaks in the distribution of the weight update values are a result of the training algorithm attempting to adjust the value of the context units in order to provide a context value that will resolve short-term memory requirements. After the algorithm had been developed, it was discovered that this aspect of the weight update values had been used in the past by Wynne-Jones (1992) and in the Meiosis Networks of Hanson (1990). The method of Wynne-Jones (1992) in particular is very closely related; in this case principal component analysis of the weight updates and the Hessian matrix is used to detect oscillating nodes in fully trained feed-forward networks. This aspect of backpropagation training is fully discussed in Wynne-Jones (1992), to which reader is referred for further details. The above assertions lead to the proposed training algorithm, which states that if there are distinct maxima in the distribution of weight update values of the weights entering a context unit, then this is an indication that the batch learning algorithm requires this context unit for the storage of more than one context. If this conflict can be resolved, the network can effectively store all the contexts required, leading to a reduction in training time and potentially an increase III performance. The training algorithm is given below (the mode of the distribution is defined as the number of distinct maxima): For all context units { Set N = modality ot the distribution ot weight update values; It N > 1 then { Add N-1 new context units to the network which are identical (in terms ot weighted inputs) to the current context unit. 282 Leerink and Jabri } } Adjust each of these N context units (including the original) by the weight update value determined by each maxima (the average value of the mode). Adjust all weights leaving these N context units so that the addition of the new units do not affect any subsequent layers (division by N). This ensures that the network retains all previously acquired knowledge. The main problem in the implementation of the above algorithm is the automatic detection of significant maxima in the distribution of weight updates. A standard statistical approach for the determination of the modality (the number of maxima) of a distribution of noisy data is to fit a curve of a certain predetermined order to the data. The maxima (and minima) are then found by setting the derivative to zero. This method was found to be unsuitable mainly because after curve fitting it was difficult to determine the significance of the detected peaks. It was decided that only instances of bi-modality and tri-modality were to be identified, each corresponding to the addition of one or two context units. The following heuristic was constructed: • Calculate the mean and standard deviation of the weight update values. • Obtain the maximum value in the distribution. • If there are any peaks larger than 60% of the maxima outside one standard deviation of the mean, regard this as significant. This heuristic provided adequate identification of the modalities. The distribution was divided into three areas using the mean ± the standard deviation as boundaries. Depending on the number of maxima detected, the average within each area is used to adjust the weights. 4 Discussion According to our algorithm it follows that if at least one weight entering a context unit has a multi-modal distribution, then that context unit is duplicated. In the case where multi-modality is detected in more than one weight, context units were added according to the highest modality. Although this algorithm increases the computational load during training, the standard deviation of the weight updates rapidly decreases as the network converges. The narrowing of the distribution makes it more difficult to determine the modality. In practice it was only found useful to apply the algorithm during the initial training epochs, typically during the first 20. During simulations in which strong multi-modalities were detected in certain nodes, frequently the multi-modalities would persist in the newly created nodes. In this Constructive Learning Using Internal Representation Conflicts 283 manner a strong bi-modality would cause one node to split into two, the two nodes to grow to four, etc. This behaviour was prevented by disabling the splitting of a node for a variable number of epochs after a multi-modality had been detected. Disabling this behaviour for two epochs provided good results. 5 Simulation Results The algorithm was evaluated empirically on two different tasks: • Phoneme recognition from continuous multi-speaker speech usmg the TIMIT (Garofolo, 1988) acoustic-phonetic database . • Sequence Recognition: Learning a finite-state grammar from examples of valid sequences. For the phoneme recognition task the algorithm decreased training times by a factor of 2 to 10, depending on the size of the network and the size of the training set. The sequence recognition task has been studied by other researchers in the past, notably Fahlman (1991). Fahlman compared the performance of the recurrent cascade correlation (RCC) network with that of previous results by Cleeremans et al (1989) who used an Elman (1990) network. It was concluded that the RCC algorithm provides the same or better performance than the Elman network with less training cycles on a smaller training set. Our simulations have shown that the recurrent error propagation network of Robinson and Fallside (1991), when trained with our constructive algorithm and a learning rate adaptation heuristic, can provide the same performance as the RCC architecture in 40% fewer training epochs using a training set of the same size. The resulting network has the same number of weights as the minimum size RCC network which correctly solves this problem. Constructive algorithms are often criticized in terms of efficiency, i.e. "Is the increase in learning speed due to the algorithm or just the additional degrees of freedom resulting from the added neuron and associated weights?". To address this question several simulations were conducted on the speech recognition task, comparing the performance and learning time of a network with N fixed context units to that of a network with small number of context units and growing a network with a maximum of N context units. Results indicate that the constructive algorithm consistently trains faster, even though both networks often have the same final performance. 6 Summary In this paper the statistical properties of the weight update values obtained during the training of a simple recurrent network using back-propagation through time have been examined. An algorithm has been presented for using these properties to detect internal representation conflicts during training and to use this information to add recurrent units to the network. Simulation results show that the algorithm decreases training time compared to networks which have a fixed number of context units. The algorithm has not been applied to feedforward networks, but can III principle be added to all training algorithms that operate in batch mode. 284 Leerink and Jabri References Chen, D., Giles, C.L., Sun, G.Z., Chen, H.H., Lee, Y.C., Goudreau, M.W. (1993). Constructive Learning of Recurrent Neural Networks. In 1993 IEEE International Conference on Neural Networks, 111:1196-1201. Piscataway, NJ: IEEE Press. Cleeremans, A., Servan-Schreiber, D., and McClelland, J.L. (1989). Finite State Automata and Simple Recurrent Networks. Neural Computation 1:372-381. Elman, J .L. (1990). Finding Structure in Time. Cognitive Science 14:179-21l. Fahlman, S.E. and C. Lebiere (1990). The Cascade Correlation Learning Architecture. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 2, 524-532. San Mateo, CA: Morgan Kaufmann. Fahlman, S.E. (1991). The Recurrent Cascade Correlation Architecture. Technical Report CMU-CS-91-100. School of Computer Science, Carnegie Mellon University. Frean, M. (1990). The Upstart Algorithm: A Method for Constructing and Training Feedforward Neural Networks. Neural Computation 2:198-209. Garofolo, J.S. (1988). Getting Started with the DARPA TIMIT CD-ROM: an Acoustic Phonetic Continuous Speech Database. National Institute of Standards and Technology (NIST), Gaithersburgh, Maryland. Hanson, S.J. (1990). Meiosis Networks. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 2, 533-541, San Mateo, CA: Morgan Kaufmann. Jordan, M.1. (1989). Serial Order: A Parallel, Distributed Processing Approach. In Advances in Connectionist Theory: Speech, eds. J.L. Elman and D.E. Rumelhart. Hillsdale: Erlbaum. Le Cun, Y., J .S. Denker, and S.A Solla (1990). Optimal Brain Damage. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 2, 598-605. San Mateo, CA: Morgan Kaufmann. Reber, A.S. (1967). Implicit learning of artificial grammars. Journal of Verbal Learning and Verbal Behavior 5:855-863. Robinson, A.J. and Fallside F. (1991). An error propagation network speech recognition system. Computer Speech and Language 5:259-274. Sietsma, J. and RJ.F Dow (1988). Neural Net Pruning-\Vhy and How. In IEEE International Conference on Neural Networks. (San Diego 1988), 1:325-333. Wynne-Jones, M. (1992) Node Splitting: A Constructive Algorithm for FeedForward Neural Networks. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 4, 1072-1079. San Mateo, CA: Morgan Kaufmann. Werbos, P.J. (1990). Backpropagation Through Time, How It Works and How to Do It. Proceedings of the IEEE, 78:1550-1560.	1993	7
825	Optimal Unsupervised Motor Learning Predicts the Internal Representation of Barn Owl Head Movements Terence D. Sanger Jet Propulsion Laboratory MS 303-310 4800 Oak Grove Drive Pasadena, CA 91109 Abstract (Masino and Knudsen 1990) showed some remarkable results which suggest that head motion in the barn owl is controlled by distinct circuits coding for the horizontal and vertical components of movement. This implies the existence of a set of orthogonal internal coordinates that are related to meaningful coordinates of the external world. No coherent computational theory has yet been proposed to explain this finding. I have proposed a simple model which provides a framework for a theory of low-level motor learning. I show that the theory predicts the observed microstimulation results in the barn owl. The model rests on the concept of "Optimal U nsupervised Motor Learning", which provides a set of criteria that predict optimal internal representations. I describe two iterative Neural Network algorithms which find the optimal solution and demonstrate possible mechanisms for the development of internal representations in animals. 1 INTRODUCTION In the sensory domain, many algorithms for unsupervised learning have been proposed. These algorithms learn depending on statistical properties of the input data, and often can be used to find useful "intermediate" sensory representations 614 Bam Owl Head Movements 615 u p y z Figure 1: Structure of Optimal Unsupervised Motor Learning. z is a reduced-order internal representation between sensory data y and motor commands u. P is the plant and G and N are adaptive sensory and motor networks. A desired value of z produces a motor command u = N z resulting in a new intermediate value z = GPNz. by extracting important features from the environment (Kohonen 1982, Sanger 1989, Linsker 1989, Becker 1992, for example). An extension of these ideas to the domain of motor control has been proposed in (Sanger 1993). This work defined the concept of "Optimal Unsupervised Motor Learning" as a method for determining optimal internal representations for movement. These representations are intended to model the important controllable components of the sensory environment, and neural networks are capable of learning the computations necessary to gain control of these components. In order to use this theory as a model for biological systems, we need methods to infer the form of biological internal representations so that these representations can be compared to those predicted by the theory. Discrepancies between the predictions and results may be due either to incorrect assumptions in the model, or to constraints on biological systems which prevent them from achieving optimality. In either case, such discrepancies can lead to improvements in the model and are thus important for our understanding of the computations involved. On the other hand, if the model succeeds in making qualitative predictions of biological responses, then we can claim that the biological system possesses the optimality properties of the model, although it is unlikely to perform its computations in exactly the same manner. 2 BARN OWL EXPERIMENTS A relevant set of experiments was performed by (Masino and Knudsen 1990) in the barn owl. These experiments involved microstimulation of sites in the optic tectum responsible for head movement. By studying the responses to stimulation at different sites separated by short or long time intervals, it was possible to infer the existence of distinct "channels" for head movement which could be made refractory by prior stimulation. These channels were oriented in the horizontal and vertical directions in external coordinates, despite the fact that the neck musculature of the barn owl is sufficiently complex that such orientations appear unrelated to any set 616 Sanger of natural motor coordinates. This result raises two related questions. First, why are the two channels orthogonal with respect to external Cartesian coordinates, and second, why are they oriented horizontally and vertically? The theory of Optimal Unsupervised Motor Learning described below provides a model which attempts to answer both questions. It automatically develops orthogonal internal coordinates since such coordinates can be used to minimize redundancy in the internal representation and simplify computation of motor commands. The selection of the internal coordinates will be based on the statistics of the components of the sensory data which are controllable, so that if horizontal and vertical movements are distinguished in the environment then these components will determine the orientation of intermediate channels. We can hypothesize that the horizontal and vertical directions are distinguished in the owl by their relation to sensory information generated from physical properties of the environment such as gravity or symmetry properties of the owl's head. In the simulation below, I show that reasonable assumptions on such symmetry properties are sufficient to guarantee horizontal and vertical orientations of the intermediate coordinate system. 3 OPTIMAL UNSUPERVISED MOTOR LEARNING Optimal Unsupervised Motor Learning (OUML) attempts to invert the dynamics of an unknown plant while maintaining control of the most important modes (Sanger 1993). Figure 1 shows the general structure of the control loop, where the plant P maps motor commands u into sensory outputs y = Pu, the adaptive sensory transformation G maps sensory data y into a reduced order intermediate representation z = Gy, and the adaptive motor transformation N maps desired values of z into the motor commands u = N z which achieve them. Let z = G P N z be the value of the intermediate variables after movement, and f) = P NGy be the resulting value of the sensory variables. For any chosen value of z we want z = z, so that we successfully control the intermediate variables. In (Sanger 1993) it was proposed that we want to choose z to have lower dimensionality than y and to represent only the coordinates which are most important for controlling the desired behavior. Thus, in general, f) =/; y and Ily - f)1I is the performance error. OUML can then be described as 1. Minimize the movement error 1If) - yll 2. Subject to accurate control z = z. These criteria lead to a choice of internal representation that maximizes the loop gain through the plant. Theorem 1: (Sanger 1993) For any sensory mapping G there exists a motor mapping N such t~at z = z, and [; _ E[lIy - f)1I] is mi1!.imized when G is chosen to minimize E[lly - G-1Gyll]' where G-l is such that GG-l = I. The function G is an arbitrary right inverse of G, and this function determines the asymptotic values of the unobserved modes. In other words, since G in general is dimensionality-reducing, z = Gy will not respond to all the modes in y so that dissimilar states may project to identical intermediate control variables z. The Barn Owl Head Movements 617 Plant 1 II Motor Sensory Linear Linear Eigenvectors of E[yy'l ] RBF Linear Eigenvectors of basis function outputs Polynomial Polynomial Eigenvectors of basis function outputs Figure 2: Special cases of Theorem 1. If the plant inverse is linear or can be approximated using a sum of radial basis functions or a polynomial, then simple closed-form solutions exist for the optimal sensory network and the motor network only needs to be linear or polynomial. function a- 1 G is a projection operator that determines the resulting plant output fJ for any desired value of y. Unsupervised motor learning is "optimal" when the projection surface determined by a- 1G is the best approximation to the statistical density of desired values of y. Without detailed knowledge of the plant, it may be difficult to find the general solution described by the theorem. Fortunately, there are several important special cases in which simple closed-form solutions exist. These cases are summarized in figure 2 and are determined by the class of functions to which the plant inverse belongs. If the plant inverse can be approximated as a sum of radial basis functions, then the motor network need only be linear and the optimal sensory network is given by the eigenvectors of the autocorrelation matrix of the basis function outputs (as in (Sanger 1991a)). If the plant inverse can be approximated as a polynomial over a set of basis functions (as in (Sanger 1991b)), then the motor network needs to be a polynomial, and again the optimal sensory network is given by the eigenvectors of the autocorrelation matrix of the basis function outputs. Since the model of the barn owl proposed below has a linear inverse we are interested in the linear case, so we know that the mappings Nand G need only be linear and that the optimal value of G is given by the eigenvectors of the autocorrelation matrix of the plant outputs y. In fact, it can be shown that the optimal Nand G are given by the matrices ofleft and right singular vectors of the plant inverse (Sanger 1993). Although several algorithms for iterative computation of eigenvectors exist, until recently there were no iterative algorithms for finding the left and right singular vectors. I have developed two such algorithms, called the "Double Generalized Hebbian Algorithm" (DGHA) and the "Orthogonal Asymmetric Encoder" (OAE). (These algorithms are described in detail elsewhere in this volume.) DGHA is described by: !J..G !J..NT while OAE is described by: !J..G !J..NT r(zyT - LT[zzT]G) r(zuT - LT[zzT]NT) r(zyT - LT[zzT]G) r( Gy - LT[GGT]z)uT where LT[ ] is an operator that sets the above diagonal elements of its matrix argument to zero, y = Pu, z = Gy, z = NT u, and r is a learning rate constant. 618 Sanger Neck Muscles Movement Sensors e u Motor Transform Sensory Transform N z Figure 3: Owl model, and simulation results. The "Sensory Transform" box shows the orientation tuning of the learned internal representation. 4 SIMULATION I use OUML to simulate the owl head movement experiments described in (Masino and Knudsen 1990), and I predict the form of the internal motor representation. I assume a simple model for the owl head using two sets of muscles which are not aligned with either the horizontal or the vertical direction (see the upper left block of figure 3). This model is an extreme oversimplification of the large number of muscle groups present in the barn owl neck, but it will serve to illustrate the case of muscles which do not distinguish the horizontal and vertical directions. I assume that during learning the owl gives essentially random commands to the muscles, but that the physics of head movement result in a slight predominance of either vertical or horizontal motion. This assumption comes from the symmetry properties of the owl head, for which it is reasonable to expect that the axes of rotational symmetry lie in the coronal, sagittal, and transverse planes, and that the moments of inertia about these axes are not equal. I model sensory receptors using a set of 12 oriented directionally-tuned units, each with a half-bandwidth at half-height of 15 degrees (see the upper right block of figure 3). Together, the Neck Muscles and Movement Sensors (the two upper blocks of figure 3) form the model of the plant which transforms motor commands u into sensory outputs y. Although this plant is nonlinear, it can be shown to have an approximately linear inverse on y Barn Owl Head Movements 619 Desired Direction Figure 4: Unsupervised Motor Learning successfully controls the owl head simulation. its range. The sensory units are connected through an adaptive linear network G to three intermediate units which will become the internal coordinate system z. The three intermediate units are then connected back to the motor outputs through a motor network N so that desired sensory states can be mapped onto the motor commands necessary to produce them. The sensory to intermediate and intermediate to motor mappings were allowed to adapt to 1000 random head movements, with learning controlled by DGHA. 5 RESULTS After learning, the first intermediate unit responded to the existence of a motion, and did not indicate its direction. The second and third units became broadly tuned to orthogonal directions. Over many repeated learning sessions starting from random initial conditions, it was found that the intermediate units were always aligned with the horizontal and vertical axes and never with the diagonal motor axes. The resulting orientation tuning from a typical session is shown in the lower right box of figure 3. Note that these units are much more broadly tuned than the movement sensors (the half-bandwidth at half-height is 45 degrees). The orientation of the internal channels is determined by the assumed symmetry properties of the owl head. This information is available to the owl as sensory data, and OUML allows it to determine the motor representation. The system has successfully inverted the plant, as shown in figure 4. (Masino and Knudsen 1990) investigated the intermediate representations in the owl by taking advantage of the refractory period of the internal channels. It was found that if two electrical stimuli which at long latency tended to move the owl's head in directions located in adjacent quadrants were instead presented at short latency, the second head movement would be aligned with either the horizontal or vertical axis. Figure 5 shows the general form of the experimental results, which are consistent with the hypothesis that there are four independent channels coding 620 Sanger Move 2a Move 1 Move 2b Long Interval Move 1 Move 2a iliL Move 2b Short Interval Figure 5: Schematic description of the owl head movement experiment. At long interstimulus intervals (lSI), moves 2a and 2b move up and to the right, but at short lSI the rightward channel is refractory from move 1 and thus moves 2a and 2b only have an upward component. ---I •• I or .. 11 I 0' a. "". -- .. ... '10 h. ~"""'tfII., Figure 6: Movements align with the vertical axis as the lSI shortens. a. Owl data (reprinted with permission from (Masino and Knudsen 1990». h. Simulation results. the direction of head movement, and that the first movement makes either the left, right, up, or down channels refractory. As the interstimulus interval (lSI) is shortened, the alignment of the second movement with the horizontal or vertical axis becomes more pronounced. This is shown in figure 6a for the barn owl and 6b for the simulation. If we stimulate sites that move in many different directions, we find that at short latency the second movement always aligns with the horizontal or vertical axis, as shown in figure 7a for the owl and figure 7b for the simulation. 6 CONCLUSION Optimal Unsupervised Motor Learning provides a model for adaptation in low-level motor systems. It predicts the development of orthogonal intermediate representations whose orientation is determined by the statistics of the controllable components of the sensory environment. The existence of iterative neural algorithms for both linear and nonlinear plants allows simulation of biological systems, and I have Barn Owl Head Movements 621 .... • I i ; ~ §~ a. h. I.ONG SHORT "TEaVAL ., .. " .. INTERVAL .,--"-Figure 7: At long lSI, the second movement can occur in many directions, but at short lSI will tend to align with the horizontal or vertical axis. a. Owl data (reprinted with permission from (Masino and Knudsen 1990)). h. Simulation results. shown that the optimal internal representation predicts the horizontal and vertical alignment of the internal channels for barn owl head movement. Acknowledgements Thanks are due to Tom Masino for helpful discussions as well as for allowing reproduction of the figures from (Masino and Knudsen 1990). 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 Becker S., 1992, An Information-Theoretic Unsupervised Learning Algorithm for Neural Networks, PhD thesis, Univ. Toronto Dept. Computer Science. Kohonen T., 1982, Self-organized formation of topologically correct feature maps, Biological Cybernetics, 43:59-69. Linsker R., 1989, How to generate ordered maps by maximizing the mutual information between input and output signals, Neural Computation, 1:402-411. Masino T ., Knudsen E. I., 1990, Horizontal and vertical components of head movement are controlled by distinct neural circuits in the barn owl, Nature, 345:434-437. Sanger T. D., 1989, Optimal unsupervised learning in a single-layer linear feedforward neural network, Neural Networks, 2:459-473. Sanger T. D., 1991a, Optimal hidden units for two-layer nonlinear feedforward neural networks, International Journal of Pattern Recognition and Artificial Intelligence, 5(4):545-561, Also appears in C. H. Chen, ed., Neural Networks in Pattern Recognition and Their Applications, World Scientific, 1991, pp. 43-59. Sanger T. D., 1991b, A tree-structured adaptive network for function approximation in high dimensional spaces, IEEE Trans. Neural Networks, 2(2):285-293. Sanger T. D., 1993, Optimal unsupervised motor learning, IEEE Trans. Neural Networks, in press.	1993	70
826	Amplifying and Linearizing Apical Synaptic Inputs to Cortical Pyramidal Cells. Ojvind Bernander, Christof Koch ... Computation and Neural Systems Program, California Institute of Technology, 139-74 Pasadena, Ca 91125, USA. Rodney J. Douglas Anatomical Neuropharmacology Unit, Dept. Pharmacology, Oxford, UK. Abstract Intradendritic electrophysiological recordings reveal a bewildering repertoire of complex electrical spikes and plateaus that are difficult to reconcile with conventional notions of neuronal function. In this paper we argue that such dendritic events are just an exuberant expression of a more important mechanism - a proportional current amplifier whose primary task is to offset electrotonic losses. Using the example of functionally important synaptic inputs to the superficial layers of an anatomically and electrophysiologically reconstructed layer 5 pyramidal neuron, we derive and simulate the properties of conductances that linearize and amplify distal synaptic input current in a graded manner. The amplification depends on a potassium conductance in the apical tuft and calcium conductances in the apical trunk. ·To whom all correspondence should be addressed. 519 520 Bemander, Koch, and Douglas 1 INTRODUCTION About half the pyramidal neurons in layer 5 of neocortex have long apical dendrites that arborize extensively in layers 1-3. There the dendrites receive synaptic input from the inter-areal feedback projections (Felleman and van Essen, 1991) that play an important role in many models of brain function (Rockland and Virga, 1989). At first sight this seems to be an unsatisfactory arrangement. In light of traditional passive models of dendritic function the distant inputs cannot have a significant effect on the output discharge of the pyramidal cell. The distal inputs are at least one to two space constants removed from the soma in layer 5 and so only a small fraction of the voltage signal will reach there. Nevertheless, experiments in cortical slices have shown that synapses located in even the most superficial cortical layers can provide excitation strong enough to elicit action potentials in the somata of layer 5 pyramidal cells (Cauller and Connors, 1992, 1994). These results suggest that the apical dendrites are active rather than passive, and able to amplify the signal en route to the soma. Indeed, electrophysiological recordings from cortical pyramidal cells provide ample evidence for a variety of voltage-dependent dendritic conductances that could perform such amplification (Spencer and Kandel, 1961; Regehr et al., 1993; Yuste and Tank, 1993; Pockberger, 1991; Amitai et al., 1993; Kim and Connors, 1993). Although the available experimental data on the various active conductances provide direct support for amplification, they are not adequate to specify the mechanism by which it occurs. Consequently, notions of dendritic amplification have been informal, usually favoring voltage gain, and mechanisms that have a binary (high gain) quality. In this paper, we formalize what conductance properties are required for a current amplifier, and derive the required form of their voltage dependency by analysis. We propose that current amplification depends on two active conductances: a voltage-dependent K+ conductance, gK, in the superficial part of the dendritic tree that linearizes synaptic input, and a voltage-dependent Ca 2+ conductance, gc a, in layer 4 that amplifies the result of the linearization stage. Spencer and Kandel (1961) hypothesized the presence of dendritic calcium channels that amplify distal inputs. More recently, a modeling study of a cerebellar Purkinje cell suggests that dendritic calcium counteracts attenuation of distal inputs so that the somatic response is independent of synaptic location (De Schutter and Bower, 1992). A gain-control mechanism involving both potassium and calcium has also been proposed in locust non-spiking interneurons (Laurent, 1993). In these cells, the two conductances counteract the nonlinearity of graded transmitter release, so that the output of the interneuron was independent of its membrane voltage. The principle that we used can be explained with the help of a highly simplified three compartment model (Fig. 1A). The leftmost node represents the soma and is clamped to -50 m V. The justification for this is that the time-averaged somatic voltage is remarkably constant and close to -50 m V for a wide range of spike rates. The middle node represents the apical trunk containing gCa, and the rightmost node represents the apical tuft with a synaptic induced conductance change gsyn in parallel with gK. For simplicity we assume that the model is in steady-state, and has an infinite membrane resistance, Rm. ~ 1 o Ul H Amplifying and Linearizing Apical Synaptic Inputs to Cortical Pyramidal Cells 521 150ma 9 9 Vsoma -Eea EK T Esyn ~----~--~I----~---B PaSSlve response and targets , , , , Linearized and amplified 100 gsyn (nS) 200 c Actlvation curves 30 10 O~--~--~--~----~L-~--~ -50 10 v (mV) Figure 1: Simplified model used to demonstrate the concepts of saturation, linearization, and amplification. (A) Circuit diagram. The somatic compartment was clamped to V$oma = -50 mV with ECa = 115 mV, EK = -95 mV, E$yn = 0 m V, and g = 40 nS. The membrane capacitance was ignored, since only steady state properties were studied, and membrane leak was not included for simplicity. (B) Somatic current, I$oma, in response to synaptic input. The passive response (thin dashed line) is sublinear and saturates for low values of gsyn. The linearized response (thick solid line) is obtained by introducing an inactivating potassium conductance, OK ("gA" in c). A persistent persistent OK results in a somewhat sub-linear response (thick dashed line; "gM" in c). The addition of a calcium conductance amplifies the response (thin solid line). (C) Analytically derived activation curves. The inactivating potassium conductance ("IA") was derived, but the persistent version (" IM") proved to be more stable. 522 Bemander, Koch, and Douglas 2 RESULTS Fig. 1B shows the computed relationship between the excitatory synaptic input conductance and the axial current, I soma , flowing into the somatic (leftmost) compartment. The synaptic input rapidly saturates; increasing gsyn beyond about 50 nS leads to little further increase in Isoma. This saturation is due to the EPSP in the distal compartment reducing the effective synaptic driving potential. We propose that the first goal of dendritic amplification is to linearize this relationship, so that the soma is more sensitive to the exact amount of excitatory input impinging on the apical tuft, by introducing a potassium conductance that provides a hyperpolarizing current in proportion to the degree of membrane depolarization. The voltage-dependence of such a conductance can be derived by postulating a linear relationship between the synaptic current flowing into the somatic node and the synaptic input, i.e. Isoma = constant· gsyn. In conjunction with Ohm's law and current conservation, this relation leads to a simple fractional polynominal for the voltage dependency of gK (labeled "gA" in Fig. 1C). As the membrane potential depolarizes, gK activates and pulls it back towards EK . At large depolarizations gK inactivates, similar to the "A" potassium conductance, resulting overall in a linear relationship between input and output (Fig. 1B). As the slope conductance of this particular form of gK can become negative, causing amplification of the synaptic input, we use a variant of gK that is monotonized by leveling out the activation curve after it has reached its maximum, similar to the "M" current (Fig. IC). Incorporating this non-inactivating K+ conductance into the distal compartment results in a slightly sublinear relationship between input and output (Fig. 1B). With gK in place, amplification of Isoma is achieved by introducing an inward current between the soma and the postsynaptic site. The voltage-dependency of the amplification conductance can be derived by postulating Isoma = gain · constant· gsyn' This leads to the non-inactivating gCa shown in Fig. 1C, in which the overall relationship between synaptic input and somatic output current (Fig. 1B) reflects the amplification. We extend this concept of deriving the form of the required conductances to a detailed model of a morphologically reconstructed layer 5 pyramidal cell from cat visual cortex (Douglas et al., 1991, Fig. 2A;). We assume a passive dendritic tree, and include a complement of eight common voltage-dependent conductances in its soma. 500 non-NMDA synapses are distributed on the dendritic tuft throughout layers 1, 2 and 3, and we assume a proportionality between the presynaptic firing frequency fin and the time-averaged synaptic induced conductance change. When fin is increased, the detailed model exhibits the same saturation as seen in the simple model (Fig. 2B). Even if an 500 synapses are activated at fin = 500 Hz only 0.65 nA of current is delivered to the soma. This saturation is caused when the synaptic input current flows into the high input resistances of the distal dendrites, thereby reducing the synaptic driving potential. Layer 1 and 2 input together can contribute a maximum of 0.25 nA to the soma. This is too little current to cause the cell to spike, in contrast with the experimental evidence (Cauller and Connors, 1994), in which spike discharge was evoked reliably. Electrotonic losses make only a minor contribution to the small somatic signal. Even when the membrane leak current is eliminated by setting Rm to infinity, Isoma only increases a mere 2% to 0.66 nA. Amplifying and Linearizing Apical Synaptic Inputs to Cortical Pyramidal Cells 523 ~8r2layer1 ·<v layer3 1100 urn l: ./ Layer 4 LayerS C Activation Curves c 1 0 M .jJ III :> M .jJ u III .-i III § M .jJ U III '"' 'H Q60 V (mV) B Current deli vered to soma 2~----~------~----~------~ § 1 til H 40 30 20 10 00 D N :r: '-" .jJ ~ 0 'H Layer 5: Passive dendrlte Layers 1-3: Llneanzed and ampllfled ~--- ~~ ~-~-~~ ~~ Layers 1-3: PaSSive dendrlte 100 fln (Hz) Input-Output behavior 50 Active, predicted Active dendrite Passive dendrlte 150 200 200 Figure 2: Amplification in the detailed model. (A) The morphology of this layer V pyramidal cell was reconstructed from a HRP-stained cell in area 17 of the adult cat (Douglas et ai., 1991). The layers are marked in alternating black and grey. The boundaries between superficial layers are not exact, but rough estimates and were chosen at branch points; a few basal dendrites may reach into layer 6. Axon not shown. (B) Current delivered to the soma by stimulation of 500 AMPA synapses throughout either layer 5 or layers 1-3. (C) Derived activation curves for gK and gCa' Sigmoidal fits of the form g(V) = 1/(1 + e(Vhcll/-V)/K), resulted in ]{K = 3.9 mY, Vhalj,K = -51 mY, KCa = 13.7 mY, Vhalj,Ca = -14 mY. (D) Output spike rate as a function of input activation rate of 500 AMPA synapses in layers 1-3, with and without the derived conductances. The dashed line shows the lout rate predicted by using the linear target Isoma as a function of lin in combination with the somatic f - I relationship. 524 Bemander, Koch, and Douglas Vrn (rnV) o -50 100 200 t (rnsec) Figure 3: Dendritic calcium spikes. All-or-nothing dendritic Ca2+ calcium spikes can be generated by adding a voltage-independent but Ca 2+ -dependent K+ conductance to the apical tree with gma~ = 11.4 nS. The trace shown is in response to sustained intradendritic current injection of 0.5 nA. For clamp currents of 0.3 nA or less, no calcium spikes are triggered and only single somatic spikes are obtained (not shown). These currents do not substantially affect the current amplifier effect. By analogy with the simple model of Fig. 1, we eliminate the saturating response by introducing a non-inactivating form of gK spread evenly throughout layers 1-3. The resulting linearized response is amplified by a Ca2+ conductance located at the base of the apical tuft, where the apical dendrite crosses from layer 4 to layer 3 (Fig. 2A). This is in agreement with recent calcium imaging experiments, which established that layer 5 neocortical pyramidal cells have a calcium hot spot in the apical tree about 500-600 pm away from the soma (Tank et ai., 1988). Although the derivation of the voltage-dependency of these two conductances is more complicated than in the three compartment model, the principle of the derivation is similar (Bernander, 1993, Fig. 2C;). We derive a Ca2+ conductance, for a synaptic current gain of two, resembling a non-inactivating, high-threshold calcium conductance. The curve relating synaptic input frequency and axial current flowing into the soma (Fig. 2B) shows both the linearized and amplified relationships. Once above threshold, the model cell has a linear current-discharge relation with a slope of about 50 spikes per second per nA, in good agreement with experimental observations in vitro (Mason and Larkman, 1990) and in vivo (Ahmed et a/., 1993). Given a sustained synaptic input frequency, the somatic f-I relationship can be used to convert the synaptic current flowing into the soma 130ma into an equivalent output frequency (Abbott, 1991; Powers et a/., 1992; Fig. 2D). This simple transformation accounts for all the relevant nonlinearities, including synaptic saturation, interaction and the threshold mechanism at the soma or elsewhere. We confirmed the validity of our transformation method by explicitly computing the expected relationship between lin and lout, without constraining the somatic potential, and comparing the two. Qualitatively, both methods lead to very similar results (Fig. 2D): in the Amplifying and Linearizing Apical Synaptic Inputs to Cortical Pyramidal Cells 525 presence of dendritic gCa superficial synaptic input can robustly drive the cell, in a proportional manner over a large input range. The amplification mechanism derived above is continuous in the input rate. It does not exhibit the slow calcium spikes described in the literature (Pockberger, 1991; Amitai et ai., 1993; Kim and Connors, 1993). However, it is straightforward to add a calcium-dependent potassium conductance yielding such spikes. Incorporating such a conductance into the apical trunk leads to calcium spikes (Fig. 3) in response to an intradendritic current injection of 0.4 nA or more, while for weaker inputs no such events are seen. In response to synaptic input to the tuft of 120 Hz or more, these spikes are activated, resulting in a moderate depression (25% or less) of the average output rate, lout (not shown). In our view, the function of the dendritic conductances underlying this all-or-none voltage event is the gradual current amplification of superficial input, without amplifying synaptic input to the basal dendrites (Bernander, 1993). Because gCa depolarizes the membrane, further activating gCa, the gain of the current amplifier is very sensitive to the density and shape of the dendritic gCa. Thus, neuromodulators that act upon gCa control the extent to which cortical feedback pathways, acting via superficial synaptic input, have access to the output of the cell. Acknowledgements This work was supported by the Office of Naval Research, the National Institute of Mental Health through the Center for Neuroscience, the Medical Research Council of the United Kingdom, and the International Human Frontier Science Program. References [1] L.F. Abbott. Realistic synaptic inputs for model neuronal networks. Network, 2:245-258, 1991. [2] B. Ahmed, J .C. Anderson, R.J. Douglas, K.A.C. Martin, and J .C. Nelson. The polyneuronal innervation of spiny steallate neurons in cat visual cortex. Submitted, 1993. [3] Y. Amitai, A. Friedman, B.W. Connors, and M.J. Gutnick. Regenerative activity in apical dendrites of pyramidal cells in neocortex. Cerebral Cortex, 3:26-38, 1993. [4] 6 Bernander. Synaptic integration and its control in neocortical pyramidal cells. May 1993. Ph.D. thesis, California Institute of Technology. [5] L.J. CauUer and B.W. Connors. Functions of very distal dendrites: experimental and computational studies of layer I synapses on neocortical pyramidal cells. In T. McKenna, J. Javis, and S.F. Zarnetzer, editors, Single Neuron Computation, chapter 8, pages 199-229. Academic Press, Boston, MA, 1992. [6] L.J. Cauller and B.W. Connors. J. Neuroscience, In Press. [7] E. De Schutter and J .M. Bower. Firing rate of purkinje cells does not depend on the dendritic location of parallel fiber inputs. Eur. J. of Neurosci., S5:17, 1992. 526 Bemander, Koch, and Douglas [8] R.J. Douglas, K.A.C. Martin, and D. Whitteridge. An intracellular analysis of the visual responses of neurones in cat visual cortex. J. Physiology, 440:659696, 1991. [9] D.J. Felleman and D.C. Van Essen. Distributed hierarchical processing in the primate cerebral cortex. Cerebral Cortex, 1:1-47, 1991. [10] H.G. Kim and B.W. Connors. Apical dendrites of the neocortex: Correlation between sodium- and calcium-dependent spiking and pyramidal cell morphology. J. Neuroscience, In press. [11] G. Laurent. A dendritic gain-control mechanism in axonless neurons of the locust, schistocerca americana. J Physiology (London), 470:45-54, 1993. [12] A. Mason and A.U. Larkman. Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology. J. Neuroscience, 10(5):1415-1428, 1990. [13] H. Pockberger. Electrophysiological and morphological properties of rat motor cortex neurons in vivo. Brain Research, 539:181-190, 1991. [14] P.K. Powers, R.F. Tobinson, and M.A. Konodi. Effective synaptic current can be estimated from measurements of neuronal discharge. J. Neurophysiology, 68(3):964-968, 1992. [15] W.G. Regehr, J. Kehoe, P. Ascher, and C.M. Armstrong. Synaptically triggered action-potentials in dendrites. Neuron, 11(1):145-151,1993. [16] K.S. Rockland and A. Virga. Terminal arbors of individual "feedback" axons projecting from area V2 to VI in the macaque monkey: a study using immunohistochemistry of anterogradely transported phaseoulus vulgarisleucoagglutinin. J. Compo Neurol., 285:54-72, 1989. [17] W.A. Spencer and E.R. Kandel. Electrophysiology of hippocampal neurons. IV fast prepotentials. J. Neurophysiology, 24:272-285, 1961. [18] D.W. Tank, M. Sugimori, J .A. Connor, and R.R. Llimis. Spatially resolved calcium dynamics of mammalian purkinje cells in cerebellar slice. Science, 242:773-777, 1988. [19] R. Yuste, K.R. Delaney, M.J. Gutnick, and D.W. Tank. Spatially localized calcium accumulations in apical dendrites of layer 5 neocortical neurons. In Neuroscience Abstr. 19, page 616.2, 1993.	1993	71
827	Decoding Cursive Scripts Yoram Singer and Naftali Tishby Institute of Computer Science and Center for Neural Computation Hebrew University, Jerusalem 91904, Israel Abstract Online cursive handwriting recognition is currently one of the most intriguing challenges in pattern recognition. This study presents a novel approach to this problem which is composed of two complementary phases. The first is dynamic encoding of the writing trajectory into a compact sequence of discrete motor control symbols. In this compact representation we largely remove the redundancy of the script, while preserving most of its intelligible components. In the second phase these control sequences are used to train adaptive probabilistic acyclic automata (PAA) for the important ingredients of the writing trajectories, e.g. letters. We present a new and efficient learning algorithm for such stochastic automata, and demonstrate its utility for spotting and segmentation of cursive scripts. Our experiments show that over 90% of the letters are correctly spotted and identified, prior to any higher level language model. Moreover, both the training and recognition algorithms are very efficient compared to other modeling methods, and the models are 'on-line' adaptable to other writers and styles. 1 Introduction While the emerging technology of pen-computing is already available on the world's markets, there is an on growing gap between the state of the hardware and the quality of the available online handwriting recognition algorithms. Clearly, the critical requirement for the success of this technology is the availability of reliable and robust cursive handwriting recognition methods. 833 834 Singer and Tishby We have previously proposed a dynamic encoding scheme for cursive handwriting based on an oscillatory model of handwriting [8, 9] and demonstrated its power mainly through analysis by synthesis. Here we continue with this paradigm and use the dynamic encoding scheme as the front-end for a complete stochastic model of cursive script. The accumulated experience in temporal pattern recognition in the past 30 years has yielded some important lessons relevant to handwriting. The first is that one can not predefine the basic 'units' of such temporal patterns due to the strong interaction, or 'coarticulation', between such units. Any reasonable model must allow for the large variability of the basic handwriting components in different contexts and by different writers. Thus true adaptability is a key ingredient of a good stochastic model of handwriting. Most, if not all, currently used models of handwriting and speech are hard to adapt and require vast amounts of training data for some robustness in performance. In this paper we propose a simpler stochastic modeling scheme, which we call Probabilistic Acyclic Automata (PAA), with the important feature of being adaptive. The training algorithm modifies the architecture and dimensionality of the model while optimizing its predictive power. This is achieved through the minimization of the "description length" of the model and training sequences, following the minimum description length (MDL) principle. Another interesting feature of our algorithm is that precisely the same procedure is used in both training and recognition phases, which enables continuous adaptation. The structure of the paper is as follows. In section 2 we review our dynamic encoding method, used as the front-end to the stochastic modeling phase. We briefly describe the estimation and quantization process, and show how the discrete motor control sequences are estimated and used, in section 3. Section 4 deals with our stochastic modeling approach and the PAA learning algorithm. The algorithm is demonstrated by the modeling of handwritten letters. Sections 5 and 6 deal with preliminary applications of our approach to segmentation and recognition of cursi ve handwriting. 2 Dynamic encoding of cursive handwriting Motivated by the oscillatory motion model of handwriting, as described e.g. by Hollerbach in 1981 [2], we developed a parameter estimation and regularization method which serves for the analysis, synthesis and coding of cursive handwriting. This regularization technique results in a compact and efficient discrete representation of handwriting. Handwriting is generated by the human muscular motor system, which can be simplified as spring muscles near a mechanical equilibrium state. When the movements are small it is justified to assume that the spring muscles operate in the linear regime, so the basic movements are simple harmonic oscillations, superimposed by a simple linear drift. Movements are excited by selecting a pair of agonist-antagonist muscles that are modeled by the spring pair. In a restricted form this simple motion is described by the following two equations, Vx(t) = x(t) = acos(wxt + f/;) + c Vy(t) = yet) = bcos(wyt) , (1) where Vx(t) and Vy(t) are the horizontal and vertical pen velocities respectively, Wx and Wy are the angular velocities, a, b are the velocity amplitudes, ¢ is the relative Decoding Cursive Scripts 835 phase lag, and c is the horizontal drift velocity. Assuming that these describe the true trajectory, the horizontal drift, c, is estimated as the average horizontal velocity, c = Jv 2:[:1 Vx(i). For fixed values of the parameters a, b,w and 1; these equations describe a cycloidal trajectory. Our main assumption is that the cycloidal trajectory is the natural (free) pen motion, which is modified only at the velocity zero crossings. Thus changes in the dynamical parameters occur only at t he zero crossings and preserve the continuity of the velocity field. This assumption implies that the angular velocities W x , Wy and amplitudes a, b can be considered constant between consecutive zero crossings. Denoting by tf and t; , the i'th zero crossing locations of the horizontal and vertical velocities, and by Li and L; , the horizontal and vertical progression during the i'th interval, then the estimated amplitudes are, a = 2(tf~ =tX) , b = 2(J~ :t Y )' Those .+1 • .+1 • amplitudes define the vertical and horizontal scales of the written letters. Examination of the vertical velocity dynamics reveals the following : (a) There is a virtual center of the vertical movement and velocity trajectory is approximately symmetric around this center. (b) The vertical velocity zero crossings occur while the pen is at almost fixed vertical levels which correspond to high, normal and low modulation values, yielding altogether 5 quantized levels. The actual pen levels achieved at the vertical velocity zero crossings vary around the quantized values, with approximately normal distribution. Let the indicator, It (It E {I , . . . , 5}), be the most probable quantized level when the pen is at the position obtained at the t'th zero crossing. \Ve need to estimate concurrently the 5 quantized levels H 1, ... , H 5, their variance (J' (assumed the same for all levels), and the indicators It. In this model the observed data is the sequence of actual pen levels L(t), while the complete data is the sequence of levels and indicators {It , L(t)} . The task of estimating the parameters {Hi , (J'} is performed via maximum likelihood estimation from incomplete data, commonly done by the EM algorithm[l] and described in [9]. The horizontal amplitude is similarly quantized to 3 levels. After performing slant equalization of the handwriting, namely, orthogonalizing the x and y motions, the velocities Vx(t) , "~(t) become approximately uncorrelated. When Wx ~ wy , the two velocities are uncorrelated if there is a ±900 phase-lag between Vx and Vy . There are also locations of total halt in both velocities (no pen movement) which we take as a zero phase lag. Considering the vertical oscillations as a 'master clock', the horizontal oscillations can be viewed as a 'slave clock ' whose phase and amplitude vary around the 'master clock'. For English cursive writing, the frequency ratio between the two clocks is limited to the set {~, 1,2}, thus Vy induces a grid for the possible Vx zero crossings. The phase-lag of the horizontal oscillation is therefore restricted to the values 00, ±900 at the zero crossings of Vy . The most likely phase-lag trajectory is determined by dynamic programming over the entire grid. At the end of this process the horizontal oscillations are fully determined by the vertical oscillations and the pen trajectory's description greatly simplified. The variations in the vertical angular velocity for a given writer are small, except in short intervals where the writer hesitates or stops. The only information that should be preserved is the typical vertical angular velocity, denoted by w. The 836 Singer and Tishby normalized discretized equations of motion now become, { ~ ai sin(wt + <Pi) + 1 hsin(wt) ai E {AI, Ai, A3} <Pj E {-90°, 0°, 90°} hE {H1 2 Hil 11::; 11 ,/2 ::; 5} . (2) We used analysis by synthesis technique in order to verify our assumptions and estimation scheme. The final result of the whole process is depicted in Fig. 1, where the original handwriting is plotted together with its reconstruction from the discrete representation. Figure 1: The original and the fully quantized cursive scripts. 3 Discrete control sequences The process described in the previous section results in a many to one mapping from the continuous velocity field, Vx(t), Vy(t), to a discrete set of symbols. This set is composed of the cartesian product of the quantized vertical and horizontal amplitudes and the phase-lags between these velocities. We treat this discrete control sequence as a cartesian product time series. Using the value (0' to indicate that the corresponding oscillation continues with the same dynamics, a change in the phase lag can be encoded by setting the code to zero for one dimension, while switching to a new value in the other dimension. A zero in both dimensions indicates no activity. In this way we can model 'pen ups' intervals and incorporate auxiliary symbols like 'dashes', 'dots', and 'crosses', that play an important role in resolving disambiguations between letters. These auxiliary are modeled as a separate channel and are ordered according to their X coordinate. We encode the control levels by numbers from 1 to 5 , for the 5 levels of vertical positions. The quantized horizontal amplitudes are coded by 5 values as well: 2 for positive amplitudes (small and large), 2 for negative amplitudes, and one for zero amplitude. Below is an example of our discrete representation for the handwriting depicted in Fig. 1. The upper and lower lines encode the vertical and horizontal oscillations respectively, and the auxiliary channel is omitted. In this example there is only one location where both symbols are (0', indicating a pen-up at the end of the word. 240204204001005002040202204020402424204020500204020402400440240220 104034030410420320401050010502425305010502041032403050033105001000 4 Stochastic modeling of the motor control sequences Existing stochastic modeling methods, such as Hidden Markov Models (HMM) [3], suffer from several serious drawbacks. They suffer from the need to 'fix' a-priory the Decoding Cursive Scripts 837 architecture of the model; they require large amounts of segmented training data; and they are very hard to adapt to new data. The stochastic model presented here is an on-line learning algorithm whose important property is its simple adaptability to new examples. We begin with a brief introduction to probabilistic automata, leaving the theoretical issues and some of the more technical details to another place. A probabilistic automaton is a 6-tuple (Q , ~ , T", qs, qe), where Q is a finite set of n states, ~ is an alphabet of size k, T : Q x ~ --+ Q is the state transition function, , : ~ x Q --+ [0,1] is the transition (output) probability where for every q E Q, LaE~ ,( O'lq) = l. qs E Q is a start state, and qe E Q is an end state. A probabilistic automaton is called acyclic if it contains no cycles. We denote such automata by PAA. This type of automaton is also known as a Markov process with a single source and a single absorbing state. The rest of the states are all transient states. Such automata induce non-zero probabilities on a finite set of strings. Given an input string a = (0'1, .. . , 0' n) if at the of end its 'run' the automaton entered the final state qe, the probability of a string a is defined to be, pea) = n{:l ,(O'ilqi-l) where qo = qs, qi = T(qi-1, O'i) . On the other hand, if qN f. qe then pea) = O. The inference of the P AA structure from data can be viewed as a communication problem. Suppose that one wants to transmit an ensemble of strings, all created by the same PAA. If both sides know the structure and probabilities of the PAA then the transmitter can optimally encode the strings by using the PAA transition probabilities. If only the transmitter knows the structure and the receiver has to discover it while receiving new strings, each time a new transition occurs, the transmitter has to send the next state index as well. Since the automaton is acyclic, the possible next states are limited to those which do not form a cycle when the new edge is added to the automaton. Let k~ be the number of legal next states from a state q known to the receiver at time t. Then the encoding of the next state index requires at least log2(k~ + 1) bits. The receiver also needs to estimate the state transition probability from the previously received strings. Let n(O'lq) be the number of times the symbol 0' has been observed by the receiver while being in state q. Then the transition probability is estimated by Laplace's rule of succession, ?(O'lq) = L n(alq )~\ 1 I' In sum, if q is the current state and ktq the number of I n(al q + ~ (7 EE possible next states known to the receiver, the number of bits required to encode the next symbol 0' (assuming optimal coding scheme) is given by: (a) if the transition T(q, 0') has already been observed: -log2(P(0'Iq)) ; (b) if the transition T(q, 0') has never occurred before: -log2(.P(0'Iq)) + log2(k~ + 1). In training such a model from empirical observations it is necessary to infer the structure of the PAA as well its parameters. We can thus use the above coding scheme to find a minimal description length (MDL) of the data, provided that our model assumption is correct. Since the true PAA is not known to us, we need to imitate the role of the receiver in order to find the optimal coding of a message. This can be done efficiently via dynamic programming for each individual string. After the optimal coding for a single string has been found , the new states are added, the transition probabilities ?(O'lq) are updated and the number of legal next states kg is recalculated. An exan~ple of the learning algorithm is given in Fig. 2, with the estimated probabilities P, written on the graph edges. 838 Singer and Tishby (b) (d) Figure 2: Demonstration of the PAA learning algorithm. Figure (a) shows the original automaton from which the examples were created. Figures (b )-( d) are the intermediate automata built by the algorithm. Edges drawn with bold, dashed, and grey lines correspond to transitions with the symbols '0', '1', and the terminating symbol, respectively. 5 Automatic segmentation of cursive scripts Since the learning algorithm of a PAA is an on-line scheme, only a small number of segmented examples is needed in order to built an initial model. For cursive handwriting we manually collected and segmented about 10 examples, for each lower case cursive letter, and built 26 initial models. At this stage the models are small and do not capture the full variability of the control sequences. Yet this set of initial automata was sufficient to gradually segment cursive scripts into letters and update the models from these segments. Segmented words with high likelihood are fed back into the learning algorithm and the models are further refined. The process is iterated until all the training data is segmented with high likelihood. The likelihood of new data might not be defined due the incompleteness of the automata, hence the learning algorithm is again applied in order to induce probabilities. Let Pi~j be the probability that a model 5 (which represents a cursive letter) generates the control symbols Si, ... , Sj -1 (j > i). The log-likelihood of a proposed segmentation (i1, i2 , ... , iN+d of a word 51,52 , ... , 5N is, N N L ((i1, . . . , iN+1)1(51, ... , 5N) , (Sl, . . . , sL)) = log(II Pi~~iJ+J = L log(Pi~~iJ+l) j=l j=l The segmentation is calculated efficiently by maintaining a layers graph and using dynamic programming to compute recursively the most likely segmentation. Formally, let M L( n, k) be the highest likelihood segmentation of the word up to the Decoding Cursive Scripts 839 n'th control symbol and the k'th letter in the word. Then, M L(n, k) = . ma~ {M L(i, k - 1) + log (Pi:~)} tk-l~t~n The best segmentation is obtained by tracking the most likely path from M(N, L) back to M(l, 1). The result of such a segmentation is depicted in Fig. 3. Figure 3: Temporal segmentation of the word impossible. The segmentation is performed by applying the automata of the letters contained in the word, and finding the Maximum-Likelihood sequence of models via dynamic programming. 6 Inducing probabilities for unlabeled words Using this scheme we automatically segmented a database which contained about 1200 frequent english words, by three different writers. After adding the segmented letters to the training set the resulting automata were general enough, yet very compact. Thus inducing probabilities and recognition of unlabeled data could be performed efficiently. The probability of locating letters in certain locations in new unlabeled words (i.e. words whose transcription is not given) can be evaluated by the automata. These probabilities are calculated by applying the various models on each sub-string of the control sequence, in parallel. Since the automata can accommodate different lengths of observations, the log-likelihood should be divided by the length of the sequence. This normalized log-likelihood is an approximation of the entropy induced by the models, and measures the uncertainty in determining the transcription of a word. The score which measures the uncertainty of the occurrence of a letter S in place n in the a word is, Score(nIS) = maxI t 10g(P:'n+l_d. The result of applying several automata to a new word is shown in Fig. 4. High probability of a given automaton indicates a beginning of a letter with the corresponding model. The probabilities for the letters k, a, e, b are plotted top to bottom. The correspondence between high likelihood points and the relevant locations in the words are shown with dashed lines. These locations occur near the 'true' occurrence of the letter and indicate that these probabilities can be used for recognition and spotting of cursive handwriting. There are other locations where the automata obtain high scores. These correspond to words with high similarity to the model letter and can be resolved by higher level models, similar to techniques used in speech. 7 Conclusions and future research In this paper we present a novel stochastic modeling approach for the analysis, spotting, and recognition of online cursive handwriting. Our scheme is based on a 840 Singer and Tishby Figure 4: The normalized log-likelihood scores induced by the automata for the letters k, a, e, and b (top to bottom). Locations with high score are marked with dashed lines and indicate the relative positions of the letters in the word. discrete dynamic representation of the handwriting trajectory, followed by training adaptive probabilistic automata for frequent writing sequences. These automata are easy to train and provide simple adaptation mechanism with sufficient power to capture the high variability of cursively written words. Preliminary experiments show that over 90% of the single letters are correctly identified and located, without any additional higher level language model. Methods for higher level statistical language models are also being investigated [6], and will be incorporated into a complete recognition system. Acknowledgments We would like to thank Dana Ron for useful discussions and Lee Giles for providing us with the software for plotting finite state machines. Y.S. would like to thank the Clore foundation for its support. References [1] A. Dempster, N. Laird, and D. Rubin. Maximum likelihood estimation from incomplete data via the EM algorithm. 1. Roy. Statist. Soc., 39(B):1-38, 1977. [2] J .M. Hollerbach. An oscillation theory of handwriting. Bio. Cyb., 39, 1981. [3] L.R. Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proc. IEEE, pages 257-286, Feb. 1989. [4] J . Rissanen. Modeling by shortest data description. Automaiica, 14, 1978. [5] J. Rissanen. Stochastic complexity and modeling. Annals of Stat., 14(3), 1986. [6] D. Ron, Y. Singer, and N. Tishby. The power of amnesia. In this volume. [7] D.E. Rumelhart. Theory to practice: a case study - recognizing cursive handwriting. In Proc. of 1992 NEC Conf. on Computation and Cognition. [8] Y. Singer and N. Tishby. Dynamical encoding of cursive handwriting. In IEEE Conference on Computer Vision and Pattern Recognition, 1993. [9] Y. Singer and N. Tishby. Dynamical encoding of cursive handwriting. Technical Report CS93-4, The Hebrew University of Jerusalem, 1993. PART VII IMPLEMENTATIONS	1993	72
828	Foraging in an Uncertain Environment Using Predictive Hebbian Learning P. Read Montague: Peter Dayan, and Terrence J. Sejnowski Computational Neurobiology Lab, The Salk Institute, 100 ION. Torrey Pines Rd, La Jolla, CA, 92037, USA read~bohr.bcm.tmc.edu Abstract Survival is enhanced by an ability to predict the availability of food, the likelihood of predators, and the presence of mates. We present a concrete model that uses diffuse neurotransmitter systems to implement a predictive version of a Hebb learning rule embedded in a neural architecture based on anatomical and physiological studies on bees. The model captured the strategies seen in the behavior of bees and a number of other animals when foraging in an uncertain environment. The predictive model suggests a unified way in which neuromodulatory influences can be used to bias actions and control synaptic plasticity. Successful predictions enhance adaptive behavior by allowing organisms to prepare for future actions, rewards, or punishments. Moreover, it is possible to improve upon behavioral choices if the consequences of executing different actions can be reliably predicted. Although classical and instrumental conditioning results from the psychological literature [1] demonstrate that the vertebrate brain is capable of reliable prediction, how these predictions are computed in brains is not yet known. The brains of vertebrates and invertebrates possess small nuclei which project axons throughout large expanses of target tissue and deliver various neurotransmitters such as dopamine, norepinephrine, and acetylcholine [4]. The activity in these systems may report on reinforcing stimuli in the world or may reflect an expectation of future reward [5, 6,7,8]. *Division of Neuroscience, Baylor College of Medicine, Houston, TX 77030 598 Foraging in an Uncertain Environment Using Predictive Hebbian Learning 599 A particularly striking example is that of the honeybee. Honeybees can be conditioned to a sensory stimulus such as a color, visual pattern, or an odorant when the sensory stimulus is paired with application of sucrose to the antennae or proboscis. An identified neuron, VUMmxl, projects widely throughout the entire bee brain, becomes active in response to sucrose, and its firing can substitute for the unconditioned odor stimulus in classical conditioning experiments [8]. Similar diffusely projecting neurons in the bee brain may substitute for reward when paired with a visual stimulus. In this paper, we suggest a role for diffuse neurotransmitter systems in learning and behavior that is analogous to the function we previously postulated for them in developmental selforganization[3, 2]. Specifically, we: (i) identify a neural substrate/architecture which is known to exist in both vertebrates and invertebrates and which delivers information to widespread regions of the brain; (ii) describe an algorithm that is both mathematically sound and biologically feasible; and (iii) show that a version of this local algorithm, in the context of the neural architecture, reproduces the foraging and decision behavior observed in bumble bees and a number of other animals. Our premise is that the predictive relationships between sensory stimuli and rewards are constructed through these diffuse systems and are used to shape both ongoing behavior and reward-dependent synaptic plasticity. We illustrate this using a simple example from the ethological literature for which constraints are available at a number of different levels. A Foraging Problem Real and colleagues [9, 10] performed a series of experiments on bumble bees foraging on artificial flowers whose colors, blue and yellow, predicted of the delivery of nectar. They examined how bees respond to the mean and variability of this reward delivery in a foraging version of a stochastic two-armed bandit problem [11]. All the blue flowers contained 2\-1l of nectar, l of the yellow flowers contained 6 \-1l, and the remaining j of the yellow flowers contained no nectar at all. In practice, 85% of the bees' visits were to the constant yield blue flowers despite the equivalent mean return from the more variable yellow flowers. When the contingencies for reward were reversed, the bees switched their preference for flower color within 1 to 3 visits to flowers. They further demonstrated that the bees could be induced to visit the variable and constant flowers with equal frequency if the mean reward from the variable flower type was made sufficiently high. This experimental finding shows that bumble bees, like honeybees, can learn to associate color with reward. Further, color and odor learning in honeybees has approximately the same time course as the shift in preference descri bed above for the bumble bees [12]. It also indicates that under the conditions of a foraging task, bees prefer less variable rewards and compute the reward availability in the short term. This is a behavioral strategy utilized by a variety of animals under similar conditions for reward [9, 10, 13] suggesting a common set of constraints in the underlying neural substrate. The Model Fig. 1 shows a diagram of the model architecture, which is based on the considerations above about diffuse systems. Sensory input drives the units 'B' and 'Y' representing blue and yellow flowers. These neurons (outputs x~ and xi respectively at time t) project 600 Montague, Dayan, and Sejnowski Action selection Lateral inhibition Motor systems Figure 1: Neural architecture showing how predictions about future expected reinforcement can be made in the brain using a diffuse neurotransmitter system [3, 2]. In the context of bee foraging [9], sensory input drives the units Band Y representing blue and yellow flowers. These units project to a reinforcement neuron P through a set of variable weights (filled circles w B and w Y) and to an action selection system. Unit S provides input to n and fires while the bee sips the nectar. R projects its output rt through a fixed weight to P. The variable weights onto P implement predictions about future reward rt (see text) and P's output is sensitive to temporal changes in its input. The output projections of P, bt (lines with arrows), influence learning and also the selection of actions such as steering in flight and landing, as in equation 5 (see text). Modulated lateral inhibition (dark circle) in the action selection layer symbolizes this. Before encountering a flower and its nectar, the output of P will reflect the temporal difference only between the sensory inputs Band Y. During an encounter with a flower and nectar, the prediction error bt is determined by the output of B or Y and R, and learning occurs at connections w B and w Y. These strengths are modified according to the correlation between presynaptic activity and the prediction error bt produced by neuron P as in equation 3 (see text). Learning is restricted to visits to flowers [14]. through excitatory connection weights both to a diffusely projecting neuron P (weights w B and w Y) and to other processing stages which control the selection of actions such as steering in flight and landing. P receives additional input rt through unchangeable wei~hts. In the absence of nectar (rt = 0), the net input to P becomes Vt = Wt ·Xt = w~x~ +wt x~. The first assumption in the construction of this model is that learning (adjustment of weights) is contingent upon approaching and landing on a flower. This assumption is supported specifically by data from learning in the honeybee: color learning for flowers is restricted to the final few seconds prior to landing on the flower and experiencing the nectar [14]. This fact suggests a simple model in which the strengths of variable connections Wt are adjusted according to a presynaptic correlational rule: (1 ) where oc is the learning rate [15]. There are two problems with this formulation: (i) learning would only occur about contingencies in the presence of a reinforcing stimulus (rt =/: 0); Foraging in an Uncertain Environment Using Predictive Hebbian Learning 601 A 1.0 0 .8 ..... :::::s 0..0.6 ..... :::::s o 0.4 0.2 0.0 '----~---~----' 0.0 5.0 10.0 Nectar volume (f-ll) B 100.0 -80.0 ~ '-' (1) :::::s 60.0 ..0 0 ..... 40.0 <:n ..... . <:n .20.0 > 0.0 0 5 10 15 20 25 30 Trial Figure 2: Simulations of bee foraging behavior using predictive Hebbian learning. A) Reinforcement neuron output as a function of nectar volume for a fixed concentration of nectar[9, 10]. B) Proportion of visits to blue flowers. Each trial represents approximately 40 flower visits averaged over 5 real bees and exactly 40 flower visits for a single model bee. Trials 1 - 15 for the real and model bees had blue flowers as the constant type, the remaining trials had yellow flowers as constant. At the beginning of each trial, wYand w B were set to 0.5 consistent with evidence that information from past foraging bouts is not used[14]. The real bees were more variable than the model bees - sources of stochasticity such as the two-dimensional feeding ground were not represented. The real bees also had a slight preference for blue flowers [21]. Note the slower drop for A = 0.1 when the flowers are switched. and (ii) there is no provision for allowing a sensory event to predict the future delivery of reinforcement. The latter problem makes equation 1 inconsistent with a substantial volume of data on classical and instrumental conditioning [16]. Adding a postsynaptic factor to equation 1 does not alter these conclusions [17]. This inadequacy suggests that another form of learning rule and a model in which P has a direct input from rt. Assume that the firing rate of P is sensitive only to changes in its input over time and habituates to constant or slowly varying input, like magnocellular ganglion cells in the retina [18]. Under this assumption, the output of P, bt. reflects a temporal derivative of its net input, approximated by: (2) where y is a factor that controls the weighting of near against distant rewards. We take y = 1 for the current discussion. In the presence of the reinforcement, the weights w B and w Y are adjusted according to the simple correlational rule: (3) This permits the weights onto P to act as predictions of the expected reward consequent on landing on a flower and can also be derived in a more general way for the prediction of future values of any scalar quantity [19]. 602 Montague, Dayan, and Sejnowski A .100.0 ~ '-' Q.) 80.0 0... ~ Q.) 60.0 ~ .~ 40.0 > 8 C'-l 20.0 ..... .C'-l .-> 0 .0 0 .0 8--£lv=2 <r-----(> v = 8 b------i!. V = 30 2.0 4.0 6.0 Mean B 30.0 8 20.0 ~ ·0 ~ > 10.0 0 .0 0.0 o A= 0 .1 + A= 0.9 2.0 4.0 Mean 6 .0 Figure 3: Tradeoff between the mean and variance of nectar delivery. A) Method of selecting indifference points. The indifference point is taken as the first mean for a given variance (bold v in legend) for which a stochastic trial demonstrates the indifference. This method of calculation tends to bias the indifference points to the left. B) Indifference plot for model and real bees. Each point represents the (mean, variance) pair for which the bee sampled each flower type equally. The circles are for A = 0.1 and the pluses are for A = 0.9. When the bee actually lands on a flower and samples the nectar, R influences the output of P through its fixed connection (Fig. 1). Suppose that just prior to sampling the nectar the bee switched to viewing a blue flower, for example. Then, since Tt-l = 0, lit would be Tt x~_1 w~_I. In this way, the term x~_1 w~_1 is a prediction of the value of Tt and the difference Tt x~_1 wt 1 is the error in that prediction. Adjusting the weight w~ according to the correlational rule in equation 3 allows the weight w~, through P's outputs, to report to the rest of the brain the amount of reinforcement Tt expected from blue flowers when they are sensed. As the model bee flies between flowers, reinforcement from nectar is not present (Tt = 0) and lit is proportional to V t - V t- 1. w B and w Y can again be used as predictions but through modulation of action choice. For example, suppose the learning process in equation 3 sets w Y less than w B• In flight, switching from viewing yellow flowers to viewing blue flowers causes lit to be positive and biases the activity in any action selection units driven by outgoing connections from B. This makes the bee more likely than chance to land on or steer towards blue flowers. This discussion is not offered as an accurate model of action choice, rather, it simply indicates how output from a diffuse system could also be used to influence action choice. The biological assumptions of this neural architecture are explicit: (i) the diffusely projecting neuron changes its firing according to the temporal difference in its inputs; (ii) the output of P is used to adjust its weights upon landing; and (iii) the output otherwise biases the selection of actions by modulating the activity of its target neurons. For the particular case of the bee, both the learning rule described in equation 3 and the biasing of action selection described above can be further simplified for the purposes of a Foraging in an Uncertain Environment Using Predictive Hebbian Learning 603 simple demonstration. As mentioned above, significant learning about a particular flower color may occur only in the 1 - 2 seconds just prior to an encounter [21, 14]. This is tantamount to restricting weight changes to each encounter with the reinforcer which allows only the sensory input just preceding the delivery or non-delivery of r t to drive synaptic plasticity. We therefore make the learning rule punctate, updating the weights on a flower by flower basis. During each encounter with the reinforcer in the environment, P produces a prediction error cSt = rt - Vt-l where rt is the actual reward at time t, and the last flower color seen by the bee at time t, say blue, causes a prediction Vt -l = wt lX~_l of future reward rt to be made through the weight w~_l and the input activity xt l' The weights are then updated using a form of the delta rule[20]: (4) where A is a time constant and controls the rate of forgetting. In this rule, the weights from the sensory input onto P still mediate a prediction of r; however, the temporal component for choosing how to steer and when to land has been removed. We model the temporal biasing of actions such as steering and landing with a probabilistic algorithm that uses the same weights onto P to choose which flower is actually visited on each trial. At each flower visit, the predictions are used directly to choose an action, according to: e~(WYxY) q(Y) = e~(wBxB) + ell(wYxY) (5) where q(Y) is the probability of choosing a yellow flower. Values of J.L > 0 amplify the difference between the two predictions so that larger values of J.L make it more likely that the larger prediction will result in choice toward the associated flower color. In the limit as J.L ---+ 00 this approaches a winner-take-all rule. In the simulations, J.L was varied from 2.8 to 6.0 and comparable results obtained. Changing J.L alters the magnitude of the weights that develop onto neuron P since different values of J.L enforce different degrees of competition between the predictions. To apply the model to the foraging experiment, it is necessary to specify how the amount of nectar in a particular flower gets reported to P. We assume that the reinforcement neuron R delivers its signal rt as a saturating function of nectar volume (Fig. 2A). Harder and Real [10] suggest just this sort of decelerating function of nectar volume and justify it on biomechanical grounds. Fig. 2B shows the behavior of model bees compared with that of real bees [9] in the experiment testing the extent to which they prefer a constant reward to a variable reward of the same long-term mean. Further details are presented in the figure legend. The behavior of the model matched the observed data for A = 0.9 suggesting that the real bee utilizes information over a small time window for controlling its foraging [9]. At this value of A, the average proportion of visits to blue was 85% for the real bees and 83% for the model bees. The constant and variable flower types were switched at trial 15 and both bees switched flower preference in 1 - 3 subsequent visits. The average proportion of visits to blue changed to 23% and 20%, respectively, for the real and model bee. Part of the reason for the real bees' apparent preference for blue may come from inherent biases. Honey bees, for instance, are known to learn about shorter wavelengths more quickly than others [21]. In our model, A is a measure of the length of time over which an observation exerts an influence on flower selection rather than being a measure of the bee's time horizon in terms of the mean rate of energy intake [9, 10]. 604 Montague, Dayan, and Sejnowski Real bees can be induced to forage equally on the constant and variable flower types if the mean reward from the variable type is made sufficiently large, as in Fig. 3B. For a given variance, the mean reward was increased until the bees appeared indifferent between the flowers. In this experiment, the constant flower type contained 0.5J.11 of nectar. The data for the real bee is shown as points connected by a solid line in order to make clear the envelope of the real data. The indifference points for A = 0.1 (circles) and A = 0.9 (pluses) also demonstrate that a higher value of A is again better at reproducing the bee's behavior. The model captured both the functional relationship and the spread of the real data. The diffuse neurotransmitter system reports prediction errors to control learning and bias the selection of actions. Distributing such a signal diffusely throughout a large set of target structures permits this prediction error to influence learning generally as a factor in a correlational or Hebbian rule. The same signal, in its second role, biases activity in an action selection system to favor rewarding behavior. In the model, construction of the prediction error only requires convergent input from sensory representations onto a neuron or neurons whose output is a temporal derivative of its input. The output of this neuron can also be used as a secondary reinforcer to associate other sensory stimuli with the predicted reward. We have shown how this relatively simple predictive learning system closely simulates the behavior of bumble bees in a foraging task. Acknowledgements This work was supported by the Howard Hughes Medical Institute, the National Institute of Mental Health, the UK Science and Engineering Research Council, and computational resources from the San Diego Supercomputer Center. We would like to thank Patricia Churchland, Anthony Dayan, Alexandre Pouget, David Raizen, Steven Quartz and Richard Zemel for their helpful comments and criticisms. References [1] Konorksi, 1. Conditioned reflexes and neuron organization, (Cambridge, England, Cambridge University Press, 1948). [2] Quartz, SR, Dayan, P, Montague, PR, Sejnowski, Tl. (1992) Society for Neurosciences Abstracts. 18, 210. [3] Montague, PR, Dayan, P, Nowlan, Sl, Pouget, A, Sejnowski, Tl. (1993) In Advances in Neural Information Processing Systems 5, Sl Hanson, ID Cowan, CL Giles, editors, (San Mateo CA: Morgan Kaufmann), pp. 969-976. [4] Morrison, IH and Magistretti, Pl. Trends in Neurosciences, 6, 146 (1983). [5] Wise, RA. Behavioral and Brain Sciences, 5,39 (1982). [6] Cole, Bl and Robbins, TW. Neuropsychopharmacology, 7, 129 (1992). [7] Schultz, W. Seminars in the Neurosciences, 4, 129 (1992). [8] Hammer, M, thesis, FU Berlin (1991). [9] Real, LA. Science, 253, pp 980 (1991). Foraging in an Uncertain Environment Using Predictive Hebbian Learning 605 [10] Real, LA. Ecology, 62,20 (1981); Harder, LD and Real, LA. Ecology, 68(4), 1104 (1987); Real, LA, Ellner, S, Harder, LD. Ecology, 71(4), 1625 (1990). [11] Berry, DA and Fristedt, B. Bandit Problems: Sequential Allocation of Experiments. (London, England: Chapman and Hall, 1985). [12] Gould, JL. In Foraging Behavior, AC Kamil, JR Krebs and HR Pulliam, editors, (New York, NY: Plenum, 1987), p 479. [13] Krebs, JR, Kacelnik, A, Taylor, P. Nature" 275, 27 (1978), Houston, A, Kacelnik, A, McNamara, J. In Functional Ontogeny, D McFarland, editor, (London: Pitman, 1982). [14] Menzel, R and Erber, 1. Scientific American, 239(1), 102. [15] Carew, TJ, Hawkins RD, Abrams 1W and Kandel ER. Journal of Neuroscience, 4(5), 1217 (1984). [16] Mackintosh, NJ. Conditioning and Associative Learning. (Oxford, England: Oxford University Press, 1983). Sutton, RS and Barto, AG. Psychological Review, 882, 135 (1981). Sutton, RS and Barto, AG. Proceedings of the Ninth Annual Conference of the Cognitive Science Society. Seattle, WA (1987). [17] Reeke, GN, Jr and Sporns, O. Annual Review of Neuroscience. 16,597 (1993). [18] Dowling, JE. The Retina. (Cambridge, MA: Harvard University Press, 1987). [19] The overall algorithm is a temporal difference (TO) learning rule and is related to an algorithm Samuel devised for teaching a checker playing program, Samuel, AL. IBM Journal of Research and Development, 3,211 (1959). It was first suggested in its present form in Sutton, RS, thesis, University of Massachusetts (1984); Sutton and Barto [1] showed how it could be used for classical conditioning; Barto, AG, Sutton, RS and Anderson, CWo IEEE Transactions on Systems, Man, and Cybernetics, 13, 834 (1983) used a variant of it in a form of instrumental conditioning task; Barto, AG, Sutton, RS, Watkins, CJCH, Technical Report 89-95, (Computer and Information Science, University of Massachusetts, Amherst, MA, 1989); Barto, AG, Bradtke, SJ, Singh, SP, Technical Report 91-57, (Computer and Information Science, University of Massachusetts, Amherst, MA, 1991) showed its relationship to dynamic programming, an engineering method of optimal control. [20] Rescorla, RA and Wagner, AR. In Classical Conditioning II: Current Research and Theory, AH Black and WF Prokasy, editors, (New York, NY: Appleton-CenturyCrofts, 1972), p 64; Widrow, B and Stearns, SD. Adaptive Signal Processing, (Englewood Cliffs, NJ: Prentice-Hall, 1985). [21] Menzel, R, Erber, J and Masuhr, J. In Experimental Analysis of Insect Behavior, LB Browne, editor, (Berlin, Germany: Springer-Verlag, 1974), p 195.	1993	73
829	Analysis of Short Term Memories for Neural Networks Jose C. Principe, Hui-H. Hsu and Jyh-Ming Kuo Computational NeuroEngineering Laboratory Department of Electrical Engineering University of Florida, CSE 447 Gainesville, FL 32611 principe@synapse.ee.ufi.edu Abstract Short term memory is indispensable for the processing of time varying information with artificial neural networks. In this paper a model for linear memories is presented, and ways to include memories in connectionist topologies are discussed. A comparison is drawn among different memory types, with indication of what is the salient characteristic of each memory model. 1 INTRODUCTION An adaptive system that has to interact with the external world is faced with the problem of coping with the time varying nature of real world signals. Time varying signals, natural or man made, carry information in their time structure. The problem is then one of devising methods and topologies (in the case of interest here, neural topologies) that explore information along time.This problem can be appropriately called temporal pattern recognition, as opposed to the more traditional case of static pattern recognition. In static pattern recognition an input is represented by a point in a space with dimensionality given by the number of signal features, while in temporal pattern recognition the inputs are sequence of features. These sequence of features can also be thought as a point but in a vector space of increasing dimensionality. Fortunately the recent history of the input signal is the one that bears more information to the decision making, so the effective dimensionality is finite but very large and unspecified a priori. How to find the appropriate window of input data 1011 1012 Principe, Hsu, and Kuo (memory depth) for a given application is a difficult problem. Likewise, how to combine the information in this time window to better meet the processing goal is also nontrivial. Since we are interested in adaptive systems, the goal is to let the system find these quantities adaptively using the output error information. These abstract ideas can be framed more quantitatively in a geometric setting (vector space). Assume that the input is a vector [u(l), ... u(n), .... ] of growing size. The adaptive processor (a neural network in our case) has a fixed size to represent this information, which we assign to its state vector [x1(n), .... xN(n)] of size N. The usefulness of xk(n) depends on how well it spans the growing input space (defined by the vector u(n», and how well it spans the decision space which is normally associated with the minimization of the mean square error (Figure 1). Therefore, in principle, the procedure can be divided into a representational and a mapping problem. The most general solution to this problem is to consider a nonlinear projection manifold which can be modified to meet both requirements. In terms of neural topologies, this translates to a full recurrent system, where the weights are adapted such that the error criterion is minimized. Experience has shown that this is a rather difficult proposition. Instead, neural network researchers have worked with a wealth of methods that in some way constrain the neural topology. Projection space Nonlinear mapping error ~ Optimal Decision space Figure 1. Projection ofu(n) and the error for the task. (for simplicity we are representing only linear manifolds) The solution that we have been studying is also constrained. We consider a linear manifold as the projection space, which we call the memory space. The projection of u(n) in this space is subsequently mapped by means of a feedforward neural network (multilayer perceptron) to a vector in decision space that minimizes the error criterion. This model gives rise to the focused topologies. The advantage of this constrained model is that it allows an analytical study of the memory structures, since they become linear filters. It is important to stress that the choice of the projection space is crucial for the ultimate performance of the system, because if the projected version of u(n) in the memory space discards valuable information about u(n), then Analysis of Short Term Memories for Neural Networks 1013 the nonlinear mapping will always produce sub-optimal results. 2 Projection in the memory space If the projection space is linear, then the representational problem can be studied with linear system concepts. The projected vector u(n) becomes Yn N Yn = L w0n-k (1) k=l where xn are the memory traces. Notice that in this equation the coefficients wk are independent of time, and their number fixed to N. What is the most general linear structure that implements this projection operation? It is the generalizedfeedfonvard structure [Principe et aI, 1992] (Figure 2), which in connectionist circles has been called the time lagged recursive network [Back and Tsoi, 1992]. One can show that the defining relation for generalized feedforward structures is gk (n) = g (n) • gk-l (n) k';? 1 where • represents the convolution operation, and go (n) = (5 (n) . This relation means that the next state vector is constructed from the previous state vector by convolution with the same function g(n), yet unspecified. Different choices of g(n) will provide different choices for the projection space axes. When we apply the input u(n) to this structure, the axes of the projection space become xk(n), the convolution of u(n) with the tap signals. The projection is obtained by linearly weighting the tap signals according to equation (1). Figure 2. The generalizedfeedfonvard structure We define a memory structure as a linear system whose generating kernel g(n) is causal g (n) = 0 fo r n < 0 and normalized, i.e. 00 L Ig(n)1 = 1 n=O We define memory depth D as the modified center of mass (first moment in time) of the last memory tap. 00 D = L ngk(n) n=O And we define the memory resolution R as the number of taps by unit time, which 1014 Principe, Hsu, and Kuo becomes liD. The purpose of the memory structure is to transform the search for an unconstrained number of coefficients (as necessary if we worked directly with u(n» into one of seeking a fixed number of coefficients in a space with time varying axis. 3 Review of connectionist memory structures The gamma memory [deVries and Principe, 1992] contains as special cases the context unit [Jordan, 1986] and the tap delay line as used in TDNN [Waibel et aI, 1989]. However, the gamma memory is also a special case of the generalized feedforward filters where g (n) = Jl (1 - Jl) n which leads to the gamma functions as the tap signals. Figure 3, adapted from [deVries and Principe, 1993], shows the most common connectionist memory structures and its characteristics. As can be seen when k=l, the gamma memory defaults to the context unit, and when Jl=1 the gamma memory becomes the tap delay line. In vector spaces the context unit represents a line, and by changing 11 we are finding the best projection of u(n) on this line. This representation is appropriate when one wants long memories but low resolution. Likewise, in the tap delay line, we are projecting u(n) in a memory space that is uniquely determined by the input signal, i.e. once the input signal u(n) is set, the axes become u(n-k) and the only degree of freedom is the memory order K. This memory structure has the highest resolution but lacks versatility, since one can only improve the input signal representation by increasing the order of the memory. In this respect, the simple context unit is better (or any memory with a recursive parameter), since the neural system can adapt the parameter 11 to project the input signal for better performance. We recently proved that the gamma memory structure in continuous time represents a memory space that is rigid [Principe et aI, 1994]. When minimizing the output mean square error, the distance between the input signal and the projection space decreases. The recursive parameter in the feedforward structures changes the span of the memory space with respect to the input signal u(n) (which can be visualized as some type of complex rotation). In terms of time domain analysis, the recursive parameter is finding the length of the time window (the memory depth) containing the relevant information to decrease the output mean square error. The recursive parameter Jl can be adapted by gradient descent learning [deVries and Principe, 1992], but the adaptation becomes nonlinear and multiple minima exists.Notice that the memory structure is stable for O<Jl<2. The gamma memory when utilized as a linear adaptive filter extends Widrow's ADALINE [de Vries et aI, 1992], and results in a more parsimonious filter for echo cancellation [Palkar and Principe, 1994]. Preliminary results with the gamma memory in speech also showed that the performance of word spotters improve when 11 is different from one (i.e. when it is not the tap delay line). In a signal such as speech where time warping is a problem, there is no need to use the full resolution provided by the tap delay line. It is more important to trade depth by resolution. Analysis of Short Term Memories for Neural Networks 1015 4 Other Memory Structures There are other memory structures that fit our definition. Back and Tsoi proposed a lattice structure that fits our definition of generalized feedforward structure. Essentially this system orthogonalizes the input, uncorrelating the axis of the vector space (or the signals at the taps of the memory). This method is known to provide the best speed of adaptation because gradient descent becomes Newton's method (after the lattice parameters converge). The problem is that it becomes more computational demanding (more parameters to adapt, and more calculations to perform). Tape delay line u(tJ -0 Delay operator: Z-l memory depth: K Memory resolution: 1. Context Unit z nnmnin yet) $ 11--1--+--. Memory depth: 1/J,l Memory resolution: J,l Delay operator: z-(1-J,l) Gamma memory G(z) Delay operator: z - (1- J,l) Memory depth: klJ,l Memory resolution: J,l Figure 3. Connectionist memory structures Laguerre memories A set of basis intimately related to the gamma functions is the Laguerre bases. The 1016 Principe, Hsu, and Kuo Laguerre bases is an orthogonal span of the gamma space [Silva, 1994], which means that the information provided by both memories is the same. The advantage of the Laguerre is that the signals at the taps (the basis) are less correlated and so the adaptation speed becomes faster for values of Jl close to 0 or 2 [Silva, 1994] (the condition number of the matrix created by the tap signals is bounded). Notice that the Laguerre memory is still very easy to compute (a lowpass filter followed by a cascade of first order all pass filters). aguerre memory Delay operator: z - (1 - Jl) Gamma II memories. -1 Z (1 - Jl) z - (1- Jl) z domain The Gamma memory has a multiple pole that can be adaptively moved along the real Z domain axis, i.e. the Gamma memory can only implement lowpass (0< Jl <1) or highpass (1 <Jl <2) transfer functions. We experimentally observed that in nonlinear prediction of chaotic time series the recursive parameter sometimes adapts to values less than one. The highpass creates an extra ability to match the prediction by alternating the signs of the samples in the gamma memory (the impulse response for 1< Jl <2 is alternating in sign). But with a single real parameter the adaptation is unable to move the poles to complex values. Two conditions come to mind that require a memory structure with complex poles. First, the information relevant for the signal processing task appears in periodic bursts, and second, the input signal is corrupted by periodic noise. A memory structure with adaptive complex poles can successfully cope with these two conditions by selecting in time the intervals where the information is concentrated (or the windows that do not provide any information for the task). Figure 3 shows one possible implementation for the Gamma II kernel. Notice that for stability, the parameter u must obey the condition Jl (1 +~) < 2 and o <Jl <2. Complex poles are obtained for u> O. These parameters can be adapted by gradient descent [Silva et aI, 1992]. In terms of versatility, the Gamma II has a pair of free complex poles, the Gamma I has a pole restricted to the real line in the Z domain, and the tap delay line has the pole set at the origin of the Z domain (z=O). A multilayer perceptron equipped with an input memory layer with the Gamma II memory structure implements a nonlinear mapping on an ARMA model of the input signal. 5 How to use Memory structures in Connectionist networks. Although we have presented this theory with the focused architectures (which Analysis of Short Term Memories for Neural Networks 1017 corresponds to a nonlinear moving average model (NMAX», the memory structures can be placed anywhere in the neural topology. Any nonlinear processing element can feed one of these memory kernels as an extension of [Wan, 1990]. If the memory structures are used to store traces of the output of the net, we obtain a nonlinear autoregressive model (NARX). If they are used both at the input and output, they represent a nonlinear ARMAX model shown very powerful for system identification tasks. When the memory layer is placed in the hidden layers, there is no corresponding linear model. Gamma II Delay operator: _Jl_[z_-_< l_-_Jl)_]_ [z - (l - Jl)] 2 + ~Jl2 One must realize that these types of memory structures are recursive (except the tap delay line), so their training will involve gradients that depend on time. In the focused topologies the network weights can still be trained with static backpropagation, but the recursive parameter must be trained with real time recurrent learning (RTRL) or backpropagation through time (BPTT). When memory structures are scattered through out the topology, training can be easily accomplished with backpropagation through time, provided a systematic way is utilized to decompose the global dynamics in local dynamics as suggested in [Lefebvre and Principe, 1993]. 6 Conclusions The goal of this paper is to present a set of memory structures and show their relationship. The newly introduced Gamma II is the most general of the memories reviewed. By adaptively changing the two parameters u,Jl the memory can create complex poles at any location in the unit circle. This is probably the most general memory mechanism that needs to be considered. With it one can model poles and zeros of the system that created the signal (if it accepts the linear model). In this paper we addressed the general problem of extracting patterns in time. We have been studying this problem by pre-wiring the additive neural model, and decomposing it in a linear part -the memory space- that is dedicated to the storage of past values of the input (output or internal states), and in a nonlinear part which is static. The memory space accepts local recursion, which creates a powerful representational structure and where stability can be easily enforced (test in a single parameter). Recursive memories have the tremendous advantage of being able to trade memory depth by resolution. In vector spaces this means changing the relative 1018 Principe, Hsu, and Kuo position between the projection space and the input signal. However, the problem of finding the best resolution is still open (this means adaptively finding k, the memory order). Likewise ways to adaptively find the optimal value of the memory depth need improvements since the gradient procedures used up to now may be trapped in local minima. It is still necessary to modify the definition of memory depth such that it applies to both of these new memory structures. The method is to define it as the center of mass of the envelope of the last kernel. Acknowledgments:This work was partially supported by NSF grant ECS #920878. 7 Iteferences Back, A. D. and A. C. Tsoi, An Adaptive Lattice Architecture for Dynamic Multilayer Perceptrons, Neural Computation, vol. 4, no. 6, pp. 922-931, November, 1992. de Vries, B. and J. C. Principe, "The gamma model - a new neural model for temporal processing," Neural Networks, vol. 5, no. 4, pp. 565-576, 1992. de Vries, B., J.C. Principe, and P.G. De Oliveira, "Adaline with adaptive recursive memory," Proc. IEEE Workshop Neural Networks on Signal Processing, Princeton, NJ, 1991. Jordan, M., "Attractor dynamics and parallelism in a connectionist sequential machine," Proc. 8th annual Conf. on Cognitive Science Society, pp. 531-546, 1986. Lefebvre, C., and J.C. Principe, "Object-oriented artificial neural network implementations", Proc. World Cong on Neural Nets, vol IV, pp436-439, 1993. Principe, J. deVries B., Oliveira P., "Generalized feedforward structures: a new class of adaptive fitlers", ICASSP92, vol IV, 244-248, San Francisco. Principe, J.C., and B. de Vries, "Short term neural memories for time varying signal classification," in Proc. 26th ASILOMAR Conf., pp. 766-770, 1992. Principe J. C., J.M. Kuo, and S. Celebi," An Analysis of Short Term Memory Structures in Dynamic Neural Networks", accepted in the special issue of recurrent networks of IEEE Trans. on Neural Networks. Palkar M., and J.e. Principe, "Echo cancellation with the gamma filter," to be presented at ICASSP, 1994. Silva, T.O., "On the equivalence between gamma and Laguerre filters," to be presented at ICASSP, 1994. Silva, T.O., J.C. Principe, and B. de Vries, "Generalized feedforward filters with complex poles," Proc. Second IEEE Conf. Neural Networks for Signal Processing, pp.503-510, 1992. Waiber, A., "Modular Construction of Time-Delay Neural Networks for Speech Recognition," Neural Computation I, pp39-46, 1989. Wan, A. E., "Temporal backpropagation: an efficient algorithm for finite impulse response neural networks," Connectionist Models, Proc. of the 1990 Summer School, pp.131-137, 1990.	1993	74
830	Credit Assignment through Time: Alternatives to Backpropagation Yoshua Bengio * Dept. Informatique et Recherche Operationnelle Universite de Montreal Montreal, Qc H3C-3J7 Paolo Frasconi Dip. di Sistemi e Informatica Universita di Firenze 50139 Firenze (Italy) Abstract Learning to recognize or predict sequences using long-term context has many applications. However, practical and theoretical problems are found in training recurrent neural networks to perform tasks in which input/output dependencies span long intervals. Starting from a mathematical analysis of the problem, we consider and compare alternative algorithms and architectures on tasks for which the span of the input/output dependencies can be controlled. Results on the new algorithms show performance qualitatively superior to that obtained with backpropagation. 1 Introduction Recurrent neural networks have been considered to learn to map input sequences to output sequences. Machines that could efficiently learn such tasks would be useful for many applications involving sequence prediction, recognition or production. However, practical difficulties have been reported in training recurrent neural networks to perform tasks in which the temporal contingencies present in the input/output sequences span long intervals. In fact, we can prove that dynamical systems such as recurrent neural networks will be increasingly difficult to train with gradient descent as the duration of the dependencies to be captured increases. A mathematical analysis of the problem shows that either one of two conditions arises in such systems. In the first case, the dynamics of the network allow it to reliably store bits of information (with bounded input noise), but gradients (with respect to an error at a given time step) vanish exponentially fast as one propagates them ·also, AT&T Bell Labs, Holmdel, NJ 07733 7S 76 Bengio and Frasconi backward in time. In the second case, the gradients can flow backward but the system is locally unstable and cannot reliably store bits of information in the presence of input noise. In consideration of the above problem and the understanding brought by the theoretical analysis, we have explored and compared several alternative algorithms and architectures. Comparative experiments were performed on artificial tasks on which the span of the input/output dependencies can be controlled. In all cases, a duration parameter was varied, from T/2 to T, to avoid short sequences on which the algorithm could much more easily learn. These tasks require learning to latch, i.e. store bits of information for arbitrary durations (which may vary from example to example). Such tasks cannot be performed by Time Delay Neural Networks or by recurrent networks whose memories are gradually lost with time constants that are fixed by the parameters of the network. Of all the alternatives to gradient descent that we have explored, an approach based on a probabilistic interpretation of a discrete state space, similar to hidden Markov models (HMMs), yielded the most interesting results. 2 A Difficult Problem of Error Propagation Consider a non-autonomous discrete-time system with additive inputs, such as a recurrent neural network a with a continuous activation function: at = M(at-d + Ut and the corresponding autonomous dynamics at = M(at-d (1) (2) where M is a nonlinear map (which may have tunable parameters such as network weights), and at E R n and Ut E R m are vectors representing respectively the system state and the external input at time t. In order to latch a bit of state information one wants to restrict the values of the system activity at to a subset S of its domain. In this way, it will be possible to later interpret at in at least two ways: inside S and outside S. To make sure that at remains in such a region, the system dynamics can be chosen such that this region is the basin of attraction of an attractor X (or of an attractor in a sub-manifold or subspace of at's domain). To "erase" that bit of information, the inputs may push the system activity at out of this basin of attraction and possibly into another one. In (Bengio, Simard, & Frasconi, 1994) we show that only two conditions can arise when using hyperbolic attractors to latch bits of information in such a system. Either the system is very sensitive to noise, or the derivatives of the cost at time t with respect to the system activations ao converge exponentially to 0 as t increases. This situation is the essential reason for the difficulty in using gradient descent to train a dynamical system to capture long-term dependencies in the input/output sequences. A first theorem can be used to show that when the state at is in a region where IM'I > 1, then small perturbations grow exponentially, which can yield to a loss of the information stored in the dynamics of the system: Theorem 1 A ssume x is a point of R n such that there exists an open sphere U (x) centered on x for which IM'(z)1 > 1 for all z E U(x). Then there exist Y E U(x) such that IIM(x) - M(y) I > Ilx - YII· Credit Assignment through Time: Alternatives to Backpropagation 77 A second theorem shows that when the state at is in a region where IM'I < 1, the gradients propagated backwards in time vanish exponentially fast: Theorem 2 If the input Ut is such that a system remains robustly latched nM'(adl < 1) on attmctor X after time 0, then g:~ -t 0 as t -t 00. See proofs in (Bengio, Simard, & Frasconi, 1994). A consequence of these results is that it is generally very difficult to train a parametric dynamical system (such as a recurrent neural network) to learn long-term dependencies using gradient descent. Based on the understanding brought by this analysis, we have explored and compared several alternative algorithms and architectures. 3 Global Search Methods Global search methods such as simulated annealing can be applied to this problem, but they are generally very slow. We implemented the simulated annealing algorithm presented in (Corana, Marchesi, Martini, & Ridella, 1987) for optimizing functions of continuous variables. This is a "batch learning" algorithm (updating parameters after all examples of the training set have been seen). It performs a cycle of random moves, each along one coordinate (parameter) direction. Each point is accepted or rejected according to the Metropolis criterion (Kirkpatrick, Gelatt, & Vecchi, 1983). The simulated annealing algorithm is very robust with respect to local minima and long plateaus. Another global search method evaluated in our experiments is a multi-grid random search. The algorithm tries random points around the current solution (within a hyperrectangle of decreasing size) and accepts only those that reduce the error. Thus it is resistant to problems of plateaus but not as much resistant to problems of local minima. Indeed, we found the multi-grid random search to be much faster than simulated annealing but to fail on the parity problem, probably because of local minima. 4 Time Weighted Pseudo-Newton The time-weighted pseudo-Newton algorithm uses second order derivatives of the cost with respect to each of the instantiations of a weight at different time steps to try correcting for the vanishing gradient problem. The weight update for a weight Wi is computed as follows: (3) where Wit is the instantiation for time t of parameter Wi, 1} is a global learning rate and C(p) is the cost for pattern p. In this way, each (temporal) contribution to ~Wi(p) is weighted by the inverse curvature with respect to Wit . Like for the pseudo-Newton algorithm of Becker and Le Cun (1988) we prefer using a diagonal approximation of the Hessian which is cheap to compute and guaranteed to be positive definite. The constant J1 is introduced to prevent ~w from becoming very large (when I &;C~p) I W.! is very small). We found the performance of this algorithm to be better than the regular pseudo-Newton algorithm, which is better than the simple stochastic backpropagation algorithm, but all of these algorithms perform worse and worse as the length of the sequences is increased. 78 Bengio and Frasconi 5 Discrete Error Propagation The discrete error propagation algorithm replaces sigmoids in the network by discrete threshold units and attempts to propagate discrete error information backwards in time. The basic idea behind the algorithm is that for a simple discrete element such as a threshold unit or a latch, one can write down an error propagation rule that prescribes desired changes in the values of the inputs in order to obtain certain changes in the values of the outputs. In the case of a threshold unit, such a rule assumes that the desired change for the output of the unit is discrete (+2, o or -2). However, error information propagated backwards to such as unit might have a continuous value. A stochastic process is used to convert this continuous value into an appropriate discrete desired change. In the case of a self-loop, a clear advantage of this algorithm over gradient back-propagation through sigmoid units is that the error information does not vanish as it is repeatedly propagated backwards in time around the loop, even though the unit can robustly store a bit of information. Details of the algorithm will appear in (Bengio, Simard, & Frasconi, 1994). This algorithm performed better than the time-weighted pseudo-Newton, pseudo-Newton and back-propagation algorithms but the learning curve appeared very irregular, suggesting that the algorithm is doing a local random search. 6 An EM Approach to Target Propagation The most promising of the algorithms we studied was derived from the idea of propagating targets instead of gradients. For this paper we restrict ourselves to sequence classification. We assume a finite-state learning system with the state qt at time t taking on one of n values. Different final states for each class are used as targets. The system is given a probabilistic interpretation and we assume a Markovian conditional independence model. As in HMMs, the system propagates forward a discrete distribution over the n states. Transitions may be constrained so that each state j has a defined set of successors Sj. Ut Stat~ L State _;_ ·· .~_.~_1 ..... 0_j1_ ••• ( .. ·• •• _ .•.. __ n_e--lt/\rK: Figure 1: The proposed architecture Learning is formulated as a maximum likelihood problem with missing data. Missing variables, over which an expectation is taken, are the paths in state-space. The Credit Assignment through Time: Alternatives to Backpropagation 79 EM (Expectation/Maximization) or GEM (Generalized EM) algorithms (Dempster, Laird., & Rubin, 1977) can be used to help decoupling the influence of different hypothetical paths in state-space. The estimation step of EM requires propagating backward a discrete distribution of targets. In contrast to HMMs, where parameters are adjusted in an unsupervised learning framework, we use EM in a supervised fashion. This new perspective has been successful in training static models (Jordan & Jacobs, 1994). Transition probabilities, conditional on the current input, can be computed by a parametric function such as a layer of a neural network with softmax units. We propose a modular architecture with one subnetwork Nj for each state (see Figure 1). Each subnetwork is feedforward, takes as input a continuous vector of features Ut and has one output for each successor state, interpreted as P(qt = i I qt-l = j, Ut; 0), (j = 1, ... , n, i E Sj). 0 is a set of tunable parameters. Using a Markovian assumption, the distribution over states at time t is thus obtained as a linear combination of the outputs of the subnetworks, gated by the previously computed distribution: P(qt = i lui; 0) = L P(qt-l = j lui-I; O)P(qt = i I qt-l = j, Ut; 0) (4) j where ui is a subsequence of inputs from time 1 to t inclusively. The training algorithm looks for parameters 0 of the system that maximize the likelihood L of falling in the "correct" state at the end of each sequence: L(O) = II P(qTp = qj,p I uip; 0) (5) p where p ranges over training sequences, Tp the length of the pth training sequence, and qj, the desired state at time Tp. p An auxiliary function Q(O, Ok) is constructed by introducing as hidden variables the whole state sequence, hence the complete likelihood function is defined as follows: Lc(O) = IIp(qip luip;O) (6) p and (7) where at the k+lth EM (or GEM) iteration, Ok+l is chosen to maximize (or increase) the auxiliary function Q with respect to O. If the inputs are quantized and the subnetworks perform a simple look-up in a table of probabilities, then the EM algorithm can be used, i.e., aQ~/k) = 0 can be solved analytically. If the networks have non-linearities, (e.g., with hidden units and a softmax at their output to constrain the outputs to sum to 1), then one can use the GEM algorithm (which simply increases Q, for example with gradient ascent) or directly perform (preferably stochastic) gradient ascent on the likelihood. An extra term was introduced in the optimization criterion when we found that in many cases the target information would not propagate backwards (or would be diffused over all the states). These experiments confirmed previous results indicating a general difficulty of training fully connected HMMs, with the EM algorithm converging very often to poor local maxima of the likelihood. In an attempt to understand better the phenomenon, we looked at the quantities propagated forward and the quantities propagated backward (representing credit or blame) in the 80 Bengio and Frasconi training algorithm. We found a diffusion of credit or blame occurring when the forward maps (i.e. the matrix of transition probabilities) at each time step are such that many inputs map to a few outputs, i.e., when the ratio of a small volume in the image of the map with respect to the corresponding volume in the domain is small. This ratio is the absolute value of the determinant of the Jacobian of the map. Hence, using an optimization criterion that incorporates the maximization of the average magnitude of the determinant of the transition matrices, this algorithm performs much better than the other algorithms. Two other tricks were found to be important to help convergence and reduce the problem of diffusion of credit. The first idea is to use whenever possible a structured model with a sparse connectivity matrix, thus introducing some prior knowledge about the state-space. For example, applications of HMMs to speech recognition always rely on such structured topologies. We could reduce connectivity in the transition matrix for the 2-sequence problem (see next section for its definition) by splitting some of the nodes into two subsets, each specializing on one of the sequence classes. However, sometimes it is not possible to introduce such constraints, such as in the parity problem. Another trick that drastically improved performance was to use stochastic gradient ascent in a way that helps the training algorithm get out of local optima. The learning rate is decreased when the likelihood improves but it is increased when the likelihood remains flat (the system is stuck in a plateau or local optimum). As the results in the next section show, the performances obtained with this algorithm are much better than those obtained with the other algorithms on the two simple test problems that were considered. 7 Experimental Results We present here results on two problems for which one can control the span of input/output dependencies. The 2-sequence problem is the following: classify an input sequence, at the end of the sequence, in one of two types, when only the first N elements (N = 3 in our experiments) of this sequence carry information about the sequence class. Uniform noise is added to the sequence. For the first 6 methods (see Tables 1 to 4), we used a fully connected recurrent network with 5 units (with 25 free parameters). For the EM algorithm, we used a 7 -state system with a sparse connectivity matrix (an initial state, and two separate left-to-right submodels of three states each to model the two types of sequences). The parity problem consists in producing the parity of an input sequence of 1 's and -l's (i.e., a 1 should be produced at the final output if and only if the number of 1 's in the input is odd). The target is only given at the end of the sequence. For the first 6 methods we used a minimal size network (1 input, 1 hidden, 1 output, 7 free parameters). For the EM algorithm, we used a 2-state system with a full connectivity matrix. Initial parameters were chosen randomly for each trial. Noise added to the sequence was also uniformly distributed and chosen independently for each training sequence. We considered two criteria: (1) the average classification error at the end of training, i.e., after a stopping criterion has been met (when either some allowed number of function evaluations has been performed or the task has been learned), (2) the average number of function evaluations needed to reach the stopping criterion. In the tables, "p-n" stands for pseudo-Newton. Each column corresponds to a value of the maximum sequence length T for a given set of trials. The sequence length for a particular training sequence was picked randomly within T/2 and T. Numbers Credit Assignment through Time: Alternatives to Backpropagation 81 reported are averages over 20 or more trials. 8 Conclusion Recurrent networks and other parametric dynamical systems are very powerful in their ability to represent and use context. However, theoretical and experimental evidence shows the difficulty of assigning credit through many time steps, which is required in order to learn to use and represent context. This paper studies this fundamental problem and proposes alternatives to the backpropagation algorithm to perform such learning tasks. Experiments show these alternative approaches to perform significantly better than gradient descent. The behavior of these algorithms yields a better understanding of the central issue of learning to use context, or assigning credit through many transformations. Although all of the alternative algorithms presented here showed some improvement with respect to standard stochastic gradient descent, a clear winner in our comparison was an algorithm based on the EM algorithm and a probabilistic interpretation of the system dynamics. However, experiments on more challenging tasks will have to be conducted to confirm those results. Furthermore, several extensions of this model are possible, for example allowing both inputs and outputs, with supervision on outputs rather than on states. Finally, similarly to the work we performed for recurrent networks trained with gradient descent, it would be very important to analyze theoretically the problems of propagation of credit encountered in training such Markov models. Acknowledgements We wish to emphatically thank Patrice Simard, who collaborated with us on the analysis of the theoretical difficulties in learning long-term dependencies, and on the discrete error propagation algorithm. References S. Becker and Y. Le Cun. (1988) Improving the convergence of back-propagation learning with second order methods, Proc. of the 1988 Connectionist Models Summer School, (eds. Touretzky, Hinton and Sejnowski), Morgan Kaufman, pp. 29-37. Y. Bengio, P. Simard, and P. Frasconi. (1994) Learning long-term dependencies with gradient descent is difficult, IEEE Trans. Neural Networks, (in press). A. Corana, M. Marchesi, C. Martini, and S. Ridella. (1987) Minimizing multimodal functions of continuous variables with the simulated annealing algorithm, A CM Transactions on Mathematical Software, vol. 13, no. 13, pp. 262-280. A.P. Dempster, N.M. Laird, and D.B. Rubin. (1977) Maximum-likelihood from incomplete data via the EM algorithm, J. of Royal Stat. Soc., vol. B39, pp. 1-38. M.1. Jordan and R.A. Jacobs. (1994) Hierarchical mixtures of experts and the EM algorithm, Neural Computation, (in press). S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchio (1983) Optimization by simulated annealing, Science 220, 4598, pp.671-680. 82 Bengio and Frasconi Table 1: Final classification error for the 2-sequence problem wrt sequence length ac -prop p-n time-weighted p-n multigrid discrete err. prop. simulated anneal. EM 2 3 10 25 o 0 9 34 2 6 1 3 6 16 29 23 6 0 7 4 o 0 0 0 29 14 6 22 11 o Table 2: # sequence presentations for the 2-sequence problem wrt sequence length ac -prop p-n time-weighted p-n multigrid discrete err. prop. simulated anneal. EM . e 5.1e2 5.4e2 4.1e3 6.6e2 2.0e5 3.2e3 . e 1.1e3 4.3e2 5.8e3 1.3e3 3.ge4 4.0e3 . e 1.ge3 2.4e3 2.5e3 2.1e3 8.2e4 2.ge3 . e 2.6e3 2.ge3 3.ge3 2.1e3 7.7e4 3.2e3 . e 2.5e3 2.7e3 6.4e3 2.1e3 4.3e4 2.ge3 Table 3: Final classification error for the parity problem wrt sequence length 3 5 10 20 50 100 500 back-prop ~ ~U 41 ~~ 43p-n 3 25 41 44 40 47 time-weighted p-n 26 39 43 44 multigrid 15 44 45 discrete err. prop. 0 0 0 5 simulated anneal. 3 10 0 EM 0 6 0 14 0 12 Table 4: # sequence presentations for the parity problem wrt sequence length 3 5 9 20 50 100 500 back-prop 3.6e3 5.5e3 8.7e3 1.6e4 1.1e4 p-n 2.5e2 8.ge3 8.ge3 7.7e4 1.1e4 1.le5 time-weighted p-n 4.5e4 7.0e4 3.4e4 8.1e4 multigrid 4.2e3 1.5e4 3.1e4 discrete err. prop. 5.0e3 7.ge3 1.5e4 5.4e4 simulated anneal. 5.1e5 1.2e6 8.1e5 EM 2.3e3 1.5e3 1.3e3 3.2e3 2.6e3 3.4e3	1993	75
831	Analyzing Cross Connected Networks and Jeffrey L. Elman Thomas R. Shultz Department of Psychology & McGill Cognitive Science Centre McGill University Montreal, Quebec, Canada H3A IB 1 shultz@psych.mcgill.ca Center for Research on Language Department of Cognitive Science University of California at San Diego LaJolla, CA 92093-0126 U.S.A. elman@crl.ucsd.edu Abstract The non-linear complexities of neural networks make network solutions difficult to understand. Sanger's contribution analysis is here extended to the analysis of networks automatically generated by the cascadecorrelation learning algorithm. Because such networks have cross connections that supersede hidden layers, standard analyses of hidden unit activation patterns are insufficient. A contribution is defined as the product of an output weight and the associated activation on the sending unit, whether that sending unit is an input or a hidden unit, multiplied by the sign of the output target for the current input pattern. Intercorrelations among contributions, as gleaned from the matrix of contributions x input patterns, can be subjected to principal components analysis (PCA) to extract the main features of variation in the contributions. Such an analysis is applied to three problems, continuous XOR, arithmetic comparison, and distinguishing between two interlocking spirals. In all three cases, this technique yields useful insights into network solutions that are consistent across several networks. 1 INTRODUCTION Although neural network researchers are typically impressed with the performance achieved by their learning networks, it often remains a challenge to explain or even characterize such performance. The latter difficulties stem principally from the complex non-linear properties of neural nets and from the fact that information is encoded in a form that is distributed across many weights and units. The problem is exacerbated by the fact that multiple nets generate unique solutions depending on variation in both starting states and training patterns. Two techniques for network analysis have been applied with some degree of success, focusing respectively on either a network's weights or its hidden unit activations. Hinton (e.g., Hinton & Sejnowski, 1986) pioneered a diagrammatic analysis that involves plotting a network's learned weights. Occasionally, such diagrams yield interesting insights but often, because of the highly distributed nature of network representations, the most notable features of such analyses are the complexity of the pattern of weights and its variability across multiple networks learning the same problem. 1117 1118 Shultz and Elman Statistical analysis of the activation patterns on the hidden units of three layered feedforward nets has also proven somewhat effective in understanding network performance. The relations among hidden unit activations, computed from a matrix of hidden units x input patterns, can be subjected to either cluster analysis (Elman, 1990) or PCA (Elman, 1989) to determine the way in which the hidden layer represents the various inputs. However, it is not clear how this technique should be extended to multi-layer networks or to networks with cross connections. Cross connections are direct connections that bypass intervening hidden layers. Cross connections typically speed up learning when used in static back-propagation networks (Lang & Witbrock, 1988) and are an obligatory and ubiquitous feature of some generative learning algorithms, such as cascade-correlation (Fahlman & Lebiere, 1990). Generative algorithms construct their own network topologies as they learn. In cascade-correlation, this is accomplished by recruiting new hidden units into the network, as needed, installing each on a separate layer. In addition to layer-to-layer connections, each unit in a cascadecorrelation network is fully cross connected to all non-adjacent layers downstream. Because such cross connections carry so much of the work load, any analysis restricted to hidden unit acti vations provides a partial picture of the network solution at best. Generative networks seem to provide a number of advantages over static networks, including more principled network design, leaner networks, faster learning, and more realistic simulations of hwnan cognitive development (Fahlman & Lebiere, 1990; Shultz, Schmidt, Buckingham, & Mareschal, in press). Thus, it is important to understand how these networks function, even if they seem impervious to standard analytical tools. 2 CONTRIBUTION ANALYSIS One analytical technique that might be adapted for multi-layer, cross connected nets is contribution analysis (Sanger, 1989). Sanger defined a contribution as the triple product of an output weight, the activation of a sending unit, and the sign of the output target for that input. He argued that contributions are potentially more informative than either weights alone or hidden unit activations alone. A large weight may not contribute much if it is connected to a sending unit with a small activation. Likewise, a large sending activation may not contribute much if it is connected via a small weight. In contrast, considering a full contribution, using both weight and sending activation, would more likely yield valid comparisons. Sanger (1989) applied contribution analysis to a small version of NETtalk, a net that learns to convert written English into spoken English (Sejnowski & Rosenberg, 1987). Sanger's analysis began with the construction of an output unit x hidden unit x input pattern array of contributions. Various two-dimensional slices were taken from this threedimensional array, each representing a particular output unit or a particular hidden unit. Each two-dimensional slice was then subjected to PCA, yielding information about either distributed or local hidden unit responsibilities, depending on whether the focus was on an individual output unit or individual hidden unit, respectively. 3 CONTRIBUTION ANALYSIS FOR MULTI· LAYER, CROSS CONNECTED NETS We adapted contribution analysis for use with multi-layered, cross connected cascadecorrelation nets. Assume a cascade-correlation network with j units (input units + hidden units) and k output units, being trained with i input patterns. There are j x k output weights in such a network, where an output weight is defined as any weight connected to Analyzing Cross-Connected Networks 1119 an output unit. A contribution c for a particular ijk combination is defined as Cijk = Wjk aij 2tki (1) where Wjk is the weight connecting sending unit j with output unit k, aij is the activation of sending unit j given input pattern i, and tki is the target for output unit k given input pattern i. The term 2tki adjusts the sign of the contribution so that it provides a measure of correctness. That is, positive contributions push the output activation towards the target, whereas negative contributions push the output activation away from the target. In cascade-correlation, sigmoid output units have targets of either -0.5 or +0.5. Hence, mUltiplying a target by 2 yields a positive sign for positive targets and a negative sign for negative targets. Our term 2tki is analogous to Sanger's (1989) term 2tik - 1, which is appropriate for targets of 0 and I, commonly used in back-propagation learning. In contrast to Sanger's (1989) three-dimensional array of contributions (output unit x hidden unit x input pattern). we begin with a two-dimensional output weight (k * j) x input pattern (i) array of contributions. This is because we want to include all of the contributions coming into the output units, including the cross connections from more than one layer away. Since we begin with a two-dimensional array. we do not need to employ the somewhat cumbersome slicing technique used by Sanger to isolate particular output or hidden units. Nonetheless. as will be seen, our technique does allow the identification of the roles of specific contributions. 4 PRINCIPAL COMPONENTS ANALYSIS Correlations among the various contributions across input patterns are subjected to PCA. PCA is a statistical technique that identifies significant dimensions of variation in a multi-dimensional space (Flury, 1988). A component is a line of closest fit to a set of points in multi-dimensional space. The goal of PCA is to summarize a multivariate data set using as few components as possible. It does this by taking advantage of possible correlations among the variables (contributions, in our case). We apply PCA to contributions, as defined in Equation I, taken from networks learning three different problems: continuous XOR, arithmetic comparisons. and distinguishing between interlocking spirals. The contribution matrix for each net, as described in section 3, is subjected to PCA using 1.0 as the minimum eigenvalue for retention. Varimax rotation is applied to improve the interpretability of the solution. Then the scree test is applied to eliminate components that fail to account for much of the variance (Cattell, 1966). In cases where components are eliminated. the analysis is repeated with the correct number of components. again with a varimax rotation. Component scores for the retained components are plotted to provide an indication of the function of the components. Finally. component loadings for the various contributions are examined to determine the roles of the contributions from hidden units that had been recruited into the networks. 5 APPLICATION TO THE CONTINUOUS XOR PROBLEM The simplicity of binary XOR and the small number of training patterns (four) renders application of contribution analysis superfluous. However, it is possible to construct a continuous version of the XOR problem that is more suitable for contribution analysis. We do this by dividing the input space into four quadrants. Input values are incremented in steps of 0.1 starting from 0.0 up to 1.0, yielding 100 x, y input pairs. Values of x up to 0.5 combined with values of y above 0.5 produce a positive output target (0.5), as do values of x above 0.5 combined with values of y below 0.5. Input pairs in the other two quadrants yield a negative output target (-0.5). 1120 Shultz and Elman Three cascade-correlation nets are trained on this problem. Each of the three nets generates a unique solution to the continuous XOR problem, with some variation in number of hidden units recruited. PCA of contributions yields different component loadings across the three nets and different descriptions of components. Yet with all of that variation in detail, it is apparent that all three nets make the same three distinctions that are afforded by the training patterns. The largest distinction is that which the nets are explicitly trained to make, between positive and negative outputs. Two components are sufficient to describe the representations. Plots of rotated component scores for the 100 training patterns cluster into four groups of 25 points, each cluster corresponding to one of the four quadrants described earlier. Component loadings for the various contributions on the two components indicate that the hidden units play an interactive and distributed role in separating the input patterns into their respective quadrants. 6 APPLICATION TO COMPARATIVE ARITHMETIC A less well understood problem than XOR in neural net research is that of arithmetic operations, such as addition and multiplication. What has a net learned when it learns to add, or to multiply, or to do both operations? The non-linear nature of multiplication makes it particularly interesting as a network analysis problem. The fact that several psychological simulations using neural nets involve problems of linear and non-linear arithmetic operations enhances interest in this sort of problem (McClelland, 1989; Shultz et al., in press). We designed arithmetic comparison tasks that provided interesting similarities to some of the psychological simulations. In particular, instead of simply adding or multiplying, the nets learn to compare sums or products to some value and then output whether the sum or product is greater than, less than, or equal to that comparative value. The addition and multiplication tasks each involve three linear input units. The first two input units each code a randomly selected integer in the range from 0 to 9, inclusive. The third input unit codes a randomly selected comparison integer. For addition problems, the comparison values are in the range of 0 to 19, inclusive; for multiplication the range is 0 to 82, inclusive. Two output units code the results of the comparison. Target outputs of 0.5 and -0.5 represent that the results of the arithmetic operation are greater than the comparison value, targets of -0.5 and 0.5 represent less than, and targets of 0.5 and 0.5 represent equal to. For problems involving both addition and multiplication, a fourth input unit codes the type of arithmetic operation to be performed: 0 for addition, 1 for multiplication. Nets trained on either addition or multiplication have 100 randomly selected training patterns, with the restriction that 45 of them have correct answers of greater than, 45 have correct answers of less than, and 10 have correct answers of equal to. The latter constraints are designed to reduce the natural skew of comparative values in the high direction on multiplication problems. Nets trained on both addition and multiplication have 100 randomly selected addition problems and 100 randomly selected multiplication problems, subject to the constraints just described. We trained three nets on addition, three on multiplication, and three on both addition and multiplication. 6.1 RESULTS FOR ADDITION PCA of contributions in all three addition nets yield two significant components. In each of the three nets, the component scores form three clusters, representing the three correct answers. In all three nets, the first component distinguishes greater than from less than answers and places equal to answers in the middle; the second component distinguishes Analyzing Cross-Connected Networks 1121 equal to from unequal to answers. The primary role of the hidden unit in these nets is to distinguish equality from inequality. The hidden unit is not required to perform addition per se in these nets, which have additive activation functions. 6.2 RESUL TS FOR MULTIPLICATION PCA applied to the contributions in the three multiplication nets yields from 3 to 4 significant components. Plots of rotated component scores show that the first component separates greater than from less than outputs, placing equal to outputs in the middle. Other components further differentiate the problems in these categories into several smaller groups that are related to the particular values being multiplied. Rotated component loadings indicate that component 1 is associated not only with contributions coming from the bias unit and the input units, but also with contributions from some hidden units. This underscores the need for hidden units to capture the non-linearities inherent to multiplication. 6.3 RESULTS FOR BOTH ADDITION AND MULTIPLICATION PCA of contributions yields three components in each of the three nets taught to do both addition and multiplication. In addition to the familiar distinctions between greater than, less than, and equal to outputs found in nets doing either addition or multiplication, it is of interest to determine whether nets doing both operations distinguish between adding and multiplying. Figure 1 shows the rotated component scores for net 1. Components 1 and 2 (accounting for 30.2% and 21.9% of the variance, respectively) together distinguish greater than answers from the rest. Component 3, accounting for 20.2% of the variance, separates equal to answers from less than answers and multiplication from addition for greater than answers. Together, components 2 and 3 separate multiplication from addition for less than answers. Results for the other two nets learning both multiplication and addition comparisons are essentially similar to those for net 1. 2 ~ ~ 0 s:: o c.. E -1 o v -2 2 Component 2 ~ .,+ ... .• -... " x> • . . ... . -':/ . .... ... x< •• , .-. . . I . .. . , .... "'" I r .• _. 'I. ••• ( +< .~ ==.1 ., 2 -3 -3 Component 3 Figure 1. Rotated component scores for a net doing both addition and multiplication. 6.4 DISCUSSION OF COMPARATIVE ARITHMETIC As with continuous XOR, there is considerable variation among networks learning comparative arithmetic problems. Varying numbers of hidden units are recruited by the networks and different types of components emerge from PCA of network contributions. In some cases, clear roles can be assigned to particular components, but in other cases, separation of input patterns relies on interactions among the various components. 1122 Shultz and Elman Yet with all of this variation, it is apparent that the nets learn to separate arithmetic problems according to features afforded by the training set. Nets learning either addition or multiplication differentiate the problems according to answer types: greater than, less than, and equal to. Nets learning both arithmetic operations supplement these answer distinctions with the operational distinction between adding and multiplying. 7 APPLICATION TO THE TWO-SPIRALS PROBLEM We next apply contribution analysis to a particularly difficult discrimination problem requiring a relatively large number of hidden units. The two-spirals problem requires the net to distinguish between two interlocking spirals that wrap around their origin three times. The standard version of this problem has two sets of 97 continuous-valued x, y pairs, each set representing one of the spirals. The difficulty of the two-spirals problem is underscored by the finding that standard back-propagation nets are unable to learn it (Wieland, unpublished, cited in Fahlman & Lebiere, 1990). The best success to date on the two-spirals problem was reported with cascade-correlation nets, which learned in an average of 1700 epochs while recruiting from 12 to 19 hidden units (Fahlman & Lebiere, 1990). The relative difficulty of the two-spirals problem is undoubtedly due to its high degree of non-linearity. It suited our need for a relatively difficult, but fairly well understood problem on which to apply contribution analysis. We ran three nets using the 194 continuous x, y pairs as inputs and a single sigmoid output unit, signaling -0.5 for spiral 1 and 0.5 for spiral 2. Because of the relative difficulty of interpreting plots of component scores for this problem, we focus primarily on the extreme component scores, defined as less than -lor greater than 1. Those x, y input pairs with extreme component scores on the first two components for net 1 are plotted in Figure 2 as filled points on the two spirals. There are separate plots for the positive and negative ends of each of the two components. The fllled points in each quadrant of Figure 2 define a shape resembling a tilted hourglass covering approximately one-half of the spirals. The positive end of component 1 can be seen to focus on the northeast sector of spiral 1 and the southwest sector of spiral 2. The negative end of component 1 has an opposite focus on the northeast sector of spiral 2 and the southwest sector of spiral 1. Component 2 does precisely the opposite of component 1: its positive end deals with the southeast sector of spiral 1 and the northwest sector of spiral 2 and its negative end deals with the southeast sector of spiral 2 and the northwest sector of spiral 1. Comparable plots for the other two nets show this same hourglass shape, but in a different orientation. The networks appear to be exploiting the symmetries of the two spirals in reaching a solution. Examination of Figure 2 reveals the essential symmetries of the problem. For each x, y pair, there exists a corresponding -x, -y pair 180 degrees opposite and lying on the other spiral. Networks learn to treat these mirror image points similarly, as revealed by the fact that the plots of extreme component scores in Figures 2 are perfectly symmetrical across the two spirals. If a point on one spiral is plotted, then so is the corresponding point on the other spiral, 180 degrees opposite and at the same distance out from the center of the spirals. If a trained network learns that a given x, y pair is on spiral 1, then it also seems to know that the -x, -y pair is on spiral 2. Thus, it make good sense for the network to represent these opposing pairs similarly. Recall that contributions are scaled by the sign of their targets, so that all of the products of sending activations and output weights for spiral 1 are multiplied by -1. This is to ensure that contributions bring output unit activations close to their targets in proportion Analyzing Cross-Connected Networks 1123 to the size of the contribution. Ignoring this scaling by target, the networks possess sufficient information to separate the two spirals even though they represent points of the two spirals in similar fashion. The plot of the extreme component scores in Figure 2 suggests that the critical information for separating the two spirals derives mainly from the signs of the input activations. Because scaling contributions by the sign of the output target appears to obscure a full picture of network solutions to the two-spirals problem, there may be some value in using unsealed contributions in network analysis. Use of unscaled contributions also could be justified on the grounds that the net has no knowledge of targets as it represents a particular problem; target information is only used in the error correction process. A disadvantage of using un scaled contributions is that one cannot distinguish contributions that facilitate vs. contributions that inhibit reaching a relatively error free solution. The symmetry of these network representations suggests a level of systematicity that is, on some accounts, not supposed to be possible in neural nets (Fodor & Pylyshyn, 1988). Whether this representational symmetry reflects systematicity in performance is another matter. One empirical prediction would be that as a net learns that x, y is on one spiral, it also learns at about the same time that -x, -y is on the other spiral. If confirmed, this would demonstrate a clear case of systematic cognition in neural nets. 8 GENERAL DISCUSSION Performing PCA on network contributions is here shown to be a useful technique for understanding the performance of networks constructed by the cascade-correlation learning algorithm. Because cascade-correlation nets typically possess multiple hidden layers and are fully cross connected, they are difficult to analyze with more standard methods emphasizing activation patterns on the hidden units alone. Examination of their weight patterns is also problematic, particularly in larger networks, because of the highly distributed nature of the net's representations. Analyzing contributions, in contrast to either hidden unit activations or weights, is a naturally appealing solution. Contributions capture the influence coming into output units both from adjacent hidden units and from distant, cross connected hidden and input units. Moreover, because contributions include both sending activations and connecting weights, they are not unduly sensitive to one at the expense of the other. In the three domains examined in the present paper, PCA of the network contributions both confirm some expected results and provide new insights into network performance. In all cases examined, the nets succeed in drawing all of the important distinctions in their representations that are afforded by the training patterns, whether these distinctions concern the type of output or the operation being performed on the input. In combination with further experimentation and analysis of network weights and activation patterns, this technique could help to provide an account of how networks accomplish whatever it is they learn to accomplish. It might be of interest to apply the present technique at various points in the learning process to obtain a developmental trace of network performance. Would all networks learning under the same constraints progress through the same stages of development, in terms of the problem distinctions they are able to make? This would be of particular interest to network simulations of human cognitive development, which has been claimed to be stage-like in its progressions. 1124 Shultz and Elman -q The present technique could also be useful in predicting the results of lesioning experiments on neural nets. If the role of a hidden unit can be identified by its association with a particular principal component, then it could be predicted that lesioning this unit would impair the function served by the component. Acknowledgments This research was supported by the Natural Sciences and Engineering Research Council of Canada and the MacArthur Foundation. Helpful comments were provided by Scott Fahlman, Denis Mareschal, Yuriko Oshima-Takane, and Sheldon Tetewsky. References Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, t, 245-276. Elman, 1. L. (1989). Representation and structure in connectionist models. CRL Technical Report 8903, Center for Research in Language, University of California at San Diego. Elman, J. L. (1990). Finding structure in time. Cognitive Science, 14, 179-211. Fahlman, S. E., & Lebiere, C. (1990.) The Cascade-Correlation learning architecture. In D. Touretzky (Ed.), Advances in neural information processing systems 2, (pp. 524532). Mountain View, CA: Morgan Kaufmann. Rury, B. (1988). Common principal components and related multivariate models. New York: Wesley. Fodor, J., & Pylyshyn, Z. (1988). Connectionism and cognitive architecture: A critical analysis. Cognition, 28,3-71. Hinton, G. E., & Sejnowski, T. J. (1986). Learning and relearning in Boltzmann machines. In D. E. Rume1hart & J. L. McClelland (Eds.), Parallel distrihuted processing: Explorations in the microstructure of cognition. Volwne 1: Foundalion.~, pp. 282-317. Cambridge, MA: MIT Press. Lang, K. J., & Witbrock, M. J. (1988). Learning to tell two spirals apart. In D. Touretzky, G. Hinton, & T. Sejnowski (Eds)., Proceedings of the Connectioni.rt Models Summer School, (pp. 52-59). Mountain View, CA: Morgan Kaufmann. McClelland, 1. L. (1989). Parallel distributed processing: Implications for cognition and development. In Morris, R. G. M. (Ed.), Para/lei distributed processing: Implications for psychology and neurobiology, pp. 8-45. Oxford University Press. Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations by error propagation. In D. E. Rumelhart & J. L. McClelland (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition. Volume 1: Foundations, pp. 318-362. Cambridge, MA: MIT Press. Sanger, D. (1989). Contribution analysis: A technique for assigning responsibilities to hidden units in connectionist networks. Connection Science, I, 115-138. Sejnowski, T. J., & Rosenberg, C. R. (1987). Parallel networks that learn to pronounce English text. Complex Systems, I, 145-168. Shultz, T. R., Schmidt, W. C., Buckingham, D., & Mareschal, D. (In press). Modeling cognitive development with a generative connectionist algorithm. In G. Halford & T. Simon (Eds.), Developing cognitive competence: New approaches to process mndeling. Hillsdale, NJ: Erlbaum. ·4 -8 ... 0 " IIII'ph'! o AlI.ph12 • El1rem. spiral 1 0 • Eldrltm. spire' 2 o Figure 2. Extreme rotated component scores for a net on the two-spirals problem.	1993	76
832	Recognition-based Segmentation of On-line Cursive Handwriting Nicholas S. Flann Department of Computer Science Utah State University Logan, UT 84322-4205 flannGnick.cs.usu.edu Abstract This paper introduces a new recognition-based segmentation approach to recognizing on-line cursive handwriting from a database of 10,000 English words. The original input stream of z, y pen coordinates is encoded as a sequence of uniform stroke descriptions that are processed by six feed-forward neural-networks, each designed to recognize letters of different sizes. Words are then recognized by performing best-first search over the space of all possible segmentations. Results demonstrate that the method is effective at both writer dependent recognition (1.7% to 15.5% error rate) and writer independent recognition (5.2% to 31.1% error rate). 1 Introduction With the advent of pen-based computers, the problem of automatically recognizing handwriting from the motions of a pen has gained much significance. Progress has been made in reading disjoint block letters [Weissman et. ai, 93]. However, cursive writing is much quicker and natural for humans, but poses a significant challenge to pattern recognition systems because of its variability, ambiguity and need to both segment and recognize the individual letters. Recent techniques employing selforganizing networks are described in [Morasso et. ai, 93] and [Schomaker, 1993]. This paper presents an alternative approach based on feed-forward networks. On-line handwriting consists of writing with a pen on a touch-terminal or digitizing 777 778 Flann (a) (b) (c) (d) (e) Figure 1: The five principal stages of preprocessing: (a) The original data, z, Y values sampled every 10mS. (b) The slant is normalized through a shear transformation; (c) Stroke boundaries are determined at points where y velocity equals 0 or pen-up or pen-down events occur; (d) Delayed strokes are reordered and associated with corresponding strokes of the same letters; (e) Each stroke is resampled in space to correspond to exactly 8 points. Note pen-down strokes are shown as thick lines, pen-up strokes as thin lines. Recognition-Based Segmentation of On-Line Cursive Handwriting 779 tablet. The device produces a regular stream of z, y coordinates, describing the positions of the pen while writing. Hence the problem of recognizing on-line cursively written words is one of mapping a variable length sequence of z, y coordinates to a variable length sequence of letters. Developing a system that can accurately perform this mapping faces two principal problems: First, because handwriting is done with little regularity in speed, there is unavoidable variability in input size. Second, because no pen-up events or spatial gaps signal the end of one letter and the beginning of the next, the system must perform both segmentation and recognition. This second problem necessitates the development of a recognition-based segmentation approach. In [Schenkel et al., 93] one such approach is described for connected block letter recognition where the system learns to recognize segmentation points. In this paper an alternative method is presented that first performs exhaustive recognition then searches the space of possible segmentations. The remainder of the paper describes the method in more detail and presents results that demonstrate its effectiveness at recognizing a variety of cursive handwriting styles. 2 Methodology The recognition system consists of three subsystems: (a) the preprocessor that maps the initial stream of z, y coordinates to a stream of stroke descriptions; (b) the letter classifier that learns to recognize individual letters of different size; and ( c) the word finder that performs recognition-based segmentation over the output of the letter classifier to identify the most likely word written. 2.1 Preprocessing The preprocessing stage follows steps outlined in [Guerfali & Plamondon, 93] and is illustrated in Figure 1. First the original data is smoothed by passing it through a low-pass filter, then reslanted to make the major stroke directions vertical. This is achieved by computing the mean angle of all the individual lines then applying a shear transformation to remove it. Second, the strokes boundaries are identified as points when if = 0 or when the pen is picked up or put down. Zero y velocity was chosen rather than minimum absolute velocity [Morasso et. ai, 93] since it was found to be more robust. Third, delayed strokes such as those that dot an i or cross a t are reordered to be associated with their corresponding letter. Here the delayed stroke is placed to immediately follow the closest down stroke and linked into the stroke sequence by straight line pen-up strokes. Fourth, each stroke is resampled in the space domain (using linear interpolation) so as to represent it as exactly eight z, y coordinates. Finally the new stream of z, y coordinates is converted to a stream of 14 feature values. Eight of these features are similar to those used in [Weissman et. ai, 93], and represent the angular acceleration (as the sin and cos of the angle), the angular velocity of the line (as the sin and cos of the angle), the z, y coordinates (z has a linear ramp removed), and first differential ox,Oy. One feature denotes whether the pen was down or up when the line was drawn. The remaining features encode more abstract information about the stroke. 780 Flann • 32 Figure 2: The pyramid-style architecture of the network used to recognize 2 stroke letters. The input size is 32 x 14; 32 is from the 4 input strokes (each represented by 8 resampled points), two central strokes from the letter and the 2 context strokes, one each side; 14 is from the number of features employed to represent each point. Not all the receptive fields are shown. The first hidden layer consists of 7 fields, 4 over each stroke and 3 more spanning the stroke boundaries. The next hidden layer consists of 5 fields, each spanning 3 x 20 inputs. The output is a 32 bit error-correcting code. J.) ~"I v~c.'fJcr/ "~lI"")c' (/" .p/ ~'l q\) /.h.l/ ..... ')"/\1\.1 Jt·z./I' l'-V..c..A.U,I A {jAAVv ....... t A ~...J)'l~.n",l1v..-t..>...,--ZUv ..... U.,.,,,( .lI\ ..,. ..n...d t...rt '( f,l.-v tV 'i> r' 1/"J1. tt I'-' V (,fJ 1\./11 \....-"\ ~ r.r S)y' U Iv' hV (..; .-Y.w r .M/l.JYV.JJ. ~ At.. ~ fA. "'"'I.t.N. ~ .I .. L.r.,.. U. f" I' ry \{\J?'\J)1 LA ~ \..0.m "Yi.IW11. ... ~ W ~.-;,(...vy..p/v~\.6\~ J..v rn ~ ~d~ AlA t bY)U> _~.bA ~ u...Yv:.)~ )AA. \.Oe!;\IVY' M1~~ /\.$\ t.W f1-~~, Figure 3: Examples of the class "other" for stroke sizes 1 though 6. Each letter is a random fragment of a word, such that it is not an alphabetic letter. Recognition-Based Segmentation of On-Line Cursive Handwriting 781 2.2 Letter Recognition The letter classifier consists of six separate pyramid-style neural-networks, each with an architecture suitable for recognizing a letter of one through six strokes. A neural network designed to recognize letters of size j strokes encodes a mapping from a sequence of j + 2 stroke descriptions to a 32 bit error-correcting code [Dietterich & Bakiri, 91]. Experiments have shown this use of a context window improves performance, since the allograph of the current letter is dependent on the allographs of the previous and following letters. The network architecture for stroke size two is illustrated in Figure 2. The architecture is similar to a time-delayed neural-network [Lang & Waibel, 90] in that the hierarchical structure enables different levels of abstract features to be learned. However, the individual receptive fields are not shared as in a TDNN, since translational variance is not problem and the sequence of data is important. The networks are trained using 80% of the raw data collected. This set is further divided into a training and a verification set. All training and verification data is preprocessed and hand segmented, via a graphical interface, into letter samples. These are then sorted according to size and assembled into distinct training and verification sets. It is often the case that the same letter will appear in multiple size files due to variability in writing and different contexts (such as when an 0 is followed by a 9 it is at least a 3 stroke allograph, while an 0 followed by an 1 is usually only a two stroke allograph). Included in these letter samples are samples of a new letter class "other," illustrated in Figure 3. Experiments demonstrated that use of an "other" class tightens decision boundaries and thus prevents spurious fragments-of which there are many during performance-from being recognized as real letters. Each network is trained using back-propagation until correctness on the verification set is maximized, usually requiring less than 100 epochs. 2.3 Word Interpreter To identify the correct word, the word interpreter explores the space of all possible segmentations of the input stroke sequence. First, the input sequence is partitioned into all possible fragments of size one through six, then the appropriately sized network is used to classify each fragment. An example of this process is illustrated as a matrix in Figure 4(a). The word interpreter then performs a search of this matrix to identify candidate words. Figure 4(b) and Figure 4(c) illustrates two sets of candidate words found for the example in Figure 4(a). Candidates in this search process are generated according to the following constraints: • A legal segmentation point of the input stream is one where no two adjacent fragments overlap or leave a gap. To impose this constraint the i'th fragment of size j may be extended by all of the i + j fragments, if they exist. • A legal candidate letter sequence must be a subsequence of a word in the provided lexicon of expected English words. 782 Flann UiL-tiollary Siz .. - (J DktioJliU)' Siu-107.a!:l 1»AAE 1)ARE 2)ARE 2)ARf 3)ARf &)QAf S)ORf Figure 4: (a) The matrix of fragments and their classifications that is generated by applying the letter recognizers to a sample of the word are. The original handwriting sample, following preprocessing, is given at the top of the matrix. The bottom row of the matrix corresponds to all fragments of size one (with zero overlap), the second row to all fragments of size two (with an overlap of one stroke) etc. The column of letters in each fragment box represents the letter classifications generated by the neural network of appropriate size. The higher the letter in the column, the more confident the classification. Those fragments with no high scoring letter were recognized as examples of the class "other." (b) The first five candidates found by the word recognizer employing no lexicon. The first column is the word recognized, the second column is the score for that word, the third is the sequence of fragments and their classifications. (c) The first five candidates found by the word recognizer employing a lexicon of 10748 words. Recognition-Based Segmentation of On-Line Cursive Handwriting 783 In a forward search, a candidate of size n consists of: (a) a legal sequence of fragments It, 12, .. . , In that form a prefix of the input stroke sequence, (b) a sequence of letters It, 12 , • •• , In that form a prefix of an English word from the given dictionary and (c) a score s for this candidate, defined as the average letter recognition error: E?-l 6(1., Ii) 8 = ==---:.,;...;.,,;.~ n where 6(/i, Ii) is the hamming distance between letter Is's code and the actual code produced by the neural network when given Ii as input. This scoring function is the same as employed in [Edelman et. ai, 90]. The best word candidate is one that conforms with the constraints and has the lowest score. Although this is a reasonable scoring function, it is easy to show that it is not admissible when used as an evaluation function in forward search. With a forward search, problems arise when the prefix of the correct word is poorly recognized. To help combat this problem without greatly increasing the size of the search space, both forward and backward search is performed. Search is initiated by first generating all one letter and one fragment prefix and suffix candidates. Then at each step in the search, the candidate with the lowest score is expanded by considering the cross product of all legal letter extensions (according to the lexicon) with all legal fragment extensions (according to the fragment-sequence constraints). The list of candidates is maintained as a heap for efficiency. The search process terminates when the best candidate satisfies: (1) the letter sequence is a complete word in the lexicon and (2) the fragment sequence uses all the available input strokes. The result of this bi-directional search process is illustrated in Figure 4(a)(b), where the five best candidates found are given for no lexicon and a large lexicon. The use of a 10,748 word lexicon eliminates meaningless fragment sequences, such as cvre, which is a reasonable segmentation, but not in the lexicon. The first two candidates are the same fragment sequence, found by the two search directions. The third candidate with a 10,748 word dictionary illustrates an alternative segmentation of the correct word. This candidate was identified by a backward search, but not a forward search, due to the poor recognition of the first fragment. 3 Evaluation To evaluate the system, 10 writers have provided samples of approximately 100 words picked by a random process, biased to better represent uncommon letters. Two kinds of experiments were performed. First, to test the ability of the system to learn a variety of writing styles, the system was tested and trained on distinct sets of samples from the same writer. This experiment was repeated 10 times, once for each writer. The error rate varied between 1.7% and 15.5%, with a mean of 6.2%, when employing a database of 10,748 English words. The second experiments tested the ability of the system to recognize handwriting of a writer not represented in the training set. Here the set of 10 samples were split into two sets, the training set of 9 writers with the remaining 1 writer being the test set. The error rate was understandably higher, varying between 5.2% and 31.1%, with a mean of 10.8%, when employing a database of 10,748 English words. 784 Flann 4 Summary This paper has presented a recognition-based segmentation approach for on-line cursive handwriting. The method is very flexible because segmentation is performed following exhaustive recognition. Hence, we expect the method to be successful with more natural unconstrained writing, which can include mixed block, cursive and disjoint letters, diverse orderings of delayed strokes, overwrites and erasures. Acknowledgements This work was supported by a Utah State University Faculty Grant. Thanks to Balaji Allamapatti, Rebecca Rude and Prashanth G Bilagi for code development. References [Dietterich & Bakiri, 91] Dietterich, T., G. & Bakiri, G. (1991). Error correcting output codes: A general method for improving multiclass inductive learning programs, in Proceedings of the Ninth National Conference on Artificial Intelligence, Vol 2, pp 572-577. [Edelman et. al,90] Edelman S., Tamar F., and Ullman S. (1990). Reading cursive handwriting by alignment of letter prototypes. International Journal of Computer Vision, 5:3, 303-331. [Guerfali & Plamondon, 93] Guerfali W. & Plamondon R. (1993). Normalizing and restoring on-line handwriting. Pattern Recognition, Vol. 26, No.3, pp. 419431. [Guyon et. ai, 90] Guyon I., Albrecht P., Le Cun Y., Denker J. & Hubbard W. (1991). Design of a neural network character recognizer for a touch terminal. Pattern Recognition, Vol. 24, No.2. pp. 105-119. [Lang & Waibel, 90] Lang K., J. & Waibel A., H. (1990). A time-delayed neural network architecture for isolated word recognition, Neural Networks, Vol 3, pp 33-43. [Morasso et. ai, 93] Morasso P., Barberis, S. Pagliano S. & Vergano, D. (1993). Recognition experiments of cursive dynamic handwriting with selforganizing networks. Pattern Recognition, Vol. 26, No.3, pp. 451-460. [Schenkel et al., 93] Schenkel M., Weissman H., Guyon I., Nohl C., & Henderson D. (1993). Recognition-based segmentation of on-line hand-printed words. In S. J. Hanson, J. D. Cowan & C. L. Giles (Eds), Advances in Neural Information Processing Systems, 5,723-730. San Mateo, CA: Morgan Kaufmann. [Schomaker, 1993] Schomaker L. (1993). Using stroke or character based selforganizing maps in the recognition of on-line connected cursive script. Pattern Recognition, Vol. 26. No.3., pp. 442-450. [Srihari & Bozinovic, 87] Srihari S. N. & Bozinovic R. M. (1987). A multi-level perception approach to reading cursive script. Artificial Intelligence, 33 217-255. [Weissman et. ai, 93] Weissman H., Schenkel M., Guyon I., Nohl C. & Henderson D. (1993). Recognition-based segmentation of on-line run-on hand printed words: input vs. output segmentation. Pattern Recognition.	1993	77
