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Active Support Vector Machine Classification o. L. Mangasarian Computer Sciences Dept. University of Wisconsin 1210 West Dayton Street Madison, WI 53706 olvi@cs.wisc.edu David R. Musicant Dept. of Mathematics and Computer Science Carleton College One North College Street Northfield, MN 55057 dmusican@carleton.edu Abstract An active set strategy is applied to the dual of a simple reformulation of the standard quadratic program of a linear support vector machine. This application generates a fast new dual algorithm that consists of solving a finite number of linear equations, with a typically large dimensionality equal to the number of points to be classified. However, by making novel use of the Sherman-MorrisonWoodbury formula, a much smaller matrix of the order of the original input space is inverted at each step. Thus, a problem with a 32-dimensional input space and 7 million points required inverting positive definite symmetric matrices of size 33 x 33 with a total running time of 96 minutes on a 400 MHz Pentium II. The algorithm requires no specialized quadratic or linear programming code, but merely a linear equation solver which is publicly available. 1 Introduction Support vector machines (SVMs) [23, 5, 14, 12] are powerful tools for data classification. Classification is achieved by a linear or nonlinear separating surface in the input space of the dataset. In this work we propose a very fast simple algorithm, based on an active set strategy for solving quadratic programs with bounds [18]. The algorithm is capable of accurately solving problems with millions of points and requires nothing more complicated than a commonly available linear equation solver [17, 1, 6] for a typically small (100) dimensional input space of the problem. Key to our approach are the following two changes to the standard linear SVM: 1. Maximize the margin (distance) between the parallel separating planes with respect to both orientation (w) as well as location relative to the origin b). See equation (7) below. Such an approach was also successfully utilized in the successive overrelaxation (SOR) approach of [15] as well as the smooth support vector machine (SSVM) approach of [12]. 2. The error in the soft margin (y) is minimized using the 2-norm squared instead of the conventional 1-norm. See equation (7). Such an approach has also been used successfully in generating virtual support vectors [4]. These simple, but fundamental changes, lead to a considerably simpler positive definite dual problem with nonnegativity constraints only. See equation (8). In Section 2 of the paper we begin with the standard SVM formulation and its dual and then give our formulation and its simpler dual. We corroborate with solid computational evidence that our simpler formulation does not compromise on generalization ability as evidenced by numerical tests in Section 4 on 6 public datasets. See Table 1. Section 3 gives our active support vector machine (ASVM) Algorithm 3.1 which consists of solving a system of linear equations in m dual variables with a positive definite matrix. By invoking the Sherman-Morrison-Woodbury (SMW) formula (1) we need only invert an (n + 1) x (n + 1) matrix where n is the dimensionality of the input space. This is a key feature of our approach that allows us to solve problems with millions of points by merely inverting much smaller matrices of the order of n. In concurrent work [8] Ferris and Munson also use the SMW formula but in conjunction with an interior point approach to solve massive problems based on our formulation (8) as well as the conventional formulation (6). Burges [3] has also used an active set method, but applied to the standard SVM formulation (2) instead of (7) as we do here. Both this work and Burges' appeal, in different ways, to the active set computational strategy of More and Toraldo [18]. We note that an active set computational strategy bears no relation to active learning. Section 4 describes our numerical results which indicate that the ASVM formulation has a tenfold testing correctness that is as good as the ordinary SVM, and has the capability of accurately solving massive problems with millions of points that cannot be attacked by standard methods for ordinary SVMs. We now describe our notation and give some background material. All vectors will be column vectors unless transposed to a row vector by a prime I. For a vector x E Rn, x+ denotes the vector in Rn with all of its negative components set to zero. The notation A E Rmxn will signify a real m x n matrix. For such a matrix A' will denote the transpose of A and Ai will