833	Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors Tadashi Shibatat Koji Kotani t Takeo Yamashitat Hiroshi Ishii Hideo Kosakat and Tadahiro Ohmi Department of Electronic Engineering Tohoku University Aza-Aoba, Aramaki, Aobaku, Sendai 980 lAP AN Abstract We will present the implementation of intelligent electronic circuits realized for the first time using a new functional device called Neuron MOS Transistor (neuMOS or vMOS in short) simulating the behavior of biological neurons at a single transistor level. Search for the most resembling data in the memory cell array, for instance, can be automatically carried out on hardware without any software manipulation. Soft Hardware, which we named, can arbitrarily change its logic function in real time by external control signals without any hardware modification. Implementation of a neural network equipped with an on-chip self-learning capability is also described. Through the studies of vMOS intelligent circuit implementation, we noticed an interesting similarity in the architectures of vMOS logic circuitry and biological systems. 1 INTRODUCTION The motivation of this work has stemmed from the invention of a new functional transistor which simulates the behavior of biological neurons (Shibata and Ohmi, 1991; 1992a). The transistor can perfOlID weighted summation of multiple input signals and squashing on the sum all at a single transistor level. Due to its functional similarity, the transistor was named Neuron MOSFET (abbreviated as neuMOS or vMOS). What is of significance with this new device is that a number of new architecture electronic circuits can be build using vMOS' which are different from conventional ones both in operational principles and functional capabilities. They are charactetized by a high degree of parallelism in hardware computation, a large flexibility in hardware configuration and a dramatic reduction in the circuit complexity as compared to conventional integrated 919 920 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi circuits. During the course of studies in exploring vMOS circuit applications an interesting similarity has been noticed between the basic vMOS logic circuit architecture and the common structure found in biological neuronal systems, i. e., the competitive processes of excitatory and inhibitory connections. The purpose of this article is to demonstrate how powerful the neuron-like functionality in an elemental device is in implementing intelligent functions in silicon integrated circuits. 2 NEURON MOSFET AND SOFT-HARDWARE LOGIC CIRCUITS The symbolic representation of a vMOS is given in Fig. 1. A vMOS is a regular MOS transistor except that its gate electrode is made electrically floating and multiple input terminals are capacitively coupled to the floating gate. The potential of the floating gate ~ is determined as a linear weighted sum of multiple input voltages where each weighting factor is given by the magnitude of a coupling capacitance. When <l>F' the weighted sum, exceeds the threshold voltage of the transistor, it turns on. Thus the function of a neuron model (McCulloch and Pitts, 1943) has been directly implemented in a simple transistor structure. vMOS transistors were fabricated using the doublepolysilicon gate technology and a CMOS process was employed for vMOS integrated circuits fabrication. It should be noted here that no floating-gate charging effect was employed in the operation of vMOS logic circuits. V, v2 vn 1.1.---------J. 4>F " _...-J,I '" ~ FLOATING GATE SOURCE DRAIN c v. +C V. +·····+C V Cl>F1 1 2 2 n n ) V ~ Cror Transistor "Turns ON" Figure 1: Schematic of a neuron MOS transistor. Since the weighting factors in a vMOS are detennilled by the overlapping areas of the first poly (floating gate) and second poly (input gate) patterns, they are not alterable. For this reason, in vMOS applications to self-learning neural network synthesis, a synapse cell circuit was provided to each input temlinal of a vMOS to represent an alterable connection strength. Here the plasticity of a synaptic weight was created by charging/discharging of the floating-gate in a vMOS synapse circuitry as described in 4. TheI-Vcharacteristics ofa two-input-gate vMOS having identical coupling capacitances are shown in Fig. 2, where one of the input gates is used as a gate terminal and the other as a threshold-control terminal. The apparent threshold voltage as seen from the gate terminal is changed from a depletion-mode to an enhancement-mode threshold by the voltage given to the control terminal. This variable threshold nature of a vMOS, we believe, is most essential in creating flexibility in electronic hardware systems. Figure 3(a) shows a two-input-variable Soft Hardu:are Logic (SHL) circuit which can represent all possible sh.1een Boolean functions for two binary inputs Xl and X2 by adjusting the control signals VA' VB and Ve. The inputs, Xl and X2, are directly coupled to the floating gate of a complementary vMOS inverter in the output stage with a 1:2 coupling ratio. The vMOS inve11er, which we call the main inve11er, deternlines the logic. Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors 921 x ,o---.,----t X2 o--~----t Figure 2: Measured characteristics of a variable threshold transistor. Voltage at the threshold-control tenninal was varied from +5V to -5V (from -2.5 0.0 2.5 5 0 left to right). GATE VOLTAGE (V) Oms 2ms/div 20ms (a) (b) Figure 3: Two-input-variable soft hardware logic circuit(a) and measured characteristics(b). The slow operation is due to the loading effect. (The test circuit has no output buffers.) The inputs are also coupled to the main inve11er via three inter-stage vMOS inverters (pre-inverters). When the analog variable represented by the binary inputs Xl and X2 increases~ the inputs tend to turn on the main inverter via direct connection~ while the indirect connection via pre-inverters tend to turn off the main invelter because preinverter outputs change from V DO to 0 when they turn on. This competitive process creates logics. The turn-on thresholds of pre-inverters are made alterable by control signals utilizing the variable threshold characteristics of vMOS'. Thus the real-time alteration of logic functions has been achieved and are demonstrated by experiments in Fig. 3(b). With the basic circuit architecture of the two-staged vMOS inverter configuration shown in Fig. 3(a)~ any Boolean function can be generated. We found the inverting connections via preinverters are most essential in logic synthesis. The structure indicates an interesting similarity to neuronal functional modules in which intramodular inhibitory connections play essential roles. Fixed function logics can be generated much more simply using the basic two-staged structure~ resulting in a dramatic reduction in transistor counts and interconnections. It has been demonstrated that a full adder~ 3-b and 4-b NO conve11ers can be constructed only with 8~ 16 and 28 transistors~ respectively, which should be compared to 30~ 174 and 398 transistors by conventional CMOS design, respectively. The details on vMOS circuit design are desClibed in Refs. (Shibata and Ohmi, 1993a; 1993b) and experimental verification in Ref. (Kotani et al.~ 1992). 922 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi Voo INPUT '" ~1 '" :::. ~ '" p p X :::. '" y 1 ( VOUT '" Analog -01 Inverter ~ ~ ~ 60 \:c N r N It) OUTPUT +-+ 2msec/dlv +--+ Smsec/dlv (a) (b) (c) Figure 4: Real-time rule-variable data matching circuit (a) and measured wave forms (b & c). In (c), l) is changed as 0.5, 1, 1.5, and 2 [V] from top to bottom. A unique vMOS circuit based on the basic structure of Fig. 3(a) is the real-time rulevariable data matching circuitry shown in Fig. 4(a). The circuit output becomes high when I X - y I < l). X is the input data, Y the template data and l) the window width for data matching where X, Y and l) are all time variables. Measured data are shown in Figs. 4(b) and 4(c), where it is seen the triple peaks are merged into a single peak as l) increases (Shibata et al., 1993c). The circuit is composed of only 10 vMOS' and can be easily integrated with each pixel on a image sensor chip. If vMOS circuitry is combined with a bipolar image sensor cell having an amplification function (fanaka et al., 1989), for instance, in situ image processing such as edge detection and variable-template matching would become possible, leading to an intelligent image sensor chip. 3 BINARY-MULTIVALUED-ANALOG MERGED HARDWARE COMPUTATION A winner-take-all circuit (WTA) implemented by vMOS circuitry is given in Fig. 5. Each cell is composed of a vMOS variable threshold inverter in which the apparent threshold is modified by an analog input signals VA - V c' When the common input signal VR is ramped up, the lowest threshold cell (a cell receiving the largest analog input) turns on firstly, at which instance a feedback loop is formed in each cell and the state of the cell is self-latched. As a result, only the winner cell yields an output of 1. The circuit has been applied to building an associative memory as demonstrated in Fig. 6. The binary data stored in a SRAM cell array are all simultaneously matched to the sample data by taking XNOR, and the number of matched bits are transfeITed to the floating gate of each WfA cell by capacitance coupling. The WI' A action finds the location of data having the largest number of matched bits. This principle has been also applied to an sorting circuitry (Yamashita et aI., 1993). In these circuits all computations are conducted by an algOlithm directly imbedded in the hardware. Such an analog-digital merged hardware computation algorithm is a key to implement intelligent data processing architecture on silicon. A multivalued DRAM cell equipped with the association function and a multivalued SRAM cell having self-quantizing and self-classification functions have been also developed based on the binary-multivalued-analog merged hardware algorithm (Rita et aI., 1994). Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors 923 INITIAL WINNER LATCH V~o ~1 V'~ VR V~o VR ~o CONTROL TIME V~o ~o SIGNAL Figure 5: Operational principle of vMOS Winner-Take-All circuit. + t t ! t o 1 SAMPLE DATA 0 0 ~ ~ 1 g] 0 a .... c: 0 =ti c: 0 ...., .... fr WINNER-TAKE-ALL NETWORK (a) (b) Figure 6: vMOS associative memory: (a) circuit diagram; (b) photomicrograph of a test chip. 4 HARDWARE SELF-LEARNING NEURAL NETWORKS Since vMOS itself has the basic function of a neuron, a neuron cell is very easily implemented by a complementary vMOS inve11er. The learning capability of a neural network is due to the plasticity of synaptic connections. Therefore its circuit implementation is a key issue. A stand-by power dissipation free synapse circuit which has been developed using vMOS circuitry is shown in Fig. 7(a). The circuit is a differential pair of N-channel and P-channel vMOS source followers sharing the same floating gate, which are both merged into CMOS inverters to cut off dc cunent paths. When the pre-synaptic neuron fires, both source followers are activated. Then the analog weight value stored as charges in the common floating gate is read out and transferred to the floating gate (dendrite) of the post-synaptic neuron by capacitance coupling as shown in Figs. 7 (b) and (c). The outputs of N-vMOS (V+) and P-vMOS (V-) source followers are averaged at the dendrite level, yielding an effective synapse output equal to (V+ + V-)/2. The synapse can represent both positive (excitatory) and negative (inhibitory) weights depending on whether the effective output is larger or smaller than Vnrj2, respectively. The operation of the synapse cell is demonstrated in Fig. 8(a). 924 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi P-UMOS • I tunneling electrode VVPrevious Neuron at Rest Previous Neuron Fired v"" v .. vT 7 Voo V =." 2 FLOATING GATE (DENDRITE) OF NEURON VDD 0 Voe Ylm. 2 0 Veo Voe 2 0 (a) (b) (c) Figure 7: Synapse cell circuit implemented by vMOS circuitry. Previous-Layer Neuron Fired VI ......... vEXCITATORY f t= V+ + VV+ 2 vINHIBITORY 0 -~ V" + VV+ 2 400nsec (a) 3~----------------~ 2 1 >,0 "';-1 :> -2 -3 -4 4 3 ~2 x 1 :> 0 -1 -2 CONVENTIONAL CELL NEW CELL _3L-~L-~--~---L--~ o 5 10 15 20 25 Number of Programming Pulses (b) Figure 8: (a) Measured synapse cell output characteristics; (b) weight updating characteristics as represented by N-vMOS threshold with (our new cell:bottom) or without (conventional EEPROM cell: top) feed back. The weight updating is conducted by giving high programming pulses to both V x and V y tenninals. (Their coupling capacitances are made much larger than others). Then the common floating gate is pulled up to the programming voltage~ allowing electrons to flow into the floating gate via Fowler-Nordheim tunneling. When either Vx or Vy is low, tunneling injection does not occur because the tunneling current is very sensitive to the electric field intensity, being exponentially dependent upon the tunnel oxide field (Hieda Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors 925 et al.~ 1985). The data updating occurs only at the crossing point of Vx and Vy lines~ allowing Hebb-rule-like learning directly implemented on the hardware (Shibata and Ohmi~ 1992b). Hardware-Backpropagation (HEP) learning algorithm~ which is a simplified version of the original BP ~ has been also developed in order to facilitate its hardware implementation (Ishii et al.~ 1992) and has been applied to build self-learning vMOS neural networks (Ishii et al.~ 1993). One of the drawbacks of programming by tunneling is the non-linearity in the data updating characteristics under constant pulses as shown in Fig. 8(b) (top). This difficulty has been beautifully resolved in our cell. With Vs high~ the output of the N-vMOS source follower is fed back to the tunneling electrode and the floating-gate potential is set to the tunneling electrode. In this manner~ the voltage across the tunneling oxide is always preset to a constant voltage (equal to the N-vMOS threshold) before a programming pulse is applied~ thus allowing constant charge to be injected or extracted at each pulse (Kosaka et al~ 1993) as demonstrated in Fig. 8(b) (bottom). A test self-learning circuit that leamed XOR is shown in Fig. 9. INPUT1 "XOR" INPUT2 ! I I INPUT1 \l JI-; ---"~ [ ; INPUT2 ! 400nsecldiv Figure 9: Test circuit of vMOS neural network and its response when XOR is learnt. 5 SUMMARY Development of intelligent electronic circuit systems using a new functional device called Neuron MOS Transistor has been described. vMOS circuitry is charactedzed by its high parallelism in computation scheme and the large flexibility in altering hardware functions and also by its great simplicity in the circuit organization. The ideas of Soft Hardware and the vMOS associative memory were not directly inspired from biological systems. However~ an interesting similarity is found in their basic structures. It is also demonstrated that the vMOS circuitry is very powerful in building neural networks in which learning algorithms are imbedded in the hardware. We conclude that the neuron-like functionality at an elementary device level is essentially imp0l1ant in implementing sophisticated information processing algorithms directly in the hardware. 926 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi ACKNOWLEDGMENT This work was paltially supported by the Grant-in-Aid for Scientific Research (04402029) and Grant-in-Aid for Developmental Scientitlc Research (05505003) from the Ministry of Education, Science and Culture, Japan. A palt of this work was carried out in the Super Clean Room of Laboratory for Microelectronics, Research Institute of Electrical Communication, Tohoku University. REFERENCES [1] T. Shibata and T. Ohmi, "An intelligent MOS transistor featuring gate-level weighted sum and threshold operations," in IEDM Tech. Dig., 1991, pp. 919-922. [2] T. Shibata and T. Ohmi, "A functional MOS transistor featuring gate-level weighted sum and threshold operations," IEEE Trans. Electron Devices, Vol. 39, No.6, pp.14441455 (1992a). [3] W. S. McCulloch and W. Pitts, "A logical calculus of the ideas immanent in nervous activity," Bull. Math. Biophys., Vol. 5, pp. 115-133, 1943. [4] T. Shibata and T. Onmi, "Neuron MOS binary-logic integrated circuits: Part I, Design fundamentals and soft-hardware-logic circuit implementation," IEEE Trans. Electron Devices, Vol. 40, No.3, pp. 570-576 (1993a). [5] T. Shibata and T. Ohmi, "Neuron MOS binary-logic integrated circuits: Palt II, Simplifying techniques of circuit configuration and their practical applications," IEEE Trans. Electron Devices, Vol. 40, No.5, 974-979 (1993b). [6] K. Kotani, T. Shibata, and T. Ohmi, "Neuron-MOS binary-logic circuits featuring dramatic reduction in transistor count and interconnections," in IEDM Tech. Dig., 1992, pp. 431-434. [7] T. Shibata, K. Kotani, and T. Ohmi, "Real-time reconfigurable logic circuits using neuron MOS transistors," in ISSCC Dig. Technical papers, 1993c, FA 15.3, pp. 238-239. [8] N. Tanaka, T. Ohmi, and Y. Nakamura, "A novel bipolar imaging device with selfnoise reduction capability," IEEE Trans. Electron Devices, VoL 36, No.1, pp. 31-38 (1989). [9] T. Yamashita, T. Shibata, and T. Ohmi, "Neuron MOS winner-take-all circuit and its application to associative memory," in ISSCC Dig. Technical papers, 1993, FA 15.2, pp. 236-237. [10] R. Au, T. Yamashita, T. Shibata, and T. Ohmi, "Neuron-MOS multiple-valued memory technology for intelligent data processing," in ISSCC Dig. Technical papers, 1994, FA 16.3. [11] K. Hieda, M. Wada, T. Shibata, and H. Iizuka, "Optimum design of dual-control gate cell for high-density EEPROM's," IEEE Trans. Electron Devices, vol. ED-32, no. 9, pp. 1776-1780, 1985. [12] T. Shibata and T. Ohmi, itA self-leaming neural-network LSI using neuron MOSFET's," in Dig. Tech. Papers, 1992 Symposium on VLSI Technology, Seattle, June, 1992, pp. 84-85. [13] H. Ishii, T. Shibata, H. Kosaka, and T. Ohmi, "Hardware-Backpropagation learning of neuron MOS neural networks," in IEDM Tech. Dig., 1992, pp. 435-438. [14] H. Ishii, T. Shibata, H. Kosaka, and T. Ohmi, "Hardware-learning neural network LSI using a highly functional transistor simulating neuron actions," in Proc. Intemational Joint Conference on Neural Networks '93, Nagoya, Oct. 25-29, 1993, pp. 907-910. [15] H. Kosaka, T. Shibata, H. Ishii, and T. Ohmi, "An excellent weight-updatinglinearity synapse memory cell for self-Ieaming neuron MOS neural networks," in IEDM Tech. Dig., 1993, pp. 626-626.	1993	78
834	Probabilistic Anomaly Detection Dynamic Systems Padhraic Smyth Jet Propulsion Laboratory 238-420 California Institute of Technology 4800 Oak Grove Drive Pasadena, CA 91109 Abstract • In This paper describes probabilistic methods for novelty detection when using pattern recognition methods for fault monitoring of dynamic systems. The problem of novelty detection is particularly acute when prior knowledge and training data only allow one to construct an incomplete classification model. Allowance must be made in model design so that the classifier will be robust to data generated by classes not included in the training phase. For diagnosis applications one practical approach is to construct both an input density model and a discriminative class model. Using Bayes' rule and prior estimates of the relative likelihood of data of known and unknown origin the resulting classification equations are straightforward. The paper describes the application of this method in the context of hidden Markov models for online fault monitoring of large ground antennas for spacecraft tracking, with particular application to the detection of transient behaviour of unknown origin. 1 PROBLEM BACKGROUND Conventional control-theoretic models for fault detection typically rely on an accurate model ofthe plant being monitored (Patton, Frank, and Clark, 1989). However, in practice it common that no such model exists for complex non-linear systems. The large ground antennas used by JPL's Deep Space Network (DSN) to track 825 826 Smyth Jet Prcpllslon Laboratory Mission Control Figure 1: Block diagram of typical Deep Space Network downlink planetary spacecraft fall into this category. Quite detailed analytical models exist for the electromechanical pointing systems. However, these models are primarily used for determining gross system characteristics such as resonant frequencies; they are known to be a poor fit for fault detection purposes. We have previously described the application of adaptive pattern recognition methods to the problem of online health monitoring of DSN antennas (Smyth and Mellstrom, 1992; Smyth, in press). Rapid detection and identification of failures in the electromechanical antenna pointing systems is highly desirable in order to minimize antenna downtime and thus minimise telemetry data loss when communicating with remote spacecraft (see Figure 1). Fault detection based on manual monitoring of the various antenna sensors is neither reliable or cost-effective. The pattern-recognition monitoring system operates as follows. Sensor data such as motor current, position encoder, tachometer voltages, and so forth are synchronously sampled at 50Hz by a data acquisition system. The data are blocked off into disjoint windows (200 samples are used in practice) and various features (such as estimated autoregressive coefficients) are extracted; let the feature vector be fl. The features are fed into a classification model (every 4 seconds) which in turn provides posterior probability estimates of the m possible states of the system given the estimated features from that window, p(wdfl). WI corresponds to normal conditions, the other Wi'S, 1 ~ i ~ m, correspond to known fault conditions. Finally, since the system has "memory" in the sense that it is more likely to remain in the current state than to change states, the posterior probabilities need to be correlated over time. This is achieved by a standard first-order hidden Markov Probabilistic Anomaly Detection in Dynamic Systems 827 model (HMM) which models the temporal state dependence. The hidden aspect of the model reflects the fact that while the features are directly observable, the underlying system states are not, i.e., they are in effect "hidden." Hence, the purpose of the HMM is to provide a model from which the most likely sequence of system states can be inferred given the observed sequence of feature data. The classifier portion of the model is trained using simulated hard ware faults. The feed-forward neural network has been the model of choice for this application because of its discrimination ability, its posterior probability estimation properties (Richard and Lippmann, 1992; Miller, Goodman and Smyth, 1993) and its relatively simple implementation in software. It should be noted that unlike typical speech recognition HMM applications, the transition probabilities are not estimated from data but are designed into the system based on prior knowledge of the system mean time between failure (MTBF) and other specific knowledge of the system configuration (Smyth, in press). 2 LIMITATIONS OF THE DISCRIMINATIVE MODEL The model described above assumes that there are m known mutually exclusive and exhaustive states (or "classes") of the system, WI, ... ,Wm . The mutually exclusive assumption is reasonable in many applications where multiple simultaneous failures are highly unlikely. However, the exhaustive assumption is somewhat impractical. In particular, for fault detection in a complex system such as a large antenna, there are thousands of possible fault conditions which might occur. The probability of occurrence of any single condition is very small, but nonetheless there is a significant probability that at least one of these conditions will occur over some finite time. While the common faults can be directly modelled it is not practical to assign model states to all the other minor faults which might occur. As discussed in (Smyth and Mellstrom, 1992; Smyth 1994) a discriminative model directly models P(Wi I~), the posterior probabilities of the classes given the feature data, and assumes that the classes WI, ... ,Wm are exhaustive. On the other hand, a generative model directly models the probability density function of the input data conditioned on each class, p(~IWi)' and then indirectly determines posterior class probabilities by application of Bayes' rule. Examples of generative classifiers include parametric models such as Gaussian classifiers and memory-based methods such as kernel density estimators. Generative models are by nature well suited to novelty detection whereas discriminative models have no built-in mechanism for detecting data which are different to that on which the model was trained. However, there is a trade-off; because generative models typically are doing more modelling than just searching for a decision boundary, they can be less efficient (than discriminant methods) in their use of the data. For example, generative models typically scale poorly with input dimensionality for fixed training sample size. 3 HYBRID MODELS A relatively simple and practical approach to the novelty detection problem is to use both a generative and discriminative classifier (an idea originally suggested to the author by R. P. Lippmann). An extra "m+ lth" state is added to the model to 828 Smyth cover "all other possible states" not accounted for by the known m states. In this framework, the posterior estimates of the discriminative classifier are conditioned on the event that the data come from one of the m known classes. Let the symbol w{1 , ... ,m} denote the event that the true system state is one of the known states, let Wm+l be the unknown state, and let p(wm+1I~) be the posterior probability that the system is in an unknown state given the data. Hence, one can estimate the posterior probability of individual known states as (1) where Pd(wd~,w{1,,, . ,m}) is the posterior probability estimate of state i as provided by a discriminative model, i.e., given that the system is in one of the known states. The calculation of p(wm+ll~) can be obtained via the usual application of Bayes' rule if P(~lwm+d, p(wm+d, and P(~IW{l, ,, . ,m}) are known: ( I(}) P(~lwm+dp(wm+d P Wm+l (I ( I ""m' P ~ wm+dp(wm+d + P ~ w{1, ... ,m}) L...Ji p(Wi) (2) Specifying the prior density P(~lwm+d, the distribution of the features conditioned on the occurrence of the unknown state, can be problematic. In practice we have used non-informative Bayesian priors for P(~lwm+d over a bounded space of feature values (details are available in a technical report (Smyth and Mellstrom, 1993)), although the choosing of a prior density for data of unknown origin is basically ill-posed. The stronger the constraints which can be placed on the features the narrower the resulting prior density and the better the ability of the overall model to detect novelty. If we only have very weak prior information, this will translate into a weaker criterion for accepting points which belong to the unknown category. The term P(Wm+l) (in Equation (2)) must be chosen based on the designer's prior belief of how often the system will be in an unknown state a practical choice is that the system is at least as likely to be in an unknown failure state as any of the known failure states. The P(~IW{l, ,, .,m}) term in Equation (2) is provided directly by the generative model. Typically this can be a mixture of Gaussian component densities or a kernel density estimate over all of the training data (ignoring class labels). In practice, for simplicity of implementation we use a simple Gaussian mixture model. Furthermore, because of the afore-mentioned scaling problem with input dimensions, only a subset of relatively significant input features are used in the mixture model. A less heuristic approach to this aspect of the problem (with which we have not yet experimented) would be to use a method such as projection pursuit to project the data into a lower dimensional subspace and perform the input density estimation in this space. The main point is that the generative model need not necessarily work in the full dimensional space of the input features. Integration of Equations (1) and (2) into the hidden Markov model scheme is straightforward and is not derived here the HMM now has an extra state, "unknown." The choice oftransition probabilities between the unknown and other states is once again a matter of design choice. For the antenna application at least, many of the unknown states are believed to be relatively brief transient phenomena which Probabilistic Anomaly Detection in Dynamic Systems 829 last perhaps no longer than a few seconds: hence, the Markov matrix is designed to reflect these beliefs since the expected duration of any state d[wd (in units of sampling intervals) must obey 1 d[wd =-I- PH (3) where Pii is the self-transition probability of state Wi. 4 EXPERIMENTAL RESULTS For illustrative purposes the experimental results from 2 particular models are compared. Each was applied to monitoring the servo pointing system of a DSN 34m antenna at Goldstone, California. The models were implemented within Lab View data acquisition software running in real-time on a Macintosh II computer at the antenna site. The models had previously been trained off-line on data collected some months earlier. 12 input features were used consisting of estimated autoregressive coefficients and variance terms from each window of 200 samples of multichannel data. For both models a discriminative feedforward neural network model (with 8 hidden units, sigmoidal hidden and output activation functions) was trained (using conjugate-gradient optimization) to discriminate between a normal state and 3 known and commonly occurring fault states (failed tachometer, noisy tachometer, and amplifier short circuit also known as "compensation loss"). The network output activations were normalised to sum to 1 in order to provide posterior class probability estimates. Model (a) used no HMM and assumed that the 4 known states are exhaustive, i.e., it just used the feedforward network. Model (b) used a HMM with 5 states, where a generative model (a Gaussian mixture model) and a flat prior (with bounds on the feature values) were used to determine the probability of the 5th state (as described by Equations (1) and (2)). The same neural network as in model (a) was used as a discriminator for the other 4 known states. The generative mixture model had 10 components and used only 2 of the 12 input features, the 2 which were judged to be the most sensitive to system change. The parameters of the HMM were designed according to the guidelines described earlier. Known fault states were assumed to be equally likely with 1 hour MTBF's and with 1 hour mean duration. Unknown faults were assumed to have a 20 minute MTBF and a 10 second mean duration. Both HMMs used 5-step backwards smoothing, i.e., the probability estimates at any time n are based on all past data up to time n and future data up to time n + 5 (using a larger number of backward steps was found empirically to produce no effect on the estimates). Figures 2 (a) and (b) show each model's estimates (as a function of time) that the system is in the normal state. The experiment consisted of introducing known hardware faults into the system in a controlled manner after 15 minutes and 45 minutes, each of 15 minutes duration. Model (a) 's estimates are quite noisy and contain a significant number of potential false alarms (highly undesirable in an operational environment). Model (b) is much more stable due to the smoothing effect of the HMM. Nonetheless, we note that between the 8th and 10th minutes, there appear to be some possible false alarms: 830 Smyth -- Discriminative model, no HMM .. ' ''I' ~ ... l' Probability of nonnal 0.6 cmditionl 0.4 0.2 0 0.8 Probability of nonnal 0.6 cmditionl 0.4 0.2 o 0 0 I l?trom1 20 ~~~~f In ~mof taclKmJc1l:r fault nonnal candiuom 40 ~ SO 60 Imrod ctiooof Time minutes) alIIUlCIl&&tim lou fault -- Hybrid model. with HMM rrl~~ 20 In ctimof tac:homcliCl' fault , Rcsum1 30 tim of nonna1 CCJnditiom ~ SO ctioo of Time minu c:om'DCllHlim la-. fault 60 tell) Figure 2: Estimated posterior probability of normal state (a) using no HMM and the exhaustive assumption (normal + 3 fault states), (b) using a HMM with a hybrid model (normal + 3 faults + other state). these data were classified into the unknown state (not shown). On later inspection it was found that large transients (of unknown origin) were in fact present in the original sensor data and that this was what the model had detected, confirming the classification provided by the model. It is worth pointing out that the model without a generative component (whether with or without the HMM) also detected a non-normal state at the same time, but incorrectly classified this state as one of the known fault states (these results are not shown). Also not shown are the results from using a generative model alone, with no discriminative component. While its ability to detect unknown states was similar to the hybrid model, its ability to discriminate between known states was significantly worse than the hybrid model. The hybrid model has been empirically tested on a variety of other conditions where various "known" faults are omitted from the discriminative training step and then Probabilistic Anomaly Detection in Dynamic Systems 831 presented to the model during testing: in all cases, the anomalous unknown state was detected by the model, i.e., classified as a state which the model had not seen before. 5 APPLICATION ISSUES The model described here is currently being integrated into an interactive antenna health monitoring software tool for use by operations personnel at all new DSN antennas. The first such antenna is currently being built at the Goldstone (California) DSN site and is scheduled for delivery to DSN operations in late 1994. Similar antennas, also equipped with fault detectors of the general nature described here, will be constructed at the DSN ground station complexes in Spain and Australia in the 1995-96 time-frame. The ability to detect previously unseen transient behaviour has important practical consequences: as well as being used to warn operators of servo problems in realtime, the model will also be used as a filter to a data logger to record interesting and anomalous servo data on a continuous basis. Hence, potentially novel system characteristics can be recorded for correlation with other antenna-related events (such as maser problems, receiver lock drop during RF feedback tracking, etc.) for later analysis to uncover the true cause of the anomaly. A long-term goal is to develop an algorithm which can automatically analyse the data which have been classified into the unknown state and extract distinct sub-classes which can be added as new explicit states to the HMM monitoring system in a dynamic fashion. Stolcke and Omohundro (1993) have described an algorithm which dynamically creates a state model for HMMs for the case of discrete-valued features. The case of continuous-valued features is considerably more subtle and may not be solvable unless one makes significant prior assumptions regarding the nature of the datagenerating mechanism. 6 CONCLUSION A simple hybrid classifier was proposed for novelty detection within a probabilistic framework. Although presented in the context of hidden Markov models for fault detection, the proposed scheme is perfectly general for generic classification applications. For example, it would seem highly desirable that fielded automated medical diagnosis systems (such as various neural network models which have been proposed in the literature) should always contain a "novelty-detection" component in order that novel data are identified and appropriately classified by the system. The primary weakness of the methodology proposed in this paper is the necessity for prior knowledge in the form of densities for the feature values given the unknown state. The alternative approach is not to explicitly model the the data from the unknown state but to use some form of thresholding on the input densities from the known states (Aitchison, Habbema, and Kay, 1977; Dubuisson and Masson, 1993). However, direct specification of threshold levels is itself problematic. In this sense, the specification of prior densities can be viewed as a method for automatically determining the appropriate thresholds (via Equation (2)). 832 Smyth As a final general comment, it is worth noting that online learning systems must use some form of novelty detection. Hence, hybrid generative-discriminative models (a simple form of which has been proposed here) may be a useful framework for modelling online learning. Acknowledgements The author would like to thank Jeff Mellstrom, Paul Scholtz, and Nancy Xiao for assistance in data acquisition and analysis. The research described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration and was supported in part by ARPA under grant number NOOOl4-92-J-1860 References R. Patton, P. Frank, and R. Clark (eds.), Fault Diagnosis in Dynamic Systems: Theory and Application, New York, NY: Prentice Hall, 1989. P. Smyth and J. Mellstrom, 'Fault diagnosis of antenna pointing systems using hybrid neural networks and signal processing techniques,' in Advances in Neural Information Processing Systems 4, J. E. Moody, S. J. Hanson, R. P. Lippmann (eds.), San Mateo, CA: Morgan Kaufmann, pp.667-674, 1992. P. Smyth, 'Hidden Markov models for fault detection in dynamic systems,' Pattern Recognition, vo1.27, no.l, in press. M. D. Richard and R. P. Lippmann, 'Neural network classifiers estimate Bayesian a posteriori probabilities,' Neural Computation, 3(4), pp.461-483, 1992. J. Miller, R. Goodman, and P. Smyth, 'On loss functions which minimize to conditional expected values and posterior probabilities,' IEEE Transactions on Information Theory, vo1.39, no.4, pp.1404-1408, July 1993. P. Smyth, 'Probability density estimation and local basis function neural networks,' in Computational Learning Theory and Natural Learning Systems, T. Petsche, M. Kearns, S. Hanson, R. Rivest (eds.), Cambridge, MA: MIT Press, 1994. P. Smyth and J. Mellstrom, 'Failure detection in dynamic systems: model construction without fault training data,' Telecommuncations and Data Acquisition Progress Report, vol. 112, pp.37-49, Jet Propulsion Laboratory, Pasadena, CA, February 15th 1993. A. Stokke and S. Omohundro, 'Hidden Markov model induction by Bayesian merging,' in Advances in Neural Information Processing Systems 5, C. L. Giles, S. J. Hanson and J. D. Cowan (eds.), San Mateo, CA: Morgan Kaufmann, pp.11-18, 1993. J. Aitchison, J. D. F. Habbema, and J. W. Kay, 'A critical comparison of two methods of statistical discrimination,' Applied Statistics, vo1.26, pp.15-25, 1977. B. Dubuisson and M. Masson, 'A statistical decision rule with incomplete knowledge about the classes,' Pattern Recognition, vo1.26 , no.l, pp.155-165, 1993.	1993	79
835	A Local Algorithm to Learn Trajectories with Stochastic Neural Networks Javier R. Movellan· Department of Cognitive Science University of California San Diego La Jolla, CA 92093-0515 Abstract This paper presents a simple algorithm to learn trajectories with a continuous time, continuous activation version of the Boltzmann machine. The algorithm takes advantage of intrinsic Brownian noise in the network to easily compute gradients using entirely local computations. The algorithm may be ideal for parallel hardware implementations. This paper presents a learning algorithm to train continuous stochastic networks to respond with desired trajectories in the output units to environmental input trajectories. This is a task, with potential applications to a variety of problems such as stochastic modeling of neural processes, artificial motor control, and continuous speech recognition. For example, in a continuous speech recognition problem, the input trajectory may be a sequence of fast Fourier transform coefficients, and the output a likely trajectory of phonemic patterns corresponding to the input. This paper was based on recent work on diffusion networks by Movellan and McClelland (in press) and by recent papers by Apolloni and de Falco (1991) and Neal (1992) on asymmetric Boltzmann machines. The learning algorithm can be seen as a generalization of their work to the stochastic diffusion case and to the problem of learning continuous stochastic trajectories. Diffusion networks are governed by the standard connectionist differential equations plus an independent additive noise component. The resulting process is governed ·Pa.rt of this work was done while a.t Ca.rnegie Mellon University. 83 84 Movellan by a set of Langevin stochastic differential equations dai(t) = Ai dri/ti(t) dt + crdBi(t); i E {I, ... , n} (1) where Ai is the processing rate of the ith unit, cr is the diffusion constant, which controls the flow of entropy throughout the network, and dBi(t) is a Brownian motion differential (Soon, 1973). The drift function is the deterministic part of the process. For consistency I use the same drift function as in Movellan and McClelland, 1992 but many other options are possible: dri/ti(t) = EJ=1 Wijaj(t) - /-l ai(t), where Wij is the weight from the jth to the ith unit, and /-1 is the inverse of a logistic function scaled in the (min - max) interval:/- 1(a) = log m-;~!~' In practice DNs are simulated in digital computers with a system of stochastic difference equations ai(t+dt)=ai(t)+Aidri/ti(t)dt+crzi(t)../Xi; iE{I, ... ,n} (2) where Zi(t) is a standard Gaussian random variable. I start the derivations of the learning algorithm for the trajectory learning task using the discrete time process (equation 2) and then I take limits to obtain the continuous diffusion expression. To simplify the derivations I adopt the following notation: a trajectory of states -input, hidden and output units- is represented as a = [a(I) ... a(t m )] = [al(I) ... an (I) ... al(tm ) .. . an(t m )]. The trajectory vector can be partitioned into 3 consecutive row vectors representing the trajectories of the input, hidden and output units a = [xhy). The key to the learning algorithm is obtaining the gradient of the probability of specific trajectories. Once we know this gradient we have all the information needed to increase the probability of desired trajectories and decrease the probability of unwanted trajectories. To obtain this gradient we first need to do some derivations on the transition probability densities. Using the discrete time approximation to the diffusion process, it follows that the conditional transition probability density functions are multivariate Gaussian (3) From equation 2 and 3 it follows that o A' ~ OWij log p(a(t + dt)1 a(t» = ; Zi(t) V dtaj(t) (4) Since the network is Markovian, the probability of an entire trajectory can be computed from the product of the transition probabilities t ",-1 p(a) = p(a(to» II p(a(t + dt)la(t» (5) t=to The derivative of the probability of a specific trajectory follows o ( ) A t",-l : a = p(a)~ L Zi(t)../Xi aj(t) Wij cr t=to (6) A Local Algorithm to Learn Trajectories with Stochastic Neural Networks 8S In practice, the above rule is all is needed for discrete time computer simulations. We can obtain the continuous time form by taking limits as ~t --+ 0, in which case the sum becomes Ito's stochastic integral of aj(t) with respect to the Brownian motion differential over a {to, T} interval. op(a) = p(a)'~i iT aj(t)dBi(t) a Wi; U to A similar equation may be obtained for the ~i parameters o;i~) = pea)! iT drifti(t)dBi(t) • U to (7) (8) For notational convenience I define the following random variables and refer to them as the delta signals . O1og pea) ~i iT 6Wij{a) = a = aj(t)dBi(t) Wij U to (9) and (10) A 1 B 1 n ~ (\ " 1\ A c: A I 0 ;; c: CIS > 0 n :::; ~ 0.5e:( 0.5~ en C) CIS ~ en .a: V V V ~ V ~ V l} V V 0 I I 0 I I 0 100 200 300 0 100 200 300 Time Steps Time Steps Figure 1: A) A sample Trajectory. B) The Average Trajectory. As Time Progresses Sample Trajectories Become Statistically Independent Dampening the Average. 86 Movellan The approach taken in this paper is to minimize the expected value of the error assigned to spontaneously generated trajectories 0 = E(p(a» where pea) is a signal indicating the overall error of a particular trajectory and usually depends only on the output unit trajectory. The necessary gradients follow (11) (12) Since the above learning rule does not require calculating derivatives of the p function, it provides great flexibility making it applicable to a wide variety of situations. For example pea) can be the TSS between the desired and obtained output unit trajectories or it could be a reinforcement signal indicating whether the trajectory is or is not desirable. Figure La shows a typical output of a network trained with TSS as the p signal to follow a sinusoidal trajectory. The network consisted of 1 input unit, 3 hidden units, and 1 output unit. The input was constant through time and the network was trained only with the first period of the sinusoid. The expected values in equations 11 and 12 were estimated using 400 spontaneously generated trajectories at each learning epoch. It is interesting to note that although the network was trained for a single period, it continued oscillating without dampening. However, the expected value of the activations dampened, as Figure l.b shows. The dampening of the average activation is due to the fact that as time progresses, the effects of noise accumulate and the initially phase locked trajectories become independent oscillators. 20,-------------__________ _ p transition = 0.2 Hidden state = 0 Hidden state = 1 ! '-J! p transition = 0.05 p(response 1) = 0.1 p(response 1) = 0.8 >18 16 == 14 :.c ~ 12 o ~ 0.10 .E g 8 g> 6 ...J 4 2 best possible performance [RuP=. J r~~J O~------,-------~------~ .0 900 1900 Learning Epoch Figure 2: A) The Hidden Markov Emitter. B) Average Error Throughout Training. The Bayesian Limit is Achieved at About 2000 Epochs. 2900 A Local Algorithm to Learn Trajectories with Stochastic Neural Networks 87 The learning rule is also applicable in reinforcement situations where we just have an overall measure of fitness of the obtained trajectories, but we do not know what the desired trajectory looks like. For example, in a motor control problem we could use as fitness signal (-p) the distance walked by a robot controlled by a DN network. Equations 11 and 12 could then be used to gradually improve the average distance walked by the robot. In trajectory recognition problems we could use an overall judgment of the likelihood of the obtained trajectories. I tried this last approach with a toy version of a continuous speech recognition problem. The "emitter" was a hidden Markov model (see Figure 2) that produced sequences of outputs - the equivalent of fast Fourier transform loads - fed as input to the receiver. The receiver was a DN network which received as input, sequences of 10 outputs from the emitter Markov model. The network's task was to guess the sequence of hidden states of the emitter given the sequence of outputs from the emitter. The DN outputs were interpreted as the inferred state of the emitter. Output unit activations greater than 0.5 were evaluated as indicating that the emitter was in state 1 at that particular time. Outputs smaller than 0.5 were evaluated as state O. To achieve optimal performance in this task the network had to combine two sources of information: top-down information about typical state transitions of the emitter, and bottom up information about the likelihood of the hidden states of the emitter given its responses. The network was trained with rules 11 and 12 using the negative log joint probability of the DN input trajectory and the DN output trajectory as error signal. This signal was calculated using the transition probabilities of the emitter hidden Markov model and did not require knowledge of its actual state trajectories. The necessary gradients for equations 11 and 12 were estimated using 1000 spontaneous trajectories at each learning epoch. As Figure 3 shows the network started producing unlikely trajectories but continuously improved. The figure also shows the performance expected from an optimal classifier. As training progressed the network approached optimal performance. Acknowledgements This work was funded through the NIMH grant MH47566 and a grant from the Pittsburgh Supercomputer Center. References B. Apolloni, & D. de Falco. (1991) Learning by asymmetric parallel Boltzmann machines. Neural Computation, 3, 402-408. R. Neal. (1992) Asymmetric Parallel Boltzmann Machines are Belief Networks, Neural Computation, 4, 832-834. J. Movellan & J. McClelland. (1992a) Learning continuous probability distributions with symmetric diffusion networks. To appear in Cognitive Science. T. Soon. (1973) Random Differential Equations in Science and Engineering, Academic Press, New York.	1993	8
836	Learning Complex Boolean Functions: Algorithms and Applications Arlindo L. Oliveira and Alberto Sangiovanni-Vincentelli Dept. of EECS UC Berkeley Berkeley CA 94720 Abstract The most commonly used neural network models are not well suited to direct digital implementations because each node needs to perform a large number of operations between floating point values. Fortunately, the ability to learn from examples and to generalize is not restricted to networks ofthis type. Indeed, networks where each node implements a simple Boolean function (Boolean networks) can be designed in such a way as to exhibit similar properties. Two algorithms that generate Boolean networks from examples are presented. The results show that these algorithms generalize very well in a class of problems that accept compact Boolean network descriptions. The techniques described are general and can be applied to tasks that are not known to have that characteristic. Two examples of applications are presented: image reconstruction and hand-written character recognition. 1 Introduction The main objective of this research is the design of algorithms for empirical learning that generate networks suitable for digital implementations. Although threshold gate networks can be implemented using standard digital technologies, for many applications this approach is expensive and inefficient. Pulse stream modulation [Murray and Smith, 1988] is one possible approach, but is limited to a relatively small number of neurons and becomes slow if high precision is required. Dedicated 911 912 Oliveira and Sangiovanni-Vincentelli boards based on DSP processors can achieve very high performance and are very flexible but may be too expensive for some applications. The algorithms described in this paper accept as input a training set and generate networks where each node implements a relatively simple Boolean function. Such networks will be called Boolean networks. Many applications can benefit from such an approach because the speed and compactness of digital implementations is still unmatched by its analog counterparts. Additionally, many alternatives are available to designers that want to implement Boolean networks, from full-custom design to field programmable gate arrays. This makes the digital alternative more cost effective than solutions based on analog designs. Occam's razor [Blumer et ai., 1987; Rissanen, 1986] provides the theoretical foundation for the development of algorithms that can be used to obtain Boolean networks that generalize well. According to this paradigm, simpler explanations for the available data have higher predictive power. The induction problem can therefore be posed as an optimization problem: given a labeled training set, derive the less complex Boolean network that is consistent I with the training set. Occam's razor, however, doesn't help in the choice of the particular way of measuring complexity that should be used. In general, different types of problems may require different complexity measures. The algorithms described in section 3.1 and 3.2 are greedy algorithms that aim at minimizing one specific complexity measure: the size of the overall network. Although this particular way of measuring complexity may prove inappropriate in some cases, we believe the approach proposed can be generalized and used with minor modifications in many other tasks. The problem of finding the smallest Boolean network consistent with the training set is NP-hard [Garey and Johnson, 1979] and cannot be solved exactly in most cases. Heuristic approaches like the ones described are therefore required. 2 Definitions We consider the problem of supervised learning in an attribute based description language. The attributes (input variables) are assumed to be Boolean and every exemplar in the training set is labeled with a value that describes its class. Both algorithms try to maximize the mutual information between the network output and these labels. Let variable X take the values {Xl, X2, ... xn } with probabilities p(Xd,P(X2) ... P(xn ). The entropy of X is given by H(X) = - Lj p(Xj) logp(xj) and is a measure of the uncertainty about the value of X. The uncertainty about the value of X when the value of another variable Y is known is given by H(XIY) = - Li p(Yi) Lj p(Xj Iyd logp(xj Iyd· The amount by which the uncertainty of X is reduced when the value of variable Y is known, I(Y, X) = H(X) - H(XIY) is called the mutual information between Y and X. In this context, Y will be a variable defined by the output of one or more nottes in the network and X will be the target value specified in the training set. 1 Up to some specified level. Learning Complex Boolean Functions: Algorithms and Applications 913 3 Algorithms 3.1 Muesli - An algorithm for the design of multi-level logic networks This algorithm derives the Boolean network by performing gradient descent in the mutual information between a set of nodes and the target values specified by the labels in the training set. In the pseudo code description of the algorithm given in figure 1, the function 'L (S) computes the mutual information between the nodes in S (viewed as a multi-valued variable) and the target output. muesli( nlist) { } nlist ;- sorLnlisLby1(nlist,1); sup;- 2; while (noLdone(nlist) /\ sup < max_sup) { act ;- 0; } do { act + +; success;- improvLmi(act, nlist, sup); } while (success = FALSE /\ act < max_act); if (success = TRUE) { sup;- 2; while (success = TRUE) success;- improve_mi(act, nlist, sup); } else sup + +; improVLmi(act, nlist, sup) { } nlist;- sorLnlisLby1(nlist, act); 1;- besLlunction(nlist, act, sup); if (I(nlist[l:act-l] U f) > I(nlist[l:act])) { nlist ;- nlist U I; return(TRUE); } else return(F ALSE) j Figure 1: Pseudo-code for the Muesli algorithm. The algorithm works by keeping a list of candidate nodes, nlist, that initially contains only the primary inputs. The act variable selects which node in nl ist is active. Initially, act is set to 1 and the node that provides more information about the output is selected as the active node. Function imp1'ove_miO tries to combine the active node with other nodes as to increase the mutual information. Except for very simple functions, a point will be reached where no further improve914 Oliveira and Sangiovanni-Vincentelli ments can be made for the single most informative node. The value of act is then increased (up to a pre-specified maximum) and improve_mi is again called to select auxiliary features using other nodes in ntist as the active node. If this fails, the value of sup (size of the support of each selected function) is increased until no further improvements are possible or the target is reached. The function sorLnlisLbyJ(nlist, act) sorts the first act nodes in the list by decreasing value of the information they provide about the labels. More explicitly, the first node in the sorted list is the one that provides maximal information about the labels. The second node is the one that will provide more additional information after the first has been selected and so on. Function improve_miO calls besLfunction(nlist, act, sup) to select the Boolean function f that takes as inputs node nlist[act] plus s'up-1 other nodes and maximizes I(nlist[l : act -1] U f). When sup is larger than 2 it is unfeasible to search all 22 s UP possible functions to select the desired one. However, given sup input variables, finding such a function is equivalent to selecting a partition2 of the 28UP points in the input space that maximizes a specific cost function. This partition is found using the Kernighan-Lin algorithm [Kernighan and Lin, 1970] for graph-partitioning. Figure 2 exemplifies how the algorithm works when learning the simple Boolean function f = ab + cde from a complete training set. In this example, the value of sup is always at 2. Therefore, only 2 input Boolean functions are generated. mi([]) = 0.0 Selecty = cd a nlist = [a,b,c,d,e] act = 1 mi([a]) = 0.16 nlist = [x,y,e,a,b,c,d] act = 2 mi([x,y]) = 0.74 Select x = ab nlist = [x,c,d,e,a,b] act = 1 mi([xD = 0.52 Select w = ye nlist = [x,y,e,a,b,c,d] act = 2 mi([x,w]) = 0.93 Fails to fmd f(x,?) with mi([f]) > 0.52 Set act = 2; nlist = [x,c,d,e,a,b] act = 2 mi([x,c]) = 0.63 Fails to find f(w,?) with mi([x,f]) > 0.93 Set act = 0; Select Z = x+w nlist = [z,x,y,a,b,c,d,e] act = 1 mi([z]) = 0.93 Figure 2: The muesli algorithm, illustrated 2 A single output Boolean function is equivalent to a partition of the input space in two sets. Learning Complex Boolean Functions: Algorithms and Applications 915 3.2 Fulfringe - a network generation algorithm based on decision trees This algorithm uses binary decision trees [Quinlan, 1986] as the basic underlying representation. A binary decision tree is a rooted, directed, acyclic graph, where each terminal node (a node with no outgoing edges) is labeled with one of the possible output labels and each non-terminal node has exactly two outgoing edges labeled 0 and 1. Each non-terminal node is also labeled with the name of the attribute that is tested at that node. A decision tree can be used to classify a particular example by starting at the root node and taking, until a terminal is reached, the edge labeled with the value of the attribute tested at the current node. Decision trees are usually built in a greedy way. At each step, the algorithm greedily selects the attribute to be tested as the one that provides maximal information about the label of the examples that reached that node in the decision tree. It then recurs after splitting these examples according to the value of the tested attribute. Fulfringe works by identifying patterns near the fringes of the decision tree and using them to build new features. The idea was first proposed in [Pagallo and Haussler, 1990]. N A 1\0 + 0 !A o + "A + 0 p&-g -p&-g p&g -p&g A A ~+ 1\ + A +~ A +1\ + + + + p+g -p+g p+-g -p+-g MMMM + + + + + + + + p(t)g Figure 3: Fringe patterns identified by fuifringe Figure 3 shows the patterns that fulfringe identifies. Dcfringe, proposed in [Yang et al., 1991], identifies the patterns shown in the first two rows. These patterns correspond to 8 Boolean functions of 2 variables. Since there are only 10 distinct Boolean functions that depend on two variables3 , it is natural to add the patterns in the third row and identify all possible functions of 2 variables. As in dcftinge and fringe, these new composite features are added (if they have not yet been generated) to the list of available features and a new decision tree is built. The 3The remaining 6 functions of 2 variables depend on only one or none of the variables. 916 Oliveira and Sangiovanni-Vincentelli process is iterated until a decision tree with only one decision node is built. The attribute tested at this node is a complex feature and can be viewed as the output of a Boolean network that matches the training set data. 3.3 Encoding multivalued outputs Both muesli and Julfringe generate Boolean networks with a single binary valued output. When the target label can have more than 2 values, some encoding must be used. The prefered solution is to encode the outputs using an error correcting code [Dietterich and Bakiri, 1991]. This approach preserves most of the compactness of a digital encoding while beeing much less sensitive to errors in one of the output variables. Additionally, the Hamming distance between an observed output and the closest valid codeword gives a measure of the certainty of the classification. This can be used to our advantage in problems where a failure to classify is less serious than the output of a wrong classification. 4 Performance evaluation To evaluate the algorithms, we selected a set of 11 functions of variable complexity. A complete description of these functions can be found in [Oliveira, 1994]. The first 6 functions were proposed as test cases in [Pagallo and Haussler, 1990] and accept compact disjoint normal form descriptions. The remaining ones accept compact multi-level representations but have large two level descriptions. The algorithms described in sections 3.1 and 3.2 were compared with the cascade-correlation algorithm [Fahlman and Lebiere, 1990] and a standard decision t.ree algorithm analog to ID3 [Quinlan, 1986]. As in [Pagallo and Haussler, 1990], the number of examples in the training set was selected to be equal to ~ times the description length of the function under a fixed encoding scheme, where f was set equal to 0.1. For each function, 5 training sets were randomly selected. The average accuracy for the 5 runs in an independent set of 4000 examples is listed in table 1. Table 1: Accuracy of the four algorithms. Function # inputs # examples Accuracy muesli fulfringe ID3 CasCor dnfl 80 3292 99.91 99.98 82.09 75.38 dnf2 40 2185 99.28 98.89 88.84 73.11 dnf3 32 1650 99.94 100.00 89.98 79.19 dnf4 64 2640 100.00 100.00 72.61 58.41 xor4_16 16 1200 98.35 100.00 75.20 99.91 xor5_32 32 4000 60.16 100.00 51.41 99.97 sm12 12 1540 99.90 lUO.OO 99.81 98.98 sm18 18 2720 100.00 99.92 91.48 91.30 str18 18 2720 100.00 100.00 94.55 92.57 str27 27 4160 98.64 99.35 94.24 93.90 carry8 16 2017 99.50 98.71 96.70 99.22 Average 95.97 99.71 85.35 87.45 The results show that the performance of muesli and fulfringe is consistently suLearning Complex Boolean Functions: Algorithms and Applications 917 perior to the other two algorithms. Muesli performs poorly in examples that have many xor functions, due the greedy nature of the algorithm. In particular, muesli failed to find a solution in the alloted time for 4 of the 5 runs of xor5_32 and found the exact solution in only one of the runs. ID3 was the fastest of the algorithms and Cascade-Correlation the slowest. Fulfringe and muesli exhibited similar running times for these tasks. 'rVe observed, however, that for larger problems the runtime for fulfringe becomes prohibitively high and muesli is comparatively much faster. 5 Applications To evaluate the techniques described in real problems, experiments were performed in two domains: noisy image reconstruction and handwritten character recognition. The main objective was to investigate whether the approach is applicable to problems that are not known to accept a compact Boolean network representation. The outputs were encoded using a 15 bit Hadamard error correcting code. 5.1 Image reconstruction The speed required by applications in image processing makes it a very interesting field for this type of approach. In this experiment, 16 level gray scale images were corrupted by random noise by switching each bit with 5% probability. Samples of this image were used to train a network in the reconstruction of the original image. The training set consisted of .5x5 pixel regions of corrupted images (100 binary variables per sample) labeled with the value of the center pixel. Figure 4 shows a detail of the reconstruction performed in an independent test image by the network obtained using fulfringe. Original image corrupted image Reconstructed image Figure 4: Image reconstruction experiment 5.2 Handwritten character recognition The NIST database of handwritten characters was used for this task. Individually segmented digits were normalized to a 16 by 16 binary grid. A set of 53629 digits was used for training and the resulting network was tested in a different set of 52467 918 Oliveira and Sangiovanni-Vincentelli digits. Training was performed using muesli. The algorithm was stopped after a prespecified time (48 hours on a DECstation 5000/260) ellapsed. The resulting network was placed and routed using the TimberWolf [Sechen and Sangiovanni-Vincentelli, 1986] package and occupies an area of 78.8 sq. mm. using 0.8fl technology. The accuracy on the test set was 93.9%. This value compares well with the performance obtained by alternative approaches that use a similarly sized training set and little domain knowledge, but falls short of the best results published so far. Ongoing research on this problem is concentrated on the use of domain knowledge to restrict the search for compact networks and speed up the training. Acknowledgements This work was supported by Joint Services Electronics Program grant F49620-93-C-0014. References [Blumer et al., 1987] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. Warmuth. Occam's razor. Information Processing Letters, 24:377-380, 1987. [Dietterich and Bakiri, 1991] T. G. Dietterich and G. Bakiri. Error-correcting output codes: A general method for improving multiclass inductive learning programs. In Proceedings of the Ninth National Conference on Artificial Intelligence (AAAI-91), pages 572-577. AAAI Press, 1991. [Fahlman and Lebiere, 1990] S.E. Fahlman and C. Lebiere. The cascade-correlation learning architecture. In D.S. Touretzky, editor, Advances in Neural Information Processing Systems, volume 2, pages 524-532, San Mateo, 1990. Morgan Kaufmann. [Garey and Johnson, 1979] M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York. 1979. [Kernighan and Lin, 1970] B. W. Kernighan and S. Lin. An efficient heuristic procedure for partitioning graphs. The Bell System Technical Journal, pages 291-307, February 1970. [Murray and Smith, 1988] Alan F. Murray and Anthony V. W. Smith. Asynchronous vlsi neural networks using pulse-stream arithmetic. IEEE Journal of Solid-State Circuits, 23:3:688-697, 1988. [Oliveira, 1994] Arlindo L. Oliveira. Inductive Learning by Selection of Minimal Representations. PhD thesis, UC Berkeley, 1994. In preparation. [Pagallo and Haussler, 1990] G. Pagallo and D. Haussler. Boolean feature discovery in empirical learning. Machine Learning, 1, 1990. [Quinlan, 1986] J. R. Quinlan. Induction of decision trees. Machine Learning, 1:81-106, 1986. [Rissanen, 1986) J. Rissanen. Stochastic complexity and modeling. Annals of Statistics, 14:1080-1100, 1986. [Sechen and Sangiovanni-Vincentelli, 1986J Carl Sechen and Alberto Sangiovanni-Vincentelli. TimberWolf3.2: A new standard cell placement and global routing package. In Proceedings of the 23rd Design Automation Conference, pages 432-439, 1986. [Yang et al., 1991] D. S. Yang, L. Rendell, and G. Blix. Fringe-like feature construction: A comparative study and a unifying scheme. In Proceedings of the Eight International Conference in Machine Learning, pages 223-227, San Mateo, 1991. Morgan Kaufmann.	1993	80
837	A Unified Gradient-Descent/Clustering Architecture for Finite State Machine Induction Sreerupa Das and Michael C. Mozer Department of Computer Science University of Colorado Boulder, CO 80309-0430 Abstract Although recurrent neural nets have been moderately successful in learning to emulate finite-state machines (FSMs), the continuous internal state dynamics of a neural net are not well matched to the discrete behavior of an FSM. We describe an architecture, called DOLCE, that allows discrete states to evolve in a net as learning progresses. DOLCE consists of a standard recurrent neural net trained by gradient descent and an adaptive clustering technique that quantizes the state space. DOLCE is based on the assumption that a finite set of discrete internal states is required for the task, and that the actual network state belongs to this set but has been corrupted by noise due to inaccuracy in the weights. DOLCE learns to recover the discrete state with maximum a posteriori probability from the noisy state. Simulations show that DOLCE leads to a significant improvement in generalization performance over earlier neural net approaches to FSM induction. 1 INTRODUCTION Researchers often try to understand-post hoc-representations that emerge in the hidden layers of a neural net following training. Interpretation is difficult because these representations are typically highly distributed and continuous. By "continuous," we mean that if one constructed a scatterplot over the hidden unit activity space of patterns obtained in response to various inputs, examination at any scale would reveal the patterns to be broadly distributed over the space. Continuous representations aren't always appropriate. Many task domains seem to require discrete representations-representations selected from a finite set of alternatives. If a neural net learned a discrete representation, the scatterplot over hidden activity space would show points to be superimposed at fine scales of analysis. Some 19 20 Das and Mozer examples of domains in which discrete representations might be desirable include: finite-state machine emulation, data compression, language and higher cognition (involving discrete symbol processing), and categorization in the context of decision making. In such domains, standard neural net learning procedures, which have a propensity to produce continuous representations, may not be appropriate. The work we report here involves designing an inductive bias into the learning procedure in order to encourage the formation of discrete internal representations. In the recent years, various approaches have been explored for learning discrete representations using neural networks (McMillan, Mozer, & Smolensky, 1992; Mozer & Bachrach, 1990; Mozer & Das, 1993; Schiitze, 1993; Towell & Shavlik, 1992). However, these approaches are domain specific, making strong assumptions about the nature of the task. In our work, we describe a general methodology that makes no assumption about the domain to which it is applied, beyond the fact that discrete representations are desireable. 2 FINITE STATE MACHINE INDUCTION We illustrate the methodology using the domain of finite-state machine (FSM) induction. An FSM defines a class of symbol strings. For example, the class (lOt consists of all strings with one or more repetitions of 10; 101010 is a positive example of the class, 111 is a negative example. An FSM consists principally of a finite set of states and a function that maps the current state and the current symbol of the string into a new state. Certain states of the FSM are designated "accept" states, meaning that if the FSM ends up in these states, the string is a member of the class. The induction problem is to infer an FSM that parsimoniously characterizes the positive and negative exemplars, and hence characterizes the underlying class. A generic recurrent net architecture that could be used for FSM emulation and induction is shown on the left side of Figure 1. A string is presented to the input layer of the net, one symbol at a time. Following the end of the string, the net should output whether or not the string is a member of the class. The hidden unit activity pattern at any point during presentation of a string corresponds to the internal state of an FSM. Such a net, trained by a gradient descent procedure, is able to learn to perform this or related tasks (Elman, 1990; Giles et al., 1992; Pollack, 1991; Servan-Schreiber, Cleeremans, & McClelland, 1991; Watrous & Kuhn, 1992). Although these models have been relatively successful in learning to emulate FSMs, the continuous internal state dynamics of a neural net are not well matched to the discrete behavior of FSMs. Roughly, regions of hidden unit activity space can be identified with states in an FSM, but because the activities are continuous, one often observes the network drifting from one state to another. This occurs especially with input strings longer than those on which the network was trained. To achieve more robust dynamics, one might consider quantizing the hidden state. Two approaches to quantization have been explored previously. In the first, a net is trained in the manner described above. After training, the hidden state space is partitioned into disjoint regions and each hidden activity pattern is then discretized by mapping it to the center of its corresponding region (Das & Das, 1991; Giles A Unified Gradient-Descent/Clustering Architecture for Finite State Machine Induction 21 Figure 1: On the left is a generic recurrent architecture that could be used for FSM induction. Each box corresponds to a layer of units, and arrows depict complete connectivity between layers. At each time step, a new symbol is presented on the input and the input and hidden representations are integrated to form a new hidden representation. On the right is the general architecture of DOLCE. et al., 1992). In a second approach, quantization is enforced during training by mapping the the hidden state at each time step to the nearest corner of a [0,1]" hypercube (Zeng, Goodman, & Smyth, 1993). Each of these approaches has its limitations. In the first approach, because learning does not consider the latter quantization, the hidden activity patterns that result from learning may not lie in natural clusters. Consequently, the quantization step may not group together activity patterns that correspond to the same state. In the second approach, the quantization process causes the error surface to have discontinuities and to be flat in local neighborhoods of the weight space. Hence, gradient descent learning algorithms cannot be used; instead, even more heuristic approaches are required. To overcome the limitations of these approaches, we have pursued an approach in which quantization is an integral part of the learning process. 3 DOLCE Our approach incorporates a clustering module into the recurrent net architecture, as shown on the right side of Figure 1. The hidden layer activities are processed by the clustering module before being passed on to other layers. The clustering module maps regions in hidden state space to a single point in the same space, effectively partitioning or clustering the hidden state space. Each cluster corresponds to a discrete internal state. The clusters are adaptive and dynamic, changing over the course of learning. We call this architecture DOLCE, for gynamic Qn-!ine £lustering and state extraction. The DOLCE architecture may be explored along two dimensions: (1) the clustering algorithm used (e.g., a Gaussian mixture model, ISODATA, the Forgy algorithm, vector quantization schemes), and (2) whether supervised or unsupervised training is used to identify the clusters. In unsupervised mode, the performance error on the FSM induction task has no effect on the operation of the clustering algorithm; instead, an internal criterion characterizes goodness of clusters. In supervised mode, the primary measure that affects the goodness of a cluster is the performance error. Regardless of the training mode, all clustering algorithms incorporate a pressure to 22 Das and Mozer o Figure 2: Two dimensions of a typical state space. The true states needed to perform the task are Cl, C3, and C3, while the observed hidden states, asswned to be corrupted by noise, are distributed about the Ci. produce a small number of clusters. Additionally, as we elaborate more specifically below, the algorithms must allow for a soft or continuous clustering during training, in order to be integrated into a gradient-based learning procedure. We have explored two possibilities for the clustering module. The first involves the use of Forgy's algorithm in an unsupervised mode. Forgy's (1965) algorithm determines both the number of clusters and the partitioning of the space. The second uses a Gaussian mixture model in a supervised mode, where the mixture model parameters are adjusted so as to minimize the performance error. Both approaches were successful, but as the latter approach obtained better results, we describe it in the next section. 4 CLUSTERING USING A MIXTURE MODEL Here we motivate the incorporation of a Gaussian mixture model into DOLCE, using an argument that gives the approach a solid theoretical foundation. Several assumptions underly the approach. First, we assume that the task faced by DOLCE is such that it requires a finite set of internal or true states, C = {Clt C2, ••. , CT}. This is simply the premise that motivates this line of work. Second, we assume that any observed hidden state-i.e., a hidden activity pattern that results from presentation of a symbol sequence-belongs to C but has been corrupted by noise due to inaccuracy in the network weights. Third, we assume that this noise is Gaussian and decreases as learning progresses (i.e., as the weights are adjusted to better perform the task). These assumptions are depicted in Figure 2. Based on these assumptions, we construct a Gaussian mixture distribution that models the observed hidden states: T p(hlC tT q) = ~ qi e-lh-c.12 /2q~ " L...J (27r0'~)H/2 i=l • where h denotes an observed hidden state, 0'; the variance of the noise that corrupts state Ci, qi is the prior probability that the true state is Ci, and H is the dimensionality of the hidden state space. For pedagogical purposes, a.ssume for the time being that the parameters of the mixture distribution-T, C, tT, and q-are all known; in a later section we discuss how these parameters are determined. A Unified Gradient-Descent/Clustering Architecture for Finite State Machine Induction 23 h o 000 0 00 0 OOOO!,~OO o 0 7 0 ~ ~O 0 A before training after successful training Figure 3: A schematic depiction of the hidden state space before and after training. The horizontal plane represents the state space. The bumps indicate the probability density under the mixture model. Observed hidden states are represented by small open circles. Given a noisy observed hidden state, h, DOLCE computes the maximum a posteriori (MAP) estimator of h in C. This estimator then replaces the noisy state and is used in all subsequent computation. The MAP estimator, h, is computed as follows. The probability of an observed state h being generated by a given true state i is p(hltrue state i) = (27rlTi)-!fe-lh-cill/2u:. Using Bayes' rule, one can compute the posterior probability of true state i, given that h has been observed: ( .Ih) p(hltrue state i)qi p true state z = =---'---'-------'---L:j p(hltrue state j)qj Finally, the MAP estimator is given by it = Cargmax,p(true state ilh). However, because learning requires that DOLCE's dynamics be differentiable, we use a soft version of MAP which involves using ii = L:i cip(true state ilh) instead of hand incorporating a "temperature" parameter into lTi as described below. An important parameter in the mixture model is T, the number of true states (Gaussians bumps). Because T directly corresponds to the number of states in the target FSM, if T is chosen too small, DOLCE could not emulate the FSM. Consequently, we set T to a large value, and the training procedure includes a technique for eliminating unnecessary true states. (If the initially selected T is not large enough, the training procedure will not converge to zero error on the training set, and the procedure can be restarted with a larger value of T.) At the start of training, each Gaussian center I Ci, is initialized to a random location in the hidden state space. The standard deviations of each Gaussian, lTi, are initially set to a large value. The priors, qi, are set to liT. The weights are set to initial values chosen from the uniform distribution in [-.25,.25]. All connection weights feeding into the hidden layer are second order. The network weights and mixture model parameters-C, iT, and q-are adjusted by gradient descent in a cost measure, C. This cost includes two components: (a) the performance error, £, which is a squared difference between the actual and target network output following presentation of a training string, and (b) a complexity 24 Das and Mozer c: 800,------~...., o II language 0600 i 400 E '0 2 400 200 NO ROLO OF DG language 2000,--------, language language S 200 100 NO ROLO OF DG o NO RO LO OF DG language 6 OL.......l.:.O=~ NO ROLO OF 00 Figure 4: Each graph depicts generalization performance on one of the Tomita languages for 5 alternative neural net approaches: no clustering [Ne), rigid quantization [RQ), learn then quantize [LQ], DOLCE in unsupervised mode using Forgy's algorithm [DF], DOLCE in supervised mode using mixture model [DG). The vertical axis shows the number of misclassification of 3000 test strings. Each bar is the average result across 10 replications with different initial weights. cost, which is the entropy of the prior distribution, q: where ..\ is a regularization parameter. The complexity cost is minimal when only one Gaussian has a nonzero prior, and maximal when all priors are equal. Hence, the cost encourages unnecessary Gaussians to drop out of the mixture model. The particular gradient descent procedure used is a generalization of back propagation through time (Rumelhart, Hinton, & Williams, 1986) that incorporates the mixture model. To better condition the search space and to avoid a constrained search, optimization is performed not over iT and q directly but rather over hyperparameters a and h, where u; = exp(ai)/,B and qi = exp( -bl)/Ej exp( -bj). The global parameter ,B scales the overall spread of the Gaussians, which corresponds to the level of noise in the model. As performance on the training set improves, we assume that the network weights are coming to better approximate the target weights, hence that the level of noise is decreasing. Thus, we tie ,B to the performance error e. We have used various annealing schedules and DOLCE appears robust to this variation; we currently use {3 ex 1/ e. Note that as £ --+ 0, {3 --+ 00 and the probability density under one Gaussian at h will become infinitely greater than the density under any other; consequently, the soft MAP estimator, h, becomes equivalent to the MAP estimator h, and the transformed hidden state becomes discrete. A schematic depiction of the probability landscape both before and after training is depicted in Figure 3. A Unified Gradient-Descent/Clustering Architecture for Finite State Machine Induction 25 5 SIMULATION STUDIES The network was trained on a set ofregular languages first studied by Tomita (1982). The languages, which utilize only the symbols 0 and 1, are: (1) 1·; (2) (10)·; (3) no odd number of consecutive 1 's is directly followed by an odd number of consecutive O's; (4) any string not containing the substring "000"; (5) , [(01110)(01110)].; (6) the difference between the number of ones and number of zeros in the string is a multiple of three; and (7) 0·1· 0·1· . A fixed training corpus of strings was generated for each language, with an equal number of positive and negative examples. The maximum string length varied from 5 to 10 symbols and the total number of examples varied from 50 to 150, depending on the difficulty of the induction task. Each string was presented one symbol at a time, after which DOLCE was given a target output that specified whether the string was a positive or negative example of the language. Training continued until DOLCE converged on a set of weights and mixture model parameters. Because we assume that the training examples are correctly classified, the error £ on the training set should go to zero when DOLCE has learned. If this did not happen on a given training run, we restarted the simulation with different initial random weights. For each language, ten replications of DOLCE (with the supervised mixture model) were trained, each with different random initial weights. The learning rate and regularization parameter .\ were chosen for each language by quick experimentation with the aim of maximizing the likelihood of convergence on the training set. We also trained a version of DOLCE that clustered using the unsupervised Forgy algorithm, as well as several alternative neural net approaches: a generic recurrent net, as shown on the left side of Figure 1, which used no clustering [NC]; a version with rigid quantization during training [RQ], comparable to the earlier work of Zeng, Goodman, and Smyth (1993); and a version in which the unsupervised Forgyalgorithm was used to quantize the hidden state following training [LQ], comparable to the earlier work of Das and Das (1991). In these alternative approaches, we used the same architecture as DOLCE except for the clustering procedure. We selected learning parameters to optimize performance on the training set, ran ten replications for each language, replaced runs which did not converge, and used the same training sets. 6 RESULTS AND CONCLUSION In Figure 4, we compare the generalization performance of DOLCE-both the unsupervised Forgy [DF] and supervised mixture model [DG]-to the NC, RQ, and LQ approaches. Generalization performance was tested using 3000 strings not in the training set, half positive examples and half negative. The two versions of DOLCE outperformed the alternative neural net approaches, and the DG version of DOLCE consistently outperformed the DF version. To summarize, we have described an approach that incorporates inductive bias into a learning algorithm in order to encourage the evolution of discrete representations during training. This approach is a quite general and can be applied to domains 26 Das and Mozer other than grammaticality judgement where discrete representations might be desirable. Also, this approach is not specific to recurrent networks and may be applied to feedforward networks. We are now in the process of applying DOLCE to a much larger, real-world problem that involves predicting the next symbol in a string. The data base comes from a case study in software engineering, where each symbol represents an operation in the software development process. This data is quite noisy and it is unlikely that the data can be parsimoniously described by an FSM. Nonetheless, our initial results are encouraging: DOLCE produces predictions at least three times more accurate than a standard recurrent net without clustering. Acknowledgements This research was supported by NSF Presidential Young Investigator award IRI9058450 and grant 90-21 from the James S. McDonnell Foundation. References S. Das & R. Das. (1991) Induction of discrete state-machine by stabilizing a continuous recurrent network using clustering. Computer Science and Informatics 21(2):35-40. Special Issue on Neural Computing. J.L. Elman. (1990) Finding structure in time. Cognitive Science 14:179-212. E. Forgy. (1965) Cluster analysis of multivariate data: efficiency versus interpretability of classifications. Biometrics 21:768-780. M.C. Mozer & J.D Bachrach. (1990) Discovering the structure of a reactive environment by exploration. Neural Computation 2( 4):447-457. C. McMillan, M.C. Mozer, & P. Smolensky. (1992) Rule induction through integrated symbolic and subsymbolic processing. In J.E. Moody, S.J. Hanson, & R.P. Lippmann (eds.), Advances in Neural Information Proceuing Sy6tems 4, 969-976. San Mateo, CA: Morgan Kaufmann. C.L. Giles, D. Chen, C.B. Miller, H.H. Chen, G.Z. Sun, & Y.C. Lee. (1992) Learning and extracting finite state automata with second-order recurrent neural network. Neural Computation 4(3):393-405. H. Schiitze. (1993) Word space. In S.J. Hanson, J.D. Cowan, & C.L. Giles (eds.), Advances in Neural Information Proceuing Systems 5, 895-902. San Mateo, CA: Morgan Kaufmann. M. Tomita. (1982) Dynamic construction of finite-state automata from examples using hillclimbing. Proceedings of the Fourth Annual Conference of the Cognitive Science Society, 105-108. G. Towell & J. Shavlik. (1992) Interpretion of artificial neural networks: mapping knowledge-based neural networks into rules. In J .E. Moody, S.J. Hanson, & R.P. Lippmann (eds.), Advances in Neural Information Proceuing Systems 4, 977-984. San Mateo, CA: Morgan Kaufmann. R.L. Watrous & G.M. Kuhn. (1992) Induction of finite state languages using second-order recurrent networks. In J.E. Moody, S.J. Hanson, & R.P. Lippmann (eds.), Advances in Neural Information Proceuing Systems 4, 969-976. San Mateo, CA: Morgan Kaufmann. Z. Zeng, R. Goodman, & P. Smyth. (1993) Learning finite state machines with selfclustering recurrent networks. Neural Computation 5(6):976-990.	1993	81
838	Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms Vijaykumar Gullapalli Department of Computer Science University of Massachusetts Amherst, MA 01003 vijay@cs.umass.edu Andrew G. Barto Department of Computer Science University of Massachusetts Amherst, MA 01003 barto@cs.umass.edu Abstract Reinforcement Learning methods based on approximating dynamic programming (DP) are receiving increased attention due to their utility in forming reactive control policies for systems embedded in dynamic environments. Environments are usually modeled as controlled Markov processes, but when the environment model is not known a priori, adaptive methods are necessary. Adaptive control methods are often classified as being direct or indirect. Direct methods directly adapt the control policy from experience, whereas indirect methods adapt a model of the controlled process and compute control policies based on the latest model. Our focus is on indirect adaptive DP-based methods in this paper. We present a convergence result for indirect adaptive asynchronous value iteration algorithms for the case in which a look-up table is used to store the value function. Our result implies convergence of several existing reinforcement learning algorithms such as adaptive real-time dynamic programming (ARTDP) (Barto, Bradtke, & Singh, 1993) and prioritized sweeping (Moore & Atkeson, 1993). Although the emphasis of researchers studying DP-based reinforcement learning has been on direct adaptive methods such as Q-Learning (Watkins, 1989) and methods using TD algorithms (Sutton, 1988), it is not clear that these direct methods are preferable in practice to indirect methods such as those analyzed in this paper. 695 696 Gullapalli and Barto 1 INTRODUCTION Reinforcement learning methods based on approximating dynamic programming (DP) are receiving increased attention due to their utility in forming reactive control policies for systems embedded in dynamic environments. In most of this work, learning tasks are formulated as Markovian Decision Problems (MDPs) in which the environment is modeled as a controlled Markov process. For each observed environmental state, the agent consults a policy to select an action, which when executed causes a probabilistic transition to a successor state. State transitions generate rewards, and the agent's goal is to form a policy that maximizes the expected value of a measure of the long-term reward for operating in the environment. (Equivalent formulations minimize a measure of the long-term cost of operating in the environment.) Artificial neural networks are often used to store value functions produced by these algorithms (e.g., (Tesauro, 1992)). Recent advances in reinforcement learning theory have shown that asynchronous value iteration provides an important link between reinforcement learning algorithms and classical DP methods for value iteration (VI) (Barto, Bradtke, & Singh, 1993). Whereas conventional VI algorithms use repeated exhaustive "sweeps" ofthe MDP's state set to update the value function, asynchronous VI can achieve the same result without proceeding in systematic sweeps (Bertsekas & Tsitsiklis, 1989). If the state ordering of an asynchronous VI computation is determined by state sequences generated during real or simulated interaction of a controller with the Markov process, the result is an algorithm called Real- Time DP (RTDP) (Barto, Bradtke, & Singh, 1993). Its convergence to optimal value functions in several kinds of problems follows from the convergence properties of asynchronous VI (Barto, Bradtke, & Singh, 1993). 2 MDPS WITH INCOMPLETE INFORMATION Because asynchronous VI employs a basic update operation that involves computing the expected value of the next state for all possible actions, it requires a complete and accurate model of the MDP in the form of state-transition probabilities and expected transition rewards. This is also true for the use of asynchronous VI in RTDP. Therefore, when state-transition probabilities and expected transition rewards are not completely known, asynchronous VI is not directly applicable. Problems such as these, which are called MDPs with incomplete information,l require more complex adaptive algorithms for their solution. An indirect adaptive method works by identifying the underlying MDP via estimates of state transition probabilities and expected transition rewards, whereas a direct adaptive method (e.g., Q-Learning (Watkins, 1989)) adapts the policy or the value function without forming an explicit model of the MDP through system identification. In this paper, we prove a convergence theorem for a set of algorithms we call indirect adaptive asynchronous VI algorithms. These are indirect adaptive algorithms that result from simply substituting current estimates of transition probabilities and expected transition rewards (produced by some concurrently executing identification 1 These problems should not be confused with MDPs with incomplete 6tate information, i.e., partially observable MDPs. Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms 697 algorithm) for their actual values in the asynchronous value iteration computation. We show that under certain conditions, indirect adaptive asynchronous VI algorithms converge with probability one to the optimal value function. Moreover, we use our result to infer convergence of two existing DP-based reinforcement learning algorithms, adaptive real-time dynamic programming (ARTDP) (Barto, Bradtke, & Singh, 1993), and prioritized sweeping (Moore & Atkeson, 1993). 3 CONVERGENCE OF INDIRECT ADAPTIVE ASYNCHRONOUS VI Indirect adaptive asynchronous VI algorithms are produced from non-adaptive algorithms by substituting a current approximate model of the MDP for the true model in the asynchronous value iteration computations. An indirect adaptive algorithm can be expected to converge only if the corresponding non-adaptive algorithm, with the true model used in the place of each approximate model, converges. We therefore restrict attention to indirect adaptive asynchronous VI algorithms that correspond in this way to convergent non-adaptive algorithms. We prove the following theorem: Theorem 1 For any finite 6tate, finite action MDP with an infinite-horizon di6counted performance measure, any indirect adaptive a6ynchronous VI algorithm (for which the corresponding non-adaptive algorithm converges) converges to the optimal value function with probability one if 1) the conditions for convergence of the non-adaptive algorithm are met, 2) in the limit, every action is executed from every 6tate infinitely often, and 3) the e6timate6 of the state-transition probabilities and the expected transition rewards remain bounded and converge in the limit to their true value6 with probability one. Proof The proof is given in Appendix A.2. 4 DISCUSSION Condition 2 of the theorem, which is also required by direct adaptive methods to ensure convergence, is usually unavoidable. It is typically ensured by using a stochastic policy. For example, we can use the Gibbs distribution method for selecting actions used by Watkins (1989) and others. Given condition 2, condition 3 is easily satisfied by most identification methods. In particular, the simple maximumlikelihood identification method (see Appendix A.l, items 6 and 7) converges to the true model with probability one under this condition. Our result is valid only for the special case in which the value function is explicitly stored in a look-up table. The case in which general function approximators such as neural networks are used requires further analysis. Finally, an important issue not addressed in this paper is the trade-off between system identification and control. To ensure convergence of the model, all actions have to be executed infinitely often in every state. On the other hand, on-line control objectives are best served by executing the action in each sta.te that is optimal according to the current value function (i.e., by using the certainty equivalence 698 Gullapalli and Barto optimal policy). This issue has received considerable attention from control theorists (see, for example, (Kumar, 1985), and the references therein). Although we do not address this issue in this paper, for a specific estimation method, it may be possible to determine an action selection scheme that makes the best trade-off between identification and control. 5 EXAMPLES OF INDIRECT ADAPTIVE ASYNCHRONOUS VI One example of an indirect adaptive asynchronous VI algorithm is ARTDP (Barto, Bradtke, & Singh, 1993) with maximum-likelihood identification. In this algorithm, a randomized policy is used to ensure that every action has a non-zero probability of being executed in each state. The following theorem for ARDTP follows directly from our result and the corresponding theorem for RTDP in (Barto, Bradtke, & Singh, 1993): Theorem 2 For any discounted MDP and any initial value junction, trial-based2 ARTDP converges with probability one. As a special case of the above theorem, we can obtain the result that in similar problems the prioritized sweeping algorithm of Moore and Atkeson (Moore & Atkeson, 1993) converges to the optimal value function. This is because prioritized sweeping is a special case of ARTDP in which states are selected for value updates based on their priority and the processing time available. A state's priority reflects the utility of performing an update for that state, and hence prioritized sweeping can improve the efficiency of asynchronous VI. A similar algorithm, Queue-Dyna (Peng & Williams, 1992), can also be shown to converge to the optimal value function using a simple extension of our result. 6 CONCLUSIONS We have shown convergence of indirect adaptive asynchronous value iteration under fairly general conditions. This result implies the convergence of several existing DP-based reinforcement learning algorithms. Moreover, we have discussed possible extensions to our result. Our result is a step toward a better understanding of indirect adaptive DP-based reinforcement learning methods. There are several promising directions for future work. One is to analyze the trade-off between model estimation and control mentioned earlier to determine optimal methods for action selection and to integrate our work with existing results on adaptive methods for MDPs (Kumar, 1985). Second, analysis is needed for the case in which a function approximation method, such as a neural network, is used instead of a look-up table to store the value function. A third possible direction is to analyze indirect adaptive versions of more general DPbased algorithms that combine asynchronous policy iteration with asynchronous 2 As in (Barto, Bradtke, & Singh, 1993), by trial-balled execution of an algorithm we mean its use in an infinite series of trials such that every state is selected infinitely often to be the start state of a trial. Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms 699 policy evaluation. Several non-adaptive algorithms of this nature have been proposed recently (e.g., (Williams & Baird, 1993; Singh & Gullapalli)). Finally, it will be useful to examine the relative efficacies of direct and indirect adaptive methods for solving MDPs with incomplete information. Although the emphasis of researchers studying DP-based reinforcement learning has been on direct adaptive methods such as Q-Learning and methods using TD algorithms, it is not clear that these direct methods are preferable in practice to indirect methods such as the ones discussed here. For example, Moore and Atkeson (1993) report several experiments in which prioritized sweeping significantly outperforms Q-learning in terms of the computation time and the number of observations required for convergence. More research is needed to characterize circumstances for which the various reinforcement learning methods are best suited. APPENDIX A.1 NOTATION 1. Time steps are denoted t = 1, 2, ... , and Zt denotes the last state observed before time t. Zt belongs to a finite state set S = {I, 2, ... , n}. 2. Actions in a state are selected according to a policy 7r, where 7r(i) E A, a finite set of actions, for 1 :::; i :::; n. 3. The probability of making a transition from state i to state j on executing action a is pa ( i, j). 4. The expected reward from executing action a in state i is r(i, a). The reward received at time t is denoted rt(Zt, at). 5. 0 :::; "y < 1 is the discount factor. 6. Let p~(i, j) denote the estimate at time t of the probability of transition from state i to j on executing action a E A. Several different methods can be used for estimating p~( i, j). For example, if n~( i, j) is the observed number of times before time step t that execution of action a when the system was in state i was followed by a transition to state j, and n~(i) = L:jEs nf(i, j) is the number of times action a was executed in state i before time step t, then, for 1 :::; i :::; n and for all a E A, the maximum-likelihood statetransition probability estimates at time t are a(' ') Aa(' ') nt~, J 1 < '< Pt ", J = a (')' _ J _ n. nt " Note that the maximum-likelihood estimates converge to their true values with probability one if nf(i) -+ 00 as t -+ 00, i.e., every action is executed from every state infinitely often. Let pa(i) = [pa(i, 1), ... , pa(i, n)] E [0,1]'\ and similarly, pf(i) = [Pf(i, I), ... , pf(i, n)] E [o,l]n. We will denote the lSI x IAI matrix of transition probabilities associated with state i by P( i) and its estimate at time t by Pt(i). Finally, P denotes the vector of matrices [P(I), ... , P(n)], and Pt denotes the vector [A(I), ... , A(n)]. 700 Gullapalli and Barto 7. Let rt(i, a) denote the estimate at time t of the e:Jq>ected reward r(i, a), and let rt denote all the lSI x IAI estimates at time t. Again, if maximumlikelihood estimation is used, " (") L:!=I rk(zk, Gk)h,(Zk, Gk) rt 'I., G = II( ") , nt 1. where fill: S x A -+ {O, 1} is the indicator function for the state-action pair 1.,G. B. ~* denotes the optimal value function for the MDP defined by the estimates A and rt of P and r at time t. Thus, Vi E S, ~(i) = max{rt(i, a) + "( '" p~(i, i)~(j)}. ilEA L..-J je S Similarly, V* denotes the optimal value function for the MDP defined by P and r. 9. B t ~ S is the subset of states whose values are updated at time t. Usually, at least Zt E Bt • A.2 PROOF OF THEOREM 1 In indirect adaptive asynchronous VI algorithms, the estimates of the MDP parameters at time step t, Pt and rt, are used in place of the true parameters, P and r, in the asynchronous VI computations at time t. Hence the value function is updated at time t as V. (.) _ { maxaeA{rt(i,a) + "(L:;Espf(i,i)vt(j)} ifi E Bt HI 1. vt(i) otherwise, where B(t) ~ S is the subset of states whose values are updated at time t. First note that because A and rt are assumed to be bounded for all t, Vi is also bounded for all t. Next, because the optimal value function given the model A and rt, l't, is a continuous function of the estimates A and rt, convergence of these estimates w.p. 1 to their true values implies that v. 1lI.p. 1 V* t ~ , where V* is the optimal value function for the original MDP. The convergence w.p. 1 of ~* to V* implies that given an € > 0 there exists an integer T > 0 such that for all t ;:::: T, 11l't* - VII < (1 - "() € w.p. 1. 2"( (1) Here, II . II can be any norm on lRn , although we will use the 1/10 or max norm. In algorithms based on asynchronous VI, the values of only the states in Bt ~ S are updated at time t, although the value of each state is updated infinitely often. For an arbitrary Z E S, let us define the infinite subsequence {tk}k=O to be the times when the value of state Z gets updated. Further, let us only consider updates at, or after, time T, where T is from equation (1) above, so that t~ ;:::: T for all Z E S. Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms 701 By the nature of the VI computation we have, for each t ;:::: 1, lVi+l(i) ~(i)1 ~ '"Yllvt ~II if i E Bt • (2) Using inequality (2), we can get a bound for Ivt-+l(Z) ~~(z)1 as ,. ,. Ivt-+dz) ~!(z)1 < '"Y1I:+lIIVi- ~!II + (1 - '"Y1I:)€ w.p.1. (3) ,. ,. 0 0 We can verify that the bound in (3) is correct through induction. The bound is clearly valid for k = o. Assuming it is valid for k, we show that it is valid for k + 1: Ivt+l(Z) ~~ (z)1 < '"Yllvt~~ II ,.+1 "+1 ,.+1 ,.+1 < '"Y(lIvt· ~~ II + II~! -~! II) ,.+1 ,. ,. ,.+1 < '"Ylvt(z) - ~!(z)1 +1' ((1- 1') €) w.p.l "+1,. l' '"Ylvt-+dz) ~!(z)1 + (1- '"Y)€ ,. ,. < '"Yb1l:+1I1vt. ~!II + (1 - '"Y1I:)€) + (1 - '"Y)€ w.p.l o 0 1'11:+211 Vi- ~! II + (1 - 1'11:+ 1)€. o 0 Taking the limit as k -t 00 in equation (3) and observing that for each z, lim1l:-.00 ~qz) = V(z) w.p. 1, we obtain ,. lim Ivt-+l(Z) - V(z)1 < € w.p.1. 11:-.00 ,. Since € and z are arbitrary, this implies that vt -t V w.p. 1. 0 Acknowledgements We gratefully acknowledge the significant contribution of Peter Dayan, who pointed out that a restrictive condition for convergence in an earlier version of our result was actually unnecessary. This work has also benefited from several discussions with Satinder Singh. We would also like to thank Chuck Anderson for his timely help in preparing this material for presentation at the conference. This material is based upon work supported by funding provided to A. Barto by the AFOSR, Bolling AFB, under Grant AFOSR-F49620-93-1-0269 and by the NSF under Grant ECS-92-14866. References [1] A.G. Barto, S.J. Bradtke, and S.P. Singh. Learning to act using real-time dynamic programming. Technical Report 93-02, University of Massachusetts, Amherst, MA, 1993. [2] D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Di8tributed Computation: Numerical Method6. Prentice-Hall, Englewood Cliffs, NJ, 1989. [3] P. R. Kumar. A survey of some results in stochastic adaptive control. SIAM Journal of Control and Optimization, 23(3):329-380, May 1985. 702 Gullapalli and Barto [4] A. W. Moore and C. G. Atkeson. Memory-based reinforcement learning: Efficient computation with prioritized sweeping. In S. J. Hanson, J. D. Cowan, and C. L. Giles, editors, Advance8 in Neural Information Proceuing Sy8tem8 5, pages 263-270, San Mateo, CA, 1993. Morgan Kaufmann Publishers. [5] J. Peng and R. J. Williams. Efficient learning and planning within the dyna framework. In Proceeding8 of the Second International Conference on Simulation of Adaptive Behavior, Honolulu, HI, 1992. [6] S. P. Singh and V. Gullapalli. Asynchronous modified policy iteration with single-sided updates. (Under review). [7] R. S. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3:9-44, 1988. [8] G. J. Tesauro. Practical issues in temporal difference learning. Machine Learning, 8(3/4):257-277, May 1992. [9] C. J. C. H. Watkins. Learning from delayed reward8. PhD thesis, Cambridge University, Cambridge, England, 1989. [10] R. J. Williams and L. C. Baird. Analysis of some incremental variants of policy iteration: First steps toward understanding actor-critic learning systems. Technical Report NU-CCS-93-11, Northeastern University, College of Computer Science, Boston, MA 02115, September 1993.	1993	82
839	Fast Pruning Using Principal Components Asriel U. Levin, Todd K. Leen and John E. Moody Department of Computer Science and Engineering Oregon Graduate Institute P.O. Box 91000 Portland, OR 97291-1000 Abstract We present a new algorithm for eliminating excess parameters and improving network generalization after supervised training. The method, "Principal Components Pruning (PCP)", is based on principal component analysis of the node activations of successive layers of the network. It is simple, cheap to implement, and effective. It requires no network retraining, and does not involve calculating the full Hessian of the cost function. Only the weight and the node activity correlation matrices for each layer of nodes are required. We demonstrate the efficacy of the method on a regression problem using polynomial basis functions, and on an economic time series prediction problem using a two-layer, feedforward network. 1 Introduction In supervised learning, a network is presented with a set of training exemplars [u(k), y(k)), k = 1 ... N where u(k) is the kth input and y(k) is the corresponding output. The assumption is that there exists an underlying (possibly noisy) functional relationship relating the outputs to the inputs y=/(u,e) where e denotes the noise. The aim of the learning process is to approximate this relationship based on the the training set. The success of the learned approximation 35 36 Levin, Leen, and Moody is judged by the ability of the network to approximate the outputs corresponding to inputs it was not trained on. Large networks have more functional flexibility than small networks, so are better able to fit the training data. However large networks can have higher parameter variance than small networks, resulting in poor generalization. The number of parameters in a network is a crucial factor in it's ability to generalize. No practical method exists for determining, a priori, the proper network size and connectivity. A promising approach is to start with a large, fully-connected network and through pruning or regularization, increase model bias in order to reduce model variance and improve generalization. Review of existing algorithms In recent years, several methods have been proposed. Skeletonization (Mozer and Smolensky, 1989) removes the neurons that have the least effect on the output error. This is costly and does not take into account correlations between the neuron activities. Eliminating small weights does not properly account for a weight's effect on the output error. Optimal Brain Damage (OBD) (Le Cun et al., 1990) removes those weights that least affect the training error based on a diagonal approximation of the Hessian. The diagonal assumption is inaccurate and can lead to the removal of the wrong weights. The method also requires retraining the pruned network, which is computationally expensive. Optimal Brain Surgeon (OBS) (Hassibi et al., 1992) removes the "diagonal" assumption but is impractical for large nets. Early stopping monitors the error on a validation set and halts learning when this error starts to increase. There is no guarantee that the learning curve passes through the optimal point, and the final weight is sensitive to the learning dynamics. Weight decay (ridge regression) adds a term to the objective function that penalizes large weights. The proper coefficient for this term is not known a priori, so one must perform several optimizations with different values, a cumbersome process. We propose a new method for eliminating excess parameters and improving network generalization. The method, "Principal Components Pruning (PCP)", is based on principal component analysis (PCA) and is simple, cheap and effective. 2 Background and Motivation PCA (Jolliffe, 1986) is a basic tool to reduce dimension by eliminating redundant variables. In this procedure one transforms variables to a basis in which the covariance is diagonal and then projects out the low variance directions. While application of PCA to remove input variables is useful in some cases (Leen et al., 1990), there is no guarantee that low variance variables have little effect on error. We propose a saliency measure, based on PCA, that identifies those variables that have the least effect on error. Our proposed Principal Components Pruning algorithm applies this measure to obtain a simple and cheap pruning technique in the context of supervised learning. Fast Pruning Using Principal Components 37 Special Case: PCP in Linear Regression In unbiased linear models, one can bound the bias introduced from pruning the principal degrees of freedom in the model. We assume that the observed system is described by a signal-plus-noise model with the signal generated by a function linear in the weights: y = Wou + e where u E ~P, Y E ~m, W E ~mxp, and e is a zero mean additive noise. The regression model is Y=Wu. The input correlation matrix is ~ = ~ L:k u(k)uT(k). It is convenient to define coordinates in which ~ is diagonal A = CT ~ C where C is the matrix whose columns are the orthonormal eigenvectors of~. The transformed input variables and weights are u = CT u and W = W C respectively, and the model output can be rewritten as Y = W u . It is straightforward to bound the increase in training set error resulting from removing subsets of the transformed input variable. The sum squared error is I = ~ L[y(k) - y(k)f[y(k) - y(k)] k Let Yl(k) denote the model's output when the last p -l components of u(k) are set to zero. By the triangle inequality h ~ L[y(k) - Yl(k)f[y(k) - Yl(k)] k < 1+ ~ L[Y(k) - Yl(k)f[Y(k) - Yl(k)] (1) k The second term in (1) bounds the increase in the training set errorl. This term can be rewritten as p ~ L[y(k) - Yl(k)f[Y(k) - lh(k)] L w; WiAi k i=l+l where Wi denotes the ith column of Wand Ai is the ith eigenvalue. The quantity w; Wi Ai measures the effect of the ith eigen-coordinate on the output error; it serves as our saliency measure for the weight Wi. Relying on Akaike's Final Prediction error (FPE) (Akaike, 1970), the average test set error for the original model is given by J[W] = ~ + pm I(W) -pm where pm is the number of parameters in the model. If p -l principal components are removed, then the expected test set is given by Jl[W] = N + lm Il(W) . N-lm 1 For y E Rl, the inequality is replaced by an equality. 38 Levin, Leen, and Moody If we assume that N» l * m, the last equation implies that the optimal generalization will be achieved if all principal components for which -T _ 2m! Wi WiAi < N are removed. For these eigen-coordinates the reduction in model variance will more then compensate for the increase in training error, leaving a lower expected test set error. 3 Proposed algorithm The pruning algorithm for linear regression described in the previous section can be extended to multilayer neural networks. A complete analysis of the effects on generalization performance of removing eigen-nodes in a nonlinear network is beyond the scope of this short paper. However, it can be shown that removing eigen-nodes with low saliency reduces the effective number of parameters (Moody, 1992) and should usually improve generalization. Also, as will be discussed in the next section, our PCP algorithm is related to the OBD and OBS pruning methods. As with all pruning techniques and analyses of generalization, one must assume that the data are drawn from a stationary distribution, so that the training set fairly represents the distribution of data one can expect in the future. Consider now a feedforward neural network, where each layer is of the form yi = r[WiUi] = r[Xi] . Here, ui is the input, Xi is the weighted sum of the input, r is a diagonal operator consisting of the activation function of the neurons at the layer, and yi is the output of the layer. 1. A network is trained using a supervised (e.g. backpropagation) training procedure. 2. Starting at the first layer, the correlation matrix :E for the input vector to the layer is calculated. 3. Principal components are ranked by their effect on the linear output of the layer. 2 4. The effect of removing an eigennode is evaluated using a validation set. Those that do not increase the validation error are deleted. 5. The weights of the layer are projected onto the l dimensional subspace spanned by the significant eigenvectors W -+ WClCr where the columns of C are the eigenvectors of the correlation matrix. 6. The procedure continues until all layers are pruned. 2If we assume that -r is the sigmoidal operator, relying on its contraction property, we have that the resulting output error is bounded by Ilell <= IIWlllle",lll where e",l IS error observed at Xi and IIWII is the norm of the matrices connecting it to the output. Fast Pruning Using Principal Components 39 As seen, the algorithm proposed is easy and fast to implement. The matrix dimensions are determined by the number of neurons in a layer and hence are manageable even for very large networks. No retraining is required after pruning and the speed of running the network after pruning is not affected. Note: A finer scale approach to pruning should be used ifthere is a large variation between Wij for different j. In this case, rather than examine w[ WiAi in one piece, the contribution of each wtj Ai could be examined individually and those weights for which the contribution is small can be deleted. 4 Relation to Hessian-Based Methods The effect of our PCP method is to reduce the rank of each layer of weights in a network by the removal of the least salient eigen-nodes, which reduces the effective number of parameters (Moody, 1992). This is in contrast to the OBD and OBS methods which reduce the rank by eliminating actual weights. PCP differs further from OBD and OBS in that it does not require that the network be trained to a local minimum of the error. In spite of these basic differences, the PCP method can be viewed as intermediate between OBD and OBS in terms of how it approximates the Hessian of the error function. OBD uses a diagonal approximation, while OBS uses a linearized approximation of the full Hessian. In contrast, PCP effectively prunes based upon a block-diagonal approximation of the Hessian. A brief discussion follows. In the special case of linear regression, the correlation matrix ~ is the full Hessian of the squared error.3 For a multilayer network with Q layers, let us denote the numbers of units per layer as {Pq : q = 0 . . . Q}.4 The number of weights (including biases) in each layer is bq = Pq(Pq-l + 1), and the total number of weights in the network is B = L:~=l bq . The Hessian of the error function is a B x B matrix, while the input correlation matrix for each of the units in layer q is a much simpler (Pq-l + 1) X (Pq-l + 1) matrix. Each layer has associated with it Pq identical correlation matrices. The combined set of these correlation matrices for all units in layers q = 1 .. . Q of the network serves as a linear, block-diagonal approximation to the full Hessian of the nonlinear network.5 This block-diagonal approximation has E~=l Pq(Pq-l + 1)2 non-zero elements, compared to the [E~=l Pq(Pq-l + 1)]2 elements of the full Hessian (used by OBS) and the L:~=l Pq(Pq-l + 1) diagonal elements (used by OBD). Due to its greater richness in approximating the Hessian, we expect that PCP is likely to yield better generalization performance than OBD. 3The correlation matrix and Hessian may differ by a numerical factor depending on the normalization of the squared error. If the error function is defined as one half the average squared error (ASE), then the equality holds. 4The inputs to the network constitute layer O. 5The derivation of this approximation will be presented elsewhere. However, the correspondence can be understood in analogy with the special case of linear regression. 40 Levin, Leen, and Moody 0.75 0 . 5 0.25 -0.25 0.25 o.~ 0 •. 75 a) -. -1 0.75 0.5 0.25 -1 b) -1 .. ...... .' ~# .. Figure 1: a) Underlying function (solid), training data (points), and 10th order polynomial fit (dashed). b) Underlying function, training data, and pruned regression fit (dotted). The computational complexities of the OBS, OBD, and PCP methods are respectively, where we assume that N 2: B. The computational cost of PCP is therefore significantly less than that of OBS and is similar to that of OBD. 5 Simulation Results Regression With Polynomial Basis Functions The analysis in section 2 is directly applicable to regression using a linear combination of basis functions y = W f (11,) • One simply replaces 11, with the vector of basis functions f(11,). We exercised our pruning technique on a univariate regression problem using monomial basis functions f(11,) = (1,u,u 2 , ... ,un f with n = 10. The underlying function was a sum of four sigmoids. Training and test data were generated by evaluating the underlying function at 20 uniformly spaced points in the range -1 ~ u ~ + 1 and adding gaussian noise. The underlying function, training data and the polynomial fit are shown in figure 1a. The mean squared error on the training set was 0.00648. The test set mean squared error, averaged over 9 test sets, was 0.0183 for the unpruned model. We removed the eigenfunctions with the smallest saliencies w2 >.. The lowest average test set error of 0.0126 was reached when the trailing four eigenfunctions were removed.6 . Figure 1 b shows the pruned regression fit. 6The FPE criterion suggested pruning the trailing three eigenfunctions. We note that our example does not satisfy the assumption of an unbiased model, nor are the sample sizes large enough for the FPE to be completely reliable. Fast Pruning Using Principal Components 41 0.9 0.85 0 . 8 '" 0 '" '" 0 . 75 r.l al 0 . 7 N ..... ..... «J 0 . 65 ~ 0 z 0.6 0 . 55 ·····················t 1 .......................... .......................... 0 . 5 0 2 4 6 8 10 12 Prediction Horizon (month) Time Series Prediction with a Sigmoidal Network Figure 2: Prediction of the IP index 1980 - 1990. The solid line shows the performance before pruning and the dotted line the performance after the application of the PCP algorithm. The results shown represent averages over 11 runs with the error bars representing the standard deviation of the spread. We have applied the proposed algorithm to the task of predicting the Index of Industrial Production (IP), which is one of the main gauges of U.S. economic activity. We predict the rate of change in IP over a set of future horizons based on lagged monthly observations of various macroeconomic and financial indicators (altogether 45 inputs). 7 Our standard benchmark is the rate of change in IP for January 1980 to January 1990 for models trained on January 1960 to December 1979. In all runs, we used two layer networks with 10 tanh hidden nodes and 6 linear output nodes corresponding to the various prediction horizons (1, 2, 3, 6, 9, and 12 months). The networks were trained using stochastic backprop (which with this very noisy data set outperformed more sophisticated gradient descent techniques). The test set results with and without the PCP algorithm are shown in Figure 2. Due to the significant noise and nonstationarity in the data, we found it beneficial to employ both weight decay and early stopping during training. In the above runs, the PCP algorithm was applied on top of these other regularization methods. 6 Conclusions and Extensions Our "Principal Components Pruning (PCP)" algorithm is an efficient tool for reducing the effective number of parameters of a network. It is likely to be useful when there are correlations of signal activities. The method is substantially cheaper to implement than OBS and is likely to yield better network performance than OBD.8 7Preliminary results on this problem have been described briefly in (Moody et al., 1993), and a detailed account of this work will be presented elsewhere. 8See section 4 for a discussion of the block-diagonal Hessian interpretation of our method. A systematic empirical comparison of computational cost and resulting network performance of PCP to other methods like OBD and OBS would be a worthwhile undertaking. 42 Levin, Leen, and Moody Furthermore, PCP can be used on top of any other regularization method, including early stopping or weight decay.9 Unlike OBD and OBS, PCP does not require that the network be trained to a local minimum. We are currently exploring nonlinear extensions of our linearized approach. These involve computing a block-diagonal Hessian in which the block corresponding to each unit differs from the correlation matrix for that layer by a nonlinear factor. The analysis makes use of GPE (Moody, 1992) rather than FPE. Acknowledgements One of us (TKL) thanks Andreas Weigend for stimulating discussions that provided some of the motivation for this work. AUL and JEM gratefully acknowledge the support of the Advanced Research Projects Agency and the Office of Naval Research under grant ONR NOOOI4-92-J-4062. TKL acknowledges the support of the Electric Power Research Institute under grant RP8015-2 and the Air Force Office of Scientific Research under grant F49620-93-1-0253. References Akaike, H. (1970). Statistical predictor identification. Ann. Inst. Stat. Math., 22:203. Hassibi, B., Stork, D., and Wolff, G. (1992). Optimal brain surgeon and general network pruning. Technical Report 9235, RICOH California Research Center, Menlo Park, CA. Jolliffe, I. T. (1986). Principal Component Analysis. Springer-Verlag. Le Cun, Y., Denker, J., and Solla, S. (1990). Optimal brain damage. In Touretzky, D., editor, Advances in Neural Information Processing Systems, volume 2, pages 598-605, Denver 1989. Morgan Kaufmann, San Mateo. Leen, T. K., Rudnick, M., and Hammerstrom, D. (1990). Hebbian feature discovery improves classifier efficiency. In Proceedings of the IEEE/INNS International Joint Conference on Neural Networks, pages I-51 to I-56. Moody, J. (1992). The effective number of parameters: An analysis of generalization and regularization in nonlinear learning systems. In Moody, J., Hanson, S., and Lippman, R., editors, Advances in Neural Information Processing Systems, volume 4, pages 847-854. Morgan Kaufmann. Moody, J., Levin, A., and Rehfuss, S. (1993). Predicting the u.s. index of industrial production. Neural Network World, 3:791-794. in Proceedings of Parallel Applications in Statistics and Economics '93. Mozer, M. and Smolensky, P. (1989). Skeletonization: A technique for trimming the fat from a network via relevance assesment. In Touretzky, D., editor, Advances in Neural Information Processing Systems, volume 1, pages 107-115. Morgan Kaufmann. Weigend, A. S. and Rumelhart, D. E. (1991). Generalization through minimal networks with application to forecasting. In Keramidas, E. M., editor, INTERFACE'91 - 23rd Symposium on the Interface: Computing Science and Statistics, pages 362-370. 9(Weigend and Rumelhart, 1991) called the rank of the covariance matrix of the node activities the "effective dimension of hidden units" and discussed it in the context of early stopping.	1993	83
840	Hidden Markov Models for Human Genes Pierre Baldi * Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109 Yves Chauvin t Net-ID, Inc. 601 Minnesota San Francisco, CA 94107 S0ren Brunak Center for Biological Sequence Analysis The Technical University of Denmark DK-2800 Lyngby, Denmark Jacob Engelbrecht Center for Biological Sequence Analysis The Technical University of Denmark DK-2800 Lyngby, Denmark Anders Krogh Electronics Institute The Technical University of Denmark DK-2800 Lyngby, Denmark .Abstract Human genes are not continuous but rather consist of short coding regions (exons) interspersed with highly variable non-coding regions (introns). We apply HMMs to the problem of modeling exons, introns and detecting splice sites in the human genome. Our most interesting result so far is the detection of particular oscillatory patterns, with a minimal period ofroughly 10 nucleotides, that seem to be characteristic of exon regions and may have significant biological implications. • and Division of Biology, California Institute of Technology. t and Department of Psychology, Stanford University. 761 762 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh 3' splice site acceptor site exon intron EXON CONSENSUS SEQUENCES 5' splice site donor site I I I I I I I I I I NC AG I G AC AG I GTAGAGT CCCCCCCC T ~------------------~ Figure 1: Structure of eukaryotic genes (not to scale: introns are typically much longer than exons). 1 INTRODUCTION The genes of higher organisms are not continuous. Rather, they consist of relatively short coding regions called exons interspersed with non-coding regions of highly variable length called introns (Fig. 1). A complete gene may comprise as many as fifty exons. Very often, exons encode discrete functional or structural units of proteins. Prior to the translation of genes into proteins, a complex set of biochemical mechanisms is responsible for the precise cutting of genes at the splice junctions, i.e. the boundaries between introns and exons, and the subsequent removal and ligation which results in the production of mature messenger RNA. The translation machinery of the cell operates directly onto the mRNA, converting a primary sequence of nucleotides into the corresponding primary sequence of amino acids, according to the rules of the genetic code. The genetic code converts every three contiguous nucIeotides, or codons, into one of the twenty amino acids (or into a stop signal). Therefore the splicing process must be exceedingly precise since a shift of only one base pair completely upsets the codon reading frame for translation. Many details of the splicing process are not known; in particular it is not clear how acceptor sites (i.e. intron/exon boundaries) and donor sites (i.e. exon/intron boundaries) are recognized with extremely high accuracy. Both acceptor and donor sites are signaled by the existence of consensus sequences, i.e. short sequences of nucleotides which are highly conserved across genes and, to some extent, across species. For instance, Hidden Markov Models for Human Genes 763 most introns start with GT and terminate with AG and additional patterns can be detected in the proximity of the splice sites. The main problem with consensus sequences, in addition to their variability, is that by themselves they are insufficient for reliable splice site detection. Indeed, whereas exons are relatively short with an average length around 150 nucleotides, introns are often much longer, with several thousand of seemingly random nucleotides. Therefore numerous false positive consensus signals are bound to occur inside the introns. The GT dinucleotide constitutes roughly 5% of the dinucleotides in human DNA, but only a very small percentage of these belongs to the splicing donor category, in the order of 1.5%. The dinucleotide AG constitutes roughly 7.5% of all the dinucleotides and only around 1% of these function as splicing acceptor sites. In addition to consensus sequences at the splice sites, there seem to exist a number of other weak signals (Senapathy (1989), Brunak et a1. (1992)) embedded in the 100 intron nucleotides upstream and downstream of an exon. Partial experimental evidence seems also to suggest that the recognition of the acceptor and donor boundaries of an exon may be a concerted process. In connection with the current exponential growth of available DNA sequences and the human genome project, it has become essential to be able to algorithmically detect the boundaries between exons and introns and to parse entire genes. Unfortunately, current available methods are far from performing at the level of accuracy required for a systematic parsing of the entire human genome. Most likely, gene parsing requires the statistical integration of several weak signals, some of which are poorly known, over length scales of a few hundred nucleotides. Furthermore, initial and terminal exons, lacking one of the splice sites, need to be treated separately. 2 HMMs FOR BIOLOGICAL PRIMARY SEQUENCES The parsing problem has been tackled with classical statistical methods and more recently using neural networks (Lapedes (1988), Brunak (1991)), with encouraging results. Conventional neural networks, however, do not seem ideally suited to handle the sort of elastic deformations introduced by evolutionary tinkering in genetic sequences. Another trend in recent years, has been the casting of DNA and protein sequences problems in terms of formal languages using context free grammars, automata and Hidden Markov Models (HMMs). The combination of machine learning techniques which can take advantage of abundant data together with new flexible representations appears particularly promising. HMMs in particular have been used to model protein families and address a number of task such as multiple alignments, classification and data base searches (Baldi et al. (1993) and (1994); Haussler et a1. (1993); Krogh et al. (1994a); and references therein). It is the success obtained with this method on protein sequences and the ease with which it can handle insertions and deletions that naturally suggests its application to the parsing problem. In Krogh et al. (1994b), HMMs are applied to the problem of detecting coding/noncoding regions in bacterial DNA (E. coli), which is characterized by the absence of true introns (like other prokaryotes). Their approach leads to a HMM that integrates both genic and intergenic regions, and can be used to locate genes fairly reliably. A similar approach for human DNA, that is not based on HMMs, but uses dynamic programming and neural networks to combine various gene finding techniques, is described in Snyder and Stormo (1993). In this paper we take a 764 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh ci Main State Entropy Values 10 20 30 40 60 60 70 80 90100110120130140160160170 Main State Position 180190200210220230240260260270280290300310320330340360 Main State Position Figure 2: Entropy of emission distribution of main states. first step towards parsing the human genome with HMMs by modeling exons (and flanking intron regions). As in the applications of HMMs to speech or protein modeling, we use left-right architectures to model exon regions, intron regions or their boundaries. The architectures typically consist of a backbone of main states flanked by a sequence of delete states and a sequence of insert states, with the proper interconnections (see Baldi et al. (1994) and Krogh et al. (1994) for more details and Fig. 4 below). The data base used in the experiments to be described consists of roughly 2,000 human internal exons, with the corresponding adjacent introns, extracted from release 78 of the GenBank data base. It is essential to remark that, unlike in the previous experiments on protein families, the exons in the data base are not directly related by evolution. As a result, insertions and deletions in the model should be interpreted in terms of formal operations on the strings rather than evolutionary events. 3 EXPERIMENTS AND RESULTS A number of different HMM training experiments have been carried using different classes of sequences including exons only, flanked exons (with 50 or 100 nucleotides on each side), introns only, flanked acceptor and flanked donor sites (with 100 nucleotides on each side) and slightly different architectures and learning algorithms. Only a few relevant examples will be given here. ~ ~ ~ ;: :::t :::: ~ :; ~ ;: :::t ~ g :~ A Hidden Markov Models for Human Genes 765 ~..J~~ 00 140 .20 .40 .00 2.0 '00 .20 In •• rt at.ta Po •. tlon C 100 .00 ,.0 G 40 120 200 •• 0 320 '40 T ... 32. Figure 3: Emission distribution from main states. In an early experiment, we trained a model of length 350 using 500 flanked exons, with 100 nucleotides on each side, using gradient descent on the negative loglikelihood (Baldi and Chauvin (1994)). The exons themselves had variable lengths between 50 and 300. The entropy plot (Fig. 2), after 7 gradient descent training cycles, reveals that the HMM has learned the acceptor site quite well but appears to have some difficulties with the donor site. One possible contributing factor is the high variability of the length of the training exons: the model seems to learn two donor sites, one for short exons and one for the other exons. The most striking pattern, however, is the greater smoothness of the entropy in the exon region. In the exon region, the entropy profile is weakly oscillatory, with a period of about 20 base pairs. Discrimination and t-tests conducted on this model show that it is definitely capable of discriminating exon regions, but the confidence level is not sufficient yet to reliably search entire genomes. A slightly different model was subsequently trained using again 500 flanked exons, with the length of the exons between 100 and 200 only. The probability of emitting each one of the four nucleotides, across the main states of the model, are plott.ed in Fig. 3, after the sixt.h gradient descent training cycle. Again the donor site seems harder to learn than the acceptor site. Even more striking are the clear 766 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh Figure 4: The repeated segment of the tied model. Note that position 15 is identical to position 5. oscillatory patterns present in the exon region, characterized by a minimal period of 10 nucleotides, with A and G in phase and C and T in anti-phase. The fact that the acceptor site is easier to learn could result from the fact that exons in the training sequences are always flanked by exactly 100 nucleotides upstream. To test this hypothesis, we trained a similar model using the same sequences but in reverse order. Surprisingly, the model still learns the acceptor site (which is now downstream from the donor site) much better than the donor site. The oscillatory pattern in the reversed exon region is still present. The oscillations we observe could also be an artifact of the method: for instance, when presented with random training sequences, oscillatory HMM solutions could appear naturally as local optima of the training procedure. To test this hypothesis, we trained a model using random sequences of similar average composition as the exons and found no distinct oscillatory patterns. We also checked that our data base of exons does not correspond prevalently to a-helical domains of proteins. To further test our findings, we trained a tied exon model with a hard-wired periodicity of 10. The tied model consists of 14 identical segments of length 10 and 5 additional positions in the beginning and end of the model, making a total length of 150. During training the segments are kept identical by tying of the parameters, i.e. the parameters are constrained to be exactly the same throughout learning, as in the weight sharing procedure for neural networks. The model was trained on 800 exon sequences of length between 100 and 200, and it was tested on 262 different sequences. The parameters of the repeated segment, after training, are shown in Fig. 4. Emission probabilities are represented by horizontal bars of corresponding proportional length. There is a lot of structure in this segment. The most prominent feature is the regular expression [AT][AT]G at position 12-14. (The regular expression means "anything but T followed by A or T followed by G".) The same pattern was often found at positions with very low entropy in the "standard models" described above. In order to test the significance, the tied model was compared to a standard model of the same length. The average negative log-likelihood (NNL) they both assign to the exon sequences and to random sequences of similar composition, as well as their number of parameters are shown in the table below. Hidden Markov Models for Human Genes 767 Model Scores NLL training NLL testing # parameters Standard model 203.2 200.3 2550 with random seqs Standard model 198.8 196.4 2550 with real seqs Tied model 198.6 195.6 340 with real seqs The tied model achieves a level of performance comparable to the standard model but with significantly less free parameters, and therefore a period of 10 in the exons seems to be a strong hypothesis. Note that the period of the pattern is not strictly 10, and we found almost equally good models with a built-in period of 9 or 11. The type of left-to-right architecture we have used is not the ideal model of an exon, because of the large length variations. It would be desirable to have a model with a loop structure such that the segment can be entered as many times as necessary for any given exon (see Krogh et al. (1994b) for a loop structure used for E. coll DNA). This is one of the future lines of research. 4 CONCLUSION In summary, we are applying HMMs and related methods to the problems of exon/intron modeling and human genome parsing. Our preliminary results show that acceptor sites are intrinsically easier to learn than donor sites and that very simple HMM models alone are not sufficient for reliable genome parsing. Most importantly, interesting statistical 10 base oscillatory patterns have been detected in the exon regions. If confirmed, these patterns could have significant biological and algorithmic implications. These patterns could be related to the superimposition of several simultaneous codes (such as triplet code and frame code), and/or to the way DNA is wrapped around histone molecules (Beckmann and Trifonov (1991)). Presently, we are investigating their relationship to reading frame effects by training several HMM models using a data base of exons with the same reading frame. References Beckmann, J.S. and Trifonov, E.N. (1991) Splice Junctions Follow a 205-base Ladder. PNAS USA, 88, 2380-2383. Baldi, P., Chauvin, Y., Hunkapiller, T. and McClure, M. A. (1994) Hidden Markov Models of Biological Primary Sequence Information. PNAS USA, 91, 3, 1059-1063. Baldi, P., Chauvin, Y., Hunkapiller, T. and McClure, M. A. (1993) Hidden Markov Models in Molecular Biology: New Algorithms and Applications. Advances in Neural Information Processing Systems 5, Morgan Kaufmann, 747-754. Baldi, P. and Chauvin, Y. (1994) Smooth On-Line Learning Algorithms for Hidden Markov Models. Neural Computation, 6, 2, 305-316. Brunak, S., Engelbrecht, J. and Knudsen, S. (1991) Prediction of Human mRNA Donor and Acceptor Sites from the DNA Sequence. Journal of Molecular Biology, 220,49-65. 768 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh Engelbrecht, J., Knudsen, S. and Brunak S., (1992) GIC rich tract in 5' end of human introns, Journal of Molecular Biology, 221, 108-113. Haussler, D., Krogh, A., Mian, I.S. and Sjolander, K. (1993) Protein Modeling using Hidden Markov Models: Analysis of Globins, Proceedings of the Hawaii International Conference on System Sciences, 1, IEEE Computer Society Press, Los Alamitos, CA, 792-802. Krogh, A., Brown, M., Mian, I. S., Sjolander, K. and Haussler, D. (1994a) Hidden Markov Models in Computational Biology: Applications to Protein Modeling. Journal of Molecular Biology, 235, 1501-153l. Krogh, A., Mian, I. S. and Haussler, D. (1994b) A Hidden Markov Model that Finds Genes in E. Coli DNA, Technical Report UCSC-CRL-93-33, University of California at San ta Cruz. Lapedes, A., Barnes, C., Burks, C., Farber, R. and Sirotkin, K. Application of Neural Networks and Other Machine Learning Algorithms to DNA Sequence Analysis. In G.I. Bell and T.G. Marr, editors. The Procceedings of the Interface Between Computation Science and Nucleic Acid Sequencing Workshop. Proceedings of the Santa Fe Institute, volume VII, pages 157-182. Addison Wesley, Redwood City, CA,1988. Senapathy, P., Shapiro, M.B., and Harris, N.1. (1990) Splice Junctions, Branch Point Sites, and Exons: Sequence Statistics, Identification and Applications to Genome Project. Patterns in Nucleic Acid Sequences, Academic Press, 252-278. Snyder, E.E. and Stormo, G.D. (1993) Identification of coding regions in genomic DNA sequences: an application of dynamic programming and neural networks. Nucleic Acids Research, 21, 607-613.	1993	84
841	Postal Address Block Location Using A Convolutional Locator Network Ralph Wolf and John C. Platt Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 Abstract This paper describes the use of a convolutional neural network to perform address block location on machine-printed mail pieces. Locating the address block is a difficult object recognition problem because there is often a large amount of extraneous printing on a mail piece and because address blocks vary dramatically in size and shape. We used a convolutional locator network with four outputs, each trained to find a different corner of the address block. A simple set of rules was used to generate ABL candidates from the network output. The system performs very well: when allowed five guesses, the network will tightly bound the address delivery information in 98.2% of the cases. 1 INTRODUCTION The U.S. Postal Service delivers about 350 million mail pieces a day. On this scale, even highly sophisticated and custom-built sorting equipment quickly pays for itself. Ideally, such equipment would be able to perform optical character recognition (OCR) over an image of the entire mail piece. However, such large-scale OCR is impractical given that the sorting equipment must recognize addresses on 18 mail pieces a second. Also, the large amount of advertising and other irrelevant text that can be found on some mail pieces could easily confuse or overwhelm the address recognition system. For both of these reasons, character recognition must occur 745 746 Wolf and Platt Figure 1: Typical address blocks from our data set. Notice the wide variety in the shape, size, justification and number of lines of text. Also notice the detached ZIP code in the upper right example. Note: The USPS requires us to preserve the confidentiality of the mail stream. Therefore, the name fields of all address block figures in this paper have been scrambled for publication. However, the network was trained and tested using unmodified images. only on the relevant portion of the envelope: the destination address block. The system thus requires an address block location (ABL) module, which draws a tight bounding box around the destination address block. The ABL problem is a challenging object recognition task because address blocks vary considerably in their size and shape (see figure 1). In addition, figures 2 and 3 show that there is often a great deal of advertising or other information on the mail piece which the network must learn to ignore. Conventional systems perform ABL in two steps (Caviglione, 1990) (Palumbo, 1990). First, low-level features, such as blobs of ink, are extracted from the image. Then, address block candidates are generated using complex rules. Typically, there are hundreds of rules and tens of thousands of lines of code. The architecture of our ABL system is very different from conventional systems. Instead of using low-level features, we train a neural network to find high-level abstract features of an address block. In particular, our neural network detects the corners of the bounding box of the address block. By finding abstract features instead of trying to detect the whole address block in one step, we build a large degree of scale and shape invariance into the system. By using a neural network, we do not need to develop explicit rules or models of address blocks, which yields a more accurate system. Because the features are high-level, it becomes easy to combine these features into object hypotheses. We use simple address block statistics to convert the corner features into object hypotheses, using only 200 lines of code. Postal Address Block Location Using a Convolutional Locator Network 747 2 SYSTEM ARCHITECTURE Our ABL system takes 300 dpi grey scale images as input and produces a list of the 5 most likely ABL candidates as output. The system consists of three parts: the preprocessor, a convolutional locator network, and a candidate generator. 2.1 PREPROCESSOR The preprocessor serves two purposes. First, it substantially reduces the resolution of the input image, therefore decreasing the computational requirements of the neural network. Second, the preprocessor enhances spatial frequencies in the image which are associated with address text. The recipe used for the preprocessing is as follows: 1: Clip the top 20% of the image. 2: Spatially filter with a passband of 0.3 to 1.4mm. 3: Take the absolute value of each pixel. 4: Low-pass filter and subsample by a factor of 16 in X and Y. 5: Perform a linear contrast stretch, mapping the darkest pixel to 1.0 and the lightest pixel to 0.0. The effect of this preprocessing can be seen in figures 2 and 3. 2.2 CONVOLUTIONAL LOCATOR NETWORK We use a convolutional locator network (CLN) to find the corners of the bounding box. Each layer of a CLN convolves its weight pattern in two dimensions over the outputs of the previous layer (LeCun, 1989) (Fukushima, 1980). Unlike standard convolutional networks, the output of a CLN is a set of images, in which regions of activity correspond to recognition of a particular object. We train an output neuron of a CLN to be on when the receptive field of that neuron is over an object or feature, and off everywhere else. CLNs have been previously used to assist in the segmentation step for optical character recognition, where a neuron is trained to turn on in the center of every character, regardless of the identity of the character (Martin, 1992) (Platt, 1992). The recognition of an address block is a significantly more difficult image segmentation problem because address blocks vary over a much wider range than printed characters (see figure 1). The output of the CLN is a set of four feature maps, each corresponding to one corner of the address block. The intensity of a pixel in a given feature map represents the likelihood that the corresponding corner of the address block is located at that pixel. Figure 4 shows the architecture of our convolutional locator network (CLN). It has three layers of trainable weights, with a total of 22,800 free parameters. The network was trained via weight-shared backpropagation. The network was trained for 23 epochs on 800 mail piece images. This required 125 hours of cpu-time on an i860 based computer. Cross validation and final testing was done with two additional 748 Wolf and Platt Figure 2: The network operating on an example from the test set. The top image is the original image. The middle image is the image that is fed to the CLN after preprocessing. The preprocessing enhances the text and suppresses the background color. The bottom image is the first candidate of the ABL system. The output of the system is shown with a white and black rectangle. In this case, the first candidate is correct. Notice that our ABL system does not get confused by the horizontal lines in the image, which would confound a line-finding-based ABL system. Postal Address Block Location Using a Convolutional Locator Network 749 Figure 3: Another example from the test set. The preprocessed image still has a large amount of background noise. In this example, the first candidate of the ABL system (shown in the lower left) was almost correct, but the ZIP code got truncated. The second candidate of the system (shown in the lower right) gives the complete address. 750 Wolf and Platt Third layer of weights 4 36x16 windows Second layer of weights 8 9x9 windows First layer of weights 6 9x9 windows Output maps Second layer feature maps 2x2 subsampled first layer feature maps First layer feature maps Input image Figure 4: The architecture of the convolutional locator network used in our ABL system. data sets of 500 mail pieces each. All together, these 1800 images represent 6 Gbytes of raw data, or 25 Mbytes of preprocessed images. 2.3 CANDIDATE GENERATOR The candidate generator uses the following recipe to convert the output maps of the CLN into a list of ABL candidates: 1: Find the top 10 local maxima in each feature map. 2: Construct all possible tBL candidates by combining pairs of local maxima from opposing corners. 3: Discard candidates which have negative length or width. 4: Compute confidence of each candidate. 6: Sort the candidates according to confidence. 6: Remove duplicate and near duplicate candidates. 7: Pad the candidates by a fixed amount on all sides. The confidence of an address block candidate is: 2 Caddress block = PsizePIocation II Ci i=l where Caddress block is the confidence of the address block candidate, Psize is the prior probability of finding an address block of the hypothesized size, I\ocation is the prior probability of finding an address block in the hypothesized location, and Postal Address Block Location Using a Convolutional Locator Network 751 Ci are the value of each of the output maxima. The prior probabilities Psize and .A.ocation were based on smoothed histograms generated from the training set and validation set truths. Steps 6 and 7 each contain 4 tuning parameters which we optimized using the validation set and then froze before evaluating the final test set. 3 SYSTEM PERFORMANCE Figures 2 and 3 show the performance of the system on two challenging mail pieces from the final test set. We examined and classified the response of the system to all 500 test images. When allowed to produce five candidates, the ABL system found 98.2% of the address blocks in the test images. More specifically, 96% of the images have a compact bounding box for the complete address block. Another 2.2% have bounding boxes which contain all of the delivery information, but omit part of the name field. The remaining 1.8% fail, either because none of the candidates contain all the delivery information, or because they contain too much non-address information. The average number of candidates required to find a compact bounding box is only 1.4. 4 DISCUSSION This paper demonstrates that using a CLN to find abstract features of an object, rather than locating the entire object, provides a reasonable amount of insensitivity to the shape and scale of the obj~ct. In particular, the completely identified address blocks in the final test set had aspect ratios which ranged from 1.3 to 6.1 and their absolute X and Y dimensions both varied over a 3:1 range. They contained anywhere from 2 to 6 lines of text. In the past, rule-based systems for object recognition 'were designed from scratch and required a great deal of domain-specific knowledge. CLNs can be trained to recognize different classes of objects without a lot of domain-specific knowledge. Therefore, CLNs are a general purpose object segmentation and recognition architecture. The basic computation of a CLN is a high-speed convolution, which can be costeffectively implemented by using parallel hardware (Sickinger, 1992). Therefore, CLNs can be used to reduce the complexity and cost of hardware recognition systems. 5 CONCLUSIONS In this paper, we have described a software implementation for an address block location system which uses a convolutional locator network to detect the corners of the destination address on machine printed mail pieces. The success of this system suggests a general approach to object recognition tasks where the objects vary considerably in size and shape. We suggest the following 752 Wolf and Platt three-step approach: use a simple preprocessing algorithm to enhance stimuli which are correlated to the object, use a CLN to detect abstract features of the objects in the preprocessed image, and construct object hypotheses by a simple analysis of the network output. The use of CLNs to detect abstract features enables versatile object recognition architectures with a reasonable amount of scale and shape invariance. Acknowledgements This work was funded by USPS Contract No. 104230-90-C-344l. The authors would like to thank Dr. Binh Phan of the USPS for his generous advice and encouragement. The images used in this work were provided by the USPS. References Caviglione, M., Scaiola, (1990), "A Modular Real-time Vision System for Address Block Location," Proc. 4th Advanced Technology Conference, USPS, 42-56. Fukushima, K., (1980), "Neocognitron: A Self-Organizing Neural Network Model for a Mechanism of Pattern Recognition Unaffected by Shift in Position." Biological Cybernetics, 36, 193-202. LeCun, Y., Boser, B., Denker, J.S., Henderson, D., Howard, R. E., Hubbard, W., Jackel, L. D., (1989), "Backpropagation Applied to Handwritten Zip Code Recognition" Neural Computation, 1, 541-55l. Martin, G., Rashid, M., (1992), "Recognizing Overlapping Hand-Printed Characters by Centered-Object Integrated Segmentation and Recognition," Advances in Neural Information Processing Systems, 4, 504-51l. Palumbo, P. W., Soh, J., Srihari, S. N., Demjanenjo, V., Sridhar, R., (1990), "RealTime Address Block Location using Pipelining and Multiprocessing," Proc. 4th Advanced Technology Conference, USPS, 73-87. Platt, J., Decker, J. E, LeMoncheck, J. E., (1992), "Convolutional Neural Networks for the Combined Segmentation and Recognition of Machine Printed Characters," Proc. 5th Advanced Technology Conference, USPS, 701-713. Sackinger, E., Boser, B., Bromley, J., LeCun, Y., Jackel, L., (1992) "Application of the ANNA neural network chip to high-speed character recognition," IEEE Trans. Neural Networks, 3, (3), 498-505.	1993	85
842	An Analog VLSI Saccadic Eye Movement System Timothy K. Horiuchi Brooks Bishofberger and Christof Koch Computation and Neural Systems Program California Institute of Technology MS 139-74 Pasadena, CA 91125 Abstract In an effort to understand saccadic eye movements and their relation to visual attention and other forms of eye movements, we in collaboration with a number of other laboratories are carrying out a large-scale effort to design and build a complete primate oculomotor system using analog CMOS VLSI technology. Using this technology, a low power, compact, multi-chip system has been built which works in real-time using real-world visual inputs. We describe in this paper the performance of an early version of such a system including a 1-D array of photoreceptors mimicking the retina, a circuit computing the mean location of activity representing the superior colliculus, a saccadic burst generator, and a one degree-of-freedom rotational platform which models the dynamic properties of the primate oculomotor plant. 1 Introduction When we look around our environment, we move our eyes to center and stabilize objects of interest onto our fovea. In order to achieve this, our eyes move in quick jumps with short pauses in between. These quick jumps (up to 750 deg/sec in humans) are known as saccades and are seen in both exploratory eye movements and as reflexive eye movements in response to sudden visual, auditory, or somatosensory stimuli. Since the intent of the saccade is to bring new objects of interest onto the fovea, it can be considered a primitive attentional mechanism. Our interest 582 An Analog VLSI Saccadic Eye Movement System 583 lies in understanding how saccades are directed and how they might interact with higher attentional processes. To pursue this goal, we are designing and building a closed-loop hardware system based on current models of the saccadic system. Using traditional software methods to model neural systems is difficult because neural systems are composed of large numbers of elements with non-linear characteristics and a wide range of time-constants. Their mathematical behavior can rarely be solved analytically and simulations slow dramatically as the number and coupling of elements increases. Thus, real-time behavior, a critical issue for any system evolved for survival in a rapidly changing world, becomes impossible. Our approach to these problems has been to fabricate special purpose hardware that reflects the organization of real neural systems (Mead, 1989; Mahowald and Douglas, 1991; Horiuchi et al., 1992.) Neuromorphic analog VLSI technology has many features in common with nervous tissue such as: processing strategies that are fast and reliable, circuits that are robust against noise and component variability, local parameter storage for the construction of adaptive systems and low-power consumption. Our analog chips and the nervous system both use low-accuracy components and are significantly constrained by wiring. The design of the analog VLSI saccadic system discussed here is part of a long-term effort of a number of laboratories ( Douglas and Mahowald at Oxford University, Clark at Harvard University, Sejnowski at UCSD and the Salk Institute, Mead and Koch at Caltech) to design and build a complete replica of the early mammalian visual system in analog CMOS VLSI. The design and fabrication of all circuits is carried out via the US-government sponsored silicon service MOSIS, using their 2 J.1.m line process. 2 An Analog VLSI Saccadic System Figure 1: Diagram of the current system. The system obtains visual inputs from a photoreceptor array, computes the target location within a model of the superior colliculus and outputs the saccadic burst command to drive the eyeball. While not discussed here, an auditory localization S84 Horiuchi, Bishofberger, and Koch input is being developed to trigger saccades to acoustical stimuli. 2.1 The Oculomotor Plant The oculomotor plant is a one degree-of-freedom turntable which is driven by a pair of antagonistic-pulling motors. In the biological system where the agonist muscle pulls against a passive viscoelastic force, the fixation position is set by balancing these two forces. In our system, the viscoelastic properties of the oculomotor plant are simulated electronically and the fixation point is set by the shifting equilibrium point of these forces. In order maintain fixation off-center, like the biological system, a tonic signal to the motor controller must be maintained. 2.2 Photoreceptors The front-end of the system is an adaptive photoreceptor array (Delbriick, 1992) which amplifies small changes in light intensity yet adapts quickly to gross changes in lighting level. The current system uses a 1-D array of 32 photoreceptors 40 microns apart. This array provides the visual input to the superior colliculus circuitry. The gain control occurs locally at each pixel of the image and thus the maximum sensitivity is maintained everywhere in the image in contrast to traditional imaging arrays which may provide washed out or blacked-out areas of an image when the contrast within an image is too large. In order to trigger reflexive, visually-guided saccades, the output of the photoreceptor array is coupled to the superior colliculus model by a luminance change detection circuit. A change in luminance somewhere in the image sends a pulse of current to the colliculus circuit which computes the center of this activity. This coupling passes a current signal which is proportional to the absolute-value of the temporal derivative of a photoreceptor's voltage output, (i.e. IIdI(x, t)/dtll where I(x,t) is the output of the photreceptor array). While we are initially building a 1-D system, 2-D photoreceptor arrays have been built in anticipation of a two degree-of-freedom system. While these photoreceptor circuits have been successfully constructed, we do not discuss the results here since the performance of these circuits are described in the literature (Delbriick 1992). 2.3 Superior Colliculus Model The superior colliculus, located on the dorsal surface of the midbrain, is a key area in the behavioral orientation system of mammals. The superficial layers have a topographic map of visual space and the deeper layers contain a motor map of saccadic vectors. Microstimulation in this area initiates saccades whose metrics are related to the location stimulated. This type of representation is known as a population coding. Many neurons in the deeper layers of superior colliculus are multisensory and will generate saccades to auditory and somatosensory targets as well as visual targets. While it is clear that the superior colliculus performs a multitude of integrative functions between sensory modalities and attentional processes, our initial model of superior colliculus simply computes the center of activity from the population code seen in the superficial layers (i.e. the photoreceptor array) using the weighted average techniques developed by DeWeerth (1991) for computing the centroid of An Analog VLSI Saccadic Eye Movement System 585 Centroid Circuit Output vs. Target Error 10 2.. 2.6 2.0 I.' .«J / .. / V .:J) ·211 ~ V / V· ./ r:r'/ ·10 o 10 20 Figure 2: Output of the centroid circuit for a flashed red LED target at different angles away from the center position. Note that the output of the circuit was sampled 1 msec. after stimulus onset to account for capacitive delays. brightness. The results of the photoreceptor/centroid circuits are shown in Fig. 2. In the case of visually-guided saccades, retinal error translates directly into motor error and thus we can use the photoreceptors directly as our inputs. This simplified retina/superior colliculus model provides the motor error which is then passed on to the burst generator. 2.4 Saccadic Burst Generator The burst generator model (Fig. 3) driving the oculomotor plant receives as its input, desired change in eye position from the superior colliculus model and creates a two-component signal, a pulse and a step (Fig. 4). A pair of these pulse/step signals drive the two muscles of the eye which in turn moves the retinal array, thus closing the loop. The burst generator model is a double integrator model based on the work by Jurgens, et al (1981) and Lisberger et al (1987) which uses initial motor error as the input to the system. This motor error is injected into the "integrating" burst neuron which has negative feedback onto itself. This arrangement has the effect of firing a number of spikes proportional to the initial value of motor error. In the circuit, this integrator is implemented by a 1.9 pF capacitor. This burst of spikes serves to drive the eye rapidly against the viscosity. The burst is also integrated by the "neural integrator" (another 1.9 pF capacitor) which holds the local estimate of the current eye position from which the tonic, or holding signal is generated. Figs. 4 and 5 show output data from the burst generator chip and the response of the physical mechanism to this output. The inputs to the burst generator chip are 1) a voltage indicating desired eye position and 2) a digital trigger signal. The outputs are a pair of asynchronous digital pulse trains which carry the pulse/step signals which drive the left and right motors. S86 Horiuchi, Bishofberger, and Koch 3 Discussion As we are still in the formative stages of our project, our first goal has been to demonstrate a closed-loop system which can fixate a particular stimulus whose image is falling onto its photoreceptor array. The first set of chips represent dramatically simplified circuits in order to capture the first-order behavior of the system while using known representations. Owing to the large number of parameters that must be set, and their sensitivity to variations, we have begun a study to investigate biologically plausible approaches to automatic parameter-setting. In the short term we intend to dramatically refine the models used at each stage, most notably the superior colliculus which is involved in the integration of non-visual sources of saccade targets (e.g. memory or audition), and in the mechanisms used for target selection or fixation. In the longer run, we plan to model the interaction of this system with other oculomotor processes such as smooth pursuit, VOR, OKR, AND vergence eye movements. While the biological microcircuits of the superior colli cui us and brainstem burst generator are not well known, more is understood about the representations found in these areas. By exploring the advantages and disadvantages of various computational models in a working system, it is hoped that a truly robust system will emerge as well as better models to explain the biological data. The construction of a compact hardware system which operates in real-time can often provide a more intuitive understanding of the closed-loop system. In addition, a visually-attentive hardware system which is physically small and low-power has numerous applications in the real world such as in mobile robotics or remote surveillance. 4 Acknowledgements Many thanks go to Prof. Carver Mead and his group for developing the foundations of this research. Our laboratory is partially supported by grants from the Office of Naval Research and the Rockwell International Science Center. Tim Horiuchi is supported by a grant from the Office of Naval Research. 5 References T. Delbriick and C. Mead, (1993) Ph.D. Thesis, California Institute of Technology. S. P. DeWeerth, (1991) Ph.D. Thesis, California Institute of Technology. T. Horiuchi, W. Bair, B. Bishofberger, A. Moore, J. Lazzaro, C. Koch, (1992) Int. J. Computer Vision 8:3,203-216. R. Jiirgens, W. Becker, and H. H. Kornhuber, (1981) BioI. Cybern. 39:87-96. S. G. Lisberger, E. J. Morris, and L. Tychsen, (1987) Ann. Rev. Neurosci. 10:97129. M. Mahowald, and R. Douglas, (1991) Nature 354:515-518. C. Mead, (1989) Analog VLSI and Neural Systems, Addison-Wesley. Left Burst Neuron Motor Error < 0 Motor Error> 0 Right Burst Neuron An Analog VLSI Saccadic Eye Movement System S87 Other inputs: VOR/OKR ---t~ Smooth Pursuit IIIIII Left Motor Neuron Neural Integrator I~II.IIII Right Motor Neuron Figure 3: Schematic diagram of the burst generator. The burst neuron "samples" the motor error when it receives a trigger signal (not shown) and begins firing as a sigmoidal function of the motor error. The spikes feedback and discharge the integrator and the burst is shut down. This "pulse" signal drives the eye against the viscosity. This signal is also integrated by the neural integrator which contributes the "step" portion of the motor command to hold the eye in its final position. The neural integrator has additional velocity inputs for other oculomotor behavior such as smooth pursuit, VOR and OKR. Note that the burst neuron for the other muscle is silent in this direction. 588 Horiuchi, Bishoiberger, and Koch -0.25 0..00 0..25 0.50. 0..75 1.00 1.25 1.50. 1.75 2.00 2.25 lime (Iecondl) (10..2 ) Figure 4: Spike signals in the circuit during a small saccade. (7.5 degrees to the right, starting from 4.8 degrees to the right.) Top: Burst neuron, Middle: Neural Integrator, Bottom: Motor neuron. (one of the outputs of the chip) Note that the "neuron" circuit currently used increases both its pulse frequency and pulse duration for large input currents, causing the voltage saturation seen in the bottom trace. Eye Position VB. Time 80. 60. 40. I 20. I!! 0. :ii ~ i -20. -40. -60. ~D+---~----~----+----+----;---~~--~----+---~----~ -0.025 0.,000 0.,025 0..050. 0..075 0..100 0..125 0.,150. 0..175 0..200 0.,225 lime (lecondl) Figure 5: Horizontal position vs. time for 21 different saccades. Peak angular velocity achieved for the 60 degree saccade to the right was approximately 870 degrees per second. The input command was changed uniformly from -60 to +60 degrees. An Analog VLSI Saccadic Eye Movement System 589 Final Eye Position VS. Burst Command Voltage 80 60 40 I 20 I :1 0 ~ ~ i ·20 it; -40 -60 ·80 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 l,.,ul Voha. to BUI'II Generator (center - 2.5v) Figure 6: Linearity of the system for the position data given in the previous figure. Final eye position was computed as the average eye position during the last 20 msec. of each trace. Average of 10 Saccades from "center" to 30 deg. R 35 .. -.. .'.- .. 30 25 I 20 8 15 :-e ! ~ 10 5 0 ·5 -0.025 0.000 0.025 O.OSO 0.075 0.100 0.125 0.150 0.175 0.200 0.225 lime (Iecondl) Figure 7: Repeatability: The solid line shows averaged eye position (relative to center) vs. time for 10 identical saccades. The dashed lines show a standard deviation on each side of the mean. Most of the variability is attributed to problems with friction.	1993	86