denote the i-th row of A. A vector of ones or zeroes in a real space of arbitrary dimension will be denoted by e or 0, respectively. The identity matrix of arbitrary dimension will be denoted by I. For two vectors x and y in Rn, x ..1 y denotes orthogonality, that is x' y = O. For U E R m, Q E Rmxm and B C {I, 2, ... , m}, UB denotes UiEB, QB denotes QiEB and QBB denotes a principal submatrix of Q with rows i E B and columns j E B. The notation argminxEs f(x) denotes the set of minimizers in the set S of the real-valued function f defined on S. We use := to denote definition. The 2-norm of a matrix Q will be denoted by IIQI12. A separating plane, with respect to two given point sets A and B in Rn , is a plane that attempts to separate Rn into two halfspaces such that each open halfspace contains points mostly of A or B. A special case of the Sherman-Morrison-Woodbury (SMW) formula [9] will be utilized: (Ilv + HH')-l = v(I - H(Ilv + H'H)-l H'), (1) where v is a positive number and H is an arbitrary m x k matrix. This formula enables us to invert a large m x m matrix by merely inverting a smaller k x k matrix. 2 The Linear Support Vector Machine We consider the problem of classifying m points in the n-dimensional real space R n , represented by the m x n matrix A, according to membership of each point Ai in the class A+ or A- as specified by a given m x m diagonal matrix D with +l's or -1 's along its diagonal. For this problem the standard SVM with a linear kernel [23, 5] is given by the following quadratic program with parameter v > 0: . 1 mm ve'y + -w'w s.t. D(Aw - e-y) + y 2:: e, y 2:: O. (2) (w,'Y,y)ERn+l+= 2 x'w = 1 + 1 x o 0 000 o 0 0 Ox A0 0000 X'W = 1-1 0 o ( M · 2 argln= IIwl12 x x x A+ x x x x x x x'w =1 Figure 1: The bounding planes (3) with a soft (i.e. with some errors) margin 2/llwI12, and the plane (4) approximately separating A+ from A-. Here w is the normal to the bounding planes: x'w = 'Y ± 1 (3) and'Y determines their location relative to the origin (Figure 1.) The plane x'w = 'Y + 1 bounds the A+ points, possibly with error, and the plane x'w = 'Y -1 bounds the A - points, also possibly with some error. The separating surface is the plane: x'w = 'Y, (4) midway between the bounding planes (3). The quadratic term in (2), is twice the reciprocal of the square of the 2-norm distance 2/llw112 between the two bounding planes of (3) (see Figure 1). This term maximizes this distance which is often called the "margin". If the classes are linearly inseparable, as depicted in Figure 1, then the two planes bound the two classes with a "soft margin". That is, they bound each set approximately with some error determined by the nonnegative error variable y: ~ 'Y + 1, for Dii = 1, ::; 'Y - 1, for Dii = - 1. (5) Traditionally the I-norm of the error variable y is minimized parametrically with weight v in (2) resulting in an approximate separation as depicted in Figure 1. The dual to the standard quadratic linear SVM (2) [13, 22, 14, 7] is the following: . 1 mill - u'DAA'Du - e'u s.t. e'Du = 0, 0 < u < ve. uER=2 (6) The variables (w, 'Y) of the primal problem which determine the separating surface (4) can be obtained from the solution of the dual problem above [15, Eqns. 5 and 7]. We note immediately that the matrix DAA'D appearing in the dual objective function (6) is not positive definite in general because typically m > > n. Also, there is an equality constraint present, in addition to bound constraints, which for large problems necessitates special computational procedures such as SMO [21]. Furthermore, a one-dimensional optimization problem [15] must be solved in order to determine the locator 'Y of the separating surface (4). In order to overcome all these difficulties as well as that of dealing with the necessity of having to essentially invert a very large matrix of the order of m x m , we propose the following simple but critical modification of the standard SVM formulation (2). We change Ily lll to Ilyll§ which makes the constraint y ~ 0 redundant. We also append the term 'Y2 to w'w. This in effect maximizes the margin between the parallel separating planes (3) with respect to both wand 'Y [15], that is with respect to both orientation and location of the planes, rather that just with respect to w which merely determines the orientation of the plane. This leads to the following reformulation of the SVM: y'y 1 min v- + -(w'w + ,2) s.t. D(Aw - er) + y ~ e. (7) (w ,'Y,y)ERn+l+", 2 2 the dual of this problem is [13]: 1 I min -u'( - + D(AA' + ee')D)u - e'u. O~uER'" 2 v (8) The variables (w,,) of the primal problem which determine the separating surface (4) are recovered directly from the solution of the dual (8) above by the relations: w=A'Du, y=u/v, ,=-e'Du. (9) We immediately note that the matrix appearing in the dual objective function is positive definite and that there is no equality constraint and no upper bound on the dual variable u. The only constraint present is a simple nonnegativity one. These facts lead us to our simple finite active set algorithm which requires nothing more sophisticated than inverting an (n + 1) x (n + 1) matrix at each iteration in order to solve the dual problem (8). 3 ASVM (Active Support Vector Machine) Algorithm The algorithm consists of determining a partition of the dual variable u into nonbasic and basic variables. The nonbasic variables are those which are set to zero. The values of the basic variables are determined by finding the gradient of the objective function of (8) with respect to these variables, setting this gradient equal to zero, and solving the resulting linear equations for the basic variables. If any basic variable takes on a negative value after solving the linear equations, it is set to zero and becomes nonbasic. This is the essence of the algorithm. In order to make the algorithm converge and terminate, a few additional safeguards need to be put in place in order to allow us to invoke the More-Toraldo finite termination result [18]. The other key feature of the algorithm is a computational one and makes use of the SMW formula. This feature allows us to invert an (n + 1) x (n + 1) matrix at each step instead of a much bigger matrix of order m x m. Before stating our algorithm we define two matrices to simplifY notation as follows: H = D[A - e], Q = I /v + HH'. (10) With these definitions the dual problem (8) becomes . 1 mm f(u):= -u'Qu - eu. (11) O~uER'" 2 It will be understood that within the ASVM Algorithm, Q- 1 will always be evaluated using the SMW formula and hence only an (n+l) x (n+l) matrix is inverted. We state our algorithm now. Note that commented (%) parts of the algorithm are not needed in general and were rarely used in our numerical results presented in Section 4. The essence of the algorithm is displayed in the two boxes below. Algorithm 3.1 Active SVM (ASVM) Algorithm for (8). (0) Start with UO := (Q- 1e)+. For i = 1,2, .. . , having u i compute Ui+1 as Ifollows. (1) Define Bi := {j I u; > a}, N i := {.i I u~ = a}. (2) Determine Ui+l .- (Q-1 e·) u i+1.- a Bi ' BiBi B ' +, Ni .. Stop if Ui+1 is the global solution, that is if a ~ Ui+1 -.l QUi+1 - e ~ a. (2a) % If f(uiH ) ~ f(ui ), then go to (4a). (2b) % If 0 :s; Ut.~l .1 QBi+1Bi+l nt.~ 1 -eBi+1 ~ 0, then UH1 is a global solution on the face of active constraints: UNi = O. Set ui := uiH and go to (4b). (3) ISet i := i + 1 and go to (1). I (4a) % Move in the direction of the global minimum on the face of act · t· t 0 S t -HI Q- l d H I .we cons razn s, UNi = . e UBi := BiBi eBi an UBi argmino95. df(uki + ).(ut.1 - nki)) I nki + ).(ut.1 - Uki ) ~ O}. If U~+1 = 0 for some j E B i , set i := i + 1 and go to (1). Otherwise UH1 is a global minimum on the face UNi = 0, and go to (4b). (4b) % Iterate a gradient projection step. Set k := 0 and uk := u i . Iterate Uk+l:= argminO<A<l f(uk - ).(uk -(Quk -e))+), k:= k + l untilf(uk) < f(11/). Set uiH ::: ilk. Set i:= i + 1 and go to (1). Remark 3.2 All commented (%) parts of the algorithm are optional and are not usually implemented unless the algorithm gets stuck, which it rarely did on our examples. Hence our algorithm is particularly simple and consists of steps (0), (1),(2) and (3). The commented parts were inserted in order to comply with the active set strategy of Morr!-Toraldo result [18] for which they give finite termination. Remark 3.3 The iteration in step (4b) is a gradient projection step which is guaranteed to converge to the global solution of (8) [2, pp 223-225] and is placed here to ensure that the strict inequality f(uk) < f(u') eventually holds as required in [18]. Similarly, the step in (4a) ensures that the function value does not increase when it remains on the same face, in compliance with [18, Algortihm BCQP(b)j. 4 Numerical Implementation and Comparisons We implemented ASVM in Visual C++ 6.0 under Windows NT 4.0. The experiments were run on the UW-Madison Data Mining Institute Locop2 machine, which utilizes a 400 MHz Pentium II Xeon Processor and a maximum of 2 Gigabytes of memory available per process. We wrote all the code ourselves except for the linear equation solver, for which we used CLAPACK [1, 6]. Our stopping criterion for ASVM is triggered when the error bound residual [16] Ilu - (u - Qu + e)+ II, which is zero at the solution of (11) , goes below O.l. The first set of experiments are