843	An Optimization Method of Layered Neural Networks based on the Modified Information Criterion Sumio Watanabe Information and Communication R&D Center Ricoh Co., Ltd. 3-2-3, Shin-Yokohama, Kohoku-ku, Yokohama, 222 Japan sumio@ipe.rdc.ricoh.co.jp Abstract This paper proposes a practical optimization method for layered neural networks, by which the optimal model and parameter can be found simultaneously. 'i\Te modify the conventional information criterion into a differentiable function of parameters, and then, minimize it, while controlling it back to the ordinary form. Effectiveness of this method is discussed theoretically and experimentally. 1 INTRODUCTION Learning in art.ificialneural networks has been studied based on a statist.ical framework, because the statistical theory clarifies the quantitative relation between t.he empirical error and the prediction error. Let us consider a function <p( w; x) from the input space R/\ to the out.put space R L with a paramet.er 'lV. "\i\Te assume that training samples {(.1:j, yd}~l are taken from t.he true probabilit.y density Q(x, y). Let us define the empirical error by (1) 293 294 Watanabe and the prediction error by E(w) == J J lIy - ip(w; x)11 2Q(x, y)dxdy. (2) If we find a parameter w* which minimizes Eemp( w), then * 2(F(w) + 1) 1 < E(w ) >== (1 + NL ) < Eemp{w ) > +o(N)' (3) where < . > is the average value for the training samples, o( 1/ N) is a small term which satisfies No(I/N) ~ 0 when N ~ 00, and F(w), N, and L are respectively the numbers of the effective parameters of w, the training samples, and output units. Although the average < . > cannot be calculated in the actual application, the optimal model for the minimum prediction error can be found by choosing the model that minimizes the Akaike informat.ion crit.erion (AIC) [1], * 2(F(w) + 1) J(w)=(I+ NL )Eemp(w). (4) This method was generalized for arbitrary distance [2]. The Bayes informat.ion criterion (BIC) [3] and the minimum descript.ion lengt.h (MDL) [4] were proposed to overcome the inconsistency problem of AIC that the true model is not always chosen even when N ~ 00. The above information criteria have been applied to the neural network model selection problem, where the maximum likelihood estimator w* was calculated for each model, and then information criteria were compared. Nevertheless, the practical problem is caused by the fact. that we can not always find the ma..ximum likelihood estimator for each model, and even if we can. it takes long calculation time. In order to improve such model selection procedures, this paper proposes a practical learning algorithm by which the optimal model and parameter can be found simultaneously. Let us consider a modified information criterion, 2(FuCw) + 1) Ju(w) == (1+ NL )Eemp(w). (5) where a > 0 is a parameter and Fa(w) is a Cl-class function which converges to F(w) when a ~ O. \Ve minimize Ja(w), while controlling a as a ~ 0, To show effectiveness of this method, we show experimental results, and discuss the theoretical background. 2 A Modified Information Criterion 2.1 A Formal Information Criterion Let us consider a conditional probability distribut.ion. 1 Ily-ip(w;x)W P{W,O";ylx) = ( 2)L/2 exp(? 2 ), 2nO" _0" (6) An Optimization Method of Layered Neural Networks 295 where a function rp( w; x) = {rpi(W; x)} is given by the three-layered perceptron, II l\" rpi(W; :1:) = p(WiO + L Wij p(WjO + L wjkxd), (7) j=1 k=l and W = {w iO, Wij} is a set of biases and weights and p(.) is a sigmoidal function. Let A1max be the full-connected neuralnctwork model with 1'1." input units, H hidden units, and L output units, and /vt be the family of all models made from A1max by pruning weights or eliminating biases. \Vhcn a sct of training samples {(Xi, vd }[~:1 is given, we define an empirical loss and the prediction loss by L(w,O") 1 N N' L log P(w, 0"; vi/xd, 1 _ 1=1 -J J Q(:l",v) 10gP(w,0"; Vlx)d:t:dy. (8) (9) Minimizing Lemp (w, 0") is equivalent to minimizing Eelllp{ w), and mIIllmlzing L(w, 0") is equivalent. to minimizing E(w). \Ye assume t.hat. t.here exists a parameter (wAI'O"AI) which minimizes Lemp{W,CT) in each modcl.H E A1. By the theory of AIC, we have the following formula, (10) Based on this property, let us define a formal information criterion I (Af) for a model Af by I{Jlf) = 2N Lemp{wAI' O"~I ) + A( Fo (wAf) + 1) (11) where A is a constant and Fo (w) is the number of nonzero parameters in w, L Jl II l\ Fo{w) = L L fO(Wij) + L L fO{Wjd· (12) i=1 j=O j=lk=O where fo (x) is 0 if x = 0, or 1 if otherwise. I{1U) is formally equal to AIC if A = 2, or l\'IDL if A = 10g{N). Notc that F(w) ~ Fo{w) for arbitrary wand that F( wAJ ) = Fo (w AI) if and only if the Fisher information mat.rix of the model !II is positive definite. 2.2 A Modified Information Criterion In order to find the optimal model and parameter simultaneously, we define a modified information critcrion. For Q' > O. 2NLemp(w,0") + A{Fo{w) + 1), (13) L Jl H I{ Fo{w) LLfO'{Wij) + LLfo{wjJ.o), (14) i=l j=O j=ll,·=O where fa-(x) satisfies the following two conditions. 296 Watanabe (1) 10.(x) -+ 10(x) when 0: -+ O. (2) If Ixl :::; Ivi then 0:::; 10.(.1:) :::; 10(Y) :::; 1. For example, 1- exp( _x2 /0:2 ) and 1-1/(1 + (x/0:)2) satisfy this condition. Based on these definitions, we have the following theorem. Theorem min 1(111) = lim min 10 (w, 0'). AI EM o,~o W,CT This theorem shows that the optimal model and parameter can be found by minimizing 1a(1O, 0') while controlling 0: as 0: -+ 0 (The parameter 0: plays the same role as the temperature in the simulated annealing). As Fo.(x) -+ Fo(x) is not uniform convergence, this theorem needs the second condition on 1 a (:t'). (For proof of the theorem, see [5]). If we choose a different.iable function for 10 (10), then its local minimum can be found by the steepest descent method, dw 0 dO' 0 dt =-o10 10 (w,0'), Tt=-oO'la(w,O'). (15) These equat.ions result in a learning dynamics, N 0 A A2 0F ~1o = -TJ 2: {ow IIvi - ';'(10; .'ri) 112 + ; Ot;'}, (16) i=l where 0'2 = (I/NL)"'£//=lllvi - ,;,(w;:rdIl 2 . and 0: is slowly controlled as 0: -+ O. This dynamics can be understood as the (,lTor backpropagation with the added term. 3 Experimental Results 3.1 The true distribution is contained in the models First, we consider a case when t.he true distribut.ion is cont.ained in the model family M. Figure 1 (1) shows the true model from which t.he training samples were taken. One thousand input samples were t.aken from the uniform probability on [-0.5,0.5] x [-0.5,0.5] x [-0.5,0.5]. The output samples were calculat.ed by the network in Figure 1 (1), and noizes were added which were taken from a normal distribution with the expectation 0 and the variance 3.33 x 10-3 . Ten thousands testing samples were t.aken from t.he same distribut.ion. "Te used 10 ('IV) = 1 exp( _w2 /20'2) as a soft.ener function, and t.he "annealing schedule" of 0 ' was set as 0:( n) = 0'0 (1 - n/ n max ) + €, where 'Il is the t.raining cycle number, 0 '0 = 3.0, nmax = 25000, and € = 0.01. Figure 1 (2) shows the full-connected nlOdel Afmax with 10 hidden units, which is the initial model. In the training, the learning speed TJ was set as 0.1. We compared the empirical errors and t.he prediction errors for several cases for A (Figure 1 (5), (6)). If A = 2, the crit.erion is AIC, and if A = 10g(N) = 6.907, it is BIC or MDL. Figure 1 (3) and (4) show the optimized models and parameters for the criteria ,vith A = 2 and A = 5. \\Then .4 = 5, t.he true model could be found. An Optimization Method of Layered Neural Networks 297 3.2 The true distribution is not contained Second, let. us consider a case that the true distribution is not contained in the model family. For t.he training samples and the testing samples, we used the same probability density as the above case except that the function was (17) Figure 2 (1) and (2) show the training error and the prediction error, respectively. In t.his case, the best generalized model was found by AIC, shown in Figure 3. In the optimized network, Xl and X2 were almost separated from X3, which means that the network could find the structure of the true model in eq.{17.) The practical application to ultrasonic image reconstruct.ion is shown in Figure 3. 4 Discussion 4.1 An information criterion and pruning weights If P(w, u; ylx) sufficiently approximates Q(YI:~ ~ ) and N is sufficiently large, we have (18) where Z N = Lemp{ 'LV j\f) - LC(iJ j\f) and 'IV j\f is the parameter which minimizes L( 'lV, u) in the model lIf. Although < ZN >= 0 resulting in equation (10), its standard deviation has the same order as (1/ VN). However, if 1111 C 1If2 or lIt!1 ~ lith, then 'Ii; 1111 and 'LV 1\12 expected to be almost common. and it doesn't essentially affect the model selection problem [2]. The model family made by pruning weights or by eliminating biases is not. a totally ordered set but a partially ordered set for the order "c". Therefore, if a model 111 E M is select.ed, it is the optimal model in a local model family M' = {1If' E Mj 1If' C 111 or 111' ~ Af}, but it may not be the optimal model in the global family M. Artificial neural networks have the local minimum problem not. only in the parameter space but also in the model family. 4.2 The degenerate Fisher information matrix. If the true probability is contained in the model and the number of hidden units is larger than necessary one, then the Fisher informat.ion matrix is degenerated, and consequently. the maximum likelihood est.imator is not. subject t.o the asympt.otically normal distribution [6]. Therefore, the prediction error is not given by eq.(3), or AIC cannot. be deriyed. However, by the proposed method, the selected model has the non-degenerated Fiher information matrix, because if it is degenerate then the modified information crit.erion is not. minimized. 298 Watanabe ~ N(O,3.33 X 10 ) 10 t output unit "~,. -2.2 ~ 2.27 , -0.7 -2.9 (1) True model (2) Initial model for learning. (3) Optimized by AIC(A=2) E (w) = 3.29 X 10 -3 emp 3 E(w~ = 3.39 X 10(4) Optimized by A=5 ~m'w) = 3.31 X 10 -3 ' -3 E(W) = 3.37XlO E (w) E(w) X 10-3 emp 3.35 initial 1 3.45 3 initial 3 3.4 initial 2 3.3 initial 3.25 A 3.35 A AIC MDL (5) The emprical error (6) The prediction error Figure I: True distribution is contained in the models. E (w) emp 3.6 3.5 3.4 3.3 AIC initial 1 initial 2 ! initial 3 SIC (1) The empirical error 3.7 3.6 3.5 3.4 E(w*) initial 2 initial 3 The empirical error 3.31 X 10-3 The prediction error 3.41 X 10 -3 <'4;2' ~ WI~ '--t-t--+--+--t-+-..... A _;~9~65 ~/O~4' 3.409 X 10-3 T T AIC SIC xl x2 x3 (2) The prediction error (3) Optimized by AI C (A=2). Figure 2: True distribution is not contained in the models. An Optimization Method of Layered Neural Networks 299 (1) An Ultrasonic Imaging System (2) Sample Objects. Reconstructed Image 15 units neighborhood .. '. : '. : . ~ltrasOnic Image 32X32 (3) Neural Net.works Images for Traiuillg Images for Tcstiug Origillal Illlagcs --~~--+--~~---+----~ Restored using LS:-'I ----1-------+--------1 Restored using .\IC-----t------+------l f-~ "" ": .,~ . Restored using :.IDL_-'--_----''---____ -'--____ ..-.J (4 )Rcstored Images Figure 3: Practical Applicat.ion t.o Image Rest.oration The propo~ed method was applied t.o ultrasonic image rest.orat.ioll. Figure 3 (1). (2), (3), (4) respectively show an ultrasonic imaging system, the sample objects, and a neural network for image restorat.ion, and the original restored images. The number of paramet.ers optimized by LS~L AIC. and ':\IDL were respect.in-Iy 166. 138. and 57. Rather noizeless images w('re obtained using the modified AIC or 1IDL. For example, the '"Tail of R" ·was clearly restored using AIC. 300 Watanabe 4.3 Relation to another generalization methods In the neural information processing field, many methods have been proposed for preventing the over-fit.ting problem. One of t.he most. famous met.hods is the weight decay method, in which we assume a priori probabilit.y distribut.ion on the parameter space and minimize El (w) = Eemp( 10) + '\C( 10), (19) where ,\ and C(w) are chosen by several heuristic methods [7]. The BIC is the information criterion for such a met.hod [3], and the proposed method may be understood as a met.hod how to cont.rol ,\ and C( w). 5 Conclusion An optimization met.hod for layered neural networks was proposed ba.<;ed on the modified informat.ion criterion, and its effectiveness was discussed theoretically and experimentally. Acknowledgements The author would like to t.hank Prof. S. Amari, Prof. S. Yoshizawa, Prof. K. Aihara in University of Tokyo, and all members of the Amari seminar for their active discussions about statistical met.hods in neural net.works. References [1] H.Akaike. (1974) A New Look at the St.atistical Model Identification. it IEEE Trans. on Automatic Control, Vol.AC-19, No.6, pp.716-723. [2] N.Murata, S.Yoshizawa, and S.Amari.(1992) Learning Curves, IvIodel Sel~ction and Complexit.y of Neural Networks. Ad·lIa.nces in Neural Injorm(£tion Processing Systems 5, San Mateo, Morgan Kaufman, pp.607-614. [3] C.Schwarz (1978) Estimating the dimension of a model. Annals of St(ttistics Vo1.6, pp.461-464. [4] J .Rissanen. (1984) Universal Coding, Information, Prediction, and Estimation. IEEE Tra:ns. on Injormation Theory, Vo1.30, pp.629-636. [5] S.Watanabe. (1993) An Optimization :r.,·1ethod of Artificial Neural Networks based on a Modified Informat.ion Criterion. IEICE technical Re1JOrt Vol.NC93-52, pp.71-78. [6] H.'iVhite. (1989) Learning in Art.ificial Neural Net.works : A Stat.istical Perspective. Neural Computation, Vol.l, pp.425-464. [7] A.S.'iVeigend, D.E.Rumelhart, and B.A.Huberman. (1991) Generalizat.ion of weight-elimination with application t.o foreca.<;t.ing. Advances in Neural Information Processing Systems, Vo1.3, pp.875-882. PART II LEARNING THEORY, GENERALIZATION, AND COMPLEXITY	1993	87
844	A Computational Model for Cursive Handwriting Based on the Minimization Principle Yasuhiro Wada * Yasuharu Koike Eric Vatikiotis-Bateson Mitsuo Kawato A TR Human Infonnation Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan ABSTRACT We propose a trajectory planning and control theory for continuous movements such as connected cursive handwriting and continuous natural speech. Its hardware is based on our previously proposed forward-inverse-relaxation neural network (Wada & Kawato, 1993). Computationally, its optimization principle is the minimum torquechange criterion. Regarding the representation level, hard constraints satisfied by a trajectory are represented as a set of via-points extracted from a handwritten character. Accordingly, we propose a via-point estimation algorithm that estimates via-points by repeating the trajectory formation of a character and the via-point extraction from the character. In experiments, good quantitative agreement is found between human handwriting data and the trajectories generated by the theory. Finally, we propose a recognition schema based on the movement generation. We show a result in which the recognition schema is applied to the handwritten character recognition and can be extended to the phoneme timing estimation of natural speech. 1 INTRODUCTION In reaching movements, trajectory formation is an ill-posed problem because the hand can move along an infinite number of possible trajectories from the starting to the target point. However, humans move an arm between two targets along consistent one of an >II Present Address: Systems Lab., Kawasaki Steel Corporation, Makuhari Techno Garden, 1-3.Nakase, Mihama-ku, Chiba 261, Japan 727 728 Wada, Koike, Vatikiotis-Bateson, and Kawato infinite number of trajectories. Therefore, the brain should be able to compute a unique solution by imposing an appropriate criterion to the ill-posed problem. Especially, a smoothness performance index was intensively studied in this context. Flash & Hogan (1985) proposed a mathematical model, the minimum-jerk model. Their model is based on the kinematics of movement, independent of the dynamics of the musculoskeletal system. On the other hand, based on the idea that the objective function must be related to dynamics, Uno, Kawato & Suzuki (1989) proposed the minimum torque-change criterion which accounts for the desired trajectory determination. The criterion is based on the theory that the trajectory of the human arm is determined so as to minimize the time integral of the square of the rate of torque change. They proposed the following quadratic measure of performance. Where -rj is the torque generated by the jth actuator of M actuators, and ljis the movement time. ( 0)2 " M d-r' CT = r L dt Jo j=l dt (1) Handwriting production is an attractive subject in human motor control studies. In cursive handwriting, a symbol must be transformed into a motor command stream. This transformation process raises several questions. How can the central nervous system (eNS) represent a character symbol for producing a handwritten letter? By what principle can motor planning be made or a motor command be produced? In this paper we propose a handwriting model whose computational theory and representation are the same as the model in reaching movements. Our proposed computational model for cursive handwriting is assumed to generate a trajectory that passes through many via-points. The computational theory is based on the minimum torque-change criterion, and a representation of a character is assumed to be expressed as a set of via-points extracted from a handwritten character. In reaching movement, the boundary condition is given by the visual information, such as the location of a cup, and the trajectory formation is based on the minimum torque-change criterion, which is completely the same as the model of handwriting (Fig. 1). However, it is quite difficult to determine the via-points in order to reproduce a cursive handwritten character. We propose an algorithm that can determine the via-points of the handwritten character, based only on the same minimization principle and which does not use any other ad hoc information such as zero-crossing velocity (Hollerbach, 1981). Reaching .(reach to the object) Handwriting -. (write a character) Representation Computational Hardware n================nTheory Location of the object t Visual Information Via-Point (representation of character) Via-poitt Estimation Algorithm r-l-, -~-~--0r;1:: "" ="" '=~="":: "'~"'!H;11~ ~ jk l~t~C( ... .. Figure 1: A handwriting model. A Computational Model for Cursive Handwriting Based on the Minimization Principle 729 2 PREVIOUS WORK ON THE HANDWRITING MODEL Several handwriting models (Hollerbach, 1981; Morasso & Mussa-Ivaldi, 1982; Edleman & Flash, 1987) have been proposed. Hollerbach proposed a handwriting model based on oscillation theory. The model basically used a vertical oscillator and a horizontal oscillator. Morasso & Mussa-Ivaldi proposed a trajectory formation model using a spline function, and realized a handwritten character using the formation model. Edleman & Flash (1987) proposed a handwriting model based on snap (fourth derivative of position) minimization. The representation of a character was four basic strokes and a handwritten character was regenerated by a combination of several strokes. However, their model was different from their theory for reaching movement. Flash & Hogan (1985) have proposed the minimum jerk criterion in the reaching movement. 3 A HANDWRITING MODEL 3.1 Trajectory formation neural network: Forward-Inverse Relaxation Model (FIRM) First, we explain the trajectory formation neural network. Because the dynamics of the human arm are nonlinear, finding a unique trajectory based on the minimum torquechange criterion is a nonlinear optimization problem. Moreover, it is rather difficult. There are several criticisms of previous proposed neural networks based on the minimum torque-change criterion: (1) their spatial representation of time, (2) back propagation is essential, and (3) much time is required. Therefore, we have proposed a new neural network, FIRM(Forward-Inverse Relaxation Model) for trajectory formation (Wada & Kawato, 1993). This network can be implemented as a biologically plausible neural network and resolve the above criticisms. 3.2 Via-point estimation model Edelman & Flash (1987) have pointed out the difficulty of finding the via-points in a handwritten character. They have argued two points: (1) the number of via-points, (2) a reason for the choice of every via-point locus. It is clear in approximation theory that a character can be regenerated perfectly if the number of extracted via-pOints is large. Appropriate via-points can not be assigned according to a regular sampling rule if the sample duration is constant and long. Therefore, there is an infinite number of combinations of numbers and via-point positions in the problem of extracting via-points from a given trajectory, and a unique solution can not be found if a trajectory reformation theory is not identified. That is, it is an ill-posed problem. The algorithm for assigning the via-points finds the via-points by iteratively activating both the trajectory formation module (FIRM) and the via-point extraction module (Fig. 2). The trajectory formation module generates a trajectory based on the minimum torquechange criterion using the via-points which are extracted by the via-point extraction module. The via-point extraction module assigns the via-points so as to minimize the square error between the given trajectory and the trajectory generated by the trajectory formation module. The via-point extraction algorithm will stop when the error between the given trajectory and the trajectory generated from the extracted via-points reaches a threshold. 730 Wada, Koike, Vatikiotis-Bateson, and Kawato Via-Points Extraction Module Minimum Torque- Trajectory Formation Module (FIRM) Change Trajectory f'r~ (~y<h f'IM ( . 5 .... • o j~l (J1 (I) -9~ta(t) dl --.. Min -Min .. o j=1 dI Via-points assignment to Via-Point - Trajectory generation decrease the above trajectory Information based on minimum torqueerror (Position . Time) change criterion Figure 2: Via-point estimation model. 9~ta(t) is the given trajectory of the j-th joint angle and ei (I) represents the generated trajectory. 3.2.1 Algorithm of via-point extraction There are a via-point extraction procedure and a trajectory production procedure in the via-point extraction module. and they are iteratively computed. Trajectory production in the module is based on the minimum-jerk model (Flash & Hogan 1985) on a joint angle space. which is equivalent to the minimum torque-change model when arm dynamics are approximated as in the following dynamic equation: ",i = [i Oi (j= 1 ..... M) (2) where Ii and iji are the inertia of the link and the acceleration of the j-th joint angle. respectively. The algorithm for via-point extraction is illustrated in Fig. 3. The procedural sequence is as follows: (Step 1) A trajectory between a starting point and a final point is generated by using the minimum torque-change principle of the linear dynamics model. (Step 2) The point with the maximum square error value between the given trajectory and the generated trajectory is selected as a via-point candidate. (Step 3) If the maximum value of the square error is less than the preassigned threshold. the procedure described above is finished. If the maximum value of the square error is greater than the threshold. the via-point candidate is assigned as via-point i and a trajectory is generated from the starting point through the via-point i to the final point. This generated trajectory is added to the trajectory that has already been generated. The time of the start point of the generated trajectory is a via-point located just before the assigned via-point i. and the time of the final point of the generated trajectory is a viapoint located just after the assigned via-point i. The position error of the start point and the final point equal O. since the compensation for the error has already been made. Thus, the boundary conditions of the generated trajectory at the start and final point become O. The velocity and acceleration constraints at the start and final point are set to O. (Step 4) By repeating Steps 2 and 3, a set of via-points is found. The j-th actuator velocity constraint 9!ia and acceleration constraint O!ia at the via-point i are set by minimizing the following equation. J(8!ia,O~a) = [p{ r:!" (lP)2 dt + r:} (8'i)2 dt} ~ Min (3) J,O J, ... A Computational Model for Cursive Handwriting Based on the Minimization Principle 731 I Step31 ~Ory by Step3 time .. Figure 3: An algorithm for extracting via-points. Finally, the via-points are fed to the FIRM, and the minimum torque change trajectory is produced. This trajectory and the given trajectory are then compared again. If the value of the square error does not reach the threshold, the procedure above is repeated. It can be mathematically shown that a given trajectory is perfectly approximated with this method (completeness), and furthermore that the number of extracted via-points for a threshold is the minimum (optimality). (Wada & Kawato, 1994) 4 PERFORMANCE OF THE VIA-POINT ESTIMATION MODEL 4.1 Performance of single via-point movement First, we examine the performance of our proposed via-point estimation model. A result of via-point estimation in a movement with a via-point is shown in Fig 4. Two movements (T3-PI-T5 and T3-P2-T5) are examined. The white circle and the solid lines show the target points and measured trajectories, respectively. PI and P2 show target via-points. The black circle shows the via-points estimated by the algorithm. The estimated via-points were close to the target via-points. Thus, our proposed via-point estimation algorithm can find a via-point on the given trajectory. 0.65 • Estimated Via-Point 0.60 0 TargetPoint 0 PI 0.55 ]: 0.50 T5 >0 0.45 0.40 0.35 0.30 '--.-----,..--...,....-~---r--r-__.-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 X[m] Figure 4: A result of via-point estimation in a movement with a via-point. 4.2 Performance of the handwriting model Fig. 5 shows the case of cursive connected handwritten characters. The handwriting model can generate trajectories and velocity curves of cursive handwritten characters that are almost identical to human data. The estimated via-points are classified into two groups. The via-points in one group are extracted near the minimum points of the 732 Wada, Koike, Vatikiotis-Bateson, and Kawato 0.$2 • Eatimar.cd Via-Point ••••• Trajeclary by IIICIdoI ~.10 0.00 X(ID) (a) 0.10 (b) Figure 5: Estimated via-points in cursive handwriting. (a) and (b) show the trajectory and tangential velocity profile, respectively. The via-point estimation algorithm extracts a viapoint (segmentation point) between characters. velocity profile. The via-points of the other group are assigned to positions that are independent of the above points. Generally, the minimums of the velocity are considered to be the feature points of the movement. However, we confirmed that a given trajectory can not be reproduced by using only the first group of via-points. This finding shows that the second group of via-points is important. Our proposed algorithm based on the minimization principle can estimate points that can not be selected by any kinematic criterion. Funhermore, it is important in handwritten character recognition that the viapoint estimation algorithm extracts via-points between characters, that is, their segmentation points. 5 FROM FORMATION TO RECOGNITION 5.1 A recognition model Next, we propose a recognition system using the trajectory formation model and the viapoint estimation model. There are several reports in the literature of psychology which suggest that the formation process is related to the recognition process. (Liberman & Mattingly, 1985; Freyd, 1983) Here, we present a pattern recognition model that strongly depends on the handwriting model and the via-point estimation model (Fig.6). (1) The features of the handwritten character are extracted by the via-point estimation algorithm. (2) Some of via-points are segmented and normalized in space and time. Then, (3) a trajectory is regenerated by using the normalized via-points. (4) A symbol is identified by comparing the regenerated trajectory with the template trajectory. QJ .... ~ E ''= ~ -E .5.: o.~ c..""" IQ .!~ ;> Recognizer ~ (Reformation & Comparison) ~Ymb' Figure 6: Movement pattern recognition using extracted via-points obtained through movement pattern generator A Computational Model for Cursive Handwriting Based on the Minimization Principle 733 1 :BAD : (0,17) (18,35) (36,52) 2 :BAD : (0,18) (18,35) (36,52) 3 :BAD : (0,17) (18,35) (35,52) rItwz1 :DEAR : (0,8) (9,18) (19,31) (30,51) 2 :DEAR : (0,8) (9,18) (19,31) (30,50) 3 :DEAR : (0,8) (9,18) (19,30) (30,51) Figure 7: Results of character recognition 5.2 Performance of the character recognition model Fig. 7 shows a result of character recognition. The right-hand side shows the recognition results for the left-hand side. The best three candidates for recognition are listed. Numerals in parentheses show the number of starting via-points and the final via-point for the recognized character. 5.3 Performance of the estimation of timing of phonemes in real speech Fig. 8 shows the acoustic waveform, the spectrogram, and the articulation movement when the sentence" Sam sat on top of the potato cooker ... " is spoken. The phonemes are identified, and the vertical lines denote phoneme midpoints. White circles show the viapoints estimated by our proposed algorithm. Rather good agreement is found between the estimated via-points and the phonemes. From this experiment, we can point out two important possibilities for the estimation model of phoneme timing. The first possibility concerns speech recognition, and the second concerns speech data compression. It seems possible to extend the via-point estimation algorithm to speech recognition if a mapping from acoustic to articulator motion is identified (Shirai & Kobayashi, 1991, Papcun et al., 1992). Furthermore, with training of a forward mapping from articulator motion to acoustic data (Hirayama et al., 1993), the via-point estimation model can be used for speech data compression. 6 SUMMARY We have proposed a new handwriting model. In experiments, good qualitative and quantitative agreement is found between human handwriting data and the trajectories generated by the model. Our model is unique in that the same optimization principle and hard constraints used for reaching are also used for cursive handwriting. Also, as opposed to previous handwriting models, determination of via-points is based on the optimization principle and does not use a priori knowledge. We have demonstrated two areas of recognition, connected cursive handwritten character recognition and the estimation of phoneme timing. We incorporated the formation model into the recognition model and realized the recognition model suggested by Freyd (1983) and Liberman and Mattingly(1985). The most important point shown by the models is that the human recognition process can be realized by specifying the human formation process. REFERENCES S. Edelman & T. Flash (1987) A Model of Handwriting. Bioi. Cybern. ,57,25-36. 734 Wada, Koike, Vatikiotis-Bateson, and Kawato ... n.~"'~fl> cooker ... Figure 8: Estimation result of phoneme time. Temporal acoustics and vertical positions of the tongue blade (TBY),tongue tip (TTY), jaw (lY), and lower lip (LL Y) are shown with overlaid via-point trajectories. Vertical lines correspond to acoustic segment centers; 0 denotes via-points. T. Flash, & N. Hogan (1985) The coordination of arm movements; An experimentally confirmed mathematical model. Journal of Neuroscience, 5, 1688-1703. J. J. Freyd (1983) Representing the dynamics of a static fonn. Memory & Cognition, 11, 342-346. M. Hirayama, E. Vatikiotis-Bateson, K. Honda, Y. Koike, & M. Kawato (1993) Physiologically based speech synthesis. In Giles, C. L., Hanson, S. J., and Cowan, J. D. (eds) Advances in Neural Information Processing Systems 5,658-665. San Mateo, CA: Morgan Kaufmann Publishers. 1. M. Hollerbach (1981) An oscillation theory of handwriting. Bioi. Cybern., 39,139-156. A. M. Liberman & 1. G. Mattingly (1985) The motor theory of speech perception revised. Cognition, 21, 1-36. P. Morasso, & F. A. Mussa-Ivaldi (1982) Trajectory formation and handwriting: A computational model. Bioi. Cybern. ,45, 131-142. J. Papcun, J. Hochberg, T. R. Thomas, T. Laroche, J. Zacks, & S. Levy (1992) Inferring articulation and recognition gestures from acoustics with a neural network trained on xray microbeam data. Journal of Acoustical Society of America, 92 (2) Pt. 1. K. Shirai, & T. Kobayashi (1991) Estimation of articulatory motion using neural networks. Journal of Phonetics, 19, 379-385. Y. Uno, M. Kawato, & R. Suzuki (1989) Formation and control of optimal trajectory in human arm movement - minimum torque-change model. BioI. Cybern. 61, 89-101. Y. Wada, & M. Kawato (1993) A neural network model for arm trajectory formation using forward and inverse dynamics models. Neural Networks, 6(7),919-932. Y. Wada, & M. Kawato (1994) Long version of this paper, in preparation. PART VI ApPLICATIONS	1993	88
845	Discontinuous Generalization in Large Committee Machines H. Schwarze Dept. of Theoretical Physics Lund University Solvegatan 14A 223 62 Lund Sweden Abstract J. Hertz Nordita Blegdamsvej 17 2100 Copenhagen 0 Denmark The problem of learning from examples in multilayer networks is studied within the framework of statistical mechanics. Using the replica formalism we calculate the average generalization error of a fully connected committee machine in the limit of a large number of hidden units. If the number of training examples is proportional to the number of inputs in the network, the generalization error as a function of the training set size approaches a finite value. If the number of training examples is proportional to the number of weights in the network we find first-order phase transitions with a discontinuous drop in the generalization error for both binary and continuous weights. 1 INTRODUCTION Feedforward neural networks are widely used as nonlinear, parametric models for the solution of classification tasks and function approximation. Trained from examples of a given task, they are able to generalize, i.e. to compute the correct output for new, unknown inputs. Since the seminal work of Gardner (Gardner, 1988) much effort has been made to study the properties of feedforward networks within the framework of statistical mechanics; for reviews see e.g. (Hertz et al., 1989; Watkin et al., 1993). Most of this work has concentrated on the simplest feedforward network, the simple perceptron with only one layer of weights connecting the inputs with a 399 400 Schwarze and Hertz single output. However, most applications have to utilize architectures with hidden layers, for which only a few general theoretical results are known, e.g. (Levin et al., 1989; Krogh and Hertz, 1992; Seung et al., 1992). As an example of a two-layer network we study the committee machine (Nilsson, 1965). This architecture has only one layer of adjustable weights, while the hiddento-output weights are fixed to + 1 so as to implement a majority decision of the hidden units. For binary weights this may already be regarded as the most general two-layer architecture, because any other combination of hidden-output weights can be gauged to + 1 by flipping the signs of the corresponding input-hidden weights. Previous work has been concerned with some restricted versions of this model, such as learning geometrical tasks in machines with local input-to-hidden connectivity (Sompolinsky and Tishby, 1990) and learning in committee machines with nonoverlapping receptive fields (Schwarze and Hertz, 1992; Mato and Parga, 1992). In this tree-like architecture there are no correlations between hidden units and its behavior was found to be qualitatively similar to the simple perceptron. Recently, learning in fully connected committee machines has been studied within the annealed approximation (Schwarze and Hertz, 1993a,b; Kang et aI, 1993), revealing properties which are qualitatively different from the tree model. However, the annealed approximation (AA) is only valid at high temperatures, and a correct description of learning at low temperatures requires the solution of the quenched theory. The purpose of this paper is to extend previous work towards a better understanding of the learning properties of multilayer networks. We present results for the average generalization error of a fully connected committee machine within the replica formalism and compare them to results obtained within the AA. In particular we consider a large-net limit in which both the number of inputs Nand the number of hidden units K go to infinity but with K ~ N. The target rule is defined by another fully connected committee machine and is therefore realizable by the learning network. 2 THE MODEL We consider a network with N inputs, K hidden units and a single output unit (j. Each hidden unit (jl, I E {I, ... , K}, is connected to the inputs 8 = (81 , .•• , 8N) through the weight vector W, and performs the mapping (j1(WI , 8) = sign (Jw W, . 8). (1) The hidden units may be regarded as outputs of simple perceptrons and will be referred to as students. The factor N- 1/ 2 in (1) is included for convenience; it ensures that in the limit N -+ 00 and for iid inputs the argument of the sign function is of order 1. The overall network output is defined as the majority vote of the student committee, given by (2) Discontinuous Generalization in Large Committee Machines 401 This network is trained from P = aK N input-output examples ({", T({")), J.I. E {1, ... , P}, ofthe desired mapping T, where the components {r ofthe training inputs are independently drawn from a distribution with zero mean and unit variance. We study a realizable task defined by another committee machine with weight vectors L (the teachers), hidden units Tz and an overall output T(S) of the form (2). We will discuss both the binary version of this model with W" L E {± l}N and the continuous version in which the W,'s and L's are normalized to VN. The goal of learning is to find a network that performs well on unknown examples, which are not included in the training set. The network quality can be measured by the generalization error €({W,}) = (0[-(T({~},S) T(S)])~, (3) the probability that a randomly chosen input is misclassified. Following the statistical mechanics approach we consider a stochastic learning algorithm that for long training times yields a Gibbs distribution of networks with the corresponding partition function Z = J dpo({W,}) e-f1Et ({W,}) , (4) where (5) " is the training error, {3 = liT is a formal temperature parameter, and po( {W,}) includes a priori constraints on the weights. The average generalization and training errors at thermal equilibrium, averaged over all representations of the training examples, are given by (( (€({W,}))T)) 1 P (( (Et({~}))T )), (6) where (( ... )) denotes a quenched average over the training examples and ( ... )T a thermal average. These quantities may be obtained from the average free energy F = - T (( In Z )), which can be calculated within the standard replica formalism (Gardner, 1988; Gyorgyi and Tishby, 1990). Following this approach, we introduce order parameters and make symmetry assumptions for their values at the saddle point of the free energy; for details of the calculation see (Schwarze, 1993). We assume replica symmetry (RS) and a partial committee symmetry allowing for a specialization of the hidden units on their respective teachers. Furthermore, a self-consistent solution of the saddle-point equations requires scaling assumptions for the order parameters. Hence, we are left with the ansatz 1 R'k = N (( ( ~)T . V k )) 1 D,k = N(((W,)T,(((Wk)T)) 1 C'k= N(((W"Wk)T)) (7) 402 Schwarze and Hertz where p, ~, d, q and c are of order 1. For ~ = q = 0 this solution is symmetric under permutations of hidden units in the student network, while nonvanishing ~ and q indicate a specialization of hidden units that breaks this symmetry. The values of the order parameters at the saddle point of the replica free energy finally allow the calculation of the average generalization and training errors. 3 THEORETICAL RESULTS In the limit of small training set sizes, Q '" 0(1/ K), we find a committee-symmetric solution where each student weight vector has the same overlap to all the teacher vectors, corresponding to ~ = q = O. For both binary and continuous weights the generalization error of this solution approaches a nonvanishing residual value as shown in figure 1. Note that the asymptotic generalization ability of the committeesymmetric solution improves with increasing noise level. 0.50 DAD 0.30 w 0.20 0.10 0.00 a) 0 • • • • • " " " " " " 10 20 30 40 50 C( = PiN ...-... E-< '-' 0 w b) 0.30 0.25 0.20 0.15 0.10 0.05 , , I , , I 0.00 , -0.0 Eg .. ~~~--- _ ....... --.-.-,--, , , , , Et , , , , , I , 0.5 1.0 1.5 2.0 T Figure 1: a) Generalization (upper curve) and training (lower curve) error as functions of 0 = K Q. The results of Monte Carlo simulations for the generalization (open symbols) and training (closed symbols) errors are shown for K = 5 (circles) and K = 15 (triangles) with T = 0.5 and N = 99. The vertical lines indicate the predictions of the large-K theory for the location of the phase transition Oc = K Q c in the binary model for K = 5 and K = 15, respectively. b) Temperature dependence of the asymptotic generalization and training errors for the committee-symmetric solution. Only if the number of training examples is sufficiently large, Q '" 0(1), can the committee symmetry be broken in favor of a specialization of hidden units. We find first-order phase transitions to solutions with ~,q > 0 in both the continuous and the binary model. While in the binary model the transition is accompanied by a perfect alignment of the hidden-unit weight vectors with their respective teachers (~ = 1), this is not possible in a continuous model. Instead, we find a close approach of each student vector to one of the teachers in the continuous model: At a critical value Q" (T) of the load parameter a second minimum of the free energy appears, corresponding to the specialized solution with ~, q > O. This solution becomes the Discontinuous Generalization in Large Committee Machines 403 global minimum at Ckc(T) > Ck.(T), and its generalization error decays algebraically. In both models the symmetric, poorly generalizing state remains metastable for arbitrarily large Ck. For increasing system sizes it will take exponentially long times for a stochastic training algorithm to escape from this local minimum (see figure 1a). Figure 2 shows the qualitative behavior of the generalization error for the continuous model, and the phase diagrams in figure 3 show the location of the transitions for both models. 1/2 a. a c €o(T) --------------------=--'=-----+f--.,...I---..---i i I I f j ~ ~----------------~/~/_--------------a", O(l/K) a'" 0(1) '" p ~ KN Figure 2: Schematic behavior of the generalization error in the large-K committee machine with continuous weights. In the binary model a region of negative thermodynamic entropy (below the dashed line in figure 3a) suggests that replica symmetry has to be broken to correctly describe the metastable, symmetric solution at large Ck. A comparison of the RS solution with the results previously obtained within the AA (Schwarze and Hertz, 1993a,b) shows that the AA gives a qualitatively correct description of the main features of the learning curve. However, it fails to predict the temperature dependence of the residual generalization error (figure 1 b) and gives an incorrect description of the approach to this value. Furthermore, the quantitative predictions for the locations of the phase transitions differ considerably (figure 3). 4 SIMULATIONS We have performed Monte Carlo simulations to check our analytical findings for the binary model (see figure 1a). The influence of the metastable, poorly generalizing state is reflected by the fact that at low temperatures the simulations do not follow the predicted phase transition but get trapped in the metastable state. Only at higher temperatures do the simulations follow the first order transition (Schwarze, 1993). Furthermore, the deviation of the training error from the theoretical result indicates the existence of replica symmetry breaking for finite Q. However, the generalization error of the symmetric state is in good quantitative agreement with the 404 Schwarze and Hertz 0.8 0.6 E-< 0.4 0.2 ; .I ;,l' / ; ; ; l , ; ; ; ; I ,. i I i i j j i , i 1.0 0.8 0.6 0.4 / .......•.•. / ... --.. // ..... l./···················· 0.2 r O.O~~~~~~! ~~~~~~~~ O.O~~~~~~~~~!~-u~~ a) 5 10 15 20 0: = P/KN 25 30 b) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0: = P /KN Figure 3: Phase diagrams of the large-K committee machine. a) continuous weights: The two left lines show the RS results for the spinodal line (--), where the specialized solution appears, and the location of the phase transition (-). These results are compared to the predictions of the AA for the spinodal line (- . -) and the phase transition ( ... ). b) binary weights: The RS result for the location of the phase transition (-) and its zero-entropy line (--) are compared to the prediction of the AA for the phase transition ( ... ) and its zero-entropy line (- . -). theoretical results. In order to investigate whether our analytical results for a Gibbs ensemble of committee machines carries over to other learning scenarios we have studied a variation of this model allowing the use of backpropagation. We have considered a 'softcommittee' whose output is given by q( {W,}. S) = tanh (t. tanh (J£, . S». (8) The first-layer weights W, of this network were trained on examples (el', r(el'», J.£ E {l, ... , P}, defined by another soft-committee with weight vectors V, using on-line backpropagation with the error function £(S) = (1/2)[0'({~}, S) - r(S)]2. (9) In general this procedure is not guaranteed to yield a Gibbs distribution of weights (Hansen et al., 1993) and therefore the above analysis does not apply to this case. However, the generalization error for a network with N = 45 inputs and K = 3 hidden units, averaged over 50 independent runs, shows the same qualitative behavior as predicted for the Gibbs ensemble of committee machines (see figure 4). After an initial approach to a nonvanishing value, the average generalization error decreases rather smoothly to zero. This smooth decrease of the average error is due to the fact that some runs got trapped in a poorly-generalizing, committeesymmetric solution while others found a specialized solution with a close approach to the teacher. Discontinuous Generalization in Large Committee Machines 405 0.18 r--.....,.----r--.....,.---r--.....,.------r'1 0.16 0.1. i 0.12 0.06 0.0. 0.02 200 600 800 1000 1200 P Figure 4: Generalization error and training error of the 'soft-committee' with N = 45 and K = 3. We have used standard on-line backpropagation for the first-layer weights with a learning rate 11 = 0.01 for 1000 epochs. the results are averaged over 50 runs with different teacher networks and different training sets. 5 CONCLUSION We have presented the results of a calculation of the generalization error of a multilayer network within the statistical mechanics approach. We have found nontrivial behavior for networks with both continuous and binary weights. In both models, phase transitions from a symmetric, poorly-generalizing solution to one with specialized hidden units occur, accompanied by a discontinuous drop of the generalization error. However, the existence of a metastable, poorly generalizing solution beyond the phase transition implies the possibility of getting trapped in a local minimum during the training process. Although these results were obtained for a Gibbs distribution of networks, numerical experiments indicate that some of the general results carryover to other learning scenarios. Acknowledgements The authors would like to thank M. Biehl and S. Solla for fruitful discussions. HS acknowledges support from the EC under the SCIENCE programme (under grant number B/SCl * /915125) and by the Danish Natural Science Council and the Danish Technical Research Council through CONNECT. References E. Gardner (1988), J. Phys. A 21, 257. G. Gyorgyi and N. Tishby (1990), in Neural Networks and Spin Glasses, edited by K. Thuemann and R. Koberle, (World scientific, Singapore). L.K. Hansen, R. Pathria, and P. Salamon (1993), J. Phys. A 26, 63. J. Hertz, A. Krogh, and R.G. Palmer (1989), Introduction to the Theory of Neural 406 Schwarze and Hertz Computation, (Addison-Wesley, Redwood City, CA). K. Kang, J.-H. Oh, C. Kwon, and Y. Park (1993), preprint Pohang Institute of Science and Technology, Korea. A. Krogh and J. Hertz (1992), in Advances in Neural Information Processing Systems IV, eds. J .E. Moody, S.J. Hanson, and R.P. Lippmann, (Morgan Kaufmann, San Mateo). E. Levin, N. Tishby, and S.A. Solla (1989), in Proc. 2nd Workshop on Computational Learning Theory, (Morgan Kaufmann, San Mateo). G. Mato and N. Parga (1992), J. Phys. A 25, 5047. N.J. Nilsson (1965), Learning Machines, (McGraw-Hill, New York). H. Schwarze (1993), J. Phys. A 26, 5781. H. Schwarze and J. Hertz (1992), Europhys. Lett. 20,375. H. Schwarze and J. Hertz (1993a), J. Phys. A 26, 4919. H. Schwarze and J. Hertz (1993b), in Advances in Neural Information Processing Systems V, (Morgan Kaufmann, San Mateo). H.S. Seung, H. Sompolinsky, and N. Tishby (1992), Phys. Rev. A 45, 6056. H. Sompolinskyand N. Tishby (1990), Europhys. Lett. 13, 567. T. Watkin, A. Rau, and M. Biehl (1993), Rev. Mod. Phys. 65, 499.	1993	89
846	Use of Bad Training Data For Better Predictions Tal Grossman Complex Systems Group (T13) and CNLS LANL, MS B213 Los Alamos N .M. 87545 Alan Lapedes Complex Systems Group (T13) LANL, MS B213 Los Alamos N.M. 87545 and The Santa Fe Institute, Santa Fe, New Mexico Abstract We show how randomly scrambling the output classes of various fractions of the training data may be used to improve predictive accuracy of a classification algorithm. We present a method for calculating the "noise sensitivity signature" of a learning algorithm which is based on scrambling the output classes. This signature can be used to indicate a good match between the complexity of the classifier and the complexity of the data. Use of noise sensitivity signatures is distinctly different from other schemes to avoid overtraining, such as cross-validation, which uses only part of the training data, or various penalty functions, which are not data-adaptive. Noise sensitivity signature methods use all of the training data and are manifestly data-adaptive and non-parametric. They are well suited for situations with limited training data. 1 INTRODUCTION A major problem of pattern recognition and classification algorithms that learn from a training set of examples is to select the complexity of the model to be trained. How is it possible to avoid an overparameterized algorithm from "memorizing" the training data? The dangers inherent in over-parameterization are typically 343 344 Grossman and Lapedes illustrated by analogy to the simple numerical problem of fitting a curve to data points drawn from a simple function. If the fit is with a high degree polynomial then prediction on new points, i.e. generalization, can be quite bad, although the training set accuracy is quite good. The wild oscillations in the fitted function, needed to acheive high training set accuracy, cause poor predictions for new data. When using neural networks, this problem has two basic aspects. One is how to choose the optimal architecture (e.g. the number oflayers and units in a feed forward net), the other is to know when to stop training. Of course, these two aspects are related: Training a large net to the highest training set accuracy usually causes overfitting. However, when training is stopped at the "correct" point (where train-set accuracy is lower), large nets are generalizing as good as, or even better than, small networks (as observed e.g. in Weigend 1994). This prompts serious consideration of methods to avoid overparameterization. Various methods to select network architecture or to decide when to stop training have been suggested. These include: (1) use of a penalty function (c.!. Weigend et al. 1991). (2) use of cross validation (Stone 1974). (3) minimum description length methods (Rissanen 1989), or (4) "pruning" methods (e.g. Le Cun et al. 1990). Although all these methods are effective to various degrees, they all also suffer some form of non-optimality: (1) various forms of penalty function have been proposed and results differ between them. Typically, using a penalty function is generally preferable to not using one. However, it is not at all clear that there exists one "correct" penalty function and hence any given penalty function is usually not optimal. (2) Cross validation holds back part of the training data as a separate valdiation set. It therefore works best in the situation where use of smaller training sets, and use of relatively small validation sets, still allows close approximation to the optimal classifier. This is not likely to be the case in a significantly data-limited regime. (3) MDL methods may be viewed as a form of penalty function and are subject to the issues in point (1) above. (4) pruning methods require training a large net, which can be time consuming, and then "de-tuning" the large network using penalty functions. The issues expressed in point(l) above apply. We present a new method to avoid overfitting that uses "noisy" training data where some of the output classes for a fraction of the data are scrambled. We describe how to obtain the "noise sensitivity signature" of a classifier (with its learning algorithm), which is based on the scrambled data. This new methodology is not computationally cheap, but neither is it prohibitively expensive. It can provide an alternative to methods (1 )-( 4) above that (i) can test any complexity parameter of any classifying algorithm (i.e. the architecture, the stopping criterion etc.) (ii) uses all the training data, and (iii) is data adaptive, in contrast to fixed penalty/pruning functions. 2 A DETAILED DESCRIPTION OF THE METHOD Define a "Learning Algorithm" L(S, P), as any procedure which produces a classifier f(~), which is a (discrete) function over a given input space X (~ E X). The input of the learning algorithm L is a Training Set S and a set of parameters P. The training set S is a set of M examples, each example is a pair of an input instance ~i Use of Bad Training Data for Better Predictions 345 and the desired output Yi associated with it (i = l..M). We assume that the desired output represents an unknown "target function" f* which we try to approximate, i.e. Yi = f(:ni). The set of parameters P includes all the relevant parameters of the specific learning algorithm and architecture used. When using a feed-forward neural network classifier this set usually includes the size of the network, its connectivity pattern, the distribution of the initial weights and the learning parameters (e.g. the learning rate and momentum term size in usual back-propagation). Some of these parameters determine the "complexity" of the classifiers produced by the learning algorithm, or the set of functions f that are realizable by L. The number of hidden units in a two layer perceptron, for example, determines the number of free parameters of the model (the weights) that the learning algorithm will fit to tbe data (the training set). In general, the output of L can be any classifier: a neural network, a decision tree, boolean formula etc. The classifier f can also depend on some random choices, like the initial choice of weights in many network lenrning algortihm. It can also depend, like in pruning algorithms on any "stopping crite~'ion" which may also influence its complexity. 2.1 PRODUCING ff The classification task is given as the training set S. The first step of our method is to prepare a set of noisy, or partially scrambled realizations of S. We define S: as one partiCUlar such realization, in which for fraction P of the M examples tne desired ou.tpu.t values (classes) are changed. In this work we consider only binary classification tasks, which means that we choose pM examples at random for which yf = 1 - Yi· For each noise level p and set of n such realizations S; (f.L = l..n) is prepared, each with a different random choice of scrambled examples. Practically, 8-10 noise levels in the range p = 0.0 - 0.4, with n "" 4 - 10 realizations of S: for each level are enough. The second step is to apply the learning algorithm to each of the different S: to produce the corresponding classifiers, which are the boolean functions ff = L(S;, P). 2.2 NOISE SENSITIVITY MEASURES Using the set of ff, three quantities are measured for each noise level p: • The average performance on the original (noise free) training set S. We define the average noise-free error as 1 n M Ej(p) = Mn I: L If;(:ni) - Yil (1) I/o i And the noise-free pereformance, or score as Qj(p) = 1 - Ej(p). • In a similar way, we define the average error on the noisy training-sets S:: 1 n M En(P) = Mn L ~ If;(:ni) - yfl (2) I/o \ Note that the error of each classifier f; is measured on the training set by which it was created. The noisy-set performance is then defined as Qn(P) = 1 - En(P)· 346 Grossman and Lapedes • The average functional distance between classifiers. The functional distance between two classifiers, or boolean functions, d(J, g) is the probability of I(z) #- g(z). For a uniform input distribution, it is simply the fraction of the input space X for which I(z) #- g(z). In order to approximate this quantity, we can use another set of examples. In contrast with validation set methods, these examples need not be classified, i.e. we only need a set of inputs z, without the target outputs y, so we can usually use an "artificial" set of m random inputs. Although, in principle at least, these z instances should be taken from the same distribution as the original task examples. The approximated distance between two classifiers is therefore 1 m d(J, g) = m ~ I/(Zi) - g(zi)1 (3) , We then calculate the average distance, D(p), between the n classifiers It obtained for each noise level p: n D(p) = n(n 2_ 1) L d(J:, I;) (4) IJ.>V 3 NOISE SENSITIVITY BEHAVIOR Observing the three quantities Q,(p), Qn(P) and D(p), can we distinguish between an overparametrized classifier and a "well tuned" one? Can we use this data in order to choose the best generalizer out of several candidates? Or to find the right point to stop the learning algorithm L in order to achieve better generalization? Lets estimate how the plots of Q" Qn and D vs. p, which we call the "Noise Sensitivity Signature" (NSS) of the algorithm L, look like in several different scenarios. 3.1 D(p) The average functional distance between realizations, D(p), measures the sensitivity of the classifier (or the model) to noise. An over-parametrized architecture is expected to be very sensitive to noise since it is capable of changing its classification boundary to learn the scrambled examples. Different realizations of the noisy training set will therefore result in different classifiers. On the other hand, an under-parametrized classifier should be stable against at least a small amount of noise. Its classification boundary will not change when a few examples change their class. Note, however, that if the training set is not very "dense", an under-parametrized architecture can still yield different classifiers, even when trained on a noise free training set (e.g. when using BP with different initial weights). Therefore, it may be possible to observe some "background variance", i.e. non-zero average distance for small (down to zero) noise levels for under-parametrized classifiers. Similar considerations apply for the two quantities Q,(p) and Qn(P). When the training set is large enough, an under-parametrized classifier cannot "follow" all Use of Bad Training Data for Better Predictions 347 the changed examples. Therefore most of them just add to the training error. Nevertheless, its performance on the noise free training set, Qf(P), will not change much. As a result, when increasing the noise level P from zero (where Qf(P) = Qn(P)), we should find Qf (p) > Qn(P) up to a high noise level - where the decision boundary has changed enough so the error on the original training set becomes larg '~r than the error on the actual noisy set. The more parameters our model has, the sooner (i.e. smaller p) it will switch to the Qf(P) < Qn(P) state. If a network starts with Qf(P) = Qn(P) and then exhibits a behavior with Qf(P) < Qn(P), this is a signature of overparameterization. 3.3 THE TRAINING SET In addition to the set of parameters P and the learning algorithm itself, there is another important factor in the learning process. This is the training set S. The dependence on M, the number of examples is evident. When M is not large enough, the training set does not provide enough data in order to capture the full complexity of the original task. In other words, there are not enough constraints - to approximate well the target function f. Therefore overfitting will occur for smaller classifier complexity and the optimal network will be smaller. 4 EXPERIMENTAL RESULTS To demonstrate the possible outcomes of the method described above in several cases, we have performed the following experiment. A random neural network "teacher" was created as the target function f*. This is a two layer percept ron with 20 inputs, 5 hidden units and one output. A set of M random binary input examples was created and the teacher network was used to classify the training examples. Namely, a desired output Yi was obtained by recording the output of the teacher net when input :l:i was presented to the network, and the output was calculated by applying the usual feed forward dynamincs: (5) This binary threshold update rule is applied to each of the network's units j, i.e the hidden and the output units. The weights of the teacher were chosen from a uniform distribution [-1,1]. No threshold (bias weights) were used. The set of scrambled training sets St was produced as explained above and different network architectures were trained on it to produce the set of classifiers jl1o. The learning networks are standard two layer networks of sigmoid units, trained by conjugate gradient back-propagation, using a quadratic error function with tolerance, i.e. if the difference between an output of the net and the desired 0 or 1 target is smaller than the tolerance (taken as 0.2 in our experiment) it does not contribute to the error. The tolerance is, of course, another parameter which may influences the complexity of the resulting network, however, in this experiment it is fixed. The quantities Qf(P), Qn(P) and D(p) were calculated for networks with 1,2,3, .. 7 hidden units (1 hidden unit means just a perceptron, trained with the same error function). In our terminology, the architecture specification is part of the set of 348 Grossman and Lapedes Training Set Size hidden units 400 700 1024 1 0.81 0.04 0.81 0.001) 0.82 0.0011 2 0.81 0.04 0.84 0.05 0.86 0.04 3 0.78 0.02 0.82 0.06 0.90 0.03 4 0.77 0.03 0.81 0.05 0.90 0.03 5 0.74 ( 0.03 0.79 0.03 0.87 0.04 6 0.74 ( 0.01 0.80 0.05 0.89 0.03 7 0.71 ( 0.01 0.76 0.02 0.85 0.05 Table 1: The prediction rate for 1..7 hidden units, averaged on 4 nets that were trained on the noisefree training set of size M = 400,700,1024 (the standard deviation is given in parenthesis). parameters P that is input to the learning algorithm L. The goal is to identify the "correct" architecture according to the behavior of QJ, Qn and D with p. The experiment was done with three training set sizes M = 400, 700 and 1024. Another set of m = 1000 random examples was used to calculate D. As an "external control" this set was also classified by the teacher network and was used to measure the generalization (or prediciton rate) of the different learning networks. The prediction rate, for the networks trained on the noise free training set (averaged over 4 networks, trained with different random initial weights) is given for the 1 to 7 hidden unit architectures, for the 3 sizes of M, in Table 1. The noise sensitivity signatures of three architectures trained with M = 400 (1,2,3 hidden units) and with M = 1024 examples (2,4,6 units) are shown in Figure 1. Compare these (representative) results with the expected behaviour of the NSS as described qualitatively in the previous section. 5 CONCLUSIONS and DISCUSSION We have introduced a method of testing a learning model (with its learning algorithm) against a learning task given as a finite set of examples, by producing and characterizing its "noise sensitivity signature". Relying on the experimental results presented here, and similar results obtained with other (less artificial) learning tasks and algorithms, we suggest some guidelines for using the NSS for model tuning: 1. If D(p) approaches zero with p -+ 0, or if QJ(p) is significantly better than Qn(P) for noise levels up to 0.3 or more - the network/model complexity can be safely inreased. 2. If QJ(p) < Qn(P) already for small levels of noise (say 0.2 or less) - reduce the network complexity. 3. In more delicate situations: a "good" model will have at least a trace of concavity in D(p). A clearly convex D(p) probably indicates an over-parametrized model. In a "good" model choice, Qn (p) will follow Q J (p) closely, from below, up to a high noise level. Use of Bad Training Data for Better Predictions 349 04 02 • I I oL-__ L-__ ~ __ ~ __ ~ __ -L __ ~ __ ~ __ ~ __ ~ o oos 01 015 02 0 25 03 035 04 045 400 IlX~. 2 hrd:len UIlIIs 08 06 04 005 01 015 02 025 03 035 04 045 1024 exa~9S 4 hidden units ...•... -.•. -.--.... -...... -,----~ ......... , -" '1 02 ~ I t oL-__ L-__ ~ __ ~ __ ~ __ -L __ ~ __ ~ __ ~ __ ~ o 005 01 015 02 025 03 035 04 04 ~' oos 0 1 015 0, 025 03 035 04 045 08 O~ 04 • 04 I i 02 • I OL_--~--~--~--~---L--~--~--~--~ °0L---O~OS---0~1---0~15--~ 02--~02~5---0~ 3 ---0~35---0~4--~045 o 005 01 015 02 025 03 OJ5 04 04~ Figure 1: The signatures (Q and D vs. p) of networks with 1,2,3 hidden units (top to bottom) trained on M=400 examples (left), and networks with 2,4,6 hidden units trained on M=1024 examples. The (noisy) training set score Qn(P) is plotted with full line, the noise free score Qf(P) with dotted line, and the average functional distance D(p) with error bars (representing the standard deviation of the distance). 350 Grossman and Lapedes 5.1 Advanatages of the Method 1. The method uses all the data for training. Therefore we can extract all the available information. Unlike validation set methods - there is no need to spare part of the examples for testing (note that classified examples are not needed for the functional distance estimation). This may be an important advantage when the data is limited. As the experiment presented here shows: taking 300 examples out of the 1024 given, may result in choosing a smaller network that will give inferior prediction (see table 1). Using "delete-1 cross-validation" will minimize this problem but will need at least as much computation as the NSS calculation in order to achieve reliable prediction estimation. 2. It is an "external" method, i.e. independent of the classifier and the training algorithm. It can be used with neural nets, decision trees, boolean circuits etc. It can evaluate different classifiers, algorithms or stopping/prunning criteria. 5.2 Disadvantages 1. Computationally expensive (but not prohibitively so). In principle one can use just a few noise levels to reduce computational cost. 2. Presently requires a subjective decision in order to identify the signature, unlike cross-validation methods which produce one number. In some situations, the noise sensitivity signature gives no clear distinction between similar architectures. In these cases, however, there is almost no difference in their generalization rate. Acknowledgements We thank David Wolpert, Michael Perrone and Jerom Friedman for many iluminating discussions and usefull comments. We also thank Rob Farber for his invaluable help with software and for his assistance with the Connection Machine. Referencess Le Cun Y., Denker J.S. and Solla S. (1990), in Adv. in NIPS 2, Touretzky D.S. ed. (Morgan Kaufmann 1990) 598. Rissanen J. (1989), Stochastic Complezity in Statistical Inquiry (World Scientific 1989). Stone M. (1974), J.Roy.Statist.Soc.Ser.B 36 (1974) 11I. Wiegend A.S. (1994), in the Proc. 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. 335-342 (Erlbaum Associates, Hillsdale NJ, 1994). Wiegend A.S., Rummelhart D. and Huberman B.A. (1991), in Adv. in NIPS 3, Lippmann et al. eds. (Morgen Kaufmann 1991) 875.	1993	9
847	Inverse Dynamics of Speech Motor Control Makoto Hirayama Eric Vatikiotis-Datesol1 Mitsuo Kawato" ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan Abstract Progress ha.s been made in comput.ational implementation of speech production based on physiological dat.a. An inverse dynamics model of the speech articulator's l1111sculo-skeletal system. which is the mapping from art.iculator t.rajectories to e\ectromyogl'aphic (EMG) signals, was modeled using the acquired forward dynamics model and temporal (smoot.hness of EMG activation) and range constraints. This inverse dynamics model allows the use of a faster speech mot.or control scheme, which can be applied to phoneme-tospeech synthesis via musclo-skeletal system dynamics, or to future use in speech recognition. The forward acoustic model, which is the mapping from articulator trajectories t.o the acoustic parameters, was improved by adding velocity and voicing information inputs to distinguish acollst.ic paramet.er differences caused by changes in source characterist.ics. 1 INTRODUCTION Modeling speech articulator dynamics is important not only for speech science, but also for speech processing. This is because many issues in speech phenomena, such as coarticulation or generat.ion of aperiodic sources, are caused by temporal properties of speech articulat.or behavior due t.o musculo-skelet.al system dynamics and const.raints on neurO-l1lotor command activation . .. Also, Laboratory of Parallel Distributed Processing, Research Institute for Electronic Science, Bokkaido University, Sapporo, Hokkaido 060, Japan 1043 1044 Hirayama, Vatikiotis-Bateson, and Kawato We have proposed using neural networks for a computational implementation of speech production based on physiological activities of speech articulator muscles. In previous works (Hirayama, Vatikiotis-Bateson, Kawato and Jordan 1992; Hirayama, Vatikiotis-Bateson, Honda, Koike and Kawato 1993), a neural network learned the forward dynamics, relating motor commands to muscles and the ensuing articulator behavior. From movement t.rajectories, the forward acoustic network generated the acoustic PARCOR parameters (Itakura and Saito, 1969) that were then used to synthesize the speech acoustics. A cascade neural network containing the forward dynamics model along with a suitable smoothness criterion was used to produce a continuous motor command from a sequence of discrete articulatory targets corresponding to the phoneme input string. Along the same line, we have extended our model of speech motor control. In this paper, WI~ focus on modeling the inverse dynamics of the musculo-skeletal system. Having an inverse dynamics model allows us to use a faster control scheme, which permits phoneme-to-speech synthesis via musculo-skeletal system dynamics, and ultimately may be useful in speech recognition. The final sectioll of this paper reports improvements in the forward acoustic model, which were made by incorporating articulator velocity and voicing information to distinguish the acoustic parameter differences caused by changes in source characteristics. 2 INVERSE DYNAMICS MODELING OF MUSCULO-SKELETAL SYSTEM From the viewpoint of control theory, an inverse dynamics model of a controlled object pla.ys an essential role in fecdfonvard cont.rol. That is, an accurate inverse dynamics model outputs an appropriate control sequence that realizes a given desired trajectory by using only fecdforward cOlltrol wi t.hout any feedback information, so long as there is no perturbation from the environment. For speech a rticulators, the main control scheme cannot rely upon feedback control because of sensory feedback delays. Thus, we believe that the inverse dynamics model is essential for biological motor control of speech and for any efficient speech synthesis algorithm based on physiological data. However, the speech articulator system is an excess-degrees-of-freedom system, thus the mapping from art.iculator t.rajectory (posit.ion, velocit.y, accelerat.ion) to electromyographic (E~fG) activity is one-to-many. That is, different EMG combinations exist for the same articulat.or traject.ory (for example, co-contraction of agonist and antagonist muscle pairs). Consequently, we applied the forward modeling approach to learning an inverse model (Jordan alld Rumelhart, 1992), i.e., constrained supervised leaming, as shown in Figure 1. The inputs of the inverse Desired Trajectory r--~--..., Control p----..., Trajectory Inverse I--__ ~ Forward t------~~ Model Model ---Error Figure 1: Inverse dynamics modeling using a forward dynamics model (Jordan and Rumelhart, 1992). 1.0 0.8 0.6 0.4 0.2 Inverse Dynamics of Speech Motor Control 1045 --- Actual EMG "optimal" EMG by 10M O.O~----------~~----------r-----------~--~----~ o 1 2 3 4 Time (s) Figure 2: After learning, the inverse model output "optimal" EMG (anterior belly of the digastric) for jaw lowering is compared with actual EMG for the tf'st trajectory. dynamics model are articulator positions, velocities, and accelerations; the outputs are rectified, integrated, and filtered EIVIG for relevant muscles. The forward dynamics model previously reported (Hirayama et al., 1993) was used for determining the error signals of the inverse dynamics model . To choose a realistic EMG patt.ern from among diverse possible sciutions, we use both temporal and range const.raints. The temporal constraint is related to the smoothnt ~ss of EMG activat.ion, i.e., minimizing EI\'1G activation change (Uno, Suzuki, and Kawat.o, 1989). The minimum and maximum values of the range constraint were chosen using valucs obt.ained from t.he experimental data. Direct inverse modeling (Albus, 1975) was uscd to det.ermine weights, which were then supplied as initial weights to t.he constrained supervised learning algorithm of Jordan and Rumelhart's (1992) inverse dynamics modeling met.hod. Figure 2 shows an example of t.he inverse dynnmics model output after learning, when a real articulator trajectory, not. included in the training set, was given as the input. Note that the net.work output cannot be exactly t.he same as the actual EMG, as the network chooses a unique "optimal" EMG from many possible EMG patterns that appear in the actual EI\IG for t.he trajectory. -0.3 E -0.4 --- Experimental data --- Direct inverse modeli ng Inverse modeling using FDM c 0 -0.5 ~ UJ 0 -0.6 Q.. -0.7 0 1 2 3 4 Time (s) Figure 3: Trajectories generated by the forward dynamics net.work for the two methods of inverse dynamics modeling compared with t.he desired trajectory (experimental da t.a). 1046 Hirayama, Vatikiotis-Bateson, and Kawato Since the inverse dynamics model was obtained by learning, when the desired trajectory is given to the inverse dynamics model, an articulator trajectory can be generated with the forward dynamics network previously reported (Hirayama et al., 1993). Figure 3 compares trajectories generated by the forward dynamics network using EMG derived from the direct inverse dynamics method or the constrained supervised learning algorithm (which uses the forward dynamics model to determine the inverse dynamics model's "opt.imal" El\IG). The latter method yielded a 30.0 % average reduction in acceleration prediction error over the direct method, thereby bringing the model output trajectory closer to the experimental data. 3 TRAJECTORY FORMATION USING FORWARD AND INVERSE RELAXATION MODEL Previously, to generate a trajectory from discrete phoneme-specific via-points, we used a cascade neural network (c.f., Hirayama. et. al., 1992). The inverse dynamics model allows us t.o use an alternative network proposed by \\fada and Kawato (1993) (Figure 4). The network uses both the forward and inverse models of the controlled object, and updates a given initial rough trajectory passing through the via-points according to t.he dYllamics of the cont.rolled object and a smoothness constraint on the control input. The computation time of the net.work is much shorter than that of the cascade neural network CWada and Kawa.to, 1993). Figure 5 shows a forward dynamics model output trajectory driven by the modelgenerated motor control signals. Unlike \Vada and Kawato's original model (1993) in which generated trajectories always pass through via-points, our tl'ajectories were generated from smoothed motor control signals (i.e., after applying the smoothness constraint) and, consequently, do not. pass through the exact via-points. In this paper, a typical value for each phoneme from experimental data was chosen as the target via-point. and was given in Cartesian coordinates relative to the maxillary incisor. Alt.hough further investigation is needed to refine the phoneme-specific target specifications (e.g. lip aperture targets), reasonable coarticulated trajectories were obtained from series of discret.e via-point t.argets (Figure 5). For engineering applications such as text-to-speech synthesizers using articulatory synthesis, this kind of technique is necessary because realistic coarticula.ted trajectories must serve as input to the articulatory synthesizer. ~ e ~ (d 'd lal luI IiI lsI It I Articulatory Targets Figure 4: Speech t.rajectory formation scheme modified from the forward and inverse relaxation neural network model (\\'ada and Kawato, 1993). -0.3 £ -0.4 c: .2 -0.5 = ~ -0.6 -0.7 0.0 . -.--''''' ~-' ..... -~.---" - ~"--.. ". 0.2 '. ..... "" '.", '. 0.4 Inverse Dynamics of Speech Motor Control 1047 Network output ....... Experimental data . • . Phoneme specific targets ............. -.....•. 0.6 Time (s) \. '" ... '. 0.8 1.0 1.2 Figure 5: Jaw trajectory generated by the forward and inverse relaxation model. The output of the forward dynamics model is used for this plot. A furthe!' advantage of this network is that. it can be llsed t.o predict phonemespecific via-point.s from t.he realized t.rajectory (vVada, Koike, Vatikiotis-Bateson and Kawato, 1993). This capability will allow us to use our forward and inverse dynamicb models for speech recognition in future, through acoustic to articulatory mapping (Shirai and Kobayashi, 1991; Papcun, Hochberg, Thomas, Laroche, Zacks and Levy, 1992) and the articulatory to phoneme specific via-points mapping discussed above. Because t.rajectories may be recovered from a small set of phoneme··specific via-points, this approach should be readily applicable to problems of speech data compression. 4 DYNAMIC MODELING OF FORWARD ACOUSTICS The secoild area of progress is t.he improvement. in t.he forward acoustic network. Previously (Hirayama et al., 1993), we demonstrat.ed that acoustic signals can be obtained using a neural network that learns the mapping between articulator positions and acoustic PARCOR coetTIcients (ltakura and Saito, 1969; See also, Markel and Gray, ] 976). However, this modeling was effective only for vowels and a limited number of consonants because the architecture of the model was basically the same as that of static articulatory synthesizers (e.g. Mermelst.ein, 1973). For nat.ural speech, aperiodic sources for plosive and sibilant consonants result. in multiple sets of acoustic parameters for the same articulator configurat.ion (i.e., the mapping is one-to-many); hence, learning did not fully converge. One approach t.o solving t his problem is to make source modeling completely separat.e from the vocal tract area modeling. However, for synthesis of natural sentences, t.he vocal tract transfer function model requires anot.her model for t.he non-glottal sources associated wit.h consonant production. Since these sources are locat.ed at. various point.s along t.he vocal tract, their interaction is extremely complex. Our approach to solving this one-to-many mapping is to have the neural network learn the acoustic parameters along with the sound source characteristic specific to each phoneme. Thus, we put articulator positions with their velocities and voiced/voiceless informat.ion (e.g., Markel and Gray, 1976) into the input (Figure 6) because the sound source characterist.ics are made not only by the articulator posi1048 Hirayama, Vatikiotis-Bateson, and Kawato Articulator Positions, Velocities & VoicedNoiceless Acoustic Wave ___ G_lot_ta_1 s_o_u_rc_e ---'I-----L--'--___ -'--.J.--~~) ) ) Figure 6: Improved forward acoustic network. Inputs to the network are articulator positions and velocities and voiced/voiceless information. tion but also by the dynamic movement of articulators. For simulations, horizontal and vertical motions of jaw, upper and lower lips, and tongue tip and blade were used for the inputs and 12 dimensional PARCOR parameters were used for the outputs of the network. Figure 7(a) shows positionvelocity-voiced/voiceless network out.put compared with posit.ion-only network and experimentally obtained PARCOR parameters for a natural test sentence. Only the first two coefficients are shown. The first part of the test sentence, "Sam sat on top of the potato cooker and waited for Tommy to cut up a bag of tiny tomatoes and pop the beat tips into the pot," is shown in this plot. Figure 7(b)( c) show a part of the synthesized speech driven by funtlamental frequency pulses for voiced sounds and random noises for voiceless sounds. By using velocity and voiced/voiceless inputs, the performance was improved for natural utterances which include many vowels and consonants. The average values of the LPC-cepstrum distance mea.<.;ure between original and synthesized, were 5.17 (dB) for the position-only network and 4.18 (dB) for the position-velocityvoiced/voiceless network. When listening to the output, the sentence can be understood, and almost all vowels and many of the consonants can be classified. The overall clarity and the classifica.tion of some consonants is about as difficult as experienced in noisy international telephone calls. Although there are other potentia.l means to achieve further improvement (e.g. adding more tongue channels, using more balanced training patterns, incorporating nasality information, implementation of better glottal and non-glottal sources), the network synthesizes quite smooth and reasonable acoustic signals by incorporating aspects of the articulator dynamics. 5 CONCLUSION We are modeling the information transfer from phoneme-specific articulatory targets to acoustic wave via the musculo-skeletal system, using a series of neural networks. Electromyographic (EMG) signals are used as the reflection of motor control commands. In this paper, we have focused on the inverse dynamics modeling of the Inverse Dynamics of Speech Motor Control 1049 a 0.4 0.6 0.8 1.0 Position+Velocity+Voiced/Voiceless Network 1 0 --- Position-only Network . j ... ~.'''._~ ....... PARCOR ~or rest C\I 0 0 \ ............. .: \ r"", ~ . . . ',' ' '. 1 0 ~··I··::~ .. ···,. .. -~ j~.;.;.: .... ~ - . '.. I'" ...... .. 0.0 0.2 0.4 0.6 0.8 1.0 b Original Source (Noise + Pulse) Synthesized -+--~ c II 1: rr .:u l ll. " I J .L CJ ill L L1. 0.0 2UlJCJ n sOCJ{J 0.5 Time (s) ---- -- -1.0 f:~I: -. __ -,--~~~!I"iI=~:.~,_ ._t,.i I t ,1 " o -{.---------'ij, LJCJ 1 C I 1((,,= (seconds) - Figure 7: (a) Model output PARCOR parameters. Only kl and k2 are shown. (b) 0.2 C: Original, source model, and synthesized acoustic signals. (c) \Videband spectrogram for the original and synthesized speech. Utterance shown is "Sam sat on top" from a test sentence. 1050 Hirayama, Vatikiotis-Bateson, and Kawato musculo-skeletal system, its control for the transform from discrete linguistic information to continuous motor control signals, and articulatory speech synthesis using the articulator dynamics. '''Ie believe that. modeling the dynamics of articulat.ory motions is a key issue both for elucidating mechanisms of speech motor control and for synthesis of nat'llr'al utterances. Acknowledgetnellts We thank Yoh'ichi Toh'kura for continuous encouragement. Further support was provided by HFSP grants to M. Kawato. References Albus, J. S. (1975) A new approach to manipulator control: The cerebellar model articulation controller (CMAC). Transactions of the ASME Journal of Dynamic System, Afeasurement, and Control, 220-227. Hirayama., M., E. Vatikiotis-Bateson, M. Kawato, and 1\1. 1. Jordan \1992) Forward dynamics modeling of speech motor control using physiological data. In Moody, J. E., Hanson, S. J., and Lippmann, R. P. (eds.) Advances in Neural Information Processing Systems 4. San Mateo, CA: I\lorgan Kaufmann Publishers, 191-198. Hirayama, M., E. Vatikiotis-Bateson, K. Honda, Y. Koike, and M. Kawato (1993) Physiologically based speech synthesis. In Giles, C. L., Hanson, S. J., and Cowan, J. D. (eds.) Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann Publishers, 658-665. Itakura, F. and S. Saito (1969) Speech analysis and synthesis by partial correlation parameters. Proceeding of Japan Acoustic Society, 2-2-6 (In Japanese). Jordan, M. I. and D. E. Rumelhart (1992) Forward models: Supervised learning with a di'3tal teacher. Cognitive Science, 16, 307-354. Mermelstein, P. (1973) Articulatory model for the study of speech production. Journal of Acoustical Society of America, 53, 1070-1082. Papcun, J., J. Hochberg, T. R. Thomas, T. Laroche, J. Zacks, and S. Levy (1992) Inferring articulation and recognizing gestures from acoustics with a neural network trained on x-ray microbeam data. Jo'urnal of Acoustical Society of America, 92 (2) Pt. 1. Shirai, K. and T. Kobayashi (1991) Estimation of articulatory motion using neural networks. Journal of Phonetics, 19, 379-385. Uno, Y., R. Suzuki, and M. Kawato (1989) The minimum muscle tension change model which reproduces arm movement t.rajectories. Pr'oceedi7l9 of the 4th Symposium on Biological and Physiological Engineering, 299-302 (In Japanese). Wada, Y. and M. Kawat.o (1993) A nemal network model for arm t.rajectory formation of using fOl'ward and inverse dynamics models. Neural Networks, 6, 919-932. Wada, Y., Y. Koike, E. Vatikiotis-Bateson, and M. Kawato (1993) Movement Pattern Recognition Based on the Minimization Principle. Tech nical RI'port of IEICE, NC93-23, 85-92 (In Japanese).	1993	90
848	Learning Temporal Dependencies in Connectionist Speech Recognition Steve Renals Mike Hocbberg Tony Robinson Cambridge University Engineering Department Cambridge CB2 IPZ, UK {sjr,mmh,ajr}@eng.cam.ac.uk Abstract Hybrid connectionistfHMM systems model time both using a Markov chain and through properties of a connectionist network. In this paper, we discuss the nature of the time dependence currently employed in our systems using recurrent networks (RNs) and feed-forward multi-layer perceptrons (MLPs). In particular, we introduce local recurrences into a MLP to produce an enhanced input representation. This is in the form of an adaptive gamma filter and incorporates an automatic approach for learning temporal dependencies. We have experimented on a speakerindependent phone recognition task using the TIMIT database. Results using the gamma filtered input representation have shown improvement over the baseline MLP system. Improvements have also been obtained through merging the baseline and gamma filter models. 1 INTRODUCTION The most common approach to large-vocabulary, talker-independent speech recognition has been statistical modelling with hidden Markov models (HMMs). The HMM has an explicit model for time specified by the Markov chain parameters. This temporal model is governed by the grammar and phonology of the language being modelled. The acoustic signal is modelled as a random process of the Markov chain and adjoining local temporal information is assumed to be independent. This assumption is certainly not the case and a great deal of research has addressed the problem of modelling acoustic context. Standard HMM techniques for handling the context dependencies of the signal have ex1051 1052 Renals, Hochberg, and Robinson plicitly modelled all the n-tuples of acoustic segments (e.g., context-dependent triphone models). Typically, these systems employ a great number of parameters and, subsequently, require massive amounts of training data and/or care in smoothing of the parameters. Where the context of the model is greater than two segments, an additional problem is that it is very likely that contexts found in testing data are never observed in the training data. Recently, we have developed state-of-the-art continuous speech recognition systems using hybrid connectionistlHMM methods (Robinson, 1994; Renals et aI., 1994). These hybrid connectionistlHMM systems model context at two levels (although these levels are not necessarily at distinct scales). As in the traditional HMM, a Markov process is used to specify the duration and lexical constraints on the model. The connectionist framework provides a conditional likelihood estimate of the local (in time) acoustic waveform given the Markov process. Acoustic context is handled by either expanding the network input to include multiple, adjacent input frames, or using recurrent connections in the network to provide some memory of the previous acoustic inputs. 2 DEPTH AND RESOLUTION Following Principe et al. (1993), we may characterise the time dependence displayed by a particular model in terms of depth and resolution. Loosely speaking, the depth tells us how far back in time a model is able to lookl , and the resolution tells us how accurately the past to a given depth may be reconstructed. The baseline models that we currently use are very different in terms of these characteristics. Multi-layer Perceptron The feed-forward multi-layer perceptron (MLP) does not naturally model time, but simply maps an input to an output. Crude temporal dependence may be imparted into the system by using a delay-lined input (figure 1 a); an extension of this approach is the time-delay neural network (TDNN). The MLP may be interpreted as acting as a FIR filter. A delay-lined input representation may be characterised as having low depth (limited by the delay line length) and high resolution (no smoothing). Recurrent Network The recurrent network (RN) models time dependencies of the acoustic signal via a fullyconnected, recurrent hidden layer (figure 1 b). The RN has a potentially infinite depth (although in practice this is limited by available training algorithms) and low resolution, and may be regarded as analogous to an IIR filter. A small amount of future context is available to the RN, through a four frame target delay. Experiments Experiments on the DARPA Resource Management (RM) database have indicated that the tradeoff between depth and resolution is important. In Robinson et al. (1993), we compared different acoustic front ends using a MLP and a RN. Both networks used 68 1 In the language of section 3, the depth may be expressed as the mean duration, relative to the target, of the last kernel in a filter that is convolved with the input. Learning Temporal Dependencies in Connectionist Speech Recognition 1053 p(q" I X~~), Vk = I , .... K u(t) y(t-4) x(t) Hidden Layer 512 - 1,024 hidden units Xn_c .,. xn_1 xn+1 ... xn+c (a) Multi-layer Perceptron (b) Recurrent Network Figure 1: Connectionist architectures used for speech recognition. outputs (corresponding to phones); the MLP used 1000 hidden units and the RN used 256 hidden units. Both architectures were trained using a training set containing 3990 sentences spoken by 109 speakers. Two different resolutions were used in the front-end computation of mel-frequency cepstral coefficients (MFCCs): one with a 20ms Hamming window and a lOms frame step (referred to as 20110), the other with a 32ms Hamming window and a 16ms frame step (referred to as 32116). A priori, we expected the higher resolution frame rate (20/10) to produce a higher performance recogniser because rapid speech events would be more accurately modelled. While this was the case for the MLP, the RN showed better results using the lower resolution front end (32/16) (see table 1). For the higher resolution front-end, both models require a greater depth (in frames) for the same context (in milliseconds). In these experiments the network architectures were constant so increasing the resolution of the front end results in a loss of depth. Word Error Rate % Net Front End feb89 oct89 feb91 sep92 RN 20/10 6.1 7.6 7.4 12.1 RN 32116 5.9 6.3 6.1 11.5 MLP 20/10 5.7 7.1 7.6 12.0 MLP 32/16 6.6 7.8 8.5 15.0 Table 1: Comparison of acoustic front ends using a RN and a MLP for continuous speech recognition on the RM task, using a wordpair grammar of perplexity 60. The four test sets (feb89, oct89, feb91 and sep92, labelled according to their date of release by DARPA) each contain 300 sentences spoken by 10 new speakers. In the case of the MLP we were able to explicitly set the memory depth. Previous experiments had determined that a memory depth of 6 frames (together with a target delayed by 3 frames) was adequate for problems relating to this database. In the case of the RN, memory 1054 Renals, Hochberg, and Robinson P(qlx) P(qlx) Output Layer Hidden Layer (1000 hidden units) Hidden Layer (1000 hidden units) x(l) x(I+2) (a) Gamma Filtered Input (b) Gamma Filter + Future Context Figure 2: Gamma memory applied to the network input. The simple gamma memory in (a) does not incorporate any information about the future, unless the target is delayed. In (b) there is an explicit delay line to incorporate some future context. depth is not determined directly, but results from the interaction between the network architecture (i.e., number of state units) and the training process (in this case, back-propagation through time). We hypothesise that the RN failed to make use of the higher resolution front end because it did not adapt to the required depth. 3 GAMMA MEMORY STRUCTURE The tradeoff between depth and resolution has led us to investigate other network architectures. The gamma filter, introduced by de Vries and Principe (1992) and Principe et al. (1993), is a memory structure designed to automatically determine the appropriate depth and resolution (figure 2). This locally recurrent architecture enables lowpass and bandpass filters to be learned from data (using back-propagation through time or real-time recurrent learning) with only a few additional parameters. We may regard the gamma memory as a generalisation of a delay line (Mozer, 1993) in which the kth tap at time t is obtained by convolving the input time series with a kernel function, g~(t), and where 11 parametrises the Kth order gamma filter, gg(t) = 8(t) l<k<K. This family of kernels is attractive, since it may be computed incrementally by dXk(t) ---;tt = -l1xk(t) + I1Xk-l (t) . This is in contrast to some other kernels that have been proposed (e.g., Gaussian kernels proposed by Bodenhausen and Waibel (1991) in which the convolutions must be performed Learning Temporal Dependencies in Connectionist Speech Recognition 1055 explicitly). In the discrete time case the filter becomes: Xk(t) = (l - Il)Xk(t - 1) + IlXk-l(t - 1) This recursive filter is guaranteed to be stable when 0 < J1 < 2. In the experiments reported below we have replaced the input delay line of a MLP with a gamma memory structure, using one gamma filter for each input feature. This structure is referred to as a "focused gamma net" by de Vries and Principe (1992). Owing to the effects of anticipatory coarticulation, information about the future is as important as past context in speech recognition. A simple gamma filtered input (figure 2a) does not include any future context. There are various ways in which this may be remedied; • Use the same architecture, but delay the target (similar to figure Ib); • Explicitly specify future context by adding a delay line from the future (figure 2b); • Use two gamma filters per feature: one forward, one backward in time. A drawback of the first approach is that the central frame corresponding to the delayed target will have been smoothed by the action of the gamma filter. The third approach necessitates two passes when either training or running the network. 4 SPEECH RECOGNITION EXPERIMENTS We have performed experiments using the standard TIMIT speech database. This database is divided into 462 training speakers and 168 test speakers. Each speaker utters eight sentences that are used in these experiments, giving a training set of 3696 sentences and a test set of 1344 sentences. We have used this database for a continuous phone recognition task: labelling each sentence using a sequence of symbols, drawn from the standard 61 element phone set. The acoustic data was preprocessed using a 12th order perceptual linear prediction (PLP) analysis to produce an energy coefficient plus 12 PLP cepstral coefficients for each frame of data. A 20ms Hamming window was used with a lOms frame step. The temporal derivatives of each of these features was also estimated (using a linear regression over ± 3 adjacent frames) giving a total of 26 features per frame. The networks we employed (table 2) were MLPs, with 1000 hidden units, 61 output units (one per phone) and a variety of input representations. The Markov process used single state phone models, a bigram phone grammar, and a Viterbi decoder was used for recognition. The feed-forward weights in each network were initialised with identical sets of small random values. The gamma filter coefficients were initialised to 1.0 (equivalent to a delay line). The feed-forward weights were trained using back-propagation and the gamma filter coefficients were trained in a forward in time back-propagation procedure equivalent to real-time recurrent learning. An important detail is that the gradient step size was substantially lower (by a factor of 10) for the gamma filter parameters compared with the feed-forward weights. This was necessary to prevent the gamma filter parameters from becoming unstable. The baseline system using a delay line (Base) corresponds to figure 1 a, with ± 3 frames of context. The basic four-tap gamma filter G4 is illustrated in figure 2a (but using 1 fewer 1056 Renals, Hochberg, and Robinson System ID Description Base Baseline delay line, ± 3 frames of context G4 Gamma filter, 4 taps G7 Gamma filter, 7 taps, delayed target G7i G7 initialised using weights from Base G4F3 Gamma filter, 4 taps, 3 frames future context G4F3i G4F3 initialised using weights from Base Table 2: Input representations used in the experiments. Note that G7i and G4F3i were initialised using a partially trained weight matrix (after six epochs) from Base. tap than the picture) and G7 is a 7 frame gamma filter with the target delayed for 3 frames, thus providing some future context (but at the expense of smoothing the "centre" frame). Future context is explicitly incorporated in G4F3, in which the three adjacent future frames are included (similar to figure 2b). Systems G7i and G4F3i were both initialised using a partially trained weight matrix for the delay line system, Base. This was equivalent to fixing the value of the gamma filter coefficients to a constant (1.0) during the first six epochs of training and only adapting the feed-forward weights, before allowing the gamma filter coefficients to adapt. The results of using these systems on the TIM IT phone recognition task are given in table 3. Table 4 contains the results of some model merging experiments, in which the output probability estimates of 2 or more networks were averaged to produce a merged estimate. System ID Depth Correct% Insert. % Subst.% Delet.% Error % Base 4.0 67.6 4.1 24.7 7.7 36.5 G4 8.5 65.8 4.1 25.9 8.3 38.2 G7 11.7 65.5 4.1 26.0 8.5 38.6 G7i 5.8 67.3 3.8 24.5 8.2 36.5 G4F3 9.6 67.8 3.8 24.2 8.0 36.0 G4F3i 4.9 68.0 3.9 24.2 7.8 35.9 Table 3: TIMIT phone recognition results for the systems defined in table 2. The Depth value is estimated as the ratio of filter order to average filter parameter KIJ.!. Future context is ignored in the estimate of depth, and the estimates for G7 and G7i are adjusted to account for the delayed target. System ID Correct% Insert.% Subst.% Delet.% Error% G4F3+Base 68.1 3.2 23.7 8.2 35.1 G4F3 +G4F3i 68.2 3.2 23.5 8.3 35.0 G7 + Base 67.0 3.2 24.4 8.6 36.2 G7+G7i 67.4 3.6 24.4 8.2 36.2 Table 4: Model merging on the TIMIT phone recognition task. Learning Temporal Dependencies in Connectionist Speech Recognition 1057 O.B PLP Coefficients Derivatives 0.6 0.4 I I 0.2 E Cl C2 C3 C4 C5 C6 C7 CB C9 Cl0 Cll C12 Feature Figure 3: Gamma filter coefficients for G4F3. The coefficients correspond to energy (E) and 12 PLP cepstral coefficients (C1-C12) and their temporal derivatives. 5 DISCUSSION Several comments may be made about the results in section 4. As can be seen in table 3, replacing a delay line with an adaptive gamma filter can lead to an improvement in performance. Knowledge of future context is important. This is shown by G4, which had no future context or delayed target information, and had poorer performance than the baseline. However, incorporating future context using a delay line (G4F3) gives better performance than a pure gamma filter representation with a delayed target (G7). Training the locally recurrent gamma filter coefficients is not trivial. Fixing the gamma filter coefficients to 1.0 (delay line) whilst adapting the feed-forward weights during the first part of training is beneficial. This is demonstrated by comparing the performance of G7 with G7i and G4F3 with G4F3i. Finally, table 4 shows that model merging generally leads to improved recognition performance relative to the component models. This also indicates that the delay line and gamma filter input representations are somewhat complementary. Figure 3 displays the trained gamma filter coefficients for G4F3. There are several points to make about the learned temporal dependencies. • The derivative parameters are smaller compared with the static PLP parameters. This indicates the derivative filters have greater depth and lower resolution compared with the static PLP filters. • If a gamma filter is regarded as a lowpass IIR filter, then lower filter coefficients indicate a greater degree of smoothing. Better estimated coefficients (e.g., static PLP coefficients Cl and C2) give rise to gamma filters with less smoothing. • The training schedule has a significant effect on filter coefficients. The depth estimates of G4F3 and G4F3i in table 3 demonstrate that very different sets of filters were arrived at for the same architecture with identical initial parameters, but with different training schedules. 1058 Renals, Hochberg, and Robinson We are investigating the possibility of using gamma filters to model speaker characteristics. Preliminary experiments in which the gamma filters of speaker independent networks were adapted to a new speaker have indicated that the gamma filter coefficients are speaker dependent. This is an attractive approach to speaker adaptation, since very few parameters (26 in our case) need be adapted to a new speaker. Gamma filtering is a simple, well-motivated approach to modelling temporal dependencies for speech recognition and other problems. It adds minimal complexity to the system (in our case a parameter increase of 0.01 %), and these initial experiments have shown an improvement in phone recognition performance on the TIM IT database. A further increase in performance resulted from a model merging process. We note that gamma filtering and model merging may be regarded as two sides of the same coin: gamma filtering smooths the input acoustic features, while model merging smooths the output probability estimates. Acknowledgement This work was supported by ESPRIT BRA 6487, WERNICKE. SR was supported by a SERC postdoctoral fellowship and a travel grant from the NIPS foundation. TR was supported by a SERC advanced fellowship. References Bodenhausen, D., & Waibel, A. (1991). The Tempo 2 algorithm: Adjusting time delays by supervised learning. In Lippmann, R. P., Moody, J. E., & Touretzky, D. S. (Eds.), Advances in Neural Information Processing Systems, Vol. 3, pp. 155-161. Morgan Kaufmann, San Mateo CA. de Vries, B., & Principe, J. C. (1992). The gamma model-a new neural model for temporal processing. Neural Networks, 5,565-576. Mozer, M. C. (1993). Neural net architectures for temporal sequence processing. In Weigend, A. S., & Gershenfeld, N. (Eds.), Predicting the future and understanding the past. Addison-Wesley, Redwood City CA. Principe, J. C., de Vries, B., & de Oliveira, P. G. (1993). The gamma filter-a new class of adaptive IIR filters with restricted feedback. IEEE Transactions on Signal Processing, 41, 649-656. Renals, S., Morgan, N., Bourlard, H., Cohen, M., & Franco, H. (1994). Connectionist probability estimators in HMM speech recognition. IEEE Transactions on Speech and Audio Processing. In press. Robinson, A. J., Almeida, L., Boite, J.-M., Bourlard, H., Fallside, F., Hochberg, M., Kershaw, D., Kohn, P., Konig, Y., Morgan, N., Neto, J. P., Renals, S., Saerens, M., & Wooters, C. (1993). A neural network based, speaker independent, large vocabulary, continuous speech recognition system: the WERNICKE project. In Proceedings European Conference on Speech Communication and Technology, pp. 1941-1944 Berlin. Robinson, T. (1994). The application of recurrent nets to phone probability estimation. IEEE Transactions on Neural Networks. In press.	1993	91
849	Odor Processing in the Bee: a Preliminary Study of the Role of Central Input to the Antennal Lobe. Christiane Linster David Marsan ESPeI, Laboratoire d'Electronique 10, Rue Vauquelin, 75005 Paris linster@neurones.espci.fr Claudine Masson Laboratoire de Neurobiologie Comparee des Invertebrees INRNCNRS (URA 1190) 91140 Bures sur Yvette, France masson@jouy.inra.fr Abstract Michel Kerszberg Institut Pasteur CNRS (URA 1284) Neurobiologie Moleculaire 25, Rue du Dr. Roux 75015 Paris, France Based on precise anatomical data of the bee's olfactory system, we propose an investigation of the possible mechanisms of modulation and control between the two levels of olfactory information processing: the antennallobe glomeruli and the mushroom bodies. We use simplified neurons, but realistic architecture. As a first conclusion, we postulate that the feature extraction performed by the antennallobe (glomeruli and interneurons) necessitates central input from the mushroom bodies for fine tuning. The central input thus facilitates the evolution from fuzzy olfactory images in the glomerular layer towards more focussed images upon odor presentation. 1. Introduction Honeybee foraging behavior is based on discrimination among complex odors which is the result of a memory process involving extraction and recall of "key-features" representative of the plant aroma (for a review see Masson et al. 1993). The study of the neural correlates of such mechanisms requires a determination of how the olfactory system successively analyses odors at each stage (namely: receptor cells, antennal lobe interneurons and glomeruli, mushroom bodies). Thus far, all experimental studies suggest the implication of both antennallobe and mushroom bodies in these processes. The signal transmitted by the receptor cells is essentially unstable and fluctuating. The antennallobe appears as the location of noise reduction and feature extraction. The specific associative components operating on the olfactory memory trace would be essentially located in the mushroom bodies. The results of neuroethological experiments indicate furthermore that both the 527 528 Linster, Marsan, Masson, and Kerszberg feed-forward connections from the antennal lobe projection neurons to the mushroom bodies and the feedback connections from the mushroom bodies to the antennal lobe neurons are crucial for the storage and the recall of odor signals (Masson 1977; Erber et al. 1980; Erber 1981). Interestingly, the antennallobe compares to the mammalian olfactory bulb. Computational models of the insect antennal lobe (Kerszberg and Masson 1993; Linster et aI. 1993) and the mammalian olfactory bulb (Anton et a1. 1991; Li and Hopfield 1989; Schild 1988) have demonstrated that feature extraction can be performed in the glomerular layer, but the possible role of central input to the glomerular layer has not been investigated (although it has been included, as a uniform signal, in the Li and Hopfield model). On the other hand, several models of the mammalian olfactory cortex (Hasselmo 1993; Wilson and Bower 1989; LiljenstrOm 1991) have investigated its associative memory function, but have ignored the nature of the input from the olfactory bulb to this system. Based on anatomical and electrophysiological data obtained for the bee's olfactory system (Fonta et aI. 1993; Sun et al. 1993), we propose in this paper to investigate of the possible mechanisms of modulation and control between the two levels of olfactory information processing in a formal neural model. In the model, the presentation of an "odor" (a mixture of several molecules) differentially activates several populations of glomeruli. Due to coupling by local interneurons, competition is triggered between the activated glomeruli, in agreement with a recent proposal (Kerszberg and Masson 1993). We investigate the role of the different types of neurons implicated in the circuitry, and study the modulation of the glomerular states by reentrant input from the upper centers in the brain (i.e. mushroom bodies). 2. Olfactory circuitry in the bee's antennal lobe and mushroom bodies 95% of sensory cells located on the bee's antenna are olfactory (Esslen and Kaissling 1976), and convey signals to the antennal lobes. In the honeybee, due to some overlap of receptor cell responses, the peripheral representation of an odor stimulus is represented in an across fiber code (Fonta et al. 1993). Sensory axons project on two categories of antennal lobe neurons, namely local interneurons (LIN) and output neurons (ON). The synaptic contacts between sensory neurons and antennal lobe neurons, as well as the synaptic contacts between antennallobe neurons are localized in areas of high synaptic density, the antennal lobe glomeruli; each glomerulus represents an identifiable morphological neuropilar sub-unit (of which there are 165 for the worker honeybee) (Arnold et aI. 1985). Local interneurons constitute the majority of antennallobe neurons, and there is evidence that a majority of the LINs are inhibitory. As receptor cells are supposed to synapse mainly with LINs, the high level of excitation observed in the responses of ONs suggests that local excitation also exists (Malun 1991), in the form of spiking or non-spiking LINs, or as a modulation of local excitatbility. All LINs are pluriglomerular, but the majority of them, heterogeneous local interneurons (or HeteroLINs), have a high density of dendrite branches in one particular glomerulus, and sparser branches distributed across other glomeruli. A second category, homogeneous local interneurons (or Homo LINs), distribute their branches more homogeneously over the whole antennal lobe. Similarly, some of the ONs have dendrites invading only one glomerulus (Uniglomerular, or Uni ON), whereas the others (PI uri ON) are pluriglomerular. The axons of both types of ON project to different areas of the protocerebrum, including the mushroom bodies (Fonta et aI. 1993). Odor Processing in the Bee 529 3. Olfactory processing in the bee's antennal lobe glomeruli Responses of antennal lobe neurons to various odor stimuli are characterized by complex temporal patterns of activation and inactivation (Sun et al. 1993). Intracellularly recorded responses to odor mixtures are in general very complex and difficult to interpret from the responses to single odor components. A tendency to select particular odor related information is expressed by the category of "localized" antennallobe neurons, both Hetero LlNs and Uni ONs. In contrast, "global" neurons, both Homo LINs and Pluri ONs are often more responsive to mixtures than to single components. This might indicate that the related localized glomeruli represent functional sub units which are particularly involved in the discrimination of some key features. An adaptation of the 2DG method to the honeybee antennallobe has permitted to study the spatial distribution of odor related activity in the antennal lobe glomeruli (Nicolas et al. 1993; Masson et al. 1993). Results obtained with several individuals indicate that a correspondence can be established between two different odors and the activity maps they induce. This suggests that in the antennal lobe, different odor qualities with different biological meaning might be decoded according to separate spatial maps sharing a number of common processing areas. 4. Model of olfactory circuitry In the model, we introduce the different categories of neurons described above (Figure 1). Glomeruli are grouped into several regions and each receptor cell projects onto all local interneurons with arborizations in one region. Interneurons corresponding to heterogeneous LlNs can be (i) excitatory, these have a dendritic arborization (input and output synapses) restricted to one glomerulus; they provide "local" excitation, or, (ii) inhibitory, these have a dense arborization (mainly input synapses) in one glomerulus and sparse arborizations (mainly output synapses) in all others; they provide "local inhibition" and "lateral inhibition" between glomeruli. Interneurons corresponding to homogeneous LINs are inhibitory and have sparse arborizations (input and output synapses) in all glomeruli; they provide "uniform inhibition" over the glomerular layer. Output neurons are postsynaptic only to interneurons, they do not receive direct input from receptor cells. Each output neuron collects information from all interneurons in one glomerulus: thus modeling uniglomerular ONs. Implementation: The different neuron populations associated with one glomerulus are represented in the program as one unit (each unit is governed by one differential equation); the output of one unit represents the average firing probability of all neurons in this population (assuming that on the average, all neurons in one population receive the same input and have the same intrinsic properties). All units have membrane constants and a non-linear output function. Connection delays and connection strengths between units are chosen randomly around an average value: this assures a "realistic spatial averaging" over populations. The differential equations associated with the units are translated into difference equations and simulated by synchronous updating (sampling step Sms). 530 Linster, Marsan, Masson, and Kerszberg Molecule spectra Receptor cell types Receptor input ~ Global inhibition . :. Glomerular region ___ Local inhibition and lateral inhibition "'-. Local modulation :;:: '.:',' Modulation of global inhibition " Global inhibitory interneuron o Localized output neuron • Localized excitatory interneuron o Localized inhibitory interneuron Figure 1: Organization of the model olfactory circuitry. In the model, we introduce receptor cells with overlapping molecule spectra; each receptor cell has its maximal spiking probability P for the presence of a particular molecule i. The axons of the receptor cells project into distinct regions of the glomerular layer. All allowed connections exist with the same probability, but with different connection strengths. The activity of each glomerulus is represented by its associated output neurons. Central input projects onto the global inhibitory interneurons (modulation of global inhibition) or on all interneurons in one glomerulus (local modulation). 5. Olfactory processing by the model circuitry In the model, odors are represented as one-dimensional arrays of molecules; each molecule can be present in varying amounts. Due to the gaussian distributions of receptor cell sensitivities, an active molecule activates more than one receptor cell (with varying degrees of activation). As each receptor cell projects into all glomeruli belonging to its target region, thus, a molecular bouquet differentially activates a number of glomeruli in different glomerular regions. This triggers several phenomena: (i) due to the excitatory elements local to each glomerulus, and activated glomerulus tends to enhance the activation it receives from the receptor cells, (ii) the local inhibitory elements are activated (with a certain delay) by the receptor cell activity and by the self-activation of the local excitatory elements, and, (iii) trend to inhibit neighboring glomeruli. These phenomena result in a competition between active glomeruli: during a number of sampling steps, the output activity of each glomerulus (represented by the firing probability of the associated output neuron), oscillates from high activity to low activity. Due to the competition provided by Odor Processing in the Bee 531 the lateral inhibition, the spatial oscillatory activity pattern changes over time, and a stable activity map is reached eventually. A number of glomeruli "win" and stay active, whereas others "loose" and are inhibited (Figure 2). The activities of individual output neurons follow the general pattern described above: oscillation of the activity during a number of sampling steps until the activity "settles" down to a stable value. A stable activity can either be a constant firing probability, or a "stable" oscillation of the firing probability. An output neuron associated to a particular glomerulus may be active for a particular odor input, and silent for others. Complex temporal patterns of excitation and inhibition may occur after stimulus presentation .. Thus, the model predicts that odor representation is performed through spatial maps of activity spanning the whole glomerular layer. Individual output neurons, representing the activity of their associated glomeruli may be either excited or inhibited by a particular odor pattern. Glomeruli 1 - 15 After stabilization Figure 2: Behavior of the model after stimulation of the receptor cells with the molecule array indicated in the figure. For several sampling steps (of 5 ms), the activity (firing probability) of the ON associated to each glomerulus is shown. At step I, all glomeruli are differentially activated by the receptor cell input. Lateral inhibition silences all glomeruli during the next sampling step. At step 3, some glomeruli are highly activated (due to their local excitation), whereas others are almost silenced. Then, t spatial activation pattern oscillates for a number of sampling steps (which depends on the strength of the lateral inhibitory connections and on the number of active molecules in the odor array), and finally stabilizes in a spatial activity map. 6. Comparison of odor processing in the Bee's antennal lobe and in the model Antennallobe neurons in the bee show various response patterns to stimulation with pure components and mixtures. Most LINs and ONs respond with simple excitation or inhibition to stimulation, often followed by a hyperpolarized (resp. depolarized) phase. Interestingly, most LINs respond with various degrees of excitation to stimulation with binary odors and mixtures, whereas ONs respond equally often by excitation than by inhibition (Sun et a1. 1993). In the model, LINs receive direct afferent input from receptor cells, and are therefore differentially activated by odor stimulation; they respond with varying degrees of excitation to stimulation with pure components and their mixtures. Output neurons in the model receive indirect input from receptor cells via local interneurons. Output neurons in the model are either activated (if their associated 532 Linster, Marsan, Masson, and Kerszberg glomerulus wins the competition) or inhibited (if their associated glomerulus looses the competition) by odor stimulation. In the simulations, output neurons which are excited for a particular odor stimulation belong to an active glomerulus in the spatial activity map associated to that odor. For each odor, a particular activity map is established. An output neuron is either excited or inhibited by a particular odor stimulation, indicating that it takes part in the representation of an activity map across glomeruli, which might be compared to the antennallobe 2DG maps. 7. Modulation of the model dynamics Odor detection by modulation of spontaneous activity At high spontaneous activity, all glomeruli in the model oscillate spontaneously (Figure 3). Odor stimulation tends to synchronize these oscillations, but no feature detection is perfonned. In the model, the underlying activity map which corresponds to the odor signal can only emerge if the spontaneous activity is decreased (Figure 3). Decreasing of the spontaneous activity can be achieved by 5i) activation of the global inhibitory interneurons by central input, or, (ii) decreasing of the spiking threshold of all antennallobe neurons. These data fit well with experimental data (see Sun et al. 1993, Figures 7 and 8). _: ; '; "'j ;' ; TT«t¥1"Ttrrt" .:&.&&~ .. &.~ . , Stimulus annlication Figure 3 .II Ii. 500ms Reduction of spontaneous activity Figure 3: Modulation of the spontaneous activity. We show the spiking probabilities of output neurons associated to different glomeruli. Arrows indicate stimulus onset. Stimulus presentation synchronizes the oscillations. A decreasing of the spontaneous activity results in the emergence of the underlying activity map: several output neurons exhibit high activities, whereas the others are silent. Contrast enhancement by modulation of lateral inhibition Presentation of an odor in the model differentially activates many or all glomeruli, which, due to the local excitation, try to enhance the activation due to the odor stimulus. Due to the competition between glomeruli, feature detection is performed in the glomerular layer, which enhances some elements of the stimulus and suppresses others. In the model, for a given odor stimulation, the number of winning glomeruli depends on the strength of the lateral inhibition between glomeruli (Figure 5). At low lateral inhibition, most glomeruli stay active for any odor; no feature extraction is perfonned. Odor Processing in the Bee 533 Increasing of the lateral inhibition focuses the odor maps, which can now differentiate different odor inputs. annat'\(')(\(')('\. ()~ .... ()ne()(). ... ....... U II1J .00.00000. 0 •• 000.00. • ••• 00.00. • ••••••••• D [l] [[] m .ooeooooo. 0 •• 000.00. • ••• 00.00. • ••••••••• DDLJ[]D eooeooooo. 0 •• 000.00. • •• eooeoo. • ........ . Figure 5: Stabilized activity maps for different odor stimuli with increasing lateral inhibition strength. At low competition, all glomeruli tend to be active due to their local excitation. Increasing of lateral inhibition permits to enhance the important features of each odor, and leads to uncorrelated activity maps for the different stimulations. Increasing of the lateral inhibition permits to focus a fuzzy olfactory image in the glomerular layer, or to "smell closer". A fuzzy sampling of an odor may be useful at first approach, whereas a more precise analysis of its important components is facilitated by increasing the competition between glomeruli increases contrast enhancement. 8. Discussion We have presented the computational abilities of the neural circuitry in the antennallobe model, based on what is known of the bee's circuitry. Single cell responses and global activity patterns are comparable to the odor processing mechanisms proposed in the insect (Linster et al. 1993; Masson et al. 1993; Kerszberg and Masson 1993) and in the vertebrate (Kauer et al. 1991; Li and Hopfield 1989; Freeman 1991) literature. As suggested by Kerszberg and Masson (1993), we show that odor preprocessing is based on spontaneous dynamics of the antennal lobe glomeruli, and that, in addition, feature detection needs competition between activated glomeruli due to global and lateral inhibition. The model is able to predict the role of the four types of neurons morphologically identified in the bee an ten nal lobe. It also predicts how intracellular recordings and 2DG data can be explained by the odor processing mechanism. Furthermore, modulation of the models dynamics opens up a number of new ideas about the respective role of the two main categories ("localized" and "global") of antennallobe neurons, and the possible role of central input to these neurons. Acknowledgements The authors are grateful to G. Dreyfus and L. Personnaz for fruitful discussions. 534 Linster, Marsan, Masson, and Kerszberg References Arnold, G., Masson, C., Budhargusa, S. 1985. Comparative study of the antennal pathway of the workerbee and the drone (Apis mellifera). Cell Tissue Res. 242: 593-605. Erber, J. 1981 Neural correlates of learning in the honeybee. TINS 4:270-273. Erber, J., Masuhr, T., Menzel, R. 1980. Localisation of shon-term memory in the brain of the bee, Apis melli/era. Physiolo. Entomol. 5: 343-358. EssIen, J., Kaissling, K.E. 1976. Zahl und Verteilung antennaler Sensillen bei der Honigbiene. Zoomorphologie 83: 227-251. Fonta, C., Sun, X., Masson, C. 1193. Morphology and spatial distribution of bee antennallobe interneurons responsive to odours. Chemical Senses, 18 (2): pp. 101119. Hasselmo, M.E. 1993. Acetycholine and Learning in a Conical Associative Memory. Neural Computation, 5: 32-44. Kauer, J.S., Neff, S.R., Hamilton, K.A., Cinelli, A.R. 1991. The Salamander Olfactory Pathway: Visualizing and Modeling Circuit Activity. in Olfaction: A Model System for Computational Neuroscience. Davis, J. and Eichenbaum, H. (eds): 4468. MIT Press. Kerszberg, M., Masson, C., 1993. Signal Induced Selection among Spontaneous Activity Patterns of Bee's Olfactory Glomeruli, submitted. Li, Z., Hopfield, J.1., 1989. Modeling the Olfactory Bulb and its Neural Oscillatory Processings. Biological Cybernetics 61:379-392. LiljenstrOm, H. 1991. Modeling the dynamics of olfactory cortex using simplified network units and realistic architecture. International Journal of Neural Systems, (1&2): 115. Linster, C., Masson, C., Kerszberg, M., Personnaz, L., Dreyfus, G. 1993 Computational Diversity in a Fonnal Model of the Insect Macroglomerulus., Neural Computation, 5:239-252. Malun, D. 1991. Inventory and distribution of synapses of identified uniglomerular projection neurons in the antennallobe of periplaneta americana. J.Comp. Neurol. 305: 348-360. Masson, C. 1977. Central olfactory pathways and plasticity of responses to odor stimuli in insects. in Olfaction and Taste VI. Le Magnen, J., Mac Leod, P. (eds) IRL, London: 305-314. Masson, C., Mustaparta, H. 1990. Chemical Information Processing in the Olfactory System of Insects. Physiol. Reviews 70(1):199-245. Masson, C., Pham-Delegue, MH., Fonta, C., Gascuel, J., Arnold, G., Nicolas, G., Kerszberg, M. 1993. Recent advances in the concept of adaptation to natural odour signals in the honeybee Apis mellifera L. Apidologie 24: 169-194. Menzel, R. 1983. Neurobiology of learning and memory: the honeybee as a model system. Naturwissenschaften 70: 504-511. Nicolas, G., Arnold, G., Patte, F., Masson, C. 1993. Distribution regionale de l'incorporation du 3H2-Desoxyglucose dans Ie lobe antennaire de l'ouvriere d'abeille. CR. Acad. Sc. Paris (Sciences de la Vie), 316: 1245-1249. Schild , D. 1988 Principles of odor coding and a neural network for odor discrimination, Biophys. J. 54:1001-101l. Sun, X., Fonta, C., Masson, C. 1993. Odour quality processing by bee antennal lobe neurons. Chemical Senses 18 (4): 355-377.	1993	92
850	Surface Learning with Applications to Lipreading Christoph Bregler .* Computer Science Division University of California Berkeley, CA 94720 Stephen M. Omohundro * **Int. Computer Science Institute 1947 Center Street Suite 600 Berkeley, CA 94704 Abstract Most connectionist research has focused on learning mappings from one space to another (eg. classification and regression). This paper introduces the more general task of learning constraint surfaces. It describes a simple but powerful architecture for learning and manipulating nonlinear surfaces from data. We demonstrate the technique on low dimensional synthetic surfaces and compare it to nearest neighbor approaches. We then show its utility in learning the space of lip images in a system for improving speech recognition by lip reading. This learned surface is used to improve the visual tracking performance during recognition. 1 Surface Learning Mappings are an appropriate representation for systems whose variables naturally decompose into "inputs" and "outputs)). To use a learned mapping, the input variables must be known and error-free and a single output value must be estimated for each input. Many tasks in vision, robotics, and control must maintain relationships between variables which don't naturally decompose in this way. Instead, there is a nonlinear constraint surface on which the values of the variables are jointly restricted to lie. We propose a representation for such surfaces which supports a wide range of queries and which can be naturally learned from data. The simplest queries are "completion queries)). In these queries, the values of certain variables are specified and the values (or constraints on the values) of remaining 43 44 Bregler and Omohundro Figure 1: Using a constraint surface to reduce uncertainty in two variables ~. Figure 2: Finding the closest point in a surface to a given point. variables are to be determined. This reduces to a conventional mapping query if the "input" variables are specified and the system reports the values of corresponding "output" variables. Such queries can also be used to invert mappings, however, by specifying the "output" variables in the query. Figure 1 shows a generalization in which the variables are known to lie with certain ranges and the constraint surface is used to further restrict these ranges. For recognition tasks, "nearest point" queries in which the system must return the surface point which is closest to a specified sample point are important (Figure 2). For example, symmetry-invariant classification can be performed by taking the surface to be generated by applying all symmetry operations to class prototypes (eg. translations, rotations, and scalings of exemplar characters in an OCR system). In our representation we are able to efficiently find the globally nearest surface point in this kind of query. Other important classes of queries are "interpolation queries" and "prediction queries". For these, two or more points on a curve are specified and the goal is to interpolate between them or extrapolate beyond them. Knowledge of the constraint surface can dramatically improve performance over "knowledge-free" approaches like linear or spline interpolation. In addition to supporting these and other queries, one would like a representation which can be efficiently learned. The training data is a set of points randomly drawn from the surface. The system should generalize from these training points to form a representation of the surface (Figure 3). This task is more difficult than mapping learning for several reasons: 1) The system must discover the dimension of the surface, 2) The surface may be topologically complex (eg. a torus or a sphere) •• • • ••• • • • • • • • •• •• • •• Surface Learning with Applications to Lipreading 45 Figure 3: Surface Learning and may not support a single set of coordinates, 3) The broader range of queries discussed above must be supported. Our approach starts from the observation that if the data points were drawn from a linear surface, then a principle components analysis could be used to discover the dimension of the linear space and to find the best-fit linear space of that dimension. The largest principle vectors would span the space and there would be a precipitous drop in the principle values at the dimension of the surface. A principle components analysis will no longer work, however, when the surface is nonlinear because even a I-dimensional curve could be embedded so as to span all the dimensions of the space. If a nonlinear surface is smooth, however, then each local piece looks more and more linear under magnification. If we consider only those data points which lie within a local region, then to a good approximation they come from a linear surface patch. The principle values can be used to determine the most likely dimension of the surface and that number of the largest principle components span its tangent space (Omohundro, 1988). The key idea behind our representations is to "glue" these local patches together using a partition of unity. We are exploring several implementations, but all the results reported here come from a represenation based on the "nearest point" query. The surface is represented as a mapping from the embedding space to itself which takes each point to the nearest surface point. K-means clustering is used to determine a initial set of "prototype centers" from the data points. A principle components analysis is performed on a specified number of the nearest neighbors of each prototype. These "local peA" results are used to estimate the dimension of the surface and to find the best linear projection in the neighborhood of prototype i. The influence of these local models is determined by Gaussians centered on the prototype location with a variance determined by the local sample density. The projection onto the surface is determined by forming a partition of unity from these Gaussians and using it to form a convex linear combination of the local linear projections: (1) This initial model is then refined to minimize the mean squared error between the 46 Bregler and Omohundro a) b) Figure 4: Learning a I-dimensional surface. a) The surface to learn b) The local patches and the range of their influence functions, c) The learned surface training samples and the nearest surface point using EM optimization and gradient descent. 2 Synthetic Examples To see how this approach works, consider 200 samples drawn from a I-dimensional curve in a two-dimensional space (Figure 4a). 16 prototype centers are chosen by kmeans clustering. At each center, a local principle components analysis is performed on the closest 20 training samples. Figure 4b shows the prototype centers and the two local principle components as straight lines. In this case, the larger principle value is several times larger than the smaller one. The system therefore attempts to construct a one-dimensional learned surface. The circles in Figure 4b show the extent of the Gaussian influence functions for each prototype. Figure 4c shows the resulting learned suface. It was generated by randomly selecting 2000 points in the neighborhood of the surface and projecting them according to the learned model. Figure 5 shows the same process applied to learning a two-dimensional surface embedded in three dimensions. To quantify the performance of this learning algorithm, we studied the effect of the different parameters on learning a two-dimensional sphere in three dimensions. It is easy to compare the learned results with the correct ones in this case. Figure 6a shows how the empirical error in the nearest point query decreases as a function of the number of training samples. We compare it against the error made by a nearest-neighbor algorithm. With 50 training samples our approach produces an error which is one-fourth as large. Figure 6b shows how the average size of the local principle values depends on the number of nearest neighbors included. Because this is a two-dimensional surface, the two largest values are well-separated from the third largest. The rate of growth of the principle values is useful for determining the dimension of the surface in the presence of noise. Surface Learning with Applications to Lipreading 47 Figure 5: Learning a two-dimensional surface in the three dimensions a) 1000 random samples on the surface b) The two largest local principle components at each of 100 prototype centers based on 25 nearest neighbors. :::~--+ ~--:+=~-+=t-+=:--+:~:+~ '0000- - j=----~~ c-t-r--t---r =:=. ~ ~f . t::- ·=t~~f·t~ :::- -~~r~l- -:=t:==t~f :::: ..::t~ L_ ~:--=- -:-:- ':::::-\ I ==-~~~-l== -- ------+IOD~ - -+--1---- ----+ 4000- - . -~:'::: 1000-'ODD 1OG OO 15000 ZOO 00 ~OOD 3000{) 3SGOO lBO.OO 160 .00 120.00 9>.00 60.00 ".00 20.00 •. oo~ __ ~~ _______ _ '.00 80.00 100.00 1110.00 Figure 6: Quantitative performance on learning a two-dimensional sphere in three dimensions. a) Mean squared error of closest point querries as function of the number of samples for the learned surface vs. nearest training point b) The mean square root of the three principle values as a function of number of neighbors included in each local PCA. 48 Bregler and Omohundro a b Figure 7: Snakes for finding the lip contours a) A correctly placed snake b) A snake which has gotten stuck in a local minimum of the simple energy function. 3 Modelling the space of lips We are using this technique as a part of system to do "lipreading". To provide features for "vise me classification" (visemes are the visual analog of phonemes), we would like the system to reliably track the shape of a speaker's lips in video images. It should be able to identify the corners of the lips and to estimate the bounding curves robustly under a variety of imaging and lighting conditions. Two approaches to this kind of tracking task are "snakes" (Kass, et. aI, 1987) and "deformable templates" (Yuille, 1991). Both of these approaches minimize an "energy function" which is a sum of an internal model energy and an energy measuring the match to external image features. For example, to use the "snake" approach for lip tracking, we form the internal energy from the first and second derivatives of the coordinates along the snake, prefering smoother snakes to less smooth ones. The external energy is formed from an estimate of the negative image gradient along the snake. Figure 7a shows a snake which has correctly relaxed onto a lip contour. This energy function is not very specific to lips, however. For example, the internal energy just causes the snake to be a controlled continuity spline. The "lip- snakes" sometimes relax onto undesirable local minima like that shown in Figure 7b. Models based on deformable templates allow a researcher to more strongly constrain the shape space (typically with hand-coded quadratic linking polynomials), but are difficult to use for representing fine grain lip features. Our approach is to use surface learning as described here to build a model of the space of lips. We can then replace the internal energy described above by a quantity computed from the distance to the learned surface in lip feature space. Our training set consists of 4500 images of a speaker uttering random wordsl . The training images are initially "labeled" with the conventional snake algorithm. Incorrectly aligned snakes are removed from the database by hand. The contour shape is parameterized by the x and y coordinates of 40 evenly spaced points along the snake. All values are normalized to give a lip width of 1. Each lip contour is IThe data was collected for an earlier lipreading system described in (Bregler, Hild, Manke, Waibel 1993) (Ja ~d Surface Learning with Applications to Lipreading 49 C7b e Figure 8: Two principle axes in a local patch in lip space. a, b, and c are configurations along the first principle axis, while d, e, and f are along the third axis. a b c Figure 9: a) Initial crude estimate of the contour b) An intermediate step in the relaxation c) The final contour. therefore a point in an 80-dimensional "lip- space". The lip configurations which actually occur lie on a lower dimensional surface embedded in this space. Our experiments show that a 5-dimensional surface in the 80-dimensional lip space is sufficient to describe the contours with single pixel accuracy in the image. Figure 8 shows some lip models along two of the principle axes in the local neighborhood of one of the patches. The lip recognition system uses this learned surface to improve the performance of tracking on new image sequences. The tracking algorithm starts with a crude initial estimate of the lip position and size. It chooses the closest model in the lip surface and maps the corresponding resized contour back onto the estimated image position (Figure 9a). The external image energy is taken to be the cumulative magnitude of graylevel gradient estimates along the current contour. This term has maximum value when the curve is aligned exactly on the lip boundary. We perform gradient ascent in the contour space, but constrain the contour to lie in the learned lip surface. This is achieved by reprojecting the contour onto the lip surface after each gradient step. The surface thereby acts as the analog of the internal energy in the snake and deformable template approaches. Figure 9b shows the result after a few steps and figure 9c shows the final contour. The image gradient is estimated using an image filter whose width is gradually reduced as the search proceeds. The lip contours in successive images in the video sequence are found by starting with the relaxed contour from the previous image and performing gradient ascent 50 Bregler and Omohundro with the altered external image energies. Empirically, surface-based tracking is far more robust than the "knowledge-free" approaches. While we have described the approach in the context of contour finding, it is much more general and we are currently extending the system to model more complex aspects of the image. The full lipreading system which combines the described tracking algorithm and a hybrid connectionist speech recognizer (MLP /HMM) is described in (Bregler and Konig 1994). Additionally we will use the lip surface to interpolate visual features to match them with the higher rate auditory features. 4 Conclusions We have presented the task of learning surfaces from data and described several important queries that the learned surfaces should support: completion, nearest point, interpolation, and prediction. We have described an algorithm which is capable of efficiently performing these tasks and demonstrated it on both synthetic data and on a real-world lip-tracking problem. The approach can be made computationally efficient using the "bumptree" data structure described in (Omohundro, 1991). We are currently studying the use of "model merging" to improve the representation and are also applying it to robot control. Acknowledgements This research was funded in part by Advanced Research Project Agency contract #NOOOO 1493 C0249 and by the International Computer Science Institute. The database was collected with a grant from Land Baden Wuerttenberg (Landesschwerpunkt Neuroinformatik) at Alex Waibel's institute. References C. Bregler, H. Hild, S. Manke & A. Waibel. (1993) Improving Connected Letter Recognition by Lipreading. In Proc. of Int. Conf. on Acoustics, Speech, and Signal Processing, Minneapolis. C. Bregler, Y. Konig (1994) "Eigenlips" for Robust Speech Recognition. In Proc. of Int. Conf. on Acoustics, Speech, and Signal Processing, Adelaide. M. Kass, A. Witkin, and D. Terzopoulos. (1987) SNAKES: Active Contour Models, in Proc. of the First Int. Conf. on Computer Vision, London. S. Omohundro. (1988) Fundamentals of Geometric Learning. University of Illinois at Urbana-Champaign Technical Report UIUCDCS-R-88-1408. S. Omohundro. (1991) Bumptrees for Efficient Function, Constraint, and Classification Learning. In Lippmann, Moody, and Touretzky (ed.), Advances in Neural Information Processing Systems 3. San Mateo, CA: Morgan Kaufmann. A. Yuille. (1991) Deformable Templates for Face Recognition, Journal of Cognitive Neuroscience, Volume 3, Number 1.	1993	93
851	Figure of Merit Training for Detection and Spotting Eric I. Chang and Richard P. Lippmann MIT Lincoln Laboratory Lexington, MA 02173-0073, USA Abstract Spotting tasks require detection of target patterns from a background of richly varied non-target inputs. The performance measure of interest for these tasks, called the figure of merit (FOM), is the detection rate for target patterns when the false alarm rate is in an acceptable range. A new approach to training spotters is presented which computes the FOM gradient for each input pattern and then directly maximizes the FOM using b ackpropagati on. This eliminates the need for thresholds during training. It also uses network resources to model Bayesian a posteriori probability functions accurately only for patterns which have a significant effect on the detection accuracy over the false alarm rate of interest. FOM training increased detection accuracy by 5 percentage points for a hybrid radial basis function (RBF) - hidden Markov model (HMM) wordspotter on the credit-card speech corpus. 1 INTRODUCTION Spotting tasks require accurate detection of target patterns from a background of richly varied non-target inputs. Examples include keyword spotting from continuous acoustic input, spotting cars in satellite images, detecting faults in complex systems over a wide range of operating conditions, detecting earthquakes from continuous seismic signals, and finding printed text on images which contain complex graphics. These problems share three common characteristics. First, the number of instances of target patterns is unknown. Second, patterns from background, non-target, classes are varied and often difficult to model accurately. Third, the performance measure of interest, called the figure of merit (FOM), is the detection rate for target patterns when the false alarm rate is over a specified range. Neural network classifiers are often used for detection problems by training on target and background classes, optionally normalizing target outputs using the background output, 1019 1020 Chang and Lippmann PUTATIVE HITS nA uS I NORMALIZATION AND THRESHOLDING I jl A Us ~~ SACKG ROUND CLASSIFIER t INPUT PATTERN Figure 1. Block diagram of a spotting system. and thresholding the resulting score to generate putative hits, as shown in Figure 1. Putative hits in this figure are input patterns which generate normalized scores above a threshold. We have developed a hybrid radial basis function (RBF) - hidden Markov model (HMM) keyword spotter. This wordspotter was evaluated using the NIST credit card speech database as in (Rohlicek, 1993, Zeppenfeld, 1993) using the same train/evaluation split of the training conversations as was used in (Zeppenfeld, 1993). The system spots 20 target keywords, includes one general filler class, and uses a Viterbi decoding backtrace as described in (Lippmann, 1993) to backpropagate errors over a sequence of input speech frames. The performance of this spotting system and its improved versions is analyzed by plotting detection versus false alarm rate curves as shown in Figure 2. These curves are generated by adjusting the classifier output threshold to allow few or many putative hits. Wordspotter putative hits used to generate Figure 2 correspond to speech frames when the difference between the cumulative log Viterbi scores in output HMM nodes of word and filler models is above a threshold. The FOM for this wordspotter is defined as the average keyword detection rate when the false alarm rate ranges from 1 to 10 false alarms per keyword per hour. The 69.7% figure of merit for this system means that 69.7% of keyword occurrences are detected on the average while generating from 20 to 200 false alarms per hour of input speech. 2 PROBLEMS WITH BACKPROPAGATION TRAINING Neural network classifiers used for spotting tasks can be trained using conventional backpropagation procedures with 1 of N desired outputs and a squared error cost function. This approach to training does not maximize the FOM because it attempts to estimate Bayesian a posteriori probability functions accurately for all inputs even if a particular input has little effect on detection accuracy at false alarm rates of interest. Excessive network resources may be allocated to modeling the distribution of common background inputs dissimilar from targets and of high-scoring target inputs which are easily detected. This problem can be addressed by training only when network outputs are above thresholds. This approach is problematic because it is difficult to set the threshold for different keywords, because using fixed target values of 1.0 and 0.0 requires careful normalization of network output scores to prevent saturation and maintain backpropagation effectiveness, and because the gradient calculated from a fixed target value does not reflect the actual impact on the FOM. 100 (J) 90 z 0 80 i= 70 0 W t60 w c t50 O 40 w a:: 30 a:: 0 20 0 ~ 0 10 0 0 Figure of Merit Training for Detection and Spotting 1021 A SPLIT OF CREDIT-CARD TRAINING DATA ","I'I .. ~.~I'iI.I: ••••••••• :::.:./.::.:: .. !!! .. ~.~ ......... u::.~ .. .::: .. l- /'-:-,/, f / ./ FOM BACK-PROP (FOM: 69.7%) EMBEDDED REESTIMATION (FOM: 64.5%) ISOLATED WORD TRAIN (FOM: 62.5%) 2 4 6 8 10 FALSE ALARMS PER KW PER HR Figure 2. Detection vs. false alarm rate curve for a 20-word hybrid wordspotter. Figure 3 shows the gradient of true hits and false alarms when target values are set to be 1.0 for true hits and 0.0 for false alarms, the output unit is sigmoidal, and the threshold for a putative hit is set to roughly 0.6. The gradient is the derivative of the squared error cost with respect to the input of the sigmodal output unit. As can be seen, low-scoring hits or false alarms that may affect the FOM are ignored, the gradient is discontinuous at the threshold, the gradient does not fall to zero fast enough at high values, and the relative sizes of the hit and false alarm gradients do not reflect the true effect of a hit or false alarm on the FOM. 3 FIGURE OF MERIT TRAINING A new approach to training a spotter system called "figure of merit training" is to directly compute the FOM and its derivative. This derivative is the change in FOM over the change in the output score of a putative hit and can be used instead of the derivative of a squarederror or other cost function during training. Since the FOM is calculated by sorting true hits and false alarms separately for each target class and forming detection versus false alarm curves, these measures and their derivatives can not be computed analytically. Instead, the FOM and its derivative are computed using fast sort routines. These routines insert a new 0.2 r--------------------, THRESHOLD L....... HIT GRADIENT !z w Ci 0 I-----------f------==-'-'""!l « a: <!) GRADIENT .0.2 L....................L .............. ~_'_'_~ .......... ..L....................J'_'_'_~ ........... ......J...... ............ .....L.<_.'-'--'-J.......o. ........... o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0,8 0.9 OUTPUT VALUE Figure 3. The gradient for a sigmoid output unit when the target value for true hits is set to 1.0 and the target value for false alarms is set to 0.0. 1022 Chang and Lippmann putative hit into an already sorted list and calculate the change in the FOM caused by that insertion. The running putative hit list used to compute the FOM is updated after every new putative hit is observed and it must contain all putative hits observed during the most recent past training cycle through all training patterns. The gradient estimate is smoothed over nearby putative hit scores to account for the quantized nature of detection versus false alarm rate curves. Figure 4 shows plots of linearly scaled gradients for the 20-word hybrid wordspotter. Each value on the curve represents the smoothed change in the FOM that occurs when a single hit or false alarm with the specified normalized log output score is inserted into the current putative hit list. Gradients are positive for putative hits corresponding to true hits and negative for false alarms. They also fall off to zero for putative hits with extremely high or low scores. Shapes of these curves vary across words. The relative importance of a hit or false alarm, the normalized output score which results in high gradient values, and the shape of the gradient curve varies. Use of a squared error or other cost function with sigmoid output nodes would not generate this variety of gradients or automatically identify the range of putative hit scores where gradients should be high. Application ofFOM training requires only the gradients shown in these curves with no supplementary thresholds. Patterns with low and high inputs will have a minimal effect during training without using thresholds because they produce gradients near zero. Different keywords have dramatically different gradients. For example, credit-card is long and the detection rate is high. The overall FOM thus doesn't change much if more true hits are found. A high scoring false alarm, however, decreases the FOM drastically. There is thus a large negative gradient for false alarms for credit-card. The keywords account and check are usually short in duration and thus more difficult to detect, thus any increase in number of true hits strongly increases the overall FOM. On the other hand, since in this database, the words account and check occur much less frequently than credit-card, a high scoring false alarm for the words account and check has less impact on the overall FOM. The gradient for false alarms for these words is thus correspondingly smaller. Comparing the curves in Figure 3 with the fixed prototypical curve in Figure 4 demonstrates the dramatic differences in gradients that occur when the gradient is calculated to maximize the FOM directly instead of using a threshold with sigmoid output nodes. "ACCOUNT" 0.3 r-------, o ~ w is -03 « ffi -D.6 HIT FA -0. 9 ~--'--.L..-L--'--.L...-L---' "CHECK' "CREDIT-CARD' -100 0 100 200 300 -100 0 100 200 300 -100 0 100 200 300 PUTATIVE HIT SCORE Figure 4. Figure of merit gradients computed for true hits (HIT) and false alarms (FA) with scores ranging from -100 to 300 for the keywords account, check, and credit-card. Figure of Merit Training for Detection and Spotting 1023 FaM training is a general technique that can applied to any "spotting" task where a set of targets must be discriminated from background inputs. FaM training was successfully tested using the hybrid radial basis function (RBF) - hidden Markov model (HMM) keyword spotter described in (Lippmann, 1993). 4 IMPLEMENTATION OF FOM TRAINING FaM training is applied to our high-performance HMM wordspotter after forward-backward training is complete. Word models in the HMM wordspotter are first used to spot on training conversations. The FaM gradient of each putative hit is calculated when this hit is inserted into the putative hit list. The speech segment corresponding to a putative hit is excised from the conversation speech file and the corresponding keyword model is used to match each frame with a particular state in the model using a Viterbi backtrace (shown in Figure 5.) The gradient is then used to adjust the location of each Gaussian component in a node as in RBF classifiers (Lippmann, 1993) and also the state weight of each state. The state weight is a penalty added for each frame assigned to a state. The weight for each individual state is adjusted according to how important each state is to the detection of the keyword. For example, many false alarms for the word card are words that sound like part of the keyword such as hard or far. The first few states of the card model represent the sound /kJ and false alarms stay in these front states only a short time. If the state weight of the first few states of the card model is large, then a true hit has a larger score than false alarms. The putative hit score which is used to detect peaks representing putative hits is generated according to S I = Sk d - S rll . tota eywor J r er (EQ 1) In this equation, Stotal is the putative hit score, Skeyword is the log Viterbi score in the RADIAL BASIS FUNCTIO NODES RAW KEYWORD SCORE • • • • • • • • • • V~TERBI ALIGNMENT Figure 5. State weights and center updates are applied to the state that is matched to each frame in a Viterbi backtrace. 1024 Chang and Lippmann last node of a specific keyword model computed using the Viterbi algorithm from the beginning of the conversation to the frame where the putative hit ended, and S Ii/Ie r is the log Viterbi score in the last node of the filler model. The filler score is used to normalize the keyword score and approximate a posterior probability. The keyword score is calculated using a modified form of the Viterbi algorithm a. (t + 1) = max(a. (t) + a .. , a. 1 (t) + a. .) + d . (t, x) + W .• / / /, / //-1, / / / (EQ2) This equation is identical to the normal Viterbi recursion for left-to-right linear word models after initialization, except the extra state score wi is added. In this equation, a i (t) is the log Viterbi score in node i at time t, aj . is the log of the transition probability from node i to node j ,and dj (t, x) is the log lik~lihood distance score for node i for the input feature vector x at time t . Word scores are computed and a peak-picking algorithm looks for maxima above a low threshold. After a peak representing a putative hit is detected, frames of a putative hit are aligned with the states in the keyword model using the Viterbi backtrace and both the means of Gaussians in each state and state weights of the keyword model are modified. State weights are modified according to (EQ 3) In this equation, Wj (t) is the state weight in node i at time t, gradient is the FOM gradient for the putative hit, llstate is the stepsize for state weight adaptation, and duration is the number of frames aligned to node i . If a true hit occurs, and the gradient is positive, the state weight is increased in proportion to the number of frames assigned to a state. If a false alarm occurs, the state weight is reduced in proportion to the number of frames assigned to a state. The state weight will thus be strongly positive if there are many more frames for a true hit that for a false alarm. It will be strongly negative if there are more frames for a false alarm than for a true hit. High state weight values should thus improve discrimination between true hits and false alarms. The center of the Gaussian components within each node, which are similar to Gaussians in radial basis function networks, are modified according to x.(t) -m .. (t) m .. (t+ 1) = miJ.(t) +gradientxllcenterX J /J V a .. /J (EQ4) In this equation, m j . (t) is the j th component of the mean vector for a Gaussian hidden node in HMM state 1 at time t, gradient is the FOM gradient, llcenter is the stepsize for moving Gaussian centers, x· (t) is the value of the j th component of the input feature vector at time t, and a j . is the'standard deviation of the j th component of the Gaussian hidden node in HMM st~te i . For each true hit, the centers of Gaussian hidden nodes in a state move toward the observation vectors of frames assigned to a particular state. For a false alarm, the centers move away from the observation vectors that are assigned to a particular state. Over time, the centers move closer to the true hit observation vectors and further away from false alarm observation vectors. Figure of Merit Training for Detection and Spotting 1025 0.95 ,---------------------., 0.9 0.85 FEMALE TRAIN 0.8 \--_,,-r 0.75 FOM 0.7 0.65 L---~~~~------~ 0.6 MALE TEST 0.55 0.5 L..-_.....J-__ .l.....-_--'--__ .J...-_---'-__ ...L...-_--'-_-----' o 20 40 60 80 100 120 140 160 NUMBER OF CONVERSATIONS Figure 6. Change in FOM vs. the number of conversations that the models have been trained with. There were 25 male training conversations and 23 female training conversations. 5 EXPERIMENTAL RESULTS Experiments were performed using a HMM wordspotter that was trained using maximum likelihood algorithm. More complicated models were created for words which occur frequently in the training set. The word models for card and credit-card were increased to four mixtures per state. The models for cash, charge, check, credit, dollar, interest, money, month, and visa were increased to two mixtures per state. All other word models had one mixture per state. The number of states per keyword is roughly 1.5 times the number of phonemes in each keyword. Covariance matrices were diagonal and variances were estimated separately for all states. All systems were trained on the first 50 talkers in the credit card training corpus and evaluated using the last 20 talkers. An initial set of models was trained during 16 passes through the training data using wholeword training and Viterbi alignment on only the excised words from the training conversations. This training provided a FOM of 62.5% on the 20 evaluation talkers. Embedded forward-backward reestimation training was then performed where models of keywords and fillers are linked together and trained jointly on conversations which were split up into sentence-length fragments. This second stage ofHMM training increased the FOM by two percentage points to 64.5%. The detection rate curves of these systems are shown in Figure 2. FOM training was then performed for six passes through the training data. On each pass, conversations were presented in a new random order. The change in FOM for the training set and the evaluation set is shown in Figure 6. The FOM on the training data for both male and female talkers increased by more than 10 percentage points after roughly 50 conversations had been presented. The FOM on the evaluation data increased by 5.2 percentage points to 69.7% after three passes through the training data, but then decreased with further training. This result suggests that the extra structure learned during the final three training passes is overfitting the training data and providing poor performance on the evaluation set. Figure 7 shows the spectrograms of high scoring true hits and false alarms for the word card generated by our wordspotter. All false alarms shown are actually the occurrences of the word car. The spectrograms of the true hits and the false alarms are very similar and the actual excised speech segments are difficult even for humans to distinguish. 1026 Chang and Lippmann A) True hits for card Figure 7. Spectrograms of high scoring true hit and false alarm for the word card. 6 SUMMARY Detection of target signals embedded in a noisy background is a common and difficult problem distinct from the task of classification. The evaluation metric of a spotting system, called Figure of Merit (FOM), is also different from the classification accuracy used to evaluate classification systems. FOM training uses a gradient which directly reflects a putative hit's impact on the FOM to modify the parameters of the spotting system. FOM training does not require careful adjustment of thresholds and target values and has been applied to improve a wordspotter's FOM from 64.5% to 69.7% on the credit card database. POM training can also be applied to other spotting tasks such as arrhythmia detection and address block location. ACKNOWLEDGEMENT This work was sponsored by the Advanced Research Projects Agency. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government. Portions of this work used the HTK Toolkit developed by Dr. Steve Young of Cambridge University. BIBLIOGRAPHY R. Lippmann & E. Singer. (1993) Hybrid HMM/Neural-NetworkApproaches to Wordspotting. In ICASSP '93, volume I, pages 565-568. J. Rohlicek et. al. (1993) Phonetic and Language Modeling for Wordspotting. In ICASSP '93, volume II, pages 459-462. T. Zeppenfeld, R. Houghton & A. Waibel. (1993) Improving the MS-TDNN for Word Spotting. In ICASSP '93, volume II, pages 475-478.	1993	94
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853	Autoencoders, Minimum Description Length and Helmholtz Free Energy Geoffrey E. Hinton Department of Computer Science University of Toronto 6 King's College Road Toronto M5S lA4, Canada Richard S. Zemel Computational Neuroscience Laboratory The Salk Institute 10010 North Torrey Pines Road La Jolla, CA 92037 Abstract An autoencoder network uses a set of recognition weights to convert an input vector into a code vector. It then uses a set of generative weights to convert the code vector into an approximate reconstruction of the input vector. We derive an objective function for training autoencoders based on the Minimum Description Length (MDL) principle. The aim is to minimize the information required to describe both the code vector and the reconstruction error. We show that this information is minimized by choosing code vectors stochastically according to a Boltzmann distribution, where the generative weights define the energy of each possible code vector given the input vector. Unfortunately, if the code vectors use distributed representations, it is exponentially expensive to compute this Boltzmann distribution because it involves all possible code vectors. We show that the recognition weights of an autoencoder can be used to compute an approximation to the Boltzmann distribution and that this approximation gives an upper bound on the description length. Even when this bound is poor, it can be used as a Lyapunov function for learning both the generative and the recognition weights. We demonstrate that this approach can be used to learn factorial codes. 1 INTRODUCTION Many of the unsupervised learning algorithms that have been suggested for neural networks can be seen as variations on two basic methods: Principal Components Analysis (PCA) 3 4 Hinton and Zemel and Vector Quantization (VQ) which is also called clustering or competitive learning. Both of these algorithms can be implemented simply within the autoencoder framework (Baldi and Hornik, 1989; Hinton, 1989) which suggests that this framework may also include other algorithms that combine aspects of both. VQ is powerful because it uses a very non-linear mapping from the input vector to the code but weak because the code is a purely local representation. Conversely, PCA is weak because the mapping is linear but powerful because the code is a distributed, factorial representation. We describe a new objective function for training autoencoders that allows them to discover non-linear, factorial representations. 2 THE MINIMUM DESCRIPfION LENGTH APPROACH One method of deriving a cost function for the activities of the hidden units in an autoencoder is to apply the Minimum Description Length (MDL) principle (Rissanen 1989). We imagine a communication game in which a sender observes an ensemble of training vectors and must then communicate these vectors to a receiver. For our purposes, the sender can wait until all of the input vectors have been observed before communicating any of them - an online method is not required. Assuming that the components of the vectors are finely quantized we can ask how many bits must be communicated to allow the receiver to reconstruct the input vectors perfectly. Perhaps the simplest method of communicating the vectors would be to send each component of each vector separately. Even this simple method requires some further specification before we can count the number of bits required. To send the value, Xi,c, of component i of input vector c we must encode this value as a bit string. If the sender and the receiver have already agreed on a probability distribution that assigns a probability p( x) to each possible quantized value, x, Shannon's coding theorem implies that x can be communicated at a cost that is bounded below by -log p( x) bits. Moreover, by using block coding techniques we can get arbitrarily close to this bound so we shall treat it as the true cost. For coding real values to within a quantization width of t it is often convenient to assume a Gaussian probability distribution with mean zero and standard deviation (1'. Provided that (1' is large compared with t, the cost of coding the value x is then -logt + 0.5 log 21r(1'2 + x 2 /2(1'2. This simple method of communicating the trainjng vectors is generally very wasteful. If the components of a vector are correlated it is generally more efficient to convert the input vector into some other representation before communicating it. The essence of the MDL principle is that the best model of the data is the one that minimizes the total number of bits required to communicate it, including the bits required to describe the coding scheme. For an autoencoder it is convenient to divide the total description length into three terms. An input vector is communicated to the receiver by sending the activities of the hidden units and the residual differences between the true input vector and the one that can be reconstructed from the hidden activities. There is a code cost for the hidden activities and a reconstruction cost for the residual differences. In addition there is a one-time model cost for communicating the weights that are required to convert the hidden activities into the output of the net. This model cost is generally very important within the MDL framework, but in this paper we will ignore it. In effect, we are considering the limit in which there is so much data that this limited model cost is negligible. PCA can be viewed as a special case of MDL in which we ignore the model cost and we limit the code cost by only using m hidden units. The question of how many bits are required Autoencoders, Minimum Description Length, and Helmhotz Free Energy 5 to code each hidden unit activity is also ignored. Thus the only remaining term is the reconstruction cost. Assuming that the residual differences are encoded using a zero-mean Gaussian with the same predetermined variance for each component, the reconstruction cost is minimized by minimizing the squared differences. Similarly, VQ is a version of MDL in which we limit the code cost to at most log m bits by using only m winner-lake-all hidden units, we ignore the model cost, and we minimize the reconstruction cost. In standard VQ we assume that each input vector is converted into a specific code. Surprisingly, it is more efficient to choose the codes stochastically so that the very same input vector is sometimes communicated using one code and sometimes using another. This type of "stochastic VQ" is exactly equivalent to maximizing the log probability of the data under a mixture of Gaussians model. Each code of the VQ then corresponds to the mean of a Gaussian and the probability of picking the code is the posterior probability of the input vector under that Gaussian. Since this derivation of the mixture of Gaussians model is crucial to the new techniques described later, we shall describe it in some detail. 2.1 The "bits-back" argument The description length of an input vector using a particular code is the sum of the code cost and reconstruction cost. We define this to be the energy of the code, for reasons that will become clear later. Given an input vector, we define the energy of a code to be the sum of the code cost and the reconstruction cost. If the prior probability of code i is 1f'i and its squared reconstruction error is d; the energy of the code is k d2 Ei = -log 1f'i - k log t + "2 log 21f'0'2 + 20'2 (1) where k is the dimensionality of the input vector, 0'2 is the variance of the fixed Gaussian used for encoding the reconstruction errors and t is the quantization width. Now consider the following situation: We have fitted a VQ to some training data and, for a particular input vector, two of the codes are equally good in the sense that they have equal energies. In a standard VQ we would gain no advantage from the fact that there are two equally good codes. However, the fact that we have a choice of two codes should be worth something. It does not matter which code we use so if we are vague about the choice of code we should be able to save one bit when communicating the code. To make this argument precise consider the following communication game: The sender is already communicating a large number of random bits to the receiver, and we want to compute the additional cost of communicating some input vectors. For each input vector we have a number of alternative codes h1 ... hi ... hm and each code has an energy, Ei. In a standard VQ we would pick the code, j, with the lowest energy. But suppose we pick code i with a probability Pi that depends on Ei. Our expected cost then appears to be higher since we sometimes pick codes that do not have the minimum value of E. < Cost >= LPiEi i (2) where < ... > is used to denote an expected value. However, the sender can use her freedom of choice in stochastically picking codes to communicate some of the random 6 Hinton and Zemel bits that need to be communicated anyway. It is easy to see how random bits can be used to stochastically choose a code, but it is less obvious how these bits can be recovered by the receiver, because he is only sent the chosen code and does not know the probability distribution from which it was picked. This distribution depends on the particular input vector that is being communicated. To recover the random bits, the receiver waits until all of the training vectors have been communicated losslessly and then runs exactly the same learning algorithm as the sender used. This allows the receiver to recover the recognition weights that are used to convert input vectors into codes, even though the only weights that are explicitly communicated from the sender to the receiver are the generative weights that convert codes into approximate reconstructions of the input. After learning the recognition weights, the receiver can reconstruct the probability distribution from which each code was stochastically picked because the input vector has already been communicated. Since he also knows which code was chosen, he can figure out the random bits that were used to do the picking. The expected number of random bits required to pick a code stochastically is simply the entropy of the probability distribution over codes H = - LPi logpi (3) So, allowing for the fact that these random bits have been successfully communicated, the true expected combined cost is (4) Note that F has exactly the form of Helmholtz free energy. It can be shown that the probability distribution which minimizes F is e-E ; Pi = Lj e-Ej (5) This is exactly the posterior probability distribution obtained when fitting a mixture of Gaussians to an input vector. The idea that a stochastic choice of codes is more efficient than just choosing the code with the smallest value of E is an example of the concept of stochastic complexity (Rissanen, 1989) and can also be derived in other ways. The concept of stochastic complexity is unnecessarily complicated if we are only interested in fitting a mixture of Gaussians. Instead of thinking in terms of a stochastically chosen code plus a reconstruction error, we can simply use Shannon's coding theorem directly by assuming that we code the input vectors using the mixture of Gaussians probability distribution. However, when we start using more complicated coding schemes in which the input is reconstructed from the activities of several different hidden units, the formulation in terms of F is much easier to work with because it liberates us from the constraint that the probability distribution over codes must be the optimal one. There is generally no efficient way of computing the optimal distribution, but it is nevertheless possible to use F with a suboptimal distribution as a Lyapunov function for learning (Neal and Hinton, 1993). In MDL terms we are simply using a suboptimal coding scheme in order to make the computation tractable. One particular class of suboptimal distributions is very attractive for computational reasons. In a factorial distribution the probability distribution over m d alternatives factors into d independent distributions over m alternatives. Because they can be represented compactly, Autoencoders, Minimum Description Length, and Helmhotz Free Energy 7 factorial distributions can be computed conveniently by a non-stochastic feed-forward recognition network. 3 FACTORIAL STOCHASTIC VECTOR QUANTIZATION Instead of coding the input vector by a single, stochastically chosen hidden unit, we could use several different pools of hidden units and stochastically pick one unit in each pool. All of the selected units within this distributed representation are then used to reconstruct the input. This amounts to using several different VQs which cooperate to reconstruct the input. Each VQ can be viewed as a dimension and the chosen unit within the VQ is the value on that dimension. The number of possible distributed codes is m d where d is the number of VQs and m is the number of units within a VQ. The weights from the hidden units to the output units determine what output is produced by each possible distributed code. Once these weights are fixed, they determine the reconstruction error that would be caused by using a particular distributed code. If the prior probabilities of each code are also fixed, Eq. 5 defines the optimal probability distribution over distributed codes, where the index i now ranges over the m d possible codes. Computing the correct distribution requires an amount of work that is exponential in d, so we restrict ourselves to the suboptimal distributions that can be factored into d independent distributions, one for each VQ. The fact that the correct distribution is not really factorial will not lead to major problems as it does in mean field approximations of Boltzmann machines (Galland, 1993). It will simply lead to an overestimate of the description length but this overestimate can still be used as a bound when learning the weights. Also the excess bits caused by the non-independence will force the generative weights towards values that cause the correct distribution to be approximately factorial. 3.1 Computing the Expected Reconstruction Error To perform gradient descent in the description length given in Eq. 4, it is necessary to compute, for each training example, the derivative of the expected reconstruction cost with respect to the activation probability of each hidden unit. An obvious way to approximate this derivative is to use Monte Carlo simulations in which we stochastically pick one hidden unit in each pool. This way of computing derivatives is faithful to the underlying stochastic model, but it is inevitably either slow or inaccurate. Fortunately, it can be replaced by a fast exact method when the output units are linear and there is a squared error measure for the reconstruction. Given the probability, hi, of picking hidden unit i in VQ v, we can compute the expected reconstructed output Yj for output unit j on a given training case (6) where bj is the bias of unit j and wji is the generative weight from ito j in VQ v. We can also compute the variance in the reconstructed output caused by the stochastic choices within the VQs. Under the assumption that the stochastic choices within different VQs are independent, the variances contributed by the different VQs can simply be added. (7) 8 Hinton and Zemel The expected squared reconstruction error for each output unit is Vi + (Yj - dj )2 where dj is the desired output. So if the reconstruction error is coded assuming a zero-mean Gaussian distribution the expected reconstruction cost can be computed exactlyl. It is therefore straightforward to compute the derivatives, with respect to any weight in the network, of all the terms in Eq. 4. 4 AN EXAMPLE OF FACTORIAL VECTOR QUANTIZATION Zemel (1993) presents several different data sets for which factorial vector quantization (FVQ) produces efficient encodings. We briefly describe one of those examples. The data set consists of 200 images of simple curves as shown in figure 1. A network containing 4 VQs, each containing 6 hidden units, is trained on this data set. After training, the final outgoing weights for the hidden units are as shown in figure 2. Each VQ has learned to represent the height of the spline segment that connects a pair of control points. By chaining these four segments together the image can be reconstructed fairly accurately. For new images generated in the same way, the description length is approximately 18 bits for the reconstruction cost and 7 bits for the code. By contrast, a stochastic vector quantizer with 24 hidden units in a single competing group has a reconstruction cost of 36 bits and a code cost of 4 bits. A set of 4 separate stochastic VQs each of which is trained on a different 8x3 vertical slice of the image also does slightly worse than the factorial VQ (by 5 bits) because it cannot smoothly blend the separate segments of the curve together. A purely linear network with 24 hidden units that performs a version of principal components analysis has a slightly lower reconstruction cost but a much higher code cost. Random y Positions Fixed x Positions -------> Figure 1: Each image in the spline dataset is generated by fitting a spline to 5 control points with randomly chosen y-positions. An image is formed by blurring the spline with a Gaussian. The intensity of each pixel is indicated by the area of white in the display. The resulting images are 8x12 pixels, but have only 5 underlying degrees of freedom. 1 Each VQ contributes non-Gaussian noise and the combined noise is also non-Gaussian. But since its variance is known, the expected cost of coding the reconstruction error using a Gaussian prior can be computed exactly. The fact that this prior is not ideal simply means that the computed reconstruction cost is an upper bound on the cost using a better prior. Autoencoders, Minimum Description Length, and Helmhotz Free Energy 9 "" -':' ". I 1""-: •• :.> : :] ':':" .. "~--~ .. Figure 2: The outgoing weights of the hidden units for a network containing 4 VQs with 6 units in each, trained on the spline dataset. Each 8x 12 weight block corresponds to a single unit, and each row of these blocks corresponds to one VQ. 5 DISCUSSION A natural approach to unsupervised learning is to use a generative model that defines a probability distribution over observable vectors. The obvious maximum likelihood learning procedure is then to adjust the parameters of the model so as to maximize the sum of the log probabilities of a set of observed vectors. This approach works very well for generative models, such as a mixture of Gaussians, in which it is tractable to compute the expectations that are required for the application of the EM algorithm. It can also be applied to the wider class of models in which it is tractable to compute the derivatives of the log probability of the data with respect to each model parameter. However. for non-linear models that use distributed codes it is usually intractable to compute these derivatives since they require that we integrate over all of the exponentially many codes that could have been used to generate each particular observed vector. The MDL principle suggest a way of making learning tractable in these more complicated generative models. The optimal way to code an observed vector is to use the correct posterior probability distribution over codes given the current model parameters. However, we are free to use a suboptimal probability distribution that is easier to compute. The description length using this suboptimal method can still be used as a Lyapunov function for learning the model parameters because it is an upper bound on the optimal description length. The excess description length caused by using the wrong distribution has the form of a Kullback-Liebler distance and acts as a penalty term that encourages the recognition weights to approximate the correct distribution as well as possible. There is an interesting relationship to statistical physics. Given an input vector, each possible code acts like an alternative configuration of a physical system. The function 10 Hinton and Zemel E defined in Eq. 1 is the energy of this configuration. The function F in Eq. 4 is the Helmholtz free energy which is minimized by the thermal equilibrium or Boltzmann distribution. The probability assigned to each code at this minimum is exactly its posterior probability given the parameters of the generative model. The difficulty of performing maximum likelihood learning corresponds to the difficulty of computing properties of the equilibrium distribution. Learning is much more tractable if we use the non-equilibrium Helmholtz free energy as a Lyapunov function (Neal and Hinton, 1993). We can then use the recognition weights of an autoencoder to compute some non-equilibrium distribution. The derivatives of F encourage the recognition weights to approximate the equilibrium distribution as well as they can, but we do not need to reach the equilibrium distribution before adjusting the generative weights that define the energy function of the analogous physical system. In this paper we have shown that an autoencoder network can learn factorial codes by using non-equilibrium Helmholtz free energy as an objective function. In related work (Zemel and Hinton 1994) we apply the same approach to learning population codes. We anticipate that the general approach described here will be useful for a wide variety of complicated generative models. It may even be relevant for gradient descent learning in situations where the model is so complicated that it is seldom feasible to consider more than one or two of the innumerable ways in which the model could generate each observation. Acknowledgements This research was supported by grants from the Ontario Information Technology Research Center, the Institute for Robotics and Intelligent Systems, and NSERC. Geoffrey Hinton is the Noranda Fellow of the Canadian Institute for Advanced Research. We thank Peter Dayan, Yann Le Cun, Radford Neal and Chris Williams for helpful discussions. References Baldi, P. and Hornik, K. (1989) Neural networks and principal components analysis: Learning from examples without local minima. Neural Networks, 2, 53-58. Galland, C. C. (1993) The limitations of deterministic Boltzmann machine learning. Network, 4, 355-379. Hinton, G. E. (1989) Connectionist learning procedures. Artificial Intelligence, 40, 185234. Neal, R., and Hinton. G. E. (1993) A new view of the EM algorithm that justifies incremental and other variants. Manuscript available/rom the authors. Rissanen.1. ( 1989) Stochastic Complexity in Statistical Inquiry. World Scientific Publishing Co .• Singapore. Zemel. R. S. (1993) A Minimum Description Length Framework/or Unsupervised Learning. PhD. Thesis. Department of Computer Science, University of Toronto. Zemel, R. S. and Hinton. G. E. (1994) Developing Population Codes by Minimizing Description Length. In I. Cowan, G. Tesauro. and I. Alspector (Eds.), Advances in Neural In/ormation Processing Systems 6, San Mateo, CA: Morgan Kaufmann.	1993	96