designed to show that our reformulation (8) of the SVM (7) and its associated algorithm ASVM yield similar performance to the standard SVM (2), referred to here as SVM-QP. For six datasets available from the UCI Machine Learning Repository [19], we performed tenfold cross validation in order to compare test set accuracies between ASVM and SVM-QP. We implemented SVM-QP using the high-performing CPLEX barrier quadratic programming solver [10], and utilized a tuning set for both algorithms to find the optimal value of the parameter v , using the default stopping criterion of CPLEX. Altering the CPLEX default stopping criterion to match that of ASVM did not result in significant change in timing relative to ASVM, but did reduce test set correctness. In order to obtain additional timing comparison information, we also ran the wellknown SVM optimized algorithm SVM1ight [11]. Joachims, the author of SVM1ight , provided us with the newest version of the software (Version 3.lOb) and advice on setting the parameters. All features for these experiments were normalized to the range [-1, + 1] as recommended in the SVM1ight documentation. We chose to use Dataset Training Testing Time Dataset Training Testing Time m x n 1\lqorithm Correctness Correctness CPU sec) m x n ~Iqorithm Correctness Correctness (CPU sec) Liver Disorders CPLEX 70.76% 68.41% 7.87 Ionosphere CPLEX 92.81% 88.60% 9.84 ~VMf~ht 70.37% 68.12% 0.26 ~VMf~ht 92.81% 88.60% 0.23 345 x 6 ",SVM 70.40% 67.25% 0.03 351 x 34 ",SVM 93.29% 87.75% 0.26 Cleveland Heart CPLEX 87.50% 64.20% 4.17 ic Tae Toe CPLEX 65.34% 65.34% 206.52 ~VMf~ht 87.50% 64.20% 0.17 ~VMf~ht 65.34% 65.34% 0.23 297 x 13 SVM 87.24% 85.56% 0.05 958 x 9 SVM 70.27% 69.72% 0.05 Pima Diabetes CPLEX 77.36% 76.95% 128.90 Votes CPLEX 96.02% 95.85% 27.26 ~VMf~ht 77.36% 76.95% 0.19 ~VMf~ht 96.02% 95.85% 0.06 768 x 8 SVM 78.04% 78.12% 0.08 435 x 16 SVM 96.73% 96.07% 0.09 Table 1: ASVM compared with conventional SVM-QP (CPLEX and SVM1ight ) on VCI datasets. ASVM test correctness is comparable to SVM-QP, with timing much faster than CPLEX and faster than or comparable to SVM1ight • #01 Training Testing Time Points Iterations Correctness Correctness (CPU min) 4 million 5 86.09% 86.06% 38.04 7 million 5 86.10% 86.28% 95.57 Table 2: Performance of ASVM on NDC generated datasets in R 32 . (1/ = 0.01) the default termination error criterion in SVM1ight of 0.001, which is actually a less stringent criterion than the one we used for ASVM. This is because the criterion we used for ASVM (see above) is an aggregate over the errors for all points, whereas the SVM1ight criterion reflects a minimum error threshold for each point. The second set of experiments show that ASVM performs well on massive datasets. We created synthetic data of Gaussian distribution by using our own NDC Data Generator [20] as suggested by Usama Fayyad. The results of our experiments are shown in Table 2. We did try to run SVM1ight on these datasets as well, but we ran into memory difficulties. Note that for these experiments, all the data was brought into memory. As such, the running time reported consists of the time used to actually solve the problem to termination excluding I/O time. This is consistent with the measurement techniques used by other popular approaches [11, 21]. Putting all the data in memory is simpler to code and results in faster running times. However, it is not a fundamental requirement of our algorithm block matrix multiplications, incremental evaluations of Q-1 using another application of the SMW formula, and indices on the dataset can be used to create an efficient disk based version of ASVM. 5 Conclusion A very fast, finite and simple algorithm, ASVM, capable of classifying massive datasets has been proposed and implemented. ASVM requires nothing more complex than a commonly available linear equation solver for solving small systems with few variables even for massive datasets. Future work includes extensions to parallel processing of the data, handling very large datasets directly from disk as well as extending our approach to nonlinear kernels. Acknow ledgements We are indebted to our colleagues Thorsten Joachims for helping us to get SVM1ight running significantly faster on the UCI datasets, and to Glenn Fung for his efforts in running the experiments for revisions of this work. Research described in this Data Mining Institute Report 00-04, April 2000, was supported by National Science Foundation Grants CCR-9729842 and CDA-9623632, by Air Force Office of Scientific Research Grant F49620-00-1-0085 and by Microsoft. References [1] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Cros, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen. LAPACK User's Guide. SIAM, Philadelphia, Pennsylvania, second edition, 1995. [2] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, MA, second edition, 1999. [3] C. J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):121-167, 1998. [4] C. J. C. Burges and B. Sch6lkopf. Improving the accuracy and speed of support vector machines. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems -9-, pages 375-381, Cambridge, MA, 1997. MIT Press. [5] V. Cherkassky and F. Mulier. Learning from Data - Concepts, Theory and Methods. John Wiley & Sons, New York, 1998. [6] CLAPACK. f2c'ed version of LAPACK. http://www.netlib.org/clapack. [7] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge University Press, Cambridge, 2000. [8] M. C. Ferris and T. S. Munson. Interior point methods for massive support vector machines. Technical Report 00-05, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, May 2000. [9] G. H. Golub and C. F. Van Loan. Matrix Computations. The John Hopkins University Press, Baltimore, Maryland, 3rd edition, 1996. [10] ILOG, Incline Village, Nevada. CPLEX 6.5 Reference Manual, 1999. [11] T. Joachims. SVMlight, 1998. http://www-ai . informatik . uni -dortmund. de/FORSCHUNG/VERFAHREN/SVM_LIGHT/sVID_light.eng.html. [12] Yuh-Jye Lee and O. L. Mangasarian. SSVM: A smooth support vector machine. Computational Optimization and Applications, 2000. [13] O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994. [14] O. L. Mangasarian. Generalized support vector machines. In A. Smola, P. Bartlett, B. Sch6lkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, pages 135- 146, Cambridge, MA, 2000. MIT Press. [15] O. L. Mangasarian and D. R. Musicant. Successive overrelaxation for support vector machines. IEEE Transactions on Neural Networks, 10:1032- 1037, 1999. [16] O. L. Mangasarian and J. Ren. New improved error bounds for the linear complementarity problem. Mathematical Programming, 66:241- 255, 1994. [17] MATLAB. User's Guide. The MathWorks, Inc., Natick, MA 01760,1992. [18] J. J. More and G. Toraldo. Algorithms for bound constrained quadratic programs. Numerische Mathematik, 55:377-400, 1989. [19] P. M. Murphy and D. W. Aha. UCI repository of machine learning databases, 1992. www.ics.uci.edu/ rvmlearn/MLRepository.html. [20] D. R. Musicant. NDC: normally distributed clustered datasets, 1998. www.cs.wisc.edu/rvmusicant/data/ndc/. [21] J. Platt. Sequential minimal optimization: A fast algorithm for training support vector machines. In Sch6lkopf et al. [22], pages 185- 208. [22] B. Sch6lkopf, C. Burges, and A. Smola (editors). Advances in Kernel Methods: Support Vector Machines. MIT Press, Cambridge, MA, 1998. [23] V. N. Vapnik. The Nature of Statistical Learning Theory. 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Decomposition of Reinforcement Learning for Admission Control of Self-Similar Call Arrival Processes Jakob Carlstrom Department of Electrical Engineering, Technion, Haifa 32000, Israel jakob@ee . technion . ac . il Abstract This paper presents predictive gain scheduling, a technique for simplifying reinforcement learning problems by decomposition. Link admission control of self-similar call traffic is used to demonstrate the technique. The control problem is decomposed into on-line prediction of near-future call arrival rates, and precomputation of policies for Poisson call arrival processes. At decision time, the predictions are used to select among the policies. Simulations show that this technique results in significantly faster learning without any performance loss, compared to a reinforcement learning controller that does not decompose the problem. 1 Introduction In multi-service communications networks, such as Asynchronous Transfer Mode (ATM) networks, resource control is of crucial importance for the network operator as well as for the users. The objective is to maintain the service quality while maximizing the operator's revenue. At the call level, service quality (Grade of Service) is measured in terms of call blocking probabilities, and the key resource to be controlled is bandwidth. Network routing and call admission control (CAC) are two such resource control problems. Markov decision processes offer a framework for optimal CAC and routing [1]. By modelling the dynamics of the network with traffic and computing control policies using dynamic programming [2], resource control is optimized. A standard assumption in such models is that calls arrive according to Poisson processes. This makes the models of the dynamics relatively simple. Although the Poisson assumption is valid for most user-initiated requests in communications