854	Dual Mechanisms for Neural Binding and Segmentation Paul Sajda and Leif H. Finkel Department of Bioengineering and Institute of Neurological Science University of Pennsylvania 220 South 33rd Street Philadelphia, PA. 19104-6392 Abstract We propose that the binding and segmentation of visual features is mediated by two complementary mechanisms; a low resolution, spatial-based, resource-free process and a high resolution, temporal-based, resource-limited process. In the visual cortex, the former depends upon the orderly topographic organization in striate and extrastriate areas while the latter may be related to observed temporal relationships between neuronal activities. Computer simulations illustrate the role the two mechanisms play in figure/ ground discrimination, depth-from-occlusion, and the vividness of perceptual completion. 1 COMPLEMENTARY BINDING MECHANISMS The "binding problem" is a classic problem in computational neuroscience which considers how neuronal activities are grouped to create mental representations. For the case of visual processing, the binding of neuronal activities requires a mechanism for selectively grouping fragmented visual features in order to construct the coherent representations (i.e. objects) which we perceive. In this paper we argue for the existence of two complementary mechanisms for neural binding, and we show how such mechanisms may operate in the constructiO:l of intermediate-level visual representations. 993 994 Sajda and Finkel Ordered cortical topography has been found in both striate and extrastriate areas and is believed to be a fundamental organizational principle of visual cortex. One functional role for this topographic mapping may be to facilitate a spatial-based binding system. For example, active neurons or neural populations within a cortical area could be grouped together based on topographic proximity while those in different areas could be grouped if they lie in rough topographic register. An advantage of this scheme is that it can be carried out in parallel across the visual field. However, a spatial-based mechanism will tend to bind overlapping or occluded objects which should otherwise be segmented. An alternative binding mechanism is therefore necessary for binding and segmenting overlapping objects and surfaces. Temporal binding is a second type of neural binding. Temporal binding differs from spatial binding in two essential ways; 1) it operates at high spatial resolutions and 2) it binds and segments in the temporal domain, allowing for the coexistence of multiple objects in the same topographic region. Others have proposed that temporal events, such as phase-locked firing or I oscillations may play a role in neural binding (von der Malsburg, 1981; Gray and Singer, 1989, Crick and Koch, 1990). For purposes of this discussion, we do not consider the specific nature of the temporal events underlying neural binding, only that the binding itself is temporally dependent. The disadvantage of operating in the temporal domain is that the biophysical properties of cortical neurons (e.g. membrane time constants) forces this processing to be resource-limited-only a small number of objects or surfaces can be bound and segmented simultaneously. 2 COMPUTING INTERMEDIATE-LEVEL VISUAL REPRESENTATIONS: DIRECTION OF FIGURE We consider how these two classes of binding can be used to compute contextdependent (non-local) characteristics about the visual scene. An example of a context-dependent scene characteristic is contour ownership or direction of figure. Direction of figure is a useful intermediate-level visual representation since it can be used to organize an image into a perceptual scene (e.g. infer relative depth and link segregated features). Figure lA illustrates the relationship between contours and surfaces implied by direction of figure. We describe a model which utilizes both spatial and temporal binding to compute direction of figure (DOF). Prior to computing the DOF, the surface contours in the image are extracted. These contours are then temporally bound by a process we call "contour binding" (Finkel and Sajda, 1992). In the model, the temporal properties of the units are represented by a temporal binding value. We will not consider the details of this process except to say that units with similar temporal binding values are bound together while those with different values are segmented. In vivo, this temporal binding value may be represented by phase of neural firing, oscillation frequency, or some other specific temporal property of neuronal activity. The DOF is computed by circuitry which is organized in a columnar structure, shown in figure 2A. There are two primary circuits which operate to compute the direction of figure; one being a temporal-dependent/spatial-independent (TDSI) circuit selective to "closure", the other a spatial-dependent/temporal-independent A Dual Mechanisms for Neural Binding and Segmentation 995 closure concavities similarity & proximity , f ~\ / .... -B /" 1',,I ' direction of line endings Figure 1: A Direction of figure as a surface representation. At point (1) the contour belongs to the surface contour of region A and therefore A owns the contour. This relationship is represented locally as a "direction of figure" vector pointing toward region A. Additional ownership relationships are shown for points (2) and (3). B Cues used in determining direction of figure . (SDTI) circuit selective to "similarity and proximity". There are also two secondary circuits which playa transient role in determining direction of figure. One is based on the observation that concave segments bounded by discontinuities are a cue for occlusion and ownership, while the other considers the direction of line endings as a potential cue. Figure IB summarizes the cues used to determine direction of figure. In this paper, we focus on the TDSI and SDTI circuits since they best illustrate the nature of the dual binding mechanisms. The perceptual consequences of attributing closure discrimination to temporal binding and similarity/proximity to spatial binding is illustrated in figure 3. 2.1 TDSI CIRCUIT Figure 2B(i) shows the neural architecture of the TDSI mechanism. The activity of the TDSI circuit selective for a direction of figure a is computed by comparing the amount of closure on either side of a contour. Closure is computed by summing the temporal dependent inputs over all directions i; T DSl Oi = [~Sf(ti) - L Sf- 180o (td] 1 I I 0 (1) The brackets ([]) indicate an implicit thresholding (if x < 0 then [xl = 0, otherwise [xl = x) and Si(ti) and sf- 180° (td are the temporal dependent inputs, computed as; snt;) = { : { (Sj > ST) if and ((ti ~t) < tj < (ti + ~t)) otherwise (2) 996 Sajda and Finkel t , to/from contour binding A .... toIfrom ~ other DOFcolumns TOSI son ex - 1800 (i) (ii) B Figure 2: A Divisions, inputs, and outputs for a DOF column. B The two primary circuits operating to compute direction of figure. (i) Top view of temporaldependent/spatial-independent (TDSI) circuit architecture. Filled square represents position of a specific column in the network. Unfilled squares represent other DOF columns serving as input to this column. Bold curve corresponds to a surface contour in the input. Shown is the pattern of long-range horizontal connections converging on the right side of the column (side ex). (ii) Top view of spatialdependent/temporal-independent (SDTI) circuit architecture. Shown is the pattern of connections converging on the right side of the column (side ex). where ex and ex - 1800 represent the regions on either side of the contour, Sj is the activation of a unit along the direction i (For simulations i varies between 00 and 315 0 by increments of 45 0 ), 6.t determines the range of temporal binding values over which the column will integrate input, and ST is the activation threshold. The temporal dependence of this circuit implies that only those DOF columns having the same temporal binding value affect the closure computation. 2.2 SDTI CIRCUIT Figure 2B(ii) illustrates the neural architecture of the SDTI mechanism. The SDTI circuit organizes elements in the scene based on "proximity" and "similarity" of orientation. Unlike the TDSI circuit which depends upon temporal binding, the SDTI circuit uses spatial binding to access information across the network. Activity is integrated from units with similar orientation tuning which lie in a direction orthogonal to the contour (i.e. from parallel line segments). The activity of the SDTI circuit selective for a direction of figure ex is computed by comparing input from similar orientations on either side of a contour; SDTl Oi = _1_ (2: sf(Od - 2: sr- 180o (Od) Smax . . t I (3) where Smax is a constant for normalizing the SDTI activity between 0 and 1 and sf (OJ) and sf- 1800 (Od are spatial dependent inputs selective for an orientation 0, Dual Mechanisms for Neural Binding and Segmentation 997 A B Figure 3: A The model predicts that a closed figure could not be discriminated in parallel search since its detection depends on resource-limited temporal binding. B Conversely, proximal parallel segments are predicted to be discriminated in parallel search due to resource-free spatial binding. computed as; (4) where ex and ex - 180° represent the regions on either side of the contour, () is the orientation of the contour, i is the direction from which the unit receives input, Cij is the connection strength (Cij falls off as a gaussian with distance), and Sj(x, y, (}j) is the activation of a unit along the direction i which is mapped to retinotopic location (x, y) and selective for an orientation (}j (For simulations i varies between the following three angles; 1- (}i,1- ((}i -45°), 1- ((}i +45°)). Since the efficacy of the connections, Cij, decrease with distance, columns which are further apart are less likely to be bound together. Neighboring parallel contours generate the greatest activation and the circuit tends to discriminate the region between the two parallel contours as the figure. 2.3 COMPUTED DOF The activity of a direction of figure unit representing a direction ex is given by the sum of the four components; DOF Ci = C1(TDSl Ci ) + C2(SDTrl:) + C3(CON Ci ) + C4(DLE Ci ) (5) where the constants define the contribution of each cue to the computed DOF. Note that in this paper we have not considered the mechanisms for computing the DOF given the two secondary cues (concavities (CO N Ci ) and direction of line endings (DLE Ci )) . The DOF activation is computed for all directions ex (For simulations ex varies between 0° and 315° by increments of 45°) with the direction producing the largest activation representing the direction of figure. 3 SIMULATION RESULTS The following are simulations illustrating the role the dual binding mechanisms play in perceptual organization. All simulations were carried out using the NEXUS Neural Simulation Environment (Sajda and Finkel, 1992). 998 Sajda and Finkel 100 ~ >l &. ~ '" "C u: '" 60 X 60 0 A B C Figure 4: A 128x128 pixel grayscale image. B Direction of figure computed by the network. Direction of figure is shown as an oriented arrowhead, where the orientation represents the preferred direction of the DOF unit which is most active. C Depth of surfaces. Direction of figure relationships (such as those in the inset of B) are used to infer relative depth. Plot shows % activity of units in the foreground network- higher activity implies that the surface is closer to the viewer. 3.1 FIGURE/GROUND AND DEPTH-FROM-OCCLUSION Figure 4A is a grayscale image used as input to the network. Figure 4B shows the direction of figure computed by the model. Note that though the surface contours are incomplete, the model is still able to characterize the direction of figure and distinguish figure/ground over most of the contour. This is in contrast to models proposing diffusion-like mechanisms for determining figure/ground relationships which tend to fail if complete contour closure is not realized. The model utilizes direction of figure to determine occlusion relationships and stratify objects in relative depth, results shown in figure 4C. This method of inferring the relative depth of surfaces given occlusion is in contrast to traditional approaches utilizing T-junctions. The obvious advantage of using direction of figure is that it is a context-dependent feature directly linked to the representation of surfaces. 3.2 VIVIDNESS OF PERCEPTUAL COMPLETION Our previous work (Finkel and Sajda, 1992) has shown that direction of figure is important for completion phenomena, such as the construction of illusory contours and surfaces. More interestingly, our model offers an explanation for differences in perceived vividness between different inducing stimuli. For example, subjects tend to rank the vividness of the illusory figures in figure 5 from left to right, with the figure on the left being the most vivid and that on the right the least. Our model accounts for this effect in terms of the magnitude of the direction of figure along the illusory contour. Figure 6 shows the individual components contributing to the direction of figure. For a typical inducer, such as the pacman in figure 6, the TDSI and SDTI circuits tend to force the direction of figure of the L-shaped segment to region 1 while the concavity/convexity transformation tries to force the direction of figure of the segment to be toward region 2. This transformation transiently overwhelms the TDSI and SDTI responses, so that the direction of figure of the L-shaped segment is toward region 2. However, the TDSI and SDTI activation will affect the magnitude of the direction of figure, as shown in figure 7. For example, A Dual Mechanisms for Neural Binding and Segmentation 999 r .. .... B r-., L..I c Figure 5: Illusory contour vividness as a function of inducer shape. Three types of inducers are arranged to generate an illusory square. A pacman inducer, B thick L inducer and C thin L inducer. Subjects rank the vividness of the illusory squares from left to right ((A) > (B) > (C)). C?C?Q TDSI component SDn component concavity/convexity component Figure 6: Processes contributing to the direction of figure of the L-shaped contour segment. The TDSI and SDTI circuits assign the contour to region 1, while the change of the concavity to a convexity assigns the segment to region 2. :, ':::::::::;:::::::1 ,. :~ ii ." ... " E ii Ii :: +E •• ...... .5 ii Itll' .' f!. i -:.: .14 ". r r A B Figure 7: A Activity of SDTI units for the upper left inducer of each stimulus, where the area of each square is proportional to unit activity. The SDTI units try to assign the L-shaped segment to the region of the pacman. Numerical values indicates the magnitude of the SDTI effect. B Magnitude of direction of figure along the L-shaped segment as a function of inducer shape. The direction of figure in all cases is toward the region of the illusory square. 1000 Sajda and Finkel the weaker the activation of the TDSI and SDTI circuits, the stronger the activation of the DOF units assigning the L-shaped segment to region 2. Referring back to the inducer types in figure 5, one can see that though the TDSI component is the same for all three inducers (i.e. all three generate the same amount of closure) the SDTI contribution differs, shown quantitatively in figure 7 A. The contribution of the SDTI circuit is greatest for the thin L inducers and least for the pacmen inducers-the L-shaped segments for the pacman stimulus are more strongly owned by the surface of the illusory square than those for the thin L inducer. This is illustrated in figure 7B, a plot of the magnitude of the direction of figure for each inducer configuration. This result can be interpreted as the model's ordering of perceived vividness, which is consistent with that of human observers. 4 CONCLUSION In this paper we have argued for the utility of binding neural activities in both the spatial and temporal domains. We have shown that a scheme consisting of these complementary mechanisms can be used to compute context-dependent scene characteristics, such as direction of figure. Finally, we have illustrated with computer simulations the role these dual binding mechanisms play in accounting for aspects of figure/ ground perception, depth-from-occlusion, and perceptual vividness of illusory contours and surfaces. It is interesting to speculate on the relationship between these complementary binding mechanisms and the traditional distinction between preattentive and attentional perception. Acknowledgements This work is supported by grants from ONR (N00014-90-J-1864, N00014-93-1-0681), The Whitaker Foundation, and The McDonnell-Pew Program in Cognitive NeuroSCIence. References F. Crick and C. Koch. Towards a neurobiological theory of consciousness. Seminars in Neuroscience, 2:263-275, 1990. L.H. Finkel and P. Sajda. Object discrimination based on depth-from-occlusion. Neural Computation, 4(6):901-921,1992. C. M. Gray and W. Singer. Neuronal oscillations in orientation columns of cat visual cortex. Proceedings of the National Academy of Science USA, 86:1698-1702, 1989. P. Sajda and L. Finkel. NEXUS: A simulation environment for large-scale neural systems. Simulation, 59(6):358-364, 1992. C. von der Malsburg. The correlation theory of brain function. Technical Report Internal Rep. No. 81-2, Max-Plank-Institute for Biophysical Chemistry, Department of Neurobiology, Gottingen, Germany, 1981.	1993	97
855	The Statistical Mechanics of k-Satisfaction Scott Kirkpatrick* Racah Institute for Physics and Center for Neural Computation Hebrew University Jerusalem, 91904 Israel kirk@fiz.huji.ac.il Geza Gyorgyi Institute for Theoretical Physics Eotvos University 1-1088 Puskin u. 5-7 Budapest, Hungary gyorgyi@ludens.elte.hu, N aft ali Tishby and Lidror Troyansky Institute of Computer Science and Center for Neural Computation The Hebrew University of Jerusalem 91904 Jerusalem, Israel {tishby, lidrort }@cs.huji.ac.il Abstract The satisfiability of random CNF formulae with precisely k variables per clause ("k-SAT") is a popular testbed for the performance of search algorithms. Formulae have M clauses from N variables, randomly negated, keeping the ratio a = M / N fixed . For k = 2, this model has been proven to have a sharp threshold at a = 1 between formulae which are almost aways satisfiable and formulae which are almost never satisfiable as N --jo 00 . Computer experiments for k = 2, 3, 4, 5 and 6, (carried out in collaboration with B. Selman of ATT Bell Labs). show similar threshold behavior for each value of k. Finite-size scaling, a theory of the critical point phenomena used in statistical physics, is shown to characterize the size dependence near the threshold. Annealed and replica-based mean field theories give a good account of the results. "Permanent address: IBM TJ Watson Research Center, Yorktown Heights, NY 10598 USA. (kirk@watson.ibm.com) Portions of this work were done while visiting the Salk Institute, with support from the McDonnell-Pew Foundation. 439 440 Kirkpatrick, Gyorgyi, Tishby, and Troyansky 1 Large-scale computation without a length scale It is increasingly possible to model the natural world on a computer. Condensed matter physics has strategies to manage the complexities of such calculations, usually depending on a characteristic length. For example, molecules or atoms with finite ranged interactions can be broken down into weakly interacting smaller parts. We may also use symmetry to identify natural modes of the system as a whole. Even in the most difficult case, continuous phase transitions correlated over a wide range of scales, the renormalization group provides a way of collapsing the problem down to its "relevant" parts by providing a generator of behavior on all scales in terms of the critical point itself. But length scales are not much help in organizing another sort of large calculation. Examples include large rule-based "expert systems" that model the particulars of complex industrial processes. Digital Equipment, for example, has used a network of three or more expert systems (originally called "R1/XCON") to check computer orders for completeness and internal consistency, to schedule production and shipping, and to aid a salesman to anticipate customers' needs. This very detailed set of tasks in 1979 required 2 programmers and 250 rules to deal with 100 parts. In the ten years described by Barker (1989), it grew 100X, employing 60 programmers and nearly 20,000 rules to deal with 30,000 part numbers. 100X in ten years is only moderate growth, and it would be valuable to understand how technical, social, and business factors have constrained it. Many important commercial and scientific problems without length scales are ready for attack by computer modelling or automatic classification, and lie within a few decades of XCON's size. Retail industries routinely track 105 - 106 distinct items kept in stock. Banks, credit card companies, and specialized information providers are building models of what 108 Americans have bought and might want to buy next. In biology, human metabolism is currently described in terms of > 1000 substances coupled through> 10,000 reactions, and the data is doubling yearly. Similarly, amino acid sequences are known for> 60,000 proteins. A deeper understanding of the computational cost of these problems of order 106±2 is needed to see which are practical and how they can be simplified. We study an idealization of XC ON-style resolution search, and find obvious collective effects which may be at the heart of its computational complexity. 2 Threshold Phenomena and Random k-SAT Properties of randomly generated combinatorial structures often exhibit sharp threshold phenomena analogous to the phase transitions studied in condensed matter physics. Recently, thresholds have been observed in randomly generated Boolean formulae. Mitchell et al. (1992) consider the k-satisfiability problem (k-SAT). An instance of k-SAT is a Boolean formula in conjunctive normal form (CNF), i.e., a conjunction (logical AND) of disjunctions or clauses (logical ORs), where each disjunction contains exactly k literals. A literal is a Boolean variable or, with equal probability, its negation. The task is to determine whether there is an assignment to the variables such that all clauses evaluate to true. Here, we will use N to denote the number of variables and M for the number of clauses in a formula. The Statistical Mechanics of k-Satisfaction 441 For randomly generated 2-SAT instances, it has been shown analytically that for large N, when the ratio a: = M / N is less than 1 the instances are almost all satisfiable, whereas for ratios larger than 1, almost all instances are unsatisfiable (Chvatal and Reed 1992; Goerdt 1992). For k ~ 3, a rigorous analysis has proven to be elusive. Experimental evidence, however, strongly suggests a threshold with a: ~ 4.3 for 3SAT (Mitchell et al. 1992; Crawford and Auton 1993; Larrabee 1993). One of the main reasons for studying randomly generated 3CNF formulae is for their use in the empirical evaluation of combinatorial search algorithms. 3CNF formulae are good candidates for the evaluation of such algorithms because determining their satisfiability is an NP-complete problem. This also holds for larger values of k. For k = 1 or 2, the satisfiability problem can be solved efficiently (Aspvall et al. 1979). Despite the worst-case complexity, simple heuristic methods can usually determine the satisfiability of random formulae. However, computationally challenging test instances are found by generating formulae at or near the threshold (Mitchell et al. 1992). Cheeseman (1991) has made a similar observation of increased computational cost for heuristic search at a boundary between two distinct phases or behaviors of a combinatorial model. We will provide a precise characterization of the N -dependence of the threshold phenomena for k-SAT with k ranging from 2 to 6. We will employ finite size scaling, a method from statistical physics in which direct observation of the width of the threshold, or "critical region" of a transition is used to characterize the "universal" behavior of quantities across the entire critical region, extending the analysis to combinatorial problems in which N characterizes the size of the model observed. For discussion of the applicability of finite-size scaling to systems without a metric, see Kirkpatrick and Selman (1993). 1. ill O . B ~ ~ '" ~ 0 . 6 ... ill § 0 . 4 ~ 0 g ',j ~ 0 . 2 ~ 0 ,': i .' if 1(/ if II! !i i! !i 11/ '1,1 ~i ii J 0 Thr •• ho~d. rOr 2SAT. 3SAT , 4SAT, 5SAT , and 6SAT 1.0 :1 /'<> / .'" :' .... I: ! / Ii' // // f/ }' ..... ; J.. .. ' ",' 20 30 M I N 40 so Fig. 1: Fraction of unsatisfiable formulae for 2-, 3- 4-, 5- and 6-SAT. 60 442 Kirkpatrick, Gyorgyi, Tishby, and Troyansky 3 Experimental data We have generated extensive data on the satisfiability of randomly generated kCNF formulae with k ranging from 2 to 6. Fig. 1 shows the fraction of random k-SAT formulae that is unsatisfiable as a function of the ratio, a. For example, the left-most curve in Fig. 1 shows the fraction of formulae that is unsatisfiable for random 2CNF formulae with 50 variables over a range of values of a. Each data point was generated using 10000 randomly generated formulae, giving 1 % accuracy. We used a highly optimized implementation of the Davis-Putnam procedure (Crawford and Auton 1993). The procedure works best on formulae with smaller k . Data was obtained for k = 2 on samples with N ~ 500, for k = 3 with N ~ 100, and for k = 5 with N ~ 40, all at comparable computing cost. Fig. 1 (for N ranging from 10 to 50) shows a threshold for each value of k. Except for the case k = 2, the curves cross at a single point and sharpen up with increasing N. For k = 2, the intersections between the curves for the largest values of N seem to be converging to a single point as well, although the curves for smaller N deviate. The point where 50% of the formulae are unsatisfiable is thought to be where the computationally hardest problems are found (Mitchell et al. 1992; Cheeseman et al. 1991). The 50% point lies consistently to the right of the scale-invariant point (the point where the curves cross each other), and shifts with N. There is a simple explanation for the rapid shift of the thresholds to the right with increasing k . The probability that a given clause is satisfied by a random input configuration is (2k 1)/2k = (1 - 2- k) _ 'k. If we treat the clauses as independent, the probability that all clauses are satisfied is ,~ = ,kN . We define the entropy, 5, per in~ut as l/N times the log2 of the expected number of satisfying configurations,2N 'k . 5 = 1 + alog2(,k) = 1- a/aann, and the vanishing of the entropy gives an estimate of the threshold, identical to the upper bound derived by several workers (see Franco (1983) and citations in Chvatal (1992)): aann = -(log2(1 - 2-k))-1 ~ (ln2)2k. This is called an annealed estimate for C¥c, because it ignores the interactions between clauses, just as annealed theories of materials (see Mezard 1986) average over many details of the disorder. We have marked aann with an arrow for each k in the figures, and tabulate it in Table 1. 4 Results of Finite-Size Scaling Analysis From Fig. 1, it is clear that the threshold "sharpens up" for larger values of N. Both the threshold shift and the increasing slope in the curves of Fig. 1 can be accounted for by finite size scaling. (See Stauffer and Aharony (1992) or Kirkpatrick and Swendsen (1985).) We plot the fraction of samples unsatisfied against the dimensionless rescaled variable, y = Nl/V(a - c¥c)/ac . Values for a c and 1I must be derived from the experimental data. First a c is determined as the crossing point of the curves for large N in Fig. 1. Then 1I is determined to make the slopes match up through the critical region. In Fig. 2 (for k = 3) we find that these two parameters capture both the threshold shift and the steepening of the curves, using a c = 4.17 and 1I = 1.5. We see that F, the fraction J i 01 a 01 f '0 Of I 02 • .. .fi' i. _>SAT.,. ",.12 • "=20 • N=24 a N=tO Il N. 50 a. N .. 100 .... . -2 -\ The Statistical Mechanics of k-Satisfaction 443 scakMf CFOuover functton, III SAT modele 2 Y 3 Fig. 2: Rescaled 3-SAT data using a c = 4.17, lJ = 1.5. Fig. 3: Rescaled data for 2-, 3-, 4-, 5-, and 6-SAT approach annealed limit. of unsatisfiable formulae, is given by F(N, a) = I(y) , where the invariant function, I, is that graphed in Fig. 2. A description of the 50% threshold shift follows immediately. If we define y' by I(y') = 0.5, then a50 = a c(1 + y' N- 1/ V ) . From Fig. 2 we find that a50 ~ 4.17 + 3.1N- 2/ 3 . Crawford and Auton (1993) fit their data on the 50% point as a function of N by arbitrarily assuming that the leading correction will be O(I/N) . They obtain a50 = 4.24 + 6/ N. However, the two expressions differ by only a few percent as N ranges from 10 to 00. We also obtained good results in rescaling the data for the other values of k. In Table 1 we give the critical parameters obtained from this analysis. The error bars are subjective, and show the range of each parameter over which the best fits were obtained. Note that v appears to be tending to 1, and aann becomes an increasingly good approximation to a c as k increases. The success of finite-size scaling with different powers, v, is strong evidence for criticality, i.e., diverging correlations, even in the absence of any length. Finally, we found that all the crossovers were similar in shape. In fact, combining the various rescaled curves in figure 3 shows that the curves for k ~ 3 all coincide in the vicinity of the 50% point, and tend to a limiting form, which can be obtained by extending the annealed arguments of the previous section. If we define then the probability that a formula remains unsatisfied for all 2N configurations is The curve for k = 2 is similar in form, but shifted to the right from the other ones. 444 Kirkpatrick, Gyorgyi, Tishby, and Troyansky k O'ann 0'2 O'c 0" V 2 2.41 1.38 1.0 2.25 2.6±.2 3 5.19 4.25 4.17±.03 0.74 1.5±.1 4 10.74 9.58 9.75±.05 0.67 1.25±.05 5 21.83 20.6 20.9±.1 0.71 1.1±.O5 6 44.01 42.8 43.2±.2 0.69 1.05±.05 Table 1: Critical parameters for random k-SAT. 5 Outline of Statistical Mechanics Analysis Space permits only a sketch of our analysis of this model. Since the N inputs are binary, we may represent them as a vector, X, of Ising spins: X={xi=±l} i=l, ... N. Each random formula, F, can be written as a sum of its M clauses, Cj, M F = LCj, j=1 where k Cj = II (1 - Jj 1X)/2. 1=1 where the vector, Jj,l, has only one non-zero element, ±1, at the input which it selects. F evaluates to the number of clauses left unsatisfied by a particular configuration. It is natural to take the value of F to be the energy. The partition function, z = tr{x.}e.6.r = tr{x.} II e.6Cj , j where f3 is the inverse of a fictitious temperature, factors into contributions from each clause. The "annealed" approximation mentioned above consists simply of taking the trace over each subproduct individually, neglecting their interactions. In this construction, we expect both energy and entropy, S, to be extensive quantities, that is, proportional to N. Fig. 4 shows that this is indeed the case for S( a). The lines in Fig. 4 are the annealed predictions S( a, k) = 1 - 0'/ aann. Expressions for the energy can also be obtained from the annealed theory, and used to compare the specific heat observed in numerical experiments with the simple limit in which the clauses do not interact. This gives evidence supporting the identification of the unsatisfied phase as a spin glass. Finally, a plausible phase diagram for the spin glass-like "unsatisfied" phase is obtained by solving for S(T) = 0 at finite temperatures. To perform the averaging over the random clauses correctly requires introducing replicas (see Mezard 1986), which are identical copies of the random formula, and defining q, the overlap between the expectation values of the spins in any two replicas, as the new order parameter. The results appear to be capable of accounting The Statistical Mechanics of k-Satisfaction 445 for the difference between experiment and the annealed predictions at finite k. For example, an uncontrolled approximation in which we consider just two replicas gives the values of a2 in Table 1, and accounts rather closely for the average overlap found experimentally between pairs of lowest energy states, as shown in Fig. 5. The 2-replica theory gives q as the solution of a(k, q) = 2k(1 + q)k-l(4k - 2k+l + (1 - ql)/ln«l + q)(l - q)) for q as a function of a. This gives the lines in Fig 5. We defined a2 (in Table 1) as the point of inflection, or the maximum in the slope of q(a). Entropy tor It- SAT. l = 2. 3, t . S o . 0 ' o. o 1 10 15 u t loO H/ N ' l i.frlk.ll'Sp·' n6k2 p' • ' n2 f.k2 p'_ ' nlO,p' D · nH .p· ....... ' n12U p2' • 'n20kf, p ' ..-....t ' nlOkS p' • ' n20kS p'25 30 07 0.1 0.5 ~ 04 03 02 01 Fig. 4: Entropy as function of a for k = 2, 3, 4, and 5. 111M ,~~:'qob.I2CM<A' • 'qob.l2_ 0 'qobNek3' x ',,1211<2' • 'qob.t2Ok2' • 'qob1121c2' • 20 2S Fig. 5: q calculated from 2-replica theory vs experimental ground state overlaps. Arrows pointing up are O'ann, arrows pointing down are a2. 6 Conclusions We have shown how finite size scaling methods from statistical physics can be used to model the threshold in randomly generated k-SAT problems. Given the good fit of our scaling analysis, we conjecture that this method can also give useful models of phase transitions in other combinatorial problems with a control parameter. Several authors have attempted to relate NP-hardness or NP-completeness to the characteristics of phase transitions in models of disordered systems. Fu and Anderson (see Fu 1989) have proposed spin glasses (magnets with 2-spin interactions of random sign) as having inherent exponential complexity. Huberman and colleagues (see Clearwater 1991) were first to focus on the diverging correlation length seen at continuous phase transitions as the root of computational complexity. In fact, both effects can play important roles, but are not sufficient and may not even be necessary. There are NP-complete problems (e.g. travelling salesman, or max-clique) which lack a phase boundary at which "hard problems" cluster. Percolation thresholds are phase transitions, yet the cost of exploring the largest cluster never exceeds N steps, Exponential search cost in k-SAT comes from the random signs of the inputs, which require that the space be searched repeatedly. Note that a satisfying 446 Kirkpatrick, Gyorgyi, TIshby, and Troyansky input configuration in 2-SAT can be determined, or its non-existence proven, in polynomial time, because it can be reduced to a percolation problem on a random directed graph (Aspvall 1979). The spin glass Hamiltonians studied by Fu and Anderson have a form close to our 2-SAT formulae, but the questions studied are different. Finding an input configuration which falsifies the minimum number of clauses is like finding the ground state in a spin glass phase, and is NP-hard when a > a c , even for k = 2. Therefore, if both diverging correlations (diverging in size if no lengths are defined) and random sign or "spin-glass" effects are present, we expect a local search like Davis-Putnam to be exponentially difficult on average. But these characteristics do not imply NP-completeness. 7 References Aspvall, B., Plass, M.F., and Tarjan, R.E. (1979) A linear-time algorithm for testing the truth of certain quantified Boolean formulae. Inform. Process. Let., Vol. 8., 1979, 289-314. Barker, V. E., and O'Connor, D. (1989). Commun. Assoc. for Computing Machinery, 32(3), 1989, 298-318. Cheeseman, P., Kanefsky, B., and Taylor, W.M. (1991). Where the really hard problems are. Proceedings IJCAI-91, 1991, 163-169. Clearwater, S.H., Huberman B.A., Hogg, T. (1991) Cooperative Solution of Constraint Satisfaction Problems. Science, Vol. 254, 1991, 1181-1183 Crawford, J.M. and Auton L.D. (1993). Experimental Results on the Crossover Point in Satisfiability Problems. Proc. of AAAI-99, 1993. Chvatal, V. and Reed, B. (1992) Mick Gets Some: The Odds are on his Side. Proc. of STOC, 1992, 620-627. Fu, Y. (1989). The Uses and Abuses of Statistical Mechanics in Computational Complexity. in Lectures in the Sciences of Complexity, ed. D. Stein, pp. 815-826, Addison-Wesley, 1989. Franco, J. and Paull, M. (1988). Probabilistic Analysis of the Davis-Putnam Procedure for solving the Satisfiability Problem. Discrete Applied Math., Vol. 5, 77-87, 1983. Goerdt, A. (1992). A threshold for unsatisfiability. Proc. 17th Int. Symp. on the Math. Foundations of Compo Sc., Prague, Czechoslovakia, 1992. Kirkpatrick, S. and Swendsen, R.H. (1985). Statistical Mechanics and Disordered Systems. CA CM, Vol. 28, 1985, 363-373. Kirkpatrick, S., and Selman, B. (1993), submitted for publication. Larrabee, T. and Tsuji, Y. (1993) Evidence for a Satisfiability Threshold for Random 3CNF Formulas, Proc. of the AAAI Spring Symposium on AI and NP-hard problems, Palto Alto, CA, 1993. Mezard, M., Parisi, G., Virasoro, M.A. (1986). Spin Glass Theory and Beyond, Singapore: World Scientific, 1986. Mitchell, D., Selman, B., and Levesque, H.J. (1992) Hard and Easy Distributions of SAT problems. Proc. of AAAI-92, 1992, 456-465. Stauffer, D. and Aharony, A. (1992) Introduction to Percolation Theory. London: Taylor and Francis, 1992. See especially Ch. 4.	1993	98
856	Lipreading by neural networks: Visual preprocessing, learning and sensory integration Gregory J. Wolff Ricoh California Research Center 2882 Sand Hill Road Suite 115 Menlo Park, CA 94025-7022 wolff@crc.ricoh.com K. Venkatesh Prasad Ricoh California Research Center 2882 Sand Hill Road Suite 115 Menlo Park, CA 94025-7022 prasad@crc.ricoh.com David G. Stork Ricoh California Research Center 2882 Sand Hill Road Suite 115 Menlo Park, CA 94025-7022 stor k@crc.ricoh.com Marcus Hennecke Department of Electrical Engineering Mail Code 4055 Abstract Stanford University Stanford, CA 94305 We have developed visual preprocessing algorithms for extracting phonologically relevant features from the grayscale video image of a speaker, to provide speaker-independent inputs for an automatic lipreading ("speechreading") system. Visual features such as mouth open/closed, tongue visible/not-visible, teeth visible/notvisible, and several shape descriptors of the mouth and its motion are all rapidly computable in a manner quite insensitive to lighting conditions. We formed a hybrid speechreading system consisting of two time delay neural networks (video and acoustic) and integrated their responses by means of independent opinion pooling the Bayesian optimal method given conditional independence, which seems to hold for our data. This hybrid system had an error rate 25% lower than that of the acoustic subsystem alone on a five-utterance speaker-independent task, indicating that video can be used to improve speech recognition. 1027 1028 Wolff, Prasad, Stork, and Hennecke 1 INTRODUCTION Automated speech recognition is notoriously hard, and thus any predictive source of information and constraints that could be incorporated into a computer speech recognition system would be desirable. Humans, especially the hearing impaired, can utilize visual information "speech reading" for improved accuracy (Dodd & Campbell, 1987, Sanders & Goodrich, 1971). Speech reading can provide direct information about segments, phonemes, rate, speaker gender and identity, and subtle information for segmenting speech from background noise or multiple speakers (De Filippo & Sims, 1988, Green & Miller, 1985). Fundamental support for the use of visual information comes from the complementary nature of the visual and acoustic speech signals. Utterances that are difficult to distinguish acoustically are the easiest to distinguish. visually, and vice versa. Thus, for example /mi/ H /ni/ are highly confusable acoustically but are easily distinguished based on the visual information of lip closure. Conversely, /bi/ H /pi/ are highly confusable visually ("visemes"), but are easily distinguished acoustically by the voice-onset time (the delay between the burst sound and the onset of vocal fold vibration). Thus automatic lipreading promises to help acoustic speech recognition systems for those utterances where they need it most; visual information cannot contribute much information to those utterances that are already well recognized acoustically. 1.1 PREVIOUS SYSTEMS The system described below differs from recent speech reading systems. Whereas Yuhas et al. (1989) recognized static images and acoustic spectra for vowel recognition, ours recognizes dynamic consonant-vowel (CV) utterances. Whereas Petajan, Bischoff & Bodoff (1988) used thresholded pixel based representations of speakers, our system uses more sophisticated visual preprocessing to obtain phonologically relevant features. Whereas Pentland and Mase (1989) used optical flow methods for estimating the motion of four lip regions (and used no acoustic subsystem), we obtain several other features from intensity profiles. Whereas Bregler et al. (1993) used direct pixel images, our recognition engine used a far more compressed visual representation; our method of integration, too, was based on statistical properties of our data. We build upon the basic recognizer architecture of Stork, Wolff and Levine (1992), but extend it to grayscale video input. 2 VISUAL PREPROCESSING The sheer quantity of image data presents a hurdle to utilizing video information for speech recognition. Our approach to video preprocessing makes use of several simple computations to reduce the large amount of data to a manageable set of low-level image statistics describing the region of interest around the mouth. These statistics capture such features as the positions of the upper and lower lip, the mouth shape, and their time derivatives. The rest of this section describes the computation of these features. Grayscale video images are captured at 30 frames/second with a standard NTSC Lipreading by Neural Networks: Visual Preprocessing, Learning, and Sensory Integration 1029 pixel posiCion pixel position Figure 1: (Left) The central bands of the automatically determined ROI from two frames of the video sequence of the utterance /ba/ and their associated luminance profiles along the central marked line. Notice that the lowest valley in this profile changes drastically in intensity as the mouth changes from closed to open. In addition, the linear separation between the peaks adjacent to the lowest valley also increases as the mouth opens. These features are identified on the ROI from a single frame (right). The position, intensity, and temporal variation of these features provide input to our recognizer. camera, and subsampled to give 150 x 150 pixel image sequence. A 64 x 64 pixel region of interest (ROI) is detected and tracked by means of the following operations on the full video images: • Convolve with 3 x 3 pixel low-pass filter • Convolve with 3 x 3 pixel edge detector • Convolve with 3 x 3 pixel low-pass filter • Threshold at (Imax - I min)/2 • Triangulate eyes with mouth (to remove spatial noise) (to detect edges) (to smooth edges) (to isolate eyes and mouth) (to obtain ROI) We also use temporal coherence in frame-to-frame correlations to reduce the effects of noise in the profile or missing data (such as "closed" eyes). Within the ROI the phonological features are found by the following steps (see Figure 1): • Convolve with 16 x 16 pixel low-pass filter • Extract a vertical intensity profile • Extract a horizontal intensity profile • Locate and label intensity peaks and valleys • Calculate interframe peak motion (to remove noise) (mouth height) (mouth width) (candidates for teeth, tongue) (speed estimates) Video preprocessing tasks, including temporal averaging, are usually complicated because they require identifying corresponding pixels across frames. We circumvent this pixel correspondence problem by matching labeled features (such as intensity extrema peaks and valleys) on successive frames. 1030 Wolff, Prasad, Stork, and Hennecke 2.1 FEATURES The seventeen video features which serve as input to our recognizer are: • Horizontal separation between the left and right mouth corners • Vertical separation between the top and bottom lips For each of the three vertically aligned positions: • Vertical position: Pv • Intensity value: I • Change in intensity versus time: !:::.I /!:::.t For both of the mouth corner positions: • Horizontal position: Ph • Intensity value: I • Change in intensity versus time: !:::.I /!:::.t For each speaker, each feature was scaled have a zero mean and unit standard deviation. 3 DATA COLLECTION AND NETWORK TRAINING We trained the modified time delay neural network (Waibel, 1989) shown in Figure 2 on both the video and acoustic data. (See Stork, Wolff and Levine (1992) for a complete description of the architecture.) For the video only (VO) network, the input layer consists of 24 samples of each of the 17 features, corresponding to roughly 0.8 seconds. Each (sigmoidal) hidden unit received signals from a receptive field of 17 features for five consecutive frames. Each of the different hidden units (there were 3 for the results reported below) is replicated to cover the entire input space with overlapping receptive fields. The next layer consisted of 5 rows of x-units (one row for each possible utterance), with exponential transfer functions. They received inputs from the hidden units for 11 consecutive frames, thus they indirectly received input from a total of 18 input frames corresponding to roughly 0.6 seconds. The activities of the x-units encode the likelihood that a given letter occurs in that interval. The final layer consists of five p-units (probability units), which encode the relative probabilities of the presence of each of the possible utterances across the entire input window. Each p-unit sums the entire row of corresponding x-units, normalized by the sum over all x-units. (Note that "weights" from the x-units to the p-units are fixed.) The acoustic only (AO) network shared the same architecture, except that the input consisted of 100 frames (1 second) of 14 mel scale coefficients each, and the x-units received fan in from 25 consecutive hidden units. In the TDNN architecture, weights are shared, i.e., the pattern of input-to-hidden weights is forced to be the same at each interval. Thus the total number of independent weights in this VO network is 428, and 593 for the AO network. These networks were trained using Backpropagation to minimize the KullbackLeibler distance (cross-entropy) between the targets and outputs, t· E = D(t II p) = Ltdn(--.!..). (1) . Pi l Here the target probability is 1 for the target category, and 0 for all other categories. In this case Equation 1 simplifies to E = -In(pc) where c is the correct category. Lipreading by Neural Networks: Visual Preprocessing. Learning. and Sensory Integration 1031 en c> ci '" Q)c> -0 . -0 c> I Outputs for utterance ma3 bada!a lama Q ~--------~~----------~ 5 10 15 20 '" Time Figure 2: Modified time delay neural network architecture (left) and unit activities for a particular pattern (right). The output probabilities are calculated by integrating over the entire input window and normalizing across categories. Note the temporal segmentation which naturally occurs in the layer of X-units. 3.1 SENSORY INTEGRATION Given the output probability distributions of the two networks, we combine them assuming conditional independence and using Bayes rule to obtain: (2) That is, the joint probability of the utterance belonging to category Ci is just the normalized product of the outputs for category Ci of each network. This "independent opinion pooling" (Berger, 1985) offers several ad vantages over other methods for combining the modalities. First, it is optimal if the two signals really are conditionally independent, which appears to be the case for our data. (Proving that two signals are not conditionally independent is difficult.) Moreover, Massaro and Cohen (1983) have shown that human recognition performance is consistent with the independence assumption. A second advantage is simplicity. The combination adds no extra parameters beyond those used to model each signal, thus generalization performance should be good. Furthermore, the independent recognizers can be developed and trained separately, the only requirement is that they both output probability estimations. A third advantage is that this system automatically compensates for noise and assigns more importance to the network which is most sure of its classification. For example, if the video data were very noisy (or missing), the video network would 1032 Wolff, Prasad, Stork, and Hennecke ma la ::J .e- fa ::J o da ba judge all utterances equally likely. In this case the video contribution would cancel out, and the final output probabilities would be determined solely by the audio network. Bregler et al. (1993) attempt to compensate for the variance between channels by using the entropy of the output of the individual networks as a weighting on their contribution to the final outputs. Their ad hoc method suffers several drawbacks. For example, it does not distinguish the case where a one category is highly likely and the rest equiprobable, from the case where several categories are moderately likely. A final advantage of Eq. 2 is that it does not require synchrony of the acoustic and visual features. The registration between the two signals could be off substantially (as long as the same utterance is present in the input to both networks). On the contrary, methods which attempt to detect cross-modal features would be very sensitive to the relative timing of the two signals. 4 RESULTS Video Test o 0 o 0 0 o 0 0 0 0 o 00 0 0 0000 0 0 0 0 0 0 ba da fa la ma Input 54% correct Audio Test maooooO o :OQ~ _ la ::J .e- fa ::J o da 00 0 0 0 baOO ooQ ba da fa la ma Input 64% correct ::J Q. ::J o AV Test maooooO :: 0 :c9~ da 00 0 0 0 baOo 0 0 ba da fa la ma Input 72% correct Figure 3: Confusion matrices for the video only (VO), acoustic only (AO), and the AV networks. Each vertical column is labeled by the spoken CV pair presented as input; each horizontal row represents the output by the network. The radius of each disk in the array is proportional to the output probability given an input letter. The recognition accuracy (measured as a percentage of novel test patterns properly classified by maximum network output) is shown. The video and audio networks were trained separately on several different consonants in the same vowel context (/ba/, Ida/, Ifa/, Ila/, Ima/) recorded from several different speakers. (For the results reported below, there were 10 speakers, repeating each of 5 CV pairs 5 times. Four of these were used for training, and one for testing generalization.) For the video only networks, the correct classification (using the Max decision rule) on unseen data is typically 40-60%. As expected, the audio networks perform better with classification rates in the 50-70% range on these small sets of similar utterances. :; Q. :; o Lipreading by Neural Networks: Visual Preprocessing, Learning, and Sensory Integration 1033 Figure 3 shows the confusion matrices for the network outputs. We see that for the video only network the confusion matrix is fairly diagonal, indicating generally good performance. However the video network does tend to confuse utterances such as /ba/ H /maj. The audio network generally makes fewer errors, but confuses other utterances, such as /ba/ H / da/. The performance for the combined outputs (the AV network) is much better than either of the individual networks, achieving classification rates above 70%. (In previous work with only 4 speakers, classifications rates of up to 95% have been achieved.) We also see a strongly diagonal confusion matrix for the AV network, indicating that complementary nature of the the confusions made by the individual networks. 5 RELATIONSHIP TO HUMAN PERCEPTION Visual '6 ·0 . . o· • o . o o . t"· :~6 · ·: · . o· •• 0 • . o ••• • . . . . 0 . o· ·0··· ··0 • 0 .•••• O~~9hj k I~ r~'~~h Input Acoustic o 0 .,# . ·0 0 ::s ::s Q. Q. :; :; 0 ~·~~;k;~;S~h 0 Input AV Input Figure 4: Confusion matrices from human recognition performance for video only, acoustic only, and combined speech for CV pairs (Massaro et aI., 1993). Interestingly, our results are qualitatively similar to findings in human perception. Massaro et aI. (1993) presented Visual only, Acoustic only, and combined speech to subjects and collected response probabilities. As can be seen in the confusion matrices of Figure 4, subjects are not so bad at lipreading. The Visual only confusion matrix shows a strong diagonal component, though confusions such as /ma/ H /ba/ are common. Performance on acoustic speech is better, of course, but there are still confusions such as /ba/ H / daj. Combined speech yields even better recognition performance, eliminating most confusions. In fact, Massaro et aI. found that the response probabilities of combined speech are accurately predicted by the product of the two single mode response probabilities. Massaro uses this and other evidence to argue quite convincingly that humans treat acoustic and visual speech channels independently, combining them only at a rather late stage of processing. 1034 Wolff, Prasad, Stork, and Hennecke 6 CONCLUSIONS AND FUTURE WORK The video pre-processing presented here represents a first pass at reducing the amount of visual data to a manageable level in order to enable on-line processing. Our results indicate that even these straightforward, computationally tractable methods can significantly enhance speech recognition. Future efforts will concentrate on refining the pre-processing to capture more information, such as rounding and f-tuck, and testing the efficacy of our recognition system on larger datasets. The complementary nature of the acoustic and visual signals lead us to believe that a further refined speech reading system will significantly improve the state-of-the-art acoustic recognizers, especially in noisy environments. References J. O. Berger. (1985) Statistical decision theory and Bayesian analysis (2nd ed'). 272-275, New York: Springer-Verlag. C. Bregler, S. Manke, H. Hild & A. Waibel. (1993) Bimodal Sensor Integration on the example of "Speech-Reading". Proc. ICNN-93, Vol. II 667-677. C. L. De Filippo & D. G. Sims (eds.), (1988) New Reflections on Speechreading (Special issue of The Volta Review). 90(5). B. Dodd & R. Campbell (eds.). (1987) Hearing by Eye: The Psychology of Lip-reading. Hillsdale, N J: Lawrence Erlbaum Press. K. P. Green & J. L. Miller. (1985) On the role of visual rate information in phonetic perception. Perception and Psychophysics 38, 269-276. D. W. Massaro & M. M. Cohen (1983) Evaluation and integration of visual and auditory information in speech perception J. Exp. Psych: Human Perception and Performance 9, 753-771. D. W. Massaro, M. M. Cohen & A. T. Gesi (1993). Long-term training, transfer, and retention in learning to lipread. Perception £3 Psychophysics, 53, 549-562. A. Pentland & K. Mase (1989) Lip reading: Automatic visual recognition of spoken words. Proc. Image Understanding and Machine Vision, Optical Society of America, June 12-14. E. D. Petajan, B. Bischoff & D. Bodoff. (1988) An improved automatic lipreading system to enhance speech recognition. ACM SIGCHI-88, 19-25. D. Sanders & S. Goodrich. (1971) The relative contribution of visual and auditory components of speech to speech intelligibility as a function. of three conditions of frequency distortion. J. Speech and Hearing Research 14, 154-159. D. G. Stork, G. Wolff & E. Levine. (1992) Neural network lipreading system for improved speech recognition. Proc. IJCNN-92, Vol. II 285-295. A. Waibel. (1989) Modular construction of time-delay neural networks for speech recognition. Neural Computation 1, 39-46. B. P. Yuhas, M. H. Goldstein, Jr., T. J. Sejnowski & R. E. Jenkins. (1988) Neural network models of sensory integration for improved vowel recognition. Proc. IEEE 78(10), 16581668.	1993	99
857	SIMPLIFYING NEURAL NETS BY DISCOVERING FLAT MINIMA Sepp Hochreiter" Jiirgen Schmidhubert Fakultat fiir Informatik, H2 Technische Universitat Miinchen 80290 Miinchen, Germany Abstract We present a new algorithm for finding low complexity networks with high generalization capability. The algorithm searches for large connected regions of so-called ''fiat'' minima of the error function. In the weight-space environment of a "flat" minimum, the error remains approximately constant. Using an MDL-based argument, flat minima can be shown to correspond to low expected overfitting. Although our algorithm requires the computation of second order derivatives, it has backprop's order of complexity. Experiments with feedforward and recurrent nets are described. In an application to stock market prediction, the method outperforms conventional backprop, weight decay, and "optimal brain surgeon" . 1 INTRODUCTION Previous algorithms for finding low complexity networks with high generalization capability are based on significant prior assumptions. They can be broadly classified as follows: (1) Assumptions about the prior weight distribution. Hinton and van Camp [3] and Williams [17] assume that pushing the posterior distribution (after learning) close to the prior leads to "good" generalization. Weight decay can be derived e.g. from Gaussian priors. Nowlan and Hinton [10] assume that networks with many similar weights generated by Gaussian mixtures are "better" a priori. MacKay's priors [6] are implicit in additional penalty terms, which embody the "hochreit@informatik. tu-muenchen .de t schmidhu@informatik.tu-muenchen.de 530 Sepp Hochreiter. Jurgen Schmidhuber assumptions made. (2) Prior assumptions about how theoretical results on early stopping and network complexity carryover to practical applications. Examples are methods based on validation sets (see [8]), Vapnik's "structural risk minimization" [1] [14], and the methods of Holden [5] and Wang et al. [15]. Our approach requires less prior assumptions than most other approaches (see appendix A.l). Basic idea of flat minima search. Our algorithm finds a large region in weight space with the property that each weight vector from that region has similar small error. Such regions are called "flat minima". To get an intuitive feeling for why ''flat'' minima are interesting, consider this (see also Wolpert [18]): a "sharp" minimum corresponds to weights which have to be specified with high precision. A ''flat'' minimum corresponds to weights many of which can be given with low precision. In the terminology of the theory of minimum description length (MDL), fewer bits of information are required to pick a ''flat'' minimum (corresponding to a "simple" or low complexity-network). The MDL principle suggests that low network complexity corresponds to high generalization performance (see e.g. [4, 13]). Unlike Hinton and van Camp's method [3] (see appendix A.3), our approach does not depend on explicitly choosing a "good" prior. Our algorithm finds "flat" minima by searching for weights that minimize both training error and weight precision. This requires the computation of the Hessian. However, by using Pearlmutter's and M~ner's efficient second order method [11, 7], we obtain the same order of complexity as with conventional backprop. A utomatically, the method effectively reduces numbers of units, weigths, and input lines, as well as the sensitivity of outputs with respect to remaining weights and units. Excellent experimental generalization results will be reported in section 4. 2 TASK / ARCHITECTURE / BOXES Generalization task. The task is to approximate an unknown relation [) c X x Z between a set of inputs X C RN and a set of outputs Z C RK. [) is taken to be a function. A relation D is obtained from [) by adding noise to the outputs. All training information is given by a finite relation Do C D. Do is called the training set. The pth element of Do is denoted by an input/target pair (xp, dp). Architecture. For simplicity, we will focus on a standard feedforward net (but in the experiments, we will use recurrent nets as well). The net has N input units, K output units, W weights, and differentiable activation functions. It maps input vectors xp E RN to output vectors op E RK. The weight from unit j to i is denoted by Wij. The W -dimensional weight vector is denoted by w. Training error. Mean squared error Eq(w, Do) := l.riol E(xp,dp)EDo II dp op 112 is used, where II . II denotes the Euclidian norm, and 1.1 denotes the cardinality of a set. To define regions in weight space with the property that each weight vector from that region has "similar small error", we introduce the tolerable error Etal, a positive constant. "Small" error is defined as being smaller than Etal. Eq(w, Do) > Etal implies "underfitting" . Boxes. Each weight W satisfying Eq(w, Do) ~ Etal defines an "acceptable minimum". We are interested in large regions of connected acceptable minima. Simplifying Neural Nets by Discovering Flat Minima 531 Such regions are called fiat minima. They are associated with low expected generalization error (see [4]). To simplify the algorithm for finding large connected regions (see below), we do not consider maximal connected regions but focus on so-called "boxes" within regions: for each acceptable minimum w, its box Mw in weight space is a W-dimensional hypercuboid with center w. For simplicity, each edge of the box is taken to be parallel to one weight axis. Half the length of the box edge in direction of the axis corresponding to weight Wij is denoted by ~wii ' which is the maximal (positive) value such that for all i, j, all positive K.ii :5 ~Wij can be added to or subtracted from the corresponding component of W simultaneously without violating Eq(. , Do) :5 Etol (~Wij gives the precision of Wij). Mw's box volume is defined by ~w := 2w ni,j ~Wij. 3 THE ALGORITHM The algorithm is designed to find a W defining a box Mw with maximal box volume ~w. This is equivalent to finding a box Mw with minimal B( w, Do) := -log(~w/2W) = Li,j -log ~Wi.j. Note the relationship to MDL (B is the number of bits required to describe the weights). In appendix A.2, we derive the following algorithm. It minimizes E(w, Do) = Eq(w, Do) + >'B(w, Do), where B = ~ (-WIOg{+ ~logL(:~~j)2 + WlogL (L: /~ ,)2) . (1) ' ,3" " ' ,3 L"(8w;) Here 0" is the activation of the kth output unit, { is a constant, and >. is a positive variable ensuring either Eq(w, Do) :5 Etol, or ensuring an expected decrease of Eq(., Do) during learning (see [16] for adjusting >.). E(w, Do) is minimized by gradient descent. To minimize B(w, Do), we compute 8B(w, Do) '"" 8B(w, Do) 820" ~ II (2) 8 = L8 " 8 8 lor a u, v . wuv I. .• 8(~) Wij wuv "',',) vw., It can be shown (see [4]) that by using Pearlmutter's and M~ller's efficient second order method [11, 7], the gradient of B( w, Do) can be computed in O(W) time (see details in [4]). Therefore, our algorithm has the same order of complexity as standard backprop. 4 EXPERIMENTAL RESULTS (see [4] for details) EXPERIMENT 1 - noisy classification. The first experiment is taken from Pearlmutter and Rosenfeld [12]. The task is to decide whether the x-coordinate of a point in 2-dimensional space exceeds zero (class 1) or does not (class 2). Noisy training examples are generated as follows: data points are obtained from a Gaussian with zero mean and stdev 1.0, bounded in the interval [-3.0,3.0]. The data points are misclassified with a probability of 0.05. Final input data is obtained by adding a zero mean Gaussian with stdev 0.15 to the data points. In a test with 2,000,000 data points, it was found that the procedure above leads to 9.27 per cent 532 Sepp Hochreiter, Jurgen Schmidhuber Backprop New approach Backprop New approach MSE dto MSE dto MSE dto MSE dto 1 0.220 1.35 0.193 0.00 6 0.219 1.24 0.187 0.04 2 0.223 1.16 0.189 0.09 7 0.215 1.14 0.187 0.07 3 0.222 1.37 0.186 0.13 8 0.214 1.10 0.185 0.01 4 0.213 1.18 0.181 0.01 9 0.218 1.21 0.190 0.09 5 0.222 1.24 0.195 0.25 10 0.214 1.21 0.188 0.07 Table 1: 10 comparisons of conventional backprop (BP) and our new method (FMS). The second row (labeled "MSE") shows mean squared error on the test set. The third row ("dto") shows the difference between the fraction (in per cent) of misclassifications and the optimal fraction (9.27). The remaining rows provide the analoguous information for the new approach, which clearly outperforms backprop. misclassified data. No method will misclassify less than 9.27 per cent, due to the inherent noise in the data. The training set is based on 200 fixed data points. The test set is based on 120,000 data points. Results. 10 conventional backprop (BP) nets were tested against 10 equally initialized networks based on our new method ("flat minima search", FMS). After 1,000 epochs, the weights of our nets essentially stopped changing (automatic "early stopping"), while backprop kept changing weights to learn the outliers in the data set and overfit. In the end, our approach left a single hidden unit h with a maximal weight of 30.0 or -30.0 from the x-axis input. Unlike with backprop, the other hidden units were effectively pruned away (outputs near zero). So was the y-axis input (zero weight to h). It can be shown that this corresponds to an "optimal" net with minimal numbers of units and weights. Table 1 illustrates the superior performance of our approach. EXPERIMENT 2 - recurrent nets. The method works for continually running fully recurrent nets as well. At every time step, a recurrent net with sigmoid activations in [0,1] sees an input vector from a stream of randomly chosen input vectors from the set {(0,0), (0,1),(1,0),(1,1)}. The task is to switch on the first output unit whenever an input (1,0) had occurred two time steps ago, and to switch on the second output unit without delay in response to any input (0,1). The task can be solved by a single hidden unit. Results. With conventional recurrent net algorithms, after training, both hidden units were used to store the input vector. Not so with our new approach. We trained 20 networks. All of them learned perfect solutions. Like with weight decay, most weights to the output decayed to zero. But unlike with weight decay, strong inhibitory connections (-30.0) switched off one of the hidden units, effectively pruning it away. EXPERIMENT 3 - stock market prediction. We predict the DAX (German stock market index) based on fundamental (experiments 3.1 and 3.2) and technical (experiment 3.3) indicators. We use strictly layered feedforward nets with sigmoid units active in [-1,1]' and the following performance measures: Confidence: output 0 > a positive tendency, 0 < -a negative tendency. Performance: Sum of confidently, incorrectly predicted DAX changes is subtracted Simplifying Neural Nets by Discovering Flat Minima 533 from sum of confidently, correctly predicted ones. The result is divided by the sum of absolute changes. EXPERIMENT 3.1: Fundamental inputs: (a) German interest rate ("Umlaufsrendite"), (b) industrial production divided by money supply, (c) business sentiments ("IFO Geschiiftsklimaindex"). 24 training examples, 68 test examples, quarterly prediction, confidence: 0: = 0.0/0.6/0.9, architecture: (3-8-1). EXPERIMENT 3.2: Fundamental inputs: (a), (b), (c) as in expo 3.1, (d) dividend rate, (e) foreign orders in manufacturing industry. 228 training examples, 100 test examples, monthly prediction, confidence: 0: = 0.0/0.6/0.8, architecture: (5-8-1). EXPERIMENT 3.3: Technical inputs: (a) 8 most recent DAX-changes, (b) DAX, (c) change of 24-week relative strength index ("RSI"), (d) difference of "5 week statistic", (e) "MACD" (difference of exponentially weighted 6 week and 24 week DAX). 320 training examples, 100 test examples, weekly predictions, confidence: 0: = 0.0/0.2/0.4, architecture: (12-9-1). The following methods are tested: (1) Conventional backprop (BP), (2) optimal brain surgeon (OBS [2]), (3) weight decay (WD [16]), (4) flat minima search (FMS). Results. Our method clearly outperforms the other methods. FMS is up to 63 per cent better than the best competitor (see [4] for details). APPENDIX - THEORETICAL JUSTIFICATION A.t. OVERFITTING ERROR In analogy to [15] and [1], we decompose the generalization error into an "overfitting" error and an "underfitting" error. There is no significant underfitting error (corresponding to Vapnik's empirical risk) if Eq(w, Do) ~ Etol . Some thought is required, however, to define the "overfitting" error. We do this in a novel way. Since we do not know the relation D, we cannot know p(o: I D), the "optimal" posterior weight distribution we would obtain by training the net on D (- "sure thing hypothesis"). But, for theoretical purposes, suppose we did know p(o: I D). Then we could use p(o: I D) to initialize weights before learning the training set Do. Using the Kullback-Leibler distance, we measure the information (due to noise) conveyed by Do, but not by D. In conjunction with the initialization above, this provides the conceptual setting for defining an overfitting error measure. But, the initialization does not really matter, because it does not heavily influence the posterior (see [4]). The overfittin~ error is the Kullback-Leibler distance of the posteriors: Eo(D, Do) = J p(o: I Do) log (p(o: I Do)/p(o: I D») do:. Eo(D, Do) is the expectation of log (p(o: I Do)/p(o: I D)) (the expected difference of the minimal description of 0: with respect to D and Do, after learning Do). Now we measure the expected overfitting error relative to Mw (see section 2) by computing the expectation of log (p( 0: I Do) / p( 0: I D» in the range Mw: Ero(w) = f3 (1M", PM", (0: I Do)Eq(O:, D)do: - Eq(Do, M w») . (3) Here PM",(O: I Do) := p(o: I Do)/ IM", p(a: I Do)da: is the posterior of Do scaled to obtain a distribution within Mw, and Eq(Do, Mw) := IM", PM", (a I Do)Eq(a, Do)do: is the mean error in Mw with respect to Do. 534 Sepp Hochreiter. JiJrgen Schmidhuber Clearly, we would like to pick W such that Ero( w) is minimized. Towards this purpose, we need two additional prior assumptions, which are actually implicit in most previous approaches (which make additional stronger assumptions, see section 1): (1) "Closeness assumption": Every minimum of E q(., Do) is "close" to a maximum of p(aID) (see formal definition in [4]). Intuitively, "closeness" ensures that Do can indeed tell us something about D, such that training on Do may indeed reduce the error on D. (2) "Flatness assumption": The peaks of p(aID)'s maxima are not sharp. This MDL-like assumption holds if not all weights have to be known exactly to model D. It ensures that there are regions with low error on D. A.2. HOW TO FLATTEN THE NETWORK OUTPUT To find nets with flat outputs, two conditions will be defined to specify B(w, Do) (see section 3). The first condition ensures flatness. The second condition enforces "equal flatness" in all weight space · directions. In both cases, linear approximations will be made (to be justified in [4]). We are looking for weights (causing tolerable error) that can be perturbed without causing significant output changes. Perturbing the weights w by 6w (with components 6Wij), we obtain ED(w,6w) := L,,(o"(w + 6w) - o"(w»)2, where o"(w) expresses o"'s dependence on w (in what follows, however, w often will be suppressed for convenience). Linear approximation (justified in [4]) gives us "Flatness Condition 1": 00" 00" ED(w, 6w) ~ L(L -;;;-:-:6Wij)2 :$ L(L 1-;;;-:-:1I6wij1)2 :$ ( , (4) .. .. uW'J .. .. uW'J .. '.J .. '.J where ( > 0 defines tolerable output changes within a box and is small enough to allow for linear approximation (it does not appear in B(w, Do)'s gradient, see section 3). Many Mw satisfy flatness condition 1. To select a particular, very flat Mw, the following "Flatness Condition 2" uses up degrees of freedom left by (4): . . ( 2 "( 00")2 ( 2" 00" 2 'VZ,),U,V: 6Wij) L-~ = 6wuv ) L-(-~-)' " uW'J " uWuv (5) Flatness Condition 2 enforces equal "directed errors" EDij(W,6wij) = L,,(O"(Wij + 6Wij) - O"(Wij)? ~ L,,(:;:;6wij?, where Ok(Wij) has the obvious meaning. It can be shown (see [4]) that with given box volume, we need flatness condition 2 to minimize the expected description length of the box center. Flatness condition 2 influences the algorithm as follows: (1) The algorithm prefers to increase the 6Wij'S of weights which currently are not important to generate the target output. (2) The algorithm enforces equal sensitivity of all output units with respect to the weights. Hence, the algorithm tends to group hidden units according to their relevance for groups of output units. Flatness condition 2 is essential: flatness condition 1 by itself corresponds to nothing more but first order derivative reduction (ordinary sensitivity reduction, e.g. [9]). Linear approximation is justified by the choice of f in equation (4). We first solve equation (5) for 16w,; I = 16W., I ( 2:, (a':::.) 2 / 2:, U.::J 2) Simplifying Neural Nets by Discovering Flat Minima 535 (fixing u, v for all i, j). Then we insert 16wij I into equation (4) (replacing the second "$" in (4) by ";;;:"). This gives us an equation for the 16wijl (which depend on w, but this is notationally suppressed): ( ook 16wi·1 = Vii "(_)2 J L- ow .. k IJ (6) The 16wijl approximate the ~Wij from section 2. Thus, B(w,Do) (see section 3) can be approximated by B(w, Do) :;;;: Ei,j -log 16wijl. This immediately leads to the algorithm given by equation (1). How can this approximation be justified? The learning process itself enforces its validity (see justification in [4]). Initially, the conditions above are valid only in a very small environment of an "initial" acceptable minimum. But during search for new acceptable minima with more associated box volume, the corresponding environments are enlarged, which implies that the absolute values of the entries in the Hessian decrease. It can be shown (see [4]) that the algorithm tends to suppress the following values: (1) unit activations, (2) first order activation derivatives, (3) the sum of all contributions of an arbitary unit activation to the net output. Since weights, inputs, activation functions, and their first and second order derivatives are bounded, it can be shown (see [4]) that the entries in the Hessian decrease where the corresponding 16wij I increase. A.3. RELATION TO HINTON AND VAN CAMP Hinton and van Camp [3] minimize the sum of two terms: the first is conventional enor plus variance, the other is the distance f p( a I Do) log (p( a I Do) I p( a» da between posterior pea I Do) and prior pea). The problem is to choose a "good" prior. In contrast to their approach, our approach does not require a "good" prior given in advance. Furthermore, Hinton and van Camp have to compute variances of weights and units, which (in general) cannot be done using linear approximation. Intuitively speaking, their weight variances are related to our ~Wij. Our approach, however, does justify linear approximation. References [1] I. Guyon, V. Vapnik, B. Boser, L. Bottou, and S. A. Solla. Structural risk minimization for character recognition. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4, pages 471-479. San Mateo, CA: Morgan Kaufmann, 1992. [2] B. Hassibi and D. G. Stork. Second order derivatives for network pruning: Optimal bra.i.n surgeon. In J. D. Cowan S. J. Hanson and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 164-171. San Mateo, CA: Morgan Kaufmann, 1993. [3] G. E. Hinton and D. van Camp. Keeping neural networks simple. In Proceedings of the International Conference on Artificial Neural Networks, Amsterdam, pa.ges 11-18. Springer, 1993. 536 Sepp Hoc/zreiter, Jiirgen Schmidhuber [4] S. Hochreiter and J. Schmidhuber. Flat Inllllma search for discovering simple nets. Technical Report FKI-200-94, Fakultiit fiir Informatik, Technische Universitiit Munchen, 1994. [5] S. B. Holden. On the Theory of Generalization and Self-Structuring in Linearly Weighted Connectionist Networks. PhD thesis, Cambridge University, Engineering Department, 1994. [6] D. J. C. MacKay. A practical Bayesian framework for backprop networks. Neural Computation, 4:448-472, 1992. [7] M. F. M~ller. Exact calculation of the product of the Hessian matrix offeed-forward network error functions and a vector in O(N) time. Technical Report PB-432, Computer Science Department, Aarhus University, Denmark, 1993. [8] J. E. Moody and J. Utans. Architecture selection strategies for neural networks: Application to corporate bond rating prediction. In A. N. Refenes, editor, Neural Networks in the Capital Markets. John Wiley & Sons, 1994. [9] A. F. Murray and P. J. Edwards. Synaptic weight noise during MLP learning enhances fault-tolerance, generalisation and learning trajectory. In J. D. Cowan S. J. Hanson and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 491-498. San Mateo, CA: Morgan Kaufmann, 1993. [10] S. J. Nowlan and G. E. Hinton. Simplifying neural networks by soft weight sharing. Neural Computation, 4:173-193, 1992. [11] B. A. Pearlmutter. Fast exact multiplication by the Hessian. Neural Computation, 1994. [12] B. A. Pearlmutter and R. Rosenfeld. Chaitin-Kolmogorov complexity and generalization in neural networks. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 925-931. San Mateo, CA: Morgan Kaufmann, 1991. [13] J. H. Schmid huber. Discovering problem solutions with low Kolmogorov complexity and high generalization capability. Technical Report FKI-194-94, Fakultiit fUr Informatik, Technische U niversitiit Munchen, 1994. [14] V. Vapnik. Principles of risk minimization for learning theory. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4, pages 831-838. San Mateo, CA: Morgan Kaufmann, 1992. [15] C. Wang, S. S. Venkatesh, and J. S. Judd. Optimal stopping and effective machine complexity in learning. In J . D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems 6, pages 303-310. Morgan Kaufmann, San Mateo, CA, 1994. [16] A. S. Weigend, D. E. Rumelhart, and B. A. Huberman. Generalization by weightelimination with application to forecasting. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 875-882. San Mateo, CA: Morgan Kaufmann, 1991. [17] P. M. Williams. Bayesian regularisation and pruning using a Laplace prior. Technical report, School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton, 1994. [18] D. H. Wolpert. Bayesian backpropagation over i-o functions rather than weights. In J. D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems 6, pages 200-207. San Mateo, CA: Morgan Kaufmann, 1994.	1994	1
858	Hierarchical Mixtures of Experts Methodology Applied to Continuous Speech Recognition Ying Zhao, Richard Schwartz, Jason Sroka: John Makhoul BBN System and Technologies 70 Fawcett Street Cambridge MA 02138 Abstract In this paper, we incorporate the Hierarchical Mixtures of Experts (HME) method of probability estimation, developed by Jordan [1], into an HMMbased continuous speech recognition system. The resulting system can be thought of as a continuous-density HMM system, but instead of using gaussian mixtures, the HME system employs a large set of hierarchically organized but relatively small neural networks to perform the probability density estimation. The hierarchical structure is reminiscent of a decision tree except for two important differences: each "expert" or neural net performs a "soft" decision rather than a hard decision, and, unlike ordinary decision trees, the parameters of all the neural nets in the HME are automatically trainable using the EM algorithm. We report results on the ARPA 5,OOO-word and 4O,OOO-word Wall Street Journal corpus using HME models. 1 Introduction Recent research has shown that a continuous-density HMM (CD-HMM) system can outperform a more constrained tied-mixture HMM system for large-vocabulary continuous speech recognition (CSR) when a large amount of training data is available [2]. In other work, the utility of decision trees has been demonstrated in classification problems by using the "divide and conquer" paradigm effectively, where a problem is divided into a hierarchical set of simpler problems. We present here a new CD-HMM system which *MIT, Cambridge MA 02139 860 Ying Zhao, Richard Schwartz, Jason Sroka, John Makhoul has similar properties and possesses the same advantages as decision trees, but has the additional important advantage of having automatically trainable "soft" decision boundaries. 2 Hierarchical Mixtures of Experts The method of Hierarchical Mixtures of Experts (HME) developed recently by Jordan [1] breaks a large scale task into many small ones by partitioning the input space into a nested set of regions, then building a simple but specific model (local expert) in each region. The idea behind this method follows the principle of divide-and-conquer which has been utilized in certain approaches to classification problems, such as decision trees. In the decision tree approach, at each level of the tree, the data are divided explicitly into regions. In contrast, the HME model makes use of "soft" splits of the data, i.e., instead of the data being explicitly divided into regions, the data may lie simultaneously in multiple regions with certain probabilities. Therefore, the variance-increasing effect of lopping off distant data in the decision tree can be ameliorated. Furthermore, the "hard" boundaries in the decision tree are fixed once a decision is made, while the "soft" boundaries in the HME are parameterized with generalized sigmoidal functions, which can be adjusted automatically using the Expectation-Maximization (EM) algorithm during the splitting. Now we describe how to apply the HME methodology to the CSR problem. For each state of a phonetic HMM, a separate HME is used to estimate the likelihood. The actual HME first computes a posterior probability P(llz, s), the probability of phoneme class I, given the input feature vector z and state s. That probability is then divided by the a priori probability of the phone class I at state s. A one-level HME performs the following computation: c P(llz, s) = L P(llci, z, s)P(cilz, s) (1) i=l where I = 1, , .. , L indicates phoneme class, Ci represents a local region in the input space, and C is the number of regions. P(cilz, s) can be viewed as a gating network, while P(lICi, z, s) can be viewed as a local expert classifier (expert network) in the region c, [1]. In a two-level HME, each region Ci is divided in turn into C subregions. The term P(IICi, z, s) is then computed in a similar manner to equation (1), and so on. If in some of these subregions there are no data available, we back off to the parent network. 3 TECHNICAL DETAILS As in Jordan's paper, we use a generalized sigmoidal function to parameterize P(cilz) as follows: (2) where z can be the direct input (in a one-layer neural net) or the hidden layer vector (in a two-layer neural net), and v,, i = 1, .. " C are weights which need to train. Similarly, the local phoneme classifier in region Ci, P(llc" z), can be parameterized with a generalized Mixtures of Experts Applied to Continuous Speech Recognition 861 sigmoidal function also: (3) where 8;i,j = 1, ... , L are weights. The whole system consists of two set of parameters: Vi, i = 1, ... , C and 8;i' j = 1, ... , L, e = {8;i' Vi}. All parameters are estimated by using the EM algorithm. The EM is an iterative approach to maximum likelihood estimation. Each iteration of an EM algorithm is composed of two steps: an Expectation (E) step and a Maximization (M) step. The M step involves the maximization of a likelihood function that is redefined in each iteration by the E step. Using the parameterizations in (2) and (3), we obtain the following iterative procedure for computing parameters e = {Vi, 8;i}: 1 . .. I' (0) d 8(0) f . I C ' 1 L . lDltIa lze Vi an ;i or 1. = , ... , ,} = , ... , . 2. E-step: In each iteration n, for each data pair (z(t), l(t», t = 1, ... ,N, compute zi(tin) = P(cilz(t), l(t), e(n~ = P(Ci Iz(t), v~n»p(l(t)lci' z(t), 8~~~,i) (4) where i = 1, ... , C. Zi(t)<n) represents the probability of the data t lying in the region i, given the current parameter estimation e(n). It will be used as a weight for this data in the region i in the M-step. The idea of "soft" splitting reflects that these weights are probabilities between 0 and 1, instead of a "hard"decision 0 or 1. 3. M-step: (5) (6) 4. Iterate until 8;i' Vi converge. The first maximization means fitting a generalized sigmoidal model (3) using the labeled data (z(t), l(t» and weighting Zi(t)<n). The second one means fitting a generalized sigmoidal model (2) using inputs z(t) and outputs Zi(t)<n). The criterion for fitting is the cross-entropy. Typically, the fitting can be solved by the Newton-Raphson method. However, it is quite expensive. Viewing this type of fitting as a multi-class classification task, we developed a technique to invert a generalized sigmoidal function more efficiently, which will be described in the following. A common method in a multi-class classification is to divide the problem into many 2-c1ass classifications. However, this method results in a positive and negative training unbalance usually. To avoid the positive and negative training unbalance, the following technique can be used to solve multi-class posterior probabilities simultaneously. Suppose we have a labeled data set, (z(t), l(t», t = 1, ... , N, where l(t) E {I, ... , L} is the label for t-th data. We use a generalized sigmoidal function to model the posterior 862 Ying Zhao, Richard Schwartz, Jason Sroka, John Makhoul probability P(llz), where 1 = I, ... , L as follows: e9'f'z P,(z) = P(llz) = 9T L:k e "z Obviously, since these probabilities sum up to one, we have L-I PL(Z) = 1 - L P,(z). '=1 Now, a training sample z(t) with a class label let) can be interpreted as: { 0.9 1 = let) P,(z(t» = 1:':1 1 =/l(t) If we define T P,(z) 9, z = log PL(z) equation (10) implies that (7) (8) (9) (10) (11) for 1 = I, ... , L with 9Lz = O. This expression is the generalized sigmoidal function in (7). This means, we can train parameters in (7) to satisfy Equation (10) from the data. Using a least squares criterion, the objective is . ,,[ P'(Z(t»] 2 mm L..J 9T z(t) - log --t PL(Z(t» for 1 = I, ... , L - 1. Denote a data matrix as x= A least squares solution to (12) is z(l) z(2) zeN) 9, = (loga)(XT X)-I [L z(t) - L Z(t)] '(t)=l '(t)=L for 1 = I, ... , L, where a = 9(L - 1). Substituting (13) into (11), we get ZT(XT X)-l ~ z(t) (12) (13) a L.JI(I~I P,(z) = zT(XT X)-l L: z(t) (14) L:k a 1(1)=" Equation (13) and (14) are very easy to compute. Basically, we only have to accumulate the matrix XT X and sum z(t) into different classes 1 = I, ... , L. We can obtain probabilities P,(z) by a single inversion of matrix XT X after a pass through the training data. Mixtures of Experts Applied to Continuous Speech Recognition 863 4 Relation to Other Work The work reported here is very different from our previous work utilizing neural nets for CSR. There, a single segmental neural network (SNN) is used to model a complete phonetic segment [3]. Here, each HME estimates the probability density for each state of a phonetic HMM. The work here is more similar to that by Cohen et al. [4], the major difference being that in [4], a single very large neural net is used to perform the probability density modeling. The training of such a large network requires the use of a specialized parallel processing machine, so that the training can be done in a reasonable amount of time. By using the HME method and dividing the problem into many smaller problems, we are able to perform the needed training computation on regular workstations. Most of the previous work on CD-HMM work has utilized mixtures of gaussians to estimate the probability densities of an HMM. Since a ' multilayer feedforward neural network is a universal continuous function approximator, we decided to explore the use of neural nets as an alternative approach for continuous density estimation. 5 Experimental Results Word Error Rate HMM 7.8 SNN 8.5 HMM+SNN 7.1 HME 7.6 HME+HMM 6.8 Prior-modified HME + HMM 6.2 Table 1: Error Rates for the ARPA WSJ 5K Development Test, Trigram Grammar Word Error Rate HMM 9.5 HME+ HMM 8.7 Table 2: Error Rates for the ARPA WSJ 40K Test Set, Trigram Grammar In our initial application of the HME method to large-vocabulary CSR, we used phonetic context-independent HMEs to estimate the likelihoods at each state of 5-state HMMs. We implemented a two-level HME, with the input space divided into 46 regions, and each of those regions is further divided into 46 subregions. The initial divisions were accomplished by supervised training, with each division trained to one of the 46 phonemes in the system. All gating and local expert networks in the HME had identical structures a two-layer generalized sigmoidal network. The whole HME system was implemented within an N-best paradigm [3], where the recognized sequence was obtained as a result of a rescoring of an N-best list obtained from our baseline BYBLOS system (tied-mixture HMM) with a statistical trigram grammar. 864 Ying Zhao, Richard Schwartz, Jason Sroka, John Makhoul We then built a context-dependent HME system based on the structure of the contextindependent HME models described above. For each state, the whole training data was divided into 46 parts according to its left or right context. Then for each context, a separate HME model was built for that context. To be computationally feasible, we used only one-level HMEs here. We first experimented using a left-context and right-context model. We tested the HME implementation on the ARPA 5,OOO-word Wall Street Journal corpus (WSJl, H2 dev set). We report the word error rates on the same test set for a number of different systems. Table 1 shows the word error rates for i) the baseline HMM system; ii) the segment-based neural net system (SNN) iii) the hybrid SNNIHMM system iv) a HME system alone. v) a HME system combined with HMM; vi) a HME +HMM system with modified priors. From Table 1, The performance of the baseline tied-mixture HMM is 7.8%. The performance of the SNN system (8.5%) is comparable to the HMM alone. We see that the performance of a HME (7.6%) is as good as the HMM system, which is better than the SNN system. When combined with the baseline HMM system, the HME and SNN both improve performance over the HMM alone about 10% from 7.8% to 6.8% and from 7.8% to 7.1% respectively. We found out that the improvement could be made larger for a hybrid HMElHMM by adjusting the context-dependent priors with the context-independent priors, and then smooth context-dependent models with a context-independent model. More specifically, in a context-dependent HME model, we usually estimate the posterior probability phoneme I, P(llc, z, s), given left or right context c and the acoustic input z in a particular state s. Because the samples may be sparse for many of context models, it is necessary to regularize (smooth) context-dependent models with a contextindependent model, where there is much more data available. However, since the two models have different priors: P(llc, s) in a context-dependent model and P(lls) in a context-independent model, a simple interpolation between the two models which is P(ll ) P(x , c,.)P(' c.) . d d od I d P(ll ) P(x , .)P(' .) c, z, s = P(x c,.) 10 a context- epen ent mean z, s = P(x .) in a context-independent model is inconsistent. To scale the context-dependent priors P(llc, s) with a context-independent prior P(lls), we weighted each input data point z with the weight :c.i'c:;) for a prior adjusting. After this modification, a context-dependent HME actually estimates P(z ~~:~('I'). It combines better with a context-independent model. For the same experiment we showed in Table 1, the word error for the HME (with HMM) droped from 6.8% to 6.2% when priors were modified. For this 5,OOO-word development set, we got a total of about 20% word error reduction over the tied-mixture HMM system using a HME-based neural network system. We then switched our experiment domain from a 5,OOO-word to 40,OOO-word the test set. During this year, the BYBLOS system has been improVed from a tied-mixture system to a continuous density system. We also switched to using this new continuous density BYBLOS in our hybrid HMElHMM system. The language model used here was a 40,OOO-word trigram grammar. The result is shown in Table 2. From Table 2, we see that there is about a 10% word error rate reduction over the continuous density HMM system by combining a context-dependent HME system. Compared with the 20% improvement over the tied-mixture system we made for the 5,OOO-word development set, the improvement over the continuous density system in this 40,OOO-word Mixtures of Experts Applied to Continuous Speech Recognition 865 development is less. This may be due to the big improvement of the HMM system itself. 6 CONCLUSIONS The method of hierarchical mixtures of experts can be used as a continous density estimator to speech recognition. Experimental results showed that estimations from this approach are consistent with the estimations from the HMM system. The frame-based neural net system using hierarchical mixtures of experts improves the performance of both the state-of-the-art tied mixture HMM system and the continuous density HMM system. The HME system itself has the same performance as the state-of-the-art tied mixture HME system. 7 Acknowledgments This work was funded by the Advanced Research Projects Agency of the Department of Defense. References [1] Michael Jordan, "Hierarchical Mixtures of Experts and the EM Algorithm," Neural Computation, 1994, in press. [2] D. Pallett, J. Fiscus, W. Fisher, J. Garofolo, B. Lund, and M. Pryzbocki, "1993 Benchmark Tests for the ARPA Spoken Language Program," Proc. ARPA Human Language Technology Workshop, Plainsboro, NJ, Morgan Kaufman Publishers, 1994. [3] G. Zavaliagkos, Y. Zhao, R. Schwartz and J. Makhoul, "A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition," in Advances in Neural Information Processing Systems 5, S. J. Hanson, J. D. Cowan and C. L. Giles, eds., Morgan Kaufmann Publishers, 1993. [4] M. Cohen, H. Franco, N. Morgan, D. Rumelhart and V. Abrash, "Context-Dependent Multiple Distribution Phonetic Modeling with MLPS," in Advances in Neural Information Processing Systems 5, S. J. Hanson, 1. D. Cowan and C. L. Giles, eds., Morgan Kaufmann Publishers, 1993.	1994	10
859	Neural Network Ensembles, Cross Validation, and Active Learning Anders Krogh" Nordita Blegdamsvej 17 2100 Copenhagen, Denmark Jesper Vedelsby Electronics Institute, Building 349 Technical University of Denmark 2800 Lyngby, Denmark Abstract Learning of continuous valued functions using neural network ensembles (committees) can give improved accuracy, reliable estimation of the generalization error, and active learning. The ambiguity is defined as the variation of the output of ensemble members averaged over unlabeled data, so it quantifies the disagreement among the networks. It is discussed how to use the ambiguity in combination with cross-validation to give a reliable estimate of the ensemble generalization error, and how this type of ensemble cross-validation can sometimes improve performance. It is shown how to estimate the optimal weights of the ensemble members using unlabeled data. By a generalization of query by committee, it is finally shown how the ambiguity can be used to select new training data to be labeled in an active learning scheme. 1 INTRODUCTION It is well known that a combination of many different predictors can improve predictions. In the neural networks community "ensembles" of neural networks has been investigated by several authors, see for instance [1, 2, 3]. Most often the networks in the ensemble are trained individually and then their predictions are combined. This combination is usually done by majority (in classification) or by simple averaging (in regression), but one can also use a weighted combination of the networks . .. Author to whom correspondence should be addressed. Email: kroghlnordita. elk 232 Anders Krogh, Jesper Vedelsby At the workshop after the last NIPS conference (December, 1993) an entire session was devoted to ensembles of neural networks ("Putting it all together", chaired by Michael Perrone). Many interesting papers were given, and it showed that this area is getting a lot of attention. A combination of the output of several networks (or other predictors) is only useful if they disagree on some inputs. Clearly, there is no more information to be gained from a million identical networks than there is from just one of them (see also [2]). By quantifying the disagreement in the ensemble it turns out to be possible to state this insight rigorously for an ensemble used for approximation of realvalued functions (regression). The simple and beautiful expression that relates the disagreement (called the ensemble ambiguity) and the generalization error is the basis for this paper, so we will derive it with no further delay. 2 THE BIAS-VARIANCE TRADEOFF Assume the task is to learn a function J from RN to R for which you have a sample of p examples, (xiJ , yiJ), where yiJ = J(xiJ) and J.t = 1, . . . ,p. These examples are assumed to be drawn randomly from the distribution p(x). Anything in the following is easy to generalize to several output variables. The ensemble consists of N networks and the output of network a on input x is called va (x). A weighted ensemble average is denoted by a bar, like V(x) = L Wa Va(x). (1) a This is the final output of the ensemble. We think of the weight Wa as our belief in network a and therefore constrain the weights to be positive and sum to one. The constraint on the sum is crucial for some of the following results. The ambiguity on input x of a single member of the ensemble is defined as aa (x) = (Va(x) - V(x))2 . The ensemble ambiguity on input x is a(x) = Lwaaa(x) = LWa(va(x) - V(x))2 . (2) a a It is simply the variance of the weighted ensemble around the weighed mean, and it measures the disagreement among the networks on input x. The quadratic error of network a and of the ensemble are (J(x) - Va(x))2 (J(x) - V(X))2 respectively. Adding and subtracting J( x) in (2) yields a(x) = L Wafa(X) - e(x) a (3) (4) (5) (after a little algebra using that the weights sum to one). Calling the weighted average of the individual errors €( x) = La Wa fa (x) this becomes e(x) = €(x) - a(x). (6) Neural Network Ensembles, Cross Validation, and Active Learning 233 All these formulas can be averaged over the input distribution. Averages over the input distribution will be denoted by capital letter, so E J dxp(xVl!(x) J dxp(x)aa(x) J dxp(x)e(x). (7) (8) (9) The first two of these are the generalization error and the ambiguity respectively for network n , and E is the generalization error for the ensemble. From (6) we then find for the ensemble generalization error (10) The first term on the right is the weighted average of the generalization errors of the individual networks (E = La waEa), and the second is the weighted average of the ambiguities (A = La WaAa), which we refer to as the ensemble ambiguity. The beauty of this equation is that it separates the generalization error into a term that depends on the generalization errors of the individual networks and another term that contain all correlations between the networks. Furthermore, the correlation term A can be estimated entirely from unlabeled data, i. e., no knowledge is required of the real function to be approximated. The term "unlabeled example" is borrowed from classification problems, and in this context it means an input x for which the value of the target function f( x) is unknown. Equation (10) expresses the tradeoff between bias and variance in the ensemble, but in a different way than the the common bias-variance relation [4] in which the averages are over possible training sets instead of ensemble averages. If the ensemble is strongly biased the ambiguity will be small, because the networks implement very similar functions and thus agree on inputs even outside the training set. Therefore the generalization error will be essentially equal to the weighted average of the generalization errors of the individual networks. If, on the other hand, there is a large variance, the ambiguity is high and in this case the generalization error will be smaller than the average generalization error. See also [5]. From this equation one can immediately see that the generalization error of the ensemble is always smaller than the (weighted) average of the ensemble errors, E < E. In particular for uniform weights: E ~ ~ 'fEcx (11) which has been noted by several authors, see e.g. [3] . 3 THE CROSS-VALIDATION ENSEMBLE From (10) it is obvious that increasing the ambiguity (while not increasing individual generalization errors) will improve the overall generalization. We want the networks to disagree! How can we increase the ambiguity of the ensemble? One way is to use different types of approximators like a mixture of neural networks of different topologies or a mixture of completely different types of approximators. Another 234 1. :~ t , .. , E o ...... -' '.-.. ' ........ ....,. > -1.k! ~ .t. -4 f. :\,'. -·-.l :--, ____ ~. , ,' If . . -2 ,', . . ..... , .. v '. --: '1 ~. __ .. -.tI" ,._ • ." • o x .' .~.--c·· . . -- --\\ .~ .. - ..... Anders Krogh, Jesper Vedelsby _._ ..... . '-._._.1 1\.1 ~~ .~ . 2 -, ~ \. • ' 0' : ~: 4 Figure 1: An ensemble of five networks were trained to approximate the square wave target function f(x). The final ensemble output (solid smooth curve) and the outputs of the individual networks (dotted curves) are shown. Also the square root of the ambiguity is shown (dash-dot line) _ For training 200 random examples were used, but each network had a cross-validation set of size 40, so they were each trained on 160 examples. obvious way is to train the networks on different training sets. Furthermore, to be able to estimate the first term in (10) it would be desirable to have some kind of cross-validation. This suggests the following strategy. Chose a number K :::; p. For each network in the ensemble hold out K examples for testing, where the N test sets should have minimal overlap, i. e., the N training sets should be as different as possible. If, for instance, K :::; piN it is possible to choose the K test sets with no overlap. This enables us to estimate the generalization error E(X of the individual members of the ensemble, and at the same time make sure that the ambiguity increases. When holding out examples the generalization errors for the individual members of the ensemble, E(X, will increase, but the conjecture is that for a good choice of the size of the ensemble (N) and the test set size (K), the ambiguity will increase more and thus one will get a decrease in overall generalization error. This conjecture has been tested experimentally on a simple square wave function of one variable shown in Figure 1. Five identical feed-forward networks with one hidden layer of 20 units were trained independently by back-propagation using 200 random examples. For each network a cross-validation set of K examples was held out for testing as described above. The "true" generalization and the ambiguity were estimated from a set of 1000 random inputs. The weights were uniform, w(X = 1/5 (non-uniform weights are addressed later). In Figure 2 average results over 12 independent runs are shown for some values of Neural Network Ensembles, Cross Validation, and Active Learning 235 Figure 2: The solid line shows the generalization error for uniform weights as a function of K, where K is the size of the cross-validation sets. The dotted line is the error estimated from equation (10) . The dashed line is for the optimal weights estimated by the use of the generalization errors for the individual networks estimated from the crossvalidation sets as described in the text. The bottom solid line is the generalization error one would obtain if the individual generalization errors were known exactly (the best possible weights). 0.08 ,-----r----,--~---r-----, o t= w 0.06 c o ~ .!::! co ... ~ 0.04 Q) (!) 0.02 '---_---1 __ ---'-__ --'-__ -----' o 20 40 60 80 Size of CV set K (top solid line). First, one should note that the generalization error is the same for a cross-validation set of size 40 as for size 0, although not lower, so it supports the conjecture in a weaker form. However, we have done many experiments, and depending on the experimental setup the curve can take on almost any form, sometimes the error is larger at zero than at 40 or vice versa. In the experiments shown, only ensembles with at least four converging networks out of five were used. If all the ensembles were kept, the error would have been significantly higher at ]{ = a than for K > a because in about half of the runs none of the networks in the ensemble converged something that seldom happened when a cross-validation set was used. Thus it is still unclear under which circumstances one can expect a drop in generalization error when using cross-validation in this fashion. The dotted line in Figure 2 is the error estimated from equation (10) using the cross-validation sets for each of the networks to estimate Ea, and one notices a good agreement. 4 OPTIMAL WEIGHTS The weights Wa can be estimated as described in e.g. [3]. We suggest instead to use unlabeled data and estimate them in such a way that they minimize the generalization error given in (10) . There is no analytical solution for the weights, but something can be said about the minimum point of the generalization error. Calculating the derivative of E as given in (10) subject to the constraints on the weights and setting it equal to zero shows that E a - A a = E or Wa = O. (12) (The calculation is not shown because of space limitations, but it is easy to do.) That is, Ea - Aa has to be the same for all the networks. Notice that Aa depends on the weights through the ensemble average of the outputs. It shows that the optimal weights have to be chosen such that each network contributes exactly waE 236 Anders Krogh, Jesper Vedelsby to the generalization error. Note, however, that a member of the ensemble can have such a poor generalization or be so correlated with the rest of the ensemble that it is optimal to set its weight to zero. The weights can be "learned" from unlabeled examples, e.g. by gradient descent minimization of the estimate of the generalization error (10). A more efficient approach to finding the optimal weights is to turn it into a quadratic optimization problem. That problem is non-trivial only because of the constraints on the weights (L:a Wa = 1 and Wa 2:: 0). Define the correlation matrix, C af3 = f dxp(x)Va(x)V f3 (x) . (13) Then, using that the weights sum to one, equation (10) can be rewritten as E = L wa Ea + L w aCaf3 w f3 - L waCaa . (14) a af3 a Having estimates of E a and Caf3 the optimal weights can be found by linear programming or other optimization techniques. Just like the ambiguity, the correlation matrix can be estimated from unlabeled data to any accuracy needed (provided that the input distribution p is known). In Figure 2 the results from an experiment with weight optimization are shown. The dashed curve shows the generalization error when the weights are optimized as described above using the estimates of Ea from the cross-validation (on K exampies). The lowest solid curve is for the idealized case, when it is assumed that the errors Ea are known exactly, so it shows the lowest possible error. The performance improvement is quite convincing when the cross-validation estimates are used. It is important to notice that any estimate of the generalization error of the individual networks can be used in equation (14). If one is certain that the individual networks do not overfit, one might even use the training errors as estimates for Ea (see [3]). It is also possible to use some kind of regularization in (14), if the cross-validation sets are small. 5 ACTIVE LEARNING In some neural network applications it is very time consuming and/or expensive to acquire training data, e.g., if a complicated measurement is required to find the value of the target function for a certain input. Therefore it is desirable to only use examples with maximal information about the function. Methods where the learner points out good examples are often called active learning. We propose a query-based active learning scheme that applies to ensembles of networks with continuous-valued output. It is essentially a generalization of query by committee [6, 7] that was developed for classification problems. Our basic assumption is that those patterns in the input space yielding the largest error are those points we would benefit the most from including in the training set. Since the generalization error is always non-negative, we see from (6) that the weighted average of the individual network errors is always larger than or equal to the ensemble ambiguity, f(X) 2:: a(x), (15) Neural Network Ensembles. Cross Validation. and Active Learning 237 2.5 r"':":'T---r--"T""--.-----r---, . . . : 0.5 o 10 20 30 40 50 o 10 20 30 40 50 Training set size Training set size Figure 3: In both plots the full line shows the average generalization for active learning, and the dashed line for passive learning as a function of the number of training examples. The dots in the left plot show the results of the individual experiments contributing to the mean for the active learning. The dots in right plot show the same for passive learning. which tells us that the ambiguity is a lower bound for the weighted average of the squared error. An input pattern that yields a large ambiguity will always have a large average error. On the other hand, a low ambiguity does not necessarily imply a low error. If the individual networks are trained to a low training error on the same set of examples then both the error and the ambiguity are low on the training points. This ensures that a pattern yielding a large ambiguity cannot be in the close neighborhood of a training example. The ambiguity will to some extent follow the fluctuations in the error. Since the ambiguity is calculated from unlabeled examples the input-space can be scanned for these areas to any detail. These ideas are well illustrated in Figure 1, where the correlation between error and ambiguity is quite strong, although not perfect. The results of an experiment with the active learning scheme is shown in Figure 3. An ensemble of 5 networks was trained to approximate the square-wave function shown in Figure 1, but in this experiments the function was restricted to the interval from - 2 to 2. The curves show the final generalization error of the ensemble in a passive (dashed line) and an active learning test (solid line). For each training set size 2x40 independent tests were made, all starting with the same initial training set of a single example. Examples were generated and added one at a time. In the passive test examples were generated at random, and in the active one each example was selected as the input that gave the largest ambiguity out of 800 random ones. Figure 3 also shows the distribution of the individual results of the active and passive learning tests. Not only do we obtain significantly better generalization by active learning, there is also less scatter in the results. It seems to be easier for the ensemble to learn from the actively generated set. 238 Anders Krogh. Jesper Vedelsby 6 CONCLUSION The central idea in this paper was to show that there is a lot to be gained from using unlabeled data when training in ensembles. Although we dealt with neural networks, all the theory holds for any other type of method used as the individual members of the ensemble. It was shown that apart from getting the individual members of the ensemble to generalize well, it is important for generalization that the individuals disagrees as much as possible, and we discussed one method to make even identical networks disagree. This was done by training the individuals on different training sets by holding out some examples for each individual during training. This had the added advantage that these examples could be used for testing, and thereby one could obtain good estimates of the generalization error. It was discussed how to find the optimal weights for the individuals of the ensemble. For our simple test problem the weights found improved the performance of the ensemble significantly. Finally a method for active learning was described, which was based on the method of query by committee developed for classification problems. The idea is that if the ensemble disagrees strongly on an input, it would be good to find the label for that input and include it in the training set for the ensemble. It was shown how active learning improves the learning curve a lot for a simple test problem. Acknowledgements We would like to thank Peter Salamon for numerous discussions and for his implementation of linear programming for optimization of the weights. We also thank Lars Kai Hansen for many discussions and great insights, and David Wolpert for valuable comments. References [1] L.K. Hansen and P Salamon. Neural network ensembles. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(10):993- 1001, Oct. 1990. [2] D.H Wolpert. Stacked generalization. Neural Networks, 5(2):241-59, 1992. [3] Michael P. Perrone and Leon N Cooper. When networks disagree: Ensemble method for neural networks. In R. J. Mammone, editor, Neural Networks for Speech and Image processing. Chapman-Hall, 1993. [4] S. Geman, E. Bienenstock, and R Doursat. Neural networks and the bias/variance dilemma. Neural Computation, 4(1):1-58, Jan. 1992. [5] Ronny Meir. Bias, variance and the combination of estimators; the case of linear least squares. Preprint (In Neuroprose), Technion, Heifa, Israel, 1994. [6] H.S. Seung, M. Opper, and H. Sompolinsky. Query by committee. In Proceedings of the Fifth Workshop on Computational Learning Theory, pages 287-294, San Mateo, CA, 1992. Morgan Kaufmann. [7] Y. Freund, H.S. Seung, E. Shamir, and N. Tishby. Information, prediction, and query by committee. In Advances in Neural Information Processing Systems, volume 5, San Mateo, California, 1993. Morgan Kaufmann.	1994	100
860	Extracting Rules from Artificial Neural Networks with Distributed Representations Sebastian Thrun University of Bonn Department of Computer Science III Romerstr. 164, D-53117 Bonn, Germany E-mail: thrun@carbon.informatik.uni-bonn.de Abstract Although artificial neural networks have been applied in a variety of real-world scenarios with remarkable success, they have often been criticized for exhibiting a low degree of human comprehensibility. Techniques that compile compact sets of symbolic rules out of artificial neural networks offer a promising perspective to overcome this obvious deficiency of neural network representations. This paper presents an approach to the extraction of if-then rules from artificial neural networks. Its key mechanism is validity interval analysis, which is a generic tool for extracting symbolic knowledge by propagating rule-like knowledge through Backpropagation-style neural networks. Empirical studies in a robot arm domain illustrate the appropriateness of the proposed method for extracting rules from networks with real-valued and distributed representations. 1 Introduction In the last few years artificial neural networks have been applied successfully to a variety of real-world problems. For example, neural networks have been successfully applied in the area of speech generation [12] and recognition [18], vision and robotics [8], handwritten character recognition [5], medical diagnostics [11], and game playing [13]. While in these and other approaches neural networks have frequently found to outperform more traditional approaches, one of their major shortcomings is their low degree of human comprehensibility. In recent years, a variety of approaches for compiling rules out of networks have been proposed. Most approaches [1, 3,4,6, 7, 16, 17] compile networks into sets of rules with equivalent structure: Each processing unit is mapped into a separate rule-or a smal1 set of rules-, and the ingoing weights are interpreted as preconditions to this rule. Sparse connectivity facilitates this type rule extraction, and so do binary activation values. In order to enforce such properties, which is a necessary prerequisite for these techniques to work effectively, some approaches rely on specialized training procedures, network initializations 506 Sebastian Thrun and/or architectures. While such a methodology is intriguing, as it draws a clear one-to-one correspondence between neural inference and rule-based inference, it is not universally applicable to arbitrary Backpropagation-style neural networks. This is because artificial neural networks might not meet the strong representational and structural requirements necessary for these techniques to work successfully. When the internal representation of the network is distributed in nature, individual hidden units typically do not represent clear, logical entities. One might argue that networks, if one is interested in extracting rules, should be constructed appropriately. But this would outrule most existing network implementation~, as such considerations have barely played a role. In addition, such an argument would suppress the development of distributed, non-discrete internal representations, which have often be attributed for the generalization properties of neural networks. It is this more general class of networks that is at stake in this paper. This paper presents a rule extraction method which finds rules by analyzing networks as a whole. The rules are of the type "if X then y," where both x and y are described by a linear set of constraints. The engine for proving the correspondence of rule and network classification is VI-Analysis. Rules extracted by VI-Analysis can be proven to exactly describe the network. 2 Validity-Interval Analysis Validity Interval Analysis (in short: VI-Analysis) is a generic tool for analyzing the inputoutput behavior of Backpropagation-style neural networks. In short, they key idea of VIAnalysis is to attach intervals to the activation range of each unit (or a subset of all units, like input and output units only), such that the network's activations must lie within these intervals. These intervals are called validity intervals. VI-Analysis checks whether such a set of intervals is consistent, i.e., whether there exists a set of network activations inside the validity intervals. It does this by iteratively refining the validity intervals, excluding activations that are provably inconsistent with other intervals. In what follows we will present the general VI-Analysis algorithm, which can be found in more detail elsewhere [14], Let n denote the total number of units in the network, and let Xi denote the (output) activation of unit i (i = 1, ... , n). If unit i is an input unit, its activation value will simply be the external input value. If not, i.e., if i refers to a hidden or an output unit, let P( i) denote the set of units that are connected to unit i through a link. The activation Xi is computed in two steps: with L WikXk + Oi kEP(i) The auxiliary variable neti is the net-input of unit i, and Wik and Oi are the weights and biases, respectively. O'j denotes the transfer function (squashing function), which usually is given by 1 + e-net , Validity intervals for activation values Xi are denoted by [ai, bi ]. If necessary, validity intervals are projected into the net-input space of unit i, where they will be denoted by [a~, b~]. Let T be a set of validity intervals for (a subset of) all units. An activation vector (XI, .. " xn) is said to be admissible with respect to T, if all activations lie in T. A set of intervals T is consistent, if there exists an admissible activation vector. Otherwise T is inconsistent. Assume an initial set of intervals, denoted by T, is given (in the next section we will present a procedure for generating initial intervals). VI-Analysis refines T iteratively using linear Extracting Rules from Artificial Neural Networks with Distributed Representations 507 non-linear .<;quashing functioll' CJ linear equations Figure 1: VI-Analysis in a single weight layer. Units in layer P are connected to the units in layer S. A validity interval [aj, bj ] is assigned to each unit j E PuS. By projecting the validity intervals for all i E S, intervals [a~, b~] for the net-inputs netj are created. These, plus the validity intervals for all units k E P, form a set of linear constraints on the activations x k in layer P. Linear programming is now employed to refine all interval bounds one-by-one. programming [9], so that those activation values which are inconsistent with other intervals are excluded. In order to simplify the presentation, let us assume without loss of ¥enerality (a) that the network is layered and fully connected between two adjacent layers, and (b) that there is an interval [aj, bj ] ~ [0,1] in I for every unit in P and S.2 Consider a single weight layer, connecting a layer of preceding units, denoted by p, to a layer of succeeding units, denoted by S (cf Fig. 1). In order to make linear programming techniques applicable, the non-linearity of the transfer function must be eliminated. This is achieved by projecting [ai, bi ] back to the corresponding net-input intervals3 [ai, biJ = {T-I([ai' biD E ~2 for all i E S. The resulting validity intervals in P and S form the foIIowing set of linear constraints on the activation values in P: Vk E P: Xk > ak and Xk < bk Vi E S: L WjkXk + ()j > a~ [by substituting neti = L WikXk + ()d , kEP kEP (1) L WikXk + ()j < b~ , [by substituting netj = L WikXk + ()i] kEP kEP Notice that all these constraints are linear in the activation values Xk (k E P). Linear programming allows to maximize or minimize arbitrary linear combinations of the variables x j while not violating a set of linear constraints [9]. Hence, linear programming can be applied to refine lower and upper bounds for validity intervals one-by-one. In VI-Analysis, constraints are propagated in two phases: 1. Forward phase. To refine the bounds aj and bj for units i E S, new bounds iii and hi are 'This assumption simplifies the description of VI-Analysis, although VI-Analysis can also be applied to arbitrary non-layered, partially connected network architectures, as well as recurrent networks not examined here. 2The canonical interval [0, I] corresponds to the state of maximum ignorance about the activation of a unit, and hence is the default interval if no more specific interval is known. 3Here ~ denotes the set of real numbers extended by ±oo. Notice that this projection assumes that the transfer function is monotonic. 508 derived: with with A' a · z min neti = max neti Sebastian Thrun min L: WikXk + Oi kE1' max L: WikXk + OJ kE1' If o'i > ai, a tighter lower bound is found and ai is updated by o'i . Likewise, bi is set to hi if hi < bi . Notice that the minimax operator is computed within the bounds imposed by Eq. I, using the Simplex algorithm (linear programming) [9]. 2. Backward phase. In the backward phase the bounds ak and bk of all units k E Pare refined. li k minxk and hk = max Xk As in the forward phase, ak is updated by o'k if lik > ak, and h is updated by hk if hk < bk. If the network has multiple weight layers, this process is applied to all weight layers one-byone. Repetitive refinement results in the propagation of interval constraints through multiple layers in both directions. The convergence of VI-Analysis follows from the fact that the update rule that intervals are changed monotonically, since they can only shrink or stay the same. Recall that the "input" of VI-Analysis is a set of intervals I ~ [0, l]n that constrain the activations of the network. VI-Analysis generates a refined set of intervals, I' ~ I, so that all admissible activation values in the original intervals I are also in the refined intervals I'. In other words, the difference between the original set of intervals and the refined set of intervals I - I' is inconsistent. In summary, VI-Analysis analyzes intervals I in order to detect inconsistencies. If I is found to be inconsistent, there is provably no admissible activation vector in I . Detecting inconsistencies is the driving mechanism for the verification and extraction of rules presented in turn. 3 Rule Extraction The rules considered in this paper are propositional if-then rules. Although VI-Analysis is able to prove rules expressed by arbitrary linear constraints [14], for the sake of simplicity we will consider only rules where the precondition is given by a set of intervals for the individual input values, and the output is a single target category. Rules of this type can be written as: !linput E some hypercube I then class is C (or short: I -- C) for some target class C. The compliance of a rule with the network can be verified through VI-Analysis. Assume, without loss of generality, the network has a single output unit, and input patterns are classified as members of class C if and only if the output activation, Xout, is larger than a threshold e (see [14] for networks with multiple output units). A rule conjecture I -- C is then verified by showing that there is no input vector i E I that falls into the opposite class, ,C. This is done by including the (negated) condition Xout E [0, e] into the set of intervals: Ineg = 1+ {xout E [0, e]}. If the rule is correct, Xout will never be in [0, e]. Hence, if VI-Analysis finds an inconsistency in Ineg, the rule I -- ,C is proven to be incorrect, and thus the original rule I -- C holds true for the network at hand. This illustrates how rules are verified using VI-Analysis. It remains to be shown how such conjectures can be generated in a systematic way. Two major classes of approaches can be distinguished, specific-to-general and general-to-specific. Extracting Rules from Artificial Neural Networks with Distributed Representations 509 Figure 2: Robot Ann. (a) Front view of two arm configurations. (b) Two-dimensional side view. The grey area indicates the workspace, which partially intersects with the table. 1. Specific-to-general. A generic way to generate rules, which forms the basis for the experimental results reported in the next section, is to start with rather specific rules which are easy to verify, and gradually generalize those rules by enlarging the corresponding validity intervals. Imagine one has a training instance that, without loss of generality, falls into a class C. The input vector of the training instance already forms a (degenerate) set of validity intervals I. VI-Analysis will, applied to I, trivially confirm the membership in C, and hence the single-point rule I ~ C. Starting with I, a sequence of more general rule preconditions I C II C I2 C ... can be obtained by enlarging the precondition of the rule (i.e., the input intervals I) by small amounts, and using VI-Analysis to verify if the new rule is still a member of its class. In this way randomly generated instances can be used as "seeds" for rules, which are then generalized via VI-Analysis. 2. General-to-specific. An alternative way to extract rules, which has been studied in more detail elsewhere [14], works from general to specific. General-to-specific rule search maintains a list of non-proven conjectures, R. R is initialized with the most general rules (like "everything is in C" and "nothing is in C"). VI-Analysis is then applied to prove rules in R. If it successfully confirms a rule, the rule and its complement is removed from R. If not, the rule is removed, too, but instead new rules are added to R. These new rules form a specialized version of the old rule, so that their disjunct is exactly the old rule. For example, new rules can be generated by splitting the hypercube spanned by the old rule into disjoint regions, one for each new rule. Then, the new set R is checked with VI-Analysis. The whole procedure continues till R is empty and the whole input domain is described by rules. In discrete domains, such a strategy amounts to searching directed acyclic graphs in breadth-first manner. Obviously, there is a variety of alternative techniques to generate meaningful rule hypotheses. For example, one might employ a symbolic learning technique such as decision tree learning [10] to the same training data that was used for training the network. The rules, which are a result of the symbolic approach, constitute hypotheses that can be checked using VI-Analysis. 4 Empirical Results In this section we will be interested in extracting rules in a real-valued robot arm domain. We trained a neural network to model the forward kinematics function of a 5 degree-of-freedom robot arm. The arm, a Mitsubishi RV-Ml, is depicted in Fig. 2. Its kinematic function determines the position of the tip of the manipulator in (x, y, z) workspace coordinates and 510 Sebastian Thrun coverage average (per rule) cumulative first 10 rules 9.79% 30.2% first 100 rules 2.59% 47.8% first 1 000 rules 1.20% 61.6% first 10000 rules 0.335% 84.4% Table 1: Rule coverage in the robot arm domain. These numbers inc1ude rules for both concepts, SAFE and UNSAFE. the angle of the manipulator h to the table based on the angles of the five joints. As can be seen in Fig. 2, the workspace intersects with the table on which the arm is mounted. Hence, some configurations of the joints are safe, namely those for which z ~ 0, whiJe others can physically not be reached without a col1ision that would damage the robot (unsafe). When operating the robot arm one has to be able to tell safe from unsafe. Henceforth, we are interested in a set of rules that describes the subspace of safe and unsafe joint configurations. A total of 8192 training examples was used for training the network (four input, five hidden and four output units), resulting in a considerably accurate model of the kinematics of the robot arm. Notice that the network operates in a continuous space. Obviously, compiling the network into logical rules node-by-node, as frequently done in other approaches to rule extraction, is difficult due to the real-valued and distributed nature of the internal representation. Instead, we applied VI-Analysis using a specific-to-general mechanism as described above. More specifically, we incrementally constructed a collection of rules that gradually covered the workspace of the robot arm. Rules were generated whenever a (random) joint configuration was not covered by a previously generated rule. Table 1 shows average results that characterize the extraction of rules. Initially, each rule covers a rather large fraction of the 5-dimensional joint configuration space. As few as 11 rules, on average, suffice to cover more than 50% (by volume) of the whole input space. However, these 50% are the easy half. As the domain gets increasingly covered by rules, gradually more specific rules are generated in regions closer to the c1ass boundary. After extracting 10,000 rules, only 84.4% of the input space is covered. Since the decision boundary between the two c1asses is highly non-linear, finitely many rules will never cover the input space completely. How general are the rules extracted by VI-Analysis? Genera])y speaking, for joint configurations c10se to the c1ass boundary, i.e., where the tip of the manipulator is close to the table, we observed that the extracted rules were rather specific. If instead the initial configuration was closer to the center of a class, VI-Analysis was observed to produce more general rules that had a larger coverage in the workspace. Here VI-Analysis managed to extract surprisingly general rules. For example, the configuration a = (300 ,800 ,200 ,600 , -200 ), which is depicted in Fig. 3, yields the rule !!.a2 ~ 90.50 and a3 ~ 27.30 then SAFE. Notice that out of 10 initial constraints, 8 were successfully removed by VI-Analysis. The rule lacks both bounds on a), a4 and as and the lower bounds on a2 and a3. Fig. 3a shows the front view of the initial arm configuration and the generalized rule (grey area). Fig. 3b shows a side view of the arm, along with a slice of the rule (the base joint a) is kept fixed). Notice that this very rule covers 17.1 % of the configuration space (by volume). Such general rules were frequently found in the robot arm domain. This conc1 udes the brief description of the experimental results. Not mentioned here are results with different size networks, and results obtained for the MONK's benchmark problems. For example, in the MONK's problems [15], VI-Analysis successfully extracted compact target Extracting Rules from Artificial Neural Networks with Distributed Representations 511 Figure 3: A single rule, extracted from the network. (a) Front view. (b) Two-dimensional side view. The grey area indicates safe positions for the tip of the manipulator. concepts using the originally published weight sets. These results can be found in [14]. 5 Discussion In this paper we have presented a mechanism for the extraction of rules from Backpropagationstyle neural networks. There are several limitations of the current approach that warrant future research. (a) Speed. While the one-to-one compilation of networks into rules is fast, rule extraction via VI-Analysis requires mUltiple runs of linear programming, each of which can be computationally expensive [9]. Searching the rule space without domain-specific search heuristics can thus be a most time-consuming undertaking. In all our experiments, however, we observed reasonably fast convergence of the VI-Algorithm, and we successfully managed to extract rules from larger networks in reasonable amounts of time. Recently, Craven and Shavlik proposed a more efficient search method which can be applied in conjunction with VI-Analysis [2]. (b) Language. Currently VI-Analysis is limited to the extraction of if-then rules with linear preconditions. While in [14] it has been shown how to generalize VI-Analysis to rules expressed by arbitrary linear constraints, a more powerful rule language is clearly desirable. (c) Linear optimization. Linear programming analyzes multiple weight layers independently, resulting in an overly careful refinement of intervals. This effect can prevent from detecting correct rules. If linear programming is replaced by a non-linear optimization method that considers multiple weight layers simultaneously, more powerful rules can be generated. On the other hand, efficient non-linear optimization techniques might find rules which do not describe the network accurately. Moreover, it is generally questionable whether there will ever exist techniques for mapping arbitrary networks accurately into compact rule sets. Neural networks are their own best description, and symbolic rules might not be appropriate for describing the input-output behavior of a complex neural network. A key feature of of the approach presented in this paper is the particular way rules are extracted. Unlike other approaches to the extraction of rules, this mechanism does not compile networks into structurally equivalent set of rules. Instead it analyzes the input output relation of networks as a whole. As a consequence, rules can be extracted from unstructured networks with distributed and real-valued internal representations. In addition, the extracted rules describe the neural network accurately, regardless of the size of the network. This makes VI-Analysis a promising candidate for scaling rule extraction techniques to deep networks, in which approximate rule extraction methods can suffer from cumulative errors. We conjecture that such properties are important if meaningful rules are to be extracted in today's and tomorrow's successful Backpropagation applications. 512 Sebastian Thrun Acknowledgment The author wishes to express his gratitude to Marc Craven, Tom Dietterich, Clayton McMillan. Tom Mitchell and Jude Shavlik for their invaluable feedback that has influenced this research. References [I] M. W. Craven and J. W. Shavlik. Learning symbolic rules using artificial neural networks. In Paul E. Utgoff, editor, Proceedings of the Tenth International Conference on Machine Learning, 1993. Morgan Kaufmann. [2] M. W. Craven and J. W. Shavlik. Using sampling and queries to extmct rules from tmined neural networks. In Proceedings of the Eleventh International Conference on Machine Learning, 1994. Morgan Kaufmann. [3] L.-M. Fu. Integration of neural heuristics into knowledge-based inference. Connection Science, 1(3):325-339,1989. [4] C. L. Giles and C. W. Omlin. Rule refinement with recurrent neural networks. In Proceedings of the IEEE International Conference on Neural Network, 1993. IEEE Neuml Network Council. [5] Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard. W. Hubbard, and L. D. Jackel. Backpropagation applied to handwritten zip code recognition. Neural Computation, 1 :541-551. 1990. [6] J. J. Mahoney and R. J. Mooney. Combining neural and symbolic learning to revise probabilistic rule bases. In J. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 5, 1993. Morgan Kaufmann. [7] C. McMillan, M. C. Mozer, and P. Smolensky. Rule induction through integrated symbolic and subsymbolic processing. In J. E. Moody. S. J. Hanson. and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 4. 1992. Morgan Kaufmann. [8] D. A. Pomerleau. ALVINN: an autonomous land vehicle in a neural network. Technical Report CMU-CS-89-1 07. Computer Science Dept. Carnegie Mellon University, Pittsburgh PA, 1989. [9] W. H. Press. Numerical recipes in C " the art of scientific computing. Cambridge University Press, Cambridge [Cambridgeshire], New York, 1988. [10] J. R. Quinlan. Induction of decision trees. Machine Learning, 1:81-106,1986. [II] J. Rennie. Cancer catcher: Neural net catches errors that slip through pap tests. Scientific American, 262, May 1990. [12] T. J. Sejnowski and C. R. Rosenberg. Nettalk: A parallel network that learns to read aloud. Technical Report JHUIEECS-86/01, Johns Hopkins University, 1986. [13] G. J. Tesauro. Practical issues in tempoml difference learning. Machine Learning. 8, 1992. [14] S. Thrun. Extracting provably correct rules from artificial neuml networks. Technical Report IAI-TR-93-5, University of Bonn. Institut flir Informatik III, D-53117 Bonn, May 1993. [15] S. Thrun, J. Bala,E. Bloedorn, I. Bmtko, B. Cestnik, J. Cheng, K. Dejong, S. Dzeroski, D. Fisher, S. E. Fahlman, R. Hamann, K. Kaufman, S. Keller. I. Kononenko, J. Kreuziger, R. S. Michalski, T.M. Mitchell, P. Pachowicz, Y. Reich, H. Vafaie, W. Van de WeIde, W. Wenzel, J. Wnek, and J. Zhang. The MONK's problems - a performance comparison of different learning algorithms. Technical Report CMU-CS-91-197, Carnegie Mellon University. Pittsburgh, PA, December 1991. [16] G. Towell and J. W. Shavlik. Interpretation of artificial neural networks: Mapping knowledgebased neural networks into rules. In J. E. Moody. S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 4. 1992. Morgan Kaufmann. [17] V. Tresp and J. Hollatz. Network structuring and training using rule-based knowledge. In J. E. Moody, S. J. Hanson. and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 5,1993. Morgan Kaufmann. [18] A. H. Waibel. Modular construction of time-delay neural networks for speech recognition. Neural Computation, 1 :39-46, 1989.	1994	101
861	A model of the hippocampus combining selforganization and associative memory function. Michael E. Hasselmo, Eric Schnell Joshua Berke and Edi Barkai Dept. of Psychology, Harvard University 33 Kirkland St., Cambridge, MA 02138 hasselmo@katla.harvard.edu Abstract A model of the hippocampus is presented which forms rapid self -organized representations of input arriving via the perforant path, performs recall of previous associations in region CA3, and performs comparison of this recall with afferent input in region CA 1. This comparison drives feedback regulation of cholinergic modulation to set appropriate dynamics for learning of new representations in region CA3 and CA 1. The network responds to novel patterns with increased cholinergic modulation, allowing storage of new self-organized representations, but responds to familiar patterns with a decrease in acetylcholine, allowing recall based on previous representations. This requires selectivity of the cholinergic suppression of synaptic transmission in stratum radiatum of regions CA3 and CAl, which has been demonstrated experimentally. 1 INTRODUCTION A number of models of hippocampal function have been developed (Burgess et aI., 1994; Myers and Gluck, 1994; Touretzky et al., 1994), but remarkably few simulations have addressed hippocampal function within the constraints provided by physiological and anatomical data. Theories of the function of specific subregions of the hippocampal formation often do not address physiological mechanisms for changing dynamics between learning of novel stimuli and recall of familiar stimuli. For example, the afferent input to the hippocampus has been proposed to form orthogonal representations of entorhinal activity (Marr, 1971; McNaughton and Morris, 1987; Eichenbaum and Buckingham, 1990), but simulations have not addressed the problem of when these representations 78 Michael E. Hasselmo. Eric Schnell. Joshua Berke. Edi Barkai should remain stable, and when they should be altered. In addition, models of autoassociative memory function in region CA3 (Marr, 1971; McNaughton and Morris, 1987; Levy, 1989; Eichenbaum and Buckingham, 1990) and heteroassociative memory function at the Schaffer collaterals projecting from region CA3 to CAl (Levy, 1989; McNaughton, 1991) require very different activation dynamics during learning versus recall. Acetylcholine may set appropriate dynamics for storing new information in the cortex (Hasselmo et aI., 1992, 1993; Hasselmo, 1993, 1994; Hasselmo and Bower, 1993). Acetylcholine has been shown to selectively suppress synaptic transmission at intrinsic but not afferent fiber synapses (Hasselmo and Bower, 1992), to suppress the neuronal adaptation of cortical pyramidal cells (Hasselmo et aI., 1994; Barkai and Hasselmo, 1994), and to enhance long-term potentiation of synaptic potentials (Hasselmo, 1994b). Models show that suppression of synaptic transmission during learning prevents recall of previously stored information from interfering with the storage of new information (Hasselmo et al., 1992, 1993; Hasselmo, 1993, 1994a), while cholinergic enhancement of synaptic modification enhances the rate of learning (Hasselmo, 1994b). Feedback regulation of cholinergic modulation may set the appropriate level of cholinergic modulation dependent upon the novelty or familiarity of a particular input pattern. We have explored possible mechanisms for the feedback regulation of cholinergic modulation in simulations of region CAl (Hasselmo and Schnell, 1994) and region CA3. Here we show that self-regulated learning and recall of self-organized representations can be obtained in a network simulation of the hippocampal formation. This model utilizes selective cholinergic suppression of synaptic transmission in stratum radiatum of region CA3, which has been demonstrated in brain slice preparations of the hippocampus. 2 METHODS 2.1. SIMPLIFIED REPRESENTA nON OF HIPPOCAMPAL NEURONS. In place of the sigmoid input-output functions used in many models, this model uses a simple representation in which the output of a neuron is not explicitly constrained, but the total network activity is regulated by feedback from inhibitory interneurons and adaptation due to intracellular calcium concentration. Separate variables represent pyramidal cell membrane potential a, intracellular calcium concentration c, and the membrane potential of inhibitory interneurons h: l1ai = Ai -l1ai - J..l.C + L Wijg(aj - e) - Hikg(hk - e h) j I1c· = 'Vg(a . - e ) - Qc I i· I C I1hk = IWkjg(aj-eo)-l1hk- IHk/g(h/-e) j / where A = afferent input, " = passive decay of membrane potential, Il = strength of calA Model of Hippocampus 79 cium-dependent potassium current (proportional to intracellular calcium), Wij = excitatory recurrent synapses (longitudinal association path tenninating in stratum radiatum), gO is a threshold linear function proportional to the amount by which membrane potential exceeds an output threshold 00 or threshold for calcium current Oc' 'Y = strength of voltagedependent calcium current, n = diffusion constant of calcium, Wki = excitatory synapses inhibitory interneurons, Hilc = inhibitory synapses from interneurons to pyramidal cells, Hk}= inhibitory synapses between interneurons. This representation gives neurons adaptation characteristics similar to those observed with intracellular recording (Barkai and Hasselmo, 1994), including a prominent afterhyperpolarization potential (see Figure 1). An B 'N~JJL ....J \..-C lO .... ~ .. -14 Figure 1. Comparison of pyramidal cell model with experimental data. In Figure I, A shows the membrane potential of a modeled pyramidal cell in response to simulated current injection. Output of this model is a continuous variable proportional to how much membrane potential exceeds threshold. This is analogous to the reciprocal of interspike interval in real neuronal recordings. Note that the model displays adaptation during current injection and afterhyperpolarization afterwards, due to the calcium-dependent potassium current. B shows the intracellularly recorded membrane potential in a pirifonn cortex pyramidal cell, demonstrating adaptation of firing frequency due to activation of calcium-dependent potassium current. The firing rate falls off in a manner similar to the smooth decrease in firing rate in the simplified representation. C shows an intracellular recording illustrating long-tenn afterhyperpolarization caused by calcium influx induced by spiking of the neuron during current injection. 2.2. NETWORK CONNECTIVITY A schematic representation of the network simulation of the hippocampal fonnation is shown in Figure 2. The anatomy of the hippocampal fonnation is summarized on the left in A, and the function of these different subregions in the model is shown on the right in B. Each of the subregions in the model contained a population of excitatory neurons with a single inhibitory interneuron mediating feedback inhibition and keeping excitatory activity bounded. Thus, the local activation dynamics in each region follow the equations presented above. The connectivity of the network is further summarized in Figure 3 in the Results section. A learning rule of the Hebbian type was utilized at all synaptic connections, with the exception of the mossy fibers from the dentate gyrus to region CA3, and the connections to and from the medial septum. Self-organization of perforant path synapses was obtained through decay of synapses with only pre or post-synaptic activity, and growth of synapses with combined activity. Associative memory function at synapses 80 Michael E. Hasse/mo, Eric Schnell, Joshua Berke, Edi Barkai arising from region CA3 was obtained through synaptic modification during cholinergic suppression of synaptic transmission. Feedback regulation of cholinergic modulation B Entorhinal cortex """""""""" • • • • • • : Self-organized : representation : r-----..L.\ • ~ Comparison .L-______ ~~.-----~~~--~~ Regulation of learning dynamics Figure 2. Schematic representation of hippocampal circuitry and the corresponding function of connections in the model. 2.3. CHOLINERGIC MODULA nON The total output from region CAl determined the level of cholinergic modulation within both region CA3 and CAl, with increased output causing decreased modulation. This is consistent with experimental evidence suggesting that activity in region CAl and region CA3 can inhibit activity in the medial septum, and thereby downregulate cholinergic modulation. This effect was obtained in the model by excitatory connections from region CAl to an inhibitory interneuron in the medial septum, which suppressed the activity of a cholinergic neuron providing modulation to the full network. When levels of cholinergic modulation were high, there was strong suppression of synaptic transmission at the excitatory recurrent synapses in CA3 and the Schaffer collaterals projecting from region CA3 to CAL This prevented the spread of activity due to previous learning from interfering with self-organization. When levels of cholinergic modulation were decreased, the strength of synaptic transmission was increased, allowing associative recall to dominate. Cholinergic modulation also increased the rate of synaptic modification and depolarized neurons. 2.4. TESTS OF SELF-REGULATED LEARNING AND RECALL Simulations of the full hippocampal network evaluated the response to the sequential presentation of a series of highly overlapping activity patterns in the entorhinal cortex. Recall was tested with interspersed presentation of degraded versions of previously presented activity patterns. For effective recall, the pattern of activity in entorhinal cortex layer IV evoked by degraded patterns matched the pattern evoked by the full learned version of these patterns. The function of the full network is illustrated in Figure 3. In simulations A Model of Hippocampus 81 focused on region CA3, activity patterns were induced sequentially in region CA3, representing afferent input from the entorhinal cortex. Different levels of external activation of the cholinergic neuron resulted in different levels of learning of new overlapping patterns. These results are illustrated in Figure 4. 2.5. BRAIN SLICE EXPERIMENTS The effects in the simulations of region CA3 depended upon the cholinergic suppression of synaptic transmission in stratum radiatum of this region The cholinergic suppression of glutamatergic synaptic transmission in region CA3 was tested in brain slice preparations by analysis of the influence of the cholinergic agonist carbachol on the size of field potentials elicited by stimulation of stratum radiatum. These experiments used techniques similar to previously published work in region CAl (Hasselmo and Schnell, 1994). 3 RESULTS In the full hippocampal simulation, input of an unfamiliar pattern to entorhinal cortex layer II resulted in high levels of acetylcholine. This allowed rapid self-organization of the perforant path input to the dentate gyrus and region CAl. Cholinergic suppression of synaptic transmission in region CAl prevented recall from interfering with self-organization. Instead, recurrent collaterals in region CA3 stored an autoassociative representation of the input from the dentate gyrus to region CA3, and connections from CA3 to CA 1 stored associations between the pattern of activity in CA3 and the associated self-organized representation in region CAl. 111111 2 I I I II j ld II r Q)2dl' r ~ 311111 4 n , n 3d I I I 4d II I ld II I 2d --I.~ £ Self-org ~ ~ I "T I' , " I "'"' I 'I I II' I I' I I I " I I I I II 1 I • iden~ity matrix If n I II , I'll n 'f n I II , 'I identity ~ " self-org "matrix ~ ,auto-" ,,~> M assoc u ..... « « 'at) u u c >. c • c • :.a.£:i .9 hetero.9 hetero8 ~ assoc ~ assoc C! ~ ~ I.Ll , r 'I I II , , U I I I U I H I I , , U I I I Neuron # .. , " • • f'l't 'I .. i " Itt r 1 I' ,I I I"'" " ~' , I I , , 1 II r 1 l 11 I J 1 't III I' , " I I I '( 1'1 I I ( U. I I I n , 'I II I II I W. , I I II I II • .1 l " I I I I III I I Jl 1 lU Figure 3. Activity in each subregion of the full network simulation of the hippocampal formation during presentation of a sequence of activity patterns in entorhinal cortex. 82 Michael E. Hasselmo, Eric Schnell, Joshua Berke, Edi Barkai In Figure 3. width of the lines represents the activity of each neuron at a particular time step. As seen here. the network forms a self-organized representation of each new pattern consisting of active neurons in the dentate gyrus and region CAL At the same time. an association is formed between the self-organized representation in region CAl and the same afferent input pattern presented to entorhinal cortex layer IV. Four overlapping patterns (1-4) are presented sequentially. each of which results in learning of a separate selforganized representation in the dentate gyrus and region CAl. with an association formed between this representation and the full input pattern in entorhinal cortex. The recall characteristics of the network are apparent when degraded versions of the afferent input patterns are presented in the sequence (ld-4d). This degraded afferent input weakly activates the same representations previously formed in the dentate gyrus. Recurrent excitation in region CA3 enhances this activity. giving robust recall of the full version of this pattern. This activity then reaches CA 1. where it causes strong activation if it matches the pattern of afferent input from the entorhinal cortex. Strong activation in region CAl decreases cholinergic modulation. preventing formation of a new representation and allowing recall to dominate. Strong activation of the representation stored in region CAl then activates the full representation of the pattern in entorhinal cortex layer IV. Thus. the network can accurately recall each of many highly overlapping patterns. The effect of cholinergic modulation on the level of learning or recall can be seen more clearly in a simulation of auto-associative memory function in region CA3 as shown in Figure 4. Each box shows the response of the network to sequential presentation of full and degraded versions of two highly overlapping input patterns. The width of the black traces represents the activity of each of 10 CA3 pyramidal cells during each simulation step. In the top row. level of cholinergic modulation (ACh) is plotted. In A. external activation of the cholinergic neuron is absent. so there is no cholinergic suppression of synaptic transmission. In this case. the first pattern is learned and recalled properly. but subsequent presentation of a second overlapping pattern results only in recall of the previously learned pattern. In B. with greater cholinergic suppression. recall is suppressed sufficiently to allow learning of a combination of the two input patterns. Finally. in C. strong cholinergic suppression prevents recall. allowing learning of the new overlapping pattern to dominate over the previously stored pattern. Stored A patterns ACh ••• Inhib Q\ •• •• .';:: ~ • ~ • 0 -.gN.g - N ACh input = 0.0 ..... ... - .. ,11",,11. . ... -..... -. N B ACh input = 0.15 C ACh input = 0.3 111 ... 111 ...... 11 .... "' ... 11 •• 111 •• , 11'_ I •• "' •• __ '111" ' ... _ .. , ....... , .. _ ...... .,~ . ",. ·' .. _"11 .. ··.'.· . .u .... .... , ........ . . 111 .. . .... . Figure 4. Increased cholinergic suppression of synaptic transmission in region CA3 causes greater learning of new aspects of afferent input patterns. A Model of Hippocampus 83 Extracellular recording in brain slice preparations of hippocampal region CA3 have demonstrated that perfusion of the cholinergic agonist carbachol strongly suppresses synaptic potentials recorded in stratum radiatum, as shown in Figure 5. In contrast, suppression of synaptic transmission at the afferent fiber synapses arising from entorhinal cortex is much weaker. At a concentration of 20J..tM, carbachol suppressed synaptic potentials in stratum radiatum on average by 54.4% (n=5). Synaptic potentials elicited in stratum lacunosum were more weakly suppressed, with an average suppression of28%. Control Carbachol (20JlM) Wash Figure 5. Cholinergic suppression of synaptic transmission in stratum radiatum of CA3. 4 DISCUSSION In this model of the hippocampus, self-organization at perforant path synapses forms compressed representations of specific patterns of cortical activity associated with events in the environment. Feedback regulation of cholinergic modulation sets appropriate dynamics for learning in response to novel stimuli, allowing predominance of self-organization, and appropriate dynamics for recall in response to familiar stimuli, allowing predominance of associative memory function. This combination of self-organization and associative memory function may also occur in neocortical structures. The selective cholinergic suppression of feedback and intrinsic synapses has been proposed to allow self-organization of feedforward synapses while feedback synapses mediate storage of associations between higher level representations and activity in primary cortical areas (Hasselmo, 1994b). This previous proposal could provide a physiological justification for a similar mechanism utilized in recent models (Dayan et al., 1995). Detailed modeling of cholinergic effects in the hippocampus provides a theoretical framework for linking the considerable behavioral evidence for a role of acetylcholine in memory function (Hagan and Morris, 1989) to the neurophysiological evidence for the effects of acetylcholine within cortical structures (Hasselmo and Bower, 1992; 1993; Hasselmo, 1994a, 1994b). 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862	Glove-TalkII: Mapping Hand Gestures to Speech Using Neural Networks S. Sidney Fels Department of Computer Science University of Toronto Toronto, ON, M5S lA4 ssfels@ai.toronto.edu Geoffrey Hinton Department of Computer Science University of Toronto Toronto, ON, M5S lA4 hinton@ai.toronto.edu Abstract Glove-TaikII is a system which translates hand gestures to speech through an adaptive interface. Hand gestures are mapped continuously to 10 control parameters of a parallel formant speech synthesizer. The mapping allows the hand to act as an artificial vocal tract that produces speech in real time. This gives an unlimited vocabulary in addition to direct control of fundamental frequency and volume. Currently, the best version of Glove-TalkII uses several input devices (including a CyberGlove, a ContactGlove, a 3space tracker, and a foot-pedal), a parallel formant speech synthesizer and 3 neural networks. The gesture-to-speech task is divided into vowel and consonant production by using a gating network to weight the outputs of a vowel and a consonant neural network. The gating network and the consonant network are trained with examples from the user. The vowel network implements a fixed, user-defined relationship between hand-position and vowel sound and does not require any training examples from the user. Volume, fundamental frequency and stop consonants are produced with a fixed mapping from the input devices. One subject has trained to speak intelligibly with Glove-TalkII. He speaks slowly with speech quality similar to a text-to-speech synthesizer but with far more natural-sounding pitch variations. 844 S. Sidney Fe Is, Geoffrey Hinton 1 Introduction There are many different possible schemes for converting hand gestures to speech. The choice of scheme depends on the granularity of the speech that you want to produce. Figure 1 identifies a spectrum defined by possible divisions of speech based on the duration of the sound for each granularity. What is interesting is that in general, the coarser the division of speech, the smaller the bandwidth necessary for the user. In contrast, where the granularity of speech is on the order of articulatory muscle movements (i.e. the artificial vocal tract [AVT]) high bandwidth control is necessary for good speech. Devices which implement this model of speech production are like musical instruments which produce speech sounds. The user must control the timing of sounds to produce speech much as a musician plays notes to produce music. The AVT allows unlimited vocabulary, control of pitch and non-verbal sounds. Glove-TalkII is an adaptive interface that implements an AVT. Translating gestures to speech using an AVT model has a long history beginning in the late 1700's. Systems developed include a bellows-driven hand-varied resonator tube with auxiliary controls (1790's [9]), a rubber-moulded skull with actuators for manipulating tongue and jaw position (1880's [1]) and a keyboard-footpedal interface controlling a set of linearly spaced bandpass frequency generators called the Yoder (1940 [3]). The Yoder was demonstrated at the World's Fair in 1939 by operators who had trained continuously for one year to learn to speak with the system. This suggests that the task of speaking with a gestural interface is very difficult and the training times could be significantly decreased with a better interface. GloveTalkII is implemented with neural networks which allows the system to learn the user's interpretation of an articulatory model of speaking. This paper begins with an overview of the whole Glove-TalkII system. Then, each neural network is described along with its training and test results. Finally, a qualitative analysis is provided of the speech produced by a single subject after 100 hours of speaking with Glove-TalkI!. Artificial Vocal Tract Phoneme Finger Syllable Word (AVT) Generator Spelling Generator Generator I > 10-30 100 130 200 600 Approximate time per gesture (msec) Figure 1: Spectrum of gesture-to-speech mappings based on the granularity of speech. 2 Overview of Glove-TalkII The Glove-TalkIl system converts hand gestures to speech, based on a gesture-toformant model. The gesture vocabulary is based on a vocal-articulator model of the hand. By dividing the mapping tasks into independent subtasks, a substantial reduction in network size and training time is possible (see [4]). Figure 2 illustrates the whole Glove-TalkIl system. Important features include the Glove-Talkll Rlabt Hand data ll.y.:r: roll.l!itch. yaw every 1/60 IIOCUld ~~ ~, k-.--'10 flell analea 4 abduction anclea thumb and pinkie rotation wrist pitch and yaw every 1/100 aecond Fllled Pitch Mappina VIC Doc:ision Network Vowel Network Ccnsonant Network Fllled SlOp Mappina 845 )))S~ I---~ Combinina Function Figure 2: Block diagram of Glove-TalkII: input from the user is measured by the Cyberglove, polhemus, keyboard and foot pedal, then mapped using neural networks and fixed functions to formant parameters which drive the parallel formant synthesizer [8]. three neural networks labeled vowel/consonant decision (V /C), vowel, and consonant. The V /C network is trained on data collected from the user to decide whether he wants to produce a vowel or a consonant sound. Likewise, the consonant network is trained to produce consonant sounds based on user-generated examples based on an initial gesture vocabulary. In contrast, the vowel network implements a fixed mapping between hand-positions and vowel phonemes defined by the user. Nine contact points measured on the user's left hand by a ContactGlove designate the nine stop consonants (B, D, G, J, P, T, K, CH, NG), because the dynamics of such sounds proved too fast to be controlled by the user. The foot pedal provides a volume control by adjusting the speech amplitude and this mapping is fixed. The fundamental frequency, which is related to the pitch of the speech, is determined by a fixed mapping from the user's hand height. The output of the system drives 10 control parameters of a parallel formant speech synthesizer every 10 msec. The 10 control parameters are: nasal amplitude (ALF), first, second and third formant frequency and amplitude (F1, A1, F2, A2, F3, A3), high frequency amplitude (AHF), degree of voicing (V) and fundamental frequency (FO). Each of the control parameters is quantized to 6 bits. Once trained, Glove-Talk II can be used as follows: to initiate speech, the user forms the hand shape of the first sound she intends to produce. She depresses the foot pedal and the sound comes out of the synthesizer. Vowels and consonants of various qualities are produced in a continuous fashion through the appropriate co-ordination of hand and foot motions. Words are formed by making the correct motions; for example, to say "hello" the user forms the "h" sound, depresses the foot pedal and quickly moves her hand to produce the "e" sound, then the "I" sound and finally the "0" sound. The user has complete control of the timing and quality of the individual sounds. The articulatory mapping between gestures and speech 846 Figure 3: Hand-position to Vowel Sound Mapping. The coordinates are specified relative to the origin at the sound A. The X and Y coordinates form a horizontal plane parallel to the floor when the user is sitting. The 11 cardinal phoneme targets are determined with the text-to-speech synthesizer. S. Sidney Fe/s, Geoffrey Hinton ... .-),~O""':--+--"'--7-... -~-~c- ---:-... --!----"~ Y(cm) I .. u is decided a priori. The mapping is based on a simplistic articulatory phonetic description of speech (5]. The X,Y coordinates (measured by the polhemus) are mapped to something like tongue position and height l producing vowels when the user's hand is in an open configuration (see figure 2 for the correspondence and table 1 for a typical vowel configuration). Manner and place of articulation for non-stop consonants are determined by opposition of the thumb with the index and middle fingers as described in table 1. The ring finger controls voicing. Only static articulatory configurations are used as training points for the neural networks, and the interpolation between them is a result of the learning but is not explicitly trained. Ideally, the transitions should also be learned, but in the text-to-speech formant data we use for training [6] these transitions are poor, and it is very hard to extract formant trajectories from real speech accurately. 2.1 The Vowel/Consonant (VIC) Network The VIC network decides, on the basis of the current configuration of the user's hand, to emit a vowel or a consonant sound. For the quantitative results reported here, we used a 10-5-1 feed-forward network with sigmoid activations [7]. The 10 inputs are ten scaled hand parameters measured with a Cyberglove: 8 flex angles (knuckle and middle joints of the thumb, index, middle and ring fingers), thumb abduction angle and thumb rotation angle. The output is a single number representing the probability that the hand configuration indicates a vowel. The output of the VIC network is used to gate the outputs of the vowel and consonant networks, which then produce a mixture of vowel and consonant formant parameters. The training data available includes only user-produced vowel or consonant sounds. The network interpolates between hand configurations to create a smooth but fairly rapid transition between vowels and consonants. For quantitative analysis, typical training data consists of 2600 examples of consonant configurations (350 approximants, 1510 fricatives [and aspirant], and 740 nasals) and 700 examples of vowel configurations. The consonant examples were obtained from training data collected for the consonant network by an expert user. The vowel examples were collected from the user by requiring him to move his hand in vowel configurations for a specified amount of time. This procedure was performed in several sessions. The test set consists of 1614 examples (1380 consonants and 234 vowels). After training,2 the mean squared error on the training and test lIn reality, the XY coordinates map more closely to changes in the first two formants, FI and F2 of vowels. From the user's perspective though, the link to tongue movement is useful. 2The V Ie network, the vowel network and the consonant network are trained using Glove-Talkll 847 ~ ~ ~ ~ '.~ '.:~:~:::. , . . .;;J(il. ''':;;1.. ., ~ • •• :$ ••• . ... ). ... .. DH F H L M 7·: .... :· , I.!:. ~ .:::~::. ~ ~:;§::. " ~':" ':: .. ' ... ~: to:.·.·. ":"~"'" N R S SH TH ~ ~ ~ ~ .. ' :<: • . ~ , ' . '.>' ":'s ~.,,> • ~ ....... . (, ~~. • 0:' :~: • ••• '<1. •• V W Z ZH vowel Table 1: Static Gesture-to-Consonant Mapping for all phonemes. Note, each gesture corresponds to a static non-stop consonant phoneme generated by the text-to-speech synthesizer. set was less than 10-4 . During normal speaking neither network made perceptual errors. The decision boundary feels quite sharp, and provides very predictable, quick transitions from vowels to consonants and back. Also, vowel sounds are produced when the user hyperextends his hand. Any unusual configurations that would intuitively be expected to produce consonant sounds do indeed produce consonant sounds. 2.2 The Vowel Network The vowel network is a 2-11-8 feed forward network. The 11 hidden units are normalized radial basis functions (RBFs) [2] which are centered to respond to one of 11 cardinal vowels. The outputs are sigmoid units representing 8 synthesizer control parameters (ALF, F1, AI, F2, A2, F3, A3, AHF). The radial basis function used is: L(Wji-O.)~ oj=e<l'j2 (1) where OJ is the (un-normalized) output of the RBF unit, Wji is the weight from unit i to unit j, 0i is the output of input unit i, and (1/ is the variance of the RBF. The normalization used is: O· nj = L J (2) mEpom where nj is the normalized output of unit j and the summation is over all the units in the group of normalized RBF units. The centres of the RBF units are fixed conjugate gradient descent and a line search. 848 S. Sidney Fels, Geoffrey Hinton according to the X and Y values of each of the 11 vowels in the predefined mapping (see figure 2). The variances of the 11 RBF's are set to 0.025. The weights from the RBF units to the output units are trained. For the training data, 100 identical examples of each vowel are generated from their corresponding X and Y positions in the user-defined mapping, providing 1100 examples. Noise is then added to the scaled X and Y coordinates for each example. The added noise is uniformly distributed in the range -0.025 to 0.025. In terms of unscaled ranges, these correspond to an X range of approximately ± 0.5 cm and a Y range of ± 0.26 cm. Three different test sets were created. Each test set had 50 examples of each vowel for a total of 550 examples. The first test set used additive uniform noise in the interval ± 0.025. The second and third test sets used additive uniform noise in the interval ± 0.05 and ± 0.1 respectively. The mean squared error on the training set was 0.0016. The MSE on the additive noise test sets (noise = ± 0.025, 0.05 and 0.01) was 0.0018, 0.0038, 0.0120 which corresponds to expected errors of 1.1 %, 3.1 % and 5.5% in the formant parameters, respectively. This network performs well perceptually. The key feature is the normalization of the RBF units. Often, when speaking, the user will overshoot cardinal vowel positions (especially when she is producing dipthongs) and all the RBF units will be quite suppressed. However, the normalization magnifies any slight difference between the activities of the units and the sound produced will be dominated by the cardinal vowel corresponding to the one whose centre is closest in hand space. 2.3 The Consonant Network The consonant network is a 10-14-9 feed-forward network. The 14 hidden units are normalized RBF units. Each RBF is centred at a hand configuration determined from training data collected from the user corresponding to one of 14 static consonant phonemes. The target consonants are created with a text-to-speech synthesizer. Figure 1 defines the initial mapping for each of the 14 consonants. The 9 sigmoid output units represent 9 control parameters of the formant synthesizer (ALF, F1, AI, F2, A2, F3, A3, AHF, V). The voicing parameter is required since consonant sounds have different degrees of voicing. The inputs are the same as for the manager V Ie network. Training and test data for the consonant network is obtained from the user. Target data is created for each of the 14 consonant sounds using the text-to-speech synthesizer. The scheme to collect data for a single consonant is: 1. The target consonant is played for 100 msec through the speech synthesizer; 2. the user forms a hand configuration corresponding to the consonant; 3. the user depresses the foot pedal to begin recording; the start of recording is indicated by the appearance of a green square; 4. 10-15 time steps of hand data are collected and stored with the corresponding formant parameter targets and phoneme identifier; the end of data collection is indicated by turning the green square red; 5. the user chooses whether to save the data to a file, and whether to redo the current target or move to the next one. Glove-Talkll 849 Using this procedure 350 approximants, 1510 fricatives and 700 nasals were collected and scaled for the training data. The hand data were averaged for each consonant sound to form the RBF centres. For the test data, 255 approximants, 960 fricatives and 165 nasals were collected and scaled. The RBF variances were set to 0.05. The mean square error on the training set was 0.005 and on the testing set was 0.01 corresponding to expected errors of 3.3% and 4.7% in the formant parameters, respectively. Listening to the output of the network reveals that each sound is produced reasonably well when the user's hand is held in a fixed position. The only difficulty is that the Rand L sounds are very sensitive to motion of the index finger. 3 Qualitative Performance of Glove-TalkII One subject, who is an accomplished pianist, has been trained extensively to speak with Glove-TalkII. We expected that his pianistic skill in forming finger patterns and his musical training would help him learn to speak with Glove-TalkII. After 100 hours of training, his speech with Glove-TalklI is intelligible and somewhat natural-sounding. He still finds it difficult to speak quickly, pronounce polysyllabic words, and speak spontaneously. During his training, Glove-TalkII also adapted to suit changes required by the subject. Initially, good performance of the VIC network is critical for the user to learn to speak. If the V Ie network performs poorly the user hears a mixture of vowel and consonant sounds making it difficult to adjust his hand configurations to say different utterances. For this reason, it is important to have the user comfortable with the initial mapping so that the training data collected leads to the VIC network performing well. In the 100 hours of practice, Glove-Talk II was retrained about 10 times. Four significant changes were made from the original system analysed here for the new subject. First, the NG sound was added to the non-stop consonant list by adding an additional hand shape, namely the user touches his pinkie to his thumb on his right hand. To accomodate this change, the consonant and VIC network had two inputs added to represent the two flex angles of the pinkie. Also, the consonant network has an extra hidden unit for the NG sound. Second, the consonant network was trained to allow the RBF centres to change. After the hidden-to-output weights were trained until little improvement was seen, the input-to-hidden weights (i.e. the RBF centres) were also allowed to adapt. This noticeably improved performance for the user. Third, the vowel mapping was altered so that the I was moved closer to the EE sound and the entire mapping was reduced to 75% of its size. Fourth, for this subject, the VIC network needed was a 10-10-1 feed-forward sigmoid unit network. Understanding the interaction between the user's adaptation and Glove-TalkII's adaptation remains an interesting research pursuit. 4 Summary The initial mapping is loosely based on an articulatory model of speech. An open configuration of the hand corresponds to an unobstructed vocal tract, which in turn generates vowel sounds. Different vowel sounds are produced by movements of the hand in a horizontal X-Y plane that corresponds to movements of the first two formants which are roughly related to tongue position. Consonants other than stops are produced by closing the index, middle, or ring fingers or flexing the thumb, representing constrictions in the vocal tract. Stop consonants are produced by 850 S. Sidney Fels, Geoffrey Hinton contact switches worn on the user's left hand. FO is controlled by hand height and speaking intensity by foot pedal depression. Glove-TaikII learns the user's interpretation of this initial mapping. The VIC network and the consonant network learn the mapping from examples generated by the user during phases of training. The vowel network is trained on examples computed from the user-defined mapping between hand-position and vowels. The FO and volume mappings are non-adaptive. One subject was trained to use Glove-TalkII. After 100 hours of practice he is able to speak intelligibly. His speech is fairly slow (1.5 to 3 times slower than normal speech) and somewhat robotic. It sounds similar to speech produced with a text-tospeech synthesizer but has a more natural intonation contour which greatly improves the intelligibility and naturalness of the speech. Reading novel passages intelligibly usually requires several attempts, especially with polysyllabic words. Intelligible spontaneous speech is possible but difficult. Acknowledgements We thank Peter Dayan, Sageev Oore and Mike Revow for their contributions. This research was funded by the Institute for Robotics and Intelligent Systems and NSERC. Geoffrey Hinton is the Noranda fellow of the Canadian Institute for Advanced Research. References [1] A. G. Bell. Making a talking-machine. In Beinn Bhreagh Recorder, pages 61-72, November 1909~ [2] D. Broomhead and D. Lowe. Multivariable functional interpolation and adaptive networks. Complex Systems, 2:321-355, 1988. [3] Homer Dudley, R. R. Riesz, and S. S. A. Watkins. A synthetic speaker. Journal of the Franklin Institute, 227(6):739-764, June 1939. [4] S. S. Fels. Building adaptive interfaces using neural networks: The Glove-Talk pilot study. Technical Report CRG-TR-90-1, University of Toronto, 1990. [5] P. Ladefoged. A course in Phonetics (2 ed.). Harcourt Brace Javanovich, New York, 1982. [6] E. Lewis. A 'C' implementation of the JSRU text-to-speech system. Technical report, Computer Science Dept., University of Bristol, 1989. [7] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by back-propagating errors. Nature, 323:533-536, 1986. [8] J. M. Rye and J. N. Holmes. A versatile software parallel-formant speech synthesizer. Technical Report JSRU-RR-1016, Joint Speech Research Unit, Malvern, UK, 1982. [9] Wolfgang Ritter von Kempelen. Mechanismus der menschlichen Sprache nebst Beschreibungeiner sprechenden Maschine. Mit einer Einleitung vonHerbert E. Brekle und Wolfgang Wild. Stuttgart-Bad Cannstatt F. Frommann, Stuttgart, 1970.	1994	103