networks, a number of studies [3, 4, 5] indicate that many types of arrival processes in wide-area networks as well as in local area networks are statistically selfsimilar. This makes it difficult to find models of the dynamics, and the models become large and complex. If the number of system states is large, straightforward application of dynamic programming is unfeasible. Nevertheless, the "fractal" burst structure of self-similar traffic should be possible to exploit in the design of efficient resource control methods. We have previously presented a method based on temporal-difference (TD) learning for CAC of self-similar call traffic, which yields higher revenue than a TD-based controller assuming Poisson call arrival processes [7]. However, a drawback of this method is the slow convergence of the control policy. This paper presents an alternative solution to the above problem, called predictive gain scheduling. It decomposes the control problem into two parts: time-series prediction of near-future call arrival rates and precomputation of a set of control policies for Poisson call arrival processes. At decision time, a policy is selected based on these predictions. Thus, the self-similar arrival process is approximated by a quasi-stationary Poisson process. The rate predictions are made by (artificial) neural networks (NNs), trained on-line. The policies can be computed using dynamic programming or other reinforcement learning techniques [6]. This paper concentrates on the link admission control problem. However, the controllers we describe can be used as building block in optimal routing, as shown in [8] and [9]. Other recent work on reinforcement learning for CAC and routing includes [10], where Marbach et al. show how to extend the use of TD learning to network routing, and [11] where Tong et al. apply reinforcement learning to routing subject to Quality of Service constraints. 2 Self-Similar Call Arrival Processes The limitations of the traditional Poisson model for network arrival processes have been demonstrated in a number of studies, e.g. [3, 4, 5], which indicate the existence of heavytailed inter-arrival time distributions and long-term correlations in the arrival processes. Self-similar (fractal-like) models have been shown to correspond better with this traffic. A self-similar arrival process has no "natural" burst length. On the contrary, its arrival intensity varies considerably over many time scales. This makes the variance of its sample mean decay slowly with the sample size, and its auto-correlation function decay slowly with time, compared to Poisson traffic [4]. The complexity of control and prediction of Poisson traffic is reduced by the memory-less property of the Poisson process: its expected future depends on the arrival intensity, but not on the process history. On the other hand, the long-range dependence of self-similar traffic makes it possible to improve predictions of the process future by observing the history. A compact statistical measure of the degree of self-similarity of a stochastic process is the Hurst parameter [4]. For self-similar traffic this parameter takes values in the interval (0.5, 1], whereas Poisson processes have a Hurst parameter of 0.5. 3 The Link Admission Control Problem In the link admission control (LAC) problem, a link with capacity C [units/s] is offered calls from K different service classes. Calls belonging to such a class j E J = {I, ... , K} have the same bandwidth requirements hj [units/s]. The per-class call holding times are assumed to be exponentially distributed with mean 1/ftj [s]. Access to the link is controlled by a policy:rc that maps states x E X to actions a EA,:rc: X -+ A. The set X contains all feasible link states, and the action set is A = ((ai, ... ,aK) : aj E {O, Il,j E J), where aj is ° for rejecting a presumptive class-j call and 1 for accepting it. The set of link states is given by X = N x H, where N is the set of feasible call number tuples, and His the Cartesian product of some representations, '1, of the history of the per-class call arrival processes (needed because of the memory of self-similar arrival processes). N is given by N = {n : nj ;:: 0, j E J; Injhj ::; C}' jEJ where nj is the number of type-j calls accepted on the link. We assume uniform call charging, which means that the reward rate p(t) at time t is equal to the carried bandwidth: pet) = p(x(t» = I n/t)bj (1) jEl Time evolves continuously, with discrete call arrival and departure events, enumerated by k = 0,1,2, ... Denote by rk+l the immediate reward obtained from entering a state Xk at time tk until entering the next state Xk+l at time tk+1• The expectation of this reward is E,,{rk+l} = E,,{P(Xk)[tk+1 t,)} = P(Xk)1:(X",:rr(Xk» (2) where t'(xk,:rr) is the expected sojourn time in state Xk under policy:rr. By taking optimal actions, the policy controls the probabilities of state transitions so as to increase the probability of reaching states that yield high long-term rewards. The objective of link: admission control is to find a policy :rr that maximizes the average reward per stage: R(,,) ~ )~"! E.