863	Learning in large linear perceptrons and why the thermodynamic limit is relevant to the real world Peter Sollich Department of Physics, University of Edinburgh Edinburgh EH9 3JZ, U.K. P.Sollich~ed.ac.uk Abstract We present a new method for obtaining the response function 9 and its average G from which most of the properties of learning and generalization in linear perceptrons can be derived. We first rederive the known results for the 'thermodynamic limit' of infinite perceptron size N and show explicitly that 9 is self-averaging in this limit. We then discuss extensions of our method to more general learning scenarios with anisotropic teacher space priors, input distributions, and weight decay terms. Finally, we use our method to calculate the finite N corrections of order 1/ N to G and discuss the corresponding finite size effects on generalization and learning dynamics. An important spin-off is the observation that results obtained in the thermodynamic limit are often directly relevant to systems of fairly modest, 'real-world' sizes. 1 INTRODUCTION One of the main areas of research within the Neural Networks community is the issue of learning and generalization. Starting from a set of training examples (normally assumed to be input-output pairs) generated by some unknown 'teacher' rule V, one wants to find, using a suitable learning or training algorithm, a student N (read 'Neural Network') which generalizes from the training set, i.e., predicts the outputs corresponding to inputs not contained in the training set as accurately as possible. 208 Peter Sollich If the inputs are N-dimensional vectors x E nN and the outputs are scalars yEn, then one of the simplest functional forms that can be assumed for the student .N is the linear perceptron, which is parametrized in terms of a weight vector W N E nN and implements the linear input-output mapping YN(X) = 7Nw~x. (1) A commonly used learning algorithm for the linear perceptron is gradient descent on the training error, i.e., the error that the student .N makes on the training set. Using the standard squared output deviation error measure, the training error for a given set ofp training examples {(xlJ , ylJ),J-l = 1 . . . p} is Et = L:IJ ~(ylJ-YN(XIJ))2 = ~ L:IJ (ylJ w~xlJ / VN)2. To prevent the student from fitting noise in the training data, a quadratic weight decay term ~AW~ is normally added to the training error, with the value of the weight decay parameter A determining how strongly large weight vectors are penalized. Gradient descent is thus performed on the function E = Et + ~AW~, and the corresponding learning dynamics is, in a continuous time approximation, dw N / dt = -"V wE. As discussed in detail by Krogh and Hertz (1992), this results in an exponential approach of W N to its asymptotic value, with decay constants given by the eigenvalues of the matrix M N , defined by (1 denotes the N x N identity matrix) MN = Al + A, A = ~ L:IJ xlJ(xlJ)T. To examine what generalization performance is achieved by the above learning algorithm, one has to make an assumption about the functional form of the teacher. The simplest such assumption is that the problem is learnable, i. e., that the teacher, like the student, is a linear perceptron. A teacher V is then specified by a weight vector Wv and maps a given input x to the output Yv(x) = w~ x/VN. We assume that the test inputs for which the student is asked to predict the corresponding outputs are drawn from an isotropic Gaussian distribution, P(x) <X exp( _~x2). The generalization error, i. e., the average error that a student .N makes on a random input when compared to teacher V, is given by fg = ~((YN(X) - yv(x))2)p(x) = 2~(WN - w v)2. (2) Inserting the learning dynamics W N = W N(t), the generalization acquires a time dependence, which in its exact form depends on the specific training set, teacher, and initial value of the student weight vector, WN(t = 0). We shall confine our attention to the average of this time-dependent generalization error over all possible training sets and teachers; to avoid clutter, we write this average simply as fg(t). We assume that the inputs x lJ in the training set are chosen independently and randomly from the same distribution as the test inputs, and that the corresponding training outputs are the teacher outputs corrupted by additive noise, ylJ = yv(xlJ) + 171J, where the 171J have zero mean and variance rr2. If we further assume an isotropic Gaussian prior on the teacher weight vectors, P(wv ) <X exp( -~w~), then the average generalization error for t -> 00 is (Krogh and Hertz, 1992) 1 [ OC] fg(t -> 00) = 2 rr2C + A(rr2 - A) OA ' (3) where G is the average of the so-called response function over the training inputs: C (Q) Q 1 t M- 1 (4) = P( {x"}), N r N· Learning in Large Linear Perceptrons 209 The time dependence of the average generalization error for finite but large t is an exponential approach to the asymptotic value (3) with decay constant oX + amin, where am in is the lowest eigenvalue occurring in the average eigenvalue spectrum of the input correlation matrix A (Krogh and Hertz, 1992). This average eigenvalue spectrum, which we denote by p(a), can be calculated from the average response function according to (Krogh, 1992) p(a) = ! lim ImGI.>.=_a-if, 11' f-O+ (5) where we have assumed p( a) to be normalized, fda p( a) = 1. Eqs. (3,5) show that the key quantity determining learning and generalization in the linear percept ron is the average response function G defined in (4). This function has previously been calculated in the 'thermodynamic limit', N -- 00 at a = piN = const., using a diagrammatic expansion (Hertz et al., 1989) and the replica method (Opper, 1989, Kinzel and Opper, 1991). In Section 2, we present what we believe to be a much simpler method for calculating G, based only on simple matrix identities. We also show explicitly that 9 is self-averaging in the thermodynamic limit, which means that the fluctuations of 9 around its average G become vanishingly small as N -> 00 . This implies, for example, that the generalization error is also self-averaging. In Section 3 we extend the method to more general cases such as anisotropic teacher space priors and input distributions, and general quadratic penalty terms. Finite size effects are considered in Section 4, where we calculate the O(IIN) corrections to G, €g(t -- (0) and p(a). We discuss the resulting effects on generalization and learning dynamics and derive explicit conditions on the perceptron size N for results obtained in the thermodynamic limit to be valid. We conclude in Section 5 with a brief summary and discussion of our results. 2 THE BASIC METHOD Our method for calculating the average response function G is based on a recursion relation relating the values of the (unaveraged) response function 9 for p and p + 1 training examples. Assume that we are given a set of p training examples with corresponding matrix M N . By adding a new training example with input x, we obtain the matrix Mt = MN + ~ xxT . It is straightforward to show that the inverse of Mt can be expressed as 1 M-1 TM-1 ( + ) - 1 _ -1 IV N XX N MN - MN 1 . 1 + 1 TMN X N x (One way of proving this identi~ is to multiply both sides by Mt .and exploit ~he fact that MtM~',l = 1 + kxx M;~} . ) TaklOg the trace, we obtalO the follow 109 recursion relation for g: 1 ..!.x™-2x 9 (p + 1) = 9 (p) - N 1 N 1 T ~ -1 . + N X N X (6) Now denote Zi = ~xTMNix (i = 1,2). With x drawn randomly from the assumed input distribution P(x) ex: exp( ~x2), the Zi can readily be shown to be random 210 Peter Sollich variables with means and (co-)variances () 1 M-i (A A) 2 M-i-j Zi = N tr N' ~Zi~Zj = N1 tr N • Combining this with the fact that tr M:vk ~ N>.-k = O(N), we have that the fluctuations LlZi of the Zi around their average values are 0(1/.JN); inserting this into (6), we obtain 9(p + 1) lIt M-2 9(p) _ N r N + 0(N-3/2) N 1 + ~trM:vl 9(p) + ~ a9(p) 1 + 0(N-3/2). N ()>. 1 + 9(p) (7) Starting from 9(0) = 1/>., we can apply this recursion p = aN times to obtain 9(p) up to terms which add up to at most 0(pN-3/2) = 0(1/.JN). This shows that 9 is self-averaging in the thermodynamic limit: whatever the training set, the value of 9 will always be the same up to fluctuations of 0(1/.JN). In fact, we shall show in Section 4 that the fluctuations of 9 are only 0(1/ N). This means that the 0(N-3/ 2) fluctuations from each iteration of (7) are only weakly correlated, so that they add up like independent random variables to give a total fluctuation for 9(p) of O«p/ N 3)1/2) = 0(1/ N). We have seen that, in the thermodynamic limit, 9 is identical to its average G because its fluctuations are vanishingly small. To calculate the value of G in the thermodynamic limit as a function of a and >., we insert the relation 9(p+1)-9(p) = -ka9(a)/aa + 0(1/ N 2 ) into eq. (7) (with 9 replaced by G) and neglect all finite N corrections. This yields the partial differential equation aG aG 1 aa a>. 1 + G = 0, (8) which can readily be solved using the method of characteristic curves (see, e.g., John, 1978). Using the initial condition Glo:o = 1/>. gives a/(l + G) = l/G - >., which leads to the well-known result (see, e.g., Hertz et al., 1989) G = ;>. (1 - a - >. + J(l - a - >.)2 + 4>') . (9) In the complex >. plane, G has a pole at >. = 0 and a branch cut arising from the root; according to eq. (5), these singularities determine the average eigenvalue spectrum pea) of A, with the result (Krogh, 1992) pea) = (1- a)8(1 - a)6(a) + -2 1 J(a+ - a)(a - a_), (10) 1ra where 8(x) is the Heaviside step function, 8(x) = 1 for x > 0 and 0 otherwise. The root in eq. (10) only contributes when its argument is non-negative, i.e., for a between the 'spectral limits' a_ and a+, which have the values a± = (1 ± fo)2. 3 EXTENSIONS TO MORE GENERAL LEARNING SCENARIOS We now discuss some extensions of our method to more general learning scenarios. First, consider the case of an anisotropic teacher space prior, P(wv ) ex: Learning in Large Linear Perceptrons 211 exp(-!w~:E~lWV)' with symmetric positive definite :Ev. This leaves the definition of the response function unchanged; eq. (3), however, has to be replaced by fg(t -- 00) = 1/2{q2G + A[q2 A(~tr :Ev)]oGloA}. As a second extension, assume that the inputs are drawn from an anisotropic distribution, P(x) oc exp(-~xT:E-1x). It can then be shown that the asymptotic value of the average generalization error is still given by eq. (3) if the response function is redefined to be 9 = ~ tr :EM~l. This modified response function can be calculated as follows: First we rewrite 9 as ~tr (A:E- 1 + A)-I, where A = ~ L:JJ(xJJ)TxJJ is the correlation matrix of the transformed input examples xl' = :E-1/2x JJ . Since the xJJ are distributed according to P(xJJ ) oc exp( _~(xJJ)2), the problem is thus reduced to finding the response function 9 = ~ tr (L + A)-1 for isotropically distributed inputs and L = A:E- 1. The recursion relations between 9(p + 1) and 9(p) derived in the previous section remain valid, and result, in the thermodynamic limit, in a ?i!f~rential ~~uat.ion for the aver are response function ~ analo.gous. t? eq. (8). The 1mt1al cond1tlon 1S now Gla=o = N tr L -1, and one obtams an 1mphc1t equatiOn for G, 1 ( a )-1 G = N tr L + 1 + G 1 , (11) where in the case of an anisotropic input distribution considered here, L = A:E- 1. If :E has a particularly simple form, then the dependence of G on a and A can be obtained analytically, but in general eq. (11) has to solved numerically. Finally, one can also investigate the effect of a general quadratic weight decay term, !W;:AWN' in the energy function E. The expression for the average generalization error becomes more cumbersome than eq. (3) in this case, but the result can still be expressed in terms of the average response function G = (9) = (~tr (A + A)-l), which can be obtained as the solution of eq. (11) for L = A. 4 FINITE N CORRECTIONS So far, we have focussed attention on the thermodynamic limit of perceptrons of infinite size N. The results are clearly only approximately valid for real, finite systems, and it is therefore interesting to investigate corrections for finite N. This we do in the present section by calculating the O(IIN) corrections to G and pea). For details of the calculations and results of computer simulations which support our theoretical analysis, we refer the reader to (Sollich, 1994). First note that, for A = 0, the exact result for the average response function is GI.~=o = (a - 1 - IIN)-l for a > 1 + liN (see, e.g., Eaton, 1983), which clearly admits a series expansion in powers of liN. We assume that a similar expansion also exists for nonzero A, and write (12) Go is the value of G in the thermodynamic limit as given by eq. (9). For finite N, the fluctuations ~9 = 9-G of9 around its average value G become relevant; for A = 0, the variance of these fluctuations is known to have a power series expansion in II N, and again we assume a similar expansion for finite A, ((~9)2) = ~2 IN + 0(11 N2), 212 Peter Sollich where the first term is 0(1/ N) and not 0(1) because, as discussed in Section 2, the fluctuations ofg for large N are no greater than 0(1/v'N). To calculate G I and A 2, one starts again from the recursion relation (6), now expanding everything up to second order in the fluctuation quantities AZi and Ag. Averaging over the training inputs and collecting orders of l/N yields after some straightforward algebra the known eq. (8) for Go and two linear partial differential equations for GI and A 2 , the latter obtained by squaring both sides of eq. (6) . Solving these, one obtains G _ G6(1 - AGO) I (1 + AG6)2 (13) In the limit A -- 0, G 1 = l/(a - 1)2 consistent with the exact result for G quoted above; likewise, the result A 2 == 0 agrees with the exact series expansion of the variance of the fluctuations of 9 for A = 0, which begins with an 0(1/ N 2 ) term (see, e.g., Barber et ai., 1994). (a) (b) 0.5 1.5 -- A=O.OOl 0.4 -------. A = 0.1 1.0 0.3 0.5 0.2 0.1 0.0 0.0 -0.5 0.0 0.5 1.0 a 1.5 2.0 0.0 0.5 1.0 a 1.5 2.0 Figure 1: Average generalization error: Result for N -- 00, (g,O, and coefficient of O(l/N) correction, (g,l. (a) Noise free teacher, (72 = O. (b) Noisy teacher, (72 = 0.5. Curves are labeled by the value of the weight decay parameter A. From the l/N expansion (12) of G we obtain, using eq. (3), a corresponding expansion of the asymptotic value of the average generalization error, which we write as (g(t -- 00) = (g,O + (g,d N + 0(1/ N2). It follows that the thermodynamic limit result for the average generalization error, (g,O, is a good approximation to the true result for finite N as long as N ~ Nc = /fg,I/(g,ol. In Figure 1, we plot (g,O and (g,l for several values of A and (j2 . It can be seen that the relative size of the first order correction l(g,I/(g,ol and hence the critical system size Nc for validity of the thermodynamic limit result is largest when A is small. Exploiting this fact, Nc can be bounded by 1/(1 - a) for a < 1 and (3a + l)/[a(a - l)J for a > l. It follows, for example, that the critical system size Nc is smaller than 5 as long as a < 0.8 or a > l.72, for all A and (j2. This bound on Nc can be tightened for non-zero A; for A > 2, for example, one has Nc < (2A - l)/(A + 1)2 < 1/3. We have thus shown explicitly that thermodynamic limit calculations of learning and generalization behaviour can be relevant for fairly small, 'real-world' systems of size N of the order of a few tens or hundreds. This is in contrast to the widespread suspicion Learning in Large Linear Perceptrons 213 among non-physicists that the methods of statistical physics give valid results only for huge system sizes of the order of N :::::: 1023 . (a) po PI (1 - 0-)0(1 - 0-) a I 4 (b) a __ ----_ I 4 a Figure 2: Schematic plot of the average eigenvalue spectrum p( a) of the input correlation matrix A. (a) Result for N -+ 00, po(a). (b) O(l/N) correction, Pl(a). Arrows indicate 6-peaks and are labeled by the corresponding heights. We now consider the 0(1/ N) correction to the average eigenvalue spectrum of the input correlation matrix A. Setting p(a) = po(a) + pda)/ N + 0(1/ N 2 ), po(a) is the N -+ 00 result given by eq. (10), and from eq. (13) one derives Figure 2 shows sketches of Po (a) and PI (a). Note that fda PI (a) = 0 as expected since the normalization of p( a) is independent of N. Furthermore, there is no O(l/N) correction to the 6-peak in po(a) at a = 0, since this peak arises from the N - p zero eigenvalues of A for c.r = p/ N < 1 and therefore has a height of 1- c.r for any finite N. The 6-peaks in PI (a) at the spectral limits a+ and a_ are an artefact of the truncated l/N expansion: p(a) is determined by the singularities of G as a function of A, and the location of these singularities is only obtained correctly by resumming the full l/N expansion. The 6-peaks in pl(a) can be interpreted as 'precursors' of a broadening of the eigenvalue spectrum of A to values which, when the whole 1/ N series is resummed, will lie outside the N -+ 00 spectral range [a_ , a+]. The negative term in PI (a) represents the corresponding 'flattening' of the eigenvalue spectrum between a_ and a+ . We can thus conclude that the average eigenvalue spectrum of A for finite N will be broader than for N -+ 00, which means in particular that the learning dynamics will be slowed down since the smallest eigenvalue amin of A will be smaller than a_. From our result for PI (a) we can also deduce when the N -+ 00 result po(a) is valid for finite N; the condition turns out to be N ~ a/[(a+ - a)(a - a_)]. Consistent with our discussion of the broadening of the eigenvalue spectrum of A, N has to be larger for a near the spectral limits a_, a+ if po(a) is to be a good approximation to the finite N average eigenvalue spectrum of A. 214 Peter Sollich 5 SUMMARY AND DISCUSSION We have presented a new method, based on simple matrix identities, for calculating the response function 9 and its average G which determine most of the properties of learning and generalization in linear perceptrons. In the thermodynamic limit, N --+ 00, we have recovered the known result for G and have shown explicitly that 9 is self-averaging. Extensions of our method to more general learning scenarios have also been discussed. Finally, we have obtained the 0(1/ N) corrections to G and the corresponding corrections to the average generalization error, and shown explicitly that the results obtained in the thermodynamic limit can be valid for fairly small, 'real-world' system sizes N . We have also calculated the 0(1/ N) correction to the average eigenvalue spectrum of the input correlation matrix A and interpreted it in terms of a broadening of the spectrum for finite N, which will cause a slowing down of the learning dynamics. We remark that the O( 1/ N) corrections that we have obtained can also be used in different contexts, for example for calculations of test error fluctuations and optimal test set size (Barber et al., 1994). Another application is in an analysis of the evidence procedure in Bayesian inference for finite N, where optimal values of 'hyperparameters' like the weight decay parameter A are determined on the basis of the training data (G Marion, in preparation). We hope, therefore, that our results will pave the way for a systematic investigation of finite size effects in learning and generalization. References D Barber, D Saad, and P Sollich (1994). Finite size effects and optimal test set size in linear perceptrons. Submitted to J. Phys. A. M LEaton (1983). Multivariate Statistics - A Vector Space Approach. Wiley, New York. J A Hertz, A Krogh, and G I Thorbergsson (1989). Phase transitions in simple learning. J. Phys. A, 22:2133-2150. F John (1978). Partial Differential Equations. Springer, New York, 3rd ed. W Kinzel and M Opper (1991). Dynamics of learning. In E Domany, J L van Hemmen, and K Schulten, editors, Models of Neural Networks, pages 149-171. Springer, Berlin. A Krogh (1992). Learning with noise in a linear perceptron. J. Phys. A, 25:11191133. A Krogh and J A Hertz (1992). Generalization in a linear percept ron in the presence of noise. J. Phys. A, 25:1135-1147. M Opper (1989). Learning in neural networks: Solvable dynamics. Europhysics Letters, 8:389-392. P Sollich (1994). Finite-size effects in learning and generalization in linear perceptrons. J. Phys. A, 27:7771-7784.	1994	104
864	Learning Saccadic Eye Movements Using Multiscale Spatial Filters Rajesh P.N. Rao and Dana H. Ballard Department of Computer Science University of Rochester Rochester, NY 14627 {rao)dana}~cs.rochester.edu Abstract We describe a framework for learning saccadic eye movements using a photometric representation of target points in natural scenes. The representation takes the form of a high-dimensional vector comprised of the responses of spatial filters at different orientations and scales. We first demonstrate the use of this response vector in the task of locating previously foveated points in a scene and subsequently use this property in a multisaccade strategy to derive an adaptive motor map for delivering accurate saccades. 1 Introduction There has been recent interest in the use of space-variant sensors in active vision systems for tasks such as visual search and object tracking [14]. Such sensors realize the simultaneous need for wide field-of-view and good visual acuity. One popular class of space-variant sensors is formed by log-polar sensors which have a small area near the optical axis of greatly increased resolution (the fovea) coupled with a peripheral region that witnesses a gradual logarithmic falloff in resolution as one moves radially outward. These sensors are inspired by similar structures found in the primate retina where one finds both a peripheral region of gradually decreasing acuity and a circularly symmetric area centmlis characterized by a greater density of receptors and a disproportionate representation in the optic nerve [3]. The peripheral region, though of low visual acuity, is more sensitive to light intensity and movement. The existence of a region optimized for discrimination and recognition surrounded by a region geared towards detection thus allows the image of an object of interest detected in the outer region to be placed on the more analytic center for closer scrutiny. Such a strategy however necessitates the existence of (a) methods to determine which location in the periphery to foveate next, and (b) fast gaze-shifting mechanisms to achieve this 894 Rajesh P. N. Rao, Dana H. Ballard foveation. In the case of humans, the "where-to-Iook-next" issue is addressed by both bottom-up strategies such as motion or salience clues from the periphery as well as topdown strategies such as search for a particular form or color. Gaze-shifting is accomplished via very rapid eye movements called saccades. Due to their high velocities, guidance through visual feedback is not possible and hence, saccadic movement is preprogrammed or ballistic: a pattern of muscle activation is calculated in advance that will direct the fovea almost exactly to the desired position [3]. In this paper, we describe an iconic representation of scene points that facilitates topdown foveal targeting. The representation takes the form of a high-dimensional vector comprised of the responses of different order Gaussian derivative filters, which are known to form the principal components of natural images [5], at variety of orientations and scales. Such a representation has been recently shown to be useful for visual tasks ranging from texture segmentation [7] to object indexing using a sparse distributed memory [11]. We describe how this photometric representation of scene points can be used in locating previously foveated points when a log-polar sensor is being used. This property is then used in a simple learning strategy that makes use of multiple corrective saccades to adaptively form a retinotopic motor map similar in spirit to the one known to exist in the deep layers of the primate superior colliculus [13]. Our approach differs from previous strategies for learning motor maps (for instance, [12]) in that we use the visual modality to actively supply the necessary reinforcement signal required during the motor learning step (Section 3.2). 2 The Multiscale Spatial Filter Representation In the active vision framework, vision is seen as subserving a larger context of the encompassing behaviors that the agent is engaged in. For these behaviors, it is often possible to use temporary, iconic descriptions of the scene which are only relatively insensitive to variations in the view. Iconic scene descriptions can be obtained, for instance, by employing a bank of linear spatial filters at a variety of orientations and scales. In our approach, we use derivative of Gaussian filters since these are known to form the dominant eigenvectors of natural images [5] and can thus be expected to yield reliable results when used as basis functions for indexingl . The exact number of Gaussian derivative basis functions used is motivated by the need to make the representations invariant to rotations in the image plane (see [11] for more details). This invariance can be achieved by exploiting the property of steerability [4] which allows filter responses at arbitrary orientations to be synthesized from a finite set of basis filters. In particular, our implementation uses a minimal basis set of two firstorder directional derivatives at 0° and 90°, three second-order derivatives at 0°, 60° and 120°, and four third-order derivatives oriented at 0°, 45°, 90°, and 135°. The response of an image patch J centered at (xo, Yo) to a particular basis filter G~j can be obtained by convolving the image patch with the filter : g . If 9 ri,j(XO, Yo) = (G/ * I)(xo, Yo) = G/ (XO - x, Yo - y)J(x, y)dx dy (1) lIn addition, these filters are endorsed by recent physiological studies [15] which show that derivative-of-Gaussians provide the best fit to primate cortical receptive field profiles among the different functions suggested in the literature. Learning Saccadic Eye Movements Using Multiscale Spatial Filters 895 The iconic representation for the local image patch centered at (xo, Yo) is formed by combining into a single high-dimensional vector the responses from the nine basis filters, each (in our current implementation) at five different scales: r(xo, Yo) = (ri,j,s) , i = 1,2, 3;j = 1, . .. , i + 1; S = Smin, . .. , Smax (2) where i denotes the order of the filter, j denotes the number of filters per order, and S denotes the number of different scales. The use of multiple scales increases the perspicuity of the representation and allows interpolation strategies for scale invariance (see [9] for more details). The entire representation can be computed using only nine convolutions done at frame-rate within a pipeline image processor with nine constant size 8 x 8 kernels on a five-level octave-separated low-passfiltered pyramid of the input image. The 45-dimensional vector representation described above shares some of the favorable matching properties that accrue to high-dimensional vectors (d. [6]). In particular, the distribution of distances between points in the 45-dimensional space of these vectors approximates a normal distribution; most of the points in the space lie at approximately the mean distance and are thus relatively uncorrelated to a given point [11]. As a result, the multiscale filter bank tends to generate almost unique location-indexed signatures of image regions which can tolerate considerable noise before they are confused with other image regions. 2.1 Localization Denote the response vector from an image point as fi and that from a previously foveated model point as Tm. Then one metric for describing the similarity between the two points is simply the square of the Euclidean distance (or the sum-of-squared-differences) between their response vectors dim = llfi - r.n 112. The algorithm for locating model points in a new scene can then be described as follows: 1. For the response vector representing a model point m, create a distance image I m defined by Im(x,y) = min [Imax - t3dim , 0] (3) where t3 is a suitably chosen constant (this makes the best match the brightest point in Im). 2. Find the best match point (Xb~, Yb~) in the image using the relation (4) Figure 1 shows the use of the localization algorithm for targeting the optical axis of a uniform-resolution sensor in an example scene. 2.2 Extension to Space-Variant Sensing The localization algorithm as presented above will obviously fail for sensors exhibiting nonuniform resolution characteristics. However, the multiscale structure of the response vectors can be effectively exploited to obtain a modified localization algorithm. Since decreasing radial resolution results in an effective reduction in scale (in addition to some 896 Rajesh P. N. Rao, Dana H. Ballard (a) (b) (c) (d) Figure 1: Using response vectors to saccade to previously foveated positions. (a) Initial gaze point. (b) New gaze point; (c) To get back to the original point, the "distance image" is computed: the brightest spot represents the point whose response vector is closest to that of the original gaze point; (d) Location of best match is marked and an oculomotor command at that location can be executed to foveate that point. other minor distortions) of previously foveated regions as they move towards the periphery, the filter responses previously occuring at larger scales now occur at smaller scales. Responses usually vary smoothly between scales; it is thus possible to establish a correspondence between the two response vectors of the same point on an object imaged at different scales by using a simple interpolate-and-eompare scale matching strategy. That is, in addition to comparing an image response vector and a model response vector directly as outlined in the previous section, scale interpolated versions of the image vector are also compared with the original model response vector. In the simplest case, interpolation amounts to shifting image response vectors by one scale and thus, responses from a new image are compared with original model responses at second, third, .. , scales, then with model responses at third, fourth, ... scales, and so on upto some threshold scale. This is illustrated in Figure 2 for two discrete movements of a simulated log-polar sensor. 3 The M ultisaccade Learning Strategy Since the high speed of saccades precludes visual guidance, advance knowledge of the precise motor command to be sent to the extraocular muscles for fixation of a desired retinal location is required. Results from neurophysiological and psychophysical studies suggest that in humans, this knowledge is acquired via learning: infants show a gradual increase in saccadic accuracy during their first year [1, 2] and adults can adapt to changes (caused for example by weakening of eye-muscles) in the interrelation between visual input and the saccades needed for centering. An adaptive mechanism for automatically learning the transfer function from retinal image space into motor space is also desirable in the context of active vision systems since an autonomous calibration of the saccadic system would (a) avoid the need for manual calibration, which can sometimes be complicated, and (b) provide resilience amidst changing circumstances caused by, for instance, changes in the camera lens mechanisms or degradation of the motor apparatus. 3.1 Motor Maps In primates, the superior eollieulus (SC), a multilayered neuron complex located in the upper regions of the brain stem, is known to playa crucial role in the saccade generation [13]. The upper layers of the SC contain a retinotopie sensory map with inputs from Learning Saccadic Eye Movements Using MuLtiscaLe SpatiaL Filters 897 (a) (b) (c) Scale I Scale 2 Scale 3 Scale 4 Scale 5 111'1"" I I " III, 'II" ,li'I"" 111'1 Scille I Scale 2 s1I3 seNe 4 Scale 5 111 '1 "'1 1" 111 " 11,, 1 " 1' ,11 ,,1 ' 11 1,1 Scale I Scale 2 Scaje 3 Scale 4 Scale 5 11"1 11 , II ' ,11'''1, ,111,, 111 11 1 ,1'11'1 I (a) III' I " (b) (c) (d) Figure 2: Using response vectors with a log-polar sensor, (a) through (c) represent a sequence of images (in Cartesian coordinates) obtained by movement of a simulated log-polar sensor from an original point (marked by '+') in the foveal region (indicated by a circle) towards the right. (d) depicts the process of interpolating (in this case, shifting) and matching response vectors of the same point as it moves towards the periphery of the sensor (Positive responses are represented by proportional upward bars and negative ones by proportional downward bars with the nine smallest scale responses at the beginning and the nine largest ones at the end). the retina while the deeper layers contain a motor map approximately aligned with the sensory map. The motor map can be visualized as a topologically-organized network of neurons which reacts to a local activation caused by an input signal with a vectorial output quantity that can be transcoded into a saccadic motor command. The alignment of the sensory and motor maps suggests the following convenient strategy for foveation: an excitation in the sensory layer (signaling a foveal target) is transferred to the underlying neurons in the motor layer which deliver the required saccade. In our framework, the excitation in the sensory layer before a goal-directed saccade corresponds to the brightest spot (most likely match) in the distance image (Figure 1 (c) for example), The formation of sensory map can be achieved using Kohonen's well-known stochastic learning algorithm by using a Gaussian input density function as described in [12]. Our primary interest lies not in the formation of the sensory map but in the development of a learning algorithm that assigns appropriate motor vectors to each location in the corresponding retinotopically-organized motor map. In particular, our algorithm employs a visual reinforcement signal obtained using iconic scene representations to determine the error vector during the learning step. 3.2 Learning the Motor Map Our multisaccade learning strategy is inspired by the following observations in [2]: During the first few weeks after birth, infants appear to fixate randomly. At about 3 months of age, infants are able to fixate stimuli albeit with a number of corrective saccades of relatively large dispersion. There is however a gradual decrease in both the dispersion 898 Rajesh P. N. Rao, Dana H. Ballard and the number of saccades required for foveation in subsequent months (Figure 3 (a) depicts a sample set of fixations). After the first year, saccades are generally accurate, requiring at most one corrective saccade2 • The learning method begins by assigning random values to the motor vectors at each location. The response vector for the current fixation point is first stored and a random saccade is executed to a different point. The goal then is to refixate the original point with the help of the localization algorithm and a limited number of multiple corrective saccades. The algorithm keeps track of the motor vector with minimum error during each run and updates the motor vectors for the neighborhood around the original unit whenever an improvement is observed. The current run ends when either the original point was successfully foveated or the limit MAX for the maximum number of allowable corrective saccades was exceeded. A more detailed outline of the algorithm is as follows: 1. Initialize the motor map by assigning random values (within an appropriate range) to the saccadic motor vectors at each location. Align the optical axis of the sensor so that a suitable salient point falls on the fovea. Initialize the run number to t := O. 2. Store in memory the filter response vector of the point p currently in the center of the foveal region. Let t := t + 1. 3. Execute a random saccade to move the fovea to a different location in the scene. 4. Use the localization algorithm described in Section 2.2 and the stored response vector to find the location [ of the previously foveated point in the current retinal image. Execute a saccade using the motor vector St stored in this location in the motor map. 5. If the currently foveated region contains the original point p, return to 2 (SI is accurate); otherwise, (a) Initialize the number of corrective saccades N := 0 and let s:= St. (b) Determine the new location /' of p in the new image as in (4) and let emin be the error vector, i.e. the vector from the foveal center to /', computed from the output of the localization algorithm. (c) Execute a saccade using the motor vector Stl stored at [' and let ebe the error vector (computed from the output of the localization algorithm) from the foveal center to the new location [II of point p found as in 4. Let N := N + 1 and let s:= s+ SI' . (d) If lie'll < lliminll, then let emin := e and update the motor vectors for the units k given by the neighborhood function N(l, t) according to the wellknown Kohonen rule: (5) where 'Y(t) is an appropriate gain function (0 < 'Y(t) < 1). (e) If the currently foveated region contains the original point p, return to 2; otherwise, if N < MAX, then determine the new location [' of p in the new image as in (4) and go to 5(c) (i.e. execute the next saccade); otherwise, return to 2. 2Large saccades in adults are usually hypometric i.e. they undershoot, necessitating a slightly slower corrective saccade. There is currently no universally accepted explanation for the need for such a two-step strategy. Learning Saccadic Eye Movements Using Multiscale Spatial Filters 899 N!.IBfIll __ (a) (b) ! " 1 I I .. • + IMX-IO f.! ...... , ' ..... 1 ",~;:;!l-;;:;;;--l-;;;; l l1I)"'---;;:ll1Ol;;;--;;"=-lI) ---;",=-,C:;;,.,:---;;:; ... ;;-;:;!,,", N_la rillelllltN (c) Figure 3: (a) Successive saccades executed by a 3-month old (left) and a 5-month old (right) infant when presented with a single illuminated stimulus (Adapted from [2]) . (b) Graph showing % of saccades that end directly in the fovea plotted against the number of iterations of the learning algorithm for different values of MAX. (c) An enlarged portion of the same graph showing points when convergence was achieved. The algorithm continues typically until convergence or the completion of a maximum number of runs. The gain term -y(t) and the neighborhood N(l, t) for any location l are gradually decreased with increasing number of iterations t. 4 Results and Discussion The simulation results for learning a motor map comprising of 961 units are shown in Figures 3 (b) and (c) which depict the variation in saccadic accuracy with the number of iterations of the algorithm for values of MAX (maximum number of corrective saccades) of 1, 5 and 10. From the graphs, it can be seen that starting with an initially random assignment of vectors, the algorithm eventually assigns accurate saccadic vectors to all units. Fewer iterations seem to be required if more corrective saccades are allowed but then each iteration itself takes more time. The localization algorithm described in Section 2.1 has been implemented on a Datacube MaxVideo 200 pipeline image processing system and takes 1-2 seconds for location of points. Current work includes the integration of the multisaccade learning algorithm described above with the Datacube implementation and further evaluation of the learning algorithm. One possible drawback of the proposed algorithm is that for large retinal spaces, learning saccadic motor vectors for every retinal location can be time-consuming and in some cases, even infeasible [1]. In order to address this problem, we have recently proposed a variation of the current learning algorithm which uses a sparse motor map in conjunction with distributed coding of the saccadic motor vectors. This organization bears some striking similarities to Kanerva's sparse distributed memory model [6] and is in concurrence with recent neurophysiological evidence [8] supporting a distributed population encoding of saccadic movements in the superior colliculus. We refer the interested reader to [10] for more details. 900 Rajesh P. N. Rao, Dana H. Ballard Acknowledgments We thank the NIPS*94 referees for their helpful comments. This work was supported by NSF research grant no. CDA-8822724, NIH/PHS research grant no. 1 R24 RR06853, and a grant from the Human Science Frontiers Program. References [1] Richard N. Aslin. Perception of visual direction in human infants. In C. Granlund, editor, Visual Perception and Cognition in Infancy, pages 91-118. Hillsdale, NJ: Lawrence Erlbaum Associates, 1993. [2] Gordon W. Bronson. The Scanning Patterns of Human Infants: Implications for Visual Learning. Norwood, NJ: Ablex, 1982. [3] Roger H.S. Carpenter. Movements of the Eyes. London: Pion, 1988. [4] William T . Freeman and Edward H. Adelson. The design and use of steerable filters. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(9):891-906, September 1991. [5] Peter J.B. Hancock, Roland J. Baddeley, and Leslie S. Smith. The principal components of natural images. Network, 3:61-70, 1992. [6] Pentti Kanerva. Sparse Distributed Memory. Bradford Books, Cambridge, MA, 1988. [7] Jitendra Malik and Pietro Perona. A computational model of texture segmentation. In IEEE Conference on Computer Vision and Pattern Recognition, pages 326-332, June 1989. [8] James T. McIlwain. Distributed spatial coding in the superior colliculus: A review. Visual Neuroscience, 6:3-13, 1991. [9] Rajesh P.N. Rao and Dana H. Ballard. An active vision architecture based on iconic representations. Technical Report 548, Department of Computer Scienc~, University of Rochester, 1995. [10] Rajesh P.N. Rao and Dana H. Ballard. A computational model for visual learning of saccadic eye movements. Technical Report 558, Department of Computer Science, University of Rochester, January 1995. [11] Rajesh P.N. Rao and Dana H. Ballard. Object indexing using an iconic sparse distributed memory. Technical Report 559, Department of Computer Science, University of Rochester, January 1995. [12] Helge Ritter, Thomas Martinetz, and Klaus Schulten. Neural Computation and SelfOrganizing Maps: An Introduction. Reading, MA: Addison-Wesley, 1992. [13] David L. Sparks and Rosi Hartwich-Young. The deep layers of the superior collicuIus. In R.H. Wurtz and M.E. Goldberg, editors, The Neurobiology of Saccadic Eye Movements, pages 213-255. Amsterdam: Elsevier, 1989. [14] Massimo Tistarelli and Giulio Sandini. Dynamic aspects in active vision. Computer Vision, Graphics, and Image Processing: Image Understanding, 56(1):108-129, 1992. [15] R.A. Young. The Gaussian derivative theory of spatial vision: Analysis of cortical cell receptive field line-weighting profiles. General Motors Research Publication GMR4920, 1985.	1994	105
865	A Charge-Based CMOS Parallel Analog Vector Quantizer Gert Cauwenberghs Johns Hopkins University ECE Department 3400 N. Charles St. Baltimore, MD 21218-2686 gert@jhunix.hcf.jhu.edu Volnei Pedroni California Institute of Technology EE Department Mail Code 128-95 Pasadena, CA 91125 pedroni@romeo.caltech.edu Abstract We present an analog VLSI chip for parallel analog vector quantization. The MOSIS 2.0 J..Lm double-poly CMOS Tiny chip contains an array of 16 x 16 charge-based distance estimation cells, implementing a mean absolute difference (MAD) metric operating on a 16-input analog vector field and 16 analog template vectors. The distance cell including dynamic template storage measures 60 x 78 J..Lm2• Additionally, the chip features a winner-take-all (WTA) output circuit of linear complexity, with global positive feedback for fast and decisive settling of a single winner output. Experimental results on the complete 16 x 16 VQ system demonstrate correct operation with 34 dB analog input dynamic range and 3 J..Lsec cycle time at 0.7 mW power dissipation. 1 Introduction Vector quantization (VQ) [1] is a common ingredient in signal processing, for applications of pattern recognition and data compression in vision, speech and beyond. Certain neural network models for pattern recognition, such as Kohonen feature map classifiers [2], are closely related to VQ as well. The implementation of VQ, in its basic form, involves a search among a set of vector templates for the one which best matches the input vector, whereby the degree of matching is quantified by a given vector distance metric. Effi780 Gert Cauwenberghs, Volnei Pedroni cient hardware implementation requires a parallel search over the template set and a fast selection and encoding of the "winning" template. The chip presented here implements a parallel synchronous analog vector quantizer with 16 analog input vector components and 16 dynamically stored analog template vectors, producing a 4-bit digital output word encoding the winning template upon presentation of an input vector. The architecture is fully scalable as in previous implementations of analog vector quantizers, e.g. [3,4,5,6], and can be readily expanded toward a larger number of vector components and template vectors without structural modification of the layout. Distinct features of the present implementation include a linear winner-take-all (WTA) structure with globalized positive feedback for fast selection of the winning template, and a mean absolute difference (MAD) metric for the distance estimations, both realized with a minimum amount of circuitry. Using a linear charge-based circuit topology for MAD distance accumulation, a wide voltage range for the analog inputs and templates is achieved at relatively low energy consumption per computation cycle. 2 System Architecture The core of the VQ consists of a 16 x 16 2-D array of distance estimation cells, configured to interconnect columns and rows according to the vector input components and template outputs. Each cell computes in parallel the absolute difference distance between one component x} of the input vector x and the corresponding component yi} of one of the template vectors y, (1) The mean absolute difference (MAD) distance between input and template vectors is accumulated along rows A • 1 ~ . d(X,y') = 16 ~ Ix} - y'}I, i = 1 ... 16 J=l and presented to the WTA, which selects the single winner kWTA = arg min d(x, f) . i (2) (3) Additional parts are included in the architecture for binary encoding of the winning output, and for address selection to write and refresh the template vectors. 3 VLSI Circuit Implementation The circuit implementation of the major components of the VQ, for MAD distance estimation and WTA selection, is described below. Both MAD distance and WTA cells operate in clocked synchronous mode using a precharge/evaluate scheme in the voltage domain. The approach followed here offers a wide analog voltage range of inputs and templates at low power weak-inversion MOS operation, and a fast and decisive settling of the winning output using a single communication line for global positive feedback. The output encoding and address decoding circuitry are implemented using standard CMOS logic. A Charge-Based CMOS Parallel Analog Vector Quantizer VVRi -r--------------------~----- VVRi 812 Xj -r----------~----------------Zi (a) 781 Vref PRE -1 4/2 (b) Figure 1: Schematic of distance estimation circuitry. (a) Absolute distance cell. ( b) Output precharge circuitry. 3.1 Distance Estimation Cell The schematic of the distance estimation cell, replicated along rows and columns of the VQ array, is shown in Figure 1 (a). The cell contains two source followers, which buffer the input voltage x j and the template voltage yi j . The template voltage is stored dynamically onto Cstore, written or refreshed by activating WRi while the y' j value is presented on the Xj input line. The WRi and WRi signal levels along rows of the VQ array are driven by the address decoder, which selects a single template vector yi to be written to with data presented at the input x when WR is active. Additional lateral transistors connect symmetrically to the source follower outputs x / and yi /- By means of resistive division, the lateral transistors construct the maximum and minimum of x l' and yi / on Zi i HI and Zi j LO, respectively. In particular, when x j is h I . I HI , . LO muc arger than y' j' the vo tage Z' j approaches x j and the voltage Z' j approaches yi /- By symmetry, the complementary argument holds in case Xj is much smaller than yi j. Therefore, the differential compo~ent of Zi j HI and Zi J LO approximately represents the absolute difference value of x J and y' j : i HI i LO , i ') . 'i ') Z j z j ~ max(xj, y j mm(xj ,y j (4) I , i 'I I i I Xj - Y j ~ K Xj - Y j , with K the MOS back gate effect coefficient [7]. The mean absolute difference (MAD) distances (2) are obtained by accumulating con782 Gert Cauwenberghs, Volnei Pedroni tributions (4) along rows of cells through capacitive coupling, using the well known technique of correlated double sampling. To this purpose, a coupling capacitor Cc is provided in every cell, coupling its differential output to the corresponding output row line. In the precharge phase, the maximum values Zi j HI are coupled to the output by activating HI, and the output lines are preset to reference voltage Vref by activating PRE, Figure 1 (b). In the evaluate phase, PRE is de-activated, and the minimum values Zi j LO are coupled to the output by activating LO. From (4), the resulting voltage outputs on the floating row lines are given by 1 L16 . HI . LO V, f (Zl . Zl . ) re 16. J J J=1 (5) 1 16 Vref- K 16~ IXj-/jl. J=1 The last term in (5) corresponds directly to the distance measure d(x, yi) in (2). Notice that the negative sign in (5) could be reversed by interchanging clocks HI and LO, if needed. Since the subsequent WTA stage searches for maximum Zi, the inverted distance metric is in the form needed for VQ. Characteristics of the MAD distance estimation (5), measured directly on the VQ array with uniform inputs x j and templates yi j' are shown in Figure 2. The magnified view in Figure 2 (b) clearly illustrates the effective smoothing of the absolute difference function (4) near the origin, x j ~ yi j' The smoothing is caused by the shift in x/and yi j' due to the conductance of the lateral coupling transistors connected to the source follower outputs in Figure 1 (a), and extends over a voltage range comparable to the thermal voltage kT /q depending on the relative geometry of the transistors and current bias level of the source followers. The observed width of the flat region in Figure 2 spans roughly 60 mY, and shows little variation for bias current settings below 0.5 f-I,A. Tuning of the bias current allows to balance speed and power dissipation requirements, since the output response is slew-rate limited by the source followers. 3.2 Winner-Take-All Circuitry The circuit implementation of the winner-take-all (WTA) function combines the compact sizing and modularity of a linear architecture as in [4,8,9] with positive feedback for fast and decisive output settling independent of signal levels, as in [6,3]. Typical positive feedback structures for WTA operation use a logarithmic tree [6] or a fully interconnected network [3], with implementation complexities of order O(n log n) and O(n2) respectively, n being the number of WTA inputs. The present implementation features an O(n) complexity in a linear structure by means of globalized positive feedback, communicated over a single line. The schematic of the WTA cell, receiving the input Zi and constructing the digital output di through global competition communicated over the COMM line, is shown in Figure 3. The global COMM line is source connected to input transistor Mi and positive feedback transistor Mf, and receives a constant bias current lb (WTA) from Mbl. Locally, the WTA operation is governed by the dynamics of d;' on (parasitic) capacitor C p' A high pulse A Charge-Based CMOS Parallel Analog Vector Quantizer 0.0 -0.2 ~ -0.4 l ~ -0.6 I '-N -0.8 -1.0 ~--~----~------~--~ 1.5 2.0 Xj (V) (a) 2.5 -0.12 :> -0.14 ..... -0.16 ~ ~ -018 I • -0.20 -0.22 -0.2 -0.1 0.0 0.1 ! Xj-Yj (V) (b) Figure 2: Distance estimation characteristics, (a) for various values of yi j; (b) magnified view. 783 0.2 on RST, resetting d/ to zero, marks the beginning of the WTA cycle. With Mf initially inactive, the total bias current n Ib (WTA) through COMM is divided over all competing WTA cells, according to the relative Zi voltage levels, and each cell fraction is locally mirrored by the Mml-Mm2 pair onto d;', charging C p' The cell with the highest Zi input voltage receives the largest fraction of bias current, and charges C p at the highest rate. The winning output is detennined by the first d;' reaching the threshold to turn on the corresponding Mf feedback transistor, say i = k. This threshold voltage is given by the source voltage on COMM, common for all cells. The positive feedback of the state dk ' through Mf, which eventually claims the entire fraction of the bias current, enhances and latches the winning output level dk ' to the positive supply and shuts off the remaining losing outputs d;' to zero, i :f:. k. The additional circuitry at the output stage of the cell serves to buffer the binary d;' value at the d; output tenninal. No more than one winner can practically co-exist at equilibrium, by nature of the combined positive feedback and global renonnalization in the WTA competition. Moreover, the output settling times of the winner and losers are fairly independent of the input signal levels, and are given mainly by the bias current level Ib (WTA) and the parasitic capacitance Cpo Tests conducted on a separate 16-element WTA array, identical to the one used on the VQ chip, have demonstrated single-winner WTA operation with response time below 0.5 /-Lsec at less than 2 /-L W power dissipation per cell. 4 Functionality Test To characterize the performance of the entire VQ system under typical real-time conditions, the chip was presented a periodic sequence of 16 distinct input vectors x(i), stored and refreshed dynamically in the 16 template locations y; by circularly incrementing the template address and activating WR at the beginning of every cycle. The test vectors rep784 Gert Cauwenberghs, Volnei Pedroni Mm2 2212 Zi -1 8/2 MI T di 11/2 11/2 COMM Mb1 VbWfA 11/2 812 RST Figure 3: Circuit schematic of winner-take-all cell. resent a single triangular pattern rotated over the 16 component indices with single index increments in sequence. The fundamental component xo(i) is illustrated on the top trace of the scope plot in Figure 4. The other components are uniformly displaced in time over one period, by a number of cycles equal to the index, x j (i) = xo(i - j mod 16). Figure 4 also displays the VQ output waveforms in response to the triangular input sequence, with the desired parabolic profile for the analog distance output ZO and the expected alternating bit pattern of the WTA least significant output bit. l The triangle test performed correctly at speeds limited by the instrumentation equipment, and the dissipated power on the chip measures 0.7 mW at 3 f.Lsec cycle time2 and 5 V supply voltage. An estimate for the dynamic range of analog input and template voltages was obtained directly by observing the smallest and largest absolute voltage difference still resolved correctly by the VQ output, uniformly over all components. By tuning the voltage range of the triangular test vectors, the recorded minimum and maximum voltage amplitudes for 5 V supply voltage are Ymin = 87.5 mV and YlII3lt = 4 V, respectively. The estimated analog dynamic range YlII3lt /Ymin is thus 45.7, or roughly 34 dB, per cell. The value obtained for Y min indicates that the dynamic range is limited mainly by the smoothing of the absolute distance measure characteristic (1) near the origin in Figure 2 (b). We notice that a similar limitation of dynamic range applies to other distance metrics with vanishing slope near the origin as well, the popular mean square error (MSE) formulation in particular. The MSE metric is frequently adopted in VQ implementations using strong inversion MOS circuitry, and offers a dynamic range typically worse than obtained here regardless of implementation accuracy, due to the relatively wide flat region of the MSE distance function near the origin. ITbe voltages on the scope plot are inverted as a consequence of the chip test setup. 2including template write operations A "Charge-Based CMOS Parallel Analog Vector Quantizer 785 Figure 4: Scope plot of VQ waveforms. Top: Analog input Xo. Center: Analog distance output zO. Bottom: Least significant bit of encoded output. Table 1: Technology Supply voltage Power dissipation VQchip Dynamic range inputs, templates Area VQchip distance cell WTAcell Features of the VQ chip 2 ~ p-well double-poly CMOS +5V 0.7 mW (3 J.lSec cycle time) 34 dB 2.2 mm X 2.25 mm 6O~X78~ 76~X80~ 786 Gert Cauwenberghs, Volnei Pedroni 5 Conclusion We proposed and demonstrated a synchronous charge-based CMOS VLSI system for parallel analog vector quantization, featuring a mean absolute difference (MAD) metric, and a linear winner-take-all (WTA) structure with globalized positive feedback. By virtue of the MAD metric, a fairly large (34 dB) analog dynamic range of inputs and templates has been obtained in the distance computations through simple charge-based circuitry. Likewise, fast and unambiguous settling of the WTA outputs, using global competition communicated over a single wire, has been obtained by adopting a compact linear circuit structure to implement the positive feedback WTA function. The resulting structure of the VQ chip is highly modular, and the functional characteristics are fairly consistent over a wide range of bias levels, including the MOS weak inversion and subthreshold regions. This allows the circuitry to be tuned to accommodate various speed and power requirements. A summary of the chip features of the 16 x 16 vector quantizer is presented in Table I. Acknowledgments Fabrication of the CMOS chip was provided through the DARPAINSF MOSIS service. The authors thank Amnon Yariv for stimulating discussions and encouragement. References [1] A. Gersho and RM. Gray, Vector Quantization and Signal Compression, Norwell, MA: Kluwer, 1992. [2] T. Kohonen, Self-Organisation and Associative Memory, Berlin: Springer-Verlag, 1984. [3] Y. He and U. Cilingiroglu, "A Charge-Based On-Chip Adaptation Kohonen Neural Network," IEEE Transactions on Neural Networks, vol. 4 (3), pp 462-469, 1993. [4] J.e. Lee, B.J. Sheu, and W.e. Fang, "VLSI Neuroprocessors for Video Motion Detection," IEEE Transactions on Neural Networks, vol. 4 (2), pp 78-191, 1993. [5] R Tawel, "Real-Time Focal-Plane Image Compression," in Proceedings Data Compression Conference" Snowbird, Utah, IEEE Computer Society Press, pp 401-409, 1993. [6] G.T. Tuttle, S. Fallahi, and A.A. Abidi, "An 8b CMOS Vector NO Converter," in ISSCC Technical Digest, IEEE Press, vol. 36, pp 38-39, 1993. [7] C.A. Mead, Analog VLSI and Neural Systems, Reading, MA: Addison-Wesley, 1989. [8] J. Lazzaro, S. Ryckebusch, M.A. Mahowald, and C.A. Mead, "Winner-Take-All Networks of O(n) Complexity," in Advances in Neural Information Processing Systems, San Mateo, CA: Morgan Kaufman, vol. 1, pp 703-711, 1989. [9] A.G. Andreou, K.A. Boahen, P.O. Pouliquen, A. Pavasovic, RE. Jenkins, and K. Strohbehn, "Current-Mode Subthreshold MOS Circuits for Analog VLSI Neural Systems," IEEE Transactions on Neural Networks, vol. 2 (2), pp 205-213, 1991.	1994	106
866	Boosting the Performance of RBF Networks with Dynamic Decay Adjustment Jay Diamond Intel Corporation 2200 Mission College Blvd. Santa Clara, CA, USA 95052 MS:SC9-15 Michael R. Berthold Forschungszentrum Informatik Gruppe ACID (Prof. D. Schmid) Haid-und-Neu-Strasse 10-14 76131 Karlsruhe, Germany eMail: berthold@fzLde eMail: jdiamond@mipos3.intel.com Abstract Radial Basis Function (RBF) Networks, also known as networks of locally-tuned processing units (see [6]) are well known for their ease of use. Most algorithms used to train these types of networks, however, require a fixed architecture, in which the number of units in the hidden layer must be determined before training starts. The RCE training algorithm, introduced by Reilly, Cooper and Elbaum (see [8]), and its probabilistic extension, the P-RCE algorithm, take advantage of a growing structure in which hidden units are only introduced when necessary. The nature of these algorithms allows training to reach stability much faster than is the case for gradient-descent based methods. Unfortunately P-RCE networks do not adjust the standard deviation of their prototypes individually, using only one global value for this parameter. This paper introduces the Dynamic Decay Adjustment (DDA) algorithm which utilizes the constructive nature of the P-RCE algorithm together with independent adaptation of each prototype's decay factor. In addition, this radial adjustment is class dependent and distinguishes between different neighbours. It is shown that networks trained with the presented algorithm perform substantially better than common RBF networks. 522 Michael R. Berthold, Jay Diamond 1 Introduction Moody and Darken proposed Networks with locally-tuned processing units, which are also known as Radial Basis Functions (RBFs, see [6]). Networks of this type have a single layer of units with a selective response for some range of the input variables. Earn unit has an overall response function, possibly a Gaussian: D_("') (11x - riW) .J.t,i X = exp 2 (Ii (1) Here x is the input to the network, ri denotes the center of the i-th RBF and (Ii determines its standard deviation. The second layer computes the output function for each class as follows: m (2) i=l with m indicating the number of RBFs and Ai being the weight for each RBF. Moody and Darken propose a hybrid training, a combination of unsupervised clustering for the centers and radii of the RBFs and supervised training of the weights. Unfortunately their algorithm requires a fixed network topology, which means that the number of RBFs must be determined in advance. The same problem applies to the Generalized Radial Basis Functions (GRBF), proposed in [12]. Here a gradient descent technique is used to implement a supervised training of the center locations, which has the disadvantage of long training times. In contrast RCE (Restricted Coulomb Energy) Networks construct their architecture dynamically during training (see [7] for an overview). This algorithm was inspired by systems of charged particles in a three-dimensional space and is analogous to the Liapunov equation: 1 m Qi ~ = - L l: II'" _ ... ·II L i=l X T, 2 (3) where ~ is the electrostatic potential induced by fixed particles with charges -Qi and locations ri. One variation of this type of networks is the so called P-RCE network, which attempts to classify data using a probabilistic distribution derived from the training set. The underlying training algorithm for P-RCE is identical to RCE training with gaussian activation functions used in the forward pass to resemble a Probabilistic Neural Network (PNN [10]). PNNs are not suitable for large databases because they commit one new prototype for each training pattern they encounter, effectively becoming a referential memory scheme. In contrast, the P-RCE algorithm introduces a new prototype only when necessary. This occurs when the prototype of a conflicting class misclassifies the new pattern during the training phase. The probabilistic extension is modelled by incrementing the a-priori rate of occurrence for prototypes of the same class as the input vector, therefore weights are only connecting RBFs and an output node of the same class. The recall phase of the P-RCE network is similar to RBFs, except that it uses one global radius for all prototypes and scales each gaussian by the a-priori rate of occurrence: (4) Boosting the Performance of RBF Networks with Dynamic Decay Adjustment 523 Figure 1: This picture shows how a new pattern results in a slightly higher activity for a prototype of the right class than for the conflicting prototype. Using only one threshold, no new prototype would be introduced in this case. where c denotes the class for which the activation is computed, me is the number of prototypes for class c, and R is the constant radius of the gaussian activation functions. The global radius of this method and the inability to recognize areas of conflict, leads to confusion in some areas of the feature space, and therefore non-optimal recognition performance. The Dynamic Decay Adjustment (DDA) algorithm presented in this paper was developed to solve the inherent problems associated with these methods. The constructive part of the P-RCE algorithm is used to build a network with an appropriate number of RBF units, for which the decay factor is computed based on information about neighbours. This technique increases the recognition accuracy in areas of conflict. The following sections explain the algorithm, compare it with others, and examine some simulation results. 2 The Algorithm Since the P-RCE training algorithm already uses an independent area of influence for each RBF, it is relatively straightforward to extract an individual radius. This results, however, in the problem illustrated in figure 1. The new pattern p of class B is properly covered by the right prototype of the same class. However, the left prototype of conflicting class A results in almost the same activation and this leads to a very low confidence when the network must classify the pattern p. To solve this dilemma, two different radii, or thresholds1 are introduced: a so-called positive threshold (0+), which must be overtaken by an activation of a prototype of the same class so that no new prototype is added, and a negative threshold (0-), which is the upper limit for the activation of conflicting classes. Figure 2 shows an example in which the new pattern correctly results in activations above the positive threshold for the correct class B and below the negative threshold for conflicting class A. This results in better classification-confidence in areas where training IThe conversion from the threshold to the radius is straightforward as long as the activation function is invertible. 524 new input pattern (class B) Michael R. Berthold, Jay Diamond x Figure 2: The proposed algorithm distinguishes between prototypes of correct and conflicting classes and uses different thresholds. Here the level of confidence is higher for the correct classification of the new pattern. patterns did not result in new prototypes. The network is required to hold the following two equations for every pattern x of class c from the training data: 3i : Rf(x) 2:: 8+ (5) Vk :/; c, 1 ~ j ~ mk : Rj(x) < 8(6) The algorithm to construct a classifier can be extracted partly from the ReE algorithm. The following pseudo code shows what the training for one new pattern x of class c looks like: / / reset weights: FORALL prototypes pf DO Af = 0.0 END FOR / / train one complete epoch FORALL training pattern (x,c) DO: IF 3pi : Ri( x) 2:: 8+ THEN Ai+ = 1.0 ELSE / / "commit": introduce new prototype add new prototype P~c+1 with: ~c+1 =x O'~ +1 = maJ:C {O' : R~ +1 (r7) < 8-} c k#cl\l~J::;mk c A~c+1 = 1.0 mc+= 1 ENDIF / / "shrink": adjust conflicting prototypes FORALL k :/; c, 1 ~ j ~ mk DO O'j = max{O' : Rj(x) < 8-} ENDFOR First, all weights are set to zero because otherwise they would accumulate duplicate information about training patterns. Next all training patterns are presented to the Boos,;ng the Peiformance of RBF Networks with Dynamic Decay Adjustment 525 pIx) pattern class A (1) (2) +2 pIx) pattern class B pattern class A x (3) (4) Figure 3: An example of the DDA- algorithm: (1) a pattern of class A is encountered and a new RBF is created; (2) a training pattern of class B leads to a new prototype for class B and shrinks the radius of the existing RBF of class A; (3) another pattern of class B is classified correctly and shrinks again the prototype of class A; (4) a new pattern of class A introduces another prototype of that class. network. If the new pattern is classified correctly, the weight of the closest prototype is increased; otherwise a new protoype is introduced with the new pattern defining its center. The last step of the algorithm shrinks all prototypes of conflicting classes if their activations are too high for this specific pattern. Running this algorithm over the training data until no further changes are required ensures that equations (5) and (6) hold. The choice of the two new parameters, (J+ and (J- are not as critical as it would initially appear2. For all of the experiments reported, the settings (J+ = 0.4 and (J- = 0.1 were used, and no major correlations of the results to these values were noted. Note that when choosing (J+ = (J- one ends up with an algorithm having the problem mentioned in figure l. Figure 3 shows an example that illustrates the first few training steps of the DDAalgorithm. 3 Results Several well-known databases were chosen to evaluate this algorithm (some can be found in the eMU Neural Network Benchmark Databases (see [13])). The DDA2Theoretically one would expect the dimensionality of the input- space to playa major role for the choice of those parameters 526 Michael R. Berthold, Jay Diamond algorithm was compared against PNN, RCE and P-RCE as well as a classic Multi Layer Perceptron which was trained using a modified Backpropagation algorithm (Rprop, see [9]). The number of hidden nodes of the MLP was optimized manually. In addition an RBF-network with a fixed number of hidden nodes was trained using unsupervised clustering for the center positions and a gradient descent to determine the weights (see [6] for more details). The number of hidden nodes was again optimized manually. • Vowel Recognition: Speaker independent recognition of the eleven steady state vowels of British English using a specified training set of Linear Predictive Coding (LPC) derived log area ratios (see [3]) resulting in 10 inputs and 11 classes to distinguish. The training set consisted of 528 tokens, with 462 different tokens used to test the network. algorithm II performance I #units I #epochs I Nearest Neighbour 56% 1 MLP (RPRUP) 57% 5 ..... 200 PNN 61% 528 RBF 59% 70 ..... 100 RCE 27% 125 3 P-RCE 59% 125 3 DDA-RBF 65_% 204 4 • Sonar Database: Discriminate between sonar signals bounced off a metal cylinder and those bounced off a roughly cylindrical rock (see [4] for more details). The data has 60 continuous inputs and is separated into two classes. For training and testing 104 samples each were used. algorithm II performance I #units I #epochs I MLP (RPROP) 90.4% 50 ..... 250 PNN 91.3% 104 RBF 90.7% 80 ..... 150 RCE 77.9% 68 3 P-RCE 90.4% 68 3 DDA-RBF 93.3% 68 3 • Two Spirals: This well-known problem is often used to demonstrate the generalization capability of a network (see [5]). The required task involves discriminating between two intertwined spirals. For this paper the spirals were changed slightly to make the problem more demanding. The original spirals radius declines linearly and can be correctly classified by RBF networks with one global radius. To demonstrate the ability of the DDAalgorithm to adjust the radii of each RBF individually, a quadratic decline was chosen for the radius of both spirals (see figure 4). The training set consisted of 194 points, and the spirals made three complete revolutions. Figure 4 shows both the results of an RBF Network trained with the DDA technique and the same problem solved with a Multi-Layer Perceptron (2-20-20-1) trained using a modified Error Back Propagation algorithm (Rprop, see [9]). Note that in both cases all training points are classified correctly. Boosting the Peifonnance of RBF Networks with Dynamic Decay Adjustment 527 Figure 4: The (quadratic) "two spirals problem" solved by a MLP (left) using Error Back Propagation (after 40000 epochs) and an RBF network (right) trained with the proposed DDA-algorithm (after 4 epochs). Note that all training patterns (indicated by squares vs. crosses) are classified correctly. In addition to these tasks, the BDG-database was used to compare the DDA algorithm to other approaches. This database was used by Waibel et al (see [11]) to introduce the Time Delay Neural Network (TDNN). Previously it has been shown that RBF networks perform equivalently (when using a similar architecture, [1], [2]) with the DDA technique used for training of the RBF units. The BDG task involves distinguishing the three stop consonants "B", "D" and "G". While 783 training sets were used, 749 data sets were used for testing. Each of these contains 15 frames of melscale coefficients, computed from a 10kHz, 12bit converted signal. The final frame frequency was 100Hz. algorithm II performance I #epochs I TDNN 98.5% ",50 TDRBF (P-RCE) 85.2% 5 TDRBF (DDA) 98.3% 6 4 Conclusions It has been shown that Radial Basis Function Networks can boost their performance by using the dynamic decay adjustment technique. The algorithm necessary to construct RBF networks based on the RCE method was described and a method to distinguish between conflicting and matching prototypes at the training phase was proposed. An increase in performance was noted, especially in areas of conflict, where standard (P-)RCE did not commit new prototypes. Four different datasets were used to show the performance of the proposed DDAalgorithm. In three of the cases, RBF networks trained with dynamic decay adjustment outperformed known RBF training methods and MLPs. For the fourth task, the BDG-recognition dataset, the TDRBF was able to reach the same level 528 Michael R. Berthold. Jay Diamond of performance as a TDNN. In addition, the new algorithm trains very quickly. Fewer than 6 epochs were sufficient to reach stability for all problems presented. Acknowledgements Thanks go to our supervisors Prof. D. Schmid and Mark Holler for their support and the opportunity to work on this project. References [1] M. R. Berthold: "A Time Delay Radial Basis FUnction Network for Phoneme Recognition" in Proc. of the IEEE International Conference on Neural Networks, 7, p.447D---4473, 1994. [2] M. R. Berthold: "The TDRBF: A Shift Invariant Radial Basis Function Network" in Proc. of the Irish Neural Network Conference, p.7-12, 1994. [3] D. Deterding: "Speaker Normalization for Automatic Speech Recognition" , PhD Thesis, University of Cambridge, 1989. [4] R. Gorman, T. Sejnowski: "Analysis of Hidden Units in a Layered Network Trained to Classify Sonar Targets" in Neural Networks 1, pp.75. [5] K. Lang, M. Witbrock: "Learning to Tell Two Spirals Apart", in Proc. of Connectionist Models Summer School, 1988. [6] J. Moody, C.J. Darken: "Fast Learning in Networks of Locally-Tuned Processing Units" in Neural Computation 1, p.281-294, 1989. [7] M.J. Hudak: "RCE Classifiers: Theory and Practice" in Cybernetics and Systems 23, p.483-515, 1992. [8] D.L. Reilly, L.N. Cooper, C. Elbaum: "A Neural Model for Category Learning" in BioI. Cybernet. 45, p.35-41, 1982. [9] M. Riedmiller, H. Braun: "A Direct Adaptive Method for Faster Backpropagation Learning: The Rprop Algorithm" in Proc. of the IEEE International Conference on Neural Networks, 1, p.586-591, 1993. [10] D.F. Specht: "Probabilistic Neural Networks" in Neural Networks 3, p.109118,1990. [11] A. Waibel, T. Hanazawa, G. Hinton, K. Shikano, K. Lang: "Phoneme Recognition Using Time-Delay Neural Networks" in IEEE Trans. in Acoustics, Speech and Signal Processing Vol. 37, No. 3, 1989. [12] D. Wettschereck, T. Dietterich: "Improving the Performance of Radial Basis Function Networks by Learning Center Locations" in Advances in Neural Information Processing Systems 4, p.1133- 1140, 1991. [13] S. Fahlman, M. White: "The Carnegie Mellon University Collection of Neural Net Benchmarks" from ftp.cs.cmu.edu in /afs/cs/project/connect/bench.	1994	107
867	An Alternative Model for Mixtures of Experts Lei Xu Dept. of Computer Science, The Chinese University of Hong Kong Shatin, Hong Kong, Emaillxu@cs.cuhk.hk Michael I. Jordan Dept. of Brain and Cognitive Sciences MIT Cambridge, MA 02139 Abstract Geoffrey E. Hinton Dept. of Computer Science University of Toronto Toronto, M5S lA4, Canada We propose an alternative model for mixtures of experts which uses a different parametric form for the gating network. The modified model is trained by the EM algorithm. In comparison with earlier models-trained by either EM or gradient ascent-there is no need to select a learning stepsize. We report simulation experiments which show that the new architecture yields faster convergence. We also apply the new model to two problem domains: piecewise nonlinear function approximation and the combination of multiple previously trained classifiers. 1 INTRODUCTION For the mixtures of experts architecture (Jacobs, Jordan, Nowlan & Hinton, 1991), the EM algorithm decouples the learning process in a manner that fits well with the modular structure and yields a considerably improved rate of convergence (Jordan & Jacobs, 1994). The favorable properties of EM have also been shown by theoretical analyses (Jordan & Xu, in press; Xu & Jordan, 1994). It is difficult to apply EM to some parts of the mixtures of experts architecture because of the nonlinearity of softmax gating network. This makes the maximiza634 Lei Xu, Michael!. Jordan, Geoffrey E. Hinton tion with respect to the parameters in gating network nonlinear and analytically unsolvable even for the simplest generalized linear case. Jordan and Jacobs (1994) suggested a double-loop approach in which an inner loop of iteratively-reweighted least squares (IRLS) is used to perform the nonlinear optimization. However, this requires extra computation and the stepsize must be chosen carefully to guarantee the convergence of the inner loop. We propose an alternative model for mixtures of experts which uses a different parametric form for the gating network. This form is chosen so that the maximization with respect to the parameters of the gating network can be handled analytically. Thus, a single-loop EM can be used, and no learning stepsize is required to guarantee convergence. We report simulation experiments which show that the new architecture yields faster convergence. We also apply the model to two problem domains. One is a piecewise nonlinear function approximation problem with smooth blending of pieces specified by polynomial, trigonometric, or other prespecified basis functions. The other is to combine classifiers developed previously-a general problem with a variety of applications (Xu, et al., 1991, 1992). Xu and Jordan (1993) proposed to solve the problem by using the mixtures of experts architecture and suggested an algorithm for bypassing the difficulty caused by the softmax gating networks. Here, we show that the algorithm of Xu and Jordan (1993) can be regarded as a special case of the single-loop EM given in this paper and that the single-loop EM also provides a further improvement. 2 MIXTURES OF EXPERTS AND EM LEARNING The mixtures of experts model is based on the following conditional mixture: K P(ylx,6) I: 9j(X, lI)P(ylx, OJ), P(ylx, OJ) where x ERn, and 6 consists of 1I,{Oj}f, and OJ consists of {wj}f,{rj}f. The vector Ij (x, Wj) ~s the output ofthe j-th expert net. The scalar 9j (x, 1I), j = 1, ... , K is given by the softmax function: 9j(X,1I) = e.B;(x,v)/I:e.B;(X,v). (2) In this equation, Pj (x, 1I), j = 1, ... ,K are the outputs of the gating network. The parameter 6 is estimated by Maximum Likelihood (ML), where the log likelihood is given by L = Lt In P(y(t) Ix(t), 6). The ML estimate can be found iteratively using the EM algorithm as follows. Given the current estimate 6(1~), each iteration consists of two steps. (1) E-step. For each pair {x(t),y(t)}, compute h~k)(y(t)lx(t») = PUlx(t),y(t»), and then form a set of objective functions: Qj(Oj) I:h;k)(y(t)lx(t») InP(y(t)lx(t), OJ), j = 1,,, ',K; An Alternative Model for Mixtures of Experts Qg(V) = L L h)k) (y(t) Ix(t») lngt) (x(t) ,v(k»). t j (2). M-step. Find a new estimate e(k+l) = {{ot+l)}f=I,V(k+l)} with: 635 (3) OJk+l) = argmax Qj(Oj), j = 1, ... , K; v(k+l) = arg max Qg(v). (4) ~ v In certain cases, for example when I; (x, Wj) is linear in the parameters Wj, maXe j Qj(Oj) can be solved by solving 8Qj/80j = O. When l;(x,Wj) is nonlinear with respect to Wj, however, the maximization cannot be performed analytically. Moreover, due to the nonlinearity of softmax, maXv Qg(v) cannot be solved analytically in any case. There are two possibilities for attacking these nonlinear optimization problems. One is to use a conventional iterative optimization technique (e.g., gradient ascent) to perform one or more inner-loop iterations. The other is to simply find a new estimate such that Qj(ot+l») ~ Qj(Ojk»), Qg(v(k+l») ~ Qg(v(k»). Usually, the algorithms that perform a full maximization during the M step are referred as "EM" algorithms, and algorithms that simply increase the Q function during the M step as "GEM" algorithms. In this paper we will further distinguish between EM algorithms requiring and not requiring an iterative inner loop by designating them as double-loop EM and single-loop EM respectively. Jordan and Jacobs (1994) considered the case of linear {3j(x, v) = vJ[x,l] with v = [VI,···, VK] and semi-linear I; (wnx, 1]) with nonlinear 1;(.). They proposed a double-loop EM algorithm by using the IRLS method to implement the inner-loop iteration. For more general nonlinear {3j(x, v) and I;(x, OJ), Jordan and Xu (in press) showed that an extended IRLS can be used for this inner loop. It can be shown that IRLS and the extension are equivalent to solving eq. (3) by the so-called Fisher Scoring method. 3 A NEW GATING NET AND A SINGLE-LOOP EM To sidestep the need for a nonlinear optimization routine in the inner loop of the EM algorithm, we propose the following modified gating network: gj(x, v) = CkjP(xlvj)/ L:i CkiP(xIVi) , L:j Ckj = 1, Ckj ~ 0, P(xIVj) = aj(vj)-lbj(x) exp{cj(Vj)Ttj(x)} (5) where v = {Ckj,Vj,j = 1,···,K}, tj(x) is a vector of sufficient statistics, and the P(xlvj)'s are density functions from the exponential family. The most common example is the Gaussian: (6) In eq. (5), gj(x, v) is actually the posterior probability PUlx) that x is assigned to the partition corresponding to the j-th expert net, obtained from Bayes' rule: gj(x, v) = PUlx) = CkjP(xIVj)/ P(x, v), P(x, v) = L CkiP(xlvi). (7) 636 Lei Xu, Michael I. Jordan, Geoffrey E. Hinton Inserting this 9j(X, v) into the model eq. (1), we get " a·P(xlv·) P(ylx, 8) = L~( ) P(ylx, OJ). . X,V 3 (8) If we do ML estimation directly on this P(ylx,8) and derive an EM algorithm, we again find that the maximization maXv Q9(v) cannot be solved analytically. To avoid this difficulty, we rewrite eq. (8) as: P(y, x) = P(ylx, 8)P(x, v) = L ajP(xlvj)P(ylx, OJ). (9) j This suggests an asymmetrical representation for the joint density. We accordingly perform ML estimation based on L' = 2:t In P(y(t), x(t») to determine the parameters a j , Vj, OJ of the gating net and the expert nets. This can be done by the following EM algorithm: (1) E-step. Compute a\k) P( x(t) Iv~k »)P(y(t) Ix(t) O<k») h(k)(y(t) Ix(t») _ 3 3 ' 3 . j - 2:i a~k) P(x(t)lv?»)P(y(t)lx(t), 0)"»)' (10) Then let Qj(Oj),j = 1,· .. , K be the same as given in eq. (3), and decompose Q9(v) further into L h)k '(y(t) Ix(t») In P( x(t) IVj), j = 1, ... , K; t LLh;k)(y(t)lx(t»)lnaj, with a= {al, .. ·,aK}. (11) j (2). M-step. Find a new estimate for j = 1,···, K O;k+l) = argmaXSj Qj(Oj), V]"+l) = argmaxllj QJ(Vj)' a(k+1) = arg maXa, QlX, s.t. 2:j aj = 1. (12) The maximization for the expert nets is the same as in eq. (4). However, for the gating net the maximization now becomes analytically solvable as long as P(xlvj) is from the exponential family. That is, we have: V~k+l) = 2:t h)")(y(t) Ix(t»)tj(x(t») 3 2:t h)k)(y(t)lx(t») , a;k+l) = ~ L h)")(y(t) Ix(t»). t (13) In particular, when P(xIVj) is a Gaussian density, the update becomes: An Alternative Model for Mixtures of Experts 637 Two issues deserve to be emphasized further: (1) The gating nets eq. (2) and eq. (5) become identical when f3j(x, v) = lnaj + In bj (x) +Cj (Vj)T tj (x) -In aj(vj). In other words, the gating net in eq. (5) explicitly uses this function family instead of the function family defined by a multilayer feedforward network. (2) It follows from eq. (9) that max In P(y, xiS) = max [In P(ylx, S) + In P(xlv)]. So, the solution given by eqs. (10) through (14) is actually different from the one given by the original eqs. (3) and (4). The former tries to model both the mapping from x to y and the input x, while the latter only models the mapping from x and y. In fact, here we learn the parameters of the gating net and the expert nets via an asymmetrical representation eq. (9) of the joint density P(y, x) which includes P(ylx) implicitly. However, in the testing phase, the total output still follows eq. (8). 4 PIECEWISE NONLINEAR APPROXIMATION The simple form /j(x, Wj) = wJ[x, 1] is not the only case to which single-loop EM applies. Whenever /j(x, Wj) can be written in a form linear in the parameters: (15) where 4>i,j(X) are prespecified basis functions, maX8j Qf!(Oj),j = 1"", K in eq. (3) is still a weighted least squares problem that can be soived analytically. One useful special case is when 4>i,j(X) are canonical polynomial terms X~l .. 'X~d, rj ~ 0. In this case, the mixture of experts model implements piecewise polynomial approximations. Another case is that 4>i,j(X) is TIi sini (jll'xt) cosi(jll'xt}, ri ~ 0, in which case the mixture of experts implements piecewise trigonometric approximations. 5 COMBINING MULTIPLE CLASSIFIERS Given pattern classes Ci, i = 1, ... , M, we consider classifiers ej that for each input x produce an output Pj(ylx): Pj(ylx) = [Pj(llx ), ... ,pj(Mlx)), pj(ilx) ~ 0, LPj(ilx) = 1. (16) The problem of Combining Multiple Classifiers (CMC) is to combine these Pj(ylx)'s to give a combined estimate of P(ylx) . Xu and Jordan (1993) proposed to solve CMC problems by regarding the problem as a special example of the mixture density problem eq. (1) with the Pj(ylx)'s known and only the gating net 9j(X, v) to be learned. In Xu and Jordan (1993), one problem encountered was also the nonlinearity of softmax gating networks, and an algorithm was proposed to avoid the difficulty. Actually, the single-loop EM given by eq. (10) and eq. (13) can be directly used to solve the CMC problem. In particular, when P(xlvj) is Gaussian, eq. (13) becomes eq. (14). Assuming that al = .. . = aK in eq. (7), eq. (10) becomes 638 Lei Xu, Michaell. Jordan, Geoffrey E. Hinton h)k) (y(t) Ix(t)) = P(x(t)lvt))P(y(t)lx(t))/ L:i P(x(t)lvi(k))p(y(t)lx(t)). If we divide both the numerator and denominator by L:i P(x(t) Ivi(k)), we get ht)(y(t) Ix(t)) = gj(x, v)P(y(t)lx(t))/ L:i gj(x, v)P(y(t)lx(t)) . Comparing this equation with eq. (7a) in Xu and Jordan (1993), we can see that the two equations are actually the same. Despite the different notation, C¥j(x) and Pj(y1.t) Ix(t)) in Xu and Jordan (1993) are the same as gj(x, v) and P(y(t)lx(t)) in Section 3. So the algorithm of Xu and Jordan (1993) is a special case of the single-loop EM given in Section 3. 6 SIMULATION RESULTS We compare the performance of the EM algorithm presented earlier with the model of mixtures of experts presented by Jordan and Jacobs (1994). As shown in Fig. l(a), we consider a mixture of experts model with K = 2. For the expert nets, each P(ylx, OJ) is Gaussian given by eq. (1) with linear !;(x,Wj) = wJ[x, 1]. For the new gating net, each P(x, Vj) in eq. (5) is Gaussian given by eq. (6). For the old gating net eq. (2), f31(X,V) = 0 and f32(X,v) = vT[x, 1]. The learning speeds of the two are significantly different. The new algorithm takes k=15 iterations for the log-likelihood to converge to the value of -1271.8. These iterations require about 1,351,383 MATLAB flops. For the old algorithm, we use the IRLS algorithm given in Jordan and Jacobs (1994) for the inner loop iteration. In experiments, we found that it usually took a large number of iterations for the inner loop to converge. To save computations, we limit the maximum number of iterations by Tmaz = 10. We found that this saved computation without obviously influencing the overall performance. From Fig. 1 (b), we see that the outer loop converges in about 16 iterations. Each inner loop takes 290498 flops and the entire process requires 5,312,695 flops. So, we see that the new algorithm yields a speedup of about 4,648,608/1,441,475 = 3.9. Moreover, no external adjustment is needed to ensure the convergence of the new algorithm. But for the old one the direct use of IRLS can make the inner loop diverge and we need to appropriately rescale the updating stepsize of IRLS. Figs. 2(a) and (b) show the results of a simulation of a piecewise polynomial approximation problem utilizing the approach described in Section 4. We consider a mixture of experts model with K = 2. For expert nets, each P(ylx, OJ) is Gaussian given by eq. (1) with !;(x,Wj) = W3,jX3+W2,jX2+Wl,jX+WD,j. In the new gating net eq. (5), each P(x, Vj) is again Gaussian given by eq. (6). We see that the higher order nonlinear regression has been fit quite well. For multiple classifier combination, the problem and data are the same as in Xu and Jordan (1993). Table 1 shows the classification results. Com-old and Com-new denote the method given in in Xu and Jordan (1993) and in Section 5 respectively. We see that both improve the classification rate of each individual network considerably and that Com - new improves on Com - old. Training set Testing set Classifer el Classifer el 89.9% 93.3% 89.2% 92.7% Com- old 98.6% 98.0% Com- new 99.4% 99.0% Table 1 A comparison of the correct classification rates An Alternative Model for Mixtures of Experts 639 o.s U 2 U 3 U 4 (a) (b) Figure 1: (a) 1000 samples from y = a1X + a2 + c, a1 = 0.8, a2 = 0.4, x E [-1,1.5] with prior a1 = 0.25 and y = ai x + a~ + c, ai = 0.8, a2 =' 2.4, x E [1,4] with prior a2 = 0.75, where x is uniform random variable and z is from Gaussian N(O, 0.3). The two lines through the clouds are the estimated models of two expert nets. The fits obtained by the two learning algorithms are almost the same. (b) The evolution of the log-likelihood. The solid line is for the modified learning algorithm. The dotted line is for the original learning algorithm (the outer loop iteration). 7 REMARKS Recently, Ghahramani and Jordan (1994) proposed solving function approximation problems by using a mixture of Gaussians to estimate the joint density of the input and output (see also Specht, 1991; Tresp, et al., 1994). In the special case of linear I;(x, Wj) = wnx,I] and Gaussian P(xlvj) with equal priors, the method given in Section 3 provides the same result as Ghahramani and Jordan (1994) although the parameterizations of the two methods are different. However, the method of this paper also applies to nonlinear l;(x,Wj) = Wn<pj(x) , 1] for piecewise nonlinear approximation or more generally I; (x, Wj) that is nonlinear with respect to Wj, and applies to cases in which P(y, xlvj, OJ) is not Gaussian, as well as the case of combining multiple classifiers. Furthermore, the methods proposed in Sections 3 and 4 can also be extended to the hierarchical mixture of experts architecture (Jacobs & Jordan, 1994) so that single-loop EM can be used to facilitate its training. References Ghahramani, Z., & Jordan, M.I. (1994). Function approximation via density estimation using the EM approach. In Cowan, J.D., Tesauro, G., and Alspector, J., (Eds.), Advances in NIPS 6. San Mateo, CA: Morgan Kaufmann. Jacobs, R.A., Jordan, M.L, Nowlan, S.J., & Hinton, G.E. (1991). Adaptive mixtures of local experts. Neural Computation, 9, 79-87. 640 Lei Xu, Michael!. Jordan, Geoffrey E. Hinton 110' 0 .as -\ I -\.5 -2 1 ~ -2.5 -2 -3 -3.5 -4 0 (a) (b) Figure 2: Piecewise 3rd polynomial approximation. (a) 1000 samples from y = alx3+a3x+a4+c, x E [-1,1.5] with prior 0'1 = 0.4 and y = a~x2+a~x2+a4+c, x E [1,4] with prior 0'2 = 0.6, where x is uniform random variable and z is from Gaussian N(0,0.15). The two curves through the clouds are the estimated models of two expert nets. (b) The evolution of the log-likelihood. Jordan, M.I., & Jacobs, R.A. (1994). Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6, 181-214. Jordan, M.I., & Xu, L. (in press). Convergence results for the EM approach to mixtures-of-experts architectures. Neural Networks. Specht, D. (1991). A general regression neural network. IEEE Trans. Neural Networks, 2, 568-576. Tresp, V., Ahmad, S., and Neuneier, R. (1994). Training neural networks with deficient data. In Cowan, J.D., Tesauro, G., & Alspector, J., (Eds.), Advances in NIPS 6, San Mateo, CA: Morgan Kaufmann. Xu, L., Krzyzak A., & Suen, C.Y. (1991). Associative switch for combining multiple classifiers. Proc. of 1991 HCNN, Vol. I. Seattle, 43-48. Xu, L., Krzyzak A., & Suen, C.Y. (1992). Several methods for combining multiple classifiers and their applications in handwritten character recognition. IEEE Thans. on SMC, Vol. SMC-22, 418-435. Xu, L., & Jordan, M.I. (1993). EM Learning on a generalized finite mixture model for combining multiple classifiers. Proceedings of World Congress on Neural Networks, Vol. IV. Portland, OR, 227-230. Xu, L., & Jordan, M.I. (1994). On convergence properties of the EM algorithm for Gaussian mixtures. Submitted to Neural Computation.	1994	108
868	Catastrophic Interference in Human Motor Learning Tom Brashers-Krug, Reza Shadmehrt, and Emanuel Todorov Dept. of Brain and Cognitive Sciences, M. I. T., Cambridge, MA 02139 tCurrently at Dept. of Biomedical Eng., Johns Hopkins Univ., Baltimore, MD 21205 Email: tbk@ai.mit.edu, reza@bme.jhu.edu, emo@aLmit.edu Abstract Biological sensorimotor systems are not static maps that transform input (sensory information) into output (motor behavior). Evidence from many lines of research suggests that their representations are plastic, experience-dependent entities. While this plasticity is essential for flexible behavior, it presents the nervous system with difficult organizational challenges. If the sensorimotor system adapts itself to perform well under one set of circumstances, will it then perform poorly when placed in an environment with different demands (negative transfer)? Will a later experience-dependent change undo the benefits of previous learning (catastrophic interference)? We explore the first question in a separate paper in this volume (Shadmehr et al. 1995). Here we present psychophysical and computational results that explore the question of catastrophic interference in the context of a dynamic motor learning task. Under some conditions, subjects show evidence of catastrophic interference. Under other conditions, however, subjects appear to be immune to its effects. These results suggest that motor learning can undergo a process of consolidation. Modular neural networks are well suited for the demands of learning multiple input/output mappings. By incorporating the notion of fast- and slow-changing connections into a modular architecture, we were able to account for the psychophysical results. 20 Tom Brashers-Krug, Reza Shadmelzr, Emanuel Todorov 1 Introduction Interacting physically with the world changes the dynamics of one's limbs. For example, when holding a heavy load, a different pattern of muscular activity is needed to move one's arm along a particular path than when not holding a load. Previous work in our laboratory has shown that humans learn a novel dynamic task by forming an internal model of the new inverse dynamics of thier limbs. (Shadmehr and Mussa-Ivaldi 1994, Shadmehr et aI, 1995). Preliminary evidence suggests that subjects can retain one of these internal models over time (BrashersKrug, et al. 1994). Humans are required, however, to move their limbs effectively under a large number of dynamic conditions. Are people able to learn and store an inverse dynamic model appropriate for each condition, or do they form such models from scratch as they need them? In particular, can learning a new inverse dynamic model overwrite or displace a previous model? We will present evidence that certain conditions must be met before a subject is able to retain more than one inverse dynamic model in a given experimental context. These conditions can be modeled as leading to a process of consolidation, whereby learning is transfered from vulnerable, low-capacity storage to a long-term, high-capacity storage. 2 Experimental Protocol We have developed a motor learning paradigm that allows us to alter the dynamics of a subject's arm and so to monitor a subject's ability to learn dynamic tasks. A subject moves the handle on the free end of a two-link planar robot arm-called a manipulandum-to guide a cursor to a series of targets displayed on a computer screen (fig la). The position and velocity of the handle of the manipulandum are recorded at ten-millisecond intervals and are used to deliver state-dependent forces to the subject's hand. In order to test a subject's ability to learn a novel dynamic task, we present the subject with a viscous force field as slhe moves from one target to the next (fig Ib). Initially, such forces perturb the subject's movements, causing them to depart from the smooth, straight-line trajectories of the baseline condition (i.e., the condition before the viscous forces were presented) (figs lc,ld). The extent of learning is measured as the degree to which a subject's movements in the force field over time come to resemble that subject's baseline movements. We have shown in previous work that subjects adapt to the imposed force fileds by forming a predictive model of the velocity-dependent forces, and that subjects use this inverse dynamic model to control their arms in what appears to be a feedforward manner (Shadmehr and Mussa-Ivaldi 1994). 3 Psychophysical Findings 3.1 Catastrophic Interference Here, we employed this paradigm to explore the consequences of learning two different dynamic tasks in a row. In an initial series of experiments, we allowed twelve subjects to learn to move the manipulandum in a first force field (Field A) for approximately 5 minutes. Immediately after this first set of movements, we presented the subjects with an anti-correlated force field (Field B). For example, if we preCatastrophic Interference in Human Motor Learning 21 <Y'/'/ . \ C D Figure 1: A: The experimental setup. B: An example of a viscous field, plotted in velocity space (mm/sec). The arrows indicate the direction and magnitude of the forces exerted by the manipulandum on the subject's hand at each location in velocity space. C: One subject's trajectories before forces were introduced. Targets are indicated by the open circles. D: Trajectories immediately after the force field in (B) was presented. sen ted the counter-clockwise curl field depicted in fig. 1 B as Field A, we would next present a clockwise curl field as Field B. Half the subjects learned the clockwise curl field first and the counter-clockwise field second; the other halflearned the two fields in the reverse order. (The first field will be referred to as Field A and the second field as Field B, whichever field was learned first.) The subjects' mean performance in Field B was worse (p< 0.0001, paired t-test) than in Field A. This phenomenon has been called negative transfer in the psychophysical literature. Negative transfer in this motor learning paradigm is explored more fully in a separate paper in this volume (Shadmehr et aI, 1995). In that paper, we suggested that this negative transfer could result from the fact that the same neural elements are learning both tasks. We predicted that, if this is the case, learning to move in Field B would interfere with a subject's ability to retain an inverse dynamic model of Field A. Learning to move in Field B would, in effect, cause subjects to "unlearn" Field A, resulting in catastrophic interference. In order to test this prediction, we compared the improvement in performance from one day to the next of two groups of subjects, with twelve subjects in each group. The subjects in the control group learned to move in one force field on Day One and were then tested on Day Two in the same force field. The subjects in the experimental group learned two separate force fields in a row on Day One and were then tested on Day Two in the first force field they learned. The experimental group retained significantly less of their learning (p< 0.01, paired t-test) from Day One to Day Two than the control group (figs 2a,2b). In other words, learning the second force field resulted in catastrophic interference. (The question of whether this represents a storage or a retrieval phenomenon is beyond the scope of this paper.) 22 Tom Brashers-Krug, Reza Shadmehr, Emanuel Todorov 3.2 Consolidation Having found evidence for catastrophic interference, we wanted to know whether there were circumstances under which dynamic motor learning was immune to being functionally erased by subsequent learning. We therefore tested two further groups of subjects. We allowed these subjects to practice longer in one field before they learned the second field. We also allowed 24 hours to pass between the time subjects first learned one field and when they learned the second field . The subjects in the experimental group (n = 10) practiced in one force field for approximately 15 minutes on Day One. They returned on Day Two and practiced in the same force field for five more minutes. They were then allowed to practice in a second force field for 15 minutes. By the end of this fifteen minutes, they were performing in the second field at a level comparable to the level they acheived in the first force field. We had the subjects return on Day Three, when they were tested for their retention of the first field they learned. We compared the difference in performance on Day Two and Day Three of this experimental group with that of a control group (n = 9) who followed the same protocol in all respects except that they never learned a second force field . In this way we could determine whether learning the second field resulted in a decrement in performance for the experimental group when compared with the control group. Under these conditions, we found no difference in the retention of learning between the experimental and control groups (fig 2c, 2d). That is, learning the second field under these conditions no longer resulted in catastrophic interference. What subjects had learned about the first field had become resistant to such interference. It had become consolidated. We can not tell from these experiments whether such consolidation is the result of the increased practice in the first field, or whether it is the result of the 24 hours that elapsed between when the first field was first learned and when the second field was learned. There is evidence that increased practice in a motor task can engage different neural circuits than those that are active during initial learning (Jenkins, et al 1994). The shift to "practiced" circuits may represent the neural substrate of consolidation. There is also evidence that time can be an important factor in the consolidation of skill learning. (Karni and Sagi 1993) In the next section, we present a model of our results that assumes that time is the key variable in the consolidation of motor learning. The model could also be applied to a practice-based model of consolidation with minor modifications. 4 Computational Modeling of the Experimental Results In order to model the results presented above we need a network that learns to compute an appropriate control signal Y given the current state and the desired next state X of the plant. More precisely, it needs to compute a mapping from joint angles 0, joint velocities iJ, and desired joint accelerations jj to torques. Several approaches for solving this problem have been proposed. One way to learn a mapping from X into Y is to use direct inverse modeling: apply a control signal, measure the next state of the plant, and use the current state, new state, and control signal as a training pair for the controller. This approach is not suitable for explaining nonCatastrophic Interference ill Human Motor Learning 23 0.95 0.95 ~ 0.85 "," ~ 0.85 0.75 0.75 A B 0.65 0.65 0 50 100 150 0 50 100 150 0.95 0.95 0.85 0.85 ~ 0.75 C 0.75 D 0 50 100 150 0 50 100 150 Figure 2: Plots of average learning curves. The correlation of trajectories in the force field to baseline trajectories (before the force field was applied) is plotted as a function of movement number. A: Learning curves for the first experimental group. Dark curve: learning curve on Day One in Field A. Light curve: learning curve in Field A on Day Two. This group learned Field B immediately after Field A on Day One (learning curve for Field B not shown). Note minimal improvement from Day One to Day Two. B: Learning curves for the first control group. Dark curve: learning curve in Field A on Day One. Light curve: learning curve in Field A on Day Two. Control group never learned Field B. Note significant improvement from Day One to Day Two. C: Second experimental group. Dark line: learning curve in Field A on Day Two immediately before learning Field B (Field B curve not shown). Light line: learning curve in Field A on Day Three. D: Second control group. Dark and light lines: learning curves in FIeld A on Day Two and Three, respectively. Note the similarity of the curves in C and D. This indicates that learning Field B did not significantly affect the experimental group's retention of Field A. All curves in C and D are significantly higher than curves for the initial learning of Field A on Day One. convex mappings, however. The learning situation we must model is non-convex: we change the dynamic environment the controller operates in by presenting force fileds, and so there will be different Y values corresponding to any X value. A different approach that solves the non convexity problem is distal supervised learning (Jordan and Rumelhart 1992): produce control signals, observe the new state of the plant, and use that information to train a forward model that maps actions into states; then learn a controller, using the forward model to backpropagate error from observable cartesian coordinates to the unknown control space. Distal supervised learning solves the non convexity problem by learning one correct value of Y for each X. But that can not explain the consolidation of learning - when the force field changes back to something already learned, our controller should rapidly recover its performance in that old field, meaning that it should retain information about all Y s that a particular X can map into. An architecture that seems to have most of the desirable properties discussed above is the Mixture of Experts (ME) model (Jordan and Jacobs 1994): several experts learn a mapping from X to Y . A separate gating network selects an expert which is most likely to be correct at each moment. Such a model has been used previously (Jacobs and Jordan 1993) to learn a feedforward controller for a 2-joint planar arm operating under different loads. In their model however they assumed that the 24 Tom Brashers-Krug, Reza Shadmelzr, Emanuel Todorov identity of the load is known to the controller. The subjects in our study were not given any explicit cues about the identity of the fields they were learning. The mixture of experts cannot be used directly here because it decides which expert to select based on a soft partitioning of the input space, and in our experiment any force field is active over any portion of the input space at different moments in time. Here we propose an extension to the ME architecture that is able to deal with mappings overlapping in X space. Another aspect of the results that is difficult to model using standard computational architectures is memory consolidation. To account for this effect we introduce two different learning rates (Alverez and Squire 1994). Some connections in the network change faster, as a result of which they can serve as short-term memory. We also introduce an off-line training phase (possibly corresponding to sleep) in which random inputs are generated, the part of the network containing the fast connections is used to produce a target output, and the resulting input-output pair is used to train the slowly changing connections. During the offline phase the faster changing connections are fixed, after that they are randomized. 4.1 Modified Mixture of Experts The ME model assumes that Y is generated from X by one of N processes (Wl, ... , W N) and therefore the likelihood function is: L(8IXt, Yt) = P(Ytlxt, 8) = I: git P(YtIWi, Xt, 8) i g/ = P(WiIXt, 8), where 8 represents the parameters of the model and gi is the prior probability. We want to use the posterior probability P(Wi IXt, Yt) instead, because the processes (different force fields) are separable in XY space, but not in X space. If we want to implement an on-line controller such a term is not available, because at time t, Yt is still unknown (the task of the controller is to produce it). We could approximate P(WiIXt, Yt) with P(WdXt-l. Yt-!) , because dynamic conditions do not change very often. Now the gating network (which computes P(Wdxt) in ME) is going to select expert i based on the previous XY pair. This approach would obviously lead to a single large error at the moment when the force field changes, but so will any model using only Xt to compute Yt. In fact such an error seems to be consistent with our psychophysics data. Thus the learning rule is: ~8i = lIih/(Yt - J.'i(Xt, 8i)) g.t _ h.t-1 I I h.t = P(W~I ~ ~ 8) = git P(YtIWi, Xt, 8) I I XII y" '" .tp( IW. 8)' L..tj g, Yt "Xt, where h: is the posterior probability and J.'i is the output of exper i. J.'i is a linear function of the inputs. In our model we used 4 experts. In order to model the process of consolidation, we gave one expert a learning rate that was 10 times higher than the learning rate of the other 3 experts. Catastrophic Interference in Human Motor Learning 25 4.2 The Model We simulated the dynamics of a 2-joint planar arm similar to the one used in our previous work (Shadmehr and Mussa-Ivaldi, 1994). The torque applied to the arm at every point in time is the sum of the outputs of a fixed controller, a PO controller, and an adaptive controller with the architecture described above. The fixed controller maps (), iJ (current state), and jj (desired next state) into a torque T. The mapping is exact when no external forces are applied. The desired trajectories are minimum-jerk trajectories (Flash and Hogan 1985) sampled at 100Hz. The desired trajectories are 10 cm long and last 0.5 seconds. The PO controller is used to compensate for the troques produced by the force field while the adaptive controller has not yet learned to do that. The adaptive part of the controller consists of a mixture of 4 linear experts (whose initial output is small) and a modified gating network described above. The system operates as follows: (), iJ, jj are sent to the fixed controller, which outputs a torque Tl; the PO controller outputs a torque T2 based on the current deviation from the desired joint position and velocity; 8 terms describing the current state of the arm (and chosen to linearize the mapping to be learned) are sent to the mixture model, which outputs a torque T3; Te = Tl + T2 T3 is applied to the plant as a control signal; the actual torque T = Te + TJ is computed. The mixture model is trained to produce the torque TJ resulting from the force field. In other words, the adaptive part of the controller learns to compensate for the force field exerted by the environment. The parameters of the mixture model are updated after every movement, so a training pair (Xt, Yt) is actually a batch of 50 points. The input, Xt, consists of the 8 terms describing the current state and the desired next state; the output, Yt, is the torque vector that the force field produces.The compensatory torques for a complete movement are computed before the movement starts. The only processing done during the movement is the computations necessary for the PO controller. 4.3 Results 4.3.1 Negative Transfer When the network was given two successive, incompatable mappings to learn (this corresponds to learning to move in two opposite force fields), the resulting performance very much resembled that of our human subjects. The performance in the second mapping was much poorer than that in the first mapping. The fast-learning expert changed its weights to learn both tasks. Since the two tasks involved anticorrelated maps, the fast expert's weights after learning the first mapping were very inappropriate for the second task, leading to the observed negative transfer. 4.3.2 Catastrophic Interference When the network was trained on two successive, opposite force fields, with no consolidation occurring between the two training sessions, the learning in second training session overwrote the learning that occurred during the first training session (fig 3A). Since the expert with the fast-changing weights attempted to learn both mappings, this catastrophic interference is not unexpected. 4.3.3 Consolidation When the network was allowed to undergo consolidation between learning the first and the second force field, the network no longer suffered from catastrophic inter26 Tom Braslzers-Krug, Reza Slzadmelzr, Emanuel Todorov A D.65~ ____ ~ ___ --:=" o 10 20 Figure 3: A: Learning curves for the ME architecture. Dark line: curve when first learning Field A. light line: curve when given Field A a second time, after learning FIeld B (no consolidation allowed between learning Field A and Field B). Note lack of retention of Field A. B: Learning curves for the same architecture in Field A before and after learning Field B. Consolidations was allowed between learning Field A and Field B. ference (fig 3B). The learning that had initially resided in the fast-learning expert was transfered to one of the slower-learning networks. Thus, when the expert with the fast-changing connections learned the second mapping, the original learning was no longer destroyed. In addition, when the network was allowed to consolidate the second force field, a different slow-learning expert stored the second mapping. In this way, the network stored multiple maps in long-term memory. 5 Conclusions We have presented psychophysical evidence for catastrophic interference. We have also shown results that suggest that motor learning can undergo a process of consolidation. By adding distinct fast- and slow-changing weights to a mixture of experts architecture, we were able to account for these psychophysical findings. We plan to investigate further the neural correlates of consolidation using both brain imaging in humans and electrophysiological studies in primates. References P. Alverez and L. Squire. (1994) Memory consolidation and the medial temporal love: a simple network model, PNAS91:15:7041-7045. T. Brashers-Krug, et al. (1994) Temporal aspects of human motor learning, Soc. Neurosci. Abstract in press. T. Flash and N. Hogan. (1985) The coordination of arm movements: an experimentally confirmed mathematical model. J. Neurosci. 5:1688-1703. R. Jacobs and M. Jordan. (1993) Learning piecewise control strategies in a modular neural network architecture. IEEE Trans. on Systems, Man and Cyber. 23:2: 337-345. I. Jenkins, et al. (1994) Motor sequence learning: a study with positron emission tomography, J. Neurosci.14:3775-3790. M. Jordan and R. Jacobs. (1994) Hierarchical mixture of experts and the EM algorithm, Neural Computation 6:2: 181-214. M. Jordan and D. Rumelhart. (1992) Forward models: supervised learning with a distal teacher Cognitive Sci. 16:307-354 A. Karni and D. Sagi. (1993) Nature 365:250. R. Shadmehr and F. Mussa-Ivaldi. (1994) Adaptive representation of dynamics during learning of a motor task, J. Neurosci.14:5:3208-3224. R. Shadmehr, T. Brashers-Krug, F. Mussa-Ivaldi, (1995) Interference in learning internal models of inverse dynamics in humans, Adv Neural Inform Proc Systvol 7, in press	1994	109
869	A Comparison of Discrete-Time Operator Models for Nonlinear System Identification Andrew D. Back, Ah Chung Tsoi Department of Electrical and Computer Engineering, University of Queensland St. Lucia, Qld 4072. Australia. e-mail: {back.act}@elec.uq.oz.au Abstract We present a unifying view of discrete-time operator models used in the context of finite word length linear signal processing. Comparisons are made between the recently presented gamma operator model, and the delta and rho operator models for performing nonlinear system identification and prediction using neural networks. A new model based on an adaptive bilinear transformation which generalizes all of the above models is presented. 1 INTRODUCTION The shift operator, defined as qx(t) ~ x(t + 1), is frequently used to provide time-domain signals to neural network models. Using the shift operator, a discrete-time model for system identification or time series prediction problems may be constructed. A common method of developing nonlinear system identification models is to use a neural network architecture as an estimator F(Y(t), X(t); 0) of F(Y(t), X(t», where 0 represents the parameter vector of the network. Shift operators at the input of the network provide the regression vectors Y(t-l) = [yet-I), ... , y(t-N)]', andX(t) = [x(t), ... , x(t-M)]' in a manner analogous to linear filters, where [.], represents the vector transpose. It is known that linear models based on the shift operator q suffer problems when used to modellightly-damped-Iow-frequency (LDLF) systems, with poles near (1,0) on the unit circle in the complex plane [5]. As the sampling rate increases, coefficient sensitivity and round-off noise become a problem as the difference between successive sampled inputs becomes smaller and smaller. 884 Andrew D. Back, Ah Chung Tsoi A method of overcoming this problem is to use an alternative discrete-time operator. Agarwal and Burrus first proposed the use of the delta operator in digital filters to replace the shift operator in an attempt to overcome the problems described above [1]. The delta operator is defined as {) = q-l ~ (1) where ~ is the discrete-time sampling interval. Williamson showed that the delta operator allows better performance in terms of coefficient sensitivity for digital filters derived from the direct form structure [19], and a number of authors have considered using it in linear filtering, estimation and control [5, 7, 8] More recently, de Vries, Principe at. al. proposed the gamma operator [2, 3] as a means of studying neural network models for processing time-varying patterns. This operator is defined by 'Y = q - (1 - c) C (2) It may be observed that it is a generalization of the delta operator with adjustable parameters c. An extension to the basic gamma operator introducing complex poles using a second order operator, was given in [18]. This raises the question, is the gamma operator capable of providing better neural network modelling capabilities for LDLF systems ? Further, are there any other operators which may be better than these for nonlinear modelling and prediction using neural networks? In the context of robust adaptive control, Palaniswami has introduced the rho operator which has shown useful improvements over the performance ofthe delta operator [9, 10]. The rho operator is defined as p = (3) where Cl, C2 are adjustable parameters. The rho operator generalizes the delta and gamma operators. For the case where Cl~ = C2~ = 1, the rho operator reduces to the usual shift operator. When c) = 0, and C2 = 1, the rho operator reduces to the delta operator [10]. For Cl ~ = C2~ = c, the rho operator is equivalent to the gamma operator. One advantage of the rho operator over the delta operator is that it is stably invertible, allowing the derivation of simpler algorithms [9]. The p operator can be considered as a stable low pass filter, and parameter estimation using the p operator is low frequency biased. For adaptive control systems, this gives robustness advantages for systems with unmodelled high frequency characteristics [9]. By defining the bilinear transformation (BLT) as an operator, it is possible to introduce an operator which generalizes all of the above operators. We can therefore define the pi operator as 11" = 2 (Clq C2) ~ (C3Q + C4) (4) with the restriction that Cl C4 f C2C3 (to ensure 11" is not a constant function [14]). The bilinear mapping produced has a pole at q = -C4/C3. By appropriate setting of the Cl, C2, C3, C4 parameters each operator, the pi operator can be reduced to each of the previous operators. In the work reported here, we consider these alternative discrete-time operators in feedforward neural network models for system identification tasks. We compare the popular A Comparison of Discrete-Time Operator Models for Nonlinear System Identification 885 gamma model [4] with other models based on the shift, delta, rho and pi operators. A framework of models and Gauss-Newton training algorithms is provided, and the models are compared by simulation experiments. 2 OPERATOR MODELS FOR NONLINEAR SIGNAL PROCESSING A model which generalizes the usual discrete-time linear moving average model, ie, a single layer network is given by yet) C(v,O) G(v,O)x(t) M L: bw-i i=O I q-~ shift operator 8-' delta operator ,-i gamma operator p - i rho operator 7r-i pi operator (5) (6) This general class of moving average model can be termed MA(v). We define uo(t) ~ x(t), and Ui(t) ~ V-IUi_l (t) and hence obtain ~Ui-l(t - 1) + Ui(t - 1) I x(t - i) CUi-l(t - 1) + (1 - C)Ui(t - 1) C2~Ui-l(t - 1) + (1- Cl~)Ui(t - 1) 2~1 (C3Ui-l(t) + C4Ui-l(t - 1») ~Ui(t - 1) shift operator delta operator gamma operator (7) rho operator pi operator A nonlinear model may be defined using a multilayer perceptron (MLP) with the v-operator elements at the input stage. The input vector ZP( t) to the network is Z?(t) = [Xi(t), V-1Xi(t), ... , V-MXi(t)]' (8) where Xi(t) is the ith input to the system. This model is termed the v-operator multilayer perceptron or MLP(v) model. An MLP(v) model having L layers with No, N I , ... , NL nodes per layer, is defined in the same manner as a usual MLP, with (9) Nz L WiiZJ-I(t) (10) i=l where each neuron i in layer 1 has an output of z!(t); a layer consists of N/ neurons (1 = 0 denotes the input layer, and 1 = L denotes the output layer, zJvz = 1.0 may be used for a bias); 10 is a sigmoid function typically evaluated as tanh(·), and a synaptic connection between unit i in the previous layer and unit k in the current layer is represented by wt. The notation t may be used to represent a discrete time or pattern instance. While the case 886 Andrew D. Back, Ah Chung Tsoi we consider employs the v-operator at the input layer only, it would be feasible to use the operators throughout the network as required. On-line algorithms to update the operator parameters in the MA(v) model can be found readily. In the case of the MLP(v) model, we approach the problem by backpropagating the error information to the input layer and using this to update the operator coefficients. de Vries and Principe et. al., proposed stochastic gradient descent type algorithms for adjusting the c operator coefficient using a least-squares error criterion [2, 12]. For brevity we omit the updating procedures for the MLP network weights; a variety of methods may be applied (see for example [13, 15]). We define an instantaneous output error criterion J(t) = !e2(t), where e(t) = y(t) - f)(t). Defining 0 as the estimated operator parameter vector at time t of the parameter vector 0, we have { c gamma operator o = [CI , C2]' rho operator [CI, C2, C3, 134 ]' pi operator A first order algorithm to update the coefficients is Oi(t + 1) Oi(t) + ~Oi(t) ~Oi (t) = -71\1 eJ ((}; t) where the adjustment in weights is found as ~Oi(t) = -71 oJ(t) o(}j M 71 I: 1/J1'(t)Oj(t) i=1 (11) (12) (13) (14) where OJ (t) is the backpropagated error at the jth node of input layer, and 1/J1' (t) is the first order sensitivity vector of the model operator parameters, defined by I &Ui(t) gamma operator ./~ (t) = [::~(t) &Ui(t)]' rho operator (15) !Pi &Cjl ' &Cj2 [ &Ui(t) &Ui(t) &Ui(t) &Ui(t)] I Pi operator &Cjl ' &Cj2 ' &Cj3 ' &Cj4 Substituting Ui(t) in from (7), the recursive equations for 1/J1 (t) (noting that 1/J1 (t)= tPHt) 'Vj) are tPi(t) = Ui_l(t - 1) - Ui(t - 1) + CitPi-l(t - 1) + (1 - C)tPi(t - 1) gamma operator [ C2~tPi-I,I(t - 1) + (1- CI~)tf;i I(t - 1) ~Ui(t - 1) ] tPi(t) = ~Ui-I(t - 1) + C2~tPi-I,2(t - 1) + (1 CI~)tPi,2(t _ 1) rho operator 2t (C3tPi-I,I(t) + C4tPi-l,1(t - 1») + ~tPi,l(t - 1) -2~2 (C3Ui-l(t) + C4Ui_l(t - 1») ~Ui(t - 1), c} C1 2~1 ~C3tPi-I,2(t) + C4tPi-I,2(t - 1») + ~tPi,2(t - 1) + tUi(t - 1), 2t (Ui_l(t) + C3tPi-l,3(t) + C4tPi-I,3(t - 1») + ~tPi~3(t - 1), 2t (C3tPi-I,4(t) + Ui-I(t - 1) + C4tPi-l,4(t - 1» + T.-tPi,4(t - 1) pi operator A Comparison of Discrete-Time Operator Models for Nonlinear System Identification 887 for the gamma, rho, and pi operators respectively, and where ""i ,j (t) refers to the jth element of the ith "" vector, with ""i ,o( t) = O. A more powerful updating procedure can be obtained by using the Gauss-Newton method [6]. In this case, we replace (14) with (omitting i subscripts for clarity), OCt + 1) = OCt) + 'Y(t)R- 1 (t)",,(t)A -16(t) (16) where 'Y(t) is the gain sequence (see [6] for details), A-I is a weighting matrix which may be replaced by the identity matrix [16], or estimated as [6] A(t) = A(t - 1) + 'Y(t) (62(t) - A(t - 1») R( t) is an approximate Hessian matrix, defined by R(t + 1) = A(t)R(t) + «(t)""(t),,,,'(t) (17) (18) where A(t) = 1 - «(t). Efficient computation of R- 1 may be performed using the matrix inversion lemma [17], factorization methods such as Cholesky decomposition or other fast algorithms. Using the well known matrix inversion lemma [6], we substitute pet) = R-l(t), where P t _1_p t _ «(t) ( P(t)",,(t)""'(t)P(t) ) () A(t) () A(t) A(t) + «(t)""'(t)P(t)",,(t) (19) The initial values of the coefficients are important in determining convergence. Principe et. al. [12] note that setting the coefficients for the gamma operator to unity provided the best approach for certain problems. 3 SIMULATION EXAMPLES We are primarily interested in the differences between the operators themselves for modelling and prediction, and not the associated difficulties of training multilayer perceptrons (recall that our models will only differ at the input layer). For the purposes of a more direct comparison, in this paper we test the models using a single layer network. Hence these linear system examples are used to provide an indication of the operators' performance. 3.1 EXPERIMENT 1 The first problem considered is a system identification task arising in the context of high bit rate echo cancellation [5]. In this case, the system is described by 0.0254- 0.0296z-1 + 0.00425z-2 H(z) = 1 _ 1.957z-1 + 0.957z-2 (20) This system has poles on the real axis at 0.9994, and 0.9577, thus it is an LDLF system. The input signal to the system in each case consisted of uniform white noise with unit variance. A Gauss-Newton algorithm was used to determine all unknown weights. We conducted Monte-Carlo tests using 20 runs of differently seeded training samples each of 2000 points to obtain the results reported. We assessed the performance of the models by using the Signal-to-NoiseRatio (SNR) defined as 1010g( E[d(t)2Jj E[e(t)2D, where E[·l is the expectation operator, and d(t) is the desired signal. For each run, we used the last 500 samples to compute a SNR figure. 888 Andrew D. Back, Ah Chung Tsoi 0.04 0.04 0.04 0.03 • • 0.03 0.03 0.02 0.02 0.02 0.01 o.ot o.ot o .~ .. 0 0 ....• -0.01 ~. -0.01 -0.01 -0.02 .~ -0.02 -0.02 -0.03 • -0.03 -0.03 -O''¥900 1920 -0·11900 1920 -0·11900 1920 (a) (b) (c) 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 0 -{l.01 -0.01 -0.02 -0.02 -0.03 -0.03 -0·11900 1920 -O''¥900 1920 (d) (e) Figure 1: Comparison of typical model output results for Experiment 1 with models based on the following operators: (a) shift, (b) delta (c) gamma, (d) rho, and (e) pi. Table 1: System Identification Experiment 1 Results For the purposes of this experiment, we conducted several trials and selected 0(0) values which provided stable convergence. The values chosen for this experiment were: 0(0) = {0.75, [0.5,0.75]' [0.75,0.7,0.35,-0.25]} for the gamma, rho and pi operator models respectively. In each case we used model order M = 8. Results for this experiment are shown in Table 1 and Figure 1. We observe that the pi operator gives the best performance overall. Some difficulties with instability occurring were encountered, thereby requiring a stability correction mechanism to be used on the operator updates. The next best performance was observed in the rho and then gamma models, with fewer instability problems occurring. 3.2 EXPERIMENT 2 The second experiment used a model described by 1 - 0.8731z- 1 - 0.8731z-2 + z-3 H(z) = 1 - 2.8653z- 1 + 2.7505z-2 - 0.8843z-3 (21) This system is a 3rd order lowpass filter tested in [11]. The same experimental procedures as used in Experiment 1 were followed in this case. For the second experiment (see Table 2), it was found that the pi operator gave the best results A Comparison of Discrete-Time Operator Models for Nonlinear System Identification 889 Table 2: System Identification Experiment 2 Results recorded over all the tests. On average however, the improvement for this identification problem is less. It is observed that that the pi model is only slightly better than the gamma and rho models. Interestingly, the gamma and rho models had no problems with stability, while the pi model still suffered from convergence problems due to instability. As before, the delta model gave a wide variation in results and performed poorly. From these and other experiments performed it appears that performance advantages can be obtained through the use of the more complex operators. As observed from the best recorded runs, the extra degrees of freedom in the rho and pi operators appear to provide the means to give better performance than the gamma model. The improvements of the more complex operators come at the expense of potential convergence problems due to instabilities occurring in the operators and a potentially multimodal mean square output error surface in the operator parameter space. Clearly, there is a need for further investigation into the performance of these models on a wider range of tasks. We present these preliminary examples as an indication of how these alternative operators perform on some system identification problems. 4 CONCLUSIONS Models based on the delta operator, rho operator, and pi operator have been presented and new algorithms derived. Comparisons have been made to the previously presented gamma model introduced by de Vries, Principe et. al. [4] for nonlinear signal processing applications. While the simulation examples considered show are only linear, it is important to realize that the derivations are applicable for multilayer perceptrons, and that the input stage of these networks is identical to what we have considered here. We treat only the linear case in the examples in order not to complicate our understanding of the results, knowing that what happens in the input layer is important to higher layers in network structures. The results obtained indicate that the more complex operators provide a potentially more powerful modelling structure, though there is a need for further work into mechanisms of maintaining stability while retaining good convergence properties. The rho model was able to perform better than the gamma model on the problems tested, and gave similar results in terms of susceptibility to convergence and instability problems. The pi model appears capable of giving the best performance overall, but requires more attention to ensure the stability of the coefficients. For future work it would be of value to analyse the convergence of the algorithms, in order to design methods which ensure stability can be maintained, while not disrupting the convergence of the model. 890 Andrew D. Back, Ah Chung Tsoi Acknowledgements The first author acknowledges financial support from the Australian Research Council. The second author acknowledges partial support from the Australian Research Council. References [1] R.C. Agarwal and C.S. Burrus, ''New recursive digital filter structures having very low sensitivity and roundoff noise", IEEE Trans. Circuits and Systems, vol. cas-22, pp. 921-927, Dec. 1975. [2] de Vries, B. Principe, J.C. "A theory for neural networks with time delays", Advances in Neural Information Processing Systems, 3, R.P. Lippmann (Ed.), pp 162 - 168, 1991. [3] de Vries, B., Principe, J. and P.G. de Oliveira "Adaline with adaptive recursive memory", Neural Networks for Signal Processing I. Juang, B.H., Kung, S.Y., Kamm, C.A. (Eds) IEEE Press, pp. 101-110, 1991. [4] de Vries, B. Principe, J. "The Gamma Model - a new neural model for temporal processing". Neural Networks. Vol 5, No 4, pp 565 - 576, 1992. [5] H. Fan and Q. Li, "A () operator recursive gradient algorithm for adaptive signal processing", Proc. IEEE Int. Conf. Acoust. Speech and Signal Proc., vol. nI, pp. 492-495, 1993. [6] L. Ljung, and T. SOderstrtlm, Theory and Practice of Recursive Identification, Cambridge, Massachusetts: The MIT Press, 1983. [7] R.H. Middleton, and G.C. Goodwin, Digital Control and Estimation, Englewood Cliffs: Prentice Hall, 1990. [8] V. Peterka, "Control of Uncertain Processes: Applied Theory and Algorithms", Kybernetika, vol. 22, pp. 1-102, 1986. [9] M. Palaniswami, "A new discrete-time operator for digital estimation and control". The Uni versity of Melbourne, Department of Electrical Engineering, Technical Report No.1, 1989. [10] M. Palaniswami, "Digital Estimation and Control with a New Discrete Time Operator", Proc. 30th IEEE Conf. Decision and Control, pp. 1631-1632, 1991. [11] J.C. Principe, B. de Vries, J-M. Kuo and P. Guedes de Oliveira, "Modeling Applications with the Focused Gamma Net", Advances in Neural Information Processing Systems, vol. 4, pp. 143-150,1991. [12] J.C. Principe, B. de Vries, and P. Guedes de Oliveira, "The Gamma Filter - a new class of adaptive IIR filters with restricted feedback", IEEE Trans. Signal Processing, vol. 41, pp. 649-656, 1993. [13] G. V. Puskorius, and L.A. Feldkamp, "Decoupled Extended Kalman Filter Training of Feedforward Layered Networks", Proc. Int Joint Conf. Neural Networks, Seattle, vol I, pp. 771-777,1991. [14] E.B. Saff and A.D. Snider, Fundamentals of Complex Analysis for Mathematics, Science and Engineering. Englewood Cliffs, NJ: Prentice-Hall, 1976. [15] S. Shah and F. Palmieri, "MEKA - A Fast Local Algorithm for Training Feedfoward Neural Networks", Proc Int Joint Conf. on Neural Networks, vol m, pp. 41-46, 1990. [16] J.J. Shynk, "Adaptive IIR filtering using parallel-form realizations", IEEE Trans. Acoust. Speech Signal Proc., vol. 37, pp. 519-533,1989. [17] Soderstrom and Stoica, "System Identification", London: Prentice Hall, 1989. [18] T.O. de Silva, P.G. de Oliveira, J.e. Principe, and B. de Vries, " Generalized feedforward filters with complex poles", Neural Networks for Signal Processing n, S.Y. Kung et. al. (Eds) Piscataway,NJ: IEEE Press, 1992. [19] D. Williamson, "Delay replacement in direct form structures", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 453-460, Aprl. 1988. PARTvm VISUAL PROCESSING	1994	11
870	On the Computational Complexity of Networks of Spiking Neurons (Extended Abstract) Wolfgang Maass Institute for Theoretical Computer Science Technische Universitaet Graz A-80lO Graz, Austria e-mail: maass@igi.tu-graz.ac.at Abstract We investigate the computational power of a formal model for networks of spiking neurons, both for the assumption of an unlimited timing precision, and for the case of a limited timing precision. We also prove upper and lower bounds for the number of examples that are needed to train such networks. 1 Introduction and Basic Definitions There exists substantial evidence that timing phenomena such as temporal differences between spikes and frequencies of oscillating subsystems are integral parts of various information processing mechanisms in biological neural systems (for a survey and references see e.g. Abeles, 1991; Churchland and Sejnowski, 1992; Aertsen, 1993). Furthermore simulations of a variety of specific mathematical models for networks of spiking neurons have shown that temporal coding offers interesting possibilities for solving classical benchmark-problems such as associative memory, binding, and pattern segmentation (for an overview see Gerstner et al., 1992). Some aspects of these models have also been studied analytically, but almost nothing is known about their computational complexity (see Judd and Aihara, 1993, for some first results in this direction). In this article we introduce a simple formal model SNN for networks of spiking neurons that allows us to model the most important timing phenomena of neural nets (including synaptic modulation), and we prove upper and lower bounds for its computational power and learning complexity. Further 184 Wolfgang Maass details to the results reported in this article may be found in Maass, 1994a,1994b, 1994c. Definition of a Spiking Neuron Network (SNN): An SNN N consists of - a finite directed graph {V, E} (we refer to the elements of V as "neurons" and to the elements of E as "synapses") - a subset Yin S; V of input neurons - a subset Vout S; V of output neurons - for each neuron v E V - Yin a threshold-function 9 v : R+ -+ R U {oo} {where R+ := {x E R : x ~ O}) - for each synapse {u, v} E E a response-function £u,v : R+ -+ R and a weight- function Wu,v : R+ -+ R . We assume that the firing of the input neurons v E Yin is determined from outside of N, i.e. the sets Fv S; R+ of firing times ("spike trains") for the neurons v E Yin are given as the input of N. Furthermore we assume that a set T S; R+ of potential firing times 7iiiSfjeen fixed. For a neuron v E V - Yin one defines its set Fv of firing times recursively. The first element of Fv is inf{t E T : Pv(t) ~ 0 v(O)} , and for any s E Fv the next larger element of Fv is inf{t E T : t > sand Pv(t) ~ 0 v(t - s)} ,where the potential function Pv : R+ -+ R is defined by Pv(t) := 0 + L L wu,v(s) . £u,v(t - s) u : {u, v} E EsE Fu : s < t The firing times ("spike trains") Fv of the output neurons v E Vout that result in this way are interpreted as the output of N. Regarding the set T of potential firing times we consider in this article the case T = R+ (SNN with continuous time) and the case T = {i· JJ : i E N} for some JJ with 1/ JJ E N (SNN with discrete time). We assume that for each SNN N there exists a bound TN E R with TN > 0 such that 0 v(x) = 00 for all x E (0, TN) and all v E V - Yin (TN may be interpreted as the minimum of all "refractory periods" Tref of neurons in N). Furthermore we assume that all "input spike trains" Fv with v E Yin satisfy IFv n [0, t]l < 00 for all t E R+. On the basis of these assumptions one can also in the continuous case easily show that the firing times are well-defined for all v E V - Yin (and occur in distances of at least TN)' Input- and Output-Conventions: For simulations between SNN's and Turing machines we assume that the SNN either gets an input (or produces an output) from {O, 1}* in the form of a spike-train (i.e. one bit per unit of time), or encoded into the phase-difference of just two spikes. Real-valued input or output for an SNN is always encoded into the phase-difference of two spikes. Remarks a) In models for biological neural systems one assumes that if x time-units have On the Computational Complexity of Networks of Spiking Neurons /85 passed since its last firing, the current threshold 0 11 (z) of a neuron v is "infinite" for z < TreJ (where TreJ = refractory period of neuron v), and then approaches quite rapidly from above some constant value. A neuron v "fires" (i.e. it sends an "action potential" or "spike" along its axon) when its current membrane potential PII (t) at the axon hillock exceeds its current threshold 0 11 . PII (t) is the sum of various postsynaptic potentials W U ,II(S). t:U ,II(t - s). Each of these terms describes an excitatory (EPSP) or inhibitory (IPSP) postsynaptic potential at the axon hillock of neuron v at time t, as a result of a spike that had been generated by a "presynaptic" neuron u at time s, and which has been transmitted through a synapse between both neurons. Recordings of an EPSP typically show a function that has a constant value c (c = resting membrane potential; e.g. c = -70m V) for some initial time-interval (reflecting the axonal and synaptic transmission time), then rises to a peak-value, and finally drops back to the same constant value c. An IPSP tends to have the negative shape of an EPSP. For the sake of mathematical simplicity we assume in the SNN-model that the constant initial and final value of all response-functions t:U ,1I is equal to 0 (in other words: t:U ,1I models the difference between a postsynaptic potential and the resting membrane potential c). Different presynaptic neurons u generate postsynaptic potentials of different sizes at the axon hillock of a neuron v, depending on the size, location and current state of the synapse (or synapses) between u an? v. This effect is modelled by the weight-factors W U ,II(S). The precise shapes of threshold-, response-, and weight-functions vary among different biological neural systems, and even within the same system. Fortunately one can prove significant upper bounds for the computational complexity of SNN's N without any assumptions about the specific shapes of these functions of N. Instead, we only assume that they are of a reasonably simple mathematical structure. b) In order to prove lower bounds for the computational complexity of an SNN N one is forced to make more specific assumptions about these functions. All lower bound results that are reported in this article require only some rather weak basic assumptions about the response- and threshold-functions. They mainly require that EPSP's have some (arbitrarily short) segment where they increase linearly, and some (arbitrarily short) segment where they decrease linearly (for details see Maass, 1994a, 1994b). c) Although the model SNN is apparently more "realistic" than all models for biological neural nets whose computational complexity has previously been analyzed, it deliberately sacrifices a large number of more intricate biological details for the sake of mathematical tractability. Our model is closely related to those of (Buhmann and Schulten, 1986), and (Gerstner, 1991, 1992). Similarly as in (Buhmann and Schulten, 1986) we consider here only the deterministic case. d) The model SNN is also suitable for investigating algorithms that involve synaptic modulation at various time-scales. Hence one can investigate within this framework not only the complexity of algorithms for supervised and unsupervised learning, but also the potential computational power of rapid weight-changes within the course of a computation. In the theorems of this paper we allow that the value of a weight WU,II(S) at a firing time s E Fu is defined by an algebraic computation tree (see van Leeuwen, 1990) in terms of its value at previous firing times s' E Fu with s' < s, some preceding firing times s < s of arbitrary other neurons, and arbitrary realvalued parameters. In this way WU,II(S) can be defined by different rational functions /86 Wolfgang Maass of the abovementioned arguments, depending on the numerical relationship between these arguments (which can be evaluated by comparing first the relative size of arbitrary rational functions of these arguments). As a simple special case one can for example increase wu •tI (perhaps up to some specified saturation-value) as long as neurons u and v fire coherently, and decrease wu •tI otherwise. For the sake of simplicity in the statements of our results we assume in this extended abstract that the algebraic computation tree for each weight wU •tI involves only 0(1) tests and rational functions of degree 0(1) that depend only on 0(1) of the abovementioned arguments. Furthermore we assume in Theorems 3, 4 and 5 that either each weight is an arbitrary time-invariant real, or that each current weight is rounded off to bit-length poly(1ogpN') in binary representation, and does not depend on the times of firings that occured longer than time 0(1) ago. Furthermore we assume in Theorems 3 and 5 that the parameters in the algebraic computation tree are rationals of bit-length O(1ogpN'). e) It is well-known that the Vapnik-Chervonenkis dimension {"VC-dimension"} of a neural net N (and the pseudo-dimension for the case of a neural net N with realvalued output, with some suitable fixed norm for measuring the error) can be used to bound the number of examples that are needed to train N (see Haussler, 1992). Obviously these notions have to be defined differently for a network with timedependent weights. We propose to define the VC-dimension (pseudo-dimension)of an SNN N with time-dependent weights as the VC-dimension (pseudo-dimension) of the class of all functions that can be computed by N with different assignments of values to the real-valued (or rational-valued) parameters of N that are involved in the definitions of the piecewise rational response-, threshold-, and weight-functions of N. In a biological neural system N these parameters might for example reflect the concentrations of certain chemical substances that are known to modulate the behavior of N. f) The focus in the investigation of computations in biological neural systems differs in two essential aspects from that of classical computational complexity theory. First, one is not only interested in single computations of a neural net for unrelated inputs z, but also in its ability to process an interrelated sequence «(z( i), y( i)} )ieN of inputs and outputs, which may for example include an initial training sequence for learning or associative memory. Secondly, exact timing of computations is allimportant in biological neural nets, and many tasks have to be solved within a specific number of steps. Therefore an analysis in terms of the notion of a real-time computation and real-time simulation appears to be more adequate for models of biological neural nets than the more traditional analysis via complexity classes. One says that a sequence «(z(i),y(i)})ieN is processed in real-time by a machine M, if for every i E N the machine M outputs y( i) within a constant number c of computation steps after having received input z(i). One says that M' simulates M in real-time (with delay factor ~), if every sequence that is processed in real-time by M (with some constant c), can also be processed in real-time by M' (with a constant ~ . c). For SNN's M we count each spike in M as a computation step. These definitions imply that a real-time simulation of M by M' is a special case of a linear-time simulation, and hence that any problem that can be solved by M with a certain time complexity ten), can be solved by M' with time complexity O(t(n» On the Computational Complexity of Networks of Spiking Neurons 187 (see Maass, 1994a, 1994b, for details). 2 Networks of Spiking Neurons with Continuous Time Theorem 1: If the response- and threshold-functions of the neurons satisfy some rather weak basic assumptions (see Maass, 1994a, 1994b), then one can build from such neurons for any given dEN an SNN NTM(d) of finite size with rational delays that can simulate with a suitable assignment of rational values from [0, 1] to its weights any Turing machine with at most d tapes in real-time. Furthermore NTM(2) can compute any function F : {0,1}* -- {0,1}* with a suitable assignment of real values from [0,"1] to its weights. The fixed SNN NTM(d) of Theorem 1 can simulate Turing machines whose tape content is much larger than the size of NTM (d), by encoding such tape content into the phase-difference between two oscillators. The proof of Theorem 1 transforms arbitrary computations of Turing machines into operations on such phase-differences. The last part of Theorem 1 implies that the VC-dimension of some finite SNN's is infinite. In contrast to that the following result shows that one can give finite bounds for the VC-dimension of those SNN's that only use a bounded numbers of spikes in their computation. Furthermore the last part of the claim of Theorem 2 implies that their VC-dimension may in fact grow linearly with the number S of spikes that occur in a computation. Theorem 2: The VC-dimension and pseudo-dimension of any SNN N with piecewise linear response- and threshold-functions, arbitrary real-valued parameters and time-dependent weights (as specified in section 1) can be bounded (even for realvalued inputs and outputs) by D(IEI . WI . S(log IVI + log S» if N uses in each computation at most S spikes. Furthermore one can construct SNN's (with any response- and threshold-functions that satisfy our basic assumptions, with fixed rational parameters and rational timeinvariant weights) whose VC-dimension is for computations with up to S spikes as large as O(IEI . S). We refer to Maass, 1994a, 1994c, for upper bounds on the computational power of SNN's with continuous time. 3 Networks of Spiking Neurons with Discrete Time In this section we consider the case where all firing times of neurons in N are multiples of some J.l with 1/ J.l EN. We restrict our attention to the biologically plausible case where there exists some tN ~ 1 such that for all z > tN all response functions £U,II(Z) have the value ° and all threshold functions ell(z) have some arbitrary constant value. If tN is chosen minimal with this property, we refer to PN := rtN/J.ll as the timing-precision ofN. Obviously for PN = 1 the SNN is equivalent to a "non-spiking" neural net that consists of linear threshold gates, whereas a SNN with continuous time may be viewed as the opposite extremal case for PN -- 00. 188 Wolfgang Maass The following result provides a significant upper bound for the computational power of an SNN with discrete time, even in the presence of arbitrary real-valued parameters and weights. Its proof is technically rather involved. Theorem 3: Assume that N is an SNN with timing-precision PJII, piecewise polynomial response- and piecewise rational threshold-functions with arbitrary real-valued parameters, and weight-functions as specified in section 1. Then one can simulate N for boolean valued inputs in real-time by a Turing machine with poly(lVl, logpJII,log l/TJII) states and poly(lVl, logpJII, tJII/TJII) tape-cells. On the other hand any Turing machine with q states that uses at most s tapecells can be simulated in real-time by an SNN N with any response- and thresholdfunctions that satisfy our basic assumptions, with rational parameters and timeinvariant rational weights, with O(q) neurons, logpJII = O(s), and tJII/TJII = 0(1). The next result shows that the VC-dimension of any SNN with discrete time is finite, and grows proportionally to logpJII. The proof of its lower bound combines a new explicit construction with that of Maass, 1993. Theorem 4: Assume that the SNN N has the same properties as in Theorem 3. Then the VC-dimension and the pseudo-dimension of N (for arbitrary real valued inputs) can be bounded by O(IEI·IVI·logpJII), independently of the number of spikes in its computations. Furthermore one can construct SNN's N of this type with any response- and threshold-functions that satisfy our basic assumptions, with rational parameters and time-invariant rational weights, so that N has (already for boolean inputs) a VCdimension of at least O(IEI(logpJII + log IE!». 4 Relationships to other Computational Models We consider here the relationship between SNN's with discrete time and recurrent analog neural nets. In the latter no "spikes" or other non-trivial timing-phenomena occur, but the output of a gate consists of the "analog" value of some squashingor activation function that is applied to the weighted sum of its inputs. See e.g. (Siegelmann and Sontag, 1992) or (Maass, 1993) for recent results about the computational power of such models. We consider in this section a perhaps more "realistic" version of such modelsN, where the output of each gate is rounded off to an integer multiple of some ~ (with a EN). We refer to a as the number of activation levels of N. It is an interesting open problem whether such analog neural nets (with gate-outputs interpreted as firing rates) or networks of spiking neurons provide a more adequate computational model for biological neural systems. Theorem 5 shows that in spite of their quite different structure the computational power of these two models is in fact closely related. On the side the following theorem also exhibits a new subclass of deterministic finite automata (DFA's) which turns out to be of particular interest in the context of neural nets. We say that a DFA M is a sparse DFA of size s if M can be realized by a Turing machine with s states and space-bound s (such that each step of M corresponds to one step of the Turing machine). Note that a sparse DFA may have exponentially in s many states, but that only poly(s) bits are needed to describe its On the Computational Complexity of Networks of Spiking Neurons 189 transition function. Sparse DFA's are relatively easy to construct, and hence are very useful for demonstrating (via Theorem 5) that a specific task can be carried out on a "spiking" neural net with a realistic timing precision (respectively on an analog neural net with a realistic number of activation levels). Theorem 5: The following classes of machines have closely related computational power in the sense that there is a polynomial p such that each computational model from any of these classes can be simulated in real-time (with delay-factor ~ p(s») by some computational model from any other class (with the size-parameter s replaced by p(s»): • sparse DFA's of size s • SNN's with 0(1) neurons and timing precision 23 • recurrent analog neural nets that consist of O( 1) gates with piecewise rational activation functions with 23 activation levels, and parameters and weights of bit-length $ s • neural nets that consist of s linear threshold gates (with recurrencies) with arbitrary real weights. The result of Theorem 5 is remarkably stable since it holds no matter whether one considers just SNN's N with 0(1) neurons that employ very simple fixed piecewise linear response- and threshold-functions with parameters of bit-length 0(1) (with tN/TN = 0(1) and time-invariant weights of bit-length $ s), or if one considers SNN's N with s neurons with arbitrary piecewise polynomial response- and piecewise rational threshold-functions with arbitrary real-valued parameters, tN/TN ~ s, and time-dependent weights (as specified in section 1). 5 Conclusion We have introduced a simple formal model SNN for networks of spiking neurons, and have shown that significant bounds for its computational power and sample complexity can be derived from rather weak assumptions about the mathematical structure of its response-, threshold-, and weight-functions. Furthermore we have established quantitative relationships between the computational power of a model for networks of spiking neurons with a limited timing precision (i.e. SNN's with discrete time) and a quite realistic version of recurrent analog neural nets (with a bounded number of activation levels). The simulations which provide the proof of this result create an interesting link between computations with spike-coding (in an SNN) and computations with frequency-coding (in analog neural nets). We also have established such relationships for the case of SNN's with continuous time (see Maass 1994a, 1994b, 1994c), but space does not permit to report these results in this article. The Theorems 1 and 5 of this article establish the existence of mechanisms for simulating arbitrary Turing machines (and hence any common computational model) on an SNN. As a consequence one can now demonstrate that a concrete task (such as binding, pattern-matching, associative memory) can be carried out on an SNN by simply showing that some arbitrary common computational model can carry out that task. Furthermore one can bound the required timing-precision of the SNN in terms of the space needed on a Turing machine. 190 Wolfgang Maass Since we have based our investigations on the rather refined notion of a real-time simulation, our results provide information not only about the possibility to implement computations, but also adaptive behavior on networks of spiking neurons. Acknowledgement I would like to thank Wulfram Gerstner for helpful discussions. References M. Abeles. (1991) Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge University Press. A. Aertsen. ed. (1993) Brain Theory: Spatio-Temporal Aspects of Brain Function. Elsevier. J. Buhmann, K. Schulten. (1986) Associative recognition and storage in a model network of physiological neurons. Bioi. Cybern. 54: 319-335. P. S. Churchland, T. J. Sejnowski. (1992) The Computational Brain. MIT-Press. W. Gerstner. (1991) Associative memory in a network of "biological" neurons. Advances in Neural Information Processing Systems, vol. 3, Morgan Kaufmann: 84-90. W. Gerstner, R. Ritz, J. L. van Hemmen. (1992) A biologically motivated and analytically soluble model of collective oscillations in the cortex. Bioi. Cybern. 68: 363-374. D. Haussler. (1992) Decision theoretic generalizations of the PAC model for neural nets and other learning applications. Inf and Comput. 95: 129-161. K. T. Judd, K. Aihara. (1993) Pulse propagation networks: A neural network model that uses temporal coding by action potentials. Neural Networks 6: 203-215. J. van Leeuwen, ed. (1990) Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity. MIT-Press. W. Maass. (1993) Bounds for the computational power and learning complexity of analog neural nets. Proc. 25th Annual ACM Symposium on the Theory of Computing, 335-344. W. Maass. (1994a) On the computational complexity of networks of spiking neurons (extended abstract). TR 393 from May 1994 of the Institutes for Information Processing Graz (for a more detailed version see the file maass.spiking.ps.Z in the neuroprose archive). W. Maass. (1994b) Lower bounds for the computational power of networks of spiking neurons. Neural Computation, to appear. W. Maass. (1994c) Analog computations on networks of spiking neurons (extended abstract). Submitted for publication. H. T. Siegelmann, E. D. Sontag. (1992) On the computational power of neural nets. Proc. 5th ACM- Workshop on Computational Learning Theory, 440-449.	1994	110
871	New Algorithms for 2D and 3D Point Matching: Pose Estimation and Correspondence Steven Goldl , Chien Ping LuI, Anand Rangarajanl , Suguna Pappul and Eric Mjolsness2 Department of Computer Science Yale University New Haven, CT 06520-8285 Abstract A fundamental open problem in computer vision-determining pose and correspondence between two sets of points in spaceis solved with a novel, robust and easily implementable algorithm. The technique works on noisy point sets that may be of unequal sizes and may differ by non-rigid transformations. A 2D variation calculates the pose between point sets related by an affine transformation-translation, rotation, scale and shear. A 3D to 3D variation calculates translation and rotation. An objective describing the problem is derived from Mean field theory. The objective is minimized with clocked (EM-like) dynamics. Experiments with both handwritten and synthetic data provide empirical evidence for the method. 1 Introduction Matching the representations of two images has long been the focus of much research in Computer Vision, forming an essential component of many machine-based ob1 E-mail address of authors: lastname-firstname@cs.yale.edu 2Department of Computer Science and Engineering, University of California at San Diego (UCSD), La Jolla, CA 92093-0114. E-mail: emj@cs.ucsd.edu 958 Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness ject recognition systems. Critical to most matching techniques is the determination of correspondence between spatially localized features within each image. This has traditionally been considered a hard problem - especially when the issues of noise, missing or spurious data, and non-rigid transformations are tackled [Grimson, 1990]. Many approaches have been tried, with tree-pruning techniques and generalized Hough transforms being the most common. We introduce anew, robust and easily implementable algorithm to find such poses and correspondences. The algorithm can determine non-rigid transformations between noisy 2D or 3D spatially located unlabeled feature sets despite missing or spurious features. It is derived by minimizing an objective function describing the problem with a combination of optimization techniques, incorporating Mean Field theory, slack variables, iterative projective scaling, and clocked (EM-like) dynamics. 2 2D with Affine Transformations 2.1 Formulating the Objective Our first algorithm calculates the pose between noisy, 2D point sets of unequal size related by an affine transformation - translation, rotation, scale and shear. Given two sets of points {Xj} and {Yk}, one can minimize the following objective to find the affine transformation and permutation which best maps Y onto X : J K J K E2D(m, t, A) = L L mjkllXj - t - AYkll2 + g(A) - aLL mjk j=l k=l j=l k=l with constraints: Vj Ef=l mjk ~ 1 , Vk Ef=l mjk ~ 1 , Vjk mjk ~ 0 and g(A) = 1a2 + /'i,b2 + AC2 A is decomposed into scale, rotation, vertical shear and oblique shear as follows: where, s(a) = ( eOa o ) ( cosh(c) b ,Sh2(c) = . h( ) esm c sinh(c) ) cosh(c) R(8) is the standard 2x2 rotation matrix. g(A) serves to regularize the affine transformation - bounding the scale and shear components. m is a fuzzy correspondence matrix which matches points in one image with corresponding points in the other image. The constraints on m ensure that each point in each image corresponds to at most one point in the other image. However, partial matches are allowed, in which case the sum of these partial matches may add up to no more than one. The inequality constraint on m permits a null match or multiple partial matches. The a term biases the objective towards matches. The decomposition of A in the above is not required, since A could be left as a 2x2 matrix and solved for directly in the algorithm that follows. The decomposition just provides for more precise regularization, i.e., specification of the likely kinds oftransformations. Also Sh2(C) could New Algorithms for 2D and 3D Point Matching 959 be replaced by another rotation matrix, using the singular value decomposition of A. We transform the inequality constraints into equality constraints by introducing slack variables, a standard technique from linear programming; K K+1 V j L mj k ::; 1 -+ V j L mj k = 1 k=l k=l and likewise for the column constraints. An extra row and column are added to the matrix m to hold the slack variables. Following the treatment in [Peterson and Soderberg, 1989; Yuille and Kosowsky, 1994] we employ Lagrange multipliers and an x log x barrier function to enforce the constraints with the following objective: J K J K E2D(m, t, A) = L L mjkllXj - t - AYkll 2 + g(A) ct L L mjk j=l k=l j=l k=l 1 J+1K+1 J K+1 K J+l +~ L: L mik(logmjk -1) + LJlj(L mjk -1) + LlIk(L mjk -1) (1) i=l k=l i=l k=l k=l j=l In this objective we are looking for a saddle point. (1) is minimized with respect to m, t, and A which are the correspondence matrix, translation, and affine transform, and is maximized with respect to Jl and 1I, the Lagrange multipliers that enforce the row and column constraints for m. m is fuzzy, with the degree of fuzziness dependent upon f3. 2.2 The Algorithm The algorithm to minimize the above objective proceeds in two phases. In phase one, while {t, A} are held fixed, m is initialized with a coordinate descent step, described below, and then iteratively normalized across its rows and columns until the procedure converges (iterative projective scaling). This phase is analogous to a softmax update, except that instead of enforcing a one-way, winner-take-all (maximum) constraint, a two-way, assignment constraint is being enforced. Therefore we describe this phase as a softassign. In phase two {t, A} are updated using coordinate descent. Then f3 is increased and the loop repeats. Let E2D be the above objective (1) without the terms that enforce the constraints (i.e. the x log x barrier function and the Lagrange parameters). In phase one (softassign) m is updated via coordinate descent: aE2D mjk = exp(-f3-a-) mjk Then m is iteratively normalized across j and k until Ef=l Ef=l Llmiajk < i : Using coordinate descent the {t, A} are updated in phase two. If a term of {A} cannot be computed analytically (because of its regularization), Newton's method 960 Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness is used to compute the root of the function. So if a is a term of {t, A} then in phase two we update a such that 8!~D = O. Finally f3 is increased and the loop repeats. By setting the partial derivatives of E2D to zero and initializing the Lagrange parameters to zero, the algorithm for phase one may be derived. Beginning with a small f3 allows minimization over a fuzzy correspondence matrix m, for which a global minimum is easier to find. Raising f3 drives the m's closer to 0 or 1, as the algorithm approaches a saddle point. 3 3D with Rotation and Translation The second algorithm solves the 3D-3D pose estimation problem with unknown correspondence. Given two sets of 3D points {Xj} and {Yk} find the rotation R, translation T, and correspondence m that minimize J K J K E3D(m,T,R) = LLmjkllRXj +T-YkI12-aLLmjk j=l k=1 j=l k=1 with the same constraint on the fuzzy correspondence matrix m as in 2D affine matching. Note that there is no regularization term for the T - R parameters. This algorithm also works in two phases. In the first, m is updated by a soft assign as was described for 2D affine matching. In the second phase, m is fixed, and the problem becomes a 3D to 3D pose estimation problem formulated as a weighted least squares problem. The rotation and translation are represented by a dual number quaternion (r, s) which corresponds to a screw coordinate transform [Walker et al. , 1991]. The rotation can be written as R(r) = W(r)tQ(r) and the translation as W(r)ts. Using these representations, the objective function becomes J K E3D = L L mjkllW(r)tQ(r)xj + W(r)t s - Ykl1 2 j=1 k=1 where Xj = (Xj, 0)1 and Yk = (Yk,O)t are the quaternion representations of Xj and Yk, respectively. Using the properties that Q(a)b = W(b)a and Q(a)tQ(a) = W(a)tW(a) = (ata)J, the objective function can be rewritten as where J K - L L mjkQ(Yk)tW(Xj) j=1k=1 1 J K 2LLmjkI j=1k=1 J K C3 = L L mjk(W(Xj) - Q(Yk)). j=1k=1 (2) New Algorithms for 2D and 3D Point Matching 961 With this new representation, all the information, including the current fuzzy estimate of the correspondence m are absorbed into the three 4-by-4 matrices Cl, C2 , C3 in (2), which can be minimized in closed-form [Walker et al., 1991]. 4 Experimental Results In this section we provide experimental results for both the 20 and 30 matching problems. As an application of the 20 matching algorithm, we present results in the context of handwritten character recognition. 4.1 Handwritten Character Data The data were generated using an X-windows tool which enables us to draw an image with the mouse on a writing pad on the screen. The contours of the images are discretized and are expressed as a set of points in the plane. In the experiments below, we generate 70 points per character on average. The inputs to the point matching algorithm are the x-y coordinates generated by the drawing program. No other pre-processing is done. The output is a correspondence matrix and a pose. In Figures 1 and 2, we show the correspondences found between several images drawn in this fashion.To make the actual point matches easier to see, we have drawn the correspondences only for every other model point. ~ : ..... -_ ........ _ ..........• Figure 1: Correspondence of digits In one experiment, we drew examples of individual digits, one as a model digit and then many different variations of it. In Figure 1, it can be seen that the 962 Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness .. ~ • • • • • • •• \. .. ..,. ... ..,.. .. I •• • • • • • • •• • •• • • .."..: .... • • • • • • • • • • • ' •• J Figure 2: Correspondence: "a" found in "cat", "0" found in "song" correspondences are good for a large variation from the model digit. For example, the correspondence is invariant to scale. Also, the correspondence is good between distorted digits, as in 3 and 6, or between different forms of a digit as in 4, 3, and 2. In another experiment (Figure 2), individual letters are correctly identified within words. Here, no pre-processing to segment the cursive word into letters is done. The correspondence returned by the point matching algorithm by itself can be good enough for identification. Even similar letters may be differentiated, for example the "a" in "cat" is correctly identified even though the "e" has a similar shape and the "0" is correctly identified in "song" , despite the similarity of the "s". 4.2 Randomly generated point sets: 2D In the second set of experiments, randomly generated dot patterns were used. In each trial a model is created by randomly generating with a uniform distribution, 50 points on a grid of unit area. Independent Gaussian noise N(O, 0-) is added to each of the points creating a jittered image. Then a fraction, Pd, of points are deleted, and a fraction, P6, of spurious points are added, randomly on the unit square. Finally, a randomly generated transformation is applied to the set to generate a new image. The objective then is to recover the transformation and correspondence between the transformed image and the original point set. The transformations we have considered are A -+ (Translation, rotation, scale) and the full affine transformation, A -+ (Translation, rotation, scale, vertical shear, oblique shear) The transformation parameters, {tz, ty , (J, a, b, c} are bounded in the following way: -0.5 < tz , ty < 0.5, -270 < (J < 270 , 0.5 ~ ea ~ 2 where a is the scale parameter, and 0.7 ~ eb,ee ~ 1/0.7 where b,c are the parameters for the two shears. Each of the parameters is chosen independently and uniformly from the possible ranges. ocrt.a.1 e • .,ima"te • We use the error measure ea = 31 a w'id~h", 1 where ea IS the error measure for parameter a and widtha is the range of permissible values for a . Dividing by widtha is preferable to dividing by aaetual, which incorrectly weights small aaetual values. The reported error (y axes of Figure 3) is the average error over all the parameters. New Algorithms for 2D and 3D Point Matching 963 The time to recover the correspondence and transformation for a problem instance of 50 points is about 50 seconds on a Silicon Graphics workstation with a R4400 processor. By varying parameters such as the annealing rate or stopping criterion, this can be reduced to about 20 seconds with some degradation in accuracy. For each trial combinations of u E {0.01, 0.02, ... , 0.08} and Pd E {O%, 10%,30%, 50%} and P6 E {O%, 10%} were used. Results are reported separately for transformations A and A. For each combination of (U,Pd,P6) 500 test instances were generated. Each data point in Figures 3.a and 3.b represents the average error measure for these 500 experiments. The noise and/or deletion-addition factor increases the error measure monotonically. As expected, the transformation A has better results than the affine transformation A. ~ oS Q) Translation, rotation, scale 0.6r---~----~----~----~ E 0.4 e JE JE JE IE JE JE JE • as ~ v ~ 0.2 + + e ~ w + + + + ~ o g ~ ~ + + )( )( ~ ~ o 0.02 0.04 0.06 0.08 Standard deviation of noise ~ oS Q) Affine 0.6r---~----~----~----~ • JE • • JE • IE • E 0.4 e as ~ + + co .. 0.2 ~ + + + + e ~ w o QS()()( o 0.02 0.04 0.06 0.08 Standard deviation of noise Figure 3: 2D Results for Synthetic Data x: Pd = 0.0,P6 = 0.0, + : Pd = 0.3,P6 = 0.1, 0: Pd = O.l,p& = 0.1 * : Pd = 0.5,p& = 0.1 4.3 Randomly generated point sets: 3D A test instance for 3D point matching involves generating a random 3D point set as a model image, and then generating a test image by applying a random transformation, adding noise and then randomly deleting points. 20 points are generated uniformly within an unit cube. The parameters for the transformation are generated as follows: The three rotation angles for R are selected from a uniform distribution U[20, 70]. Translation parameters T~, Ty, Tz are selected from a uniform distribution U[2.5,7.5]. Gaussian noise N(O, u) is added to the points. The objective then is to recover the three translation and three rotation parameters and to find the correspondence between this and the original point set. The results are summarized in Figure 4. 964 Steven Gold, Chien Ping Lu, Anand Rangarajan, Suguna Pappu, Eric Mjolsness 30 X ... 0.8 X X X 0 X X ... 0 X X X ... X ... t:: 20 WO.6 0 W 0 c 0 0 0 0 C 0 0 0 0 0 0 • .0:; '.0:; J2 0.4 ~ 10 en ~ ~ ~ c a: ~ as ~ ~ ~ ~ .= 0.2 ~ + ~ + + + 0 -+ + + + 00 + 0 0.2 0.4 0.6 0.2 0.4 0.6 Standard deviation of noise Standard deviation of noise Figure 4: 3D Results for Synthetic Data x: Pd = 0.0,P8 = 0.0, + : Pd = 0.2,P8 = 0.2, 5 Conclusion 0 : Pd = 0.1,P8 = 0.1 *: Pd = 0.3,ps = 0.3 We have developed an algorithm for solving 2D and 3D correspondence problems. The algorithm handles significant noise, missing or spurious features, and nonrigid transformations. Moreover it works with point feature data alone; inclusion of other types of feature information could improve its accuracy and speed. This approach may also be extended to solve multi-level problems. Additionally, the affine transform might be modified to include higher order transformations. It may also be used as a distance measure in learning [Gold et al.,1994] . Acknowledgements This work has been supported by AFOSR grant F49620-92-J-0465, ONR/DARPA grant N00014-92-J-4048, and the Yale Center for Theoretical and Applied Neuroscience (CTAN). Jing Yan developed the handwriting interface. References S. Gold, E. Mjolsness and A. Rangarajan. (1994) Clustering with a domain-specific distance measure. In J.D. Cowan et al., (eds.), NIPS 6. Morgan Kaufmann. E. Grimson, (1990) Object Recognition by Computer, Cambridge, MA: MIT Press C. Peterson and B. Soderberg. (1989) A new method for mapping optimization problems onto neural networks, Int. Journ. of Neural Sys., 1(1):3:22. M. W. Walker, L. Shoo and R. Volz. (1991) Estimating 3-D location parameters using dual number quaternions, CVGIP: Image Understanding 54(3):358-367. A. L. Yuille and J. J . Kosowsky. (1994). Statistical physics algorithms that converge. Neural Computation, 6:341-356.	1994	111
872	PCA-Pyramids for Image Compression* Horst Bischof Department for Pattern Recognition and Image Processing Technical University Vienna Treitlstraf3e 3/1832 A-1040 Vienna, Austria bis@prip.tuwien.ac.at Kurt Hornik Institut fur Statistik und Wahrscheinlichkeitstheorie Technische UniversiUit Wien Wiedner Hauptstraf3e 8-10/1071 A-1040 Vienna, Austria Kurt.Hornik@ci.tuwien.ac.at Abstract This paper presents a new method for image compression by neural networks. First, we show that we can use neural networks in a pyramidal framework, yielding the so-called PCA pyramids. Then we present an image compression method based on the PCA pyramid, which is similar to the Laplace pyramid and wavelet transform. Some experimental results with real images are reported. Finally, we present a method to combine the quantization step with the learning of the PCA pyramid. 1 Introduction In the past few years, a lot of work has been done on using neural networks for image compression, d . e.g. (Cottrell et al., 1987; Sanger, 1989; Mougeot et al., 1991; Schweizer et al., 1991)). Typically, networks which perform a Principal Component Analysis (PCA) were employed; for a recent overview of PCA networks, see (Baldi and Hornik, 1995). A well studied and thoroughly understood PCA network architecture is the linear autoassociative network, see (Baldi and Hornik, 1989; Bourlard and Kamp, 1988). This network consists of N input and output units and M < N hidden units, and is *This work was supported in part by a grant from the Austrian National Fonds zur Forderung der wissenschaftlichen Forschung (No. S7002MAT) to Horst Bischof. 942 Horst Bischof, Kurt Hornik trained (usually by back-propagation) to reproduce the input at the output units. All units are linear. Bourlard & Kamp (Bourlard and Kamp, 1988) have shown that at the minimum of the usual quadratic error function £, the hidden units project the input on the space spanned by the first M principal components of the input distribution. In fact, as long as the output units are linear, nothing is gained by using non-linear hidden units. On average, all hidden units have equal variance. However, peA is not the only method for image compression. Among many others, the Laplace Pyramid (Burt and Adelson, 1983) and wavelets (Mallat, 1989) have successfully been used to compress images. Of particular interest is the fact that these techniques provide a hierarchical representation of the image which can be used for progressive image transmission. However, these hierarchical methods are not adaptive. In this paper, we present a combination of autoassociative networks with hierarchical methods. We propose the so-called peA pyramids, which can be seen as an extension of image pyramids with a learning algorithm as well as cascaded locally connected autoassociative networks. In other words, we combine the structure of image pyramids and neural network learning algorithms, resulting in learning pyramids. The structure of this paper is as follows. We first present image pyramids and, in particular, the peA pyramid. Then, we discuss how these pyramids can be used for image compression, and present some experimental results. Next, we discuss a method to combine the quantization step of compression with the transformation. Finally, we give some conclusions and an outline of further research. 2 The peA Pyramid Before we introduce the peA pyramid, let us describe regular image pyramids. For a discussion of irregular pyramids and their relation to neural networks, see (Bischof, 1993). In the simplest case, each successive level ofthe pyramid is obtained from the previous level by a filtering operation followed by a sampling operator. More general functions can be used to achieve the desired reduction. We therefore call them reduction functions. The structure of a pyramid is determined by the neighbor relations within the levels of the pyramid and by the "father-son" relations between adjacent levels. A cell (if it is not at the base level) has a set of children (sons) at the level directly below which provide input to the cell, a set of neighbors (brothers/sisters) at the same level, and (if it is not the apex of the pyramid) a set of parents (fathers) at the level directly above. We denote the structure of a (regular) pyramid by the expression n x nlr, where n x n (the number of sons) is the size of the reduction window and r the reduction factor which describes how the number of cells decreases from level to level. 2.1 peA Pyramids Since a pyramid reduces the information content of an image level by level, an objective for the reduction function would be to preserve as much information as possible, given the restrictions imposed by the structure of the pyramid, or equivalently, to minimize the information loss by the reduction function. This naturally PCA-Pyramids for Image Compression 943 leads to the idea of representing the pyramid by a suitable peA network. Among the many alternatives for such networks, we have chosen the autoassociative networks for two reasons. First, the analysis of Hornik & Kuan (Hornik and Kuan, 1992) shows that these networks are more stable than competing models. Second, autoassociative networks have the nice feature that they automatically provide us with the expansion function (weights from the hidden layer to output layer). Since the neural network should have the same connectivity as the pyramid (i.e., the same father-son relations), its topology is determined by the structure of the pyramid. In this paper, we confine ourselves to the 4 x 4/4 pyramid for two reasons. First, the 4 x 4/4 pyramid has the nice property that every cell has the same number of fathers, which results in homogeneous networks. Second, as experiments have shown (Bischof, 1993) the results achieved with this pyramid are similar to other structures, e.g. the 5 x 5/4 pyramid, using fewer weights. E R Ii. = E(I n+l ) = E(R(ln» (a) General Setting L-..J r n (b) 4/2 pyramid (c) Corresponding network Figure 1: From the structure of the pyramid to the topology of the network Figure 1 depicts the one-dimensional situation of a 4/2 pyramid (this is the onedimensional counterpart of the two-dimensional 4 x 4/2 pyramid). Figure 1a shows the general goal to be achieved and the notations employed; Figure 1 b shows a 4/2 pyramid. When constructing the corresponding network, we start at the output layer (Le., I~). For an n/r pyramid we typically choose the size of the output layer as n. Next, we have to include all fathers of the cells in the output layer as hidden units. Finally, we have to include all sons of the hidden layer cells in the input layer. For the 4/2 pyramid, this results in an 8-3-4 network as shown in Figure 1c. A similar construction yields an 8 x 8-3 x 3-4 x 4 network for the 4 x 4/4 pyramid. The next thing to consider are the constraints on the network weights due to the overlaps in the pyramid. To completely cover the input image with output units, we can shift the network only by four cells in each direction. Therefore, the hidden units at the borders overlap. For the 4/2 pyramid, the left and right hidden units must have identical weights. In the case of the 4 x 4/4 pyramid, the network has four independent units. The thus constructed network can be trained by some suitable learning algorithm, typically of the back-propagation type, using batches of an image as input for trai944 Horst Bischof, Kurt Hornik ning the first pyramid level. After that, the second level of the pyramid can be trained in the same way using the first pyramid level as training data, and so on. 2.2 PeA-Laplace Pyramid and Image Compression Thus far, we have introduced a network which can learn the reduction function R and the expansion function E of a pyramid. Analogously to the Laplace pyramid and the wavelet transform we can now introduce the level Li of the PCA-Laplace pyramid, given by Li = Ii - I: = Ii - E(R(Ii)) It should be noted that during learning we exactly minimize the squared Laplace (a) First 2 levels of a Laplace pyramid (upper half) and peA-Laplace pyramid (lower half) (grey = 0) (9) Reconstruction error of house image with quantization of 3 bits, 4 bits, 7 bits, and reconstructed image Figure 2: Results of PCA-Laplace-Pyramid level. The original image 10 can be completely recovered from level In and the Laplace levels Lo, ... ,Ln - 1 by 10 = E(··· E(E(In) + Ln- 1 ) + Ln- 2 )···) + Lo· Since the level In is rather small (e.g., 32 x 32 pixels) and the levels of the PCALaplace pyramid are typically sparse (i.e., many pixels are zero, see Figure 2a) and can therefore be compressed considerably by a conventional compression algorithm PCA-Pyramids for Image Compression 945 (e.g. Lempel-Ziv (Ziv and Lempel, 1977)), this image representation results in a lossless image compression algorithm. In order to achieve higher compression ratios we can quantize the levels of the PCALaplace pyramid. In this case, the compression is lossy, because the original image cannot be recovered exactly. The compression ratio and the amount of loss can be controlled by the number of bits used to quantize the levels of the PCA-Laplacian. To measure the difference between the compressed and the original image, we use the normalized mean squared error (NMSE) as in (Cottrell et al., 1987; Sanger, 1989). The NMSE is given by the mean squared error divided by the average squared intensity of the image, i.e., NMSE = MSE = ((10 - C(10))2) (I~) (I~)' where 10 and C(lo) are the original and the compressed image, respectively. The compression ratio is measured by the amount of bits used to store 10 , divided by the amount of bits used to store C(1o). 2.3 Results For the results reported here we trained the networks by a conjugate gradient algorithm for 100 steps! and used a uniform quantization which is fixed for all levels of the pyramid. As was shown in (Burt and Adelson, 1983; Mayer and Kropatsch, 1989), the results could be improved by gradually increasing the quantization from bottom to top. Figure 2b shows the error images when the levels of the PCA-Laplacian pyramid are quantized with 3, 4, and 7 bits and the reconstructed image from the 7 bit Laplacian. Note that we used the same lookup-table for the error images. To compress the levels of the PCA-Laplacian pyramid, we employed the standard UNIX compress program which implements a Lempel-Ziv algorithm. iFrom these images one can see that the results with the 4 and 7 bit quantization are very good. Visually, no difference between the reconstructed and the original image can be perceived. Table 1 shows the compression ratios and the NMSEs on these images. We have performed experiments on 20 different images, the results on these images are comparable to the ones reported here. These results compare favorably with the results in the literature (see Table 1). We have also applied a 5 x 5/4 Laplace pyramid to the house image which gave a compression ratio of 3.42 with an NMSE of 0.000087 for quantization with four bits of the Laplace levels. We have also included results achieved with JPEG. One can see that our method gives considerably better results. We have also demonstrated experimentally what happens if we train a pyramid on one image and then apply this pyramid to another image without retraining. These experiments indicate that the errors are only a little bit larger for images not trained on. With five additional steps of training the errors are almost the same. iFrom lIn all our experiments the training algorithm converged (i.e. usually after 200 steps, however the improvements between steps 20 and convergence are negligible). 946 Horst Bischof, Kurt Hornik Quant. Compression ratio Bits/Pixel NMSE 3 Bit 37.628 0.212 0.0172 4 Bit 24.773 0.323 0.0019 7 Bit 8.245 0.970 0.0000215 no Quant. 3.511 2.279 0.0 Cottrell,~Cottrell et al., 1987) 8.0 1.000 0.0059 Sanger (Sanger, 1989) 22.0 0.360 0.043 5 x 5/4 Laplace 3.420 2.339 0.000087 JPEG 8.290 0.965 0.00139 JPEG 15.774 0.507 0.00348 Table 1: Compression ratios and NMSE for various compression methods this results we can conclude that we do not need to retrain the pyramid for each new image. 3 Integration of Quantization For the results reported in the previous section we have used a fixed and uniform quantization scheme which can be improved by using adaptive quantizers like the Lloyd I algorithm, Kohonen's Feature Maps, learning vector quantization, or something similar. Such an approach as taken by Schweizer (Schweizer et al., 1991) who combined a Cottrell-type network with self-organizing feature maps. However, we can go further. With the PCA network we minimize the squared Laplace level which does not necessarily yield low compression errors. What we really want to minimize are the quantized Laplace levels. Usually, the Laplace levels have an unimodally shaped histogram centered at zero. However, for the result of the compression (i.e., compression ratio and NMSE), it is irrelevant if we shift the histogram to the left or the right as long as we shift the quantization intervals in the same way. The best results could be achieved if we have a multimodal histogram with peaks centered at the quantization points. Using neural networks for both PCA and quantization, this goal could e.g. be achieved by a modular network as in Figure 3 for the 4/2 pyramid. For quantization, we could either apply a vector quantizer to a whole patch of the Laplace level, or use a scalar quantizer (as depicted in Figure 3) for each pixel of the Laplace level. In the second case, we have to constrain the weights of the quantization network to be identical for every Laplace pixel. Since scalar quanti.zation is simpler to analyze and uses less free parameters, we only consider this case. As each quantization subnetwork can be treated separately (we only have to average the weight changes over all subnetworks), the following only considers the case of one output unit of the PCA network. PCA-Pyramids for Image Compression 947 Quantization peA Figure 3: PCA network and Quantization network The error to be minimized is the squared quantization error where p refers to the patterns in the training set, Ck is the kth weight of the quantization network, and 1 is the output of the PCA-Laplace unit. Changing the weights of the quantization network by gradient descent leads to the LVQl rule of Kohonen Ac - { 2a(lp - Ck), if k = kp is the winning unit, k 0, otherwise. For the PCA network we can proceed similarly to back-propagation to obtain the rule AWij = -K 8Ep = _K 8Ep alp = _K 8Ep 8~p 8i~ = -2K(lp _ Ck) 8i~ . 8Wij alp 8Wij alp 8z~ 8Wij 8Wij Of course, this is only one out of many possible algorithms. More elaborate minimization techniques than gradient descent could be used; similarly, LVQl could be replaced by a different quantization algorithm. But the basic idea of letting the quantization step and the the compression step adapt to each other remains unchanged. 4 Conclusions In this paper, we presented a new image compression scheme based on neural networks. The PCA and PCA-Laplace pyramids were introduced, which can be seen as both an extension of image pyramids to learning pyramids and as cascaded, locally connected autoassociators. The results achieved are promising and compare favorably to work reported in the literature. A lot of work remains to be done to analyze these networks analytically. The convergence properties of the PCA pyramid are not known; we expect results similar 948 Horst Bischof, Kurt Hornik to the ones (Baldi and Hornik, 1989) for the autoassociative network. Also, for the PCA network it would be desirable to characterize the features which are extracted. Similarly, the integrated network needs to be analyzed. It is clear that for such networks, the usual error function has local minima, but maybe they can be avoided by a proper training regime (i.e. start training the PCA pyramid, then train the vector quantizer, and finally train them together). References Baldi, P. and Hornik, K. (1989). Neural Networks and principal component analysis: Learning from examples without local minima. Neural Networks, 2:53-58. Baldi, P. and Hornik, K. (1995). Learning in Linear Neural Networks: a Survey. IEEE Transactions on Neural Networks, to appear. Bischof, H. (1993). Pyramidal Neural Networks. PhD thesis, TU-Vienna, Inst. f. Automation, Dept. f. Pattern Recognition and Image Processing. Bourlard, H. and Kamp, Y. (1988). Auto-Association by Multilayer Perceptrons and Singular Value Decomposition. Biological Cybernetics, 59:291-294. Burt, P. J. and Adelson, E. H. (1983). The Laplacian pyramid as a compact image code. IEEE Transactions on Communications, Vol. COM-31(No.4):pp.532-540. Cottrell, G., Munro, P., and Zipser, D. (1987). Learning Internal Representations from Grey-Scale Images: An Example of Extensional Programming. In Ninth Annual Conference of the Cognitive Science Society, pages 462-473. Hillsdale Erlbaum. Hornik, K. and Kuan, C. (1992). Convergence analysis of local feature extraction algorithms. Neural Networks, 5(2):229-240. Mallat, S. G. (1989). A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-ll(No. 7):pp. 674-693. Mayer, H. and Kropatsch, W. G. (1989). Progressive Bildubertragung mit der 3x3/2 Pyramide. In Burkhardt, H., H6hne, K., and Neumann, B., editors, Informatik Fachberichte 219: Mustererkennung 1989, pages 160-167, Hamburg. l1.DAGM - Symposium, Springer Verlag. Mougeot, M., Azencott, R., and Angeniol, B. (1991). Image Compression with Back Propagation: Improvement of the Visual Restoration using different Cost Functions. Neural Networks, 4:467-476. Sanger, T. (1989). Optimal Unsupervised learning in a Single-Layer Linear Feedforward Neural Network. Neural Networks, 2:433-459. Schweizer, L., Parladori, G., Sicranza, G., and Marsi, S. (1991). A fully neural approach to image compression. In Kohonen, T., Makissara, K., Simula, 0., and Kangas, J., editors, Artificial Neural Networks, volume I, pages 815-820. Ziv, J. and Lempel, A. (1977). A universal algorithm for sequential data compression. IEEE Trans. on Information Theory, 23(5):337 - 343.	1994	112
873	Morphogenesis of the Lateral Geniculate Nucleus: How Singularities Affect Global Structure Svilen Tzonev Beckman Institute University of Illinois Urbana, IL 61801 svilen@ks.uiuc.edu Klaus Schulten Beckman Institute University of Illinois Urbana, IL 61801 kschulte@ks.uiuc.edu Joseph G. Malpeli Psychology Department University of Illinois Champaign, IL 61820 jmalpeli@uiuc.edu Abstract The macaque lateral geniculate nucleus (LGN) exhibits an intricate lamination pattern, which changes midway through the nucleus at a point coincident with small gaps due to the blind spot in the retina. We present a three-dimensional model of morphogenesis in which local cell interactions cause a wave of development of neuronal receptive fields to propagate through the nucleus and establish two distinct lamination patterns. We examine the interactions between the wave and the localized singularities due to the gaps, and find that the gaps induce the change in lamination pattern. We explore critical factors which determine general LGN organization. 1 INTRODUCTION Each side of the mammalian brain contains a structure called the lateral geniculate nucleus (LGN), which receives visual input from both eyes and sends projections to 134 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli the primary visual cortex. In primates the LG N consists of several distinct layers of neurons separated by intervening layers ofaxons and dendrites. Each layer of neurons maps the opposite visual hemifield in a topographic fashion. The cells comprising these layers differ in terms of their type (magnocellular and parvocellular), their input (from ipsilateral (same side) and contralateral (opposite side) eyes), and their receptive field organization (ON and OFF center polarity). Cells in one layer receive input from one eye only (Kaas et al., 1972), and in most parts of the nucleus have the same functional properties (Schiller & Malpeli, 1978). The maps are in register, i.e., representations of a point in the visual field are found in all layers, and lie in a narrow column roughly perpendicular to the layers (Figure 1). A prominent a projection column Figure 1: A slice along the plane of symmetry of the macaque LGN. Layers are numbered ventral to dorsal. Posterior is to the left, where foveal (central) parts of the retinas are mapped; peripheral visual fields are mapped anteriorly (right). Cells in different layers have different morphology and functional properties: 6-P /C/ON; 5-P /I/ON; 4-P /C/OFF; 3-P /I/OFF; 2-M/I/ON&OFF; 1-M/C/ON&OFF, where P is parvocellular, M is magnocellular, C is contralateral, I is ipsilateral, ON and OFF refer to polarities of the receptive-field centers. The gaps in layers 6, 4, and 1 are images of the blind spot in the contralateral eye. Cells in columns perpendicular to the layers receive input from the same point in the visual field. feature in this laminar organization is the presence of cell-free gaps in some layers. These gaps are representations of the blind spot (the hole in the retina where the optic nerve exits) of the opposite retina. In the LGN of the rhesus macaque monkey (Macaca mulatta) the pattern of laminar organization drastically changes at the position of the gaps foveal to the gaps there are six distinct layers, peripheral to the gaps there are four layers. The layers are extended two-dimensional structures whereas the gaps are essentially localized. However, the laminar transition occurs in a surface that extends far beyond the gaps, cutting completely across the main axis of the LGN (Malpeli & Baker., 1975). We propose a developmental model of LGN laminar morphogenesis. In particular, we investigate the role of the blind-spot gaps in the laminar pattern transition, and their extended influence over the global organization of the nucleus. In this model a wave of development caused by local cell interactions sweeps through the system (Figure 2). Strict enforcement of retinotopy maintains and propagates an initially localized foveal pattern. At the position of the gaps, the system is in a metastable Morphogenesis of the Lateral Geniculate Nucleus wavefront maturing cells ~;r ..... . . , . F ...... I •••• ~ 1 • I : I . ~/!.. ..".,. .... ~ ... .,Y I I X .. -->immature cells blind spot gap (no cells) 135 Figure 2: Top view of a single layer. As a wave of development sweeps through the LGN the foveal part matures first and the more peripheral parts develop later. The shape of the developmental wave front is shown schematically by lines of "equal development" . state, and the perturbation in retinotopy caused by the gaps is sufficient to change the state of the system to its preferred four-layered pattern. We study the critical factors in this model, and make some predictions about LGN morphogenesis. 2 MODEL OF LGN MORPHOGENESIS We will consider only the upper four (parvocellular) layers since the laminar transition does not involve the other two layers. This transition results simply from a reordering of the four parvocellular strata (Figure 1). Foveal to the gaps, the strata form four morphologically distinct layers (6, 5, 4 and 3) because adjacent strata receive inputs from opposite eyes, which "repel" one another. Peripheral to the gaps, the reordering of strata reduces the number of parvocellular eye alternations to one, resulting in two parvocellular layers (6+4 and 5+3). 2.1 GEOMETRY AND VARIABLES LGN cells Ci are labeled by indices i = 1,2, ... , N. The cells have fixed, quazi-random and uniformly distributed locations ri EVe R3, where V = {(x,y,z) 10 < x < Sz,O < y < Sy,O < z < Sz}, and belong to one projection column Cab, a = 1,2, ... ,A and b = 1,2, ... ,B, (Figure 3). Functional properties of the neurons change in time (denoted by T), and are described by eye specificity and receptive-field polarity, ei(T), and Pi(T), respectively: ei (T), pdT) E [-1,1] C R, i = 1,2, ... ,N, T = 0, 1, ... , Tmaz • The values of eye specificity and polarity represent the proportions of synapses from competing types of retinal ganglion cells (there are four type of ganglion cells from different eyes and with ON or OFF polarity). ei = -1 (ei = 1) denotes that the i-th cell is receiving input solely from the opposite (same side) retina. Similarly, Pi = -1 (Pi = 1) denotes that the cell input is pure ON (OFF) center. Intermediate values of ei and Pi imply that the cell does not have pure properties (it receives 136 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli Figure 3: Geometry of the model. LGN cells Ci (i = 1,2, ... , N) have fixed random, and uniformly-distributed locations Ti within a volume VcR , and belong to one projection column Cab. input from retinal ganglion cells of both eyes and with different polarities). Initially, at r = 0, all LGN cells are characterized by ei, Pi = O. This corresponds to two possibilities: no retinal ganglion cells synapse on any LGN cell, or proportions of synapses from different ganglion cells on all LGN neurons are equal, i.e., neurons possess completely undetermined functionality because of competing inputs of equal strength. As the neurons mature and acquire functional properties, their eye specificity and polarity reach their asymptotic values, ±l. Even when cells are not completely mature, we will refer to them as being of four different types, depending on the signs of their functional properties. Following accepted anatomical notation, we will label them as 6, 5, 4, and 3. We denote eye specificity of cell types 6 and 4 as negative, and cell types 5 and 3 as positive. Polarity of cell types 6 and 5 is negative, while polarity of types 4 and 3 is positive. Cell functional properties are subject to the dynamics described in the following section. The process of LGN development starts from its foveal part, since in the retina it is the fovea that matures first. As more peripheral parts of the retina mature, their ganglion cells start to compete to establish permanent synapses on LGN cells. In this sorting process, each LGN cell gradually emerges with permanent synapses that connect only to several neighboring ganglions of the same type. A wave of gradual development of functionality sweeps through the nucleus. The driving force for this maturation process is described by localized cell interactions modulated by external influences. The particular pattern of the foveal lamination is shaped by external forces, and later serves as a starting point for a "propagation of sameness" of cell properties. Such a sameness propagation produces clustering of similar cells and formation of layers. It should be stressed that cells do not move, only their characteristics change. 2.2 DYNAMICS The variables describing cell functional properties are subject to the following dynamics edr + 1) Pi (r + 1) ei (r) + ~ei (r) + 1Je Pi (r) + ~Pi (r) + 1Jp, 1,2, ... ,N. (1) Morphogenesis of the Lateral Geniculate Nucleus 137 In Eq. (1), there are two contributions to the change of the intermediate variables ei ( T) and Pi ( T). The first is deterministic, given by 6.e;( r) = ,,(r,) [ (t. e; (r)! (Ie; - r;ll) + E", (r,)] (1- c; (r)) f3;. 6.p,(r) ,,(r,) [(t,p;(r)!(lr,_r;ll) +P", (r,)] (I-p;(r)) f3; •. (2) The second is a stochastic contribution corresponding to fluctuations in the growth of the synapses between retinal ganglion cells and LGN neurons. This noise in synaptic growth plays both a driving and a stabilizing role to be explained below. We explain the meaning of the variables in Eq. (2) only for the eye specificity variable ei. The corresponding parameters for polarity Pi have similar interpretations. The parameter a (ri) is the rate of cell development. This rate is the same for eye specificity and polarity. It depends on the position ri of the cell in order to allow for spatially non-uniform development. The functional form of a (ri) is given in the Appendix. The term Eint (ri) = 2:7=1 ejl (h - rjl) is effectively a cell force field. This field influences the development of nearby cells and promotes clustering of same type of cells. It depends on the maturity of the generating cells and on the distance between cells through the interaction function 1(8). We chose for 1(8) a Gaussian form, i.e., 1 (8) = exp ( _62 / (T2), with characteristic interaction distance (T. The external influences on cell development are incorporated in the term for the external field Eext(ri). This external field plays two roles: it launches a particular laminar configuration of the system (in the foveal part of the LGN), and determines its peripheral pattern. It has, thus, two contributions Eext (ri) = E!xt (ri) + E~xt (ri). The exact forms of E!xt(ri) and E~xt(ri) are provided in the Appendix. The nonlinear term (1 e~) in Eq. (2) ensures that ±1 are the only stable fixed points of the dynamics. The neuronal properties gradually converge to either of these fixed points capturing the maturation process. This term also stabilizes the dynamic variables and prevents them from diverging. The last term f3~b ( T) reflects the strict columnar organization of the maps. At each step of the development the proportion of all four types of LGN cells is calculated within a single column Cab, and f3~b(T) for different types t is adjusted such that all types are equally represented. Without this term, the cell organization degenerates to a non-laminar pattern (the system tries to minimize the surfaces between cell clusters of different type). The exact form of f3!b(T) is given in the Appendix. At each stage of LGN development, cells receive input from retinal ganglion cells of particular types. This means that eye specificity and polarity of LGN cells are not independent variables. In fact, they are tightly coupled in the sense that lei ( T) I = IPi ( T) I should hold for all cells at all times. This gives rise to coupled dynamics described by 138 mi(7+1) ei (7 + 1) Pi (7 + 1) Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli min ( I ei (7 + 1) I, IPi (7 + 1) I ) = mi (7 + 1) sgn ( ei (7 + 1) ) md7+ 1) sgn(Pi(7+ 1)), i = 1,2, ... ,N. (3) The blind spot gaps are modeled by not allowing cells in certain columns to acquire types of functionality for which retinal projections do not exist, e.g., from the blind spot of the opposite eye. Accordingly, ei is not allowed to become negative. Thus, some cells never reach a pure state ei, Pi = ± 1. It is assumed that in reality such cells die out. Of all quantities and parameters, only variables describing the neuronal receptive fields (ei and Pi) are time-dependent. 3 RESULTS We simulated the dynamics described by Eqs. (1, 2, 3), typically for 100,000 time steps. Depending on the rate of cell development, mature states were reached in about 10,000 steps. The maximum value of Q: was 0.0001. We used an interaction function with u = 1. First, we considered a two-dimensional LGN, V = {(x,z)IO < x < Sx,O < z < Sz} with Sx = 10 and Sz = 6. There were ten projection columns (with equal size) along the x axis. An initial pattern was started in the foveal part by the external field. The size of the gaps 9 measured in terms of the interaction distance u was crucial for pattern development. When the developmental wave reached the gaps, layer 6 could" jump" its gap and continued to spread peripherally if the gap was sufficiently narrow (g/u < 1.5). If its gap was not too narrow (g/u > 0.5), layer 4 completely stopped (since cells in the gaps were not allowed to acquire negative eye specificity) and so layers 5 and 3 were able to merge. Cells of type 4 reappeared after the gaps (Figure 4, right side, shows behavior similar to the two-dimensional model) because of the required equal representation of all cell types in the projection columns, and because of noise in cell development. Energetically, the most favorable position of cell type 4 would be on top of type 6, which is inconsistent with experimental observations. Therefore, one must assume the existence of an external field in the peripheral part that will drive the system away from its otherwise preferred state. If the gaps were too large (g/u > 1.5), cells of type 6 and 4 reappeared after the gaps in a more or less random vertical position and caused transitions of irregular nature. On the other hand, if the gaps were too narrow (g/u < 0.5), both layers 6 and 4 could continue to grow past their gaps, and no transition between laminar patterns occurred at all. When g/u was close to the above limits, the pattern after the gaps differed from trial to trial. For the two-dimensional system, a realistic peripheral pattern always occurred for 0.7 < g/u < 1.2. We simulated a three-dimensional system with size Sx = 10, Sy = 10, and Sz = 6, and projection columns ordered in a 10 by 10 grid. The topology of the system is different in two and three dimensions: in two dimensions the gaps interrupt the layers completely and, thus, induce perturbations which cannot be by-passed. In three dimensions the gaps are just holes in a plane and generate localized perturbations: the layers can, in principle, grow around the gaps maintaining the initial laminar pattern. Nevertheless, in the three-dimensional case, an extended transiMorphogenesis of the Lateral Geniculate Nue/ells 6 5 4~~UIIii 3 139 Figure 4: Left: Mature state of the macaque LG~ result of the three-dimensional model with 4,800 cells. Spheres with different shades represent cells with different properties. Gaps in strata 6 and 4 (this gap is not visible) are coded by the darkest color, and coincide with the transition surface between 4- and 2-layered patterns. Right: A cut of the three-dimensional structure along its plane of symmetry. A two-dimensional system exhibits similar organization. Compare with upper layers in Figure 1. Spatial segregation between layers is not modeled explicitly. tion was triggered by the gaps. The transition surface, which passed through the gaps and was oriented roughly perpendicularly to the x axis, cut completely across the nucleus (Figure 4). Several factors were critical for the general behavior of the system. As in two dimensions, the size of the gap~ must be within certain limits: typically 0.5 < g / (J" < 1.0. These limits depend on the curvature of the wavefront. The gaps must lie in a certain "inducing" interval along the x axis. If they were too close to the origin, the foveal pattern was still more stable, so no transition could be induced there. However, a spontaneous transition might occur downstream. If the gaps were too far from the origin, a ~pontaneou~ transition might occur before them. The occurrence and location of a spontaneous transition, (therefore, the limits of the "inducing" interval) depended on the external-field parameters. A realistic transition was observed only when the front of the developmental wave had sufficient curvature when it reached the gaps. Underlying anatomical reasons for a sufficiently curved front along the main axis could be the curvature of the nucleus, differences in layer thickness, or differences in ganglion-cell densities in the retinas. Propagation of the developmental wave away from the gaps was quite stable. Before and after the gaps, the wave simply propagated the already established patterns. In a system without gaps, transitions of variable shape and location occurred when the peripheral contribution to the external fields was sufficiently large; a weaker contribution allowed the foveal pattern to propagate through the entire nucleus. 4 SUMMARY \Ve present a model that successfully captures the most important features of macaque LG~ morphogenesis. It produces realistic laminar patterns and supports 140 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli the hypothesis (Lee & Malpeli, 1994) that the blind spot gaps trigger the transition between patterns. It predicts that critical factors in LGN development are the size and location of the gaps, cell interaction distances, and shape of the front of the developmental wave. The model may be general enough to incorporate the LGN organizations of other primates. Small singularities, similar to the blind spot gaps, may have an extended influence on global organization of other biological systems. Acknowledgements This work has been supported by a Beckman Institute Research Assistantship, and by grants PHS 2P41 RR05969 and NIH EY02695. References J.H. Kaas, R.W. Guillery & J.M. Allman. (1972) Some principles of organization in the dorsal lateral geniculate nucleus, Brain Behav. Evol., 6: 253-299. D.Lee & J.G.Malpeli. (1994) Global Form and Singularity: Modeling the Blind Spot's Role in Lateral Geniculate Morphogenesis, Science, 263: 1292-1294. J.G. Malpeli & F.H. Baker. (1975) The representation of the visual field in the lateral geniculate nucleus of Macaca mulatta, 1. Comp. Neural., 161: 569-594. P.H. Schiller & J.G. Malpeli. (1978) Functional specificity ofLGN ofrhesus monkey, 1. Neurophysiol., 41: 788-797. APPENDIX The form of 0 (x, y, z) (with 00 = 0.0001) was chosen as o(x,y,z) = 00(0.1+exp(-(y-Sy/2)2)). (4) Foveal external fields of the following form were used: E!:z:t(x,y,z) = lO[O(z-d)-20(z-2d)+20(z-3d)-O(d-z)] exp(-x) p!:z:t(x,y,z) = 10[20(z-2d)-1]exp(-x), (5) where d = Sz/4 is the layers' thickness and the "theta" function is defined as O( x) = 1, x > 0 and O( x) = 0, x < O. Peripheral external fields (in fact they are present everywhere but determine the pattern in the peripheral part only) were chosen as E::z:t(x,y,z) = 5[20(z-2d)-1] P::z:t(x,y,z) = 5 [O(z-d)-20(z-2d)+20(z-3d)-O(d-z)]. (6) f3!b ( 1') was calculated in the following way: at any given time 1', within the column Cab, we counted the number N~b( 1') of cells, that could be classified as one of the four types t = 3,4,5,6. Cells with ei( 1') or Pi( 1') exactly zero were not counted. The total number of classified cells is then Nab( 1') = 2:~=3 N~b( 1'). If there were no classified cells (Nab( 1') = 0), then f3~b( 1') was set to one for all t. Otherwise the ratio of different types was calculated: n~b = N~b(1')/Nab(1'). In this way we calculated f3~b (1') = 4 - 12 nab, t = 3,4,5,6. (7) If f3~b ( 1') was negative it was replaced by zero.	1994	113
874	Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping in the Barn Owl Alexandre Pougett alex@salk.edu Cedric Deffayett Terrence J. Sejnowskit cedric@salk.edu terry@salk.edu tHoward Hughes Medical Institute The Salk Institute La Jolla, CA 92037 Department of Biology University of California, San Diego and tEcole Normale Superieure 45 rue d'Ulm 75005 Paris, France Abstract The auditory system of the barn owl contains several spatial maps. In young barn owls raised with optical prisms over their eyes, these auditory maps are shifted to stay in register with the visual map, suggesting that the visual input imposes a frame of reference on the auditory maps. However, the optic tectum, the first site of convergence of visual with auditory information, is not the site of plasticity for the shift of the auditory maps; the plasticity occurs instead in the inferior colliculus, which contains an auditory map and projects into the optic tectum. We explored a model of the owl remapping in which a global reinforcement signal whose delivery is controlled by visual foveation. A hebb learning rule gated by reinforcement learned to appropriately adjust auditory maps. In addition, reinforcement learning preferentially adjusted the weights in the inferior colliculus, as in the owl brain, even though the weights were allowed to change throughout the auditory system. This observation raises the possibility that the site of learning does not have to be genetically specified, but could be determined by how the learning procedure interacts with the network architecture. 126 Alexandre Pouget, Cedric Deffayet, Te"ence J. Sejnowski c:::======:::::» • Visual System Optic Tectum c,an ~_m .t Inferior Colllc-ulus External nucleua Forebrain Field L U~ a~ Inferior Colltculu. Cenlnll Nucleus C1ec) t Cochlea t Ovold.H. Nucleull ·"bala:m.ic Relay Figure 1: Schematic view of the auditory pathways in the barn owl. 1 Introduction The barn owl relies primarily on sounds to localize prey [6] with an accuracy vastly superior to that of humans. Figure 1A illustrates some of the nuclei involved in processing auditory signals. The barn owl determines the location of sound sources by comparing the time and amplitude differences of the sound wave between the two ears. These two cues are combined together for the first time in the shell and core of the inferior colliculus (ICc) which is shown at the bottom of the diagram. Cells in the ICc are frequency tuned and subject to spatial aliasing. This prevents them from unambiguously encoding the position of objects. The first unambiguous auditory map is found at the next stage, in the external capsule of the inferior colliculus (ICx) which itself projects to the optic tectum (OT). The OT is the first subforebrain structure which contains a multimodal spatial map in which cells typically have spatially congruent visual and auditory receptive fields. In addition, these subforebrain auditory pathways send one major collateral toward the forebrain via a thalamic relay. These collaterals originate in the ICc and are thought to convey the spatial location of objects to the forebrain [3]. Within the forebrain, two major structures have been involved in auditory processing: the archistriatum and field L. The archistriatum sends a projection to both the inferior colliculus and the optic tectum. Knudsen and Knudsen (1985) have shown that these auditory maps can adapt to systematic changes in the sensory input. Furthermore, the adaptation appears to be under the control of visual input, which imposes a frame of reference on the incoming auditory signals. In owls raised with optical prisms, which introduce a systematic shift in part of the visual field, the visual map in the optic tectum was identical to that found in control animals, but the auditory map in the ICx was shifted by the amount of visual shift introduced by the prisms. This plasticity ensures that the visual and auditory maps stay in spatial register during growth Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 127 and other perturbations to sensory mismatch. Since vision instructs audition, one might expect the auditory map to shift in the optic tectum, the first site of visual-auditory convergence. Surprisingly, Brainard and Knudsen (1993b) observed that the synaptic changes took place between the ICc and the ICx, one synapse before the site of convergence. These observations raise two important questions: First, how does the animal knows how to adapt the weights in the ICx in the absence of a visual teaching signal? Second, why does the change take place at this particular location and not in the aT where a teaching signal would be readily available? In a previous model [7], this shift was simulated using backpropagation to broadcast the error back through the layers and by constraining the weights changes to the projection from the ICc to ICx. There is, however, no evidence for a feedback projection between from the aT to the ICx that could transmit the error signal; nor is there evidence to exclude plasticity at other synapses in these pathways. In this paper, we suggest an alternative approach in which vision guides the remapping of auditory maps by controlling the delivery of a scalar reinforcement signal. This learning proceeds by generating random actions and increasing the probability of actions that are consistently reinforced [1, 5] . In addition, we show that reinforcement learning correctly predicts the site of learning in the barn owl, namely at the ICx-ICc synapse, whereas backpropagation [8] does not favor this location when plasticity is allowed at every synapse. This raises a general issue: the site of synaptic adjustment might be imposed by the combination of the architecture and learning rule, without having to restrict plasticity to a particular synapse. 2 Methods 2.1 Network Architecture The network architecture of the model based on the barn owl auditory system, shown in figure 2A, contains two parallel pathways. The input layer was an 8x21 map corresponding to the ICc in which units responded to frequency and interaural phase differences. These responses were pooled together to create auditory spatial maps at subsequent stages in both pathways. The rest of the network contained a series of similar auditory maps, which were connected topographically by receptive fields 13 units wide. We did not distinguish between field L and the archistriatum in the forebrain pathways and simply used two auditory maps, both called FBr. We used multiplicative (sigma-pi) units in the aT whose activities were determined according to: Yi = L,. w~Br yfBr WfkBr yfc:c (1) j The multiplicative interaction between ICx and FBr activities was an important assumption of our model. It forced the ICx and FBr to agree on a particular position before the aT was activated. As a result, if the ICx-aT synapses were modified during learning, the ICx-FBr synapses had to be changed accordingly. 128 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski Figure 2: Schematic diagram of weights (white blocks) in the barn owl auditory system. A) Diagram of the initial weights in the network. B) Pattern of weights after training with reinforcement learning on a prism-induced shift offour units. The remapping took place within the ICx and FBr. C) Pattern of weights after training with backpropagation. This time the ICx-OT and FBr-OT weights changed. Weights were clipped between 5.0 and 0.01, except for the FBr-ICx connections whose values were allowed to vary between 8.0 and 0.01. The minimum values were set to 0.01 instead of zero to prevent getting trapped in unstable local minima which are often associated with weights values of zero. The strong coupling between FBr and ICx was another important assumption of the model whose consequence will be discussed in the last section. Examples were generated by simply activating one unit in the ICc while keeping the others to zero, thereby simulating the pattern of activity that would be triggered by a single localized auditory stimulus. In all simulations, we modeled a prism-induced shift of four units. 2.2 Reinforcement learning We used stochastic units and trained the network using reinforcement learning [1]. The weighted sum of the inputs, neti, passed through a sigmoid, f(x) , is interpreted as the probability, Pi, that the unit will be active: Pi = f(neti) * 0.99 + 0.01 (2) were the output of the unit Yi was: . _ {a with probability 1 - Pi y, 1 with probability Pi (3) Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 129 Because of the form of the equation for Pi, all units in the network had a small probability (0.01) of being spontaneously active in the absence of any inputs. This is what allowed the network to perform a stochastic search in action space to find which actions were consistently associated with positive reinforcement. We ensured that at most one unit was active per trial by using a winner-take-all competition in each layer. Adjustable weights in the network were updated after each training examples with hebb-like rule gated by reinforcement: (4) A trial consisted in choosing a random target location for auditory input (ICc) and the output of the OT was used to generate a head movement. The reinforcement, r , was then set to 1 for head movements resulting in the foveation of the stimulus and to -0.05 otherwise. 2.3 Backpropagation For the backpropagation network, we used deterministic units with sigmoid activation functions in which the output of a unit was given by: (5) where neti is the weighted sum of the inputs as before. The chain rule was used to compute the partial derivatives of the squared error, E , with respect to each weights and the weights were updated after each training example according to: (6) The target vectors were similar to the input vectors, namely only one OT units was required to be activated for a given pattern, but at a position displaced by 4 units compared to the input. 3 Results 3.1 Learning site with reinforcement In a first set of simulation we kept the ICc-ICx and ICc-FBr weights fixed. Plasticity was allowed at these site in later simulations. Figure 2A shows the initial set of weights before learning starts. The central diagonal lines in the weight diagrams illustrate the fact that each unit receives only one non-zero weight from the unit in the layer below at the same location. 130 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski There are two solutions to the remapping: either the weights change within the ICx and FBr, or from the ICx and the FBr to the ~T. As shown in figure 2B, reinforcement learning converged to the first solution. In contrast, the weights between the other layers were unaltered, even though they were allowed to change. To prove that the network could have actually learned the second solution, we trained a network in which the ICc-ICx weights were kept fixed. As we expected, the network shifted its maps simultaneously in both sets of weights converging onto the OT, and the resulting weights were similar to the ones illustrated in figure 2C. However, to reach this solution, three times as many training examples were needed. The reason why learning in the ICx and FBr were favored can be attributed to probabilistic nature of reinforcement learning. If the probability of finding one solution is p, the probability of finding it twice independently is p2. Learning in the ICx and FBR is not independent because of the strong connection from the FBr to the ICx. When the remapping is learned in the FBR this connection automatically remapped the activities in the ICx which in turn allows the ICx-ICx weights to remap appropriately. In the OT on the other hand, the multiplicative connection between the ICx and FBr weights prevent a cooperation between this two sets of weights. Consequently, they have to change independently, a process which took much more training. 3.2 Learning at the ICc-ICx and ICc-FBr synapses The aliasing and sharp frequency tuning in the response of ICc neurons greatly slows down learning at the ICc-ICx and ICc-FBr synapses. We found that when these synapses were free to change, the remapping still took place within the ICx or FBr (figure 3). 3.3 Learning site with backpropagation In contrast to reinforcement learning, backpropagation adjusted the weights in two locations: between the ICx and the OT and between the Fbr and OT (figure 2C). This is the consequence of the tendency of the backpropagation algorithm to first change the weights closest to where the error is injected. 3.4 Temporal evolution of weights Whether we used reinforcement or supervised learning, the map shifted in a very similar way. There was a simultaneous decrease of the original set of weights with a simultaneous increase of the new weights, such that both sets of weights coexisted half way through learning. This indicates that the map shifted directly from the original setting to the new configuration without going through intermediate shifts. This temporal evolution of the weights is consistent the findings of Brainard and Knudsen (1993a) who found that during the intermediate phase of the remapping, cells in the inferior colli cuI us typically have two receptive fields. More recent work however indicates that for some cells the remapping is more continuous(Brainard and Knudsen, personal communication), a behavior that was not reproduced by either of the learning rule. Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 131 Figure 3: Even when the ICc-ICx weights are free to change, the network update the weights in the ICx first. A separate weight matrix is shown for each isofrequency map from the ICc to ICx. The final weight matrices were predominantly diagonal; in contrast, the weight matrix in ICx was shifted. 4 Discussion Our simulations suggest a biologically plausible mechanism by which vision can guide the remapping of auditory spatial maps in the owl's brain. Unlike previous approaches, which relied on visual signals as an explicit teacher in the optic tectum [7], our model uses a global reinforcement signal whose delivery is controlled by the foveal representation of the visual system. Other global reinforcement signals would work as well. For example, a part of the forebrain might compare auditory and visual patterns and report spatial mismatch between the two. This signal could be easily incorporated in our network and would also remap the auditory map in the inferior colli cuI us. Our model demonstrates that the site of synaptic plasticity can be constrained by the interaction between reinforcement learning and the network architecture. Reinforcement learning converged to the most probably solution through stochastic search. In the network, the strong lateral coupling between ICx and FBr and the multiplicative interaction in the OT favored a solution in which the remapping took place simultaneously in the ICx and FBr. A similar mechanism may be at work in the barn owl's brain. Colaterals from FBr to ICx are known to exist, but the multiplicative interaction has not been reported in the barn owl optic tectum. Learning mechanisms may also limit synaptic plasticity. NMDA receptors have been reported in the ICx, but they might not be expressed at other synapses. There may, however, be other mechanisms for plasticity. The site of remapping in our model was somewhat different from the existing observations. We found that the change took place within the ICx whereas Brainard and Knudsen [3] report that it is between the ICc and the ICx. A close examination of their data (figure 11 in [3]) reveals that cells at the bottom of ICx were not 132 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski remapped, as predicted by our model, but at the same time, there is little anatomical or physiological evidence for a functional and hierarchical organization within the ICx. Additional recordings are need to resolve this issue. We conclude that for the barn owl's brain, as well as for our model, synaptic plasticity within ICx was favored over changes between ICc and ICx. This supports the hypothesis that reinforcement learning is used for remapping in the barn owl auditory system. Acknowledgments We thank Eric Knudsen and Michael Brainard for helpful discussions on plasticity in the barn owl auditory system and the results of unpublished experiments. Peter Dayan and P. Read Montague helped with useful insights on the biological basis of reinforcement learning in the early stages of this project. References [1] A.G. Barto and M.1. Jordan. Gradient following without backpropagation in layered networks. Proc. IEEE Int. Conf. Neural Networks, 2:629-636, 1987. [2] M.S. Brainard and E.1. Knudsen. Dynamics of the visual calibration of the map of interaural time difference in the barn owl's optic tectum. In Society For Neuroscience Abstracts, volume 19, page 369.8, 1993. [3] M.S. Brainard and E.!. Knudsen. Experience-dependent plasticity in the inferior colliculus: a site for visual calibration of the neural representation of auditory space in the barn owl. The journal of Neuroscience, 13:4589-4608, 1993. [4] E. Knudsen and P. Knudsen. Vision guides the adjustment of auditory localization in the young barn owls. Science, 230:545-548, 1985. [5] P.R. Montague, P. Dayan, S.J. Nowlan, A. Pouget, and T.J. Sejnowski. U sing aperiodic reinforcement for directed self-organization during development. In S.J. Hanson, J.D. Cowan, and C.L. Giles, editors, Advances in Neural Information Processing Systems, volume 5. Morgan-Kaufmann, San Mateo, CA, 1993. [6] R.S. Payne. Acoustic location of prey by barn owls (tyto alba). Journal of Experimental Biology, 54:535-573, 1970. [7] D.J. Rosen, D.E. Rumelhart, and E.I. Knudsen. A connectionist model of the owl's sound localization system. In Advances in Neural Information Processing Systems, volume 6. Morgan-Kaufmann, San Mateo, CA, 1994. [8] D.E. Rumelhart, G.E. Hinton, and R.J . Williams. Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing, volume 1, chapter 8, pages 318-362. MIT Press, Cambridge, MA, 1986.	1994	114
875	A Lagrangian Formulation For Optical Backpropagation Training In Kerr-Type Optical Networks James E. Steck Mechanical Engineering Wichita State University Wichita, KS 67260-0035 Alvaro A. Cruz-Cabrara Electrical Engineering Wichita State University Wichita, KS 67260-0044 Abstract Steven R. Skinner Electrical Engineering Wichita State University Wichita, KS 67260-0044 Elizabeth C. Behrman Physics Department Wichita State University Wichita, KS 67260-0032 A training method based on a form of continuous spatially distributed optical error back-propagation is presented for an all optical network composed of nondiscrete neurons and weighted interconnections. The all optical network is feed-forward and is composed of thin layers of a Kerrtype self focusing/defocusing nonlinear optical material. The training method is derived from a Lagrangian formulation of the constrained minimization of the network error at the output. This leads to a formulation that describes training as a calculation of the distributed error of the optical signal at the output which is then reflected back through the device to assign a spatially distributed error to the internal layers. This error is then used to modify the internal weighting values. Results from several computer simulations of the training are presented, and a simple optical table demonstration of the network is discussed. 772 Elizabeth C. Behrman 1 KERR TYPE MATERIALS Kerr-type optical networks utilize thin layers of Kerr-type nonlinear materials, in which the index of refraction can vary within the material and depends on the amount of light striking the material at a given location. The material index of refraction can be described by: n(x)=no+nzI(x), where 110 is the linear index of refraction, ~ is the nonlinear coefficient, and I(x) is the irradiance of a applied optical field as a function of position x across the material layer (Armstrong, 1962). This means that a beam of light (a signal beam carrying information perhaps) passing through a layer of Kerr-type material can be steered or controlled by another beam of light which applies a spatially varying pattern of intensity onto the Kerr-type material. Steering of light with a glass lens (having constance index of refraction) is done by varying the thickness of the lens (the amount of material present) as a function of position. Thus the Kerr effect can be loosely thought of as a glass lens whose geometry and therefore focusing ability could be dynamically controlled as a function of position across the lens. Steering in the Kerr material is accomplished by a gradient or change in the material index of refraction which is created by a gradient in applied light intensity. This is illustrated by the simple experiment in Figure 1 where a small weak probe beam is steered away from a straight path by the intensity gradient of a more powerful pump beam. lex) Pump I~ > /-..... x Figure 1: Light Steering In Kerr Materials 2 OPTICAL NETWORKS USING KERR MATERIALS The Kerr optical network, shown in Figure 2, is made up of thin layers of the Kerr- type nonlinear medium separated by thick layers of a linear medium (free space) (Skinner, 1995). The signal beam to be processed propagates optically in a direction z perpendicular to the layers, from an input layer through several alternating linear and nonlinear layers to an output layer. The Kerr material layers perform the nonlinear processing and the linear layers serve as connection layers. The input (l(x)) and the weights (W\(x),W2(x) ... Wn(x)) are irradiance fields applied to the Kerr type layers, as functions of lateral position x, thus varying the A Lagrangian Formulation for Optical Backpropasation 773 refractive index profile of the nonlinear medium. Basically, the applied weight irradiences steer the signal beam via the Kerr effect discussed above to produce the correct output. The advantage of this type of optical network is that both neuron processing and weighted connections are achieved by uniform layers of the Kerr material. The all optical nature eliminates the need to physically construct neurons and connections on an individual basis. Plane Wave (Eo) Figure 2: Kerr Optical Neural Network Architecture O(x,y) • If E;(ex) is the light entering the itlt nonlinear layer at lateral position ex, then the effect of the nonlinear layer is given by (1) where W;( ex) is the applied weight field. Transmission of light at lateral location ex at the beginning of the itlt linear layer to location p just before the i+ 1 tit nonlinear layer is given by where ko c =-I 2!:lL I 3 OPTICAL BACK-PROPAGATION TRAINING (2) Traditional feed-forward artificial neural networks composed of a finite number of discrete neurons and weighted connections can be trained by many techniques. Some of the most successful techniques are based upon the well known training method called backpropagation which results from minimizing the network output error, with respect to the network weights by a gradient descent algorithm. The optical network is trained using a form of continuous optical back-propagation which is developed for a nondiscrete network. Gradient descent is applied to minimize the error over the entire output region of the optical network. This error is a continuous distribution of error calculated over the output region. 774 Elizabeth C. Behrman Optical back-propagation is a specific technique by which this error distribution is optically propagated backward through the linear and nonlinear optical layers to produce error signals by which the light applied to the nonlinear layers is modified. Recall that this applied light Wi controls what serves as connection "weights" in the optical network. Optical backpropagation minimizes the error Lo over an output region 0 0 > a subdomain of the fmal or nth layer of the network, where ~ = 'Y r O(u'fJ '(uliu )co (3) subject to the constraint that the propagated light, Ei( ex), satisfies the equations of forward propagation (1) and (2). O(P) = En+I(P) and is the network output, y is a scaling factor on the output intensity. Lo then is the squared error between the desired output value D and the average intensity 10 of the output distribution O( P). This constrained minimization problem is posed in a Lagrange formulation similar to the work of (Ie Cunn, 1988) for conventional feedforward networks and (pineda, 1987) for conventional recurrent networks; the difference being that for the optical network of this paper the Electric field E and the Lagrange multiplier are complex and also continuous in the spatial variable thus requiring the Lagrangian below. A Lagrangian is defmed as; L = 4, + :t fA; tu ) [ EI+I(U) - fFI~)~ Ie -jctP ... )z dP ] ax 0. 0. + it. JA/+~U) [Ei+~U) fF~~)~/e-jC~13-«)Z dP r ax 0. 0. (4) Taking the variation ofL with respect to Ei, the Lagrange multipliers Ai, and using gradient descent to minimize L with respect to the applied weight fields Wi gives a set of equations that amount to calculating the error at the output and propagating the error optically backwards through the network. The pertinent results are given below. The distributed assigrunent of error on the output field is calculated by A (R.) = ~ 0 '(R.) [ D - 10 ] 1f+1 ... ... 'Y (5) This error is then propagated back through the nth or final linear optical layer by the equation ·c z 6 (~) = ~ r A + (u) e -jC,/..13-u) dx " 1t ) Co " 1 (6) which is used to update the "weight" light applied to the nth nonlinear layer. Optical backpropagation, through the ith nonlinear layer (giving AlP» followed by the linear layer (giving ~i-I(P» is performed according to the equations A Lagrangian Formulation for Optical Backpropagation 775 (7) This gives the error signal ~j'I(P) used to update the "weight" light distribution Wj.I(P) applied to the i-I nonlinear layer. The "weights" are updated based upon these errors according to the gradient descent rule Wi-(~) = w/,/d(P) +l'lt~)ktPNLin2W/"t~) 2 IM[ ~(~) 6,(~) e-~ANL.nZ<lw ,ClId(p)f.IE.(I\)I'>] (8) where ,,;CP) is a learning rate which can be, but usually is not a function of layer number i and spatial position p. Figure 3 shows the optical network (thick linear layers and thin nonlinear layers) with the unifonn plane wave Eo, the input signal distribution I, forward propagation signals EI E2 ... En' the weighting light distributions at the nonlinear layers WI W2 .. , Wn. Also shown are the error signal An+1 at the output and the back-propagated error signals ~n ... ~2 ~I for updating the nonlinear layers. Common nonlinear materials exist for which the material constants are such that the second term in the first of Equations 7 becomes small. Ignoring this second term gives an approximate fonn of optical backpropagation which amounts to calculating the error at the output of the network and then reversing its direction to optically propagate this error backward through the device. This can be easily seen by comparing Equations 6 and 7 (with the second tenn dropped) for optical back-propagation of the output error An with Equations I and 2 for the forward propagation of the signal Ej. This means that the optical back-propagation training calculations potentially can be implemented in the same physical device as the forward network calculations. Equation (8) then becomes; Wi-(~) = Wio/d(~) + (2t'l,(~'>kot:.NL"2) w//d(~) [ (Et~) At~» - ~t~) ~(~)r] (9) which may be able to be implemented optically. 4 SIMULATION RESULTS To prove feasibility, the network was then trained and tested on several benchmark classification problems, two of which are discussed here. More details on these and other simulations of the optical network can be found in (Skinner, 1995). In the first (Using Nworks, 1991), iris species were classified into one of three categories: Setosa, Versicolor or Virginica. Classification was based upon length and width of the sepals and 776 Elizabeth C. Behrman petals. The network consisted of an input self-defocusing layer with an applied irradiance field which was divided into 4 separate Gaussian distributed input regions 25 microns in width followed by a linear layer. This pattern is repeated for 4 more groups composed of a nonlinear layer (with applied weights) followed by a linear layer. The final linear layer has three separate output regions 10 microns wide for binary classification as to species. The nonlinear layers were all 20 microns thick with n2=-.05 and the linear layers were 100 microns thick. The wavelength of applied light was 1 micron and the width of the network was 512 microns discretized into 512 pixels. This network was trained on a set of 50 training pairs to produce correct classification of all 50 training pairs. The network was then used to classify 50 additional pairs of test data which were not used in the training phase. The network classified 46 of these correctly for a 92 % accuracy level which is comparable to a standard feedforward network with discrete sigmoidal neurons. In the second problem, we tested the performance of the network on a set of data from a dolphin sonar discrimination experiment (Roitblat, 1991). In this study a dolphin was presented with one of three different types of objects (a tube, a sphere, and a cone), allowed to echolocate, and rewarded for choosing the correct one from a comparison array. The Fourier transforms of his click echoes, in the form of average amplitudes in each of 30 frequency bins, were then used as inputs for a neural network. Nine nonlinear layers were used along with 30 input regions and 3 output region \.. -\ output plane \1l'Y). • • O(x,y) t t t t t t t t t t t T ALn °n(x,y) • • • • • • • • • ttl Wn(x,y) I .ili'Ln En(x,y) t t t t. t t t. t t t t • • 01 (X,y) • • • ••• • t·. t t t W 1 (x,y) I .ili'L1 El(X,y) t t t t t t t t t t t T ALo ...----__ ----,1 ~ ____________ I(_X_,y_) ____________ ~1 .ili'Lo Eo t t t t t t t t t t t Plane Wave Figure 3: Optical Network Forward Data and Backward Error DataFlow A Lagrangian Formulatio1l for Optical Backpropagation 777 output regions, the remainder of the network physical parameters were the same as above for the iris classification. Half the data (13 sets of clicks) was used to train the network, with the other half of the data (14 sets) used to test the training. After training, classification of the test data set was 100% correct. 5 EXPERIMENTAL RESULTS As a proof of the concept, the optical neural network was constructed in the laboratory to be trained to perform various logic functions. Two thermal self-defocusing layers were used, one for the input and the other for a single layer of weighting. The nonlinear coefficient of the index of refraction (nJ was measured to be -3xlO'" cm21W. The nonlinear layers had a thickness (~NLo and ~NL.) of 630llm and were separated by a distance (~Lo) of 15cm. The output region was 100llm wide and placed 15cm (~L.) behind the weighting layer. The experiment used HeNe laser light to provide the input plane wave and the input and weighting irradiances. The spatial profiles of the input and weighting layers were realized by imaging a LCD spatial light modulator onto the respective nonlinear layers. The inputs were two bright or dark regions on a Gaussian input beam producing the intensity profile: whereIo= 12.5 mW/cm2, leo = 900llm, Xo = 600Ilffi, K. = 400Ilffi, and Qo and Q. are the logic inputs taking on a value of zero or one. The weight profile W.(x) = Ioexp[-(xIKo)2][1 +w.(x)] where w.(x) can range from zero to one and is found through training using an algorithm which probed the weighting mask in order update the training weights. Table 1 shows the experimental results for three different logic gates. Given is the normalized output before and after training. The network was trained to recognize a logic zero for a normalized output ~ 0.9 and a logic one or a normalized output ~ 1.1. An output value greater than I is considered a logic one and an output value less than one is a logic zero. RME is the root mean error. 6 CONCLUSIONS Work is in progress to improve the logic gate results by increasing the power of propagating signal beam as well as both the input and weighting beams. This will effectively increase the nonlinear processing capability of the network since a higher power produces more nonlinear effect. Also, more power will allow expansion of all of the beams thereby increasing the effective resolution of the thermal materials. Thisreduces the effect of heat transfer within the material which tends to wash out or diffuse benificial steep gradients in temperature which are what produce the gradients in the index of refraction. In addition, the use of photorefractive crystals for optical weight storage shows promise for being able to optically phase conjugate and backpropagate the output errror as well as implement the weight update rule for all optical network training. This appears to be simpler than optical networks using volume hologram weight storage because the Kerr network requires only planar hologram storage. 778 Elizabeth C. Behrman Inputs 0 0 0 1 1 0 1 1 Rl\1E Start 1.001 .802 .698 .807 7.3% NOR Finish 1.110 .884 .772 .896 0 Change .109 .082 .074 .089 -7.3% Output 1 0 0 0 Start .998 1.092 1.148 1.440 16.4% AND Finish .757 .855 .894 1.124 0 Change -.241 -.237 -.254 -.316 -16.4% Output 0 0 0 1 Start .998 .880 .893 .994 7.3% XNOR Finish 1.084 .933 .928 1.073 2.7% Change .086 .053 .035 .079 -4.6% Output 1 0 0 1 Table 1: Preliminary Experimental Logic Gate Results References Armstrong, J.A., Bloembergen, N., Ducuing, J., and Pershan, P.S., (1962) "Interactions Between Light Waves in a Nonlinear Dielectric", Physical Review, Vol. 127, pp. 1918-1939. Ie Cun, Yann, (1988) itA Theoretical Framework for Back -Propagation", Proceedings of the 1988 Connectionist Models Summer School, Morgan Kaufmann, pp. 21-28. Pineda, F.J., (1987) "Generalization of backpropagation to recurrent and higher order neural networks", Proceedings of IEEE Conference on Neural information Processing Systems, November 1987, IEEE Press. Roitblat., Moore, Nachtigall, and Penner, (1991) "Natural dolphin echo recognition using an integrator gateway network," in Advances in Neural Processing Systems 3 Morgan Kaufmann, San Mateo, CA, 273-281. Skinner, S.R, Steck, J.E., Behnnan, E.C., (1995) "An Optical Neural Network Using Kerr Type Nonlinear Materials", To Appear in Applied Optics. "Using Nworks, (1991) An Extended Tutorial for NeuralWorks Professional /lIPlus and NeuralWorks Explorer, NeuralWare, Inc. Pittsburgh, PA, pg. UN-18.	1994	115
876	Instance-Based State Identification for Reinforcement Learning R. Andrew McCallum Department of Computer Science University of Rochester Rochester, NY 14627-0226 mccallumCcs.rochester.edu Abstract This paper presents instance-based state identification, an approach to reinforcement learning and hidden state that builds disambiguating amounts of short-term memory on-line, and also learns with an order of magnitude fewer training steps than several previous approaches. Inspired by a key similarity between learning with hidden state and learning in continuous geometrical spaces, this approach uses instance-based (or "memory-based") learning, a method that has worked well in continuous spaces. 1 BACKGROUND AND RELATED WORK When a robot's next course of action depends on information that is hidden from the sensors because of problems such as occlusion, restricted range, bounded field of view and limited attention, the robot suffers from hidden state. More formally, we say a reinforcement learning agent suffers from the hidden state problem if the agent's state representation is non-Markovian with respect to actions and utility. The hidden state problem arises as a case of perceptual aliasing: the mapping between states of the world and sensations of the agent is not one-to-one [Whitehead, 1992]. If the agent's perceptual system produces the same outputs for two world states in which different actions are required, and if the agent's state representation consists only of its percepts, then the agent will fail to choose correct actions. Note that even if an agent's state representation includes some internal state beyond its 378 R. Andrew McCallum immediate percepts, the agent can still suffer from hidden state if it does not keep enough internal state to uncover the non-Markovian-ness of its environment. One solution to the hidden state problem is simply to avoid passing through the aliased states. This is the approach taken in Whitehead's Lion algorithm [Whitehead, 1992]. Whenever the agent finds a state that delivers inconsistent reward, it sets that state's utility so low that the policy will never visit it again. The success of this algorithm depends on a deterministic world and on the existence of a path to the goal that consists of only unaliased states. Other solutions do not avoid aliased states, but do as best they can given a nonMarkovian state representation [Littman, 1994; Singh et al., 1994; Jaakkola et al., 1995]. They involve either learning deterministic policies that execute incorrect actions in some aliased states, or learning stochastic policies with action choice probabilities matching the proportions of the different underlying aliased world states. These approaches do not depend on a path of unaliased states, but they have other limitations: when faced with many aliased states, a stochastic policy degenerates into random walk; when faced with potentially harmful results from incorrect actions, deterministically incorrect or probabilistically incorrect action choice may prove too dangerous; and when faced with performance-critical tasks, inefficiency that is proportional to the amount of aliasing may be unacceptable. The most robust solution to the hidden state problem is to augment the agent's state representation on-line so as to disambiguate the aliased states. State identification techniques uncover the hidden state information-that is, they make the agent's internal state space Markovian. This transformation from an imperfect state information model to a perfect state information model has been formalized in the decision and control literature, and involves adding previous percepts and actions to the definition of agent internal state [Bertsekas and Shreve, 1978]. By augmenting the agent's perception with history information-.short-term memory of past percepts, actions and rewards-the agent can distinguish perceptually aliased states, and can then reliably choose correct actions from them. Predefined, fixed memory representations such as order n Markov models (also known as constant-sized perception windows, linear traces or tapped-delay lines) are often undesirable. When the length of the window is more than needed, they exponentially increase the number of internal states for which a policy must be stored and learned; when the length of the memory is less than needed, the agent reverts to the disadvantages of undistinguished hidden state. Even if the agent designer understands the task well enough to know its maximal memory requirements, the agent is at a disadvantage with constant-sized windows because, for most tasks, different amounts of memory are needed at different steps of the task. The on-line memory creation approach has been adopted in several reinforcement learning algorithms. The Perceptual Distinctions Approach [Chrisman, 1992] and Utile Distinction Memory [McCallum, 1993] are both based on splitting states of a finite state machine by doing off-line analysis of statistics gathered over many steps. Recurrent-Q [Lin, 1993] is based on training recurrent neural networks. Indexed Memory [Teller, 1994] uses genetic programming to evolve agents that use load and store instructions on a register bank. A chief disadvantage of all these techniques is that they require a very large number of steps for training. Instance-Based State Identification for Reinforcement Learning 379 2 INSTANCE-BASED STATE IDENTIFICATION This paper advocates an alternate solution to the hidden state problem we term instance-based state identification. The approach was inspired by the successes of instance-based (also called "memory-based") methods for learning in continuous perception spaces, (i.e. [Atkeson, 1992; Moore, 1992]). The application of instance-based learning to short-term memory for hidden state is driven by the important insight that learning in continuous spaces and learning with hidden state have a crucial feature in common: they both begin learning without knowing the final granularity of the agent's state space. The former learns which regions of continuous input space can be represented uniformly and which areas must be finely divided among many states. The later learns which percepts can be represented uniformly because they uniquely identify a course of action without the need for memory, and which percepts must be divided among many states each with their own detailed history to distinguish them from other perceptually aliased world states. The first approach works with a continuous geometrical input space, the second works with a percept-act ion-reward "sequence" space, (or "history" space). Large continuous regions correspond to less-specified, small memories; small continuous regions correspond to more-specified, large memories. Furthermore, learning in continuous spaces and sequence spaces both have a lot to gain from instance-based methods. In situations where the state space granularity is unknown, it is especially useful to memorize the raw previous experiences. If the agent tries to fit experience to its current, flawed state space granularity, it is bound to lose information by attributing experience to the wrong states. Experience attributed to the wrong state turns to garbage and is wasted. When faced with an evolving state space, keeping raw previous experience is the path of least commitment, and thus the most cautious about losing information. 3 NEAREST SEQUENCE MEMORY There are many possible instance-based techniques to choose from, but we wanted to keep the first application simple. With that in mind, this initial algorithm is based on k-nearest neighbor. We call it Nearest Sequence Memory, (NSM). It bears emphasizing that this algorithm is the most straightforward, simple, almost naive combination of instance-based methods and history sequences that one could think of; there are still more sophisticated instance-based methods to try. The surprising result is that such a simple technique works as well as it does. Any application of k-nearest neighbor consists of three parts: 1) recording each experience, 2) using some distance metric to find neighbors of the current query point, and 3) extracting output values from those neighbors. We apply these three parts to action-percept-reward sequences and reinforcement learning by Q-Iearning lWatkins, 1989] as follows: 1. For each step the agent makes in the world, it records the action, percept and reward by adding a new state to a single, long chain of states. Thus, each state in the chain contains a snapshot of immediate experience; and all the experiences are laid out in a time-connected history chain. 380 Learning in a Geometric Space k-nearest neighbor, k = 3 • • • Learning in a Sequence Space k-nearest neighbor, k = 3 • • 00014001301201 R. Andrew McCallum • • match length 3~ 0 1 " action. percept. rewanl Figure 1: A continuous space compared with a sequence space. In each case, the "query point" is indicated with a gray cross, and the three nearest neighbors are indicated with gray shadows. In a geometric space, the neighborhood metric is defined by Euclidean distance. In a sequence space, the neighborhood metric is determined by sequence match length-the number of preceding states that match the states preceding the query point. 2. When the agent is about to choose an action, it finds states considered to be similar by looking in its state chain for states with histories similar to the current situation. The longer a state's string of previous experiences matches the agent's most recent experiences, the more likely the state represents where the agent is now. 3. Using the states, the agent obtains Q-values by averaging together the expected future reward values associated with the k nearest states for each action. The agent then chooses the action with the highest Q-value. The regular Q-Iearning update rule is used to update the k states that voted for the chosen action. Choosing to represent short-term memory as a linear trace is a simple, wellestablished technique. Nearest Sequence Memory uses a linear trace to represent memory, but it differs from the fixed-sized window approaches because it provides a variable memory-length-like k-nearest neighbor, NSM can represent varying resolution in different regions of state space. 4 DETAILS OF THE ALGORITHM A more complete description of Nearest Sequence Memory, its performance and its possible improvements can be found in [McCallum, 1995]. The interaction between the agent and its environment is described by actions, percepts and rewards. There is a finite set of possible actions, A = {al,a2, ... ,am }, Instance-Based State Identification for Reinforcement Learning 381 a finite set of possible percepts, () = {Ol, 02, ... , On}, and scalar range of possible rewards, n = [x, y], x, Y E~. At each time step, t, the agent executes an action, at E A, then as a result receives a new percept, Ot E (), and a reward, rt E n. The agent records its raw experience at time t in a "state" data point, St. Also associated with St is a slot to hold a single expected future discounted reward value, denoted q(st). This value is associated with at and no other action. 1. Find the k nearest neighbor (most similar) states for each possible future action. The state currently at the end of the chain is the "query point" from which we measure all the distances. The neighborhood metric is defined by the number of preceding experience records that match the experience records preceding the "query point" state. (Here higher values of n(s;, sJ) indicate that S; and Sj are closer neighbors.) ( _ _ )_ { 1+n(s;_1,Sj-I), n S" SJ 0 , if (a;-1 = aj-I) A (0;-1 = OJ-I) A (r;-1 = rj_I) otherwise (1) Considering each of the possible future actions ill turn, we find the k nearest neighbors and give them a vote, v(s;). v(S;) = { ~: if n(st, s;) is among the k maxv$jlaj=a; n(st, Sj)'s otherwise (2) 2. Determine the Q-value for each action by averaging individual the q-values from the k voting states for that action. Qt(a;) = L (v(s;)/k)q(sj) (3) V$jlaj=a; 3. Select an action by maximum Q-value, or by random exploration. According to an exploration probability, e, either let at+1 be randomly chosen from A, or (4) 4. Execute the action chosen in step 3, and record the resulting experience. Do this by creating a new "state" representing the current state of the environment, and storing the action-percept-reward triple associated with it: Increment the time counter: t ~ t + 1. Create St; record in it at, Ot, rt. The agent can limit its storage and computational load by limiting the number of instances it maintains to N (where N is some reasonably large number). Once the agent accumulates N instances, it can discard the oldest instance each time it adds a new one. This also provides a way to handle a changing environment. 5. Update the q-values by vote. Perform the dynamic programming step using the standard Q-Iearning rule to update those states that voted for the chosen action. Note that this actually involves performing steps 1 and 2 to get the next Q-values needed for calculating the utility of the agent's current state, Ut . (Here (3 is the learning rate.) Ut = max Qt(a) (5) a (Vsda; = at-I) q(s;) ~ (1- (3v(s;))q(s;) + (3v(s;)(r; + "YUt) (6) 382 20 15 ~ 5 10 5 Performance during learning Nearest Sequence Memory Perceptual Distinctions Approach . 1000 2000 3000 4000 5000 6000 7000 8000 Steps lD ~: t StejM 10 LMm 74 1500 _of ...... fBI StepslOLMm ~:; 153 2500 .... borof ...... 70 60 50 ., c. 40 li 30 20 10 0 R. Andrew McCallum Steps per Trial during learning Nearest Sequence Memory Recurrent-a -. ~ 20 40 60 80 100 Trials StejM 10 L.", ~: -, .... _._' 238 Ii· iii :: =L-.'-...:39==5=-___ Steps __ I0_~..,."I0000 .....borof ...... Figure 2: Comparing Nearest Sequence Memory with three other algorithms: Perceptual Distinction Approach, Recurrent-Q and Utile Distinction Memory. In each case, NSM learns with roughly an order of magnitude fewer steps. 5 EXPERIMENTAL RESULTS The performance of NSM is compared to three other algorithms using the tasks chosen by the other algorithms' designers. In each case, NSM learns the task with roughly an order of magnitude fewer steps. Although NSM learns good policies quickly, it does not always learn optimal policies. In section 6 we will discuss why the policies are not always optimal and how NSM could be improved. The Perceptual Distinctions Approach [Chrisman, 1992] was demonstrated in a space ship docking application with hidden state. The task was made difficult by noisy sensors and actions. Some of the sensors returned incorrect values 30% of the time. Various actions failed 70, 30 or 20% of the time, and when they failed, resulted in random states. NSM used f3 = 0.2, I = 0.9, k = 8, and N = 1000. PDA takes almost 8000 steps to learn the task. NSM learns a good policy in less than 1000 steps, although the policy is not quite optimal. Utile Distinction Memory [McCallum, 1993] was demonstrated on several local perception mazes. Unlike most reinforcement learning maze domains, the agent perceives only four bits indicating whether there is a barrier to the immediately adjacent north, east, south and west. NSM used f3 = 0.9, I = 0.9, k = 4, and N = 1000. In two of the mazes, NSM learns the task in only about 1/20th the time required by UDM; in the other two, NSM learns mazes that UDM did not solve at all. Instance-Based State Identification for Reinforcement Learning 383 Recurrent-Q [Lin, 1993] was demonstrated on a robot 2-cup retrieval task. The env,jronment is deterministic, but the task is made difficult by two nested levels of hidden state and by providing no reward until the task is completely finished. NSM used {3 = 0.9, I = 0.9, k = 4, and N = 1000. NSM learns good performance in about 15 trials, Recurrent-Q takes about 100 trials to reach equivalent performance. 6 DISCUSSION Nearest Sequence Memory offers much improved on-line performance and fewer training steps than its predecessors. Why is the improvement so dramatic? I believe the chief reason lies with the inherent advantage of instance-based methods, as described in section 2: the key idea behind Instance-Based State Identification is the recognition that recording raw experience is particularly advantageous when the agent is learning a policy over a changing state space granularity, as is the case when the agent is building short-term memory for disambiguating hidden state. If, instead of using an instance-based technique, the agent simply averages new experiences into its current, flawed state space model, the experiences will be applied to the wrong states, and cannot be reused when the agent reconfigures its state space. Furthermore, and perhaps even more detrimentally, incoming data is always interpreted in the context of the flawed state space, always biased in an inappropriate way-not simply recorded, kept uncommitted and open to easy reinterpretation in light of future data. The experimental results in this paper bode well for instance-based state identification. Nearest Sequence Memory is simple-if such a simplistic implementation works as well as it does, more sophisticated approaches may work even better. Here are some ideas for improvement: The agent should use a more sophisticated neighborhood distance metric than exact string match length. A new metric could account for distances between different percepts instead of considering only exact matches. A new metric could also handle continuous-valued inputs. Nearest Sequence Memory demonstrably solves tasks that involve noisy sensation and action, but it could perhaps handle noise even better if it used some technique for explicitly separating noise from structure. K-nearest neighbor does not explicitly discriminate between structure and noise. If the current query point has neighbors with wildly varying output values, there is no way to know if the variations are due to noise, (in which case they should all be averaged), or due to fine-grained structure of. the underlying function (in which case only the few closest should be averaged). Because NSM is built on k-nearest neighbor, it suffers from the same inability to methodically separate history differences that are significant for predicting reward and history differences that are not. I believe this is the single most important reason that NSM sometimes did not find optimal policies. Work in progress addresses the structure/noise issue by combining instance-based state identification with the structure/noise separation method from Utile Distinction Memory [McCallum, 1993]. The algorithm, called Utile Suffix Memory, uses a tree-structured representation, and is related to work with Ron, Singer and Tishby's Prediction Suffix Trees, Moore's Parti-game, Chapman and Kaelbling's 384 R. Andrew McCallum G-algorithm, and Moore's Variable Resolution Dynamic Programming. See [McCallum, 1994] for more details as well as references to this related work. Aclmowledgments This work has benefited from discussions with many colleagues, including: Dana Ballard, Andrew Moore, Jeff Schneider, and Jonas Karlsson. This material is based upon work supported by NSF under Grant no. IRI-8903582 and by NIH/PHS under Grant no. 1 R24 RR06853-02. References [Atkeson, 1992] Christopher G. Atkeson. Memory-based approaches to approximating continuous functions. In M. Casdagli and S. Eubank, editors, Nonlinear Modeling and Forecasting, pages 503-521. Addison Wesley, 1992. [Bertsekas and Shreve, 1978] Dimitri. P. Bertsekas and Steven E. Shreve. Stochastic Optimal Control. Academic Press, 1978. [Chrisman, 1992] Lonnie Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In Tenth Nat'l Conf. on AI, 1992. [Jaakkola et al., 1995] Tommi Jaakkola, Satinder Pal Singh, and Michael 1. Jordan. Reinforcement learning algorithm for partially observable markov decision problems. In Advances of Neural Information Processing Systems 7. Morgan Kaufmann, 1995. [Lin, 1993] Long-Ji Lin. Reinforcement Learning for Robots Using Neural Networks. PhD thesis, Carnegie Mellon, School of Computer Science, January 1993. [Littman, 1994] Michael Littman. Memoryless policies: Theoretical limitations and practical results. In Proceedings of the Third International Conference on Simulation of Adaptive Behavior: From Animals to Animats, 1994. [McCallum, 1993] R. Andrew McCallum. Overcoming incomplete perception with utile distinction memory. In The Proceedings of the Tenth International Machine Learning Conference. Morgan Kaufmann Publishers, Inc., 1993. [McCallum, 1994] R. Andrew McCallum. Utile suffix memory for reinforcement learning with hidden state. TR 549, U. of Rochester, Computer Science, 1994. [McCallum, 1995] R. Andrew McCallum. Hidden state and reinforcement learning with instance-based state identification. IEEE Trans. on Systems, Man, and Cybernetics, 1995. (In press) [Earlier version available as U. of Rochester TR 502]. [Moore, 1992] Andrew Moore. Efficient Memory-based Learning for Robot Control. PhD thesis, University of Cambridge, November 1992. [Singh et al., 1994] Satinder Pal Singh, Tommi Jaakkola, and Michael 1. Jordan. Modelfree reinforcement learning for non-markovian decision problems. In The Proceedings of the Eleventh International Machine Learning Conference, 1994. [Teller, 1994] Astro Teller. The evolution of mental models. In Kim Kinnear, editor, Advances in Genetic Programming, chapter 9. MIT Press, 1994. [Watkins, 1989] Chris Watkins. Learning from delayed rewards. PhD thesis, Cambridge University, 1989. [Whitehead, 1992] Steven Whitehead. Reinforcement Learning for the Adaptive Control of Perception and Action. PhD thesis, Department of Computer Science, University of Rochester, 1992.	1994	116
877	Nonlinear Image Interpolation using Manifold Learning Christoph Bregler Computer Science Division University of California Berkeley, CA 94720 bregler@cs.berkeley.edu Stephen M. Omohundro'" Int. Computer Science Institute 1947 Center Street Suite 600 Berkeley, CA 94704 om@research.nj .nec.com Abstract The problem of interpolating between specified images in an image sequence is a simple, but important task in model-based vision. We describe an approach based on the abstract task of "manifold learning" and present results on both synthetic and real image sequences. This problem arose in the development of a combined lip-reading and speech recognition system. 1 Introduction Perception may be viewed as the task of combining impoverished sensory input with stored world knowledge to predict aspects of the state of the world which are not directly sensed. In this paper we consider the task of image interpolation by which we mean hypothesizing the structure of images which occurred between given images in a temporal sequence. This task arose during the development of a combined lipreading and speech recognition system [3], because the time windows for auditory and visual information are different (30 frames per second for the camera vs. 100 feature vectors per second for the acoustic information). It is an excellent visual test domain in general, however, because it is easy to generate large amounts of test and training data and the performance measure is largely "theory independent" . The test consists of simply presenting two frames from a movie and comparing the "'New address: NEe Research Institute, Inc., 4 Independence Way, Princeton, NJ 08540 974 Christoph Bregler. Stephen M. Omohundro Figure 1: Linear interpolated lips. Figure 2: Desired interpolation. hypothesized intermediate frames to the actual ones. It is easy to use footage of a particular visual domain as training data in the same way. Most current approaches to model-based vision require hand-constructed CADlike models. We are developing an alternative approach in which the vision system builds up visual models automatically by learning from examples. One of the central components of this kind of learning is the abstract problem of inducing a smooth nonlinear constraint manifold from a set of examples from the manifold. We call this "manifold learning" and have developed several approaches closely related to neural networks for doing it [2]. In this paper we apply manifold learning to the image interpolation problem and numerically compare the results of this "nonlinear" process with simple linear interpolation. We find that the approach works well when the underlying model space is low-dimensional. In more complex examples, manifold learning cannot be directly applied to images but still is a central component in a more complex system (not discussed here). We present several approaches to using manifold learning for this task. We compare the performance of these approaches to that of simple linear interpolation. Figure 1 shows the results of linear interpolation of lip images from the lip-reading system. Even in the short period of 33 milliseconds linear interpolation can produce an unnatural lip image. The problem is that linear interpolation of two images just averages the two pictures. The interpolated image in Fig. 1 has two lower lip parts instead of just one. The desired interpolated image is shown in Fig. 2, and consists of a single lower lip positioned at a location between the lower lip positions in the two input pictures. Our interpolation technique is nonlinear, and is constrained to produce only images from an abstract manifold in "lip space" induced by learning. Section 2 describes the procedure, Section 4 introduces the interpolation technique based on the induced manifold, and Sections 5 and 6 describe our experiments on artificial and natural images. 2 Manifold Learning Each n * m gray level image may be thought of as a point in an n * m-dimensional space. A sequence of lip-images produced by a speaker uttering a sentence lie on a Nonlinear Image Interpolation Using Manifold Learning Graylevel Dimensions -\ -----(l6x16 pixel = 256 dim. space) Figure 3: Linear vs nonlinear interpolation. 975 1-dimensional trajectory in this space (figure 3). If the speaker were to move her lips in all possible ways, the images would define a low-dimensional submanifold (or nonlinear surface) embedded in the high-dimensional space of all possible graylevel images. If we could compute this nonlinear manifold, we could limit any interpolation algorithm to generate only images contained in it. Images not on the manifold cannot be generated by the speaker under normal circumstances. Figure 3 compares a curve of interpolated images lying on this manifold to straight line interpolation which generally leaves the manifold and enters the domain of images which violate the integrity of the model. To represent this kind of nonlinear manifold embedded in a high-dimensional feature space, we use a mixture model of local linear patches. Any smooth nonlinear manifold can be approximated arbitrarily well in each local neighborhood by a linear "patch" . In our representation, local linear patches are "glued" together with smooth "gating" functions to form a globally defined nonlinear manifold [2]. We use the "nearest-point-query" to define the manifold. Given an arbitrary point near the manifold, this returns the closest point on the manifold. We answer such queries with a weighted sum of the linear projections of the point to each local patch. The weights are defined by an "influence function" associated with each linear patch which we usually define by a Gaussian kernel. The weight for each patch is the value of its influence function at the point divided by the sum of all influence functions ("partition of unity"). Figure 4 illustrates the nearest-point-query. Because Gaussian kernels die off quickly, the effect of distant patches may be ignored, improving computational performance. The linear projections themselves consist of a dot product and so are computationally inexpensive. For learning, we must fit such a mixture of local patches to the training data. An initial estimate of the patch centers is obtained from k-means clustering. We fit a patch to each local cluster using a local principal components analysis. Fine tuning 976 PI l:,Gi(X) ' Pi(x) P(x) = ...:...' -=--l:,Gi(X) , Christoph Bregler. Stephen M. Omohundro Influence Function P2 \ Linear Patch Figure 4: Local linear patches glued together to a nonlinear manifold. of the model is done using the EM (expectation-maximization) procedure. This approach is related to the mixture of expert architecture [4], and to the manifold representation in [6]. Our EM implementation is related to [5], which uses a hierarchical gating function and local experts that compute linear mappings from one space to another space. In contrast, our approach uses a "one-level" gating function and local patches that project a space into itself. 3 Linear Preprocessing Dealing with very high-dimensional domains (e.g. 256 * 256 gray level images) requires large memory and computational resources. Much of this computation is not relevant to the task, however. Even if the space of images is nonlinear, the nonlinearity does not necessarily appear in all of the dimensions. Earlier experiments in the lip domain [3] have shown that images projected onto a lO-dimensional linear subspace still accurately represents all possible lip configurations. We therefore first project the high-dimensional images into such a linear subspace and then induce the nonlinear manifold within this lower dimensional linear subspace. This preprocessing is similar to purely linear techniques [7, 10, 9]. 4 Constraint Interpolation Geometrically, linear interpolation between two points in n-space may be thought of as moving along the straight line joining the two points. In our non-linear approach to interpolation, the point moves along a curve joining the two points which lies in the manifold of legal images. We have studied several algorithms for estimating the shortest manifold trajectory connecting two given points. For the performance results, we studied the point which is halfway along the shortest trajectory. Nonlinear Image Interpolation Using Manifold Learning 977 4.1 "Free-Fall" The computationally simplest approach is to simply project the linearly interpolated point onto the nonlinear manifold. The projection is accurate when the point is close to the manifold. In cases where the linearly interpolated point is far away (i.e. no weight of the partition of unity dominates all the other weights) the closest-pointquery does not result in a good interpolant. For a worst case, consider a point in the middle of a circle or sphere. All local patches have same weight and the weighted sum of all projections is the center point itself, which is not a manifold point. Furthermore, near such "singular" points, the final result is sensitive to small perturbations in the initial position. 4.2 "Manifold-Walk" A better approach is to "walk" along the manifold itself rather than relying on the linear interpolant. Each step of the walk is linear and in the direction of the target point but the result is immediately projected onto the manifold. This new point is then moved toward the target point and projected onto the manifold, etc. When the target is finally reached, the arc length of the curve is approximated by the accumulated lengths of the individual steps. The point half way along the curve is chosen as the interpolant. This algorithm is far more robust than the first one, because it only uses local projections, even when the two input points are far from each other. Figure 5b illustrates this algorithm. 4.3 "Manifold-Snake" This approach combines aspects of the first two algorithms. It begins with the linearly interpolated points and iteratively moves the points toward the manifold. The Manifold-Snake is a sequence of n points preferentially distributed along a smooth curve with equal distances between them. An energy function is defined on such sequences of points so that the energy minimum tries to satisfy these constraints (smoothness, equidistance, and nearness to the manifold): (1) E has value 0 if all Vi are evenly distributed on a straight line and also lie on the manifold. In general E can never be 0 if the manifold is nonlinear, but a minimum for E represents an optimizing solution. We begin with a straight line between the two input points and perform gradient descent in E to find this optimizing solution. 5 Synthetic Examples To quantify the performance of these approaches to interpolation, we generated a database of 16 * 16 pixel images consisting of rotated bars. The bars were rotated for each image by a specific angle. The images lie on a one-dimensional nonlinear manifold embedded in a 256 dimensional image space. A nonlinear manifold represented by 16 local linear patches was induced from the 256 images. Figure 6a shows 978 Christoph Bregler, Stephen M. Omohundro a) "Free Fall" b) "Surface Walk" c) "Surface Snake" Figure 5: Proposed interpolation algorithms. -~/ Figure 6: a) Linear interpolation, b) nonlinear interpolation. two bars and their linear interpolation. Figure 6b shows the nonlinear interpolation using the Manifold- Walk algorithm. Figure 7 shows the average pixel mean squared error of linear and nonlinear interpolated bars. The x-axis represents the relative angle between the two input points. Figure 8 shows some iterations of a Manifold-Snake interpolating 7 points along a 1 dimensional manifold embedded in a 2 dimensional space . .... pi ......... --/' ./ ~/ VV ,/ --.t. __ ---- --- --- Figure 7: Average pixel mean squared error of linear and nonlinear interpolated bars. Nonlinear Image Interpolation Using Manifold Learning 979 QQOOOO o Iter. I Iter. 2 Iter. 5 Iter. to Iter. 30 Iter. Figure 8: Manifold-Snake iterations on an induced 1 dimensional manifold embedded in 2 dimensions. Figure 9: 16x16 images. Top row: linear interpolation. Bottom row: nonlinear "manifold-walk" interpolation. 6 Natural Lip Images We experimented with two databases of natural lip images taken from two different subjects. Figure 9 shows a case of linear interpolated and nonlinear interpolated 16 * 16 pixel lip images using the Manifold- Walk algorithm. The manifold consists of 16 4-dimensional local linear patches. It was induced from a training set of 1931 lip images recorded with a 30 frames per second camera from a subject uttering various sentences. The nonlinear interpolated image is much closer to a realistic lip configuration than the linear interpolated image. Figure 10 shows a case of linear interpolated and nonlinear interpolated 45 * 72 pixel lip images using the Manifold-Snake algorithm. The images were recorded with a high-speed 100 frames per second cameral . Because of the much higher dimensionality of the images, we projected the images into a 16 dimensional linear subspace. Embedded in this subspace we induced a nonlinear manifold consisting of 16 4-dimensionallocallinear patches, using a training set of 2560 images. The linearly interpolated lip image shows upper and lower teeth, but with smaller contrast, because it is the average image of the open mouth and closed mouth. The nonlinearly interpolated lip images show only the upper teeth and the lips half way closed, which is closer to the real lip configuration. 7 Discussion We have shown how induced nonlinear manifolds can be used to constrain the interpolation of gray level images. Several interpolation algorithms were proposed IThe images were recorded in the UCSD Perceptual Science Lab by Michael Cohen 980 Christoph Bregler. Stephen M. Omohundro Figure 10: 45x72 images projected into a 16 dimensional subspace. Top row: linear interpolation. Bottom row: nonlinear "manifold-snake" interpolation. and experimental studies have shown that constrained nonlinear interpolation works well both in artificial domains and natural lip images. Among various other nonlinear image interpolation techniques, the work of [1] using a Gaussian Radial Basis Function network is most closely related to our approach. Their approach is based on feature locations found by pixelwise correspondence, where our approach directly interpolates graylevel images. Another related approach is presented in [8]. Their images are also first projected into a linear subspace and then modelled by a nonlinear surface but they require their training examples to lie on a grid in parameter space so that they can use spline methods. References [1] D. Beymer, A. Shahsua, and T. Poggio Example Based Image Analysis and Synthesis M.I.T. A.1. Memo No. 1431, Nov. 1993. [2] C. Bregler and S. Omohundro, Surface Learning with Applications to Lip-Reading, in Advances in Neural Information Processing Systems 6, Morgan Kaufmann, 1994. [3] C. Bregler and Y. Konig, "Eigenlips" for Robust Speech Recognition in Proc. ofIEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Adelaide, Australia, 1994. [4] R.A. Jacobs, M.1. Jordan, S.J. Nowlan, and G.E. Hinton, Adaptive mixtures of local experts in Neural Compuation, 3, 79-87. [5] M.1. Jordan and R. A. Jacobs, Hierarchical Mixtures of Experts and the EM Algorithm Neural Computation, Vol. 6, Issue 2, March 1994. [6] N. Kambhatla and T.K. Leen, Fast Non-Linear Dimension Reduction in Advances in Neural Information Processing Systems 6, Morgan Kaufmann, 1994. [7] M. Kirby, F. Weisser, and G. DangeImayr, A Model Problem in Represetation of Digital Image Sequences, in Pattern Recgonition, Vol 26, No.1, 1993. [8] H. Murase, and S. K. Nayar Learning and Recognition of 3-D Objects from Brightness Images Proc. AAAI, Washington D.C., 1993. [9] P. Simard, Y. Le Cun, J. Denker Efficient Pattern Recognition Using a New Transformation Distance Advances in Neural Information Processing Systems 5, Morgan Kaufman, 1993. [10] M. Turk and A. Pentland, Eigenfaces for Recognition Journal of Cognitive Neuroscience, Volume 3, Number 1, MIT 1991.	1994	117
878	A Growing Neural Gas Network Learns Topologies Bernd Fritzke Institut fur Neuroinformatik Ruhr-Universitat Bochum D-44 780 Bochum Germany Abstract An incremental network model is introduced which is able to learn the important topological relations in a given set of input vectors by means of a simple Hebb-like learning rule. In contrast to previous approaches like the "neural gas" method of Martinetz and Schulten (1991, 1994), this model has no parameters which change over time and is able to continue learning, adding units and connections, until a performance criterion has been met. Applications of the model include vector quantization, clustering, and interpolation. 1 INTRODUCTION In unsupervised learning settings only input data is available but no information on the desired output. What can the goal of learning be in this situation? One possible objective is dimensionality reduction: finding a low-dimensional subspace of the input vector space containing most or all of the input data. Linear subspaces with this property can be computed directly by principal component analysis or iteratively with a number of network models (Sanger, 1989; Oja, 1982). The Kohonen feature map (Kohonen, 1982) and the "growing cell structures" (Fritzke, 1994b) allow projection onto non-linear, discretely sampled subspaces of a dimensionality which has to be chosen a priori. Depending on the relation between inherent data dimensionality and dimensionality of the target space, some information on the topological arrangement of the input data may be lost in the process. 626 Bernd Fritzke This is not astonishing since a reversible mapping from high-dimensional data to lower-dimensional spaces (or structures) does not exist in general. Asking how structures must look like to allow reversible mappings directly leads to another possible objective of unsupervised learning which can be described as topology learning: Given some high-dimensional data distributionp(e), find a topological structure which closely reflects the topology of the data distribution. An elegant method to construct such structures is "competitive Hebbian learning" (CHL) (Martinetz, 1993). CHL requires the use of some vector quantization method. Martinetz and Schulten propose the "neural gas" (NG) method for this purpose (Martinetz and Schulten, 1991). We will briefly introduce and discuss the approach of Martinetz and Schulten. Then we propose a new network model which also makes use of CHL. In contrast to the above-mentioned CHL/NG combination, this model is incremental and has only constant parameters. This leads to a number of advantages over the previous approach. 2 COMPETITIVE HEBBIAN LEARNING AND NEURAL GAS CHL (Martinetz, 1993) assumes a number of centers in R n and successively inserts topological connections among them by evaluating input signals drawn from a data distribution p(e). The principle of this method is: For each input signal x connect the two closest centers (measured by Euclidean distance) by an edge. The resulting graph is a subgraph of the Delaunay triangulation (fig. 1a) corresponding to the set of centers. This subgraph (fig. 1b), which is called the "induced Delaunay triangulation", is limited to those areas of the input space R n where p(e» O. The "induced Delaunay triangulation" has been shown to optimally preserve topology in a very general sense (Martinetz, 1993). Only centers lying on the input data submanifold or in its vicinity actually develop any edges. The others are useless for the purpose of topology learning and are often called dead units. To make use of all centers they have to be placed in those regions of R n where P (e) differs from zero. This could be done by any vector quantization (VQ) procedure. Martinetz and Schulten have proposed a particular kind of VQ method, the mentioned NG method (Martinetz and Schulten, 1991). The main principle of NG is the following: For each input signal x adapt the k nearest centers whereby k is decreasing from a large initial to a small final value. A large initial value of k causes adaptation (movement towards the input signal) of a large fraction of the centers. Then k (the adaptation range) is decreased until finally only the nearest center for each input signal is adapted. The adaptation strength underlies a similar decay schedule. To realize the parameter decay one has to define the total number of adaptation steps for the NG method in advance. A Growing Neural Gas Network Learns Topologies 627 a) Delaunay triangulation b) induced Delaunay triangulation Figure 1: Two ways of defining closeness among a set of points. a) The Delaunay triangulation (thick lines) connects points having neighboring Voronoi polygons (thin lines). Basically this reduces to points having small Euclidean distance w.r.t. the given set of points. b) The induced Delaunay triangulation (thick lines) is obtained by masking the original Delaunay triangulation with a data distribution P(~) (shaded). Two centers are only connected if the common border of their Voronoi polygons lies at least partially in a region where P(~» 0 (closely adapted from Martinetz and Schulten, 1994) For a given data distribution one could now first run the NG algorithm to distribute a certain number of centers and then use CHL to generate the topology. It is, however, also possible to apply both techniques concurrently (Martinetz and Schulten, 1991). In this case a method for removing obsolete edges is required since the motion of the centers may make edges invalid which have been generated earlier. Martinetz and Schulten use an edge aging scheme for this purpose. One should note that the CHL algorithm does not influence the outcome of the NG method in any way since the adaptations in NG are based only on distance in input space and not on the network topology. On the other hand NG does influence the topology generated by CHL since it moves the centers around. The combination of NG and CHL described above is an effective method for topology learning. A problem in practical applications, however, may be to determine a priori a suitable number of centers. Depending on the complexity of the data distribution which one wants to model, very different numbers of centers may be appropriate. The nature of the NG algorithm requires a decision in advance and, if the result is not satisfying, one or several new simulations have to be performed from scratch. In the following we propose a method which overcomes this problem and offers a number of other advantages through a flexible scheme for center insertion. 628 Bernd Fritzke 3 THE GROWING NEURAL GAS ALGORITHM In the following we consider networks consisting of • a set A of units (or nodes). Each unit c E A has an associated reference vector We E Rn. The reference vectors can be regarded as positions in input space of the corresponding units. • a set N of connections (or edges) among pairs of units. These connections are not weighted. Their sole purpose is the definition of topological structure. Moreover, there is a (possibly infinite) number of n-dimensional input signals obeying some unknown probability density function P(~). The main idea of the method is to successively add new units to an initially small network by evaluating local statistical measures gathered during previous adaptation steps. This is the same approach as used in the "growing cell structures" model (Fritzke, 1994b) which, however, has a topology with a fixed dimensionality (e.g., two or three). In the approach described here, the network topology is generated incrementally by CHL and has a dimensionality which depends on the input data and may vary locally. The complete algorithm for our model which we call "growing neural gas" is given by the following: o. Start with two units a and b at random positions Wa and Wb in Rn. 1. Generate an input signal ~ according to P(~). 2. Find the nearest unit 81 and the second-nearest unit 82. 3. Increment the age of all edges emanating from 81. 4. Add the squared distance between the input signal and the nearest unit in input space to a local counter variable: Aerror(8t} = IIWSl - ell2 5. Move 81 and its direct topological neighbors1 towards ~ by fractions Eb and En, respectively, of the total distance: AWs1 = Eb(e W S1 ) AWn = En(~ w n ) for all direct neighbors n of 81 6. If 81 and 82 are connected by an edge, set the age of this edge to zero. If such an edge does not exist, create it.2 7. Remove edges with an age larger than amaz • If this results in points having no emanating edges, remove them as well. IThroughout this paper the term neighbors denotes units which are topological neighbors in the graph (as opposed to units within a small Euclidean distance of each other in input space). 2This step is Hebbian in its spirit since correlated activity is used to decide upon insertions. A Growing Neural Gas Network Learns Topologies 629 8. If the number of input signals generated so far is an integer multiple of a parameter A, insert a new unit as follows: • Determine the unit q with the maximum accumulated error. • Insert a new unit r halfway between q and its neighbor f with the largest error variable: Wr = 0.5 (wq + wf)' • Insert edges connecting the new unit r with units q and f, and remove the original edge between q and f. • Decrease the error variables of q and f by multiplying them with a constant 0:. Initialize the error variable of r with the new value of the error variable of q. 9. Decrease all error variables by multiplying them with a constant d. 10. If a stopping criterion (e.g., net size or some performance measure) is not yet fulfilled go to step 1. How does the described method work? The adaptation steps towards the input signals (5.) lead to a general movement of all units towards those areas of the input space where signals come from (P(~) > 0). The insertion of edges (6.) between the nearest and the second-nearest unit with respect to an input signal generates a single connection of the "induced Delaunay triangulation" (see fig. 1b) with respect to the current position of all units. The removal of edges (7.) is necessary to get rid of those edges which are no longer part of the "induced Delaunay triangulation" because their ending points have moved. This is achieved by local edge aging (3.) around the nearest unit combined with age re-setting of those edges (6.) which already exist between nearest and second-nearest units. With insertion and removal of edges the model tries to construct and then track the "induced Delaunay triangulation" which is a slowly moving target due to the adaptation of the reference vectors. The accumulation of squared distances (4.) during the adaptation helps to identify units lying in areas of the input space where the mapping from signals to units causes much error. To reduce this error, new units are inserted in such regions. 4 SIMULATION RESULTS We will now give some simulation results to demonstrate the general behavior of our model. The probability distribution in fig. 2 has been proposed by Martinetz and Schulten (1991) to demonstrate the non-incremental "neural gas" model. It can be seen that our model quickly learns the important topological relations in this rather complicated distribution by forming structures of different dimensionalities. The second example (fig. 3) illustrates the differences between the proposed model and the original NG network. Although the final topology is rather similar for both models, intermediate stages are quite different. Both models are able to identify the clusters in the given distribution. Only the "growing neural gas" model, however, 630 Bernd Fritzke Figure 2: The "growing neural gas" network adapts to a signal distribution which has different dimensionalities in different areas of the input space. Shown are the initial network consisting of two randomly placed units and the networks after 600, 1800, 5000, 15000 and 20000 input signals have been applied. The last network shown is not the necessarily the "final" one since the growth process could in principle be continued indefinitely. The parameters for this simulation were: A = 100, Eb = 0.2, En = 0.006, a = 0.5, amaz = 50, d = 0.995. could continue to grow to discover still smaller clusters (which are not present in this particular example, though). 5 DISCUSSION The "growing neural gas" network presented here is able to make explicit the important topological relations in a given distribution pee) of input signals. An advantage over the NG method of Martinetz and Schulten is the incremental character of the model which eliminates the need to pre-specify a network size. Instead, the growth process can be continued until a user-defined performance criterion or network size is met. All parameters are constant over time in contrast to other models which heavily rely on decaying parameters (such as the NG method or the Kohonen feature map). It should be noted that the topology generated by CHL is not an optional feature A Growing Neural Gas Network Learns Topologies "neural gas" and "competitive Hebbian learning" o 0 0 00 0 000 0 o~oo 000 ·~~r~: ;-.; .. {]J'. 0900 0 00 8 o V ~ j o co 00 0 o 631 "growing neural gas" (uses "competitive Hebbian learning") Figure 3: The NG/CHL network of Martinetz and Schulten (1991) and the author's "growing neural gas" model adapt to a clustered probability distribution. Shown are the respective initial states (top row) and a number of intermediate stages. Both the number of units in the NG model and the final number of units in the "growing neural gas" model are 100. The bottom row shows the distribution of centers after 10000 adaptation steps (the edges are as in the previous row but not shown). The center distribution is rather similar for both models although the intermediate stages differ significantly. 632 Bernd Fritzke of our method (as it is for the NG model) but an essential component since it is used to direct the (completely local) adaptation as well as insertion of centers. It is probably the proper initialization of new units by interpolation from existing ones which makes it possible to have only constant parameters and local adaptations. Possible applications of our model are clustering (as shown) and vector quantization. The network should perform particularly well in situations where the neighborhood information (in the edges) is used to implement interpolation schemes between neighboring units. By using the error occuring in early phases it can be determined where to insert new units to generate a topological look-up table of different density and different dimensionality in particular areas of the input data space. Another promising direction of research is the combination with supervised learning. This has been done earlier with the "growing cell structures" (Fritzke, 1994c) and recently also with the "growing neural gas" described in this paper (Fritzke, 1994a). A crucial property for this kind of application is the possibility to choose an arbitrary insertion criterion. This is a feature not present, e.g., in the original "growing neural gas". The first results of this new supervised network model, an incremental radial basis function network, are very promising and we are further investigating this currently. References Fritzke, B. (1994a). Fast learning with incremental rbf networks. Neural Processing Letters, 1(1):2-5. Fritzke, B. (1994b). Growing cell structures - a self-organizing network for unsupervised and supervised learning. Neural Networks, 7(9):1441-1460. Fritzke, B. (1994c). Supervised learning with growing cell structures. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6, pages 255-262. Morgan Kaufmann Publishers, San Mateo, CA. Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43:59-69. Martinetz, T. M. (1993). Competitive Hebbian learning rule forms perfectly topology preserving maps. In ICANN'93: International Conference on Artificial Neural Networks, pages 427-434, Amsterdam. Springer. Martinetz, T. M. and Schulten, K J. (1991). A "neural-gas" network learns topologies. In Kohonen, T., Makisara, K, Simula, 0., and Kangas, J., editors, Artificial Neural Networks, pages 397-402. North-Holland, Amsterdam. Martinetz, T. M. and Schulten, K J. (1994). Topology representing networks. Neural Networks, 7(3):507-522. Oja, E. (1982). A simplified neuron model as a principal component analyzer. Journal of Mathematical Biology, 15:267-273. Sanger, T. D. (1989). An optimality principle for unsupervised learning. In Touretzky, D. S., editor, Advances in Neural Information Processing Systems 1, pages 11-19. Morgan Kaufmann, San Mateo, CA.	1994	118
879	Transformation Invariant Autoassociation with Application to Handwritten Character Recognition Holger Schwenk Maurice Milgram PARC Universite Pierre et Marie Curie tour 66-56, boite 164 4, place Jussieu, 75252 Paris cedex 05, France. e-mail: schwenk@robo.jussieu.fr Abstract When training neural networks by the classical backpropagation algorithm the whole problem to learn must be expressed by a set of inputs and desired outputs. However, we often have high-level knowledge about the learning problem. In optical character recognition (OCR), for instance, we know that the classification should be invariant under a set of transformations like rotation or translation. We propose a new modular classification system based on several autoassociative multilayer perceptrons which allows the efficient incorporation of such knowledge. Results are reported on the NIST database of upper case handwritten letters and compared to other approaches to the invariance problem. 1 INCORPORATION OF EXPLICIT KNOWLEDGE The aim of supervised learning is to learn a mapping between the input and the output space from a set of example pairs (input, desired output). The classical implementation in the domain of neural networks is the backpropagation algorithm. If this learning set is sufficiently representative of the underlying data distributions, one hopes that after learning, the system is able to generalize correctly to other inputs of the same distribution. 992 Holger Schwenk, Maurice Milgram It would be better to have more powerful techniques to incorporate knowledge into the learning process than the choice of a set of examples. The use of additional knowledge is often limited to the feature extraction module. Besides simple operations like (size) normalization, we can find more sophisticated approaches like zernike moments in the domain of optical character recognition (OCR). In this paper we will not investigate this possibility, all discussed classifiers work directly on almost non preprocessed data (pixels). In the context of OCR interest focuses on invariance of the classifier under a number of given transformations (translation, rotation, ... ) of the data to classify. In general a neural network could extract those properties of a large enough learning set, but it is very hard to learn and will probably take a lot of time. In the last years two main approaches for this invariance problem have been proposed: tangent-prop and tangent-distance. An indirect incorporation can be achieved by boosting (Drucker, Schapire and Simard, 1993). In this paper we briefly discuss these approaches and will present a new classification system which allows the efficient incorporation of transformation invariances. 1.1 TANGENT PROPAGATION The principle of tangent-prop is to specify besides desired outputs also desired changes jJJ. of the output vector when transforming the net input x by the transformations tJJ. (Simard, Victorri, LeCun and Denker, 1992). For this, let us define a transformation of pattern p as t(p, a) : P --t P where P is the space of all patterns and a a parameter. Such transformations are in general highly nonlinear operations in the pixel space P and their analytical expressions are seldom known. It is therefore favorable to use a first order approximation: tp _at(p, a) I t(p, a) :::::: p + atp with aa a=O tp is called the tangent vector. This definition can be generalized to c transformations: (1) t(p, Q) :::::: p + a] tp] + ... + a c tpc = P + TpQ (2) where Tp is a n x c matrix, each column corresponding to a tangent vector. Let us define R (x) the function calculated by the network. The desired behavior of the net outputs can be obtained by adding a regularization term Er to the objective function: E, = ~ k -aR(t;~.a)tJ '" ~ k -a~~x) t~ II' (3) ti is the tangent vector for transformation tJJ. of the input vector x and a R (x) / ax is the gradient of the network with respect to the inputs. Transformation invariance of the outputs is obtained by setting jJJ. = 0, so we want that aR (x) / ax is orthogonal to ti. Tangent-prop improved the learning time and the generalization on small databases, but its applicability to highly constraint networks (many shared weights) trained on large databases remains unknown. 1.2 TANGENT DISTANCE Another class of classifiers are memory based learning methods which rely on distance metrics. The incorporation of knowledge in such classifiers can be done by a distance Transformation Invariant Autoassociation 993 measure which is (locally) invariant under a set of specified transformations. (Simard, LeCun and Denker, 1993) define tangent distance as the minimal distance between the two hyperplanes spanned up by the tangent vectors Tp in point P and Tq in point q: Dpq(p, q) = ~i!J (p + Tpii - q - TqiJ) 2 = (p + Tpii* _ q _ TqiJ) 2 (4) 01,/3 The optimality condition is that the partial derivatives 8Dpq/8ii and 8Dpq/8iJ* should be zero. The values ii* and iJ* minimizing (4) can be computed by solving these two linear systems numerically. (Simard, LeCun and Denker, 1993) obtained very good results on handwritten digits and letters using tangent distance with a I-nearest-neighborclassifier (I-nn) . A big problem of every nn-classifier, however, is that it uses no compilation of the data and it needs therefore numerous reference vectors resulting in long classification time and high memory usage. Like reported in (Simard, 1994) and (Sperdutti and Stork, 1995) important improvements are possible, but often a trade-off between speed and memory usage must be made. 2 ARCHITECTURE OF THE CLASSIFIER The main idea of our approach is to use an autoassociative multilayer perceptron with a low dimensional hidden layer for each class to recognize. These networks, called diabolo network in the following, are trained only with examples of the corresponding class. This can be seen as supervised learning without counter-examples. Each network learns a hidden layer representation which preserves optimally the information of the examples of one class. These learned networks can now be used like discriminant functions: the reconstruction error is in general much lower for examples of the learned class than for the other ones. In order to build a classifier we use a decision module which interprets the distances between the reconstructed output vectors and the presented example. In our studies we have used until now a simple minimum operator which associates the class of the net with the smallest distance (Fig. 1). The figure illustrates also typical classification behavior, here when presenting a "D" out of the test set. One can see clearly that the distance of the network "D" is much lower than for the two other ones. The character is therefore correctly classified. It is also interesting to analyze the outputs of the two networks with the next nearest distances: the network "0" tries to output a more round character and the network "B" wants to add a horizontal bar in the middle. The basic classification architecture can be adapted in two ways to the problem to be solved. One on hand we can imagine different architectures for each diabolo network, e.g. several encoding/decoding layers which allow nonlinear dimension reduction. It is even possible to use shared weights realizing local feature detectors (see (Schwenk and Milgram, 1994) for more details). One the other hand we can change the underlying distance measure, as long as the derivatives with respect to the weights can be calculated. This offers a powerful and efficient mechanism to introduce explicit knowledge into the learning algorithm of a neural network. In the discussed case, the recognition of characters represented as pixel images, we can use a 994 character to classify Holger Schwenk, Maurice Milgram score B 8.07 A scoreD I--t-~ U. --~ input diabolo output vector networks vectors 4.49 score 0 8.54 distance measures Figure 1: Basic Architecture of a Diabolo Classifier decision module transformation invariant distance measure between the net output 0 and the desired output d (that is of course identical with the net input). The networks do not need to learn each example separately any more, but they can use the set of specified transformations in order to find a common non linear model of each class. The advantage of this approach, besides a better expected generalization behavior of course, is a very low additional complexity. In comparison to the original k-nn approach, and supposedly any possible optimization, we need to calculate only one distance measure for each class to recognize, regardless of the number of learning data. We used two different versions of tangent distance with increasing complexity: 1. single sided tangent distance: Dd(d,o) =mjn~ (d+Tda - o f = ~ ( d+Tda* -of (5) This is the minimal distance between the hyperplane spanned up by the tangent vectors Td in input vector d and the untransformed output vector o. 2. double sided tangent distance: Ddo(d,o) = mi!?! (d + Tda 0 * g - ToiJ) 2 (6) Ci,/3 2 The convolution of the net output with a Gaussian 9 is necessary for the computation of the tangent vectors To (the net input d is convolved during preprocessing). Figure 2 shows a graphical comparison of Euclidean distance with the two tangent distances. Transformation Invariant Autoassociation 995 d : desired output td : tangent vector in d o : net output to : tangent vector in 0 D : Euclidean distance D d : single sided tangent distance (only d is transformed) Ddo : double sided tangent distance (both points are transformed) V' D d : gradient of D d Figure 2: Comparison of Euclidean Distance with the Different Tangent Distances The major advantage of the single sided version is that we can now calculate easily the optimal multipliers &* and therefore the whole distance (the double sided version demands expensive matrix multiplications and the numerical solution of a system of linear equations). The optimality condition 8Dd(d, 0)/8&* ~ OT gives: (7) The tangent vectors Td and the matrix Ti} = (TITd)-1 can be precomputed and stored in memory. Note that it is the same for all diabolo networks, regardless of their class. 2.1 LEARNING ALGORITHM When using a tangent distance with an autoencoder we must calculate its derivatives with respect to the weights, i.e. after application of the chain rule with respect to the output vector o. In the case of the single sided tangent distance we get: (8) The resulting learning algorithm is therefore barely more complicated than with standard Euclidean error. Furthermore it has a pleasant graphical interpretation: the net output doesn't approach directly the desired output any more, but it takes the shortest way towards the tangent hyperplane (see also fig. 2). The derivation of the double sided tangent distance with respect to the net output is more complicated. In particular we must derivate the convolution of the net output with a Gaussian as well as the tangent vectors To. These equations will be published elsewhere. Training of the whole system is stopped when the error on the cross validation set reaches a minimum. Using stochastic gradient descent convergence is typically achieved after some ten iterations. 996 Holger Schwenk, Maurice Milgram 3 APPLICATION TO CHARACTER RECOGNITION In 1992 the National Institute of Standards and Technology provided a Database of handwritten digits and letters, known under the name NIST Special-Database 3. This database contains about 45000 upper case segmented characters which we have divided into learning and cross-validation set (60%) and test set (40%) respectively. We only applied a very simple preprocessing: the binary characters were centered and sizenormalized (the aspect-ratio was kept). The net input is 16 x 16 pixels with real-values. 3.1 EXPERIMENTAL RESULTS All the following results were obtained by fully connected diabolo networks with one low dimensional hidden layer, and a set of eight transformations (x- and y-translation, rotation, scaling, axis-deformation, diagonal-deformation, x- and y-thickness). Figure 3 illustrates how the networks use the transformations. , input Euclidean distance: 11 .1 optimally ttansformed input output '" tangent distance: 0.61 , output , input Euclidean distance: 20.0 optimally ttansformed input output ., tangent distance: 0.94 , output Figure 3: Reconstruction Examples (test set). The left side of each screen dump depicts the input character and the right side the one reconstructed by the network. In the middle, finally, one can see the optimally transformed patterns as calculated when evaluating the double sided tangent distance, i.e. transformed by a* and jJ* respectively. Although the "I.:' in the first example has an unusual short horizontal line, the network reconstructs a normally sized character. It is clearly visible how the input transformation lengthens and the output transformation shortens this line in order to get a small tangent distance. The right side shows a very difficult classification problem: a heavily deformed "T". Nevertheless we get a small tangent distance, so that the character is correctly classified. In summary we note a big difference between the Euclidean and the tangent distances, this is a good indicator that the networks really use the transformations. The performances on the whole test set of about 18 000 characters are summarized in figure 4. For comparison we give also the results of a one nearest neighbor classifier on the same test set. The incorporation of knowledge improved in both cases dramatically the performances. The diabolo classifier, for instance, achieves an error rate of 4.7 % with simple Euclidean distance which goes down to 3.7 % with the single sided and to only 2.6 % with the double sided tangent distance. In order to get the same results with the I-nn approach, the whole set of 27000 reference vectors had to be used. It's worth to note the results with less references: when using only 18 000 reference vectors the error rates increased to 3.7% for the single sided and to 2.8% for the double sided version respectively. Transformation Invariant Autoassociation % 55.3 F 2.5 I-nn (27 000 refs) 4.7 r;; 11 3.7 Diabolo 4.0 LeNet c Euclidean one sided two sided Figure 4: Raw Error Rate with NIST Upper Case Letters (test set) 997 In practical applications we are not only interested in low error rates, but we need also low computationally costs. An important factor is the recognition speed. The overall processing time of a diabolo classifier using the full tangent distance corresponds to the calculation of about 7 000 Euclidean and to less than 50 tangent distances. This should be less than for any algorithm of the k-nn family. If we assume the precalculation of all the tangent vectors and other expensive matrix multiplications, we could evaluate about 80 tangent distances, but the price would be exploding memory requirements. A diabolo classifier, on the other hand, needs only few memory: the storage of the weights corresponds to about 60 reference vectors per class. On a HP 715/50 workstation we obtained a recognition rate of 7.5 chis with the single sided and of more than 2.5 chis with the double sided tangent distance. We have also a method to combine both by rejection, resulting in up to 4 chis at the same low error rates (corresponds to the calculation of 32 double sided tangent distances). The table contains also the results of a large multilayer perceptron with extensive use of shared weights, known as LeNet. (Drucker, Schapire and Simard, 1993) give an error rate of 4.0% when used alone and of 2.4% for an ensemble of three such networks trained by boosting. The networks were trained on a basic set of 10 000 examples, the cross validation and test set consisted of 2 000 and 3 000 examples respectively (Drucker, personal communication). Due to the different number of examples, the results are perhaps not exactly comparable, but we can deduce nevertheless that the state of the art on this database seems to be around 2.5 %. 4 DISCUSSION We have proposed a new classification architecture that allows the efficient incorporation of knowledge into the learning algorithm. The system is easy to train and only one structural parameter must be chosen by the supervisor: the size of the hidden layer. It achieved state of the art recognition rates on the NIST database of handwritten upper case letters at a very low computational complexity. Furthermore we can say that a hardware implementation seems to be promising. Fully connected networks with only two layers are easy to put into standardized hardware chips. We could even propagate all diabolo networks in parallel. Speedups of several orders of magnitude should therefore be possible. 998 Holger Schwenk, Maurice Milgram On this year NIPS conference several authors presented related approaches. A comparable classification architecture was proposed by (Hinton, Revow and Dayan, 1995). Instead of one non-linear global model per class, several local linear models were used by performing separately principal component analysis (PCA) on subsets of each class. Since diabolo networks with one hidden layer and linear activation functions perform PCA, this architecture can be interpreted as an hierarchical diabolo classifier with linear nets and Euclidean distance. Such an hierarchisation could also be done with our classifier, i.e. with tangent distance and sigmoidal units, and might improve the results even further. (Hastie, Simard and Sackinger, 1995) developed an iterative algorithm that learns optimal reference vectors in the sense of tangent distance. An extension allows also to learn typical invariant transformations, i.e. tangent vectors, of each class. These two algorithms allowed to reduce drastically the number of reference vectors, but the results of the original approach couldn't be achieved no longer. Acknowledgements The first author is supported by the German Academic Exchange Service under grant HSP II 516.006.512.3. The simulations were performed with the Aspirin/ MIGRAINES neural network simulator developed by the MITRE Corporation. References H. Drucker, R. Schapire, and P. Simard (1993), "Boosting performance in neural networks," Int. Journal of Pattern Recognition and Artificial Intelligence, vol. 7, no. 4, pp. 705-719. T. Hastie, P. Simard and E. Sackinger (1995), "Learning prototype models for tangent distance," in NIPS 7 (G. Tesauro, D. Touretzky, and T. Leen, eds.), Morgan Kaufmann. G. Hinton, M. Revow, and P. Dayan (1995), "Recognizing handwritten digits using mixtures of linear models," in NIPS 7 (G. Tesauro, D. Touretzky, and T. Leen, eds.), Morgan Kaufmann. H. Schwenk and M. Milgram (1994), "Structured diabolo-networks for handwritten character recognition," in International Conference on Artificial Neural Networks, pp. 985-988, Springer-Verlag. P. Simard, B. Victorri, Y. LeCun, and 1. Denker (1992), "Tangent prop - a formalism for specifying selected invariances in an adaptive network," in NIPS 4 (1. Moody, S. Hanson, and R. Lippmann, eds.), pp. 895-903, Morgan Kaufmann. P. Simard, Y. LeCun, and J. Denker (1993), "Efficient pattern recognition using a new transformation distance," in NIPS 5 (S. Hanson, J. Cowan, and C. Giles, eds.), pp. 50-58, Morgan Kaufmann. P. Simard (1994), "Efficient Computation of complex distance measures using hierarchical filtering," in NIPS 6 (J.D. Cowan, G. Tesauro, and J. Alspector, eds.), pp. 50-58, Morgan Kaufmann. A. Sperdutti and D.G. Stork (1995), ''A rapid graph-based method for arbitrary transformation invariant pattern classification," in NIPS 7 (G. Tesauro, D. Touretzky, and T. Leen, eds.), Morgan Kaufmann.	1994	119
880	Learning Many Related Tasks at the Same Time With Backpropagation Rich Caruana School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 caruana@cs.cmu.edu Abstract Hinton [6] proposed that generalization in artificial neural nets should improve if nets learn to represent the domain's underlying regularities. Abu-Mustafa's hints work [1] shows that the outputs of a backprop net can be used as inputs through which domainspecific information can be given to the net. We extend these ideas by showing that a backprop net learning many related tasks at the same time can use these tasks as inductive bias for each other and thus learn better. We identify five mechanisms by which multitask backprop improves generalization and give empirical evidence that multi task backprop generalizes better in real domains. 1 INTRODUCTION You and I rarely learn things one at a time, yet we often ask our programs to-it must be easier to learn things one at a time than to learn many things at once. Maybe not. The things you and I learn are related in many ways. They are processed by the same sensory apparatus, controlled by the same physical laws, derived from the same culture, ... Perhaps it is the similarity between the things we learn that helps us learn so well. What happens when a net learns many related functions at the same time? Will the extra information in the teaching signal of the related tasks help it learn better? Section 2 describes five mechanisms that improve generalization in backprop nets trained simultaneously on related tasks. Section 3 presents empirical results from a road-following domain and an object-recognition domain where backprop with multiple tasks improves generalization 10-40%. Section 4 briefly discusses when and how to use multitask backprop. Section 5 cites related work and Section 6 outlines directions for future work. 658 Rich Caruana 2 MECHANISMS OF MULTITASK BACKPROP We identified five mechanisms that improve generalization in backprop nets trained simultaneously on multiple related tasks. The mechanisms all derive from the summing of error gradient terms at the hidden layer from the different tasks. Each exploits a different relationship between the tasks. 2.1 Data Amplification Data amplification is an effective increase in sample size due to extra information in the training signal of related tasks. There are two types of data amplification. 2.1.1 Statistical Data Amplification Statistical amplification, occurs when there is noise in the training signals. Consider two tasks, T and T', with independent noise added to their training signals, that both benefit from computing a feature F of the inputs. A net learning both T and T' can, if it recognizes that the two tasks share F, use the two training signals to learn F better by averaging F through the noise. The simplest case is when T = T', i.e., when the two outputs are independently corrupted versions of the same signal. 2.1.2 Blocking Data Amplification The 2nd form of data amplification occurs even if there is no noise. Consider two tasks, T and T', that use a common feature F computable from the inputs, but each uses F for different training patterns. A simple example is T = A OR F and T' = NOT(A) OR F. T uses F when A = 0 and provides no information about F when A = 1. Conversely, T' provides information about F only when A = 1. A net learning just T gets information about F only on training patterns for which A = 0, but is blocked when A = 1. But a net learning both T and T' at the same time gets information about F on every training pattern; it is never blocked. It does not see more training patterns, it gets more information for each pattern. If the net learning both tasks recognizes the tasks share F, it will see a larger sample of F. Experiments with blocked functions like T and T' (where F is a hard but learnable function of the inputs such as parity) indicate backprop does learn common subfeatures better due to the larger effective sample size. 2.2 Attribute Selection Consider two tasks, T and T', that use a common subfeature F. Suppose there are many inputs to the net, but F is a function of only a few of the inputs. A net learning T will,_ if there is limited training data and/or significant noise, have difficulty distinguishing inputs relevant to F from those irrelevant to it. A net learning both T and T', however, will better select the attributes relevant to F because data amplification provides better training signals for F and that allows it to better determine which inputs to use to compute F. (Note: data amplification occurs even when there is no attribute selection problem. Attribute selection is a consequence of data amplification that makes data amplification work better when a selection problem exists.) We detect attribute selection by looking for connections to relevant inputs that grow stronger compared to connections for irrelevant inputs when multiple tasks are trained on the net. Learning Many Related Tasks at the Same Time with Backpropagation 659 2.3 Eavesdropping Consider a feature F, useful to tasks, T and T', that is easy to learn when learning T, but difficult to learn when learning T' because T' uses F in a more complex way. A net learning T will learn F, but a net learning just T' may not. If the net learning T' also learns T, T' can eavesdrop on the hidden layer learned for T ( e.g., F) and thus learn better. Moreover, once the connection is made between T' and the evolving representation for F, the extra information from T' about F will help the net learn F better via the other mechanisms. The simplest case of eavesdropping is when T = F. Abu-Mostafa calls these catalytic hints[l]. In this case the net is being told explicitly to learn a feature F that is useful to the main task. Eavesdropping sometimes causes non-monotonic generalization curves for the tasks that eavesdrop on other tasks. This happens when the eavesdropper begins to overtrain, but then finds something useful learned by another task, and begins to perform better as it starts using this new information. 2.4 Representation Bias Because nets are initialized with random weights, backprop is a stochastic search procedure; multiple runs rarely yield identical nets. Consider the set of all nets (for fixed architecture) learnable by backprop for task T. Some of these generalize better than others because they better "represent" the domain's regularities. Consider one such regularity, F, learned differently by the different nets. Now consider the set of all nets learnable by backprop for another task T' that also learns regularity F. If T and T' are both trained on one net and the net recognizes the tasks share F, search will be biased towards representations of F near the intersection of what would be learned for T or T' alone. We conjecture that representations of F near this intersection often better capture the true regularity of F because they satisfy more than one task from the domain. Representations of F Findable by Backprop A form of representation bias that is easier to experiment with occurs when the representations for F sampled by the two tasks are different minima. Suppose there are two minima, A and B, a net can find for task T. Suppose a net learning task T' also has two minima, A and C. Both share the minima at A (i.e., both would perform well if the net entered that region of weight space), but do not overlap at Band C. We ran two experiments. In the first, we selected the minima so that nets trained on T alone are equally likely to find A or B, and nets trained on T' alone are equally likely to find A or C. Nets trained on both T and T' usually fall into A for both tasks. 1 Tasks prefer hidden layer representations that other tasks prefer. In the second experiment we selected the minima so that T has a strong preference lIn these experiments the nets have sufficient capacity to find independent minima for the tasks. They are not forced to share the hidden layer representations. But because the initial weights are random, they do initially share the hidden layer and will separate the tasks (i.e., use independent chunks of the hidden layer for each task) only if learning causes them to. 660 Rich Caruana for B over A: a net trained on T always falls into B. T', however, still has no preference between A or C. When both T and T' are trained on one net, T falls into B as expected: the bias from T' is unable to pull it to A. Surprisingly, T' usually falls into C, the minima it does not share with T! T creates a "tide" in the hidden layer representation towards B that flows away from A. T' has no preference for A or C, but is subject to the tide created by T. Thus T' usually falls into C; it would have to fight the tide from T to fall into A. Tasks prefer NOT to use hidden layer representations that other tasks prefer NOT to use. 2.5 How the Mechanisms are Related The "tide" mentioned while discussing representation bias results from the aggregation of error gradients from multiple tasks at the hidden layer. It is what makes the five mechanisms tick. It biases the search trajectory towards better performing regions of weight space. Because the mechanisms arise from the same underlying cause, it easy for them to act in concert. Their combined effect can be substantial. Although the mechanisms all derive from gradient summing, they are not the same. Each emphasizes a different relationship between tasks and has different effects on what is learned. Changes in architecture, representation, and the learning procedure affect the mechanisms in different ways. One particularly noteworthy difference between the mechanisms is that if there are minima, representation bias affects learning even with infinite sample size. The other mechanisms work only with finite sample size: data amplification (and thus attribute selection) and eavesdropping are beneficial only when the sample size is too small for the training signal for one task to provide enough information to the net for it to learn good models. 3 EMPIRICAL RESULTS Experiments on carefully crafted test problems verify that each of the mechanisms can work. 2 These experiments, however, do not indicate how effective multitask backprop is on real problems: tweaking the test problems alters the size of the effects. Rather than present results for contrived problems, we present a more convincing demonstration of multi task backprop by testing it on two realistic domains. 3.1 1D-ALVINN ID-ALVINN uses a road image simulator developed by Pomerleau. It was modified to generate I-D road images comprised of a single 32-pixel horizontal scan line instead of the original 2-D 30x32-pixel image. This was done to speed learning to allow thorough experimentation. ID-ALVINN retains much of the complexity of the original 2-D domain-the complexity lost is road curvature and that due to the smaller input (960 pixels vs. 32 pixels). The principle task in ID-ALVINN is to predict steering direction. Eight additional tasks were used for multitask backprop: • whether the road is one or two lanes • location of centerline (2-lane roads only) • location of left edge of road • location of right edge of road • location of road center • intensity of road surface • intensity of region bordering road • intensity of centerline (2-lane roads only) 2We have yet to determine how to directly test the hypothesis that representations for F in the intersection of T and T' perform better. Testing this requires interpreting representations learned for real tasks; experiments on test problems demonstrate only that search is biased towards the intersection, not that the intersection is the right place to be. Learning Many Related Tasks at the Same Time with Backpropagation 661 Table 1 shows the performance of single and multitask backprop (STB and MTB, respectively) on 1D-ALVINN using nets with one hidden layer. The MTB net has 32 inputs, 16 hidden units, and 9 outputs. The 36 STB nets have 32 inputs, 2, 4, 8 or 16 hidden units, and 1 output. A similar experiment using nets with 2 hidden layers containing 2, 4, 8, 16, or 32 hidden units per layer for STB and 32 hidden units per layer for MTB yielded comparable results. The size of the MTB nets is not optimized in either experiment. Table 1: Performance of STB and MTB with One Hidden Layer on 1D-ALVINN II ROOT-MEAN SQUARED ERROR ON TEST SET TASK Single Task Backprop MTB % Change % Change 2HU T 4HU I 8HU f 16HU 16HU Best STB Mean STB 1 or 2 Lanes .201 .209 .207 .178 .156 14.1 27.4 Left Edge .069 .071 .073 .073 .062 11.3 15.3 Right Edge .076 .062 .058 .056 .051 9.8 23.5 Line Center .153 .152 .152 .152 .151 0.7 0.8 Road Center .038 .037 .039 .042 .034 8.8 14.7 Road Greylevel .054 .055 .055 .054 .038 42.1 43.4 Edge Greylevel .037 .038 .039 .038 .038 -2.6 0.0 Line Greylevel .054 .054 .054 .054 .054 0.0 0.0 Steering .093 .069 .087 .072 .058 19.0 38.4 The entries under the STB and MTB headings are the peak generalization error for nets of the specified size. The italicized STB entries are the STB runs that yielded best performance. The last two columns compare STB and MTB. The first is the percent difference between MTB and the best STB run. Positive percentages indicate MTB performs better. This test is biased towards STB because it compares a single run of MTB on an unoptimized net size with several independent runs of STB that use different random seeds and are able to find near-optimal net size. The last column is the percent difference between MTB and the average STB. Note that on the important steering task, MTB outperforms STB 20-40%. 3.2 ID-DOORS To test multitask backprop on a real problem, we created an object recognition domain similar in some respects to 1D-ALVINN. In 1D-DOORS the main tasks are to locate doorknobs and recognize door types (single or double) in images of doors collected with a robot-mounted camera. Figure 1 shows several door images. As with 1D-ALVINN, the problem was simplified by using horizontal stripes from image, one for the green channel and one for the blue channel. Each stripe is 30 pixels wide (accomplished by applying Gaussian smoothing to the original 150 pixel wide image) and occurs at the vertical location of the doorknob. 10 tasks were used: • horizontal location of doorknob • single or double door • horizontal location of doorway center • width of doorway • horizontal location of left door jamb • horizontal location of right door jamb • width of left door jamb • width of right door jamb • horizontal location of left edge of door • horizontal location of right edge of door The difficulty of 1D-DOORS precludes running as exhaustive a set of experiments as with 1D-ALVINN: runs were done only for the two most important and difficult tasks: doorknob location and door type. STB was tested on nets using 6, 24, and 96 hidden units. MTB was tested on a net with 120 hidden units. The results 662 Rich Caruana Figure 1: Sample Doors from the ID-DOORS Domain are in Table 2. STB generalizes 35-45% worse than MTB on these tasks. Less thorough experiments on the other eight tasks suggest MTB probably always yields performance equal to or better than STB. Table 2: Performance of STB and MTB on ID-DOORS. RMS ERROR ON TEST SET TASK Doorknob Loc Door Type 4 DISCUSSION In our experience, multitask backprop usually generalizes better than single task backprop. The few cases where STB has been better is on simpler tasks, and there the difference between MTB and STB was small. Multitask backprop appears to provide the most benefit on hard tasks. MTB also usually learns in fewer epochs than STB. When all tasks must be learned, MTB is computationally more efficient than training single nets. When few tasks are important, however, STB is usually more efficient (but also less accurate). Tasks do not always learn at the same rate. It is important to watch the training curve of each MTB task individually and stop training each task when its performance peaks. The easiest way to do this to take a snapshot of the net when performance peaks on a task of interest. MTB does not mean one net should be used to predict all tasks, only that all tasks should be trained on one net so they may benefit each other. Do not treat tasks as one task just because they are being trained on one net! Balancing tasks (e.g., using different learning rates for different outputs) sometimes helps tasks learn at similar rates, thus maximizing the potential benefits of MTB. Also, because the training curves for MTB are often more complex due to interactions between tasks (MTB curves are frequently multimodal), it is important to train MTB nets until all tasks appear to be overtraining. Restricting the capacity of MTB nets to force sharing or prevent overtraining usually hurts performance instead of helping it. MTB does not depend on restricted net capacity. We created the extra tasks in ID-ALVINN and ID-DOORS specifically because we thought they would improve performance on the important tasks. Multitask backprop can be used in other ways. Often the world gives us related tasks to learn. For example, the Calendar Apprentice System (CAP)[4] learns to predict the Location, Time_Of _Day, Day_Of _Week, and Duration of the meetings it schedules. These tasks are functions of the same data, share many common features, Learning Many Related Tasks at the Same Time with Backpropagation 663 and would be easy to learn together. Sometimes the world gives us related tasks in mysterious ways. For example, in a medical domain we are examining where the goal is to predict illness severity, half of the lab tests are cheap and routinely measured before admitting a patient (e.g., blood pressure, pulse, age). The rest are expensive tests requiring hospitalization. Users tell us it would be useful to predict if the severity of the illness warrants admission (and further testing) using just the pre-admission tests. Rather than ignore the most diagnostically useful information in the database, we use the expensive tests as additional tasks the net must learn. They are not very predictable from the simple pre-admission tests, but providing them to the net as outputs helps it learn illness severity better. Multitask backprop is one way of providing to a net information that at run time would only be available in the future. The training signals are needed only for the training set because they are outputs-not inputs-to the net. 5 RELATED WORK Training nets with many outputs is not new; NETtalk [9] used one net to learn phonemes and stress. This approach was natural for NETtalk where the goal was to control a synthesizer that needed both phoneme and stress commands at the same time. No analysis, however, was made of the advantages of using one net for all the tasks3 , and the different outputs were not treated as independent tasks. For example, the NETtalk stress task overtrains badly long before the phoneme task is learned well, but NETtaik did not use different snapshots of the net for different tasks. NETtalk also made no attempt to balance tasks so that they would learn at a similar rate, or to add new tasks that might improve learning but which would not be useful for controlling the synthesizer. Work has been done on serial transfer between nets [8]. Improved learning speed was reported, but not improved generalization. The key difficulties with serial transfer are that it is difficult to scale to many tasks, it is hard to prevent catastrophic interference from erasing what was learned previously, the learning sequence must be defined manually, and serial learning precludes mutual benefit between tasks. This work is most similar to catalytic hints [1][10] where extra tasks correspond to important learnable features of a main task. This work extends hints by showing that tasks can be related in more diverse ways, by expanding the class of mechanisms responsible for multitask backprop, by showing that capacity restriction is not an important mechanism for multitask backprop [2], and by demonstrating that creating many new related tasks may be an efficient way of providing domain-specific inductive bias to backprop nets. 6 FUTURE WORK We used vanilla backprop to show the benefit of training many related tasks on one net. Additional techniques may enhance the effects. Regularization and incremental net growing procedures might improve performance by promoting sharing without restricting capacity. New techniques may also be necessary to enhance the benefit of multitask backprop. Automatic balancing of task learning rates would make MTB easier to use. It would also be valuable to know when the different MTB mechanisms are working-they might be useful in different kinds of domains and might benefit from different regularization or balancing techniques. Finally, although MTB usually seems to help and rarely hurts, the only way to know it 3See [5] for evidence that NETtalk is harder to learn using separate nets. 664 Rich Caruana helps is to try it. It would be better to have a predictive theory of how tasks should relate to benefit MTB, particularly if new tasks are to be created only to provide a multitask benefit for the other important tasks in the domain. 7 SUMMARY Five mechanisms that improve generalization performance on nets trained on multiple related tasks at the same time have been identified. These mechanisms work without restricting net capacity or otherwise reducing the net's VC-dimension. Instead, they exploit backprop's ability to combine the error terms for related tasks into an aggregate gradient that points towards better underlying represen tations. Multitask backprop was tested on a simulated domain, ID-ALVINN, and on a real domain, ID-DOORS. It improved generalization performance on hard tasks in these domains 20-40% compared with the best performance that could be obtained from multiple trials of single task backprop. Acknowledgements Thanks to Tom Mitchell, Herb Simon, Dean Pomerleau, Tom Dietterich, Andrew Moore, Dave Touretzky, Scott Fahlman, Sebastian Thrun, Ken Lang, and David Zabowski for suggestions that have helped shape this work. This research is sponsored in part by the Advanced Research Projects Agency (ARPA) under grant no. F33615-93-1-1330. References [1] Y.S. Abu-Mostafa, "Learning From Hints in Neural Networks," Journal of Complexity 6:2, pp. 192-198,1989. [2] Y.S. Abu-Mostafa, "Hints and the VC Dimension," Neural Computation, 5:2, 1993. [3] R. Caruana, "Multitask Connectionist Learning," Proceedings of the 1993 Connectionist Models Summer School, pp. 372-379, 1993. [4] L. Dent, J. Boticario, J. McDermott, T. Mitchell, and D. Zabowski, "A Personal Learning Apprentice," Proceedings of 1992 National Conference on Artificial Intelligence, 1992. [5] T.G. Dietterich, H. Hild, and G. Bakiri, "A Comparative Study of ID3 and Backpropagation for English Text-to-speech Mapping," Proceedings of the Seventh International Conference on Artificial Intelligence, pp. 24-31, 1990. [6] G.E. Hinton, "Learning Distributed Representations of Concepts," Proceedings of the Eight International Conference of The Cognitive Science Society, pp. 112, 1986. [7] D.A. Pomerleau, "Neural Network Perception for Mobile Robot Guidance," Carnegie Mellon University: CMU-CS-92-115, 1992. [8] L.Y. Pratt, J. Mostow, and C.A. Kamm, "Direct Transfer of Learned Information Among Neural Networks," Proceedings of AAAI-91, 1991. [9] T.J. Sejnowski and C.R. Rosenberg, "NETtalk: A Parallel Network that Learns to Read Aloud," John Hopkins: JHU/EECS-8'6/01, 1986. [10] S.C. Suddarth and A.D.C. Holden, "Symbolic-neural Systems and the Use of Hints for Developing Complex Systems," International Journal of MaxMachine Studies 35:3, pp. 291-311, 1991.	1994	12
881	Learning To Play the Game of Chess Sebastian Thrun University of Bonn Department of Computer Science III Romerstr. 164, 0-53117 Bonn, Germany E-mail: thrun@carbon.informatik.uni-bonn.de Abstract This paper presents NeuroChess, a program which learns to play chess from the final outcome of games. NeuroChess learns chess board evaluation functions, represented by artificial neural networks. It integrates inductive neural network learning, temporal differencing, and a variant of explanation-based learning. Performance results illustrate some of the strengths and weaknesses of this approach. 1 Introduction Throughout the last decades, the game of chess has been a major testbed for research on artificial intelligence and computer science. Most oftoday's chess programs rely on intensive search to generate moves. To evaluate boards, fast evaluation functions are employed which are usually carefully designed by hand, sometimes augmented by automatic parameter tuning methods [1]. Building a chess machine that learns to play solely from the final outcome of games (win/loss/draw) is a challenging open problem in AI. In this paper, we are interested in learning to play chess from the final outcome of games. One of the earliest approaches, which learned solely by playing itself, is Samuel's famous checker player program [10]. His approach employed temporal difference learning (in short: TO) [14], which is a technique for recursively learning an evaluation function. Recently, Tesauro reported the successful application of TO to the game of Backgammon, using artificial neural network representations [16]. While his TO-Gammon approach plays grandmaster-level backgammon, recent attempts to reproduce these results in the context of Go [12] and chess have been less successful. For example, Schafer [11] reports a system just like Tesauro's TO-Gammon, applied to learning to play certain chess endgames. Gherrity [6] presented a similar system which he applied to entire chess games. Both approaches learn purely inductively from the final outcome of games. Tadepalli [15] applied a lazy version of explanation-based learning [5, 7] to endgames in chess. His approach learns from the final outcome, too, but unlike the inductive neural network approaches listed above it learns analytically, by analyzing and generalizing experiences in terms of chess-specific knowledge. 1070 Sebastian Thrun The level of play reported for all these approaches is still below the level of GNU-Chess, a publicly available chess tool which has frequently been used as a benchmark. This illustrates the hardness of the problem of learning to play chess from the final outcome of games. This paper presents NeuroChess, a program that learns to play chess from the final outcome of games. The central learning mechanisms is the explanation-based neural network (EBNN) algorithm [9, 8]. Like Tesauro's TD-Gammon approach, NeuroChess constructs a neural network evaluation function for chess boards using TO. In addition, a neural network version of explanation-based learning is employed, which analyzes games in terms of a previously learned neural network chess model. This paper describes the NeuroChess approach, discusses several training issues in the domain of chess, and presents results which elucidate some of its strengths and weaknesses. 2 Temporal Difference Learning in the Domain of Chess Temporal difference learning (TO) [14] comprises a family of approaches to prediction in cases where the event to be predicted may be delayed by an unknown number of time steps. In the context of game playing, TD methods have frequently been applied to learn functions which predict the final outcome of games. Such functions are used as board evaluation functions. The goal of TO(O), a basic variant of TO which is currently employed in the NeuroChess approach, is to find an evaluation function, V, which ranks chess boards according to their goodness: If the board S is more likely to be a winning board than the board Sf, then V(s) > V(Sf). To learn such a function, TO transforms entire chess games, denoted by a sequence of chess boards So, SI, s2, . . . , StunaJ' into training patterns for V. The TO(O) learning rule works in the following way. Assume without loss of generality we are learning white's evaluation function. Then the target values for the final board is given by { I, if Stu.»tI is a win for white 0, if StUnaJ is a draw -1, if StonaJ is a loss for white and the targets for the intermediate chess boards So, SI , S2, . .. , Stu.»tI-2 are given by Vt.1fget( St) = I· V (St+2) (1) (2) This update rule constructs V recursively. At the end of the game, V evaluates the final outcome of the game (Eq. (l ». In between, when the assignment of V -values is less obvious, V is trained based on the evaluation two half-moves later (Eq. (2». The constant I (with o ~ I ~ 1) is a so-called discount factor. It decays V exponentially in time and hence favors early over late success. Notice that in NeuroChess V is represented by an artificial neural network, which is trained to fit the target values vtarget obtained via Eqs. (l) and (2) (cj [6, 11, 12, 16]). 3 Explanation-Based Neural Network Learning In a domain as complex as chess, pure inductive learning techniques. such as neural network Back-Propagation, suffer from enormous training times. To illustrate why, consider the situation of a knight fork. in which the opponent's knight attacks our queen and king simultaneously. Suppose in order to save our king we have to move it, and hence sacrifice our queen. To learn the badness of a knight fork, NeuroChess has to discover that certain board features (like the position of the queen relative to the knight) are important, whereas Learning to Play the Game of Chess 1071 Figure 1: Fitting values and slopes in EBNN: Let V be the target function for which three examples (s\, V(S\)), (S2' V(S2)), and (S3, V(S3)) are known. Based on these points the learner might generate the hypothesis V'. If the slopes a~;:I), ar S2)OS2, and a~;:3) are also known, the learner can do much better: V". others (like the number of weak pawns) are not. Purely inductive learning algorithms such as Back-propagation figure out the relevance of individual features by observing statistical correlations in the training data. Hence, quite a few versions of a knight fork have to be experienced in order to generalize accurately. In a domain as complex as chess, such an approach might require unreasonably large amounts of training data. Explanation-based methods (EBL) [5, 7, 15] generalize more accurately from less training data. They rely instead on the availability of domain knowledge, which they use for explaining and generalizing training examples. For example, in the explanation of a knight fork, EBL methods employ knowledge about the game of chess to figure out that the position of the queen is relevant, whereas the number of weak pawns is not. Most current approaches to EBL require that the domain knowledge be represented by a set of symbolic rules. Since NeuroChess relies on neural network representations, it employs a neural network version of EBL, called explanation-based neural network learning (EBNN) [9]. In the context of chess, EBNN works in the following way: The domain-specific knowledge is represented by a separate neural network, called the chess model M. M maps arbitrary chess boards St to the corresponding expected board St+2 two half-moves later. It is trained prior to learning V, using a large database of grand-master chess games. Once trained, M captures important knowledge about temporal dependencies of chess board features in high-quality chess play. EBNN exploits M to bias the board evaluation function V. It does this by extracting slope constraints for the evaluation function V at all non-final boards, i.e., all boards for which V is updated by Eq. (2). Let with t E {a, 1,2, ... , tlioa\ - 2} (3) denote the target slope of V at St, which, because vtarget( St) is set to 'Y V (St+2) according Eq. (2), can be rewritten as oV target ( St) oV( St+2) OSt+2 = 'Y. ._OSt+2 OSt (4) using the chain rule of differentiation. The rightmost term in Eq. (4) measures how infinitesimal small changes of the chess board St influence the chess board St+2. It can be approximated by the chess model M: ovtarget(St) OV(St+2) oM(st) ~ 'Y. . (5) OSt OSt+2 OSt The right expression is only an approximation to the left side, because M is a trained neural 1072 ~ bmrd at time f (W"T"~) board attime 1+ I (black to move) predictive model network M 165 hidden unit, Sebastian Thrun ~ ~ board at time 1+2 (w"'·ro~) V(1+2) Figure 2: Learning an evaluation function in NeuroChess. Boards are mapped into a high-dimensionalJeature vector, which forms the input for both the evaluation network V and the chess model M. The evaluation network is trained by Back-propagation and the TD(O) procedure. Both networks are employed for analyzing training example in order to derive target slopes for V. network and thus its first derivative might be erroneous. Notice that both expressions on the right hand side of Eq. (5) are derivatives of neural network functions, which are easy to compute since neural networks are differentiable. The result of Eq. (5) is an estimate of the slope of the target function V at 8t . This slope adds important shape information to the target values constructed via Eq. (2). As depicted in Fig. 1, functions can be fit more accurately if in addition to target values the slopes of these values are known. Hence, instead of just fitting the target values vtarget( 8t), NeuroChess also fits these target slopes. This is done using the Tangent-Prop algorithm [13]. The complete NeuroChess learning architecture is depicted in Fig. 2. The target slopes provide a first-order approximation to the relevance of each chess board feature in the goodness of a board position. They can be interpreted as biasing the network V based on chess-specific domain knowledge, embodied in M . For the relation ofEBNN and EBL and the accommodation of inaccurate slopes in EBNN see [8]. 4 Training Issues In this section we will briefly discuss some training issues that are essential for learning good evaluation functions in the domain of chess. This list of points has mainly been produced through practical experience with the NeuroChess and related TD approaches. It illustrates the importance of a careful design of the input representation, the sampling rule and the Learning to Play the Game of Chess 1073 parameter setting in a domain as complex as chess. Sampling. The vast majority of chess boards are, loosely speaking, not interesting. If, for example, the opponent leads by more than a queen and a rook, one is most likely to loose. Without an appropriate sampling method there is the danger that the learner spends most of its time learning from uninteresting examples. Therefore, NeuroChess interleaves selfplay and expert play for guiding the sampling process. More specifically, after presenting a random number of expert moves generated from a large database of grand-master games, NeuroChess completes the game by playing itself. This sampling mechanism has been found to be of major importance to learn a good evaluation function in a reasonable amount of time. Quiescence. In the domain of chess certain boards are harder to evaluate than others. For example, in the middle of an ongoing material exchange, evaluation functions often fail to produce a good assessment. Thus, most chess programs search selectively. A common criterion for determining the depth of search is called quiescence. This criterion basically detects material threats and deepens the search correspondingly. NeuroChess' search engine does the same. Consequently, the evaluation function V is only trained using quiescent boards. Smoothness. Obviously, using the raw, canonical board description as input representation is a poor choice. This is because small changes on the board can cause a huge difference in value, contrasting the smooth nature of neural network representations. Therefore, NeuroChess maps chess board descriptions into a set of board features. These features were carefully designed by hand. Discounting. The variable 'Y in Eq. (2) allows to discount values in time. Discounting has frequently been used to bound otherwise infinite sums of pay-off. One might be inclined to think that in the game of chess no discounting is needed, as values are bounded by definition. Indeed, without discounting the evaluation function predicts the probability for winning-in the ideal case. In practice, however, random disturbations of the evaluation function can seriously hurt learning, for reasons given in [4, 17]. Empirically we found that learning failed completely when no discount factor was used. Currently, NeuroChess uses 'Y = 0.98. Learning rate. TO approaches minimize a Bellman equation [2]. In the NeuroChess domain, a close-to-optimal approximation of the Bellman equation is the constant function V(s) == O. This function violates the Bellman equation only at the end of games (Eq. (1», which is rare if complete games are considered. To prevent this, we amplified the learning rate for final values by a factor of20, which was experimentally found to produce sufficiently non-constant evaluation functions. Software architecture. Training is performed completely asynchronously on up to 20 workstations simultaneously. One of the workstations acts as a weight server, keeping track of the most recent weights and biases of the evaluation network. The other workstations can dynamically establish links to the weight server and contribute to the process of weight refinement. The main process also monitors the state of all other workstations and restarts processes when necessary. Training examples are stored in local ring buffers (1000 items per workstation). 5 Results In this section we will present results obtained with the NeuroChess architecture. Prior to learning an evaluation function, the model M (175 input, 165 hidden, and 175 output units) is trained using a database of 120,000 expert games. NeuroChess then learns an evaluation 1074 Sebastian Thrun I. e2e3 b8c6 16. b2b4 a5a4 31 . a3f8 f2e4 46. d I c2 b8h2 61 . e4f5 h3g4 65. a8e8 e6d7 2. dlf3 c6e5 17. b5c6 a4c6 32. c3b2 h8f8 47. c2c3 f6b6 62. f5f6 h6h5 66. e8e7 d7d8 3. f3d5 d7d6 18. gl f3 d8d6 33. a4d7 f3f5 48. e7e4 g6h6 63. b7b8q g4f5 67. f4c7 4. flb5 c7c6 19. d4a7 f5g4 34. d7b7 f5e5 49. d4f5 h6g5 64. b8f4 f5e6 5. b5a4 g8f6 20. c2c4 c8d7 35. b2cl f8e8 50. e4e7 g5g4 6. d5d4 c8f5 21. b4b5 c6c7 36. b7d5 e5h2 51. f5h6 g7h6 final board 7. f2f4 e5d7 22. d2d3 d6d3 37. ala7 e8e6 52. e7d7 g4h5 8. ele2d8a5 23. b5b6 c7c6 38. d5d8 f6g6 53. d7d I h5h4 9. a4b3 d7c5 24. e2d3 e4f2 39. b6b7 e6d6 54. d I d4 h4h3 10. b I a3 c5b3 25. d3c3 g4f3 40. d8a5 d6c6 55. d4b6 h2e5 11 . a2b3 e7e5 26. g2f3 f2h 1 41 . a5b4 h2b8 56. b6d4 e5e6 12. f4e5 f6e4 27. clb2 c6f3 42. a7a8 e4c3 57. c3d2 e6f5 13. e5d6 e8c8 28. a7a4 d7e7 43. c2d4 c6f6 58. e3e4 f5 g5 14. b3b4 a5a6 29. a3c2 hi f2 44. b4e7 c3a2 59. d4e3 g5e3 15. b4b5 a6a5 30. b2a3 e7f6 45. cldl a2c3 60. d2e3 f7f5 Figure 3: NeuroChess against GNU-Chess. NeuroChess plays white. Parameters: Both players searched to depth 3, which could be extended by quiescence search to at most 11. The evaluation network had no hidden units. Approximately 90% of the training boards were sampled from expert play. network V (175 input units, 0 to 80 hidden units, and one output units). To evaluate the level of play, NeuroChess plays against GNU-Chess in regular time intervals. Both players employ the same search mechanism which is adopted from GNU-Chess. Thus far, experiments lasted for 2 days to 2 weeks on I to 20 SUN Sparc Stations. A typical game is depicted in Fig. 3. This game has been chosen because it illustrates both the strengths and the shortcomings of the NeuroChess approach. The opening of NeuroChess is rather weak. In the first three moves NeuroChess moves its queen to the center of the board.' NeuroChess then escapes an attack on its queen in move 4, gets an early pawn advantage in move 12, attacks black's queen pertinaciously through moves 15 to 23, and successfully exchanges a rook. In move 33, it captures a strategically important pawn, which, after chasing black's king for a while and sacrificing a knight for no apparent reason, finally leads to a new queen (move 63). Four moves later black is mate. This game is prototypical. As can be seen from this and various other games, NeuroChess has learned successfully to protect its material, to trade material, and to protect its king. It has not learned, however, to open a game in a coordinated way, and it also frequently fails to play short. endgames even if it has a material advantage (this is due to the short planning horizon). Most importantly, it still plays incredibly poor openings, which are often responsible for a draw or a loss. Poor openings do not surprise, however, as TD propagates values from the end of a game to the beginning. Table I shows a performance comparison of NeuroChess versus GNU-Chess, with and without the explanation-based learning strategy. This table illustrates that NeuroChess wins approximately 13% of all games against GNU-Chess, if both use the same search engine. It 'This is because in the current version NeuroChess still heavily uses expert games for sampling. Whenever a grand-master moves its queen to the center of the board, the queen is usually safe, and there is indeed a positive correlation between having the queen in the center and winning in the database. NeuroChess falsely deduces that having the queen in the center is good. This effect disappears when the level of self-play is increased, but this comes at the expense of drastically increased training time, since self-play requires search. Learning to Play the Game of Chess 1075 GNU depth 2, NeuroChess depth 2 GNU depth 4, NeuroChess depth 2 # of games Back-propagation EBNN Back-propagation EBNN 100 1 0 0 0 200 6 2 0 0 500 35 13 I 0 1000 73 85 2 1 1500 130 135 3 3 2000 190 215 3 8 2400 239 316 3 II Table 1: Performance ofNeuroChess vs. GNU-Chess during training. The numbers show the total number of games won against GNU-Chess using the same number of games for testing as for training. This table also shows the importance of the explanation-based learning strategy in EBNN. Parameters: both learners used the original GNU-Chess features, the evaluation network had 80 hidden units and search was cut at depth 2, or 4, respectively (no quiescence extensions). also illustrates the utility of explanation-based learning in chess. 6 Discussion This paper presents NeuroChess, an approach for learning to play chess from the final outcomes of games. NeuroChess integrates TD, inductive neural network learning and a neural network version of explanation-based learning. The latter component analyzes games using knowledge that was previously learned from expert play. Particular care has been taken in the design of an appropriate feature representation, sampling methods, and parameter settings. Thus far, NeuroChess has successfully managed to beat GNU-Chess in several hundreds of games. However, the level of play still compares poorly to GNU-Chess and human chess players. Despite the initial success, NeuroChess faces two fundamental problems which both might weB be in the way of excellent chess play. Firstly, training time is limited, and it is to be expected that excellent chess skills develop only with excessive training time. This is particularly the case if only the final outcomes are considered. Secondly, with each step of TO-learning NeuroChess loses information. This is partially because the features used for describing chess boards are incomplete, i.e., knowledge about the feature values alone does not suffice to determine the actual board exactly. But, more importantly, neural networks have not the discriminative power to assign arbitrary values to all possible feature combinations. It is therefore unclear that a TD-like approach will ever, for example, develop good chess openmgs. Another problem of the present implementation is related to the trade-off between knowledge and search. It has been well recognized that the ul timate cost in chess is determi ned by the ti me it takes to generate a move. Chess programs can generally invest their time in search, or in the evaluation of chess boards (search-knowledge trade-off) [3]. Currently, NeuroChess does a poor job, because it spends most of its time computing board evaluations. Computing a large neural network function takes two orders of magnitude longer than evaluating an optimized linear evaluation function (like that of GNU-Chess). VLSI neural network technology offers a promising perspective to overcome this critical shortcoming of sequential neural network simulations. 1076 Sebastian Thrun Acknowledgment The author gratefully acknowledges the guidance and advise by Hans Berliner, who provided the features for representing chess boards, and without whom the current level of play would be much worse. He also thanks Tom Mitchell for his suggestion on the learning methods, and Horst Aurisch for his help with GNU-Chess and the database. References [I] Thomas S. Anantharaman. A Statistical Study of Selective Min-Max Search in Computer Chess. PhD thesis, Carnegie Mellon University, School of Computer Science, Pittsburgh, PA, 1990. Technical Report CMU-CS-90-173. [2] R. E. Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, 1957. [3] Hans J. Berliner, Gordon Goetsch, Murray S. Campbell, and Carl Ebeling. Measuring the performance potential of chess programs. Artificial Intelligence, 43:7-20, 1990. [4] Justin A. Boyan. Generalization in reinforcement learning: Safely approximating the value function. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Systems 7, San Mateo, CA, 1995. Morgan Kaufmann. (to appear). [5] Gerald Dejong and Raymond Mooney. Explanation-based learning: An alternative view. Machine Learning, 1(2): 145-176, 1986. [6] Michael Gherrity. A Game-Learning Machine. PhD thesis, University of California, San Diego, 1993. [7] Tom M. Mitchell, Rich Keller, and Smadar Kedar-Cabelli. Explanation-based generalization: A unifying view. Machine Learning, 1 (1 ):47-80, 1986. [8] Tom M. Mitchell and Sebastian Thrun. Explanation based learning: A comparison of symbolic and neural network approaches. In Paul E. Utgoff, editor, Proceedings of the Tenth International Conference on Machine Learning, pages 197-204, San Mateo, CA, 1993. Morgan Kaufmann. [9] Tom M. Mitchell and Sebastian Thrun. Explanation-based neural network learning for robot control. In S. J. Hanson, J. Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 287-294, San Mateo, CA, 1993. Morgan Kaufmann. [10] A. L. Samuel. Some studies in machine learning using the game of checkers. IBM Journal on research and development, 3:210-229, 1959. [11] Johannes Schafer. Erfolgsorientiertes Lemen mit Tiefensuche in Bauemendspielen. Technical report, UniversiUit Karlsruhe, 1993. (in German). [12] Nikolaus Schraudolph, Pater Dayan, and Terrence J. Sejnowski. Using the TD(lambda) algorithm to learn an evaluation function for the game of go. In Advances in Neural Information Processing Systems 6, San Mateo, CA, 1994. Morgan Kaufmann. [13] Patrice Simard, Bernard Victorri, Yann LeCun, and John Denker. Tangent prop -a formalism for specifying selected invariances in an adaptive network. In J. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 895-903, San Mateo, CA, 1992. Morgan Kaufmann. [14] Richard S. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3,1988. [15] Prasad Tadepalli. Planning in games using approximately learned macros. In Proceedings of the Sixth International Workshop on Machine Learning, pages 221-223, Ithaca, NY, 1989. Morgan Kaufmann. [16] Gerald J. Tesauro. Practical issues in temporal difference learning. Machine Learning, 8, 1992. [17] Sebastian Thrun and Anton Schwartz. Issues in using function approximation for reinforcement learning. In M. Mozer, P. Smolensky, D. Touretzky, J. Elman, and A. Weigend, editors, Proceedings of the 1993 Connectionist Models Summer School, Hillsdale, NJ, 1993. Erlbaum Associates.	1994	120
882	Interior Point Implementations of Alternating Minimization Training Michael Lemmon Dept. of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 lemmon@maddog.ee.nd.edu Peter T. Szymanski Dept. of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 pszymans@maddog.ee.nd.edu Abstract This paper presents an alternating minimization (AM) algorithm used in the training of radial basis function and linear regressor networks. The algorithm is a modification of a small-step interior point method used in solving primal linear programs. The algorithm has a convergence rate of O( fo,L) iterations where n is a measure of the network size and L is a measure of the resulting solution's accuracy. Two results are presented that specify how aggressively the two steps of the AM may be pursued to ensure convergence of each step of the alternating minimization. 1 Introduction In recent years, considerable research has investigated the use of alternating minimization (AM) techniques in the supervised training of radial basis function networks. AM techniques were first introduced in soft-competitive learning algorithms[l]. This training procedure was later shown to be closely related to Expectation-Maximization algorithms used by the statistical estimation community[2]. Alternating minimizations search for optimal network weights by breaking the search into two distinct minimization problems. A given network performance functional is extremalized first with respect to one set of network weights and then with respect to the remaining weights. These learning procedures have found applications in the training of local expert systems [3], and in Boltzmann machine training [4]. More recently, convergence rates have been derived by viewing the AM 570 Michael Lemmon. Peter T. Szymanski method as a variable metric algorithm [5]. This paper examines AM as a perturbed linear programming (LP) problem. Recent advances in the application of barrier function methods to LP problems have resulted in the development of "path following" or "interior point" (IP) algorithms [6]. These algorithms are characterized by a fast convergence rate that scales in a sublinear manner with problem size. This paper shows how a small-step IP algorithm for solving a primal LP problem can be modified into an AM training procedure .. The principal results of the paper are bounds on how aggressively the legs of the alternating minimization can be pursued so that the AM algorithm maintains the sublinear convergence rate characteristic of its LP counterpart and so that both legs converge to an optimal solution. 2 Problem Statement Consider a function approximation problem where a stochastic approximator learns a mapping f : IRN -+ IR. The approximator computes a predicted output, Y E IR, given the input z E IRN. The prediction is computed using a finite set of M regressors. The mth regressor is characterized by the pair (Om,wm) E IRN X IRN (m = 1, ... , M). The output of the mth regressor, Ym E IR, in response to an input, z E IRN is given by the linear function Ym = O~z. (1) The mth regressor (m = 1, ... , M) has an associated radial basis function (RBF) with parameter vector Wm E IRN. The mth RBF weights the contribution of the mth output in computing Y and is defined as a normal probability density function Q(mlz) = km exp(-u- 2 1Iwm - zW) (2) where km is a normalizing constant. The set of all weights or gating probabilities is denoted by Q. The parameterization of the mth regressor is em = (O~ ,w~)T E IR2N (m = 1, ... , M) and the parameterization of the set of M linear regressors is e - (eT eT)T I"", M . (3) The preceding stochastic approximator can be viewed as a neural network. The network consists of M + 1 neurons. M of the neurons are agent neurons while the other neuron is referred to as a gating neuron. The mth a~ent neuron is parameterized by Om, the first element of the pair em = (O?'n, w?'n) (m = 1, ... , M). The agent neurons receive as input, the vector z E IRN. The output of the mth agent neuron in response to an input z is Ym = O?'nz (m = 1, ... , M). The gating neuron is parameterized by the conditional gating probabilities, Q. The gating probabilities are defined by the set of vectors, Ii; = {WI, ... ,W M }. The gating neuron receives the agent neurons' outputs and the vector z as its input. The gating neuron computes the network's output, Y, as a hard (4) or soft (5) choice Y = Ym; m = arg max Q(mlz) (4) m=I .... ,M A L:~-I Q(mlz)Ym (5) Y= M L:m=1 Q(mlz) Interior Point Implementations of Alternating Minimization Training 571 The network will be said to be "optimal" with respect to a training set T = {( Zi, Yi) : Yi = f(zd, i = 1, ... , R} if a mean square error criterion is minimized. Define the square output error of the mth agent neuron to be em(zi) = (Yi O~Zi)2 and the square weighting or classifier error of the mth RBF to be Cm(Zi) = Ilwm - Zi 112. Let the combined square approximation error of the m,th neuron be dm (Zi) = Keem (Zi) + KcCm(Zi) and let the average square approximation error of the network be M R d(Q, e, T) = L LP(zi)Q(mlzddm(zi)' (6) m=l i=l Minimizing (6) corresponds to minimizing both the output error in the M linear regressors and the classifier error in assigning inputs to the M regressors.: Since T is a discrete set and the Q are gating probabilities, the minimization of d(Q, e, T) is constrained so that Q is a valid conditional probability mass function over the training set, T. Network training can therefore be viewed as a constrained optimization problem. In particular, this optimization problem can be expressed in a form very similar to conventional LP problems. The following notational conventions are adopted to highlight this connection. Let x E lR,MR be the gating neuron's weight vector where (7) Let em = (O~,w~)T E lR,2N denote the parameter vectors associated with the mth regressor and define the cost vector conditioned on e = (er, ... , et)T as c(e) = (p(zt}d1(zt), ... ,p(zR)d1(ZR),p(Z2)d2(Z2), ... ,p(zi)dm(Zi), ... )T (8) With this notation, the network training problem can be stated as follows, mmlmlze cT(e)x with respect to x, e (9) subject to Ax = b, x 2:: 0 where b = (1, ... ,1? E lR,R, A = [IRxR· .. IRxR] E IRRXMR, and x 2:: 0 implies Xi 2:: 0 for i = 1, ... , MR. One approach for solving this problem is to break up the optimization into two steps. The first step involves minimizing the above cost functional with respect to x assuming a fixed e. This is the Q-update of the algorithm. The second leg of the algorithm minimizes the functional with respect to e assuming fixed x. This leg is called the e-update. Because the proposed optimization alternates between two different subsets of weights, this training procedure is often referred to as alternating minimization. Note that the Q-update is an LP problem while the e-update is a quadratic programming problem. Consequently, the AM training procedure can be viewed as a perturbed LP problem. 3 Proposed Training Algorithm The preceding section noted that network training can be viewed as a perturbed LP problem. This observation is significant for there exist very efficient LP solvers 572 Michael Lemmon, Peter T. Szymanski based on barrier function methods used in non-linear optimization. Recent advances in path following or interior point (IP) methods have developed LP solvers which exhibit convergence rates which scale in a sublinear way with problem size [6]. This section introduces a modification of a small-step primal IP algorithm that can be used for neural network training. The proposed modification is later shown to preserve the computational efficiency enjoyed by its LP counterpart. To see how such a modification might arise, we first need to examine path following LP solvers. Consider the following LP problem. mmlmlze cTx with respect to x E IRn (10) subject to Ax = b, x 2: 0 This problem can be solved by solving a sequence of augmented optimization problems arising from the primal parameterization of the LP problem. mmlmlze O'O:)cTx(k) - 2:i logx~k) with respect to x(k) E IRn (11) subject to Ax(k) = b, x(k) 2: 0 where a(k) 2: 0 (k = 1,· .. ,1<) is a finite length, monotone increasing sequence of real numbers. x(a(k» denotes the solution for the kth optimization problem in the sequence and is referred to as a central point. The locus of all points, x (O'( k» where a(k) 2: 0 is called the central path. The augmented problem takes the original LP cost function and adds a logarithmic barrier which keeps the central point away from the boundaries of the feasible set. As a increases, the effect of the barrier is decreased, thereby allowing the kth central point to approach the LP problem's solution in a controlled manner. Path following (IP) methods solve the LP problem by approximately solving the sequence of augmented problems shown in (11). The parameter sequence, 0'(0),0'(1), ... , a(K), is chosen to be a monotone increasing sequence so that the central points, x(a(k», of each augmented optimization approach the LP solution in a monotone manner. It has been shown that for specific choices of the a sequence, that the sequence of approximate central points will converge to an f-neighborhood of the LP solution after a finite number of iterations. For primal IP algorithms, the required condition is that successive approximations of the central points lie within the region of quadratic convergence for a scaling steepest descent (SSD) algorithm [6]. In particular, it has been shown that if the kth approximating solution is sufficiently close to the kth central point and if O'(k+ 1) = O'(k)(l + v/..{ii) where v ~ 0.1 controls the distance between successive central points, then the "closeness" to the (k + lyt central point is guaranteed and the resulting algorithm will converge in O( foL) iterations where L = n + p+ 1 specifies the size of the LP problem and p is the total number of bits used to represent the data in A, b, and c. If the algorithm takes small steps, then it is guaranteed to converge efficiently. The preceding discussion reveals that a key component to a path following method's computational efficiency lies in controlling the iteration so that successive central points lie within the SSD algorithm's region of quadratic convergence. If we are to successfully extend such methods to (9), then this "closeness" of successive solutions must be preserved by the 6-update of the algorithm. Due to the quadratic nature Interior Point Implementations of Alternating Minimization Training 573 of the e-update, this minimization can be done exactly using a single NewtonRaphson iteration. Let e denote e-update's minimizer. If we update e to e, it is quite possible that the cost vector, c(e), will be rotated in such a way that the current solution, x(k), no longer lies in the region of quadratic convergence. Therefore, if we are to preserve the IP method's computational efficiency it will be necessary to be less "aggressive" in the e-update. In particular, this paper proposes the following convex combination as the e-update e~+l) = (1 "Y(k»e~) + "Y(k)e~+l), (12) where e~) is the mth parameter vector at time k and 0 < "Y(k) < 1 controls the size of the update. This will ensure convergence of the Q-update. Convergence of the AM algorithm also requires convergence of the e-update. For the e-update to converge, "Y(k) in (12) must go to unity as k increases. Convergence of "Y(k) to unity requires that the sequence Ile~+l),* -e~)11 be monotone decreasing. As the e-update minimizer, e(k+l),, depends upon the current weights, Q(k)(mlz), large changes to Q can prevent the sequence from being monotone decreasing. Thus, it is necessary to also be less "aggressive" in the Q-update. An appropriate bound on v is the proposed solution to guarantee convergence of the e-update. Algorithm 1 (Proposed Training Algorithm) Initialize k = O. Choose X~k). e~). and O'(k) for i= 1,···,(MR). and for m= 1,···,M. repeat O'(k+l) = O'(k)(1 + v/Vn). where v ~ 0.1. Q-update: Xo = X(k) for i = 0, ... , P - 1 Xi+l = ScalingSteepestD.escent(x~k+l), O'(k+l), e(k») X(k+l) = xp e-update: For m = 1, ... , M k=k+l until(.6. < c) e~+l) = (1 r(k»)e~) + r(k)e~+l), 4 Theoretical Results This section provides bounds on the parameter, ,(k), controlling the AM algorithm's e-update so that successive x(k) vector solutions lie within the SSD algorithm's region of quadratic convergence and on v controlling the Q-update so that successive central points are not too far apart, thus allowing convergence of the e-update. Theorem 1 Let e~) and e~ ),* be the current and minimizing parameter vectors at time k. Let c(k) = c(e(k» and c(k),* = c(e(k),*). Let 6(x, a, e) = 574 Michael Lemmon. Peter T. Szymanski IIPAXX (O'c(8) - x-I) II be the step size of the SSD update where PA = 1AT (AAT)-l A and X = diag(xl, ... , xn ) . Assume that 6(xCHl), O'CHl), 8 Ck ) = 61 < 0.5 and let e~+1) = (1 ')'Ck)e~) + ')'Ck)e~+l), •. If ')'Ck) is chosen as Ck) < 62 - 61 (13) ')' - n(I + 1I/y'n)llcCHl),. - C<k)11 where 61 < 62 = 0.5, then 6(x(Hl), O'(Hl), 8 CH1) ~ 62 = 0.5. Proof: The proof must show that the choice of ')'(k) maintains the nearness ofx(k+l) to the central path after the 8-update. Let h(x,0', 8) = PAxX(O'c(8)-x-l), hI = h(xCH1), O'CHl), 8 Ck ) and h2 = h(x(Hl) , O'(Hl), 8(Hl). Using the triangle inequality produces IIh211 ~ IIh2 - hIli + Ilhlll· Ilhlll = 61 by assumption, so Ilh211::; 1100(Hl)PAXX(c(Hl) - c(k)11 + 61. Using the convexity of the cost vectors produces (c(Hl) - c(k) ::; (1 - ')'(k)c(k) + ,),(k)cCk+1),. - c(k) resulting in IIh211 ~ II00CHl),),Ck)PAXX(cCHl),. - c(k)11 + 61 . Using the fact that IIPAXXII ~ ~ = n/O'(k) (~ is the duality gap), IIh211 ~ ,),Ck)O'(Hl)IIPAXXllllc(Hl),. - c(k)11 + 61. ~ ')'(k)n(I + 1I/y'n)llc(Hl),. - cCk)11 + 61. Plugging in the value of ')'(k) from (13) and simplifying produces the desired result IIh211 ~ 62 ~ 0.5, guaranteeing that x(Hl) remains close to x(Hl),. after the 8update. 0 Theorem 1 shows that the non-linear optimization can be embedded within the steps of the path following algorithm without it taking solutions too far from successive central points. The following two results, found in [7], provide a bound on II to guarantee convergence of the 8-update. The bound on II forces successive central points to be close and ensures convergence of the 8-update. Proposition 1 Let B = LzpzQ(mlz)zzT , E = Lzpz6Q(mlz)zzT , w = Lz pzQ(mlz)y(z)z, and 6w = Lz pz6Q(mlz)y(z)z, where Q(mlz) = Q(k)(mlz) and 6Q(mlz) = Q(Hl)(mlz) - Q(Ck)mlz). Assume that ly(z)1 ~ Y, llzll ::; (, and that B is of full rank for all valid Q 'So Finally, let J1.max = sUPQ IIB- II. Then, 118~+1),. 8~)'·11 ~ 2(112 + 211)K (14) where I( = J1.maxY( (1 + Cr!'ru ) and r = IIB-lIIIIEII < 1. Theorem 2 Assume that Ilzll ~ (, ly(z)1 ~ Y, 118m ll ~ 8 max , and that B = Lz pzQ(mlz)zzT is of full rank for all valid Q's and that liB-III IIEII = r < 1. If II < min {O.I, -1 + ..jI7"I-+-r-;-/ (=2"'7>(2:-J1.-m-ax""") , -1 + JI + ')'min f e/(2K)} (15) Interior Point Implementations of Alternating Minimization Training 575 where 1\ = ILmax Y«l + (2 ILmax/(l- r», 1min = (62 - 6d/(n(1 + 0.1/ y'n)llc(l),· c(O)11) and f0 is the largest lIer!+1),· - er!)11 such that 1(.1:) = 1, then the e-update will converge with Iler!+1),· - er!)II-+ 0 and 1(.1:) -+ 1 as k increases. The preceding results guarantee the convergence of the component minimizations separately. Convergence of the total algorithm relies on the simultaneous convergence of both steps. This is currently being addressed using contraction mapping concepts and stability results from nonlinear stability analysis [8]. The convergence rate of the algorithm is established using the LP problem's duality gap. The duality gap is the difference between the current solutions for the primal and dual formulations of the LP problem. Path following algorithms allow the duality gap to be expressed as follows A( (k» _ n + 0.5y'n L..). a a(k) . (16) and thus provide a convenient stopping criterion for the algorithm. Note that a(k) = a(O) /{3k where {3 ~ (1+11/ y'n). This implies that ~(k) = 13k ~(O) ~ {3k2L. If k is chosen so that {3k2L :::; 2- L , then ~(k) :::; 2-L which implies that k ~ 2L/ log(l/ {3). Inserting our choice of {3 one finds that k ~ (2y'nL/II)+2L. The preceding argument establishes that the .proposed convergence rate of O( y'nL) iterations. In other words, the procedure's training time scales in a sublinear manner with network size. 5 Simulation Example Simulations were run on a time series prediction task to test the proposed algorithm. The training set is T = {(Zi,Yi) : Yi = y(iT),Zi = (Yi-l,Yi-2, ... ,Yi-Nf E lR,N} for i = 0,1, ... ,100, N = 4, and T = 0.04 where the time series is defined as y(t) = sin(1I"t) - sin(211"t) + sin(311"t) - sin(1I"t/2) (17) The results describe the convergence of the algorithm. These experiments consisted of 100 randomly chosen samples with N = 4 and a number of agent neurons ranging from M = 4 to 20. This corresponds to an LP problem dimension of n = 404 to 2020. The stopping criteria for the tests was to run until the solution was within f = 10-3 of a local minimum. The number of iterations and floating point operations (FLOPS) for the AM algorithm to converge are shown in Figures l(a) and l(b) with AM results denoted by "0" and the theoretical rates by a solid line. The algorithm exhibits approximately O( y'nL) iterations to converge as predicted. The computational cost, however is O(n2 L) FLOPS which is better than the predicted O( n3.5 L). The difference is due to the use of sparse matrix techniques which reduce the number of computations. The resulting AM algorithm then has the complexity of a matrix multiplication instead of a matrix inversion. The use of the algorithm resulted in networks having mean square errors on the order of 10-3 . 6 Discussion This paper has presented an AM algorithm which can be proven to converge in O( foL) iterations. The work has established a means by which IP methods can be 576 G) ~ > c::: 8 '0 ... Z E ::J 104r---------~--________ ~ Zl~~--------~----------~ 102 103 104 LP Problem Size (n) (a) Michael Lemmon, Peter T. Szymanski ~ 1015 ,....-________ -----------, o ~ G) 0-o C ~ 01 1010 c::: :; .2 LL o ... G) .0 E ~105 ~--------~---------~ 102 103 10· LP Problem Size (n) (b) Figure 1: Convergence rates as a function of n applied to NN training in a way which preserves the computational efficiency of IP solvers. The AM algorithm can be used to solve off-line problems such as codebook generation and parameter identification in colony control applications. The method is currently being used to solve hybrid control problems of the type in [9]. Areas of future research concern the study of large-step IP methods and extensions of AM training to other EM algorithms. References [1] S. Nowlan, "Maximum likelihood competitive learning," in Advances in Neural Information Processing Systems 2, pp. 574-582, San Mateo, California: Morgan Kaufmann Publishers, Inc., 1990. [2] M. Jordan and R. Jacobs, "Hierarchical mixtures of experts and the EM algorithm," Tech. Rep. 9301, MIT Computational Cognitive Science, Apr. 1993. [3] R. Jacobs, M. Jordan, S. Nowlan, and G. Hinton, "Adaptive mixtures oflocal experts," Neural Computation, vol. 3, pp. 79-87, 1991. [4] W. Byrne, "Alternating minimization and Boltzmann machine learning," IEEE Transactions on Neural Networks, vol. 3, pp. 612-620, July 1992. [5] M. Jordan and 1. Xu, "Convergence results for the EM approach to mixtures of experts architectures," Tech. Rep. 9303, MIT Computational Cognitive Science, Sept. 1993. [6] C. Gonzaga, "Path-following methods for linear programming," SIAM Review, vol. 34, pp. 167-224, June 1992. [7] P. Szymanski and M. Lemmon, "A modified interior point method for supervisory controller design," in Proceedings of the 99rd IEEE Conference on Decision and Control, pp. 1381-1386, Dec. 1994. [8] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, New Jersey: PrenticeHall, Inc., 1993. [9] M. Lemmon, J. Stiver, and P. Antsaklis, "Event identification and intelligent hybrid control," in Hybrid Systems (R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel, eds.), vol. 736 of Lecture Notes in Computer Science, pp. 265-296, Springer-Verlag, 1993.	1994	121
883	A Convolutional Neural Network Hand Tracker Steven J. Nowlan Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 nowlan@synaptics.com Abstract John C. Platt Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 platt@synaptics.com We describe a system that can track a hand in a sequence of video frames and recognize hand gestures in a user-independent manner. The system locates the hand in each video frame and determines if the hand is open or closed. The tracking system is able to track the hand to within ±10 pixels of its correct location in 99.7% of the frames from a test set containing video sequences from 18 different individuals captured in 18 different room environments. The gesture recognition network correctly determines if the hand being tracked is open or closed in 99.1 % of the frames in this test set. The system has been designed to operate in real time with existing hardware. 1 Introduction We describe an image processing system that uses convolutional neural networks to locate the position of a (moving) hand in a video frame, and to track the position of this hand across a sequence of video frames. In addition, for each frame, the system determines if the hand is currently open or closed. The input to the system is a sequence of black and white, 320 by 240 pixel digitized video frames. We designed the system to operate in a user-independent manner, using video frames from indoor scenes with natural clutter and variable lighting conditions. For ease of hardware implementation, we have restricted the system to use only convolutional networks and simple image filtering operations, such as smoothing and frame differencing. 902 Steven J. Nowlan, John C. PIau Figure 1: Average over all examples of each of the 10 classes of handwritten digits, after first aligning all of the examples in each class before averaging. Our motivation for investigating the hand tracking problem was to explore the limits of recognition capability for convolutional networks. The structure of convolutional networks makes them naturally good at dealing with translation invariance, and with coarse representations at the upper layers, they are also capable of dealing with some degree of size variation. Convolutional networks Qave been successfully applied to machine print OCR (Platt et aI, 1992), machine print address block location (Wolf and Platt, 1994), and hand printed OCR (Le Cun et aI, 1990; Martin and Rashid, 1992). In each of these problems, convolutional networks perform very well on simultaneously segmenting and recognizing two-dimensional objects. In these problems, segmentation is often the most difficult step, and once accomplished the classification is simplified. This can be illustrated by examining the average of all of the examples for each class after alignment and scaling. For the case of hand-printed OCR (see Fig. 1), we can see that the average of all of the examples is quite representative of each class, suggesting that the classes are quite compact, once the issue of translation invariance has been dealt with. This compactness makes nearest neighbor and non-linear template matching classifiers reasonable candidates for good performance. If you perform the same trick of aligning and averaging the open and closed hands from our training database of video sequences, you will see a quite different result (Fig. 2). The extreme variability in hand orientations in both the open and closed cases means that the class averages, even after alignment, are only weakly characteristic of the classes of open and closed hands. This lack of clean structure in the class average images suggested that hand tracking is a challenging recognition problem. This paper examines whether convolutional networks are extendable to hand tracking, and hence possibly to other problems where classification remains difficult even after segmentation and alignment. 2 System Architecture The overall architecture of the system is shown in Fig. 3. There are separate hand tracking and gesture recognition subsystems. For the hand tracking subsystem, each video frame is first sub-sampled and then the previous video frame (stored) is subtracted from the current video frame to produce a difference frame. These difference frames provide a crude velocity signal to the system, since the largest signals in the difference frames tend to occur near objects that are moving (Fig. 5). Independent predictions of hand locations are made by separate convolutional networks, which look at either the intensity frame or the difference frame. A voting scheme then combines the predictions from the intensity and difference networks along with predictions based on the hand trajectory computed from 3 previous frames. A Convolutional Neural Network Hand Tracker 903 Figure 2: Average over all examples of open and closed hands from the database of training video sequences, after first aligning all of the examples in each class before averaging. Previous Subsampled Video Frame Current Video Frame ~ Frame Cropping Difference Frame Intensity 1-'----"'" Hand Locator Network .. Open/Close , Network Difference Hanel Locator Network Position Hypotheses , Is Hand Open? Position in Previous Frames Voting Procedure Hand PosHlan Figure 3: Architecture of object recognition system for hand tracking and openversus-closed hand identification.	1994	122
884	Temporal Dynamics of Generalization Neural Networks • In Changfeng Wang Department of Systems Engineering University Of Pennsylvania Philadelphia, PA 19104 fwang~ender.ee.upenn.edu Santosh S. Venkatesh Department of Electrical Engineering University Of Pennsylvania Philadelphia, PA 19104 venkateshGee.upenn.edu Abstract This paper presents a rigorous characterization of how a general nonlinear learning machine generalizes during the training process when it is trained on a random sample using a gradient descent algorithm based on reduction of training error. It is shown, in particular, that best generalization performance occurs, in general, before the global minimum of the training error is achieved. The different roles played by the complexity of the machine class and the complexity of the specific machine in the class during learning are also precisely demarcated. 1 INTRODUCTION In learning machines such as neural networks, two major factors that affect the 'goodness of fit' of the examples are network size (complexity) and training time. These are also the major factors that affect the generalization performance of the network. Many theoretical studies exploring the relation between generalization performance and machine complexity support the parsimony heuristics suggested by Occam's razor, to wit that amongst machines with similar training performance one should opt for the machine of least complexity. Multitudinous numerical experiments (cf. [5]) suggest, however, that machines of larger size than strictly necessary to explain the data can yield generalization performance similar to that of smaller machines (with 264 Changieng Wang. Santosh S. Venkatesh similar empirical error) if learning is optimally stopped at a critical point before the global minimum of the training error is achieved. These results seem to fly in contradiction with a learning theoretic interpretation of Occam's razor. In this paper, we ask the following question: How does the gradual reduction of training error affect the generalization error when a machine of given complexity is trained on a finite number of examples? Namely, we study the simultaneous effects of machine size and training time on the generalization error when a finite sample of examples is available. Our major result is a rigorous characterization of how a given learning machine generalizes during the training process when it is trained using a learning algorithm based on minimization of the empirical error (or a modification of the empirical error). In particular, we are enabled to analytically determine conditions for the existence of a finite optimal stopping time in learning for achieving optimal generalization. We interpret the results in terms of a time-dependent, effective machine size which forms the link between generalization error and machine complexity during learning viewed as an evolving process in time. Our major results are obtained by introducing new theoretical tools which allow us to obtain finer results than would otherwise be possible by direct applications of the uniform strong laws pioneered by Vapnik and Cervonenkis (henceforth refered to as VC-theory). The different roles played by the complexity of the machine class and the complexity of the specific machine in the class during learning are also precisely demarcated in our results. Since the generalization error is defined in terms of an abstract loss function, the results find wide applicability including but not limited to regression (square-error loss function) and density estimation (log-likelihood loss) problems. 2 THE LEARNING PROBLEM We consider the problem of learning from examples a relation between two vectors x and y determined by a fixed but unknown probability distribution P(x, y). This model includes, in particular, the input-output relation described by y = g(x, e), (1) where g is some unknown function of x and e, which are random vectors on the same probability space. The vector x can be viewed as the input to an unknown system, e a random noise term (possibly dependent on x) , and y the system's output. The hypothesis class from which the learning procedure selects a candidate function (hypothesis) approximating g is a parametric family of functions 1{d = {!(x, fJ) : fJ E 8d ~ ?Rd } indexed by a subset 8d of d-dimensional Euclidean space. For example, if x E ?Rm and y is a scalar, 1{d can be the class of functions computed by a feedforward neural network with one hidden layer comprised of h neurons and activation function 1/;, viz., Temporal Dynamics of Generalization in Neural Networks 265 In the above, d = (m + 2)h denotes the number of adjustable parameters. The goal of learning within the hypothesis class 1ld is to find the best approximation of the relation between x and y in 1ld from a finite set of n examples 'Dn = {(Xl , yd, · .. , (xn , Yn)} drawn by independent sampling from the distribution P(x , y). A learning algorithm is simply a map which, for every sample 'Dn (n ~ 1), produces a hypothesis in 1ld. In practical learning situations one first selects a network of fixed structure (a fixed hypothesis class Jid), and then determines the "best" weight vector e* (or equivalently, the best function I(x, e) in this class) using some training algorithm. The proximity of an approximation I( x, e) to the target function g( x, e) at each point X is measured by a loss function q : (J(x,e),g(x,e) I--t ~+. For a given hypothesis class, the function 1(·, e) is completely determined by the parameter vector e. With g fixed, the loss function may be written, with a slight abuse of notation, as a map q( x, y, e) from ~m x ~ x 8 into ~+ . Examples of the forms of loss functions are the familiar square-law loss function q(x,y,e) = (g(x,e) - f(x,e))2 commonly used in regression and learning in neural networks, and the KulbackLeibler distance (or relative entropy) q( x, y, e) = In P t f t: ) for density estimation, P y 9 x, where p(Y I i(x, e») denotes the conditional density function of y given i(x, e). The closeness of 1(-, e) to g(.) is measured by the expected (ensemble) loss or error £(e, d) ~ J q(x, y, e)p(dx, dy) . The optimal approximation Ie, e) is such that £(e, d) = min9EE>d £(e, d). In similar fashion, we define the corresponding empirical loss (or training error) by J 1 n £n(e, d) = q(x, y, e)Pn(dx, dy) = ~ ?= q(Xi, Yi , e) . • =1 where Pn denotes the joint empirical distribution of input-output pairs (x, y). The global minimum of the empirical error over 8d is denoted by 0, namely, e = arg min9EE> £n(e, d). An iterative algorithm for minimizing £n(e, d) (or a modification of it) over 8 d generates at each epoch t a random vector et :'Dn -+ 8 d . The quantity £(et , d) = E f q(x, et)p(dx, de) is referred to as the generalization error of et . We are interested in the properties of the process { et : t = 1, 2, ... }, and the time-evolution of the sequence {£(et, d) : t = 1,2, .. . }. Note that each et is a functional of Pn. When P = Pn, learning reduces to an optimization problem. Deviations from optimality arise intrinsically as a consequence of the discrepancy between Pn and P. The central idea of this work is to analyze the consequence of the deviation .6.n ~ Pn - P on the generalization error. To simplify notation, we henceforth suppress d and write simply 8 , £(e) and £n(e) instead of 8 d , £(e, d), and £n(e, d), respectively. 2.1 RegUlarity Conditions We will be interested in the local behavior of learning algorithms. Consequently, we assume that 8 is a compact set, and e is the unique global minimum of £(e) on 8. 266 Changfeng Wang, Santosh S. Venkatesh It can be argued that these assumptions are an idealization of one of the following situations: • A global algorithm is used which is able to find the global minimum of En (9), and we are interested in the stage of training when 9t has entered a region 8 where 9* is the only global minimum of E(9); • A local algorithm is used, and the algorithm has entered a region 8 which contains 9* as the unique global minimum of E(9) or as a unique local minimum with which we are content. In the sequel, we write () / {)9 to denote the gradient operator with respect to the vector 9, and likewise write ()2/{)02 to denote the matrix of operators L:l{/~;{/J:,j=l. In the rest of the development we assume the following regularity conditions: Al. The loss function q(x, y,.) is twice continuously differential for all 0 E 8 and for almost all (x, Y)j A2. P(x, y) has compact support; A3. The optimal network 9* is an interior point of 8; A4. The matrix ~(9) = ~E(O) is nonsingular. These assumptions are typically satisfied in neural network applications. We will also assume that the learning algorithm converges to the global minimum of En (0) over 8 (note that 8 may not be a true global minimum, so the assumption applies to gradient descent algorithms which converge locally). It is easy to demonstrate that for each such algorithm, there exists an algorithm which decreases the empirical error monotonically at each step of iteration. Thus, without loss of generality, we also assume that all the algorithms we consider have this monotonicity property. 3 GENERALIZATION DYNAMICS 3.1 First Phase of Learning The quality of learning based on the minimization of the empirical error depends on the value of the quantity sUPe IEn(9) - E(9)1. Under the above assumptions, it is shown in [3] that ( Inn) E(O) = En((}) + Op ..;n A) (In n) and E(O = E(9) + 0 -;;- . Therefore, for any iterative algorithm for minimizing En(O), in the initial phase of learning the reduction of training error is essentially equivalent to the reduction of generalization error. It can be further shown that this situation persists until the estimates ()t enter an n-6,. neighborhood of 8, where bn -t 1/2. The basic tool we have used in arriving at this conclusion is the VC-method. The characterization of the precise generalization properties of the machine after Ot enters an n-6 neighborhood of the limiting solution needs a more precise language than can be provided by the VC-method, and is the main content of the rest of this work. Temporal Dynamics of Generalization in Neural Networks 267 3.2 Learning by Gradient Descent In the following, we focus on generalization properties when the machine is trained using the gradient descent algorithm (Backpropagation is a Gauss-Seidel implementation of this algorithm); in particular, the adaptation is governed by the recurrence (t ~ 0), (2) where the positive quantity ( governs the rate of learning. Learning and generalization properties for other algorithms can be studied using similar techniques. Replace fn by f in (2) and let { (); , t ~ O} denote the generated sequence of vectors. We can show (though we will not do so here) that the weight vector ()t is asymptotically normally distributed with expectation (); and covariance matrix with all entries of order 0 ( k). It is precisely the deviation of ()t from (); caused by the perturbation of amount ~n = Pn P to the true distribution P which results in interesting artifacts such as a finite optimal stopping time when the number of examples is finite. 3.3 The Main Equation of Generalization Dynamics Under the regularity conditions mentioned in the last section, we can find the generalization error at each epoch of learning as an explicit function of the number of iterations, machine parameters, and the initial error. Denote by Al 2: A2 2: ... ~ Ad the eigenvalues of the matrix <1>( ()) and suppose T is the orthogonal diagonalizing matrix for <1>(()), viz., T'¢(())T = diag(Al, ... ,Ad). Set 8 = (8l , ... ,8d)' ~ T( ()o - ()) and for each i let Vi denote the ith diagonal element of the d x d matrix T'E { (:0 q( x, ()») (:0 q( x, ()») '} T. Also let S( (), p) denote the open ball of radius p at (). MAIN THEOREM Under Assumptions Al-A4, the generalization error of the machine trained according to (2) is governed by the following equation for all starting points ()o E S((), n- r ) (0 < r ~ t), and uniformly for all t 2: 0: d f(()t) = f(()) + -.!.. ,,{ Vi [1 - (1- (Ai)t] 2 + 8; Ai(l- (Ai)2t} + O(n- 3r ) . (3) 2n L...J A' i=l ' If ()o (/:. S((), n-t), then the generalization dynamics is governed by the following equation valid for all r > 0: where tl is the smallest t such that E 1[1 - (<1>]t11 = An-r for some A > 0, and C(tl)' Ci(tl) ~ 0 are constants depending on network parameters and tlIn the special case when the data is generated by the following additive noise model y=g(x)+e, (5) 268 Changfeng Wang. Santosh S. Venkatesh with E [~Ix] = 0, and E [elx] = 0-2 = constant, if g(x) = f(x,O) and the loss function q(x, 0) is given by the square-error loss function, the above equation reduces to the following form: 2 d £(Ot) = £(0) + ;n ?= {[1 - (1- f.~i)t] 2 + 61.Ai(1- f.Ai)2t} + O(n-3r ). ,=1 In particular, if f(x , 0) is linear in 0, we obtain our previous result [4] for linear machines. The result (3) is hence a substantive extension of the earlier result to very general settings. It is noted that the extension goes beyond nonlinearity and the original additive noise data generating model-we no longer require that the 'true' model be contained in the hypothesis class. 3.4 Effective Complexity Write Ci ~ Vi /.Ai . The effective complexity of the nonlinear machine Ot at t is defined to be d 2 C(O, d, t) ~ 2: ci (1 - (1- f.Ad) . i=1 Analysis shows that the term Ci indicates the level of sensitivity of output of the machine to the ith component of the normalized weight vector, 0; C(O, d, t) denotes the degree to which the approximation power of the machine is invoked by the learning process at epoch t. Indeed, as t ---+ 00, C(O, d, t) ---+ Cd = '2:.1=1 Ci, which is the complexity of the limiting machine {} which represents the maximal fitting of examples to the machine (i.e., minimized training error). For the additive noise data generating model (5) and square-error loss function, the effective complexity becomes, d C(O, d, t) = 2:(1- (1- cAif)2. i=1 The sum can be interpreted as the effective number of parameters used at epoch t. At the end of training, it becomes exactly the number of parameters of the machine. Now write 0; ~ 0* + (00 - 0* )(1- f.Ad. With these definitions, (3) can be rewritten to give the following approximation error and complexity error decomposition of generalization error in the learning process: (t ~ 0). (6) The first term on the right-hand-side, £(0;), denotes the approximation error at epoch t and is the error incurred in using 0; as an approximation of the 'truth.' Note that the approximation error depends on time t and the initial value 00 , but not the examples. Clearly, it is the error one would obtain at epoch t in minimizing the function £(0) (as opposed to £n(O» using the same learning algorithm and starting with the same step length f and initial value 00 . The second term on the righthand-side is the complexity error at epoch t. This is the part of the generalization error at t due to the substitution of £n(O) for £(0). Temporal Dynamics of Generalization in Neural Networks 269 The overfitting phenomena in learning is often intuitively attributed to the 'fitting of noise.' We see that is only partly correct: it is in fact due to the increasing use of the capacity of the machine, that the complexity penalty becomes increasingly large, this being true even when the data is clean, i.e., when e == O! Therefore, we see that (6) gives an exact trade-off of the approximation error and complexity error in the learning process. For the case of large initial error, we see from the main theorem that the complexity error is essentially the same as that at the end of training, when the initial error is reduced to about the same order as before. The reduction of the training error leads to monotone decrease in generalization error in this case. 3.5 Optimal Stopping Time We can phrase the following succinct open problem in learning in neural networks: When should learning be ideally stopped? The question was answered for linear machines which is a special form of neural networks in [4]. This section extends the result to general nonlinear machines (including neural networks) in regular cases. For this purpose, we write the generalization error in the following form: where 2 d <p(t) ~ : ?= {li(l- dd 2t - dj (1- fAdt}, s=l and dj and Ii are machine parameters. The time-evolution of generalization error during the learning process is completely determined by the function <p(t). Define tmin E { T ~ 0 : £(OT) ~ £(Ot) for all t ~ O}, that is tmin denotes an epoch at which the generalization error is minimized. The smallest such number will be referred to as the optimal stopping time of learning. In general we have Cj > a for all i. In this case, it is possible to determine that there is a finite optimal stopping time. More specifically, there exists two constants t/ and tu which depend on the machine parameters such that t/ ~ tmin ~ tu. Furthermore, it can be shown that the function <p(t) decreases monotonically for t ~ t/ and increases monotonically for all t ~ tu. Finally, we can relate the generalization performance when learning is optimally stopped to the best achievable performance by means of the following inequality: £(0 . ) < £(0.) + (1 - IC)Cd tmm 2n ' where IC = O(nO) is a constant depending on Ii and dj's, and is in the interval (O,~], and Cd denotes, as before, the limiting value of the effective machine complexity C(O·, d, t) as t -+ 00. In the pathological case where there exists i such that Cj = 0, there may not exist a finite optimal stopping time. However, even in such cases, it can be shown that if In(l - fAdl In(l - fAd) < 2, a finite optimal stopping time still exists. 270 Changfeng Wang, Santosh S. Venkatesh 4 CONCLUDING REMARKS This paper describes some major results of our recent work on a rigorous characterization of the generalization process in neural network types of learning machines. In particular, we have shown that reduction of training error may not lead to improved generalization performance. Two major techniques involved are the uniform weak law (VC-theory) and differentiable statistical functionals, with the former delivering an initial estimate, and the latter giving finer results. The results shows that the complexity (e.g. VC-dimension) of a machine class does not suffice to describe the role of machine complexity in generalization during the learning process; the appropriate complexity notion required is a time-varying and algorithm-dependent concept of effective machine complexity. Since results in this work contain parameters which are typically unknown, they cannot be used directly in practical situations. However, it is possible to frame criteria overcoming such difficulties. More details of the work described here and its extensions and applications can be found in [3]. The methodology adopted here is also readily adapted to study the dynamical effect of regularization on the learning process [3]. Acknowledgements This research was supported in part by the Air Force Office of Scientific Research under grant F49620-93-1-0120. References [1] Kolmogorov, A. and V. Tihomirov (1961). f-entropy and f-capacity of sets in functional spaces. Amer. Math. Soc. Trans. (Ser. 2), 17:277-364. [2] Vapnik, V. (1982). Estimation of Dependences Based on Empirical Data. Springer-Verlag, New York. [3] Wang, C. (1994). A Theory of Generalization in Learning Machines. Ph. D. Thesis, University of Pennsylvania. [4] Wang, C., S. S. Venkatesh, and J. S. Judd (1993). Optimal stopping and effective machine size in learning. Proceedings of NIPS'93. [5] Weigend, A. (1993). On overtraining and the effective number of hidden units. Proceedings of the 1993 Connectionist Models Summer School. 335-342. Ed. Mozer, M. C. et al. Hillsdale, NJ: Erlbaum Associates.	1994	123
885	Learning Stochastic Perceptrons Under k-Blocking Distributions Mario Marchand Ottawa-Carleton Institute for Physics University of Ottawa Ottawa, Ont., Canada KIN 6N5 mario@physics.uottawa.ca Saeed Hadjifaradji Ottawa-Carleton Institute for Physics University of Ottawa Ottawa, Ont., Canada KIN 6N5 saeed@physics.uottawa.ca Abstract We present a statistical method that PAC learns the class of stochastic perceptrons with arbitrary monotonic activation function and weights Wi E {-I, 0, + I} when the probability distribution that generates the input examples is member of a family that we call k-blocking distributions. Such distributions represent an important step beyond the case where each input variable is statistically independent since the 2k-blocking family contains all the Markov distributions of order k. By stochastic percept ron we mean a perceptron which, upon presentation of input vector x, outputs 1 with probability fCLJi WiXi - B). Because the same algorithm works for any monotonic (nondecreasing or nonincreasing) activation function f on Boolean domain, it handles the well studied cases of sigmolds and the "usual" radial basis functions. 1 INTRODUCTION Within recent years, the field of computational learning theory has emerged to provide a rigorous framework for the design and analysis of learning algorithms. A central notion in this framework, known as the "Probably Approximatively Correct" (PAC) learning criterion (Valiant, 1984), has recently been extended (Hassler, 1992) to analyze the learn ability of probabilistic concepts (Kearns and Schapire, 1994; Schapire, 1992). Such concepts, which are stochastic rules that give the probability that input example x is classified as being positive, are natural probabilistic 280 Mario Marchand, Saeed Hadjifaradji extensions of the deterministic concepts originally studied by Valiant (1984). Motivated by the stochastic nature of many "real-world" learning problems and by the indisputable fact that biological neurons are probabilistic devices, some preliminary studies about the PAC learnability of simple probabilistic neural concepts have been reported recently (Golea and Marchand, 1993; Golea and Marchand, 1994). However, the probabilistic behaviors considered in these studies are quite specific and clearly need to be extended. Indeed, only classification noise superimposed on a deterministic signum function was considered in Golea and Marchand (1993). The probabilistic network, analyzed in Golea and Marchand (1994), consists of a linear superposition of signum functions and is thus solvable as a (simple) case of linear regression. What is clearly needed is the extension to the non-linear cases of sigmolds and radial basis functions. Another criticism about Golea and Marchand (1993, 1994) is the fact that their learn ability results was established only for distributions where each input variable is statistically independent from all the others (sometimes called product distributions). In fact, very few positive learning results for non-trivial p-concepts classes are known to hold for larger classes of distributions. Therefore, in an effort to find algorithms that will work in practice, we introduce in this paper a new family of distributions that we call k-blocking. As we will argue, this family has the dual advantage of avoiding malicious and unnatural distributions that are prone to render simple concept classes unlearnable (Lin and Vitter, 1991) and of being likely to contain several distributions found in practice. Our main contribution is to present a simple statistical method that PAC learns (in polynomial time) the class of stochastic perceptrons with monotonic (but otherwise arbitrary) activation functions and weights Wi E { -1,0, + 1} when the input examples are generated according to any distribution member of the k-blocking family. Due to space constraints, only a sketch of the proofs is presented here. 2 DEFINITIONS The instance (input) space, In, is the Boolean domain {-1, + 1 } n. The set of all input variables is denoted by X. Each input example x is generated according to some unknown distribution D on In. We will often use PD(X), or simply p(x), to denote the probability of observing the vector value x under distribution D. If U and V are two disjoint subsets of X, Xu and Xv will denote the restriction (or projection) of x over the variables of U and V respectively and p D (xu I xv) will denote the probability, under distribution D, of observing the vector value Xu (for the variables in U) given that the variables in V are set to the vector value xv. Following Kearns and Schapire (1994), a probabilistic concept (p-concept) is a map c : In ~ [0, 1] for which c(x) represents the probability that example X is classified as positive. More precisely, upon presentation of input x, an output of a = 1 is generated (by an unknown target p-concept) with probability c(x) and an output of a = 0 is generated with probability 1 - c(x). A stochastic perceptron is a p-concept parameterized by a vector of n weights Wi and a activation function fO such that, the probability that input example X is Learning Stochastic Perceptrons under k-Blocking Distributions 281 classified as positive is given by (1) We consider the case of a non-linear function fO since the linear case can be solved by a standard least square approximation like the one performed by Kearns in Schapire (1994) for linear sums of basis functions. We restrict ourselves to the case where fO is monotonic i.e. either nondecreasing or nonincreasing. But since any nonincreasing f (.) combined with a weight vector w can always be represented by a nondecreasing f(·) combined with a weight vector -w, we can assume without loss of generality that the target stochastic perceptron has a nondecreasing f (. ). Hence, we allow any sigmoid-type of activation function (with arbitrary threshold). Also, since our instance space zn is on an-sphere, eq. 1 also include any nonincreasing radial basis function of the type ¢(z2) where z = Ix - wi and w is interpreted as the "center" of ¢. The only significant restriction is on the weights where we allow only for Wi E {-I, 0, +1}. As usual, the goal of the learner is to return an hypothesis h which is a good approximation of the target p-concept c. But, in contrast with decision rule learning which attempts to "filter out" the noisy behavior by returning a deterministic hypothesis, the learner will attempt the harder (and more useful) task of modeling the target p-concept by returning a p-concept hypothesis. As a measure of error between the target and the hypothesis p-concepts we adopt the variation distance dv (·,·) defined as: err(h,c) = dv(h,c) ~f LPD(X) Ih(x) - c(x)1 (2) x Where the summation is over all the 2n possible values of x. Hence, the same D is used for both training and testing. The following formulation of the PAC criterion (Valiant, 1984; Hassler, 1992) will be sufficient for our purpose. Definition 1 Algorithm A is said to PAC learn the class C of p-concepts by using the hypothesis class H (of p-concepts) under a family V of distributions on instance space In, iff for any c E C, any D E V, any 0 < t,8 < 1, algorithm A returns in a time polynomial in (l/t, 1/8, n), an hypothesis h E H such that with probability at least 1 - 8, err(h, c) < t. 3 K-BLOCKING DISTRIBUTIONS To learn the class of stochastic perceptrons, the algorithm will try to discover each weight Wi that connects to input variable Xi by estimating how the probability of observing a positive output (0" = 1) is affected by "hard-wiring" variable Xi to some fixed value. This should clearly give some information about Wi when Xi is statistically independent from all the other variables as was the case for Golea and Marchand (1993) and Schapire (1992). However, if the input variables are correlated, then the process of fixing variable Xi will carry over neighboring variables which in turn will affect other variables until all the variables are perturbed (even in the simplest case of a first order Markov chain). The information about Wi will 282 Mario Marchand, Saeed HadjiJaradji then be smeared by all the other weights. Therefore, to obtain information only on Wi, we need to break this "chain reaction" by fixing some other variables. The notion of blocking sets serves this purpose. Loosely speaking, a set of variables is said to be a blocking set1 for variable Xi if the distribution on all the remaining variables is unaffected by the setting of Xi whenever all the variables of the blocking set are set to a fixed value. More precisely, we have: Definition 2 Let B be a subset of X and let U = X - (B U {Xi}). Let XB and Xu be the restriction of X on Band U respectively and let b be an assignment for XB. Then B is said to be a blocking set for variable Xi (with respect to D), iff: PD(xulxB = b,Xi = +1) = PD(xulxB = b,Xi = -1) for all b and Xu In addition, if B is not anymore a blocking set when we remove anyone of its variables, we then say that B is a minimal blocking set for variable Xi. We thus adopt the following definition for the k-blocking family. Definition 3 Distribution D on rn is said to be k-blocking iff IBil < k for i = 1,2·· . n when each Bi is a minimal blocking set for variable Xi. The k-blocking family is quite a large class of distributions. In fact we have the following property: Property 1 All Markov distributions of kth order are members of the 2k-blocking family. Proof: By kth order Markov distributions, we mean distributions which can be exactly written as a Chow(k) expansion (see Hoeffgen, 1993) for some permutation of the variables. We prove it here (by using standard techniques such as in Abend et. al, 1965) for first order Markov distributions, the generalization for k > 1 is straightforward. Recall that for Markov chain distributions we have: p(XjIXj-b··· xI) = p(XjIXj_l) for 1 < j ~ n. Hence: P(XI ... Xj-2, Xj+2· .. XnlXj-b Xj, Xj+!) = p(Xl)p(X2Ixl)··· p(Xj IXj-l)p(Xj+llxj)··· P(XnIXn-l)!p(Xj-b Xj, Xj+!) = p(xI)p(x2Ixd··· p(xj-llxj-2)P(Xj+2Ixj+l)· .. P(XnIXn-l)!p(Xj-I) = P(Xl·· ·Xj-2,Xj+2· ··XnIXj-bXj,Xj+!) where Xj denotes the negation of Xj. Thus, we see that Markov chain distributions are a special case of 2-blocking distributions: the blocking set of each variable consisting only of the two first-neighbor variables. D. The proposed algorithm for learning stochastic perceptrons needs to be provided with a blocking set (of at most k variables) for each input variable. Hoeffgen (1993) has recently proven that Chow(l) and Chow(k > 1) expansions are efficiently learnable; the latter under some restricted conditions. We can thus use these algorithms IThe wording "blocking set" was also used by Hancock & Mansour (Proc. of COLT'91 , 179-183, Morgan Kaufmann Publ.) to denote a property of the target concept. In contrast, our definition of blocking set denotes a property of the input distribution only. Learning Stochastic Perceptrons under k-Blocking Distributions 283 to discover the blocking sets for such distributions. However, the efficient learnability of unrestricted Chow(k > 1) expansions and larger classes of distributions, such as the k-blocking family, is still unknown. In fact, from the hardness results of Hoeffgen (1993), we can see that it is definitely very hard (perhaps NP-complete) to find the blocking sets if the learner has no information available other than the fact that the distribution is k-blocking. On the other hand, we can argue that the "natural" ordering of the variables present in many "real-world" situations is such that the blocking set of any given variable is among the neighboring variables. In vision for example, we expect that the setting of a pixel will directly affect only those located in it's neighborhood; the other pixels being affected only through this neighborhood. In such cases, the neighborhood of a variable "naturally" provides its blocking set. 4 LEARNING STOCHASTIC PERCEPTRONS We first establish (the intuitive fact) that, without making much error, we can always consider that the target p-concept is defined only over the variables which are not almost always set to the same value. Lemma 1 Let V be a set of v variables Xi for which Pr(xi = ai) > 1 - a. Let c be a p-concept and let c' be the same p-concept as c except that the reading of each variable Xi E V is replaced by the reading of the constant value ai. Then err( c' , c) < v . a. Proof: Let a be the vector obtained from the concatenation of all ais and let Xv be the vector obtained from X by keeping only the components Xi which are in V. Then err(c', c) ~ Pr(xv =I- a) ~ L:iEVPr(Xi =I- ai). D. For a given set of blocking sets {Bi }f=l' the algorithm will try to discover each weight Wi by estimating the blocked influence of Xi defined as: Binf(xilhi) ~f Pr(O' = 11xBi = hi, Xi = +1) - Pr(O' = 11xBi = hi, Xi = -1) where XB i denotes the restriction of x on the blocking set Bi for variable Xi and hi is an assignment for XB i • The following lemma ensures the learner that Binf(xilhi) contains enough information about Wi. Lemma 2 Let the taryet p-concept be a stochastic perceptron on In having a nondecreasing activation function and weights taken from {-1, 0, + 1 }. Then, for any assignment hi for the variables in the blocking set Bi of variable Xi, we have: { ~ 0 if Wi = +1 Binf(xilhi) = 0 if Wi = 0 ~ 0 if Wi = -1 (3) Proof sketch: Let U = X - (Bi U {Xi}), s = L:jEUWjXj and ( = L:kEBi wkbk· Let pes) denote the probability of observing s (under D). Then Binf(xilhi) = L:sp(s) [f(s + (+ Wi) - f(s + (- Wi)]; from which we find the desired result for a nondecreasing f(·) . D. 284 Mario Marchand, Saeed Hadjifaradji In principle, lemma 2 enables the learner to discover Wi from Binf(xilbi). The learner, however, has only access to its empirical estimate Bfnf(xilbi) from a finite sample. Hence, we will use Hoeffding's inequality (Hoeffding, 1963) to find the number of examples needed for a probability p to be close to its empirical estimate ft with high probability. Lemma 3 (Hoeffding, 1963) Let YI. ... , Ym be a sequence of m independent Bernoulli trials, each succeeding with probability p. Let ft = L:l ~/m. Then: Pr (1ft - pi > E) ~ 2 exp (-2mE2) Hence, by writing Binf(xilbi) in terms of (unconditional) probabilities that can be estimated from all the training examples, we find from lemma 3 that the number mO(E,8,n) of examples needed to have IBfnf(xilbi) - Binf(xilbi)I < E with probability at least 1 - 8 is given by: mO(E,8,n) ~ ~ (:E) 2 ln (~) where /'i, = a k +1 is the lowest permissible value for PD(bi , Xi) (see lemma 1). So, if the minimal nonzero value for IBinf(xilbi)1 is (3, then the number of examples needed to find, with confidence at least 1 - 8, the exact value for Wi among { -1,0, + I} is such that we need to have: Pr(IBfnf(xilbi) - Binf(xilbi)1 < (3/2) > 1 - 8. Thus, whenever (3 is oH2(e-n ), we will need of O(e2n ) examples to find (with prob > 1-8) the value for Wi. So, in order to be able to PAC learn from a polynomial sample, we must arrange ourselves so that we do not need to worry about such low values for IBinf(xilbi)l. We therefore consider the maximum blocked influence defined as: Binf(xi) ~f Binf(xilbn where b; is the vector value for which IBinf(xilbi)1 is the largest. We now show that the learner can ignore all variables Xi for which IBinf(xi)1 is too small (without making much error). Lemma 4 Let c be a stochastic perceptron with nondecreasing activation function f (.) and weights taken from { -1, 0, + 1 }. Let V c X and let cv be the same stochastic perceptron as c except that Wi = 0 for all Xi E V and its activation function is changed to f ( . + e). Then, there always exists a value for e such that: err(cv, c) :::; 2: IBinf(xi)I iEV Proof sketch: By induction on IVI. To first verify the lemma for V = {Xl}, let b be a vector of values for the setting of all Xi E Bl and let Xu be a vector of values for the setting of all Xj E U = X - (Bl U {Xl}). Let s = LjEU WjXj and ( = LjEBl WjXj, then for e = WI, we have: err(cv, c) = 2: 2:PD(xulb )PD(blxl = -l)PD(XI = -1) Xu b x If(s + ( + WI) - f(s + ( - wl)1 ~ IBinf(xdl Learning Stochastic Perceptrons under k-Blocking Distributions 285 We now assume that the lemma holds for V = {Xl. X2·· . Xk} and prove it for W = V U {Xk+1}. Let S = {Xk+1} and let f(· + Ow), f(· + Ov) and f(· + Os) denote respectively the activation function for cw, Cv and cs. By inspecting the expressions for err(cv, c) and err(cw, cs), we can see that there always exist a value for Ow E {Ov + Wk+1,OV - Wk+l} and Os E {Wk+l. -wk+d such that err( cw, cs) ::; err( cv, c). And since dv (-, .) satisfies the triangle inequality, err(cw,c)::; err(cv,c) + IBinf(xk+1)I. D. After discovering the weights, the hypothesis p-concept h returned by the learner will simply be the table look-up of the estimated probabilities of observing a positive classification given that ~~= 1 Wi Xi = s for all s values that are observed with sufficient probability (the hypothesis can output any value for the values of s that are observed very rarely). We thus have the following learning algorithm for stochastic perceptrons. Algorithm LearnSP(n, €, 6, {Bi}i=l) 1. Call m = 128 e: ) 2kHIn e~n) training examples (where k = maXi I Bi I). 2. Compute Pr(xi = +1) for each variable Xi. Neglect Xi whenever we have Pr(xi = +1) < €/(4n) or Pr(xi = +1) > 1 - €/(4n). 3. For each variable Xi and for each of its blocking vector value hi, compute Bfnf(xilhi). Let h; be the value of hi for which IBfnf(xilhi)1 is the largest. Let Bfnf(xi) = Bfnf(xilh;). 4. For each variable Xi: (a) Let Wi = +1 whenever Bfnf(xi) > €/(4n). (b) Let Wi = -1 whenever Bfnf(xi) < €/(4n). (c) Otherwise let Wi = 0 5. Compute Pr(~~=l Wi Xi = s) for s = -n, ... + n. 6. Return the hypothesis p-concept h formed by the table look-up: h(x) = h'(s) = Pr (0' = 1 t WiXi = s) ~=l for all s for which Pr(~~=l WiXi = s) > €/(8n + 8). For the other s values, let h'(s) = 0 (or any other value). Theorem 1 Algorithm LearnSP PAC learns the class of stochastic perceptrons on In with monotonic activation functions and weights Wi E {-1, 0, + 1} under any k-blocking distribution (when a blocking set for each variable is known). The number of examples required is m = 128 (2:) 2kHIn (l~n) (and the time needed is O(n x m)) for the returned hypothesis to make error at most € with confidence at least 1 - 6. Proof sketch: From Hoeffding's inequality (lemma 3) we can show that this sample size is sufficient to ensure that: 286 Mario Marchand, Saeed Hadjifaradji • IPr(Xi = +1) - Pr(xi = +1)1 < ~/(4n) with confidence at least 1 - 6/(4n) • IBfnf(xi) - Binf(xi)1 < ~/(4n) with confidence at least 1 - 6/(4n) • IPr(I:~=l WiXi = s) - Pr(I:~=l WiXi = s)1 < ~2/[64(n + 1)] with confidence at least 1- 6/(4n + 4) • IPr(O" = llI:~=l WiXi = s) - Pr(O" = llI:~=l WiXi = s)1 < ~/4 with confidence at least 1 - 6/4 From this and from lemma 1,2 and 4, it follows that returned hypothesis will make error at most ~ with confidence at least 1 - 6. D. Acknowledgments We thank Mostefa Golea, Klaus-U. Hoeffgen and Stefan Poelt for useful comments and discussions about technical points. M. Marchand is supported by NSERC grant OGP0122405. Saeed Hadjifaradji is supported by the MCHE of Iran. References Abend K., Hartley T.J. & Kanal L.N. (1965) "Classification of Binary Random Patterns", IEEE Trans. Inform. Theory vol. IT-II, 538-544. Golea, M. & Marchand M. (1993) "On Learning Perceptrons with Binary Weights", Neural Computation vol. 5, 765-782. Golea, M. & Marchand M. (1994) "On Learning Simple Deterministic and Probabilistic Neural Concepts", in Shawe-Talor J. , Anthony M. (eds.), Computational Learning Theory: EuroCOLT'93, Oxford University Press, pp. 47-60. Haussler D. (1992) "Decision Theoritic Generalizations of the PAC Model for Neural Net and Other Learning Applications", Information and Computation vol. 100,78150. Hoeffgen K.U. (1993) "On Learning and Robust Learning of Product Distributions", Proceedings of the 6th ACM Conference on Computational Learning Theory, ACM Press, 77-83. Hoeffding W. (1963) "Probability inequalities for sums of bounded random variabIes", Journal of the American Statistical Association, vol. 58(301), 13-30. Kearns M.J. and Schapire R.E. (1994) "Efficient Distribution-free Learning ofProbabilistic Concepts", Journal of Computer and System Sciences, Vol. 48, pp. 464-497. Lin J.H. & Vitter J.S. (1991) "Complexity Results on Learning by Neural Nets", Machine Learning, Vol. 6, 211-230. Schapire R.E. (1992) The Design and Analysis of Efficient Learning Algorithms, Cambridge MA: MIT Press. Valiant L.G. (1984) "A Theory of the Learnable", Comm. ACM, Vol. 27, 11341142.	1994	124
886	Recurrent Networks: Second Order Properties and Pruning Morten With Pedersen and Lars Kai Hansen CONNECT, Electronics Institute Technical University of Denmark B349 DK-2800 Lyngby, DENMARK emails:with.lkhansen@ei.dtu.dk Abstract Second order properties of cost functions for recurrent networks are investigated. We analyze a layered fully recurrent architecture, the virtue of this architecture is that it features the conventional feedforward architecture as a special case. A detailed description of recursive computation of the full Hessian of the network cost function is provided. We discuss the possibility of invoking simplifying approximations of the Hessian and show how weight decays iron the cost function and thereby greatly assist training. We present tentative pruning results, using Hassibi et al.'s Optimal Brain Surgeon, demonstrating that recurrent networks can construct an efficient internal memory. 1 LEARNING IN RECURRENT NETWORKS Time series processing is an important application area for neural networks and numerous architectures have been suggested, see e.g. (Weigend and Gershenfeld, 94). The most general structure is a fully recurrent network and it may be adapted using Real Time Recurrent Learning (RTRL) suggested by (Williams and Zipser, 89). By invoking a recurrent network, the length of the network memory can be adapted to the given time series, while it is fixed for the conventional lag-space net (Weigend et al., 90). In forecasting, however, feedforward architectures remain the most popular structures; only few applications are reported based on the Williams&Zipser approach. The main difficulties experienced using RTRL are slow convergence and 674 Morten With Pedersen, Lars Kai Hansen lack of generalization. Analogous problems in feedforward nets are solved using second order methods for training and pruning (LeCun et al., 90; Hassibi et al., 92; Svarer et al., 93). Also, regularization by weight decay significantly improves training and generalization. In this work we initiate the investigation of second order properties for RTRL; a detailed calculation scheme for the cost function Hessian is presented, the importance of weight decay is demonstrated, and preliminary pruning results using Hassibi et al.'s Optimal Brain Surgeon (OBS) are presented. We find that the recurrent network discards the available lag space and constructs its own efficient internal memory. 1.1 REAL TIME RECURRENT LEARNING The fully connected feedback nets studied by Williams&Zipser operate like a state machine, computing the outputs from the internal units according to a state vector z(t) containing previous external inputs and internal unit outputs. Let x(t) denote a vector containing the external inputs to the net at time t, and let y(t) denote a vector containing the outputs of the units in the net. We now arrange the indices on x and y so that the elements of z(t) can be defined as , k E I , k E U where I denotes the set of indices for which Zk is an input, and U denotes the set of indices for which Zk is the output of a unit in the net. Thresholds are implemented using an input permanently clamped to unity. The k'th unit in the net is now updated according to where Wkj denotes the weight to unit k from input/unit j and "'0 is the activation function of the k'th unit. When used for time series prediction, the input vector (excluding threshold) is usually defined as x(t) = [x(t), . .. , x(t - L + 1)] where L denotes the dimension of the lag space. One of the units in the net is designated to be the output unit Yo, and its activating function 10 is often chosen to be linear in order to allow for arbitrary dynamical range. The prediction of x(t + 1) is x(t + 1) = lo[so(t»). Also, if the first prediction is at t = 1, the first example is presented at t = 0 and we 'set y(O) = O. We analyse here a modification of the standard Williams&Zipser construction that is appropriate for forecasting purposes. The studied architecture is layered. Firstly, we remove the external inputs from the linear output unit in order to prevent the network from getting trapped in a linear mode. The output then reads x(t + 1) = Yo(t + 1) = L WojYj(t) + Wthres,o (1) jeU Since y(O) = 0 we obtain a first prediction yielding x(l) = Wthres,o which is likely to be a poor prediction, and thereby introducing a significant error that is fed back into the network and used in future predictions. Secondly, when pruning Recurrent Networks: Second Order Properties and Pruning 675 a fully recurrent feedback net we would like the net to be able to reduce to a simple two-layer feedforward net if necessary. Note that this is not possible with the conventional Williams&Zipser update rule, since it doesn't include a layered feedforward net as a special case. In a layered feedforward net the output unit is disconnected from the external inputs; in this case, cf. (1) we see that x(t + 1) is based on the internal 'hidden' unit outputs Yk(t) which are calculated on the basis of z(t - 1) and thereby x(t -1). Hence, besides the startup problems, we also get a two-step ahead predictor using the standard architecture. In order to avoid the problems with the conventional Williams&Zipser update scheme we use a layered updating scheme inspired by traditional feedforward nets, in which we distinguish between hidden layer units and the output unit. At time t, the hidden units work from the input vector zh(t) , k E I , kE U , k=O where I denotes the input indices, U denotes the hidden layer units and 0 the output unit. Further, we use superscripts hand 0 to distinguish between hidden unit and output units. The activation of the hidden units is calculated according to y~(t) = fr[s~(t)] = fr [ L Wki zJ (t)] , k E U ie1uUuO (2) The hidden unit outputs are forwarded to the output unit, which then sees the input vector zkCt) OCt) _ { y~(t) Zk yO(t-1) and is updated according to , k E U k=O (3) The cost function is defined as C = E + wTRw. R is a regularization matrix, w is the concatenated set of parameters, and the sum of squared errors is 1 T E = 2 L[e(t)F , e(t) = x(t) - yO(t), t=l (4) where T is the size of the training set series. RTRL is based on gradient descent in the cost function, here we investigate accelerated training using Newton methods. For that we need to compute first and second derivatives of the cost function. The essential difficulty is to determine derivatives of the sum of squared errors: aE = _ {-.. e(t) ayO(t) aw·· L...J aw .. '3 t=l '3 (5) 676 Morten With Pedersen, Lars Kai Hansen The derivative of the output unit is computed as 8yO(t) 8r[sO(t)] 8s0(t) --._8Wij 8so(t) 8Wij (6) where 8s0(t) _ 1: . O(t) "" . 8yjl(t) 8yO(t - 1) -8-- - UO,Zj + L- WOJI 8 + woo 8 Wij j/EU Wij Wij (7) where 6j k is the Kronecker delta. This expression contains the derivative of the hidden units (8) where (9) ... 132 ... 132 <> <> ~!:'::.3-.a~.2:;:-5 ---:'.a.'::-2 ---:.a7..15'--.a-:':.1'--.a7..0:;:-5-~-::0~.05~-::-0.1;---::'!0.1·5 ~.3 .a.25 .a.2 .a.15 .a.1 .a.OS 0.05 0.1 0.15 WEIGHT VAlUE WEIGHT VALUE Figure 1: Cost function dependence of a weight connecting two hidden units for the sunspot benchmark series. Left panel: Cost function with small weight decay, the (local) optimum chosen is marked by an asterix. Right panel: The same slice through the cost function but here retrained with higher weight decay. The complexity of the training problem for the recurrent net using RTRL is demonstrated in figure 1. The important role of weight decay (we have used a simple weight decay R = at) in controlling the complexity of the cost function is evident in the right panel of figure 1. The example studied is the sunspot benchmark problem (see e.g. (Weigend et al., 90) for a definition). First, we trained a network with the small weight decay and recorded the left panel result. Secondly, the network was retrained with increased weight decay and the particular weight connecting two hidden units was varied to produce the right panel result. In both cases all other weights remained fixed at their optimal values for the given weight decay. In addition to the complexity visible in these one-parameter slices of the cost function, the cost function is highly anisotropic in weight space and consequently the network Hessian is ill-conditioned. Hence, gradient descent is hampered by slow con vergen ce. Recurrent Networks; Second Order Properties and Pruning 2 SECOND ORDER PROPERTIES OF THE COST FUNCTION 677 To improve training by use of Newton methods and for use in OBS-pruning we compute the second derivative of the error functional: 82 E = _ t [e(t) 82yO(t) _ 8yO(t) . 8yO(t)] 8Wij8wpq t=l 8Wij8wpq 8Wij 8wpq (10) The second derivative of the output is 82yO(t) _ 82 r[sO(t)] 8s0(t) 8s0(t) 8r[sO(t)] 82 SO(t) --....:.....:.- . -- . -- + . ----:...;....;..8wij 8wpq 8so(t)2 8Wij 8wpq 8so(t) 8Wij8wpq (11) with 82so(t) _, 8zJ(t) ~ 82yj,(t) 82yO(t - 1) 8z~(t) 8 8 - Ooi-O-- + ~ Woj' + woo + Dop-Wij Wpq Wpq j'EU 8wij8wpq 8wij 8wpq 8Wij (12) This expression contains the second derivative of the hidden unit outputs 82yi(t) _ 82 fr[si(t)] . 8si(t) . 8si(t) + 8fr[si(t)]. 02si(t) (13) OWijOWpq osi(t)2 OWij OWpq osi(t) OWijOWpq with 02si(t) _ ozj(t) ~ 02yj,(t - 1) 02yO(t - 1) oz~(t) (14) - Dki -0-- + L..J Wkj I + Wko + Dkp 0 8WijOWpq Wpq j'EU OWijOWpq OWijOWpq Wij Recursion in the five index quantity (14) imposes a significant computational burden; in fact the first term of the Hessian in (10), involving the second derivative, is often neglected for computational convenience (LeCun et al., 90). Here we start by analyzing the significance of this term during training. We train a layered architecture to predict the sunspot benchmark problem. In figure 2 the ratio between the largest eigenvalue of the second derivative term in (10) and the largest eigenvalue of the full Hessian is shown. The ratio is presented for two different magnitudes of weight decay. In line with our observations above the second order properties of the "ironed" cost function are manageable, and we can simplify the Hessian calculation by neglecting the second derivative term in (10), i.e., apply the Gauss-Newton approximation. 3 PRUNING BY THE OPTIMAL BRAIN SURGEON Pruning of recurrent networks has been pursued by (Giles and Omlin, 94) using a heuristic pruning technique, and significant improvement in generalization for a sequence recognition problem was demonstrated. Two pruning schemes are based on systematic estimation of weight saliency: the Optimal Brain Damage (OBD) scheme of (LeCun et al., 90) and OBS by (Hassibi et al., 93). OBD is based on the diagonal approximation of the Hessian and is very robust for forecasting (Svarer et al., 93). If an estimate of the full Hessian is available OBS can be used 678 Morten With Pedersen, Lars Kai Hansen 10' .. :.::.: .... :::::;:;::: ... :::: . . '" 10 ... '----!-10--f::20--:30~----! .. :---~50--.. ~---:::70:----=1O 1040 ''-----:'10--f::20--:30~----! .. :---~50,----.. ~---:70~---:!IO ITERATION. ITERATION. Figure 2: Ratio between the largest magnitude eigenvalue of the second derivative term of the Hessian (c.f. equation (10)) and the largest magnitude .eigenvalue of the complete Hessian as they appeared during ten training sessions. The connected circles represent the average ratio. Left panel: Training with small weight decay. Right panel: Training with a high weight decay. for estimation of saliencies incorporating linear retraining. In (Hansen and With Pedersen, 94) OBS was generalized to incorporate weight decays; we use these modifications in our experiments. Note that OBS in its standard form only allows for one weight to be eliminated at a time. The result of a pruning session is a nested family of networks. In order to select the optimal network within the family it was suggested in (Svarer et al., 93.) to use the estimated test error. In particular we use Akaike's Final Prediction Error (Akaike, 69) to estimate the network test error Etest = «(T + N)/(T - N» . 2E/T 1, and N is the number of parameters in the network. In figure 3 we show the results of such a pruning session on the sunspot data starting from a (4-4-1) network architecture. The recurrent network was trained using a damped Gauss-Newton scheme. Note that the training error increases as weights are eliminated, while the test error and the estimated test error both pass through shallow minima showing that generalization is slightly improved by pruning. In fact, by retraining the optimal architecture with reduced weight decay both training and test errors are decreased in line with the observations in (Svarer et al., 93). It is interesting to observe that the network, though starting with access to a lag-space of four delay units, has lost three of the delayed inputs; hence, rely solely on its internal memory, as seen in the right panel of. figure 3. To further illustrate the memory properties of the optimal network, we show in figure 4 the network response to a unit impulse. It is interesting that the response of the network extends for approximately 12 time steps corresponding to the "period" of the sunspot series. lThe use of Akaike's estimate is not well justified for a feedback net, test error estimates for feedback models is a topic of current research. Recurrent Networks: Second Order Properties and Pruning 0.25 . ~0.'5 w 0.1 ,' : .. "': .- - .... ~:.. __ .... ~ . .; .... ~ ........ _~_ . __ ~ ...... ~~ .-: :7 0.05 ~~-='0--~'5~~~~~~~~~--~$~-~~~"~~ NUMSER OF PARAMETERS 679 OUTPUT X(I-I) X(I-2) X(I-3) X(I-4) Figure 3: Left panel: OBS pruning of a (4-4-1) recurrent network trained on sunspot benchmark. Development of training error, test error, and Akaike estimated test error (FPE). Right panel: Architecture of the FPE-optimal network. Note that the network discards the available lag space and solely predicts from internal memory. 0.8 0.1 0.5 ! .. 0.8 0.4 .-.. , 0.3 .. w ., ~ 0.2 ... II! 0.' V"I . \ .,: " : I \ : .0.4 t orf" I , . .\./ I : \ .0.8 .' , .0.' , I , , , ,--.0.1 ... .0.20 '0 15 ~ '0 '5 ~ TIME TIME Figure 4: Left panel: Output of the pruned network after a unit impulse input at t = O. The internal memory is about 12 time units long which is, in fact, roughly the period of the sunspot series. Right panel: Activity of the four hidden units in the pruned network after a unit impUlse at time t = O. 4 CONCLUSION A layered recurrent architecture, which has a feedforward net as a special case, has been investigated. A scheme for recursive estimation of the Hessian of the fully recurrent neural net is devised. It's been shown that weight decay plays a decisive role when adapting recurrent networks. Further, it is shown that the' second order information may be used to train and prune a recurrent network and in this process the network may discard the available lag space. The network builds an efficient 680 Morten With Pedersen, Lars Kai Hansen internal memory extending beyond the lag space that was originally available. Acknowledgments We thank J an Larsen, Sara Solla, and Claus Svarer for useful discuss~ons, and Lee Giles for providing us with a preprint of (Giles and amlin, 94). We thank the anonymous reviewers for valuable comments on the manuscript. This research is supported by the Danish Natural Science and Technical Research Councils through the Computational Neural Network Center (CONNECT). References H. Akaike: Fitting Autoregressive M ode/s for Prediction. Ann. Inst. Stat. Mat. 21, 243-247, (1969). Y. Le Cun, J.S. Denker, and S.A. Solla: Optimal Brain Damage. In Advances in Neural Information Processing Systems 2, (Ed. D.S. Touretzsky) Morgan Kaufmann, 598-605, (1990). C.L. Giles and C.W. amlin: Pruning of Recurrent Neural Networks for Improved Generalization Performance. IEEE Transactions on Neural Networks, to appear. Preprint NEC Research Institute (1994). L.K. Hansen and M. With Pedersen: Controlled Growth of Cascade Correlation Nets, International Conference on Artificial Neural Networks ICANN'94 Sorrento. (Eds. M. Marinaro and P.G. Morasso) Springer, 797-801, (1994). B. Hassibi, D. G. Stork, and G. J. Wolff, Optimal Brain Surgeon and General Network Pruning, in Proceedings of the 1993 IEEE International Conference on Neural Networks, San Francisco (Eds. E.H. Ruspini et al. ) IEEE, 293-299 (1993). C. Svarer, L.K. Hansen, and J. Larsen: On Design and Evaluation of . Tapped Delay Line Networks, In Proceedings ofthe 1993 IEEE International Conference on Neural Networks, San Francisco, (Eds. E.H. Ruspini et al. ) 46-51, (1993). A.S. Weigend, B.A. Huberman, and D.E. Rumelhart: Predicting the future: A Connectionist Approach. Int. J. of Neural Systems 3, 193-209 (1990). A.S. Weigend and N.A. Gershenfeld, Eds.: Times Series Prediction: Forecasting the Future and Understanding the Past. Redwood City, CA: Addison-Wesley (1994). R.J. Williams and D. Zipser: A Learning Algorithm for Continually Running Fully Recurrent Neural Networks, Neural Computation 1, 270-280, (1989).	1994	125
887	Unsupervised Classification of 3D Objects from 2D Views Satoshi Suzuki Hiroshi Ando A TR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan satoshi@hip.atr.co.jp, ando@hip.atr.co.jp Abstract This paper presents an unsupervised learning scheme for categorizing 3D objects from their 2D projected images. The scheme exploits an auto-associative network's ability to encode each view of a single object into a representation that indicates its view direction. We propose two models that employ different classification mechanisms; the first model selects an auto-associative network whose recovered view best matches the input view, and the second model is based on a modular architecture whose additional network classifies the views by splitting the input space nonlinearly. We demonstrate the effectiveness of the proposed classification models through simulations using 3D wire-frame objects. 1 INTRODUCTION The human visual system can recognize various 3D (three-dimensional) objects from their 2D (two-dimensional) retinal images although the images vary significantly as the viewpoint changes. Recent computational models have explored how to learn to recognize 3D objects from their projected views (Poggio & Edelman, 1990). Most existing models are, however, based on supervised learning, i.e., during training the teacher tells which object each view belongs to. The model proposed by Weinshall et al. (1990) also requires a signal that segregates different objects during training. This paper, on the other hand, discusses unsupervised aspects of 3D object recognition where the system discovers categories by itself. 950 Satoshi Suzuki, Hiroshi Ando This paper presents an unsupervised classification scheme for categorizing 3D objects from their 2D views. The scheme consists of a mixture of 5-layer auto-associative networks, each of which identifies an object by non-linearly encoding the views into a representation that describes transformation of a rigid object. A mixture model with linear networks was also studied by Williams et al. (1993) for classifying objects under affine transformations. We propose two models that employ different classification mechanisms. The first model classifies the given view by selecting an auto-associative network whose recovered view best matches the input view. The second model is based on the modular architecture proposed by Jacobs et al. (1991) in which an additional 3-layer network classifies the views by directly splitting the input space. The simulations using 3D wire-frame objects demonstrate that both models effectively learn to classify each view as a 3D object. This paper is organized as follows. Section 2 describes in detail the proposed models for unsupervised classification of 3D objects. Section 3 describes the simulation results using 3D wire-frame objects. In these simulations, we test the performance of the proposed models and examine what internal representations are acquired in the hidden layers. Finally, Section 4 concludes this paper. 2 THE NETWORK MODELS This section describes an unsupervised scheme that classifies 2D views into 3D objects. We initially examined classical unsupervised clustering schemes, such as the k-means method or the vector quantization method, to see whether such methods can solve this problem (Duda & Hart, 1973). Through simulations using the wire-frame objects described in the next section, we found that these methods do not yield satisfactory performance. We, therefore, propose a new unsupervised learning scheme for classifying 3D objects. The proposed scheme exploits an auto-associative network for identifying an object. An auto-associative network finds an identity mapping through a bottleneck in the hidden layer, i.e., the l network approximates functions . F and F-1 such that Rn ~Rm r ) Rn where m < n. The network, thus, compresses the input into a low dimensional representation by eliminating redundancy. If we use a five-layer perceptron network, the network can perform nonlinear dimensionality reduction, which is a nonlinear analogue to the principal component analysis (Oja, 1991; DeMers & Cottrell, 1993). The proposed classification scheme consists of a mixture of five-layer auto-associative networks which we call the identification networks, or the I-Nets. In the case where the inputs are the projected views of a rigid object, the minimum dimension that constrains the input variation is the degree of freedom of the rigid object, which is six in the most general case, three for rotation and three for translation. Thus, a single I-Net can compress the views of an object into a representation whose dimension is its degree of freedom. The proposed scheme categorizes each view of a number of 3D objects into its class through selecting an appropriate I-Net. We present the following two models for different selection and learning methods. Model I: The model I selects an I-Net whose output best fits the input (see Fig. 1). SpecIfically, we assume a classifier whose output vector is given by the softmax function of a negative squared difference between the input and the output of the I-Nets, i.e., (1) Unsupervised CLassification of 3D Objects from 2D Views I-Net I-Net ••• I-Net 2D Projected Images of 3D Objects Modell I-Net I-Net... I-Net 2D Projected Images of 3D Objects Model II 951 Figure 1: Model I and Model II. Each I-Net (identification net) is a 5-layer auto-associative network and the C-Net (classification net) is a 3-layer network. where Y * and Yi denote the input and the output of the i th I-Net, respectively. Therefore, if only one of the I-Nets has an output that best matches the input, then the output value of the corresponding unit in the classifier becomes nearly one and the output values of the other units become nearly zero. For training the network, we maximize the following objective function: L exp[ -ally * _YiI12 ] In-'~' --~------~ L exp[ -Ily * - Yi 112 ] i (2) where a (>1) denotes a constant. This function forces the output of at least one I-Net to fit the input, and it also forces the rest of I-Nets to increase the error between the input and the output. Since it is difficult for a single I-Net to learn more than one object, we expect that the network will eventually converge to the state where each I-Net identifies only one object. Model II: The model II, on the other hand, employs an additional network which we call the classification network or the C-Net, as illustrated in Fig. 1. The C-Net classifies the given views by directly partitioning the input space. This type of modular architecture has been proposed by Jacobs et al. (1991) based on a stochastic model (see also Jordan & Jacobs, 1992). In this architecture, the final output, Y, is given by (3) where Yi denotes the output of the i th I-Net, and gi is given by the softmax function gi = eXP[SiVtexP[Sj] (4) where Si is the weighted sum arriving at the i th output unit of the C-Net. For the C-Net, we use three-layer perceptron, since a simple perceptron with two layers did not provide a good performance for the objects used for our simulations (see Section 952 Satoshi Suzuki, Hiroshi Ando 3). The results suggest that classification of such objects is not a linearly separable problem. Instead of using MLP (multi-layer perceptron), we could use other types of networks for the C-Net, such as RBF (radial basis function) (Poggio & Edelman, 1990). We maximize the objective function In LgjO'-1 exp[-lly-yJ /(20'2)] (5) j where 0'2 is the variance. This function forces the C-Net to select only one I-Net, and at the same time, the selected I-Net to encode and decode the input information. Note that the model I can be interpreted as a modified version of the model II, since maximizing (2) is essentially equivalent to maximizing (5) if we replace Sj of the C-Net in (4) with a ne&ative s~uared difference between the input and the output of the i th I-Net, i.e., Sj = -Ily -yj Ir . Although the model I is a more direct classification method that exploits auto-associative networks, it is interesting to examine what information can be extracted from the input for classification in the model II (see Section 3.2). 3 SIMULATIONS We implemented the network models described in the previous section to evaluate their performance. The 3D objects that we used for our simulations are 5-segment wire-frame objects whose six vertices are randomly selected in a unit cube, as shown in Fig. 2 (a) (see also Poggio & Edelman, 1990). Various views of the objects are obtained by orthographically projecting the objects onto an image plane whose position covers a sphere around the object (see Fig. 2 (b». The view position is defined by the two parameters, 8 and fj). In the simulations, we used x, y image coordinates of the six vertices of three wire-frame objects for the inputs to the network. The models contain three I-Nets, whose number is set equal to the number of the objects. The number of units in the third layer of the five-layer I-Nets is set equal to the number of the view parameters, which is two in our simulations. We used twenty units in the second and fourth layers. To train the network efficiently, we initially limited the ranges of 8 and fj) to 1r /8 and 1r /4 and gradually increased the range until it covered the whole sphere. During the training, objects were randomly selected among the three and their views were randomly selected within the view range. The steepest ascent method was used for maximizing the objective functions (2) and (5) in our simulations, but more efficient methods, such as the conjugate gradient method, can also be used. z (a) (b) View y Figure 2: (a) 3D wire-frame objects. (b) Viewpoint defined by two parameters, 8 and fj). Unsupervised Classification of 3D Objects from 2D Views 953 3.1 SIMULATIONS USING THE MODEL I This section describes the simulation results using the model!. As described in Section 2, the classifier of this model selects an I-Net that produces minimum error between the output and the input. We test the classification performance of the model and examine internal representations of the I-Nets after training the networks. The constant a in the objective function (2) was set to 50 during the training. Fig. 3 shows the output of the classifier plotted over the view directions when the views of an object are used for the inputs. The output value of a unit is almost equal to one over the entire range of the view direction, and the outputs of the other two units are nearly zero. This indicates that the network effectively classifies a given view into an object regardless of the view directions. We obtained satisfactory results for classification if more than five units are used in the second and fourth layers of the I-Nets. Fig. 4 shows examples of the input views of an object and the views recovered by the corresponding I-Net. The recovered views are significantly similar to the input views, indicating that each auto-associative I-Net can successfully compress and recover the views of an object. In fact, as shown in Fig. 5, the squared error between the input and the output of an I-Net is nearly zero for only one of the objects. This indicates that each I-Net can be used for identifying an object for almost entire view range. UNIT 1 UNIT 2 UNIT 3 Figure 3: Outputs of the classifier in the model I. The output value of the second unit is almost equal to one over the full view range, and the outputs of the other two units are nearly zero for one of the 3D objects. Recovered views Input views Figure 4: Examples of the input and recovered views of an object. The recovered views are significantly similar to the input views. 954 Satoshi Suzuki, Hiroshi Ando We further analyzed what information is encoded in the third layer of the I-Nets. Fig. 6 (a) illustrates the outputs of the third layer units plotted as a function of the view direction ( (}, ¢) of an object. Fig. 6 (b) shows the view direction ( (}, ¢) plotted as a function of the outputs of the third layer units. Both figures exhibit single-valued functions, i.e. the view direction of the object uniquely determines the outputs of the hidden units, and at the same time the outputs of the hidden units uniquely determine the view direction. Thus, each I-Net encodes a given view of an object into a representation that has one-to-one correspondence with the view direction. This result is expected from the condition that the dimension of the third layer is set equal to the degree of freedom of a rigid object. Object 1 Object 2 Object 3 Figure 5: Error between the input view and the recovered view of an I-Net for each object. The figures show that the I-Net recovers only the views of Object 3. (a) unit! unit2 (b) a Wlit2 unit2 Figure 6: (a) Outputs of the third layer units of an I-Net plotted over the view direction ( (}, ¢) of an object. (b) The view direction plotted over the outputs of the third layer units. Figure (b) was obtained by inversely replotting Figure (a). 3.2 SIMULATIONS USING THE MODEL n In this section, we show the simulation results using the model II. The C-Net in the model learns to classify the views by splitting the input space nonlinearly. We examine internal representations of the C-Net that lead to view invariant classification in its output. Unsupervised Classification of 3D Objects from 2D Views 955 In the simulations, we used the same 3 wire-frame objects used in the previous simulations. The C-Net contains 20 units in the hidden layer. The parameter cr in the objective function (5) was set to 0.1. Fig. 7 (a) illustrates the values of an output unit in the C-Net for an object. As in the case of the model I, the model correctly classified the views into their original object for almost entire view range. Fig. 7 (b) illustrates the outputs of two of the hidden units as examples, showing that each hidden unit has a limited view range where its output is nearly one. The C-Net, thus, combines these partially invariant representations in the hidden layer to achieve full view invariance at the output layer. To examine a generalization ability of the model, we limited the view range in the training period and tested the network using the images with the full view range. Fig. 8 (a) and (b) show the values of an output unit of the C-Net and the error of the corresponding I-Net plotted over the entire view range. The region surrounded by a rectangle indicates the range of view directions where the training was done. The figures show that the correct classification and the small recovery error are not restricted within the training range but spread across this range, suggesting that the network exhibits a satisfactory capability of generalization. We obtained similar generalization results for the model I as well. We also trained the networks with a sparse set of views rather than using randomly selected views. The results show that classification is nearly perfect regardless of the viewpoints if we use at least 16 training views evenly spaced within the full view range. Figure 7: (a) Output values of an output unit of the C-Net when the views of an object are given (cf. Fig.3). (b) Output values of two hidden units ofthe C-Net for the same object. OUTPUT ERROR Figure 8: (a) Output values of an output unit of the C-Net. (b) Errors between the input views and the recovered views of the corresponding I-Net. The region surrounded by a rectangle indicates the view range where the training was done. 956 Satoshi Suzuki, Hiroshi Ando 4 CONCLUSIONS We have presented an unsupervised classification scheme that classifies 3D objects from their 2D views. The scheme consists of a mixture of non-linear auto-associative networks each of which identifies an object by encoding an input view into a representation that indicates its view direction. The simulations using 3D wire-frame objects demonstrated that the scheme effectively clusters the given views into their original objects with no explicit identification of the object classes being provided to the networks. We presented two models that utilize different classification mechanisms. In particular, the model I employs a novel classification and learning strategy that forces only one network to reconstruct the input view, whereas the model II is based on a conventional modular architecture which requires training of a separate classification network. Although we assumed in the simulations that feature points are already identified in each view and that their correspondence between the views is also established, the scheme does not, in principle, require the identification and correspondence of features, because the scheme is based solely on the existence of non-linear mappings between a set of images of an object and its degree of freedom. Therefore, we are currently investigating how the proposed scheme can be used to classify real gray-level images of 3D objects. Acknowledgments We would like to thank Mitsuo Kawato for extensive discussions and continuous encouragement, and Hiroaki Gomi and Yasuharu Koike for helpful comments. We are also grateful to Tommy Poggio for insightful discussions. References DeMers, D. and Cottrell, G. (1993). Non-linear dimensionality reduction. In Hanson, S. 1., Cowan, 1. D. & Giles, C. L., (eds), Advances in Neural Information Processing Systems 5. Morgan Kaufmann Publishers, San Mateo, CA. 580-587. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. John Wiley & Sons, NY. Jacobs, R. A., Jordan, M. I., Nowlan, S. 1. and Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation, 3,79-87. Jordan, M. I. and Jacobs, R. A. (1992). Hierarchies of adaptive experts. In Moody, J. E., Hanson, S. J. & Lippmann, R. P., (eds), Advances in Neural Information Processing Systems 4. Morgan Kaufmann Publishers, San Mateo, CA. 985-992. Oja, E. (1991). Data compression, Feature extraction, and autoassociation in feedforward neural networks. In Kohonen, K. et al. (eds), Anificial Neural Networks. Elsevier Science publishers B.V., North-Holland. Poggio, T. and Edelman, S. (1990). A network that learns to recognize three-dimensional objects. Nature, 343, 263. Weinshall, D., Edelman, S. and Btilthoff, H. H. (1990). A self-organizing multiple-view representation of 3D objects. In Touretzky, D. S., (eds), Advances in Neural Information Processing Systems 2. Morgan Kaufmann Publishers, San Mateo, CA. 274-281. Williams, C. K. I., Zemel, R. S. and Mozer, M. C. (1993). Unsupervised learning of object models. AAAI Fall 1993 Symposium on Machine Learning in Computer Vision.	1994	126
888	Hyperparameters, Evidence and Generalisation for an Unrealisable Rule Glenn Marion and David Saad glennyGed.ac.uk, D.SaadGed.ac.uk Department of Physics, University of Edinburgh, Edinburgh, EH9 3JZ, U.K. Abstract Using a statistical mechanical formalism we calculate the evidence, generalisation error and consistency measure for a linear perceptron trained and tested on a set of examples generated by a non linear teacher. The teacher is said to be unrealisable because the student can never model it without error. Our model allows us to interpolate between the known case of a linear teacher, and an unrealisable, nonlinear teacher. A comparison of the hyperparameters which maximise the evidence with those that optimise the performance measures reveals that, in the non-linear case, the evidence procedure is a misleading guide to optimising performance. Finally, we explore the extent to which the evidence procedure is unreliable and find that, despite being sub-optimal, in some circumstances it might be a useful method for fixing the hyperparameters. 1 INTRODUCTION The analysis of supervised learning or learning from examples is a major field of research within neural networks. In general, we have a probabilistic1 teacher, which maps an N dimensional input vector x to output Yt(x) according to some distribution P(Yt I x). We are supplied with a data set v= ({Yt(xlJ), xlJ} : J.' = l..p) generated from P(Yt I x) by independently sampling the input distribution, P(x), p times. One attempts to optimise a model mapping (a student), parameterised by lThis accommodates teachers with deterministic output corrupted by noise. 256 Glenn Marion, David Saad some vector w, with respect to the underlying teacher. The training error Ew (V) is some measure of the difference between the student and the teacher outputs over the set V. Simply minimising the training error leads to the problem of over-fitting. In order to make successful predictions out-with the set V it is essential to have some prior preference for particular rules. Occams razor is an expression of our preference for the simplest rules which account for the data. Clearly Ew(V) is an unsatisfactory performance measure since it is limited to the training examples. Very often we are interested in the students ability to model a random example drawn from P(Yt I x)P(x), but not necessarily in the training set, one measure of this performance is the generalisation error. It is also desirable to predict, or estimate, the level of this error. The teacher is said to be an unrealisable rule, for the student in question, if the minimum generalisation error is non-zero. One can consider the Supervised Learning Paradigm within the context of Bayesian Inference. In particular MacKay [MacKay 92(a)] advocates the evidence procedure as a 'principled' method which, in some situations, does seem to improve performance [Thodberg 93]. However, in others, as MacKay points out the evidence procedure can be misleading [MacKay 92(b )]. In this paper we do not seek to comment on the validity of of the evidence procedure as an approximation to Hierarchical Bayes (see for example [Wolpert and Strauss 94]). Rather, we ask which performance measures do we seek to optimise and under what conditions will the evidence procedure optimise them? Theoretical results have been obtained for a linear percept ron trained on data produced by a linear perceptron [Bruce and Saad 94]. They suggest that the evidence procedure is a useful guide to optimising the learning algorithm's performance. In what follows we examine the evidence procedure for the case of a linear perceptron learning a non linear teacher. In the next section we review the Bayesian scheme and define the evidence and the relevant performance measures. In section 3 we introduce our student and teacher and discuss the calculation. Finally, in section 4 we examine the extent to which the evidence procedure optimises performance. 2 BAYESIAN FORMALISM 2.1 THE EVIDENCE If we take Ew(V) to be the usual sum squared error and assume that our data is corrupted by Gaussian noise with variance 1/2/3 then the probability, or likelihood, ofthe data(V) being produced given the model wand /3 is P(D 1/3, w) ex: e-~Ew(1)). In order to incorporate Occams Razor we also assume a prior distribution on the teacher rules, that is, we believe a priori in some rules more strongly than others. Specifically we believe that pew I ,) ex: e-"'(C(w). MUltiplying the likelihood by the prior we obtain the post training or student distribution2 P( w I V", /3) ex: e-~Ew(1))-''YC(w). It is clear that the most probable model w· is given by minimising the composite cost function /3Ew(V)+,C(w) with respect to the weights (w). This formalises the trade off between fitting the data and minimising student complexity. In this sense the Bayesian viewpoint coincides with the usual backprop standpoint. 2Integrating this over f3 and 'Y gives us the posterior P(w I 1». Hyperparameters, Evidence and Generalisation for an Unrealisable Rule 257 In fact, it should be noted that stochastic minimisation can also give rise to the same post training distribution [Seung et aI92). The parameters (3 and, are known as the hyperparameters. Here we consider C(w) = wtw in which case, is termed the weight decay. The evidence is the normalisation constant in the above expression for the post training distribution. P('D I 'Y,(3) = J n dWjP('D I (3, w)P(w I,) J That is, the probability of the data set ('D) given the hyperparameters. The evidence procedure fixes the hyperparameters to the values that maximise this probability. 2.2 THE PERFORMANCE MEASURES Many performance measures have been introduced in the literature (See e.g., [Krogh and Hertz 92) and [Seung et aI92)). Here, we consider the squared difference between the average (over the post training distribution) of the student output (y.(x)}w and that of the teacher, Yt(x) , averaged over all possible test questions and teacher outputs, P(Yt, x) and finally over all possible sets of data, 'D. fg = ((Yt(x) - (Y. (x»)w ?}P(X,Yf).'l) This is equivalent to the generalisation error given by Krogh and Hertz. Another factor we can consider is the variance of the output over the student distribution ({y.(x) - (y.(x)}wP}w,P(x)' This gives us a measure of the confidence we should have in our post training distribution and could possibly be calculated if we could estimate the input distribution P(x). Here we extend Bruce and Saad's definition [Bruce and Saad 94] of the consistency measure Dc to include unrealisable rules by adding the asymptotic error fr: = IiIIlp_oo fg, Dc = ({y.(x) - (y.(x)}w}2}w,p(x),'P - fg + fr;' We regard Dc = 0 as optimal since then the variance over our student distribution is an accurate prediction of the decaying part of the generalisation error. We can consider both these performance measures as objective functions measuring the students ability to mimic the underlying teacher. Clearly, they can only be calculated in theory and perhaps, estimated in practice. In contrast, the evidence is only a function of our assumptions and the data and the evidence procedure is, therefore, a practical method of setting the hyperparameters. 3 THE MODEL In our model the student is simply a linear perceptron. The output for an input vector xl' is given by Y: = w .xl' / v'N. The examples, against which the student is trained and tested, are produced by sampling the input distribution, P(x) and then generating outputs from the distribution, P(Yt I x) = t P(y~ I x, O)P(x I O)PA 0=1 2:0=1 P(O)P(x I 0) 258 Glenn Marion, David Saad I I I -1.0 -0.6 0.0 0.6 1.0 • Figure 1: A 2-teacher in 1D : The average output (Yt}P(yl%') (i) for Dw = 0 , (ii) for Dw > 0 (0'%'1 = 0'%',) and (iii) with Dw > 0 (0'%'1 ¥- 0'%,,). where P(Yt I x, n) <X exp([Yt - wn.xF /20'2), P(x I n) is N(an,O',;o) 3 and PA is chosen such that I:~=l PA=1. Thus, each component in the sum is a linear perceptron, whose output is corrupted by Gaussian noise of variance 0'2, and we refer to this teacher as an n-teacher. In what follows, for simplicity, we consider a two teacher (n=2) with an = O. The parameter Dw = Jv Iw1-w212 and the input distribution determine the form of the teacher. This is shown in Figure 1. which displays the average output of a 2-teacher with one dimensional input vector. For 0' Xl =0' X2' Dw controls the variance about a linear mean output, and for fixed O'XI ¥- 0'%'2' Dw controls the nonlinearity of the teacher. In the latter case, in the large N limit the variance of P(Yt I x) is zero. We can now explicitly write the evidence and perform the integration over the student parameters (over weights). Taking the logarithm of the resulting expression leads to In P(1) I >",13) = - N 1(1) where the 1 is analogous to a free energy in statistical physics. 1 >.. a 13 1 1 1 - 1(1) = -In - + -In - + -lndetg + -ln211' + -P'g'kPk - e 2 11' 2 11' 2N 2 N J J and, n gjk1 = L Afk + >"Ojk n=1 p a=N Here we are using the convention that summations are implied where repeated indices occur. 3Where N(x, 0'2) denotes a normal distribution with mean x and variance 0'2 . Hyperparameters, Evidence and Generalisation for an Un realisable Rule The performance measures for this model are 2 {g = (~'x PA{w?w? - 2w?(Wj}w + (Wj}!}}'V u 2 Oc = Nt (trg}'V {g + {r; where, (Wj}w = Pl:gl:j ' and u;eff = PAu;o 259 In order to pursue the calculation we consider the average of I(V) over all possible data sets just as, earlier, we defined our performance measures as averages over all data sets. This is some what artificial as we would normally be able to calculate I(V) and be interested in the generalisation error for our learning algorithm given a particular instance of the data. However, here we consider the thermodynamic limit (i.e., N,p 00 s.t. 0 = piN = const.) in which, due to our sampling assumptions, the behaviours for typical examples of V coincide with that of the average. Details of the calculation will be published else where [Marion and Saad 95]. 4 RESULTS AND DISCUSSION We can now examine the evidence and the performance measures for our unlearnable problem. We note that in two limits we recover the learnable, linear teacher, case. Specifically if the probability of picking one of the component teachers is zero or if both component teacher vectors are aligned. In what follows we set Pi = P~ and normalise the components of the teacher such that Iwol = l. Firstly let us consider the performance measures. The asymptotic value of both {g and loci for large 0 is PiP~u;lu;:lDwlu;eff' This is the minimum generalisation error attainable and reflects the effective noise level due to the mismatch between student and teacher. We note here that the generalisation error is a function of ~ rather than f3 and 'Y independently. Figure 2a shows the generalisation error plotted against o. The addition of unlearn ability (Dw > 0) has a similar effect to the addition of noise on the examples. The appearance of the hump can be easily understood; If there is no noise or ~ is large enough then there is a steady reduction in {g. However, if this is not so then for small 0 the student learns this effective noise and the generalisation error increases with o . As the student gets more examples the effects of the noise begin to average out and the student starts to learn the rule. The point at which the generalisation error starts to decrease is influenced by the effective noise level and the prior constraint. Figure 2b shows the absolute value of the consistency measure v's 0 for non-optimal f3. Again we see that unlearn ability acts as an effective noise. For a few examples with ~ small or with large effective noise the student distribution is narrowed until the Oc is zero. However, the generalisation error is still increasing (as described above) and loci increases to a local maximum, it then asymptotically tends to { ,q. If there is no noise or ~ is large enough then loci steadily reduces as the number of examples increases. We now examine the evidence procedure. Firstly we define f3ev ( 'Y) and 'Yev (f3) to be the hyperparameters which maximise the evidence. The evidence procedure 260 4,-------------------, 3 r, (Ui) , , l " .... : ""' ... I " l -... ""_ I .... I --..... _, 1 ,~. . ........... . . ... (ii) .....................................•... O~---r~-,----~--~ o 1 2 3 (a) Generalisation error Glenn Marion, David Saad 4 3 18J2 (iii) --I (i) 0 0 1 2 3 " tl (b) Consistency Measure Figure 2: The performance measures: Graph a shows (g for finite lambda. a(i) and a(ii) are the learnable case with noise in the latter case. a(iii) shows that the effect of adding unlearn ability is qualitatively the same as adding noise. Graph b. shows the modulus of the consistency error v's a. Curves b(i) and b(ii) are the learnable case without and with noise respectively. Curve b(iii) is an unlearnable case with the same noise level. picks the point in hyperparameter space where these curves coincide. We denote the asymptotic values of 13ev(-y) and 'Yev(13) in the limit of large a by 1300 and 'Yoo respectively. In the linear case (Dw = 0) the evidence procedure assignments of the hyperparameters (for finite a) coincide with 1300 and 'Yoo and also optimise (g and 6c in agreement with [Bruce and Saad 94] . This is shown in Figure 3a where we plot the 13 which optimises the evidence (13ev) , the consistency measure (136c) and the generalisation error (13!g) versus 'Y. The point at which the three curves coincide is the point in the 13-'Y plane identified by the evidence procedure. However, we note here that, if one of the hyperparameters is poorly determined then maximising the evidence with respect to the other is a misleading guide to optimising performance even in the linear case. The results for an unrealisable rule in the linear regime (Dw > 0, lrXI = lrX:l) are similar to the learnable case but with an increased noise due to the unlearn ability. The evidence procedure still optimises performance. In the non-linear regime (Dw > 0 , lrXI ¥lrX:l) the evidence procedure fails to minimise either performance measure. This is shown in Figure 3b where the evidence procedure point does not lie on 13!g ('Y) or 136c (-y). Indeed, its hyperparameter assignments do not coincide with 1300 and 'Yoo but are a dependent. How badly does the evidence procedure fail? We define the percentage degradation in generalisation performance as I'\, = 100 * «( 9 (Aev) - (;Pt) / (;pt. Where Aev is the evidence procedure assignment and (;pt is the optimal generalisation error with respect to A. This is plotted in Figure 4a. We also define 1'\,6 = 100* 16c(Aev)1 /(g(Aev ). This measures the error in using the variance of the Hyperparameters, Evidence and Generalisation for an Unrealisable Rule 1.0.....---------..,. .. / .....•.. 0.8/Jopt. o.e ./ ... - ... _ ............ ", (i) 0.4. ,. ". ~-"":: " '--1' "", \" ".::r0.2 (Ii) ". \..(~) ,',',,, 0.0 I I I I 0.0 0.2 0.4. o.e 0.8 .., (a) Linear Case 1.0 0.5.....-----------, /Jopt. 0.3 , ___ .... (1) \~~" -/~ . 0.1 (11),','/~; ,.' 0.2 0.0-f---'T"---,---r---1 0.0 0.6 1.0 2.0 .., (b) Non-Linear Case 261 Figure 3: The evidence procedure:Optimal f3 v's /. In both graphs for (i) the evidence(f3ev), (ii) the generalisation error (f3f g ) and (iii) the consistency measure (f36J. The point which the evidence procedure picks in the linear case is that where all three curves coincide, whereas in the non linear case it coincides only with f3ev . post training distribution to estimate the generalisation error as a percentage of the generalisation error itself. Examples of this quantity are plotted in Figure 4b. There are three important points to note concerning I'\, and 1'\,6 . Firstly, the larger the deviation from a linear rule the greater is the error. Secondly, that it is the magnitude of the effective noise due to unlearnability relative to the real noise which determines this error. In other words, if the real noise is large enough to swamp the non-linearity of the rule then the evidence procedure will not be very misleading. Finally, the magnitude of the error for relatively large deviations from linearity is only a few percent and thus the evidence procedure might well be a reasonable, if not optimal, method for setting the hyperparameters. However, clearly it would be preferable to improve our student space to enable it to model the teacher. 5 CONCLUSION We have examined the generalisation error, the consistency measure and the evidence procedure within a model which allows us to interpolate between a learnable and an unlearnable scenario. We have seen that the unlearnability acts like an effective noise on the examples. Furthermore, we have seen that for a linear student the evidence procedure breaks down, in that it fails to optimise performance, when the teacher output is non-linear. However, even for relatively large deviations of the teacher from linearity the evidence procedure is close to optimal. Bayesian methods, such as the evidence procedure, are based on the assumption that the student or hypothesis space contains the teacher generating the data. In our case, in the non-linear regime, this is clearly not true and so it is perhaps not surprising that the evidence procedure is sub-optimal. Whether or not such a breakdown of the evidence procedure is a generic feature of a mismatch between the hypothesis space and the teacher is a matter for further study. 262 o~,-----------------~ f\ (i) 0.4-/ \ J \ (Ii) J \ 0.3- J \ (iii) IC \ 0.2- _. \ . ' " , ! ",. " 0.1-1 ., .. , " ... ~., .......................... --0.0 _ .. -. ----0.0 0.5 (a) t.o Cl 2.0 5 4 3 IC, 2 0 0 Glenn Marion, David Saad , (I) , , , (Ii) , , , (iii) \ \ ,.0\0., / ,'.. , " . ! \ , .. , , ! \\ """ I . .... V "- 1 2 3 4 Cl (b) Figure 4: The relative degradation in performance compared to the optimal when using the evidence procedure to set the hyperparameters. Graph (a) shows the percentage degradation in generalisation performance K, • a(i) has Dw = 1 with the real noise level u = 1. a(ii) has this noise level reduced to u = 0.1 and a(iii) has increased non-linearity, Dw = 3, and u = 1. Graph (b) shows the error made in predicting the generalisation error from the variance of the post training distribution as a percentage of the generalisation error itself, "'6 . b(i) and b(ii) have the same parameter values as a(i) and a(ii), whilst b(iii) has Dw = 3 and u = 0.1 Acknowledgments We are very grateful to Alastair Bruce and Peter Sollich for useful discussions. GM is supported by an E.P.S.R.C. studentship. References Bruce, A.D. and Saad, D. (1994) Statistical mechanics of hypothesis evaluation. J. of Phys. A: Math. Gen. 27:3355-3363 Krogh, A. and Hertz, J. (1992) Generalisation in a linear perceptron in the presence of noise. J. of Phys. A: Math. Gen. 25:1135-1147 MacKay, D.J.C. (1992a) Bayesian interpolation. Neural Compo 4:415-447 MacKay, D.J.C. (1992b) A practical Bayesian framework for backprop networks. Neural Compo 4:448-472 Marion, G. and Saad, D. (1995) A statistical mechanical analysis of a Bayesian inference scheme for an unrealisable rule. To appear in J. of Phys. A: Math. Gen. Seung, H. S, Sompolinsky, H., Tishby, N. (1992) Statistical mechanics of learning from examples. Phys. Rev. A, 45:6056-6091 Thodberg, H.H. (1994) Bayesian backprop in action:pruning, ensembles, error bars and application to spectroscopy. Advances in Neural Information Processing Systems 6:208-215. Cowan et al.(Eds.), Morgan Kauffmann, San Mateo, CA Wolpert, D. H and Strauss, C. E. M. (1994) What Bayes has to say about the evidence procedure. To appear in Maximum entropy and Bayesian methods. G. Heidbreder (Ed.), Kluwer.	1994	127
889	Adaptive Elastic Input Field for Recognition Improvement Minoru Asogawa C&C Research Laboratories, NEe Miyamae, Miyazaki, Kawasaki Kanagawa 213 Japan asogawa~csl.cl.nec.co.jp Abstract For machines to perform classification tasks, such as speech and character recognition, appropriately handling deformed patterns is a key to achieving high performance. The authors presents a new type of classification system, an Adaptive Input Field Neural Network (AIFNN), which includes a simple pre-trained neural network and an elastic input field attached to an input layer. By using an iterative method, AIFNN can determine an optimal affine translation for an elastic input field to compensate for the original deformations. The convergence of the AIFNN algorithm is shown. AIFNN is applied for handwritten numerals recognition. Consequently, 10.83% of originally misclassified patterns are correctly categorized and total performance is improved, without modifying the neural network. 1 Introduction For machines to accomplish classification tasks, such as speech and character recognition, appropriately handling deformed patterns is a key to achieving high performance [Simard 92] [Simard 93] [Hinton 92] [Barnard 91]. The number of reasonable deformations of patterns is enormous, since they can be either linear translations (an affine translation or a time shifting) or non-linear deformations (a set of combinations of partial translations), or both. Although a simple neural network (e.g. a 3-layered neural network) is able to adapt 1102 Minoru Asogawa ~'JU""'Y'CeUs j-th Input Cell /!/" .... -I,ncul Field .;_-;)oun~e Image Figure 1: AIFNN ne~~----~----~~-------------~----~--------------------~s Position Figure 2: Delta Force non-linear deformations and to discriminate noises, it is still necessary to have additional methods or data to appropriately process deformations. This paper presents a new type of classification system, an Adaptive Input Field Neural Network (AIFNN), which includes a simple pre-trained neural network and an elastic input field attached to an input layer. The neural network is applied to non-linear deformation compensations and the elastic input field to linear deformations. The AIFNN algorithm can determine an optimal affine translation for compensating for the original patterns' deformations, which are misclassified by the pre-trained neural network. As the result, those misclassified patterns are correctly classified and the final classification performance is improved, compared to that for the original neural network, without modifying the neural network. 2 Adaptive Input Field Neural Network (AIFNN) AIFNN includes a pre-trained neural network and an elastic input field attached to an input layer (Fig. 1). The elastic input field contains receptors sampling input patterns at each location. Each receptor connects to a cell in the input layer. Each receptor links to its adjacent receptors with an elastic constraint and can move over Adaptive Elastic Input Field for Recognition Improvement 1103 the input pattern independently, as long as its relative elastic constraint is satisfied. The affine translation of the whole receptor (e.g. a shift, rotation, scale and slant translation) satisfies an elastic constraint, since a constraint violation is induced by the receptors' relative locations. 1 Partial deformations are also allowed with a little constraint violation. This feature of the elastic constraint is similar to that of the Elastic Net method [Durbin 87], which can solve NP-hard problems. Although this elastic net method is directly applicable to the template matching method, the performance is highly dependent on the template selection. Therefore, an elaborated feature space for non-linear deformations is mandatory [Hinton 92]. AIFNN utilizes something like an elastic net constraint, but does not require any prominent templates. The AIFNN algorithm is a repeated sequence of a bottom-up process (calculating a guess and comparing with the presumption) and a top-down process (modifying receptor's location to decrease the error and to satisfy the input field constraints). For applying AIFNN as a classifier, a parallel search is performed; all categories are chosen as presumption categories and the AIFNN algorithm is executed. After hundreds of repetitions, an L score is calculated, which is the sum of the error and the constraint violation in the elastic input field. A category which produces the lowest L score is chosen as a plausible category. In Section 3, it is proved that all receptors will settle to an equilibrium state. In the following sections, details about the bottom-up and top-down processes are described. Bottom-Up Process: When a novel pattern is presented, each receptor samples activation corresponding to a pattern intensity at each location. Each receptor activation is directly transmitted to a corresponding neural network input cell. Those input values are forwarded through a pre-trained neural network and an output guess is obtained. This guess is compared to the presumption category, and the negative of this error is defined as the presumption certainty. 2 For example, using the mean squared error criterion, the error ED is defined as follows; ED = ~ I)dk - Ok?, (1) k where Ok is the output value, and dk is the desired value determined by the presumption category. The presumption certainty is defined as _ED. Top-Down Process: To minimize the error and to maximize the presumption certainty, each receptor modifies the activation by moving its location over the input pattern. The new location for each receptor is determined by two elements; a direction which yields less error and a direction which satisfies the input field elastic constraint. The former element is called a Delta Force, since it relates to a delta value of an input layer cell. The latter element is named an Address Force. Each receptor moves to 1 In previous papers, [Asogawa 90] and [Asogawa 91], a shift and rotation translation was taken into account. In those models, a scale and slant translation violated the elastic constraint. 2 Although another category coding schema is also possible, for simplicity, it is presumed that each output cell corresponds to one certain category. 1104 Minoru Asogawa a new location, which is determined by a sum of those two forces. The sum force is called the Combined Force. In the next two sections, details about these forces are described. Delta Force: The Delta Force, which reduces ED by altering receptors' locations, is determined by two elements: a partial derivative for the input value to the error, and a local pattern gradient at each receptor location (Fig. 1). To decrease the error ED, the value divergence for the j-th cell is computed as, D {)ED D !lnet · == -a -- = a 6· , (2) } onetj } where aD is small positive number and 6j is a delta value for the j-th input cell and computed by the back-propagation [Yamada 91]. !lnetj and a local pattern g~adient \1,pj are utilized to calculate a Delta Force !lsf; a scalar value of !lsf is given as, D !lnetj l!lsj I = 1\1,pj I . (3) The direction of the Delta Force !lsf is chosen as being parallel to that of \1,pj . Consequently, !lsf is given as, D _ !lnetj \1,pj _ D 6j !lSj - 1\1,pjll\1,pjl - a 1\1,pjI2 \1,pj . (4) To avoid !lsf becoming infinity, when 1\1,pi I is almost equal to 0, a small constant c( = t) is added to the denominator; therefore, !lsf is defined as, D _ D 6j . !lSj - a 1\1,pj 12 + c \1,p} . (5) Address Force: If each receptor is moved iteratively following only the Delta Force, the error becomes its minimum. However, receptors may not satisfy the input field constraint and induce a large constraint violation EA. Here, EA is defined by a distance between a receptor's lattice S and a lattice which is derived by an affine translation from the original lattice. Therefore, EA is defined as follows; A 1 N 1", N 2 E 2"d(S, S) = 2" L.,.-Ilsi - sill i 1 0 "2d(T(S ; t), S), (6) where d(·,·) is a distance measure for two receptor's lattices. S is a current receptor lattice. SN is the receptor lattice given by the affine translation T (.) with parameters t and SO. SO is the original receptor lattice. Therefore, as long as the receptor's lattice can be driven by some affine translation, there is no constraint violation. The affine parameters t are estimated so as to minimize EA; {)EA -{)- = 0 for i = 1, · ··,6. t i (7) Adaptive Elastic Input Field for Recognition Improvement 1105 Since EA is quadratic with respect to ti, computing ti is moderate. The Address Force for the j-th receptor ~st is defined as the partial derivative to EA with respect to the receptor's location Sj; A _ A aEA ~Sj = -(): a;:-' (8) 1 where (}:A is a small positive constant. Combined Force: Here, all receptors are moved by a Combined Force ~s, which is a sum of the Delta Force ~sD and the Address Force ~sA. After one hundred iterations, all receptors are moved to the location which produces the minimum output error and the minimum constraint violation. Final states are evaluated with a new measurement L score, which is the sum of the error ED and the constraint violation EA; i.e. L = ED + EA. This L score is utilized to choose the correct category in a parallel search. In a parallel search, each category is temporarily chosen as a presumption and converged L scores are calculated. Those scores are compared and the category yielding the smallest L score is chosen as the correct category. This method fully exploits the features of AIFNN, but it requires a large amount of computation, which can fortunately be processed totally in parallel. In the following section, convergence of the AIFNN is shown. 3 Convergence Convergence is shown by proving that the L is a Lyapunov function. When the L is a Lyapunov function, all receptors converge to some locations after iterations. The necessary and sufficient conditions for a Lyapunov function are (1) L has a lower bound and (2) L monotonically decreases by applying the Combined Forces. (12 Lower Bound: E is the squared error at the output layer. Therefore, ED ~ O. EA is the constraint violation, which is defined with a distance between two lattices. Therefore, EA ~ O. Since the L is a sum of ED and EA, the existence of a lower bound for the L is proved. 0 (2) Monotonically Decrease: The derivative of the L is calculated to show that the L decreases monotonically. d L d ED d EA dt a:t+dT I: { aED ~} + I: {aEA ~} . aSi d t . aSi d t , , ~{(a:: + aa~:) ~~i}, (9) , h d Si . h C b' d d . were dt IS t e om me Force an gIven as, d s · __ I = ~sD + ~SA. dt (10) lJ06 Minoru Asogawa When a source image is smooth and 1'V¢d is smaller than c, the following approximation is satisfied; (11) By using Eq. (11), the Delta Force is approximated as follows; 6.sD = aD 'V ¢i 6i ~ _aD OED. 1\7 ¢d 2 + c 'V ¢i OSi (12) By using Eqs. (8) and (12), and by letting aD = aA, the L derivative is computed as follows; dL dt ( OED OEA)2 ...... _aA '" + ~ O. Las· as· i I I With Eq. (13), it is proved that L decreases monotonically.D 4 Experiments and Results (13) Hand-written numerals recognition is chosen as one of the applications of AIFNN, since performance improvement is shown by compensating for deformations [Simard 92] [Simard 93] [Hinton 92]. The numeral inputs are bi-Ievel images of 32x40. They are blurred with a 5x5 Gaussian kernel and resampled to 14x 18 pixel gray level images. To calculate an intensity and a local gradient between grids, bi-linear Lagrange interpolation is utilized. A neural network is 3 layered. The numbers of cells for the input layer, the hidden layer and the output layer are 252, 20 and 10, respectively. To obtain a simpler weight configuration, two techniques are utilized; a constant weight decay [Ishikawa 89] and a small constant addition to output function derivatives [Fahlman 88]. Training is repeated for 180 epochs with 2500 numerals, and tested with another 2500. Since image edges are almost blank, about 2400 connections between the input layer and the hidden layer are equal to 0; therefore, the number of parameters is reduced to 2870. In this experiment, a simple decision method is used; the maximum output cell is chosen as a guess and patterns are rejected when the error of the guess is greater than a threshold value. Naturally, a low threshold yields a low misclassification rate, but also yields a high rejection rate [Martin 92]. With the maximum threshold, the rates of rejection, correct classification and misclassification are 0.00%(0 patterns), 95.20%(2380 patterns) and 4.80%(120 patterns), respectively. For the 2500 numerals learning data, these rates are 0.00%(0 patterns), 99.40%(2485 patterns) and 0.60%(15 patterns). When a threshold is 0.001, the rates of rejection, correct classification and misclassification are 43.52%(1088 patterns), 56.40%(1410 patterns) and 0.08%(2 patterns), respectively. AIFNN is applied to these 1088 rejected patterns. and classifies 997 patterns correctly. Therefore, total performances for rejection, correct classification and misclassification become 0.00%(0 patterns), 95.72%(2393 patterns) and 4.28%(107 patterns), respectively. As the classification performance is improved; the number of Adaptive Elastic Input Field for Recognition Improvement 1107 misclassified patterns reduces from 120 to 107 without modifying the neural network. 10.83% of the originally misclassified patterns are correctly categorized. Fig. 3 shows an input field after one hundred iterations. I:: I ... y ........... . Ilttput Activation I... ....... .... 1 .. . 0123456789 . ..... ••••• • • .. " • III! 6 • 0.111 • It .... ....... ...... ...... •••••• ·atOl" •• •••••• ••••• • ••• ...... A .... 18 • )0 ,. ••• 6 ·iiI" • , it .. . '. . . .. " .... •••••• ...... In the figure on the left, receptors are located at each grid point in a gray lattice. The circle diameter corresponds to the pattern intensity at each receptor's location. The bottom right figure indicates the source image, and the top right figure indicates the neural network input. This image was initially misclassified as 3 instead of 8. After iteration with presumption as 8, category 8 gets the highest activation and the receptor's lattice is rotated to compensate for the initial deformation. Figure 3: Input Field After Adaptation 5 Discussion It is shown that the AIFNN can improve the classification performance for the original neural network, without modifications. This performance improvement is caused by an optimal affine translations estimation for rejected patterns. Although an affine translation is discussed in this paper, the algorithm is applicable to any deformation mechanism; such as a gain and offset equalization and 3D perspective deformation. 1108 Minoru Asogawa The requirement for a neural network in AIFNN is the capability of calculating partial derivatives for an input layer, so a layered neural network is utilized in this paper. Since partial derivative can be computed by numerical approximation, practically any neural network is applicable for AIFNN. Moreover, any differentiable error criterion is applicable; such as, a KL information and a likelihood. To reduce computation, a sequential searching is also possible; a presumption is chosen as the most plausible category, e.g. the smallest error category. If the L score falls behind a threshold, this presumption is regarded as correct. If it's not, another plausible category is chosen as a presumption and tested [Asogawa 91]. References [As ogawa 90] M. Asogawa, "Adaptive Input Field Neural Network - that can recognize rotated and/or shifted character -", Proceedings of IJCNN '90 at San Diego, vol. 3. pp. 733-738. June 1990. [As ogawa 91] M. Asogawa, "Adaptive Input Field Neural Network", Proceedings of IJCNN '91 at Singapore, vol. 1. pp. 83-88. November 1991. [Barnard 91] E. Barnard et aI., "Invariance and Neural Nets" , IEEE trans. on Neural Networks, vol. 2. no. 5, pp. 498-508. 1992. [Durbin 87] R. Durbin et al., "An analogue approach to the traveling salesman problem using an elastic net method", Nature, vol. 326. pp. 689-691. 1987. [Fahlman 88] S. Fahlman, "An empirical study of learning speed in backpropagation networks", CMU-CS-88-162, 1988. [Hinton 92] G.E. Hinton et al., "Adaptive Elastic Models for Hand-Printed Character Recognition", Advances in Neural Information Processing Systems, vol. 4. pp. 512-519. 1992. [Ishikawa 89] M. Ishikawa, "A structural learning algorithm with forgetting of link weights" , Proceedings of IJCNN '89 at Washington DC., vol. 2, pp. 626, 1989. [Martin 92] G. L. Martin et al., "Recognizing Overlapping Hand-Printed Characters by Centered-Object Integrated Segmentation and Recognition", Advances in Neural Information Processing Systems, vol. 4. pp. 504-511. 1992. [Simard 92] P. Simard et al., "Tangent Prop - A Formalism for Specifying Selected Invariances in an Adaptive Network", Advances in Neural Information Processing Systems, vol. 4. pp. 895-903. 1992. [Simard 93] P. Simard et al., "Efficient Pattern Recognition Using a New Transformation Distance" , Advances in Neural Information Processing Systems, vol. 5. pp. 50-58. 1993. [Yamada 91] K. Yamada, "Learning of category boundaries based on inverse recall by multilayer neural network", Proceedings of IJCNN '91 at Seattle, pp. 7-12 vol.2 1991.	1994	128
890	A Model for Chemosensory Reception Rainer MalakaJ Thomas Ragg Institut fUr Logik, Komplexitat und Oeduktionssysteme Universitat Karlsruhe, PO Box 0-76128 Karlsruhe, Germany e-mail: malaka@ira.uka.de.ragg@ira.uka.de Martin Hammer Institut fur Neurobiologie Freie Universitat Berlin 0-14195 Berlin, Germany e-mail: mhammer@castor.zedat.fu-berlin.de Abstract A new model for chemosensory reception is presented. It models reactions between odor molecules and receptor proteins and the activation of second messenger by receptor proteins. The mathematical formulation of the reaction kinetics is transformed into an artificial neural network (ANN). The resulting feed-forward network provides a powerful means for parameter fitting by applying learning algorithms. The weights of the network corresponding to chemical parameters can be trained by presenting experimental data. We demonstrate the simulation capabilities of the model with experimental data from honey bee chemosensory neurons. It can be shown that our model is sufficient to rebuild the observed data and that simpler models are not able to do this task. 1 INTRODUCTION Terrestrial animals, vertebrates and invertebrates, have developed very similar solutions for the problem of recognizing volatile substances [Vogt et ai., 1989]. Odor molecules bind to receptor proteins (receptor sites) at the cell membrane of the sensory cell. This interaction of odor molecules and receptor proteins activates a G-protein mediated second 62 Rainer Malaka, Thomas Ragg, Martin Hammer _____ odor molecules 'f! " 4! /2~ ___ Ins" second messengers ----"----. action potentials Ionic Influx Figure 1: Reaction cascade in chemosensory neurons. Volatile odor molecules reach receptor proteins at the surface of the chemosensory neuron. The odor bound binding proteins activate second messengers (e.g. G-proteins). The activated second messengers cause a change in the conductivity of ion channels. Through ionic influx a depolarization can build an action potential. messenger process. The concentrations of cAMP or IP3 rise rapidly and activate cyclicnucleotide-gated ion channels or IP3-gated ion channels [Breer et al., 1989, Shepherd, 1991]. As a result of this second messenger reaction cascade the conductivity of ion channels is changed and the cell can be hyperpolarized or depolarized, which can cause the generation of action potentials. It has been shown that one odor is able to activate different second messenger processes and that there is some interaction between the different second messenger processes [Breer & Boekhoff, 1992]. Figure 1 shows schematically the cascade of reactions from odor molecules over receptor proteins and second messengers up to the changing of ion channel conductance and the generation of action potentials. Responses of sensory neurons can be very complex. The response as a function of the odor concentration is highly non-linear. The response to mixtures can be synergistic or inhibitory relative to the response to the components of the compound. A synergistic effect occurs, if the response of one sensory cell to a binary mixture of two odors AI and A2 with concentrations [AI]' [A2l is higher than the sum of the responses to the odors AI, A2 at concentrations [A tl, [A2] alone. An inhibitory effect occurs, if the response to the mixture is smaller than either response to the single odors. In bee subplacode and placode recordings both effects can be observed [Akers & Getz, 1993]. Models of chemosensory reception should be complex enough to simulate the inhibitory and synergistic effects observed in sensory neurons, and they must provide a means for parameter fitting. We want to introduce a computational model which is constructed analogously to the chemical reaction cascade in the sensory neuron. The model can be expressed as an ANN and all unknown parameters can be trained with a learning algorithm. A Model for Chemosensory Reception 63 2 THE RECEPTOR TRANSDUCER MODEL The first step of odor reception is done by receptor proteins located at the cell membrane. There may be many receptor protein types in sensory cells at different concentrations and with different sensitivity to various odors. There is the possibility for different odors ligands Ai to react with a receptor protein Rj, but it is also possible for a single odor to react with different receptor proteins. The second step is the activation of second messengers. Ennis proposed a modelling of these complex reactions by a reaction step of activated odor-receptor complexes with transducer mechanisms [Ennis, 1991]. These transducers are a simplification of the second messengers processes. In Ennis' model transducers and receptor proteins are odor specific. We generalize Ennis' model by introducing transducer mechanisms T" that can be activated by odor-receptor complexes, and as with odors and receptor proteins we allow receptor proteins and transducers to react in any combination. Receptor proteins and transducer proteins are not required to be odor specific. The kinetics of the two reactions are given by Ai + Rj ~ AiRj AiRj + T" ~ AiRjT". (1) In a first reaction odor ligands Ai bind to receptor proteins Rj and build odor-receptor complexes AiRj, which can activate transducer mechanisms T" in a second reaction. Affinities lcij and Ij" describe the possibility of reactions between odor ligands Ai and receptor proteins Rj or between odor-receptor complexes Ai Rj and transducers T", respectively. The mass action equations are [AiRj] = lcij[Ad[Rj] [Ai RjT,,] = Ij,,[AiRj][T,,]. (2) The binding of odor-receptor complexes with transducer mechanisms is not dependent on the specific odor which is bound to the receptor protein, i.e. Ij" does not depend on i. It is only necessary that the receptor protein is bound. A sensory neuron can now be defined by the total concentration (or amount) of receptor proteins [H] and transducers [1']. The total concentration of either type corresponds to the sum of the free sites and the bound sites: [Hj] [Rj] + I)AiRj] (3) [T,,] + I)AiRjT,,] .1 (4) i,j Activated transducer mechanisms may elicit an excitatory or inhibitory effect depending on the kind of ion channel they open. Thus we divide the transducers T" into two types: inhibitory and excitatory transducers. With { + 1 , if transducer T" is excitatory 8" = -1 , if transducer T" is inhibitory (5) IWe use the simplification [flj] = [Rj] + L:i[AiRJ'] instead of [flj] = [Rj] + L:.[AiRj] + L:i,k [AiRjTk], which is sufficient for [flj] > [tk], see also [Malaka & Ragg, 1993]. 64 Rainer Malaka, Thomas Ragg, Martin Hammer Figure 2: ANN equivalent to the full receptor-transducer model. The input layer corresponds to the concentration of odor ligands [Ad, the first hidden layer to activated receptor protein types, the second to activated transducer mechanisms. The output neuron computes the effect E of the sensory cell. the effect can be set to the sum of all activated excitatory transducers minus the sum of all inhibitory transducers relative to the total amount of transducers. An additive constant (J is used to model spontaneous reactions. With this the effect of an odor can be set to (6) With Eqs.(2,4) and the hyperbolic function hyp(x) = x/(l + x) the effect E defined in Eq.(6) can be reformulated to E = 1 A 2: hyp (2: Ijk[AiRj ]) 15k ['h] + (J . (7) 2::L[Tk] L •. '" '" t ,) Analogously, we eliminate [AiRj] and [Rj): E= 1 A 2:hyp (2:Ijk[Rj]hYP(2:kij[Ad)) 8k[Tk]+(J . (8) Lk[n] k j i Equation (8) can now be regarded as an ANN with 4 feed-forward layers. The concentrations of the odor ligands [Ad represent the input layer, the two hidden layers correspond to activated receptor proteins and activated transducers, and one output element in layer 4 represents the effect caused by the input odor. The weight between the i-th element of the input layer to the j-th element of the first hidden layer is kij and from there to the k-th neuron of the second hidden layer Ij k [Hj]. The weight from element k of hidden layer 2 to the output element is 15k [1'k]/ 2::k[1'k]. The adaptive elements ofthe hidden layers have the hyperbolic activation functions hypo The network structure is shown in Figure 2. A Model for Chemosensory Reception 65 6 5 4 3 6 10 20 50 receptor protein types mecanisms Figure 3: Mean error in spikes per output neuron for the model with different network sizes. Network sizes are varied in the number ofreceptor protein types and the number of transducing mechanisms. 3 SIMULATION RESULTS Applying learning algorithms like backpropagation or RProp to the model network, it is possible to find parameter settings for optimal (or local optimal) simulations of chemosensory cell responses with given response characteristics. In our simulations the best training results were achieved by using the fast learning algorithm RProp, which is an imprOVed version of back propagation [Riedmiller & Braun, 1993]. For our simulations we used extracellular recordings made by Akers and Getz from single sensilla placodes of honey bee workers applying different stimuli and their binary mixtures to the antenna (see [Akers & Getz, 1992] for material and methods). The data set for training the ANNs consists of responses of 54 subplacodes to the four odors, geraniol, citral, limonene, linalool, their binary mixtures, and a mixture of all of four odors each at two concentration levels and to a blank stimulus, i.e. 23 responses to different odor stimulations for each subplacode. In a series of training runs with varying numbers of receptor protein types and transducer types the full model was trained to fit the data set. The networks were able to simulate the responses of the subplacodes, dependent on the network size. The size of the first hidden layer corresponds to the number of receptor protein types (R) in the model, the size of the second hidden layer corresponds to the number of transducing mechanisms (T). Figure 3 shows the mean error per output neuron in spikes for all combinations of one to six receptor types and one to six transducer mechanisms and for combinations with ten, twenty and fifty receptor protein types. The mean response over all subplacode responses is 18.15 spikes. The best results with errors less than two spikes per response were achieved with models with at least three receptor protein types and at least three transducer mechanisms. A model with only two transducer types is not sufficient to simulate the data. For generalization tests we generated a larger pattern set with our model. This training set 66 Rainer Malaka, Thomas Ragg, Martin Hammer spikes spi kes Figure 4: Simulation results of our model (a.b) and the Ennis model (c.d). The responses of simulated sensory cells is given in spikes. The left column (a,c) represents receptor neuron responses to mixtures of geraniol and citral, the right column (b,d) represents sensory cell responses to mixtures of limonene and linalool. The concentrations of the odorants are depicted on a logarithmic scale from 2- 5 to 26 micrograms (0.03 to 64 micrograms). Measurement points and deviations from simulated data are given by crosses in the diagrams. was divided in a set of 23 training patterns and 88 test patterns. The training set had the same structure as the experimental data. Training of new randomly initialized networks provided a mean error on the test set that was approximately 1.6 times higher than on the training set. An overfitting effect was not observable during the training sequence of 10000 A Model for Chemosensory Reception 67 learning epochs. It is also possible to transform many other models for chemosensory perception into ANN s. We fitted the stimulus summation model and the stimulus substitution model [Carr & Derby, 1986] as well as the models proposed by Ennis [Ennis, 1991]. All of the other models were not able to reproduce the complex response functions observed in honey bee sensory neurons. Some of them are able to simulate synergistic responses to binary mixtures, but none were able to produce inhibitory effects. Figure 4 shows the simulation of a sensory neuron that shows very similar spike rates for the single odors geraniol and citral and to their binary mixture at the same concentration, while the mixture interaction of limonene and linalool shows a strong synergistic effect, i.e. the response to mixture of both odors is much higher than the responses to the single odors. As shown in Figure 4a) and b) our model is able to simulate this behavior, while the Ennis model is not sufficient to show the two different types of interaction for the binary mixtures geraniol-citral and limonene-linalool, as shown in Figure 4c) and d). The error for the Ennis model is greater than four spikes per output neuron and the error for our model with six receptor types and four transducer mechanisms is smaller than one spike per output neuron. The stimulus summation and stimulus substitution model have very similar results as the Ennis model, Figure 4 e) and t) show the simulation of the stimulus summation. 4 CONCLUSIONS Artificial neural networks are a powerful tool for the simulation of the responses of chemosensory cells. The use of ANNs is consistent with theoretical modelings. Many previously proposed models are expressible as ANNs. The new receptor transducer model described in this paper is also expressible as an ANN. The use of learning algorithms is a means to fit parameters for the simulation with given experimental response data. With this method it is possible to create simulation models of chemosensory cells, that can be used in further modelings of olfactory and chemosensory systems. Applying data from honey bee placode recordings we could also investigate the necessary complexity of chemosensory models. It could be shown that only the full receptor transducer model is able to simulate the complex response characteristics observed in honey bee chemosensory cells. Most other models can show only low synergistic mixture interactions and none of the other models is able to simulate inhibitory effects in odor perception. The found parameters of the ANN do not have to correspond to physiological entities, such as affinities between molecules. The learning or parameter fitting optimizes the parameters in a way that the difference between experimental data and simulation results is minimized. If there are several solutions to this task, one solution will be found, which might differ from the actual values. But it can be said, that a model is not sufficient if the learning algorithm is not able to fit the experimental data This implies that the smallest model, which is able to simulate the given data covers the minimum of complexity necessary. For honey bees this means that a competitive receptor transducer model is necessary with at least two transducer mechanisms and three receptor protein types. Any other model, such as the stimulus summation model, the stimulus substitution model and the Ennis model, is not sufficient. The model is not restricted to insect olfactory receptor neurons and can also be applied to many types of olfactory or gustatory receptor neurons in invertebrates and vertebrates. 68 Rainer Malaka, Thomas Ragg, Martin Hammer Acknowledgments We want to thank Pat Akers and Wayne Getz for giving us subplacode response data to train the ANNs used in our model, Heinrich Braun and Wayne Getz for fruitful discussions on our work. This work was supported by grants of the Deutsche Forschungsgemeinschaft (DFG), SPP Physiologie und Theorie neuronaler Netze, and the State of Baden-WUrttemberg. References [Akers & Getz, 1992] R.P. Akers & W.M. Getz. A test of identified response classes among olfactory receptor neurons in the honeybee worker. Chemical Senses, 17(2):191-209, 1992. [Akers & Getz, 1993] RP. Akers & W.M. Getz. Response of olfactory receptor neurons in honey bees to odorants and their binary mixtures. 1. Compo Physiol. A, 173:169-185, 1993. [Breer & Boekhoff, 1992] H. Breer & I. Boekhoff. Second messenger signalling in olfaction. Current Opinion in Neurobiology, 2:439-443,1992. [Breer et al., 1989] H. Breer, I. Boekhoff, J. Strotmann, K. Rarning, & E. Tareilus. Molecular elements of olfactory signal transduction in insect antennae. In D. Schild, editor, Chemosensory Information Processing, pages 75-86. Springer, 1989. [Carr & Derby, 1986] W.E.S. Carr & C.D. Derby. Chemically stimulated feeding behavior in marine animals: the importance of chemical mixtures and the involvement of mixture interactions. 1.Chem.Ecol., 12:987-1009,1986. [Ennis,1991] D.M. Ennis. Molecular mixture models based on competitive and noncompetitive agonism. Chemical Senses, 16(1):1-17,1991. [Malaka & Ragg, 1993] R. Malaka & T. Ragg. Models for chemosensory receptors: An approach using artificial neural networks. Interner Bericht 18/93, Institut fUr Logik, Komplexitlit und Deduktionssysteme, Universitlit Karlsruhe, 1993. [Riedmiller & Braun, 1993] M. Riedmiller & H. Braun. A direct adaptive method for faster backpropagation learning: The rprop algorithm. In Proceedings of the ICNN, 1993. [Shepherd, 1991] G.M. Shepherd. Computational structure of the olfactory system. In J.L. Davis & H. Eichenbaum, editors, Olfaction A Model System for Computational Neuroscience, chapter 1, pages 3-41. MIT Press, 1991. [Yogt et al., 1989] RG. Yogt, R Rybczynski, & M.R. Lerner. The biochemistry of odorant reception and transduction. In D. Schild, editor, Chemosensory Information Processing, pages 33-76. Springer, 1989.	1994	129
891	Multidimensional Scaling and Data Clustering Thomas Hofmann & Joachim Buhmann Rheinische Friedrich-Wilhelms-U niversitat Institut fur Informatik ill, Romerstra6e 164 D-53117 Bonn, Germany email:{th.jb}@cs.uni-bonn.de Abstract Visualizing and structuring pairwise dissimilarity data are difficult combinatorial optimization problems known as multidimensional scaling or pairwise data clustering. Algorithms for embedding dissimilarity data set in a Euclidian space, for clustering these data and for actively selecting data to support the clustering process are discussed in the maximum entropy framework. Active data selection provides a strategy to discover structure in a data set efficiently with partially unknown data. 1 Introduction Grouping experimental data into compact clusters arises as a data analysis problem in psychology, linguistics, genetics and other experimental sciences. The data which are supposed to be clustered are either given by an explicit coordinate representation (central clustering) or, in the non-metric case, they are characterized by dissimilarity values for pairs of data points (pairwise clustering). In this paper we study algorithms (i) for embedding non-metric data in a D-dimensional Euclidian space, (ii) for simultaneous clustering and embedding of non-metric data, and (iii) for active data selection to determine a particular cluster structure with minimal number of data queries. All algorithms are derived from the maximum entropy principle (Hertz et al., 1991) which guarantees robust statistics (Tikochinsky et al., 1984). The data are given by a real-valued, symmetric proximity matrix D E R NXN , 'Dkl being the pairwise dissimilarity between the data points k, l. Apart from the symmetry constraint we make no further assumptions about the dissimilarities, i.e., we do not require D being a metric. The numbers 'Dkl quite often violate the triangular inequality and the dissimilarity of a datum to itself could be finite. 2 Statistical Mechanics of Multidimensional Scaling Embedding dissimilarity data in a D-dimensional Euclidian space is a non-convex optimization problem which typically exhibits a large number of local minima. Stochastic search methods like simulated annealing or its deterministic variants have been very successfulJy 460 Thomas Hofmann. Joachim Buhmann applied to such problems. The question in multidimensional scaling is to find coordinates {Xi }i~1 in a D-dimensional Euclidian space with minimal embedding costs N MDS 1 '"' [I 12 ]2 H = 2N L., Xi - Xk - 'Dik . (1) i,k=1 Without loss of generality we shift the center of mass in the origin <2::= I Xk = 0). In the maximum entropy framework the coordinates {Xi} are regarded as random variables which are distributed according to the Gibbs distribution P ( { Xj} ) = exp( - f3 (H MDS - F). The inverse temperature f3 = 1 /T controls the expected embedding costs (HMDS) (expectation values are denoted by (.). To calculate the free energy F for H MDS we approximate the coupling term 2 2:~"k=1 'DikxiXk/N ~ 2:[:1 xihi with the mean fields hi = 4 2:~= 1 'Dik(Xk}/N. Standard t~chniques to evaluate the free energy F yield the equations ' 00 00 D Z(HMDS) rv J dy J II dR.d,d' exp (-f3NF), (2) -'00 - 00 d,d'=1 f) N 00 F(HMDS ) 2 L R.~d' f3~ Lin J dXjexp (-f3f(Xi)) ' (3) d,d'=1 i=1 - 00 N IXil4 ~IXiI2 L 'Dik + 4xTR.xi + xT (hi - 4Y)· k=1 f(Xi) (4) The integral in Eq. (2) is dominated by the absolute minimum of F in the limit N ~ 00. Therefore, we calculate the saddle point equations N R. = ~ L ((Xjxf) + l(lx iI2)I) and 0 i=1 I Xi exp( -f3f(Xj)dxi I exp( -f3 f(Xj)dxi . (5) (6) Equation (6) has been derived by differentiating F with respect to hi. I denotes the D x D unit matrix. In the low temperature limit f3 ~ 00 the integral in (3) is dominated by the minimum of f(Xi) . Therefore, a new estimate of (Xi) is calculated minimizing f with respect to Xi. Since all explicit dependencies between the Xi have been eliminated, this minimization can be performed independently for all i, 1 ~ i ~ N. In the spirit of the EM algorithm for Gaussian mixture models we suggest the following algorithm to calculate a meanfield approximation for the multidimensional scaling problem. initialize (Xi)(O) randomly; t = O. while 2:::':1 I(Xi )(t) (Xi)(t-I)I > t: E- step: estimate (Xi) (t+l) as a function of (Xi)(t ) , RY) , y(t), h ~ t) M-step: calculate n (t), h~t) and determine y (t) such that the centroid condition is satisfied. Multidimensional Scaling and Data Clustering 461 This algorithm was used to determine the embedding of protein dissimilarity data as shown in Fig. 1 d. The phenomenon that the data clusters are arranged in a circular fashion is explained by the lack of small dissimilarity values. The solution in Fig. Id is about a factor of two better than the embedding found by a classical MDS program (Gower, 1966). This program determines a (N - 1)- space where the ranking of the dissimilarities is preserved and uses principle component analysis to project this tentative embedding down to two dimensions. Extensions to other MDS cost functions are currently under investigation. 3 Multidimensional Scaling and Pairwise Clustering Embedding data in a Euclidian space precedes quite often a visual inspection by the data analyst to discover structure and to group data into clusters. The question arises how both problems, the embedding problem and the clustering problem, can be solved simultaneously. The second algorithm addresses the problem to embed a data set in a Euclidian space such that the clustering structure is approximated as faithfully as possible in the maximum entropy sense by the clustering solution in this embedding space. The coordinates in the embedding space are the free parameters for this optimization problem. Clustering of non-metric dissimilarity data, also called pairwise clustering (Buhmann, Hofmann, 1994a), is a combinatorial optimization problem which depends on Boolean assignments Miv E {a, I} of datum i to cluster lJ. The cost function for pairwise clustering with J( clusters is If 1 N N E~:(M) = L 2 N L L MkvMlv'Dkl with v=1 Pv k=! 1=1 (7) In the meanfield approach we approximate the Gibbs distribution P( Ej;) corresponding to the original cost function by a family of approximating distributions. The distribution which represents most accurately the statistics of the original problem is determined by the minimum of the Kullback-Leibler divergence to the original Gibbs distribution. In the pairwise clustering case we introduce potentials {Ekv } for the effective interactions, which define a set of cost functions with non-interacting assignments. K N £<).; (M, {Ekv }) = L L Mk 1jEkl;. v=1 k=1 The optimal potentials derived from this minimization procedure are {£kv} = arg min 'DKL (PO(E~' )IIP(E~)), {£kv} (8) (9) where PO(E9{) is the Gibbs distribution corresponding to E~., and 'DKL(·II·) is the KLdivergence. This method is equivalent to minimizing an upper bound on the free energy (Buhmann, Hofmann, 1994b), F(E~:) ::; Fo(E~. ) + (VK)o, with VA" = Ej; £~" (10) (')0 denoting the average over all configurations of the cost function without interactions. Correlations between assignment variables are statistically independent for PO( E9(), i.e., (MkvA11v)0 = (Mkv )0(A11v)0. The averaged potential VI\, therefore, amounts to K N 1 K N (Vrd = L L (Mkl;) (Mlv) 2 vN'Dk1 - L L(A1kv)Eklj, (11) v=1 k ,I=1 P v=1 k=1 462 Thomas Hofmann. Joachim Buhmann the subscript of averages being omitted for conciseness. The expected assignment variables are (12) Minimizing the upper bound yields (13) The "optimal" potentials 1 N ( IN) [i~' = J N L(Mkv ) 'Dik - 2 N L(M1v)Dkl 1v k=1 Pv 1=1 (14) depend on the given distance matrix, the averaged assignment variables and the cluster probabilities. They are optimal in the sense, that if we set (15) the N * K stationarity conditions (13) are fulfilled for every i E {I, ... , N}, 11 E {I, ... , K}. A simultaneous solution ofEq. (15) with (12) constitutes a necessary condition for a minimum of the upper bound for the free energy :F. The connection between the clustering and the multidimensional scaling problem is established, if we restrict the potentials [iv to be of the form IXi - Yvf with the centroids YII = 2:~=1 Mkl/Xv/ 2::=1 Mkv. We consider the coordinates Xi as the variational parameters. The additional constraints restrict the family of approximating distributions, defined by £9". to a subset. Using the chain rule we can calculate the derivatives of the upper bound (10), resulting in the exact stationary conditions for Xi, K N K '" (M )(M ) co co '" '" (MjoJ(Mjv) ~ ia ja (~Cia -~Civ)Ya = ~ ~ N x a,v=1 j=1 a,v=1 Pa [ N ( a(Mka) T) 1 (~[ia ~[ir/) (Mia)! + ~ (Xk - Ya) Oxi (Xj - Ya), (16) where ~[iOt = £ia - [tao The derivatives a(Mka) /Oxi can be exactly calculated, since they are given as the solutions of an linear equation system with N x K unknowns for every Xi. To reduce the computational complexity an approximation can be derived under the assumption ay 0/ / aXj ~ O. In this case the right hand side of (16) can be set to zero in a first order approximation yielding an explicit formula for Xi, K K KiXi ~ ~ L(Miv) (11Yv1l2 - [tv) (Yv - L(Mia)Ya) , v=1 a=1 (17) with the covariance matrix Ki = ((yyT)j - (Y)i(Y)T) and (Y)i = 2:~=1 (Miv)Y v' The derived system of transcendental equations given by (12), (17) and the centroid condition explicitly reflects the dependencies between the clustering procedure and the Euclidian representation. Solving these equations simultaneously leads to an efficient algorithm which Multidimensional Scaling and Data Clustering a HB HA GGI MY HBX, HF, HE GP HG~~~ ••• •• [l}?,faitt\tvJqJ~!;t ...•..•. , .' •... , .••..... c ~ llt GP GGI x~ 0 ~GGG HAfo x x + + HB • ++ + MY ---+ HG,HE,HF 463 b 4tHB HG,H~ . ~ ~ HBX,HF,HE ~. GP~ ~ GGI~ ~ GGGI 420 d Random Selection 380 £re 340 300 # of selected Do, Figure 1: Similarity matrix of 145 protein sequences of the globin family (a): dark gray levels correspond to high similarity values; (b): clustering with embedding in two dimensions; (c): multidimensional scaling solution for 2-dimensional embedding; (d): quality of clustering solution with random and active data selection of 'D ik values. eKe has been calculated on the basis of the complete set of 'Di k values. interleaves the multidimensional scaling process and the clustering process and which avoids an artificial separation into two uncorrelated processes. The described algorithm for simultaneous Euclidian embedding and data clustering can be used for dimensionality reduction, e.g., high dimensional data can be projected to a low dimensional subspace in a nonlinear fashion which resembles local principle component analysis (Buhmann, Hofmann, 1994b). Figure (l) shows the clustering result for a real-world data set of 145 protein sequences. The similarity values between pairs of sequences are determined by a sequence alignment program which takes biochemical and structural information into account. The sequences belong to different protein families like hemoglobin, myoglobin and other globins; they are abbreviated with the displayed capital letters. The gray level visualization of the dissimilarity matrix with dark values for similar protein sequences shows the formation of distinct "squares" along the main diagonal. These squares correspond to the discovered partition after clustering. The embedding in two dimensions shows inter-cluster distances which are in consistent agreement with the similarity values of the data. In three and four dimensions the error between the 464 Thomas Hofmann. Joachim Buhmann given dissimilarities and the constructed distances is further reduced. The results are in good agreement with the biological classification. 4 Active Data Selection for Data Clustering Active data selection is an important issue for the analysis of data which are characterized by pairwise dissimilarity values. The size of the distance matrix grows like the square of the number of data 'points'. Such a O(N2) scaling renders the data acquisition process expensive. It is, therefore, desirable to couple the data analysis process to the data acquisition process, i.e., to actively query the supposedly most relevant dissimilarity values. Before addressing active data selection questions for data clustering we have to discuss the problem how to modify the algorithm in the case of incomplete data. If we want to avoid any assumptions about statistical dependencies, it is impossible to infer unknown values and we have to work directly with the partial dissimilarity matrix. Since the data enters only in the (re-)ca1culation of the potentials in (14), it is straightforward to appropriately modify these equations. All sums are restricted to terms with known dissimilarities and the normalization factors are adjusted accordingly. Alternatively we can try to explicitly estimate the unknown dissimilarity values based on a statistical model. For this purpose we propose two models, relying on a known group structure of the data. The first model (I) assumes that all dissimilarities between a point i and points j belonging to a group G ~ are i.i.d. random variables with the probability density Pi/1 parameterized by eiw In this scheme a subset of the known dissimilarities of i and j to other points k are used as samples for the estimation of Vij . The selection of the specific subset is determined by the clustering structure. In the second model (II) we assume that the dissimilarities between groups G v, G ~ are i.i.d. random variables with density PV/1 parameterized by e,IW The parameters ev~ are estimated on the basis of all known dissimilarities {Vij E V} between points from Gv and G~. The assignments of points to clusters are not known a priori and have to be determined in the light of the (given and estimated) data. The data selection strategy becomes self-consistent if we interpret the mean fields (.I"vfiv) of the clustering solution as posterior probabilities for the binary assignment variables. Combined with a maximum likelihood estimation for the unknown parameters given the posteriors, we arrive at an EM-like iteration scheme with the E-step replaced by the clustering algorithm. The precise form of the M-Step depends on the parametric form of the densities Pi~ or PI/~' respectively. In the case of Gaussian distributions the M-Step is described by the following estimation equations for the location parameters (I), (II), (18) with 1T:j~ = 1+~vl' ((Mil/){Mj~) + (l\tfi~)(Mjv)). Corresponding expressions are derived for the standard deviations at) or a~'~, respectively. In the case of non-normal distributions the empirical mean might still be a good estimator of the location parameter, though not necessarily a maximum likelihood estimator. The missing dissimilarities are estimated by the following statistics, derived from the empirical means. K i\[ - (I) + N - (I) - (I) '"" J. i~mi~ jvmjv Dij = ~ (l\tfiv)(JVfj~) JY + N. 1/,11=) 1~ }V (I), D~~) = '"" .".ij m- (I) (II) !} ~ "11/1 'v~ , 11-:5:~ (19) Multidimensional Scaling and Data Clustering 2600 r--r-........ -.----,--........ -.-----,., 2400 2200 2000 o L, '\., \ c, ~---,--... \ ""-! 1 \ \ t: Ac t i ve Da t~:L--, Se 1 ec t ion \ _______________________ _ 400 BOO 1200 # of selected dissimilarities 465 Figure 2: Similarity matrix of 54 word fragments generated by a dynamic programming algorithm. The clustering costs in the experiment with active data selection requires only half as much data as a random selection strategy. with Nil' = E'D.kE'D(i11k11)' For model (I) we have used a pooled estimator to exploit the data symmetry. The iteration scheme finally leads to estimates (jill or (j'lt' respectively for the parameters and Dij for all unknown dissimilarities. Criterion for Active Data Selection: We will use the expected reduction in the variance of the free energy Fo as a score, which should be maximized by the selection criterion. Fo is given by Fo(D) = -~ E;;:', log E;~l exp( -{3£i/l(D)). If we query a new dissimilarity D ij the expected reduction of the variance of the free energy is approximated by ~ .. = 2 [ aFO]2 V [D .. _ D .. ] t) aDij tJ tJ (20) The partial derivatives can be calculated exactly by solving a system of linear equations with N x [ .. : unknowns. Alternatively a first order approximation in f /I = O( 1/ N P,/) yields (21) This expression defines a relevance measure of Dij for the clustering problem since a Dij value contributes to the clustering costs only if the data i and j belong to the same cluster. Equation (21) summarizes the mean-field contributions aFo/aDij ~ a(H)o/aDjj. To derive the final form of our scoring function we have to calculate an approximation of the variance in Eq. (20) which measures the expected squared error for replacing the true value Dij with our estimate Dij . Since we assumed statistical independence the variances are additive V [Dij - Dij] = V [Dij] + V [Dij]. The total population variance is a sum of inner- and inter-cluster variances, that can be approximated by the empirical means and by the empirical variances instead of the unknown parameters of Pill or P'lt'. The sampling variance of the statistics Dij is estimated under the assumption, that the empirical means ifl'ill 466 Thomas Hofmann, Joachim Buhmann or mVJ.l respectively are uncorrelated. This holds in the hard clustering limit. We arrive at the following final expression for the variances of model (II) v [Vij-Dij] ~ L1TYJl[(Dij-mvJl)2+(I+I: 1T:JJ.l 1Tkl(j~Jl)l (22) V~Jl Vk1EV VJl For model (I) a slightly more complicated formula can be derived. Inserting the estimated variances into Eq. (20) leads to the final expression for our scoring function. To demonstrate the efficiency of the proposed selection strategy, we have compared the clustering costs achieved by active data selection with the clustering costs resulting from randomly queried data. Assignments int the case of active selection are calculated with statistical model (I). Figure 1 d demonstrates that the clustering costs decrease significantly faster when the selection criterion (20) is implemented. The structure of the clustering solution has been completely inferred with about 3300 selected V ik values. The random strategy requires about 6500 queries for the same quality. Analogous comparison results for linguistic data are summarized in Fig. 2. Note the inconsistencies in this data set reflected by smallVik values outside the cluster blocks (dark pixels) or by the large Vik values (white pixels) inside a block. Conclusion: Data analysis of dissimilarity data is a challenging problem in molecular biology, linguistics, psychology and, in general, in pattern recognition. We have presented three strategies to visualize data structures and to inquire the data structure by an efficient data selection procedure. The respective algorithms are derived in the maximum entropy framework for maximal robustness of cluster estimation and data embedding. Active data selection has been shown to require only half as much data for estimating a clustering solution of fixed quality compared to a random selection strategy. We expect the proposed selection strategy to facilitate maintenance of genome and protein data bases and to yield more robust data prototypes for efficient search and data base mining. Acknowledgement: It is a pleasure to thank M. Vingron and D. Bavelier for providing the protein data and the linguistic data, respectively. We are also grateful to A. Polzer and H.J. Warneboldt for implementing the MDS algorithm. This work was partially supported by the Ministry of Science and Research of the state Nordrhein-Westfalen. References Buhmann, J., Hofmann, T. (l994a). Central and Pairwise Data Clustering by Competitive Neural Networks. Pages 104-111 of" Advances in Neural Infonnation Processing Systems 6. Morgan Kaufmann Publishers. Buhmann, J., Hofmann, T. (1994b). A Maximum Entropy Approach to Pairwise Data Clustering. Pages 207-212 of" Proceedings of the International Conference on Pattern Recognition, Hebrew University, Jerusalem, vol. II. IEEE Computer Society Press. Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53, 325-328. Hertz, J., Krogh, A., Palmer, R. G. (1991). Introduction to the Theory of Neural Computation. New York: Addison Wesley. Tikochinsky, y, Tishby, N.Z., Levine, R. D. (1984). Alternative Approach to MaximumEntropy Inference. Physical Review A, 30, 2638-2644.	1994	13
892	A Real Time Clustering CMOS Neural Engine T. Serrano-Gotarredona, B. Linares-Barranco, and J. L. Huertas Dept. of Analog Design, National Microelectronics Center (CNM), Ed. CICA, Av. Reina Mercedes sIn, 41012 Sevilla, SPAIN. Phone: (34)-5-4239923, Fax: (34)-5-4624506, E-mail: bernabe@cnm.us.es Abstract We describe an analog VLSI implementation of the ARTI algorithm (Carpenter, 1987). A prototype chip has been fabricated in a standard low cost 1.5~m double-metal single-poly CMOS process. It has a die area of lcm2 and is mounted in a 12O-pins PGA package. The chip realizes a modified version of the original ARTI architecture. Such modification has been shown to preserve all computational properties of the original algorithm (Serrano, 1994a), while being more appropriate for VLSI realizations. The chip implements an ARTI network with 100 F 1 nodes and 18 F2 nodes. It can therefore cluster 100 binary pixels input patterns into up to 18 different categories. Modular expansibility of the system is possible by assembling an NxM array of chips without any extra interfacing circuitry, resulting in an F 1 layer with l00xN nodes, and an F2 layer with 18xM nodes. Pattern classification is performed in less than 1.8~s, which means an equivalent computing power of 2.2x109 connections and connection-updates per second. Although internally the chip is analog in nature, it interfaces to the outside world through digital signals, thus having a true asynchrounous digital behavior. Experimental chip test results are available, which have been obtained through test equipments for digital chips. 1 INTRODUCTION The original ARTI algorithm (Carpenter, 1987) proposed in 1987 is a massively parallel architecture for a self-organizing neural binary-pattern recognition machine. In response to arbitrary orderings of arbitrarily many and complex binary input patterns, ARTI is 756 Initialize weights: zij = 1 Read input pattern: I = (/1' 12, ... I N) T. Serrano-Gotarredona, B. Linares-Barranco, 1. L. Huertas N N Tj = LA L Z/i-LB L Zij+LM i = 1 i = 1 Winner-Take-All: Y, = 1 if T, = maxj {Tj } y . = 0 if j:t;l J '--------------------l z,1 = In z,1 new old Fig.!: Modified Fast Learning or Type-3 ART! implementation algorithm capable of learning, in an unsupervised way, stable recognition codes. The ARTl architecture is described by a set of Short Term Memory (STM) and another set of Long Term Memory (LTM) time domain nonlinear differential equations. It is valid to assume that the STM equations settle much faster (instantaneously) than the LTM equations, so that the STM differential equations can be substituted by nonlinear algebraic equations that describe the steady-state of the STM differential equations. Furthermore, in the fast-learning mode (Carpenter, 1987), the LTM differential equations are as well substituted by their corresponding steady-state nonlinear algebraic equations. This way, the ARTI architecture can be behaviorally modelled by the sequential application of nonlinear algebraic equations. Three different levels of ARTI implementations (both in software and in hardware) can therefore be distinguished: Type-I: Full Model Implementation: both STM and LTM time-domain differential equations are realized. This implementation is the most expensive (both in software and in hardware), and requires a large amount of computational power. Type-2: STM steady-state Implementation: only the LTM time-domain differential equations are implemented. The STM behavior is governed by nonlinear algebraic equations. This implementation requires less resources than the previous one. However, a proper sequencing of STM events has to be introduced artificially, which is architecturally implicit in the Type-I implementation. Type-3: Fast Learning Implementation: STM and LTM is implemented with algebraic equations. This implementation is computationally the less expensive one. In this case an artificial sequencing of STM and LTM events has to be done. The implementation presented in this paper realizes a modified version of the original ARTI Type-3 algorithm, more suitable for VLSI implementations. Such modified ARTI system has been shown to preserve all computational properties of the original ARTI architecture (Serrano, 1994a). The flow diagram that describes the modified ARTI architecture is shown in Fig. I. Note that there is only one binary-valued weight template (Zij)' instead ofthe two weight templates (one binary-valued and the other real-valued) of the original ARTl. For a more detailed discussion of the modified ARTI algorithm refer to (Serrano, 1994a, 1994b). In the next Section we will provide an analog current-mode based circuit that implements in hardware the flow diagram of Fig. 1. Note that, although internally this circuit is analog in nature, from its input and output signals point of view it is a true asynchronous digital A Real Time Clustering CMOS Neural Engine 757 circuit, easy to interface with any conventional digital machine. Finally, in Section 3 we will provide experimental results measured from the chip using a digital data acquisition test equipment. 2 CIRCUIT DESCRIPTION The ART! chip reported in this paper has an F J layer with 100 neurons and an F2 layer with 18 neurons. This means that it can handle binary input patterns of 100 pixels each, and cluster them into up to 18 different categories; according to a digitally adjustable vigilance parameter p. The circuit architecture of the chip is shown in Fig. 2(a). It consists of an array of 18x100 synapses, a lx100 array of "vigilance synapses", a unity gain 18-outputs current mirror, an adjustable gain 18-outputs current mirror (with p=O.O, 0.1, ... 0.9)1, 18 current-comparator-controlled switches and an 18-input-currents Winner-Take-All (WTA) (Serrano, 1994b). The inputs to the circuit are the 100 binary digital input voltages Ii ' and the outputs of the circuit are the 18 digital output voltages Yj . External control signals allow to change parameters p, LA' LB , and LM . Also, extra circuitry has been added for reading the internal weights Zij while the system is learning. Each row of synapses generates two currents, 100 100 Tj = LA LZ/i-LBLZij+LM i = 1 i= 1 100 Vj = LA LZ/i i = 1 while the row of the "vigilance synapses" generates the current 100 Vp = LALli i= 1 (1) (2) Each of the current comparators compares the current V. versus pVp ' and allows current T. to reach the WTA only if pV ~ V .. This way dompetition and vigilance occur srmultaneously and in parallel, speeding tip significantly the search process. Fig. 2(b) shows the content of a synapse in the 18x 1 00 array. It consists of three current sources with switches, two digital AND gates and a flip-flop. Each synapse receives two input voltages Ii and y . , and two global control voltages <1>/ (to enable/disable learning) and reset (to initializ6 all weights zj" to '1'). Each synapse generates two currents LAlizij-LBzij and LAlizij ,which will be1summed up for all the synapses in the same row to generate the currents T. and V.. If learning is enabled (<I> / = 1) the value of Z j" will change to Iizi . if y . = f . The ,lvigilance synapses" consist each of a current-sou~ce of value L A witI~ a s~itch controlled by the input voltage Ii' The current comparators are those proposed in (Dominguez-Castro, 1992), the WTA used is reported in (Lazzaro, 1989), and the digitally adjustable current mirror is based on (Loh, 1989), while its continuous gain fine tuning mechanism has been taken from (Adams, 1991). 1. An additional pin ofthe chip can fine-tune p between 0.9 and 1.0. 758 11 12 . . . (a) T. Serrano-Gotarredona, B. Linares-Barranco, J. L. Huertas 1\00 (b) -------....... -------..... --- 1 1 , , , , :l,t.ti~j Fig. 2: (a) System Diagram of Current-Mode ART! Chipt (b) Circuit Diagram of Synapse Fig. 3: Tree based current-mirror scheme for matched current sources The circuit has been designed in such a way that the WTA operates with a precision around 1.5% (--6 bits). This means that all LA and L8 current sources have to match within an error of less than that. From a circuit implementation point of view this is not easy to achieve, since there are 5500 current sources spread over a die area of lcm2. Typical mismatch between reasonable size MOS transistors inside such an area extension can be expected to be above 10% (pelgrom, 1989). To overcome this problem we implemented a tree-based current mirror scheme as is shown in Fig. 3. Starting from a unique current reference, and using high-precision lO(or less)-outputs current mirrors (each yielding a precision around 0.2%), only up to four cascades are needed. This way, the current mismatch attained at the synapse current sources was around 1 % for currents between LA/8 = 5J.LA and LA/8 = 10~A. This is shown in Fig. 4, where the measured dc output current-error (in %) versus input current of the tree based configuration for 18 of the 3600 LA synapse sources is depicted. A Real Time Clustering CMOS Neural Engine 759 Fig. 4: Measured current mirror cascade missmatch (1 %/div) for LA for currents below 10~A 3 EXPERIMENTAL RESULTS Fig. 5 shows a microphotograph of a prototype chip fabricated in a standard digital double-metal, single-poly 1.5~m low cost CMOS process. The chip die area is Icm2, and it is mounted in a 120-pins PGA package. Fig. 6 shows a typical training sequence accomplished by the chip and obtained experimentally using a test equipment for digital chips. The only task performed by the test equipment was to provide the input data patterns I (first column in Fig. 6), detect which of the output nodes became 'I' (pattern with a vertical bar to its right), and extract the learned weights. Each IOxlO square in Fig. 6 represents either a lOa-pixels input vector I, or one row of lOa-pixels synaptic weights z. == ezi" zp ... zIOO·) . Each row of squares in Fig. 6 represents the input pattern (first square)l and the 181vectors Zj after learning has been performed for this input pattern. The sequence shown in Fig. 6 has been obtained for p = 0.7, LA = JO/lA, LB = 9.5/lA , and LM = 950/lA. Only two iterations of input patterns presentations were necessary, in this case, for the system to learn and self-organize in response to these 18 input patterns. The last row in Fig. 6 shows the final learned templates. Fig. 7 shows final learned templates for different values of p. The numbers below each square indicate the input patterns that have been clustered into each Zj category. Delay time measurements have been performed for the feedforward action of the chip (establishment of currents T., V. , and Vp ' and their competitions until the WTA settles), and for the updating of weights: The feedforward delay is pattern and bias currents (LA' LB , LM ) dependent, but has been measured to be always below 1.6/ls. The learning time is constant and is around 180ns. Therefore, throughput time is less than 1.8/ls. A digital neuroprocessor able to perform a connections/s, b connection-updates/s, and with a dedicated WTA section with a c seconds delay, must satisfy 760 T. Serrano-Gotarredona. B. Linares-Barranco. J. L Huertas Fig. 5: Microphotograph of ARTl chip 3700 + 100 + c = 1.8J..lS a b (3) to meet the performance of our ~rototype chip. If a = band c = lOOns, the equivalent speed would be a = b = 2.2 x 10 connections and connection-updates per second. 4 CONCLUSIONS A high speed analog current-mode categorizer chip has been built using a standard low cost digital CMOS process. The high performance of the chip is achieved thanks to a simplification of the original ARTl algorithm. The simplifications introduced are such that all the original computational properties are preserved. Experimental chip test results are provided. A Real Time Clustering CMOS Neural Engine Z, Z2 Z3 Z4 Zs z6 Z7 Zg Z9 Z10 Zll Z12 Z13 Z14 ZlS Z16 Z17 Zig 1 • •••••••••••••••••• 2 • •••••••••••••••••• 3. •••••••••••••••••• 4 • •••••••••••••••••• 5 • •••••••••••••••••• 6 • •••••••••••••••••• 7 • •••••••••••••••••• 8 • •••••••••••••••••• 9 • •••••••••••••••••• 10 • •••••••••••••••••• 11 • •••••••••••••••••• 12 • •••••••••••••••••• 13. •••••••••••••••••• 14 • •••••••••••••••••• 15 • •••••••••••••••••• 16 • •••••••••••••••••• 17 • •••••• U ••••••••••• 18 • •••••••••••••••••• 1 • •••••••••••••••••• 2 • •••••••••••••••••• 3 • •••••••••••••••••• 4 • •••••••••••••••••• 5 • •••••••••••••••••• 6 • •••••••••••••••••• 7 • •••••••••••••••••• 8 • •••••••••••••••••• 9_ •••••••••••••••••• 10 • •••••••••••••••••• 11 • •••••••••••••••••• 12 • •••••••••••••••••• 13 • •••••••••••••••••• 14 II •••••••••••••••••• 15 • •••••••••••••••••• 16 • •••••••••••••••••• 17 • •••••••••••••••••• 18 • •••••••••••••••••• Fig. 6: Test sequence obtained experimentally for p=O.7, LA=lO/-lA, LB=9.5/-lA, and LM=950/-lA 761 762 T. Serrano-Gotarredona, B. Linares-Barranco, J. L. Huertas Fig. 7: Categorization of the input patterns for LA=3.2~A, LB=3.0~, LM=400~A, and different values of p References W. J. Adams and J. Ramimez-Angulo. (1991) "Extended Transconductance AdjustmentlLinearisation Technique," Electronics Letters, vol. 27, No. 10, pp. 842-844, May 1991. G. A. Carpenter and S. Grossberg. (1987) "A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine," Computer VIsion, Graphics, and Image Processing, vol. 37, pp. 54-115, 1987. R. Dominguez-castro, A. Rodrfguez-Vazquez, F. Medeiro, and 1. L. Huertas. (1992) "High Resolution CMOS Current Comparators," Proc. of the 1992 European Solid-State Circuits Conference (ESSCIRC'92), pp. 242-245, 1992. J. Lazzaro, R. Ryckebusch, M. A. Mahowald, and C. Mead. (1989) "Winner-Take-All Networks of O(n) Complexity," in Advances in Neural Information Processing Systems, vol. 1, D. S. Touretzky (Ed.), Los Altos, CA: Morgan Kaufmann, 1989, pp. 703-711. K. Loh, D. L. Hiser, W. J. Adams, and R. L. Geiger. (1989) "A Robust Digitally Programmable and ReconfigurabJe Monolithic Filter Structure," Proc. of the 1989 Int. Symp. on Circuits and Systems (ISCAS'89), Portland, Oregon, vol. 1, pp. 110-113, 1989. M. J. Pelgrom, A. C. J. Duinmaijer, and A. P. G. Welbers. (1989) "Matching Properties of MOS Transistors," IEEE Journal of Solid-State Circuits, vol. 24, No.5, pp. 1433-1440, October 1989. T. Serrano-Gotarredona and B. Linares-Barranco. (1994a) "A Modified ARTl Algorithm more suitable for VLSI Implementations," submitted for publication (journal paper). T. Serrano-Gotarredona and B. Linares-Barranco. (1994b) "A Real-Time Clustering Microchip Neural Engine," submitted for publication (journal paper).	1994	130
893	Learning from queries for maximum information gain in imperfectly learnable problems Peter Sollich David Saad Department of Physics, University of Edinburgh Edinburgh EH9 3JZ, U.K. P.Sollich~ed.ac.uk. D.Saad~ed.ac.uk Abstract In supervised learning, learning from queries rather than from random examples can improve generalization performance significantly. We study the performance of query learning for problems where the student cannot learn the teacher perfectly, which occur frequently in practice. As a prototypical scenario of this kind, we consider a linear perceptron student learning a binary perceptron teacher. Two kinds of queries for maximum information gain, i.e., minimum entropy, are investigated: Minimum student space entropy (MSSE) queries, which are appropriate if the teacher space is unknown, and minimum teacher space entropy (MTSE) queries, which can be used if the teacher space is assumed to be known, but a student of a simpler form has deliberately been chosen. We find that for MSSE queries, the structure of the student space determines the efficacy of query learning, whereas MTSE queries lead to a higher generalization error than random examples, due to a lack of feedback about the progress of the student in the way queries are selected. 1 INTRODUCTION In systems that learn from examples, the traditional approach has been to study generalization from random examples, where each example is an input-output pair 288 Peter Sollich, David Saad with the input chosen randomly from some fixed distribution and the corresponding output provided by a teacher that one is trying to approximate. However, random examples contain less and less new information as learning proceeds. Therefore, generalization performance can be improved by learning from queries, i. e., by choosing the input of each new training example such that it will be, together with its expected output, in some sense 'maximally useful'. The most widely used measure of 'usefulness' is the information gain, i.e., the decrease in entropy of the post-training probability distributions in the parameter space of the student or the teacher. We shall call the resulting queries 'minimum (student or teacher space) entropy (MSSE/MTSE) queries'; their effect on generalization performance has recently been investigated for perfectly learnable problems, where student and teacher space are identical (Seung et al., 1992, Freund et al., 1993, Sollich, 1994), and was found to depend qualitatively on the structure of the teacher. For a linear perceptron, for example, one obtains a relative reduction in generalization error compared to learning from random examples which becomes insignificant as the number of training examples, p, tends to infinity. For a perceptron with binary output, on the other hand, minimum entropy queries result in a generalization error which decays exponentially as p increases, a marked improvement over the much slower algebraic decay with p in the case of random examples. In practical situations, one almost always encounters imperfectly learnable problems, where the student can only approximate the teacher, but not learn it perfectly. Imperfectly learnable problems can arise for two reasons: Firstly, the teacher space (i.e., the space of models generating the data) might be unknown. Because the teacher space entropy is then also unknown, MSSE (and not MTSE) queries have to be used for query learning. Secondly, the teacher space may be known, but a student of a simpler structure might have deliberately been chosen to facilitate or speed up training, for example. In this case, MTSE queries could be employed as an alternative to MSSE queries. The motivation for doing this would be strongest if, as in the learning scenario that we consider below, it is known from analyses of perfectly learnable problems that the structure of the teacher space allows more significant improvements in generalization performance from query learning than the structure of the student space. With the above motivation in mind, we investigate in this paper the performance of both MSSE and MTSE queries for a prototypical imperfectly learnable problem, in which a linear perceptron student is trained on data generated by a binary perceptron teacher. Both student and teacher are specified by an N-dimensional weight vector with real components, and we will consider the thermodynamic limit N -+ 00, p -+ 00,0:' = piN = const. In Section 2 below we calculate the generalization error for learning from random examples. In Sections 3 and 4 we compare the result to MSSE and MTSE queries. Throughout, we only outline the necessary calculations; for details, we refer the reader to a forthcoming publication. We conclude in Section 5 with a summary and brief discussion of our results. 2 LEARNING FROM RANDOM EXAMPLES We denote students and teachers by N (for 'Neural network') and V (for 'element of the Version space', see Section 4), respectively, and their corresponding weight Learning from Queries for Maximum Information Gain 289 vectors by W Nand w v . For an input vector x, the outputs of a given student and teacher are YN = 7N XTWN' Yv = sgn( 7N xTwv) . Assuming that inputs are drawn from a uniform distribution over the hypersphere x 2 = N, and taking as our error measure the standard squared output difference ! (YN - Yv)2, the generalization error, i. e., the average error between student Nand teacher V when tested on random test inputs, is given by 1 [ R (2) 1/2] €g(N, V) = '2 QN + 1- 2..ftTv;: , (1) where we have set R = ];WrrWv,QN = ];w~,Qv = ];w~. As our training algorithm we take stochastic gradient descent on the trammg error Et , which for a training set e(p) = {( x~ ,Y~ = Yv( x~)), J.l = 1 ... p} is Et = ! 2:~(Y~ YN(X~))2. A weight decay term !AW~ is added for regularization, i.e., to prevent overfitting. Stochastic gradient descent on the resulting energy function E = Et + !AW~ yields a Gibbs post-training distribution of students, p(wNle(p)) ex exp(-E/T), where the training temperature T measures the amount of stochasticity in the training algorithm. For the linear perceptron students considered here, this distribution is Gaussian, with covariance matrix TM,N-l, where (IN denotes the N x N identity matrix) MN = AlN + ~ 2:~=1 x~(x~f· Since the length ofthe teacher weight vector Wv does not affect the teacher outputs, we assume a spherical prior on teacher space, P(wv ) ex 6(w~-N), for which Qv = 1. Restricting attention to the limit of zero training temperature, it is straightforward to calculate from eq. (1) the average generalization error obtained by training on random examples 1 [ OG] €g €g,min =;: AoptG + A(Aopt - A) OA ' with the function G = (]; tr M,N-1) P( {xl'}) given by (Krogh and Hertz, 1992) G = 2\ [1 - a - A + )(1- a - A)2 + 4A] . (2) (3) In eq. (2) we have explicitly subtracted the minimum achievable generalization error, €g,min = !(1-2/11'), which is nonzero since a linear perceptron cannot approximate a binary percept ron perfectly. At finite a, the generalization error is minimized when the weight decay is set to its optimal value A = Aopt = 11'/2 - 1. Note that since both G and OG/OA tend to zero as a --+ 00, the generalization error for random examples approaches the minimum achievable generalization error in this limit. 3 MINIMUM STUDENT SPACE ENTROPY QUERIES We now calculate the generalization performance resulting from MSSE queries. For the training algorithm introduced in the last section, the student space entropy (normalized by N) is given by 290 Peter Sollich, David Saad 3.0 ""T""----r----r---------------, -- A== 0.01 2.5 ---- A == 0.1 -------- A == 1 2.0 1.5 -------------------::~-::-=.:--~~.J .. _-----------1.0-+--.::..::......-.-----,----.-----r-----I o 2 a 3 4 5 Figure 1: Relative improvement K, in generalization error due to MSSE queries, for weight decay A = 0.01,0.1,1. 1 S'" = - 2N lndet M N , where we have omitted an unimportant constant which depends on the training temperature only. This entropy is minimized by choosing each new query along the direction corresponding to the minimal eigenvalue of the existing MN (Sollich, 1994). The expression for the resulting average generalization error is given by eq. (2) with G replaced by its analogue for MSSE queries (Sollich, 1994) G ~a 1- ~a Q+----:A + [a] + 1 A + [a] , where [a] is the greatest integer less than or equal to a and ~o' = a-raj. We define the improvement factor K, as the ratio of the generalization error (with the minimum achievable generalization error subtracted as in eq. (2)) for random examples to that for MSSE queries. Figure 1 shows K(O') for several values of the weight decay A. Comparing with existing results (Sollich, 1994), we find that K, is exactly the same as if our linear student were trying to approximate a linear teacher with additive noise of variance Aopt on the outputs. For large a, one can show (Sollich, 1994) that K, = 1 + I/O' + O(I/a2) and hence the relative reduction in generalization error due to querying tends to zero as a -I- 00. We investigate in the next section whether it is possible to improve generalization performance more significantly by using MTSE queries. 4 MINIMUM TEACHER SPACE ENTROPY QUERIES We now consider the generalization performance achieved by MTSE queries. We remind readers that such queries could be used if the teacher space is known, but a student of a simpler functional form has deliberately chosen. The aim in using MTSE rather than MSSE queries would be to exploit the structure of the teacher space if this is known (for perfectly learnable problems) to make query learning very efficient compared to random examples. For the case of noise free training data under consideration, the posterior probability distribution in teacher space given a certain training set is proportional to the prior Learning from Queries for Maximum Information Gain 291 distribution on the version space (the set of all teachers that could have produced the training set without error) and zero everywhere else. From this the (normalized) teacher space entropy can be derived to be, up to an additive constant, 1 Sv = N In V(p), where the version space volume V(p) is given by (8(z) = 1 for z > 0 and 0 otherwise) V(p) = JdwvP(wv) n~=l 8(JNy/Jw~x/J). It can easily be verified that this entropy is minimizedl by choosing queries x which (bisect' the existing version space, i. e., for which the hyperplane perpendicular to x splits the version space into two equal halves (Seung et al., 1992, Freund et al., 1993). Such queries lead to an exponentially shrinking version space, V(p) = 2-P, and hence a linear decrease of the entropy, Sv = -a In 2. We consider instead queries which achieve qualitatively the same effect, but permit a much simpler analysis of the resulting student performance. They are similar to those studied in the context of a learnable problem by Watkin and Rau (1992), and are defined as follows. The (p + 1 )th query is obtained by first picking a random teacher vector wp from the version space defined by the existing p training examples, and then picking the new training input Xp+l from the distribution of random inputs but under the constraint that x;+1wp = O. For the calculation of the student performance, i. e., the average generalization error, achieved by the approximate MTSE queries described above, we use an approximation based on the following observation. As the number of training examples, p, increases, the teacher vectors wp from the version space will align themselves with the true teacher w~; their components along the direction of w~ will increase, whereas their components perpendicular to we will decrease, varying widely across the N - 1 dimensional hyperplane perpendicular to we . Following Watkin and Rau (1992), we therefore assume that the only significant effect of choosing queries xp+1 with X;+lWP = 0 is on the distribution of the component ofxp+l along we . Writing this component as x~+1 = x;+1 w~/lwel, its probability distribution can readily be shown to be P(x~+1) ex: exp (-~(x~+1lsp)2) , (4) where sp is the sine of the angle between w p and we. For finite N, the value of sp is dependent on the p previous training examples that define the existing version space and on the teacher vector wp sampled randomly from this version space. In the thermodynamic limit, however, the variations of sp become vanishingly small and we can thus replace sp by its average value, which is a function of palone. In the thermodynamic limit, this average value becomes a continuous function of a = pIN, the number of training examples per weight, which we denote simply by sea). The calculation can then be split into two parts: First, the function sea) is obtained from a calculation of the teacher space entropy using the replica method, generalizing the results of Gyorgi and Tishby (1990). The average generalization 1 More precisely, what is minimized is the value of the entropy after a new training example (x, y) is added, averaged over the distribution of the unknown new training output y given the new training input x and the existing training set; see Sollich (1994). 292 0 -1 -2 -3 -4 -5 0 Peter Sollich, David Saad ................• , ....... ,., ..... ,. --':''''''::-.::, ---..... -------............ ' . ...•.....••...... 2 3 4 5 a 6 7 8 9 10 Figure 2: MTSE queries: Teacher space entropy, Sv (with value for random examples plotted for comparison), and In s, the log of the sine of the angle between the true teacher and a random teacher from the version space. error can then be calculated by using an extension of the response function method described in (Sollich, 1994b) or by another replica calculation (now in student space) as in (Dunmur and Wallace, 1993). Figure 2 shows the effects of (approximate) MTSE queries in teacher space. For large a values, the teacher space entropy decreases linearly with a, with gradient c ::::::: 0.44, whereas the entropy for random examples, also shown for comparison, decreases much more slowly (asymptotically like -In a, see (Gyorgi and Tishby, 1990)). The linear a-dependence of the entropy for queries corresponds to an average reduction of the version space volume with each new training example by a factor of exp( -c) ::::::: 0.64, which is reasonably close to the factor ~ for proper bisection of the version space. This justifies our choice of analysing approximate MTSE queries rather than true MTSE queries, since the former achieve qualitatively the same results as the latter. Before discussing the student performance achieved by (approximate) MTSE queries, we note from figure 2 that In s( a) decreases linearly with a for large a, with the same gradient as the teacher space entropy. Hence s( a) ex: exp( -ca) for large a, and MTSE queries force the average teacher from the version space to approach the true teacher exponentially quickly. It can easily be shown that if we were learning with a binary perceptIOn student, i. e., if the problem were perfectly learnable, then this would result in an exponentially decaying generalization error, €g ex: exp( -ca). MTSE queries would thus lead to a marked improvement in generalization performance over random examples (for which €g ex: l/a, see (Gyorgi and Tishby, 1990)). It is this significant benefit (in teacher space) of query learning that provides the motivation for using MTSE queries in imperfectly learnable problems such as the one considered here. The results plotted in Figure 3 for the average generalization error achieved by the linear perceptron student show, however, that MTSE queries do not have the desired effect. Far from translating the benefits in teacher space into improvements in generalization performance for the linear student, they actually lead to a deterioration of generalization performance, i. e., a larger generalization error than that Learning from Queries for Maximum Information Gain 0.6 0.5 fg 0.4 0.3 0.2 o ... \-, " , '.. , ". , '. , ...... .. -~ -2 3 -- >. = 0.01 ---- >. = 0.1 -------- >. = 1 - --- ---------------......-=---=--=---=--d-4 5 a 6 7 8 9 10 293 Figure 3: Generalization error for MTSE queries (higher curves of each pair) and random examples (lower curves), for weight decay>. = 0.01,0.1,1. The curves for random examples (which are virtually indistinguishable from one another already at a = 10) converge to the minimum achievable generalization error fg,min (dotted line) as a -+ 00. obtained for random examples. Worse still, they 'mislead' the student to such an extent that the minimum achievable generalization error is not reached even for an infinite number of training examples, a -+ 00. How does this happen? It can be verified that the angle between the student and teacher weight vectors tends to zero for a -+ 00 as expected, while Q N, the normalized squared length of the student weight vector, approaches 2 ( S )2 QN(a -+ 00) = ---= , 11' >. + s2 (5) where s = Jooo da s( a), s2 = Jooo da s2 (a). Unless the weight decay parameter >. happens to be equal to s - s2, this is different from the optimal asymptotic value, which is 2/11'. This is the reason why in general the linear student does not reach the minimum possible generalization error even as a -+ 00. The approach of QN to its non-optimal asymptotic value can cause an increase in the generalization error for large a and a corresponding minimum of the generalization error at some finite a, as can be seen in the plots for>. = 0.01 and 0.1 in Figure 3. For>. = 0, eq. (5) has the following intuitive interpretation: As a increases, the version space shrinks around the true teacher w~, and hence MTSE queries become 'more and more orthogonal' to w~ . As a consequence, the distribution of training inputs along the direction of we is narrowed down progressively (compare eq. (4)). Trying to find a best fit to the teacher's binary output function over this narrower range of inputs, the linear student learns a function which is steeper than the best fit over the range of random inputs (which would give minimum generalization error). This corresponds to a suboptimally large length of the student weight vector in agreement with eq. (5): QN(a -+ 00) > 2/11' for>. = a because s2 < s. Summarizing the results of this section, we have found that although MTSE queries are very beneficial in teacher space, they are entirely misleading for the linear student, to the extent that the student does not learn to approximate the teacher optimally even for an infinite number of training examples. 294 Peter Sollich, David Saad 5 SUMMARY AND DISCUSSION We have found in our study of an imperfectly learnable problem with a linear student and a binary teacher that queries for minimum student and teacher space entropy, respectively, have very different effects on generalization performance. Minimum student space entropy (MSSE) queries essentially have the same effect as for a linear student learning a noisy linear teacher, apart from a nonzero minimum value of the generalization error due to the unlearnability of the problem. Hence the structure of the student space is the dominating influence on the efficacy of query learning. Minimum teacher space entropy queries (MTSE), on the other hand, perform worse than random examples, leading to a higher generalization error even for an infinite number of training examples. With the benefit of hindsight, we note that this makes intuitive sense since the teacher space entropy, according to which MTSE queries are selected, contains no feedback about the progress of the student in learning the required generalization task, and thus MTSE queries cannot be guaranteed to have a positive effect. Our results, then, are a mixture of good and bad news for query learning for maximum information gain in imperfectly learnable problems: The bad news is that MTSE queries, due to a lack of feedback information about student progress, are not enough to translate significant benefits in teacher space into similar improvements of student performance and may in fact yield worse performance than random examples. The good news is that for MSSE queries, we have found evidence that the structure of the student space is the key factor in determining the efficacy of query learning. If this result holds more generally, then statements about the benefits of query learning can be made on the basis of how one is trying to learn only, independently of what one is trying to learn-a result of great practical significance. References A P Dunmur and D J Wallace (1993). Learning and generalization in a linear perceptron stochastically trained with noisy data. J. Phys. A, 26:5767-5779. Y Freund, H S Seung, E Shamjr, and N Tishby (1993). Information, prediction, and query by committee. In S J Hanson, J D Cowan, and C Lee Giles, editors, NIPS 5, pages 483-490, San Mateo, CA, Morgan Kaufmann. G Gyorgi and N Tishby (1990). Statistical theory of learning a rule. In W Theumann and R Koberle, editors, Neu.ral Networks and Spin Glasses, pages 3-36. Singapore, World Scientific. A Krogh and J A Hertz (1992). Generalization in a linear perceptron in the presence of noise. J. Phys. A,25:1135-1147. P Sollich (1994). Query construction, entropy, and generalization in neural network models. Phys. Rev. E,49:4637-4651. P Sollich (1994b). Finite-size effects in learning and generalization in linear perceptrons. J. Phys. A, 27:7771-7784. H S Seung, M Opper, and H Sompolinsky (1992). Query by committee. In Proceedings of COLT '92, pages 287-294, New York, ACM. T L H Watkin and A Rau (1992). Selecting examples for perceptrons. J. Phys. A, 25:113-121.	1994	131
894	Factorial Learning by Clustering Features Joshua B. Tenenbaum and Emanuel V. Todorov Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 {jbt.emo}~psyche . mit.edu Abstract We introduce a novel algorithm for factorial learning, motivated by segmentation problems in computational vision, in which the underlying factors correspond to clusters of highly correlated input features. The algorithm derives from a new kind of competitive clustering model, in which the cluster generators compete to explain each feature of the data set and cooperate to explain each input example, rather than competing for examples and cooperating on features, as in traditional clustering algorithms. A natural extension of the algorithm recovers hierarchical models of data generated from multiple unknown categories, each with a different, multiple causal structure. Several simulations demonstrate the power of this approach. 1 INTRODUCTION Unsupervised learning is the search for structure in data. Most unsupervised learning systems can be viewed as trying to invert a particular generative model of the data in order to recover the underlying causal structure of their world. Different learning algorithms are then primarily distinguished by the different generative models they embody, that is, the different kinds of structure they look for. Factorial learning, the subject of this paper, tries to find a set of independent causes that cooperate to produce the input examples. We focus on strong factorial learning, where the goal is to recover the actual degrees of freedom responsible for generating the observed data, as opposed to the more general weak approach, where the goal 562 Joshua B. Tenenbaum, Emmanuel V. Todorov Figure 1: A simple factorial learning problem. The learner observes an articulated hand in various configurations, with each example specified by the positions of 16 tracked features (shown as black dots). The learner might recover four underlying factors, corresponding to the positions of the fingers, each of which claims responsiblity for four features of the data set. is merely to recover some factorial model that explains the data efficiently. Strong factorial learning makes a claim about the nature of the world, while weak factorial learning only makes a claim about the nature of the learner's representations (although the two are clearly related). Standard subspace algorithms, such as principal component analysis, fit a linear, factorial model to the input data, but can only recover the true causal structure in very limited situations, such as when the data are generated by a linear combination of independent factors with significantly different variances (as in signal-from-noise separation). Recent work in factorial learning suggests that the general problem of recovering the true, multiple causal structure of an arbitrary, real-world data set is very difficult, and that specific approaches must be tailored to specific, but hopefully common, classes of problems (Foldiak, 1990; Saund, 1995; Dayan and Zemel, 1995). Our own interest in multiple cause learning was motivated by segmentation problems in computational vision, in which the underlying factors correspond ideally to disjoint clusters of highly correlated input features. Examples include the segmentation of articulated objects into functionally independent parts, or the segmentation of multiple-object motion sequences into tracks of individual objects. These problems, as well as many other problems of pattern recognition and analysis, share a common set of constraints which makes factorial learning both appropriate and tractable. Specifically, while each observed example depends on some combination of several factors, anyone input feature always depends on only one such factor (see Figure 1). Then the generative model decomposes into independent sets of functionally grouped input features, or functional parts (Tenenbaum, 1994). In this paper, we propose a learning algorithm that extracts these functional parts. The key simplifying assumption, which we call the membership constmint, states that each feature belongs to at most one functional part, and that this membership is constant over the set of training examples. The membership constraint allows us to treat the factorial learning problem as a novel kind of clustering problem. The cluster generators now compete to explain each feature of the data set and cooperate to explain each input example, rather than competing for examples and cooperating on features, as in traditional clustering systems such as K-means or mixture models. The following sections discuss the details of the feature clustering algorithm for extracting functional parts, a simple but illustrative example, and extensions. In particular, we demonstrate a natural way to relax the strict membership constraint and thus learn hierarchical models of data generated from multiple unknown categories, each with a different multiple causal structure. Factorial Learning by Clustering Features 563 2 THE FEATURE CLUSTERING ALGORITHM Our algorithm for extracting functional parts derives from a statistical mechanics formulation of the soft clustering problem (inspired by Rose, Gurewitz, and Fox, 1990; Hinton and Zemel, 1994). We take as input a data set {xt}, with I examples of J real-valued features. The best K-cluster representation of these J features is given by an optimal set of cluster parameters, {Ok}, and an optimal set of assignments, {Pi/.J. The assignment Pjk specifies the probability of assigning feature j to cluster k, and depends directly on Ejk = 2:i(Xj'l fj~)2, the total squared difference (over the I training examples) between the observed feature values x}il and cluster k's predictions fj~. The parameters Ok define cluster k's generative model, and thus determine the predictions fj~(Ok). If we limit functional parts to clusters of linearly correlated features, then the appropriate generative model has fj~ = WjkYkil + Uj, with cluster parameters Ok = {Ykil , Wjk, Uj} to be estimated. That is, for each example i, part k predicts the value of input feature j as a linear function of some part-specific factor Ykil (such as finger position in Figure 1). For the purposes of this paper, we assume zero-mean features and ignore the Uj terms. Then Ejk = 2:i(Xj'l - WjkYkil )2. The optimal cluster parameters and assignments can now be found by maximizing the complete log likelihood of the data given the K-cluster representation, or equivalently, in the framework of, statistical mechanics, by minimizing the free energy 1 1 F = E - -H = LLPjk(Ejk + -logpjk) (1) (3 j k (3 subject to the membership constraints, 2:k pjk = 1, (\lj). Minimizing the energy, E = L LPjkEjk, (2) j k reduces the expected reconstruction error, leading to more accurate representations. Maximizing the entropy, H = - LLPjklogPjk, (3) j k distributes responsibility for each feature across many parts, thus decreasing the independence of the parts and leading to simpler representations (with fewer degrees of freedom). In line with Occam's Razor, minimizing the energy-entropy tradeoff finds the representation that, at a particular temperature 1/(3, best satisfies the conflicting requirements of low error and low complexity. We minimize the free energy with a generalized EM procedure (Neal and Hinton, 1994), setting derivatives to zero and iterating the resulting update equations: e-(3Ejk Pjk = 2:k e-(3Ejk Ykil LPjkWjkXj'l j Wjk = Lxj'lYtl. (4) (5) (6) 564 Joshua B. Tenenbaum. Emmanuel V. Todorov This update procedure assumes a normalization step Ykil = Ykil /(Eil (y(l)2)1/2 in each iteration, because without some additional constraint on the magnitudes of ytl (or Wjk), inverting the generative model fj~ = WjkYkil is an ill-posed problem. This algorithm maps naturally onto a simple network architecture. The hidden unit activities, representing the part-specific factors Ykil , are computed from the observations xyl via bottom-up weights PjkWjk, normalized, and multiplied by top-down weights Wjk to generate the network's predictions fj~. The weights adapt according to a hybrid learning rule, with Wjk determined by a Hebb rule (as in subspace learning algorithms), and pjk determined by a competitive, softmax function of the reconstruction error Ejk (as in soft mixture models). 3 LEARNING A HIERARCHY OF PARTS The following simulation illustrates the algorithm's behavior on a simple, part segmentation task. The training data consist of 60 examples with 16 features each, representing the horizontal positions of 16 points on an articulated hand in various configurations (as in Figure 1). The data for this example were generated by a hierarchical, random process that produced a low correlation between all 16 features, a moderate correlation between the four features on each finger, and a high correlation between the two features on each joint (two joints per finger). To fully explain this data set, the algorithm should be able to find a corresponding hierarchy of increasingly complex functional part representations. To evaluate the network's representation of this data set, we inspect the learned weights PjkWjk, which give the total contribution of feature j to part k in (5). In Figure 2, these weights are plotted for several different values of /3, with gray boxes indicating zero weights, white indicating strong positive weights, and black indicating strong negative weights. The network was configured with K = 16 part units, to ensure that all potential parts could be found. When fewer than K distinct parts are found, some of the cluster units have identical parameters (appearing as identical columns in Figure 2). These results were generated by deterministic annealing, starting with /3 « 1, and perturbing the weights slightly each time /3 was increased, in order to break symmetries. Figure 2 shows that the number of distinct parts found increases with /3, as more accurate (and more complex) representations become favored. In (4), we see that /3 controls the number of distinct parts via the strength of the competition for features. At /3 = 0, every part takes equal responsibility for every feature. Without competition, there can be no diversity, and thus only one distinct part is discovered at low /3, corresponding to the whole hand (Figure 2a). As /3 increases, the competition for features gets stiffer, and parts split into their component subparts. The network finds first four distinct parts (with four features each), corresponding to individual fingers (Figure 2c), and then eight distinct parts (with two features each), corresponding to individual joints (Figure 2d). Figure 2b shows an intermediate representation, with something between one and four parts. Four distinct columns are visible, but they do not cleanly segregate the features. Figure 3 plots the decrease in mean reconstruction error (expressed by the energy E) Factorial Learning by Clustering Features (a) (e) ~=1 ~ ~~+-~~+-~~~~~~ ~ ~~+-r+~+-~-r~r+-r~ ~ r4~+-r+~4=P+-r~r+~~ Part k ~= 1000 Part k 565 (b) ~= 100 Part k (d) ~=2oo00 Part k Figure 2: A hierarchy of functional part representations, parameterized by {3. F---e------___ b d 2 3 4 log ~ Figure 3: A phase diagram distinguishes true parts (a, c, d) from spurious ones (b). 566 Joshua B. Tenenbaum. Emmanuel V. Todorov as (3 increases and more distinct parts emerge. Notice that within the three stable phases corresponding to good part decompositions (Figures 2a, 2c, 2d), E remains practically constant over wide variations in (3. In contrast, E varies rapidly at the boundaries between phases, where spurious part structure appears (Figure 2b). In general, good representations should lie at stable points of this phase diagram, where the error-complexity tradeoff is robust. Thus the actual number of parts in a particular data set, as well as their hierarchical structure, need not be known in advance, but can be inferred from the dynamics of learning. 4 LEARNING MULTIPLE CATEGORIES Until this point, we have assumed that each feature belongs to at most one part over the entire set of training examples, and tried to find the single K -part model that best explains the data as a whole. But the notion that a single model must explain the whole data set is quite restrictive. The data may contain several categories of examples, each characterized by a different pattern of feature correlations, and then we would like to learn a set of models, each capturing the distinctive part structure of one such category. Again we are motivated by human vision, which easily recognizes many categories of motion defined by high-level patterns of coordinated part movement, such as hand gestures and facial expressions. If we know which examples belong to which categories, learning multiple models is no harder than learning one, as in the previous section. A separate model can be fit to each category m of training examples, and the weights P'j'k wjk are frozen to produce a set of category templates. However, if the category identities are unknown, we face a novel kind of hierarchical learning task. We must simultaneously discover the optimal clustering of examples into categories, as well as the optimal clustering of features into parts within each category. We can formalize this hierarchical clustering problem as minimizing a familiar free energy, (7) in which gim. specifies the probability of assigning example i to category m, and Tim. is the associated cost. This cost is itself the free energy of the mth K -part model on the ith example, Tim = L LP'j'k(Ejk' + _(31 logp'j'k) , j k (8) in which P'j'k specifies the probability of assigning feature j to part k within category m, and Ejk' = (xj Wjky~m.)2 is the usual reconstruction error from Section 2. This algorithm was tested on a data set of 256 hand configurations with 20 features each (similar to those in Figure 1), in which each example expresses one of four possible "gestures", i.e. patterns of feature correlation. As Table 1 indicates, the five features on each finger are highly correlated across the entire data set, while variable correlations between the four fingers distinguish the gesture categories. Note that a single model with four parts explains the full variance of the data just as well as the actual four-category generating process. However, most of the data Factorial Learning by Clustering Features 567 Table 1: The 20 features are grouped into either 2, 3, or 4 functional parts. Examples No. of parts Part composition 1- 64 2 1 10 11----20 65 - 127 3 1 10 11-15 16--20 128 - 192 3 1-5 6-10 11 20 193 - 256 4 1-5 6--10 11-15 16--20 can also be explained by one of several simpler models, making the learner's task a challenging balancing act between accuracy and simplicity. Figure 4 shows a typical representation learned for this data set. The algorithm was configured with M = 8 category models (each with K = 8 parts), but only four distinct categories of examples are found after annealing on a (holding {3 constant), and their weights prk wjk are depicted in Figure 4a. Each category faithfully captures one of the actual generating categories in Table 1, with the correct number and composition of functional parts. Figure 4b depicts the responsibility gi'm that each learned category m takes for each feature i. Notice the inevitable effect of a bias towards simpler representations. Many examples are misassigned relative to Table 1, when categories with fewer degrees of freedom than their true generating categories can explain them almost as accurately. 5 CONCLUSIONS AND FUTURE DIRECTIONS The notion that many data sets are best explained in terms of functionally independent clusters of correlated features resonates with similar proposals of Foldiak (1990), Saund (1995), Hinton and Zemel (1994), and Dayan and Zemel (1995). Our approach is unique in actually formulating the learning task as a clustering problem and explicitly extracting the functional parts of the data. Factorial learning by clustering features has three principal advantages. First, the free energy cost function for clustering yields a natural complexity scale-space of functional part representations, parameterized by {3. Second, the generalized EM learning algorithm is simple and quick, and maps easily onto a network architecture. Third, by nesting free energies, we can seamlessly compose objective functions for quite complex, hierarchical unsupservised learning problems, such as the multiple category, multiple part mixture problem of Section 4. The primary limitation of our approach is that when the generative model we assume does not in fact apply to the data, the algorithm may fail to recover any meaningful structure. In ongoing work, we are pursuing a more flexible generative model that allows the underlying causes to compete directly for arbitrary featureexample pairs ij, rather than limiting competition only to features j, as in Section 2, or only to examples i, as in conventional mixture models, or segregating competition for examples and features into hierarchical stages, as in Section 4. Because this introduces many more degrees of freedom, robust learning will require additional constraints, such as temporal continuity of examples or spatial continuity of features. 568 (a) (b) 1 E Category 1 Part k Joshua B. Tenenbaum, Emmanuel V. Todorov category 2 category 3 category 4 Part k Part k Part k 521 .. --------~ .. ~w.~· Cl * () 31------~····· .... ..... . 64 128 Example i 192 256 Figure 4: Learning multiple categories, each with a different part structure. Acknowledgements Both authors are Howard Hughes Medical Institute Predoctoral Fellows. We thank Whitman Richards, Yair Weiss, and Stephen Gilbert for helpful discussions. References Dayan, P. and Zemel, R. S. (1995). Competition and multiple cause models. Neural Computation, in press. Foldiak, P. (1990). Forming sparse representations by local anti-hebbian learning. Biological Cybernetics 64, 165-170. Hinton, G. E. and Zemel, R. S. (1994). Autoencoders, minimum description length and Helmholtz free energy. In J. D. Cowan, G. Tesauro, & J. Alspector (eds.), Advances in Neural Information Processing Systems 6. San Mateo, CA: Morgan Kaufmann, 3-10. Neal, R. M., and Hinton, G. E. (1994). A new view of the EM algorithm that justifies incremental and other variants. Rose, K., Gurewitz, F., and Fox, G. (1990). Statistical mechanics and phase transitions in clustering. Physical Review Letters 65, 945-948. Saund, E. (1995). A mUltiple cause mixture model for unsupervised learning. Neural Computation 7,51-71. Tenenbaum, J. (1994). Functional parts. In A. Ram & K. Eiselt (eds.), Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society. Hillsdale, N J: Lawrence Erlbaum, 864-869.	1994	132
895	Generalization in Reinforcement Learning: Safely Approximating the Value Function Justin A. Boyan and Andrew W. Moore Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 jab@cs.cmu.edu, awm@cs.cmu.edu Abstract A straightforward approach to the curse of dimensionality in reinforcement learning and dynamic programming is to replace the lookup table with a generalizing function approximator such as a neural net. Although this has been successful in the domain of backgammon, there is no guarantee of convergence. In this paper, we show that the combination of dynamic programming and function approximation is not robust, and in even very benign cases, may produce an entirely wrong policy. We then introduce Grow-Support, a new algorithm which is safe from divergence yet can still reap the benefits of successful generalization. 1 INTRODUCTION Reinforcement learning-the problem of getting an agent to learn to act from sparse, delayed rewards-has been advanced by techniques based on dynamic programming (DP). These algorithms compute a value function which gives, for each state, the minimum possible long-term cost commencing in that state. For the high-dimensional and continuous state spaces characteristic of real-world control tasks, a discrete representation of the value function is intractable; some form of generalization is required. A natural way to incorporate generalization into DP is to use a function approximator, rather than a lookup table, to represent the value function. This approach, which dates back to uses of Legendre polynomials in DP [Bellman et al., 19631, has recently worked well on several dynamic control problems [Mahadevan and Connell, 1990, Lin, 1993] and succeeded spectacularly on the game of backgammon [Tesauro, 1992, Boyan, 1992]. On the other hand, many sensible implementations have been less successful [Bradtke, 1993, Schraudolph et al., 1994]. Indeed, given the well-established success 370 Justin Boyan, Andrew W. Moore on backgammon, the absence of similarly impressive results appearing for other games is perhaps an indication that using function approximation in reinforcement learning does not always work well. In this paper, we demonstrate that the straightforward substitution of function approximators for lookup tables in DP is not robust and, even in very benign cases, may diverge, resulting in an entirely wrong control policy. We then present Grow-Support, a new algorithm designed to converge robustly. Grow-Support grows a collection of states over which function approximation is stable. One-step backups based on Bellman error are not used; instead, values are assigned by performing "rollouts" -explicit simulations with a greedy policy. We discuss potential computational advantages of this method and demonstrate its success on some example problems for which the conventional DP algorithm fails. 2 DISCRETE AND SMOOTH VALUE ITERATION Many popular reinforcement learning algorithms, including Q-Iearning and TD(O), are based on the dynamic programmin~ algorithm known as value iteration [Watkins, 1989, Sutton, 1988, Barto et al., 1989J, which for clarity we will call discrete value iteration. Discrete value iteration takes as input a complete model of the world as a Markov Decision Task, and computes the optimal value function J: J (x) = the minimum possible sum of future costs starting from x To assure that J* is well-defined, we assume here that costs are nonnegative and that some absorbing goal state-with all future costs O-is reachable from every state. For simplicity we also assume that state transitions are deterministic. Note that J* and the world model together specify a "greedy" policy which is optimal for the domain: optimal action from state x = argmin(CosT(x, a) + J(NEXT-STATE(X, a))) aEA We now consider extending discrete value iteration to the continuous case: we replace the lookup table over all states with a function approximator trained over a sample of states. The smooth value iteration algorithm is given in the appendix. Convergence is no longer guaranteed; we instead recognize four possible classes of behavior: good convergence The function approximator accurately represents the intermediate value functions at each iteration (that is, after m iterations, the value function correctly represents the cost of the cheapest m-step path), and successfully converges to the optimal J value function. lucky convergence The function approximator does not accurately represent the intermediate value functions at each iteration; nevertheless, the algorithm manages to converge to a value function whose greedy policy is optimal. bad convergence The algorithm converges, i.e. the target J-values for the N training points stop changing, but the resulting value function and policy are poor. divergence Worst of all: small fitter errors may become magnified from one iteration to the next, resulting in a value function which never stops changing. The hope is that the intermediate value functions will be smooth and we will achieve "good convergence." Unfortunately, our experiments have generated all four of these behaviors-and the divergent behavior occurs frequently, even for quite simple problems. Generalization in Reinforcement Learning: Safely Approximating the Value Function 37 J 2.1 DIVERGENCE IN SMOOTH VALUE ITERATION We have run simulations in a variety of domains-including a continuous gridworld, a car-on-the-hill problem with nonlinear dynamics, and tic-tac-toe versus a stochastic opponent-and using a variety of function approximators, including polynomial regression, backpropagation, and local weighted regression. In our experiments, none of these function approximators was immune from divergence. The first set ofresults is from the 2-D continuous gridworld, described in Figure 1. By quantizing the state space into a 100 x 100 grid, we can compute J* with discrete value iteration, as shown in Figure 2. The optimal value function is exactly linear: J(x, y) = 20 - lOx - lOy. Since J is linear, one would hope smooth value iteration could converge to it with a function approximator as simple as linear or quadratic regression. However, the intermediate value functions of Figure 2 are not smooth and cannot be fit accurately by a low-order polynomial. Using linear regression on a sample of 256 randomly-chosen states, smooth value iteration took over 500 iterations before "luckily" converging to optimal. Quadratic regression, though it always produces a smaller fit error than linear regression, did not converge (Figure 3). The quadratic function, in trying to both be flat in the middle of state space and bend down toward 0 at the goal corner, must compensate by underestimating the values at the corner opposite the goal. These underestimates then enlarge on each iteration, as the one-step DP lookaheads erroneously indicate that points can lower their expected cost-to-go by stepping farther away from the goal. The resulting policy is anti-optimal. fontinuous Gridworld 0.8 0.6 >. 0.4 0.2 0L-0~.~2-0~.~4-0~.~6~0~.~8~1 x J(x,y) Figure 1: In the continuous gridworld domain, the state is a point (x, y) E [0,1]2. There are four actions corresponding to short steps (length 0.05, cost 0.5) in each compass direction, and the goal region is the upper right-hand corner. l(x, y) is linear. Iteration 12 Iteration 25 Iteration 40 .8 1 Figure 2: Computation of 1* by discrete value iteration 372 Justin Boyan, Andrew W. Moore Iteration 17 Iteration 43 Iteration 127 1 . 8 .8 .8 1 Figure 3: Divergence of smooth value iteration with quadratic regression (note z-axis). J(x, y) Iteration 144 o. o. >. o. .8 o. 0.20 . 40.60 . 8 1 x 1 Figure 4: The 2-D continuous gridworld with puddles, its optimal value function, and a diverging approximation of the value function by Local Weighted Regression (note z-axis). car-on-the-Hill J (pa s, vel) 0.5 pas Figure 5: The car-on-the-hill domain. When the velocity is below a threshold, the car must reverse up the left hill to gain enough speed to reach the goal, so r is discontinuous. Iteration 11 Iteration 101 Iteration 201 Figure 6: Divergeri'ce oYsmooth value iteration wit~' for car-on-th~~hill~ The neural net, a 2-layer MLP with 80 hidden units, was trained for 2000 epochs per iteration. It may seem as though the divergence of smooth value iteration shown above can be attributed to the global nature of polynomial regression. In fact, when the domain is made slightly less trivial, the same types of instabilities appear with even a highly Generalization in Reinforcement Learning: Safely Approximating the Value Function 373 Table 1: Summary of convergence results: Smooth value iteration Domain Linear Quadratic LWR Backprop 2-D grid world lucky diverge good lucky 2-D puddle world diverge diverge Car-on-the-hill good diverge local memory-based function approximator such as local weighted regression (LWR) [Cleveland and Delvin, 1988]. Figure 4 shows the continuous gridworld augmented to include two oval "puddles" through which it is costly to step. Although LWR can fit the corresponding J* function nearly perfectly, smooth value iteration with LWR nonetheless reliably diverges. On another two-dimensional domain, the car-on-the-hill (Figure 5), smooth value iteration with LWR did converge, but a neural net trained by backpropagation did not (see Figure 6) . Table 1 summarizes our results. In light of such experiments, we conclude that the straightforward combination of DP and function approximation is not robust. A general-purpose learning method will require either using a function approximator constrained to be robust during DP [Yee, 1992], or an algorithm which explicitly prevents divergence even in the face of imperfect function approximation, such as the Grow-Support algorithm we present in Section 3. 2.2 RELATED WORK Theoretically, it is not surprising that inserting a smoothing process into a recursive DP procedure can lead to trouble. In [Thrun and Schwartz, 1993] one case is analyzed with the assumption that errors due to function approximation bias are independently distributed. Another area of theoretical analysis concerns inadequately approximated J* functions. In [Singh and Yee, 1994] and [Williams, 1993] bounds are derived for the maximum reduction in optimality that can be produced by a given error in function approximation. If a basis function approximator is used, then the reduction can be large [Sabes, 1993]. These results assume generalization from a dataset containing true optimal values; the true reinforcement learning scenario is even harder because each iteration of DP requires its own function approximation. 3 THE GROW-SUPPORT ALGORITHM The Grow-Support algorithm is designed to construct the optimal value function with a generalizing function approximator while being robust and stable. It recognizes that function approximators cannot always be relied upon to fit the intermediate value functions produced by DP. Instead, it assumes only that the function approximator can represent the final J* function accurately. The specific principles of Grow-Support are these: 1. We maintain a "support" set of states whose final J* values have been computed, starting with goal states, and growing this set out from the goal. The fitter is trained only on these values, which we assume it is capable of fitting. 2. Instead of propagating values by one-step DP backups, we use simulations with the current greedy policy, called "rollouts". They explicitly verify the achievability of a state's cost-to-go estimate before adding that state to the 374 Justin Boyan, Andrew W. Moore support. In a rollout, the J values are derived from costs of actual paths to the goal, not from the values of the previous iteration's function approximation. This prevents divergence. 3. We take maximum advantage of generalization. Each iteration, we add to the support set any sample state which can, by executing a single action, reach a state that passes the rollout test. In a discrete environment, this would cause the support set to expand in one-step concentric "shells" back from the goal. But in our continuous case, the function approximator may be able to extrapolate correctly well beyond the support region-and when this happens, we can add many points to the support set at once. This leads to the very desirable behavior that the support set grows in big jumps in regions where the value function is smooth. Iteration 1, I Support I =4 Iteration 2, 1 Support 1=12 Iteration 3, ISupportl=256 Figure 7: Grow-Support with quadratic regression on the gridworld. (Compare Figure 3.) Iteration 1, I Support I =3 Iteration 2, ISupportl=213 Iteration 5, ISupportl=253 Figure 8: Grow-Support with LWR on the two-puddle gridworld. (Compare Figure 4.) Iteration 3, I Support I =79 Iteration 8, ISupportl=134 Iteration 14, ISupportl=206 3 O. 2 O. -2 o. Figure 9: Grow-Support with backprop on car-on-the-hill. (Compare Figure 6.) The algorithm, again restricted to the deterministic case for simplicity, is outlined in the appendix. In Figures 7-9, we illustrate its convergence on the same combinations of domain and function approximator which caused smooth value iteration to diverge. In Figure 8, all but three points are added to the support within only five iterations, Generalization in Reinforcement Learning: Safely Approximating the Value Function 375 and the resulting greedy policy is optimal. In Figure 9, after 14 iterations, the algorithm terminates. Although 50 states near the discontinuity were not added to the support set, the resulting policy is optimal within the support set. Grow-support converged to a near-optimal policy for all the problems and fitters in Table 1. The Grow-Support algorithm is more robust than value iteration. Empirically, it was also seen to be no more computationally expensive (and often much cheaper) despite the overhead of performing rollouts. Reasons for this are (1) the rollout test is not expensive; (2) once a state has been added to the support, its value is fixed and it needs no more computation; and most importantly, (3) the aggressive exploitation of generalization enables the algorithm to converge in very few iterations. However, with a nondeterministic problem, where multiple rollouts are required to assess the accuracy of a prediction, Grow-Support would become more expensive. It is easy to prove that Grow-Support will always terminate after a finite number of iterations. If the function approximator is inadequate for representing the J* function, Grow-Support may terminate before adding all sample states to the support set. When this happens, we then know exactly which of the sample states are having trouble and which have been learned. This suggests potential schemes for adaptively adding sample states to the support in problematic regions. Investigation of these ideas is in progress. In conclusion, we have demonstrated that dynamic programming methods may diverge when their tables are replaced by generalizing function approximators. Our Grow-Support algorithm uses rollouts, rather than one-step backups, to assign training values and to keep inaccurate states out of the training set. We believe these principles will contribute substantially to producing practical, robust, reinforcement learning. Acknowledgements We thank Scott Fahlman, Geoff Gordon, Mary Lee, Michael Littman and Marc Ringuette for their suggestions, and the NDSEG fellowship and NSF Grant IRI-9214873 for their support. APPENDIX: ALGORITHMS Smooth Value Iteration(X, G, A, NEXT-STATE, COST, FITJ): Given: _ a finite collection of states X = {Xl, X2, .. . XN} sampled from the iter := 0 continuous state space X C fRn , and goal region G C X _ a finite set of allowable actions A _ a deterministic transition function NEXT-STATE: X x A -+ X _ the I-step cost function COST: X x A -+ fR _ a smoothing function approximator FIT J ]<0) [i] := 0 Vi = 1 ... N {X ·(iter) [1] } repeat I t-+ J !rain ~ITJ(iter) to approximate the training set: : Iter .:= Iter + 1; XN t-+ /iter)[N] for ~ := 1 ... N do .(iter) [.] ._ { 0 . J 1.minaEA (COST(Xi,a) + FITJ(lter-I)(NEXT-STATE(xi,a))) until j array stops changing if Xi E G otherwise 376 Justin Boyan, Andrew W. Moore subroutine RoIloutCost(x, J): Starting from state x , follow the greedy policy defined by value function J until either reaching the goal, or exceeding a total path cost of J(x) + £. Then return: --t the actual total cost of the path, if goal is reached from x with cost ~ J(x) + e --t 00, if goal is not reached in cost J(x) + £. Grow-Support(X,G,A, NEXT-STATE, COST, FITJ): Given: • exactly the same inputs as Smooth Value Iteration. SUPPORT := {(Xi t-+ 0) I Xi E G} repeat Train FIT J to approximate the training set SUPPORT for each Xi ~ SUPPORT do c := minaEA [COsT(xi,a) + RolloutCost(NEXT-STATE(Xi, a), FITJ)] if c < 00 then add (Xi t-+ c) to the training set SUPPORT until SUPPORT stops growing or includes all sample points. References [Barto et al., 1989] A. Barto, R. Sutton, and C . Watkins. Learning and sequential decision making. Technical Report COINS 89-95, Univ. of Massachusetts, 1989. [Bellman et al., 1963] R . Bellman, R . Kalaba, and B. Kotkin. Polynomial approximation-a new computational technique in dynamic programming: Allocation processes. Mathematics of Computation, 17, 1963. [Boyan, 1992] J. A. Boyan. Modular neural networks for learning context-dependent game strategies. Master's thesis, Cambridge University, 1992. [Bradtke, 1993] S. J. Bradtke. Reinforcement learning applied to linear quadratic regulation. In S. J . Hanson, J . Cowan, and C. L. Giles, editors, NIPS-5. Morgan Kaufmann, 1993. [Cleveland and Delvin, 1988] W . S. Cleveland and S. J. Delvin. Locally weighted regression: An approach to regression analysis by local fitting. JASA , 83(403):596-610, September 1988. [Lin, 1993] L.-J. Lin. Reinforcement Learning for Robots Using Neural Networks. PhD thesis, Carnegie Mellon University, 1993. [Mahadevan and Connell, 1990] S. Mahadevan and J. Connell. Automatic programming of behavior-based robots using reinforcement learning. Technical report, IBM T. J . Watson Research Center, NY 10598, 1990. [Sabes, 1993] P. Sabes. Approximating Q-values with basis function represent ations. In Proceedings of the Fourth Connectionist Models Summer School, 1993. [Schraudolph et al., 1994] N. Schraudolph, P. Dayan, and T. Sejnowski. Using TD(>.) to learn an evaluation function for the game of Go. In J. D. Cowan, G . Tesauro, and J . Alspector, editors, NIPS-6. Morgan Kaufmann, 1994. [Singh and Yee, 1994] S. P. Singh and R. Yee. An upper bound on the loss from approximate optimal-value functions. Machine Learning, 1994. Technical Note (to appear) . [Sutton, 1988] R . Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3,1988. [Tesauro, 1992] G. Tesauro. Practical issues in temporal difference learning. Machine Learning, 8(3/4), May 1992. [Thrun and Schwartz, 1993] S. Thrun and A. Schwartz. Issues in using function approximation for reinforcement learning. In Proceedings of the Fourth Connectionist Models Summer School, 1993. [Watkins, 1989] C . Watkins. Learning from Delayed Rewards. PhD thesis, Cambridge University, 1989. [Williams, 1993] R. Williams. Tight performance bounds on greedy policies based on imperfect value functions . Technical Report NU-CCS-93-13, Northeastern University, 1993. [Yee, 1992] R . Yee. Abstraction in control learning. Technical Report COINS 92-16, Univ. of Massachusetts, 1992.	1994	133
896	A Rapid Graph-based Method for Arbitrary Transformation-Invariant Pattern Classification Alessandro Sperduti Dipartimento di Informatica U niversita di Pisa Corso Italia 40 56125 Pisa, ITALY perso~di.unipi.it David G. Stork Machine Learning and Perception Group Ricoh California Research Center 2882 Sand Hill Road # 115 Menlo Park, CA USA 94025-7022 stork~crc.ricoh.com Abstract We present a graph-based method for rapid, accurate search through prototypes for transformation-invariant pattern classification. Our method has in theory the same recognition accuracy as other recent methods based on ''tangent distance" [Simard et al., 1994], since it uses the same categorization rule. Nevertheless ours is significantly faster during classification because far fewer tangent distances need be computed. Crucial to the success of our system are 1) a novel graph architecture in which transformation constraints and geometric relationships among prototypes are encoded during learning, and 2) an improved graph search criterion, used during classification. These architectural insights are applicable to a wide range of problem domains. Here we demonstrate that on a handwriting recognition task, a basic implementation of our system requires less than half the computation of the Euclidean sorting method. 1 INTRODUCTION In recent years, the crucial issue of incorporating invariances into networks for pattern recognition has received increased attention, most especially due to the work of 666 Alessandro Sperduti, David G. Stork Simard and his colleagues. To a regular hierachical backpropagation network Simard et al. [1992] added a Jacobian network, which insured that directional derivatives were also learned. Such derivatives represented directions in feature space corresponding to the invariances of interest, such as rotation, translation, scaling and even line thinning. On small training sets for a function approximation problem, this hybrid network showed performance superior to that of a highly tuned backpropagation network taken alone; however there was negligible improvement on large sets. In order to find a simpler method applicable to real-world problems, Simard, Le Cun & Denker [1993] later used a variation of the nearest neighbor algorithm, one incorporating "tangent distance" (T-distance or DT ) as the classification metric the smallest Euclidean distance between patterns after the optimal transformation. In this way, state-of-the-art accuracy was achieved on an isolated handwritten character task, though at quite high computational complexity, owing to the inefficient search and large number of Euclidean and tangent distances that had to be calculated. Whereas Simard, Hastie & Saeckinger [1994] have recently sought to reduce this complexity by means of pre-clustering stored prototypes, we here take a different approach, one in which a (graph) data structure formed during learning contains information about transformations and geometrical relations among prototypes. Nevertheless, it should be noted that our method can be applied to a reduced (clustered) training set such as they formed, yielding yet faster recognition. Simard [1994] recently introduced a hierarchical structure of successively lower resolution patterns, which speeds search only if a minority of patterns are classified more accurately by using the tangent metric than by other metrics. In contrast, our method shows significant improvement even if the majority or all of the patterns are most accurately classified using the tangent distance. Other methods seeking fast invariant classification include Wilensky and Manukian's scheme [1994]. While quite rapid during recall, it is more properly considered distortion (rather than coherent transformation) invariant. Moreover, some transformations such as line thinning cannot be naturally incorporated into their scheme. Finally, it appears as if their scheme scales poorly (compared to tangent metric methods) as the number of invariances is increased. It seems somewhat futile to try to improve significantly upon the recognition accuracy of the tangent metric approach for databases such as NIST isolated handwritten characters, Simard et al. [1993] reported accuracies matching that of humans! Nevertheless, there remains much that can be done to increase the computational efficiency during recall. This is the problem we address. 2 TRANSFORMATION INVARIANCE In broad overview, during learning our method constructs a labelled graph data structure in which each node represents a stored prototype (labelled by its category) as given by a training set, linked by arcs representing the T-distance between them. Search through this graph (for classification) takes advantage of the graph structure and an improved search criterion. To understand the underlying computations, we must first consider tangent space. Graph-Based Method for Arbitrary Transformation-Invariant Pattern Classification 667 Figure 1: Geometry of tangent space. Here, a three-dimensional feature space contains the "current" prototype, Pc, and the subspace consisting of all patterns obtainable by performing continuous transformations of it (shaded). Two candidate prototypes and a test pattern, T, as well as their projections onto the T-space of Pc are shown. The insert (above) shows the progression of search through the corresponding portion of the recognition graph. The goal is to rapidly find the prototype closest to T (in the T-distance sense), and our algorithm (guided by the minimum angle OJ in the tangent space) finds that P 2 is so closer to T than are either PI or Pc (see text). Figure 1 illustrates geometry of tangent space and the relationships among the fundamental entities in our trained system. A labelled ("current") trained pattern is represented by Pc, and the (shaded) surface corresponds to patterns arising under continuous transformations of Pc. Such transformations might include rotation, translation, scaling, line thinning, etc. Following Simard et al. [1993], we approximate this surface in the vicinity of Pc by a subspace the tangent space or T -space of Pc which is spanned by "tangent" vectors, whose directions are determined by infinitessimally transforming the prototype Pc. The figure shows an ortho-normal basis {TVa, TV b}, which helps to speed search during classification, as we shall see. A test pattern T and two other (candidate) prototypes as well as their projections onto the T-space of Pc are shown. 668 Alessandro Sperduti, David G. Stork 3 THE ALGORITHMS Our overall approach includes constructing a graph (during learning), and searching it (for classification). The graph is constructed by the following algorithm: Graph construction Initialize N = # patterns; k = # nearest neighbors; t = # invariant transformations Begin Loop For each prototype Pi (i = 1 ~ N) • Compute a t-dimensional orthonormal basis for the T -space of Pi • Compute ("one-sided") T-distance of each of the N - 1 prototypes P j (j i- i) using Pi'S T-space • Represent Pj.l (the projection of P j onto the T-space of Pi) in the tangent orthonormal frame of Pi • Connect Pi to each of its k T-nearest neighbors, storing their associated normalized projections Ph End Loop During classification, our algorithm permits rapid search through prototypes. Thus in Figure 1, starting at Pc we seek to find another prototype (here, P2) that is closer to the test point T . After P2 is so chosen, it becomes the current pattern, and the search is extended using its T-space. Graph search ends when the closest prototype to T is found (Le., closest in a T-distance sense). We let D~ denote the current minimum tangent distance. Our search algorithm is: Graph search Input Test pattern T Initialize Do • Choose initial candidate prototype, Po • SetPc~Po • Set D~ ~ DT(Pc, T), i.e., the T-distance ofT from Pc T.L·P~ • For each prototype P j connected to Pc compute cos(Oj) = IT.Ll.L • Sort these prototypes by increasing values of OJ and put them into a candidate list • Pick P j from the top of the candidate list • In T-space of Pj, compute DT(Pj , T) If DT(Pj , T) < D~ then Pc ~ P j and D~ ~ DT(Pj , T) otherwise mark P j as a "failure" (F), and pick next prototype from the candidate list Until Candidate list empty Return D~ or the category label of the optimum prototype found Graph-Based Method for Arbitrary Transformation-Invariant Pattern Classification 669 Dr 4.91 3.70 3.61 3.03 2.94 Figure 2: The search through the "2" category graph for the T-nearest stored prototype to the test pattern is shown (N = 720 and k = 15 nearest neighbors). The number of T-distance calculations is equal to the number of nodes visited plus the number offailures (marked F); Le., in the case shown 5 + 26 = 31. The backward search step attempt is thwarted because the middle node has already been visited (marked M). Notice in the prototypes how the search is first a downward shift, then a counter-clockwise rotation a mere four steps through the graph. Figure 2 illustrates search through a network of "2" prototypes. Note how the Tdistance of the test pattern decreases, and that with only four steps through the graph the optimal prototype is found. There are several ways in which our search technique can be incorporated into a classifier. One is to store all prototypes, regardless of class, in a single large graph and perform the search; the test pattern is classified by the label of the optimal prototype found. Another, is to employ separate graphs, one for each category, and search through them (possibly in parallel); the test is classified by the minimum T-distance prototype found. The choice of method depends upon the hardware limitations, performance speed requirements, etc. Figure 3 illustrates such a search through a "2" category graph for the closest prototype to a test pattern "5." We report below results using a single graph per category, however. 3.1 Computational complexity If a graph contains N prototypes with k pointers (arcs) each, and if the patterns are of dimension m, then the storage requirement is O(N((t + 1) . m2 + kt)). The time complexity of training depends upon details of ortho-normalization, sorting, etc., and is of little interest anyway. Construction is more than an order of magnitude faster than neural network training on similar problems; for instance construction of a graph for N = 720 prototypes and k = 100 nearest neighbors takes less than 670 Alessandro Sperduti, David G. Stork [ZJ[ZJ[2J[2] Dr 5.10 5.09 5.01 4.93 4.90 Figure 3: The search through a "2" category graph given a "5" test pattern. Note how the search first tries to find a prototype that matches the upper arc of the "5," and then one possessing skew or rotation. For this test pattern, the minimum T-distance found for the "5" category (3.62) is smaller than the one found for the "2" category shown here (4.22), and indeed for any other category. Thus the test pattern is correctly classified as a "5." 20 minutes on a Sparc 10. The crucial quantity of interest is the time complexity for search. This is, of course, problem related, and depends upon the number of categories, transformation and prototypes and their statistical properties (see next Section). Worst case analyses (e.g., it is theoretically conceivable that nearly all prototypes must be visited) are irrelevant to practice. We used a slightly non-obvious search criterion at each step, the function cos(Oj), as shown in Figure 1. Not only could this criterion be calculated very efficiently in our orthonormal basis (by using simple inner products), but it actually led to a slightly more accurate search than Euclidean distance in the T-space perhaps the most natural choice of criterion. The angle OJ seems to guide the "flow" of the search along transformation directions toward the test point. 4 Simulations and results We explored the search capabilities of our system on the binary handwritten digit database of Guyon, et al. [1991J. We needed to scale all patterns by a linear factor (0.833) to insure that rotated versions did not go outside the 16 x 16 pixel grid. As required in all T-space methods, the patterns must be continuous valued (Le., here grayscale); this was achieved by convolution with a spatially symmetric Gaussian having a = .55 pixels. We had 720 training examples in each of ten digit categories; the test set consisted of 1320 test patterns formed by transforming independent prototypes in all meaningful combinations of the t = 6 transformations (four spatial directions and two rotation senses). We compared the Euclidean sorting method of Simard et al. [1993J to our graph Graph-Based Method for Arbitrary Transformation-Invariant Pattern Classification 671 1.00 ______ -----:::::::::::::==---10. 6 ... ' ',-. error - ---~. " .. - ................ -§ 0.4 u .c ~ u '" ~ 0.2 e ~ o 0 50 100 150 200 250 300 350 400 Computational complexity (equivalent number of T -distance calculations) Figure 4: Comparison of graph-based (heavy lines) and standard Euclidean sorting searches (thin lines). Search accuracy is the percentage of optimal prototypes found on the full test set of 1320 patterns in a single category (solid lines). The average search error is the per pattern difference between the global optimum T -distance and the one actually found, averaged over the non-optimal prototypes found through the search (dashed lines). Note especially that for the same computational complexity, our method has the same average error, but that this average is taken over a much smaller number of (non-optimal) prototypes. For a given criterion search accuracy, our method requires significantly less computation. For instance, if 90% of the prototypes must be found for a requisite categorization accuracy (a typical value for asymptotically high recognition accuracy), our graph-based method requires less than half the computation of the Euclidean sorting method. based method using the same data and transformations, over the full range of relevant computational complexities. Figure 4 summarizes our results. For our method, the computational complexity is adjusted by the number of neighbors inspected, k. For their Euclidean sorting method, it is adjusted by the percentage of Euclidean nearest neighbors that were then inspected for T -distance. We were quite careful to employ as many computational tricks and shortcuts on both methods we could think of. Our results reflect fairly on the full computational complexity, which was dominated by tangent and Euclidean distance calculations. We note parenthetically that many of the recognition errors for both methods could be explained by the fact that we did not include the transformation of line thinning (solely because we lacked the preprocessing capabilities); the overall accuracy of both methods will increase when this invariance is also included. 5 CONCLUSIONS AND FUTURE WORK We have demonstrated a graph-based method using tangent distance that permits search through prototypes significantly faster than the most popular current approach. Although not shown above, ours is also superior to other tree-based 672 Alessandro Sperduli. David G. Stork methods, such as k-d-trees, which are less accurate. Since our primary concern was reducing the computational complexity of search (while matching Simard et al.'s accuracy), we have not optimized over preprocessing steps, such as the Gaussian kernel width or transformation set. We note again that our method can be applied to reduced training sets, for instance ones pruned by the method of Simard, Hastie & Saeckinger [1994]. Simard's [1994] recent method in which low-resolution versions of training patterns are organized into a hierarchical data structure so as to reduce the number of multiply-accumulates required during search is in some sense "orthogonal" to ours. Our graph-based method will work with his lowresolution images too, and thus these two methods can be unified into a hybrid system. Perhaps most importantly, our work suggests a number of research avenues. We used just a single ("central") prototype Po to start search; presumably having several candidate starting points would be faster. Our general method may admit gradient descent learning of parameters of the search criterion. For instance, we can imagine scaling the different tangent basis vectors according to their relevance in guiding correct searches as determined using a validation set. Finally, our approach may admit elegant parallel implementations for real-world applications. Acknowledgements This work was begun during a visit by Dr. Sperduti to Ricoh CRC. We thank I. Guyon for the use of her database of handwritten digits and Dr. K. V. Prasad for assistance in image processing. References 1. Guyon, P. Albrecht, Y. Le Cun, J. Denker & W. Hubbard. (1991) "Comparing different neural network architectures for classifying handwritten digits," Proc. of the Inter. Joint Conference on Neural Networks, vol. II, pp. 127-132, IEEE Press. P. Simard. (1994) "Efficient computation of complex distance metrics using hierarchical filtering," in J. D. Cowan, G. Tesauro and J. Alspector (eds.) Advances in Neural Information Processing Systems-6 Morgan Kaufmann pp. 168-175. P. Simard, B. Victorrio, Y. Le Cun & J. Denker. (1992) "Tangent Prop A formalism for specifying selected invariances in an adaptive network," in J. E. Moody, S. J. Hanson and R. P. Lippmann (eds.) Advances in Neural Information Processing Systems-4 Morgan Kaufmann pp. 895-903. P. Y. Simard, Y. Le Cun & J. Denker. (1993) "Efficient Pattern Recognition Using a New Transformation Distance," in S. J. Hanson, J. D. Cowan and C. L. Giles (eds.) Advances in Neural Information Processing Systems-5 Morgan Kaufmann pp.50-58. P. Y. Simard, T. Hastie & E. Saeckinger. (1994) "Learning Prototype Models for Tangent Distance," Neural Networks for Computing Snowbird, UT (April, 1994). G. D. Wilensky & N. Manukian. (1994) "Nearest Neighbor Networks: New Neural Architectures for Distortion-Insensitive Image Recognition," Neural Networks for Computing Snowbird, UT (April, 1994).	1994	134
897	A solvable connectionist model of immediate recall of ordered lists Neil Burgess Department of Anatomy, University College London London WC1E 6BT, England (e-mail: n.burgessOucl.ac.uk) Abstract A model of short-term memory for serially ordered lists of verbal stimuli is proposed as an implementation of the 'articulatory loop' thought to mediate this type of memory (Baddeley, 1986). The model predicts the presence of a repeatable time-varying 'context' signal coding the timing of items' presentation in addition to a store of phonological information and a process of serial rehearsal. Items are associated with context nodes and phonemes by Hebbian connections showing both short and long term plasticity. Items are activated by phonemic input during presentation and reactivated by context and phonemic feedback during output. Serial selection of items occurs via a winner-take-all interaction amongst items, with the winner subsequently receiving decaying inhibition. An approximate analysis of error probabilities due to Gaussian noise during output is presented. The model provides an explanatory account of the probability of error as a function of serial position, list length, word length, phonemic similarity, temporal grouping, item and list familiarity, and is proposed as the starting point for a model of rehearsal and vocabulary acquisition. 1 Introduction Short-term memory for serially ordered lists of pronounceable stimuli is well described, at a crude level, by the idea of an 'articulatory loop' (AL). This postulates that information is phonologically encoded and decays within 2 seconds unless refreshed by serial rehearsal, see (Baddeley, 1986). It successfully accounts for (i) 52 Neil Burgess the linear relationship between memory span s (the number of items s such that 50% of lists of s items are correctly recalled) and articulation rate r (the number of items that can be said per second) in which s ~ 2r + c, where r varies as a function of the items, language and development; (ii) the fact that span is lower for lists of phonemically similar items than phonemically distinct ones; (iii) unattended speech and articulatory distract or tasks (e.g. saying blah-blah-blah ... ) both reduce memory span. Recent evidence suggests that the AL plays a role in the learning of new words both during development and during recovery after brain traumas, see e.g. (Gathercole & Baddeley, 1993). Positron emission tomography studies indicate that the phonological store is localised in the left supramarginal gyrus, whereas subvocal rehearsal involves Broca's area and some of the motor areas involved in speech planning and production (Paulesu et al., 1993). However, the detail of the types of errors committed is not addressed by the AL idea. Principally: (iv) the majority of errors are 'order errors' rather than 'item errors', and tend to involve transpositions of neighbouring or phonemically similar items; (v) the probability of correctly recalling a list as a function of list length is a sigmoid; (vi) the probability of correctly recalling an item as a function of its serial position in the list (the 'serial position curve') has a bowed shape; (vii) span increases with the familiarity of the items used, specifically the c in s ~ 2r + c can increase from 0 to 2.5 (see (Hulme et al., 1991)), and also increases if a list has been previously presented (the 'Hebb effect'); (viii) 'position specific intrusions' occur, in which an item from a previous list is recalled at the same position in the current list. Taken together, these data impose strong functional constraints on any neural mechanism implementing the AL. Most models showing serial behaviour rely on some form of 'chaining' mechanism which associates previous states to successive states, via recurrent connections of various types. Chaining of item or phoneme representations generates errors that are incompatible with human data, particularly (iv) above, see (Burgess & Hitch, 1992, Henson, 1994). Here items are maintained in serial order by association to a repeatable time-varying signal (which is suggested by position specific intrusions and is referred to below as 'context'), and by the recovery from suppression involved in the selection process - a modification of the 'competitive queuing' model for speech production (Houghton, 1990). The characteristics of STM for serially ordered items arise due to the way that context and phoneme information prompts the selection of each item. 2 The model The model consists of 3 layers of artificial neurons representing context, phonemes and items respectively, connected by Hebbian connections with long and short term plasticity, see Fig. 1. There is a winner-take-all (WTA) interaction between item nodes: at each time step the item with the greatest input is given activation 1, and the others o. The winner at the end of each time step receives a decaying inhibition that prevents it from being selected twice consecutively. During presentation, phoneme nodes are activated by acoustic or (translated) visual input, activation in the context layer follows the pattern shown in Fig. 1, item nodes receive input from phoneme nodes via connections Wij. Connections A Solvable Connectionist Model of Immediate Recall of Ordered Lists A) B) ..... 1---- nc ---I.~ context •••••• 00000 0 t=l 0 •••••• 0000 0 t=2 00 •••••• 0000 t=3 phonemes . translated .. ,/ visual input ~ 0000000 Wij(t) \ 0000000 acoustic input buffer items (WTA + suppression) [0 output 53 Figure 1: A) Context states as a function of serial position t; filled circles are active nodes, empty circles are inactive nodes. B) The architecture of the model. Full lines are connections with short and long term plasticity; dashed lines are routes by which information enters the model. Wij (t) learn the association between the context state and the winning item, and Wij and Wij learn the association with the active phonemes. During recall, the context layer is re-activated as in presentation, activation spreads to the item layer (via Wij(t)) where one item wins and activates its phonemes (via Wij(t». The item that now wins, given both context and phoneme inputs, is output, and then suppressed. As described so far, the model makes no errors. Errors occur when Gaussian noise is added to items' activations during the selection of the winning item to be output. Errors are likely when there are many items with similar activation levels due to decay of connection weights and inhibition since presentation. Items may then be selected in the wrong order, and performance will decrease with the time taken to present or recall a list. 2.1 Learning and familiarity Connection weights have both long and short term plasticity: Wij (t) (similarly Wij(t) and Wij(t)) have an incremental long term component Wi~(t), and a oneshot short term component Wl,(t) which decays by a factor b.. per second. The net weight of the connection is the sum of the two components: Wij(t) = Wi~(t)+W/i(t). Learning occurs according to: if Cj(t)ai(t) > Wij(t)j otherwise, 54 Neil Burgess { Wil(t) + eCj(t)Uoi(t) if Cj(t)Uoi(t) > 0; Wij (t) otherwise, (1) where Cj(t) and Uoi(t) are the pre- and post-connection activations, and e decreases with IW/i(t)1 so that the long term component saturates at some maximum value. These modifiable connection weights are never negative. An item's cfamiliarity' is reflected by the size of the long term components wfj and wfj of the weights storing the association with its phonemes. These components increase with each (error-free) presentation or recall of the item. For lists of totally unfamiliar items, the item nodes are completely interchangeable having only the short-term connections w!j to phoneme nodes that are learned at presentation. Whereas the presentation of a familiar item leads to the selection of a particular item node (due to the weights wfj) and, during output, this item will activate its phonemes more strongly due to the weights w! '. Unfamiliar items that are phonemically similar to a familiar item will tend to be represented by the familiar item node, and can take advantage of its long-term item-phoneme weights wfj. Presentation of a list leads to an increase in the long term component of the contextitem association. Thus, if the same list is presented more than once its recall improves, and position specific intrusions from previous lists may also occur. Notice that only weights to or from an item winning at presentation or output are increased. 3 Details There are nw items per list, np phonemes per item, and a phoneme takes time lp seconds to present or recall. At time t, item node i has activation Uoi(t) , context node i has activation Ci(t), Ct is the set of nc context nodes active at time t, phoneme node i has activation bi(t) and Pi is the set of np phonemes comprising item i. Context nodes have activation 0 or J3/2nc , phonemes take activation 0 or 1/ y'n;, so Wij(t) ~ J3/2nc and wlj(t) = Wji(t) ~ 1/ h' see (1). This sets the relative effect that context and phoneme layers have on items' activation, and ensures that items of neither few nor many phonemes are favoured, see (Burgess & Hitch, 1992). The long-term components of phoneme-item weights wfj(t) and wji(t) are 0.45/ y'n; for familiar items, and 0.15/ y'n; for unfamiliar items (chosen to match the data in Fig. 3B). The long-term components of context-item weights Wi~(t) increase by 0.15/.Jn; for each ofthe first few presentations or recalls of a list. Apart from the WTA interaction, each item node i has input: (2) where Ii(t) < 0 is a decaying inhibition imposed following an item's selection at presentation or output (see below), TJi is a (0, u) Gaussian random variable added at output only, and Ei(t) is the excitatory input to the item from the phoneme layer during presentation and the context and phoneme layers during recall: during presentation; during recall. During recall phoneme nodes are activated according to bi(t) = 2:j Wij(t)aj(t). (3) A Solvable Connectionist Model of Immediate Recall of Ordered Lists 55 One time step refers to the presentation or recall of an item and has duration nplp. The variable t increases by 1 per time step, and refers to both time and serial position. Short term connection weights and inhibition Ii(t) decay by a factor .6. per second, or .6. nplp per time step. The algorithm is as follows; rehearsal corresponds to repeating the recall phase. Presentation o. Set activations, inhibitions and short term weights to zero, t = 1. 1. Set the context layer to state Ct : Ci(t) = J3/2nc if i E Ct; Ci(t) = 0 otherwise. 2. Input items, i.e. set the phoneme layer to state 1't : bi(t) = 1/..;n; if i E 1't; bi(t) = 0 otherwise. 3. Select the winning item, i.e. ak(t) = 1 where hk(t) = maJC.i{hi(t)}; ai(t) = 0, for i =1= k. 4. Learning, i.e. increment all connection weights according to (1). 5. Decay, i.e. multiply short-term connection weights Wl;(t), w[j(t) and w[j(t), and inhibitions Ii(t) by a factor .6.n plp. 6. Inhibit winner, i.e. set Ik(t) = -2, where k is the item selected in 3. 7. t ---+ t + 1, go to 1. Recall o. t = 1. 1. Set the context layer to state Ct , as above. 2. Set all phoneme activations to zero. 3. Select the winning item, as above. 4. Output. Activate phonemes via Wji(t), select the winning item (in the presence of noise). 5. Learning, as above. 6. Decay, as above. 7. Inhibit winner, i.e. set Ik(t) = -2, where k is the item selected in 4. 8. t ---+ t + 1, go to 1. 4 Analysis The output of the model, averaged over many trials, depends on (i) the activation values of all items at the output step for each time t and, (ii) given these activations and the noise level, the probability of each item being the winner. Estimation is necessary since there is no simple exact expression for (ii), and (i) depends on which items were output prior to time t. I define "Y(t, i) to be the time elapsed, by output at time t, since item i was last selected (at presentation or output), i.e. in the absence of errors: . {(t-i)lpnp ifi<t; "Y(t, l) = (nw - (i - t))lpnp if i 2: t. (4) If there have been no prior errors, then at time t the inhibition of item l IS Ii(t) = -2(.6.)7(t,i+l), and short term weights to and from item i have decayed by a factor .6. 7(t,i). For a novel list of familiar items, the excitatory input to item i during output at time t is, see (3): Ei(t) = 3.6. 7(t,i)IICi n Ct 11/2nc + (0.45 + .6. 7(t,i))21I1'i n 1't II/np, (5) 56 A) 0.90 0.85 0.80 0.75 2 4 6 Neil Burgess s 0.6 2 4 6 Figure 2: Serial position curves. Full lines show the estimation, extra markers are error bars at one standard deviation of 5 simulations of 1,000 trials each, see §5 for parameter values. A) Rehearsal. Four consecutive recalls of a list of 7 digits ('1', .. ,'4'). B) Phonemic similarity. SPCs are shown for lists of dissimilar letters ('d'), similar letters ('s'), and alternating similar and dissimilar letters with the similar ones in odd ('0') and even ('e') positions. C.f. (Baddeley, 1968, expt. V). where IIX II is the number of elements in set X. The probability p(t, i) that item i wins at time t IS estimated by the softmax function(Brindle, 1990): ( .) '" exp (TrI.i (t)/ 0") p t, 1, '" ntu (), , Lj=1 exp (mj t /0' ) (6) where TrI.i(t) is hi(t) without the noise term, see (2-3), and 0" = 0.750'. For 0' = 0.5 (the value used below), the r.m.s. difference between p(t, i) estimated by simulation (500 trials) and by (6) is always less than 0.035 for -1 < TrI.i(t) < 1 with 2 to 6 items. Which items have been selected prior to time t affects Ii(t) in hi(t) via "I(t, i). p(t, i) is estimated for all combinations of up to two prior errors using (6) with appropriate values of TrI.i(t), and the average, weighted by the probability of each error combination, is used. The 'missing' probability corresponding to more than two prior errors is corrected for by normalising p(t, i) so that Li p(t, i) = 1 for t = 1, .. , nw' This overestimates the recency effect, especially in super-span lists. 5 Performance The parameter values used are Do = 0.75, nc = 6, 0' = 0.5. Different types of item are modelled by varying (np,lp) : 'digits' correspond to (2,0.15), 'letters' to (2,0.2), and 'words' to (5,0.15-0.3). 'Similar' items all have 1 phoneme in common, dissimilar items have none. Unless indicated otherwise, items are dissimilar and familiar, see §3 for how familiarity is modelled. The size of 0' relative to Do is set so that digit span ~ 7. np and lp are such that approximately 7 digits can be said in 2 seconds. The model's performance is shown in Figs. 2 and 3. Fig. 2A: the increase in the long-term component of context-item connections during rehearsal brings stability after a small number of rehearsals, i.e. no further errors are committed. Fig. 2B: serial position curves show the correct effect of phonemic similarity among items. A Solvable Connectionist Model of Immediate Recall of Ordered Lists 57 B) r 4 w T 3 -u 2 n 0 5 10 0.0 0.5 1.0 1.5 Figure 3: Item span. Full lines show the estimation, extra markers (A only) are error bars at one standard deviation of 3 simulations of 1,000 trials each, see §5 and §3 for parameter values. A) The probability of correctly recalling a whole list versus list length. Lists of digits ('d'), unfamiliar items (of the same length, 'u'), and experimental data on digits (adapted from (Guildford & Dallenbach, 1925), 'x') are shown. B) Span versus articulation rate (rate= 1/ipnp, with np = 5 and ip =0.15,0.2, and 0.3). Calculated curves are shown for novel lists of familiar ('f') and unfamiliar ('u') words and lists of familiar words after 5 repetitions ('r'). Data on recall of words ('w') and non-words ('n') are also shown, adapted from (Hulme et al., 1991). Fig. 3A: the probability of recalling a list correctly as a function of list length shows the correct sigmoidal relationship. Fig. 3B: item span shows the correct, approximately linear, relationship to articulation rate, with span for unfamiliar items below that for familiar items. Span increases with repeated presentations of a list in accordance with the 'Hebb effect'. Note that span is slightly overestimated for short lists of very long words. 5.1 Discussion and relation to previous work This model is an extension of (Burgess & Hitch, 1992), primarily to model effects of rehearsal and item and list familiarity by allowing connection weights to show plasticity over different timescales, and secondly to show recency and phonemic similarity effects simultaneously by changing the way phoneme nodes are activated during recall. Note that the 'context' timing signal varies with serial position: reflecting the rhythm of presentation rather than absolute time (indeed the effect of temporal grouping can be modelled by modifying the context representations to reflect the presence of pauses during presentation (Hitch et al., 1995)), so presentation and recall rates cannot be varied. The decaying inhibition that follows an items selection increases the locality of errors, i.e. if item i + 1 replaces item i, then item i is most likely to replace item i + 1 in turn (rather than e.g. item i+ 2). The model has two remaining problems: (i) selecting an item node to form the long term representation of a new item, without taking over existing item nodes, and (ii) learning the correct order of the phonemes within an item - a possible extension to address this problem is presented in (Hartley & Houghton, 1995). The mechanism for selecting items is a modification of competitive queuing 58 Neil Burgess (Houghton, 1990) in that the WTA interaction occurs at the item layer, rather than in an extra layer, so that only the winner is active and gets associated to context and phoneme nodes (this avoids partial associations of a context state to all items similar to the winner, which would prevent the zig-zag curves in Fig. 2B). The basic selection mechanism is sufficient to store serial order in itself, since items recover from suppression in the same order in which they were selected at presentation. The model ma.ps onto the articulatory loop idea in that the selection mechanism corresponds to part of the speech production ('articulation') system and the phoneme layer corresponds to the 'phonological store', and predicts that a 'context' timing signal is also present. Both the phoneme and context inputs to the item layer serve to increase span, and in addition, the former causes phonemic similarity effects and the latter causes recency, position specific intrusions and temporal grouping effects. 6 Conclusion I have proposed a simple mechanism for the storage and recall of serially ordered lists of items. The distribution of errors predicted by the model can be estimated mathematically and models a very wide variety of experimental data. By virtue of long and short term plasticity of connection weights, the model begins to address familiarity and the role of rehearsal in vocabulary acquisition. Many of the predicted error probabilities have not yet been checked experimentally: they are predictions. However, the major prediction of this model, and of (Burgess & Hitch, 1992), is that, in addition to a short-term store of phonological information and a process of sub-vocal rehearsal, STM for ordered lists of verbal items involves a third component which provides a repeatable time-varying signal reflecting the rhythm of the items' presentation. Acknowledgements: I am grateful for discussions with Rik Henson and Graham Hitch regarding data, and with Tom Hartley and George Houghton regarding error probabilities, and to Mike Page for suggesting the use of the softmax function. This work was supported by a Royal Society University Research Fellowship. References Baddeley AD (1968) Quarterly Journal of Ezperimental Pllychology 20 249-264. Baddeley AD (1986) Working Memory, Clarendon Press. Brindle, J S (1990) in: D S Tourebky (ed.) Advancell in Neural Information ProcelJlling Syatemll ! . San Mateo, CA: Morgan Kaufmann. Burgess N & Hitch G J (1992) J. Memory and Language 31 429-460. Gathercole S E & Baddeley A D (1993) Working memory and language, Erlbaum. Guildford J P & Dallenbach K M (1925) American J. of Pllychology 36 621-628. Hartley T & Houghton G (1995) J. Memory and Language to be published. Henson R (1994) Tech. Report, M.R.C. Applied Psychology Unit, Cambridge, U.K. Hitch G, Burgess N, Towse J & Culpin V (1995) Quart. J. of Ezp. Pllychology, submitted. Houghton G (1990) in: R Dale, C Mellish & M Zock (eds.), Current Rellearch in Natural Language Generation 287-319. London: Academic Press. Hulme C, Maughan S & Brown G D A (1991) J. Memory and Language 30685-701. Paulesu E, Frith C D & Frackowiak R S J (1993) Nature 362 342-344. PART II NEUROSCIENCE	1994	135
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899	U sing a neural net to instantiate a deformable model Christopher K. I. Williams; Michael D. Revowand Geoffrey E. Hinton Department of Computer Science, University of Toronto Toronto, Ontario, Canada M5S lA4 Abstract Deformable models are an attractive approach to recognizing nonrigid objects which have considerable within class variability. However, there are severe search problems associated with fitting the models to data. We show that by using neural networks to provide better starting points, the search time can be significantly reduced. The method is demonstrated on a character recognition task. In previous work we have developed an approach to handwritten character recognition based on the use of deformable models (Hinton, Williams and Revow, 1992a; Revow, Williams and Hinton, 1993). We have obtained good performance with this method, but a major problem is that the search procedure for fitting each model to an image is very computationally intensive, because there is no efficient algorithm (like dynamic programming) for this task. In this paper we demonstrate that it is possible to "compile down" some of the knowledge gained while fitting models to data to obtain better starting points that significantly reduce the search time. 1 DEFORMABLE MODELS FOR DIGIT RECOGNITION The basic idea in using deformable models for digit recognition is that each digit has a model, and a test image is classified by finding the model which is most likely to have generated it. The quality of the match between model and test image depends on the deformation of the model, the amount of ink that is attributed to noise and the distance of the remaining ink from the deformed model. ·Current address: Department of Computer Science and Applied Mathematics, Aston University, Birmingham B4 7ET, UK. 966 Christopher K. T. Williams, Michael D. Revow, Geoffrey E. Hinton More formally, the two important terms in assessing the fit are the prior probability distribution for the instantiation parameters of a model (which penalizes very distorted models), and the imaging model that characterizes the probability distribution over possible images given the instantiated modell . Let I be an image, M be a model and z be its instantiation parameters. Then the evidence for model M is given by P(IIM) = J P(zIM)P(IIM, z)dz (1) The first term in the integrand is the prior on the instantiation parameters and the second is the imaging model i.e., the likelihood of the data given the instantiated model. P(MII) is directly proportional to P(IIM), as we assume a uniform prior on each digit. Equation 1 is formally correct, but if z has more than a few dimensions the evaluation of this integral is very computationally intensive. However, it is often possible to make an approximation based on the assumption that the integrand is strongly peaked around a (global) maximum value z. In this case, the evidence can be approximated by the highest peak of the integrand times a volume factor ~(zII, M), which measures the sharpness of the peak2 . P(IIM) ~ P(zIM)P(Ilz, M)~(zII, M) (2) By Taylor expanding around z to second order it can be shown that the volume factor depends on the determinant of the Hessian of 10gP(z, 11M) . Taking logs of equation 2, defining EdeJ as the negative log of P(zIM), and EJit as the corresponding term for the imaging model, then the aim of the search is to find the minimum of Etot = EdeJ + EJit . Of course the total energy will have many local minima; for the character recognition task we aim to find the global minimum by using a continuation method (see section 1.2). 1.1 SPLINES, AFFINE TRANSFORMS AND IMAGING MODELS This section presents a brief overview of our work on using deformable models for digit recognition. For a fuller treatment, see Revow, Williams and Hinton (1993). Each digit is modelled by a cubic B-spline whose shape is determined by the positions of the control points in the object-based frame. The models have eight control points, except for the one model which has three, and the seven model which has five. To generate an ideal example of a digit the control points are positioned at their "home" locations. Deformed characters are produced by perturbing the control points away from their home locations. The home locations and covariance matrix for each model were adapted in order to improve the performance. The deformation energy only penalizes shape deformations. Affine transformations, i.e., translation, rotation, dilation, elongation, and shear, do not change the underlying shape of an object so we want the deformation energy to be invariant under them. We achieve this by giving each model its own "object-based frame" and computing the deformation energy relative to this frame. lThis framework has been used by many authors, e.g. Grenander et al (1991) . 2The Gaussian approximation has been popularized in the neural net community by MacKay (1992). Using a Neural Net to Instantiate a Deformable Model 967 The data we used consists of binary-pixel images of segmented handwritten digits. The general flavour of a imaging model for this problem is that there should be a high probability of inked pixels close to the spline, and lower probabilities further away. This can be achieved by spacing out a number of Gaussian "ink generators" uniformly along the contour; we have found that it is also useful to have a uniform background noise process over the area of the image that is able to account for pixels that occur far away from the generators. The ink generators and background process define a mixture model. Using the assumption that each data point is generated independently given the instantiated model, P(Ilz, M) factors into the product of the probability density of each black pixel under the mixture model. 1.2 RECOGNIZING ISOLATED DIGITS For each model, the aim of the search is to find the instantiation parameters that minimize E tot . The search starts with zero deformations and an initial guess for the affine parameters which scales the model so as to lie over the data with zero skew and rotation. A small number of generators with the same large variance are placed along the spline, forming a broad, smooth ridge of high ink-probability along the spline. We use a search procedure similar to the (iterative) Expectation Maximization (EM) method of fitting an unconstrained mixture of Gaussians, except that (i) the Gaussians are constrained to lie on the spline (ii) there is a deformation energy term and (iii) the affine transformation must be recalculated on each iteration. During the search the number of generators is gradually increased while their variance decreases according to predetermined "annealing" schedule3 . After fitting all the models to a particular image, we wish to evaluate which of the models best "explains" the data. The natural measure is the sum of Ejit, Edej and the volume factor. However, we have found that performance is improved by including four additional terms which are easily obtained from the final fits of the model to the image. These are (i) a measure which penalizes matches in which there are beads far from any inked pixels (the "beads in white space" problem), and (ii) the rotation, shear and elongation of the affine transform. It is hard to decide in a principled way on the correct weightings for all of these terms in the evaluation function. We estimated the weightings from the data by training a simple postprocessing neural network. These inputs are connected directly to the ten output units. The output units compete using the "softmax" function which guarantees that they form a probability distribution, summing to one. 2 PREDICTING THE INSTANTIATION PARAMETERS The search procedure described above is very time consuming. However, given many examples of images and the corresponding instantiation parameters obtained by the slow method, it is possible to train a neural network to predict the instantiation parameters of novel images. These predictions provide better starting points, so the search time can be reduced. 3The schedule starts with 8 beads increasing to 60 beads in six steps, with the variance decreasing from 0.04 to 0.0006 (measured in the object frame). The scale is set in the object-based frame so that each model is 1 unit high. 968 Christopher K. I. Williams, Michael D. Revow, Geoffrey E. Hinton 2.1 PREVIOUS WORK Previous work on hypothesizing instantiation parameters can be placed into two broad classes, correspondence based search and parameter space search. In correspondence based search, the idea is to extract features from the image and identify corresponding features in the model. Using sufficient correspondences the instantiation parameters of the model can be determined. The problem is that simple, easily detectable image features have many possible matches, and more complex features require more computation and are more difficult to detect. Grimson (1990) shows how to search the space of possible correspondences using an interpretation tree. An alternative approach, which is used in Hough transform techniques, is to directly work in parameter space. The Hough transform was originally designed for the detection of straight lines in images, and has been extended to cover a number of geometric shapes, notably conic sections. Ballard (1981) further extended the approach to arbitrary shapes with the Generalized Hough Transform. The parameter space for each model is divided into cells ("binned"), and then for each image feature a vote is added to each parameter space bin that could have produced that feature. After collecting votes from all image features we then search for peaks in the parameter space accumulator array, and attempt to verify pose. The Hough transform can be viewed as a crude way of approximating the logarithm of the posterior distribution P(zII, M) (e.g. Hunt et al , 1988). However, these two techniques have only been used on problems involving rigid models, and are not readily applicable to the digit recognition problem. For the Hough space method, binning and vote collection is impractical in the high dimensional parameter space, and for the correspondence based approach there is a lack of easily identified and highly discriminative features. The strengths of these two techniques, namely their ability to deal with arbitrary scalings, rotations and translations of the data, and their tolerance of extraneous features, are not really required for a task where the input data is fairly well segmented and normalized. Our approach is to use a neural network to predict the instantiation parameters for each model, given an input image. Zemel and Hinton (1991) used a similar method with simple 2-d objects, and more recently, Beymer et al (1993) have constructed a network which maps from a face image to a 2-d parameter space spanning head rotations and a smile/no-smile dimension. However, their method does not directly map from images to instantiation parameters; they use a computer vision correspondence algorithm to determine the displacement field of pixels in a novel image relative to a reference image, and then use this field as the input to the network. This step limits the use of the approach to images that are sufficiently similar so that the correspondence algorithm functions well. 2.2 INSTANTIATING DIGIT MODELS USING NEURAL NETWORKS The network which is used to predict the model instantiation parameters is shown in figure 1. The (unthinned) binary images are normalized to give 16 x 16 8-bit greyscale images which are fed into the neural network. The network uses a standard three-layer architecture; each hidden unit computes a weighted sum of its inputs, and then feeds this value through a sigmoidal nonlinearity u(x) = 1/(1 + e- X ). The Using a Neural Net to Instantiate a Deformable Model 969 cps for 0 model cps for I model cps for 9 model o Figure 1: The prediction network architecture. "cps" stands for control points. output values are a weighted linear combination of the hidden unit activities plus output biases. The targets are the locations of the control points in the normalized image, found from fitting models as described in section 1.2. The network was trained with backpropagation to minimize the squared error, using 900 training images and 200 validation images of each digit drawn from the br set of the CEDAR CDROM 1 database of Cities, States, ZIP Codes, Digits, and Alphabetic Characters4 . Two test sets were used; one was obtained from data in the br dataset, and the other was the (official) bs test set. After some experimentation we chose a network with twenty hidden units, which means that the net has over 8,000 weights. With such a large number of weights it is important to regularize the solution obtained by the network by using a complexity penalty; we used a weight penalty A L:j wJ and optimized A on a validation set. Targets were only set for the correct digit at the output layer; nothing was backpropagated from the other output units. The net took 440 epochs to train using the default conjugate gradient search method in the Xerion neural network simulator5. It would be possible to construct ten separate networks to carry out the same task as the net described above, but this would intensify the danger of overfitting, which is reduced by giving the network a common pool of hidden units which it can use as it decides appropriate. For comparison with the prediction net described above, a trivial network which just consisted of output biases was trained; this network simply learns the average value of the control point locations. On a validation set the squared error of the prediction net was over three times smaller than the trivial net. Although this is encouraging, the acid test is to compare the performance of elastic models settled from the predicted positions using a shortened annealing schedule; if the predictions are good, then only a short amount of settling will be required. 4Made available by the Unites States Postal Service Office of Advanced Technology. 5Xerion was designed and implemented by Drew van Camp, Tony Plate and Geoffrey Hinton at the University of Toronto. 970 Christopher K. I. Williams, Michael D. Revow, Geoffrey E. Hinton Figure 2: A comparision of the initial instantiations due to the prediction net (top row) and the trivial net (bottom row) on an image of a 2. Notice that for the two model the prediction net is much closer to the data. The other digit models mayor may not be greatly affected by the input data; for example, the predictions from both nets seem essentially the same for the zero, but for the seven the prediction net puts the model nearer to the data. The feedforward net predicts the position of the control points in the normalized image. By inverting the normalization process, the positions of the control points in the un-normalized image are determined. The model deformation and affine transformation corresponding to these image control point locations can then be determined by running a part of one iteration of the search procedure. Experiments were then conducted with a number of shortened annealing schedules; for each one, data obtained from settling on a part of the training data was used to train the postprocessing net. The performance was then evaluated on the br test set. The full annealing schedule has six stages. The shortened annealing schedules are: 1. No settling at all 2. Two iterations at the final variance of 0.0006 3. One iteration at 0.0025 and two at 0.0006 4. The full annealing schedule (for comparison) The results on the br test set are shown in table 1. The general trends are that the performance obtained using the prediction net is consistently better than the trivial net, and that longer annealing schedules lead to better performance. A comparison of schedules 3 and 4 in table 1 indicates that the performance of the prediction net/schedule 3 combination is similar to (or slightly better than) that obtained with the full annealing schedule, and is more than a factor of two faster. The results with the full schedule are almost identical to the results obtained with the default "box" initialization described in section 1.2. Figure 2 compares the outputs of the prediction and trivial nets on a particular example. Judging from the weight Using a Neural Net to Instantiate a Deformable Model 971 Schedule number Trivial net Prediction net A verage time required to settle one model (s) 1 427 200 0.12 2 329 58 0.25 3 160 32 0.49 4 40 36 1.11 Table 1: Errors on the internal test set of 2000 examples for different annealing schedules. The timing trials were carried out on a R-4400 machine. vectors and activity patterns of the hidden units, it does not seem that some of the units are specialized for a particular digit class. A run on the bs test set using schedule 3 gave an error rate of 4.76 % (129 errors), which is very similar to the 125 errors obtained using the full annealing schedule and the box initialization. A comparison of the errors made on the two runs shows that only 67 out of the 129 errors were common to the two sets. This suggests that it would be very sensible to reject cases where the two methods do not agree. 3 DISCUSSION The prediction net used above can be viewed as an interpolation scheme in the control point position space of each digit z(I) = Zo + 2:i ai(I)zi, where z(I) is the predicted position in the control point space, Zo is the contribution due to the biases, ai is the activity of hidden unit i and Zi is its location in the control point position space (learned from the data). If there are more hidden units than output dimensions, then for any particular image there are an infinite number of ways to make this equation hold exactly. However, the network will tend to find solutions so that the ai(I)'s will vary smoothly as the image is perturbed. The nets described above output just one set of instantiation parameters for a given model. However, it may be preferable to be able to represent a number of guesses about model instantiation parameters; one way of doing this is to train a network that has multiple sets of output parameters, as in the "mixture of experts" architecture of Jacobs et aI (1991). The outputs can be interpreted as a mixture distribution in the control point position space, conditioned on the input image. Another approach to providing more information about the posterior distribution is described in (Hinton, Williams and Revow, 1992b), where P(zlI) is approximated using a fixed set of basis functions whose weighting depends on the input image I. The strategies descriped above directly predict the instantiation parameters in parameter space. It is also possible to use neural networks to hypothesize correspondences, i.e. to predict an inked pixel's position on the spline given a local window of context in the image. With sufficient matches it is then possible to compute the instantiation parameters of the model. We have conducted some preliminary experiments with this method (described in Williams, 1994), which indicate that good performance can be achieved for the correspondence prediction task. 972 Christopher K. I. Williams, Michael D. Revow, Geoffrey E. Hinton We have shown that the we can obtain significant speedup using the prediction net. The schemes outlined above which allow multimodal predictions in instantiation parameter space may improve performance and deserve further investigation. We are also interested in improving the performance of the prediction net, for example by outputting a confidence measure which could be used to adjust the length of the elastic models' search appropriately. We believe that using machine learning techniques like neural networks to help reduce the amount of search required to fit complex models to data may be useful for many other problems. Acknowledgements This research was funded by Apple and by the Ontario Information Technology Research Centre. We thank Allan Jepson, Richard Durbin, Rich Zemel, Peter Dayan, Rob Tibshirani and Yann Le Cun for helpful discussions. Geoffrey Hinton is the Noranda Fellow of the Canadian Institute for Advanced Research. References Ballard, D. H. (1981). Generalizing the Hough transfrom to detect arbitrary shapes. Pattern Recognition, 13(2):111-122. Beymer, D., Shashua, A., and Poggio, T . (1993). Example Based Image Analysis and Synthesis. AI Memo 1431, AI Laboratory, MIT. Grenander, U., Chow, Y., and Keenan, D. M. (1991). Hands: A pattern theoretic study of biological shapes. Springer-Verlag. Grimson, W. E. 1. (1990). Object recognition by computer. MIT Press, Cambridge, MA. Hinton, G. E., Williams, C. K. 1., and Revow, M. D. (1992a). Adaptive elastic models for hand-printed character recognition. In Moody, J. E., Hanson, S. J., and Lippmann, R. P., editors, Advances in Neural Information Processing Systems 4. Morgan Kauffmann. Hinton, G. E., Williams, C. K. 1., and Revow, M. D. (1992b). Combinining two methods of recognizing hand-printed digits. In Aleksander, 1. and Taylor, J., editors, Artificial Neural Networks 2. Elsevier Science Publishers. Hunt, D. J., Nolte, L. W., and Ruedger, W. H. (1988). Performance of the Hough Transform and its Relationship to Statistical Signal Detection Theory. Computer Vision, Graphics and Image Processing, 43:221- 238. Jacobs, R. A., Jordan, M. 1., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation, 3(1). MacKay, D. J. C. (1992). Bayesian Interpolation. Neural Computation, 4(3):415-447. Revow, M. D., Williams, C. K. 1., and Hinton, G. E. (1993). Using mixtures of deformable models to capture variations in hand printed digits. In Srihari, S., editor, Proceedings of the Third International Workshop on Frontiers in Handwriting Recognition, pages 142-152, Buffalo, New York, USA. Williams, C. K. 1. (1994). Combining deformable models and neural networks for handprinted digit recognition. PhD thesis, Dept. of Computer Science, University of Toronto. Zemel, R. S. and Hinton, G. E. (1991). Discovering viewpoint-invariant relationships that characterize objects. In Lippmann, R. P., Moody, J. E., and Touretzky, D. S., editors, Advances In Neural Information Processing Systems 3, pages 299-305. Morgan Kaufmann Publishers.	1994	137

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