{~ ~ 'He I X, ~ x}. x E X (3) Note that the average reward does not depend on the initial state x, as the contribution from this state to the average reward tends to zero as N -+ 00 (assuming, for example, that the probability of reaching any other state y E X from every state x E X is positive). Certain states are of special interest for the optimal policy. These are the states that are candidates for intelligent blocking. The set of such states X ib C X is given by X ib = Nib X H, where Nib is the set of call number tuples for which the available bandwidth is a multiple of the bandwidth of a wideband call. In the states of X ib, the long-term reward may be increased by rejecting narrowband calls to reserve bandwidth for future, expected wideband calls. 4 Solution by Predictive Gain Scheduling Gain scheduling is a control theory technique, where the parameters of a controller are changed as a function of operating conditions [12]. The approach taken here is to look up policies in a table from predictions of the near-future per-class call arrival rates. For Poisson call arrival processes, the optimal policy for the link: admission control problem does not depend on the history, H, of the arrival processes. Due to the memory-less property, only the (constant) per-class arrival rates Aj , j E J, matter. In our gain scheduled control of self-similar call arrival processes, near-future Aj are predicted from hj- The selfsimilar call arrival processes are approximated by quasi-stationary Poisson processes, by selecting precomputed polices (for Poisson arrival processes) based on predicted A/s. One radial-basis function (REF) NN per class is trained to predict its near-future arrival rate. 4.1 Solving the Link Admission Control problem for Poisson Traffic For Poisson call arrival processes, dynamic programming offers well-established techniques for solving the LAC problem [1]. In this paper, policy iteration is used. It involves two steps: value determination and policy improvement. The value determination step makes use of the objective function (3), and the concept of relative values [1]. The difference v(x,:rr) v(y,:rr) between two relative values under a policy :rr is the expected difference in accumulated reward over an infinite time interval, starting in state X instead of state y. In this paper, the relative values are computed by solving a system of linear equations, a method chosen for its fast convergence. The dynamics of the system are characterized by state transition probabilities, given by the policy, the perclass call arrival intensities, (,q, and mean holding times, (1/,ll J The policy improvement step consists of finding the action that maximizes the relative value at each state. After improving the policy, the value determination and policy improvement steps are iterated until the policy does not change [9]. 4.2 Determining The Prediction Horizon Over what future time horizon should we predict the rates used to select policies? In this work, the prediction horizon is set to an average of estimated mean first passage times from states back to themselves, in the following referred to as the mean return time. The arrival process is approximated by a quasi-stationary Poisson process within this time interval. The motivation for this choice of prediction horizon is that the effects of a decision (action) in a state Xd influence the future probabilities of reaching other states and receiving the associated rewards, until the state Xd is reached the next time. When this happens, a new decision can be made, where the previous decision does no longer influence the future expected reward. In accordance with the assumption of quasi-stationarity, the mean return time can be estimated for call tuples n instead of the full state descriptor, x. In case of Poisson call arrival processes, the mean first passage times E,.{ Tin} from other states to a state n are the unique solution of the linear system of equations E,,{TmJ = T(m, a) + I E,,{Tln }, m E N\{n}, a = n(m) (4) IEN\!n} The limiting probability qn of occupying state n is determined for all states that are candidates for intelligent blocking, by solving a linear system of equations qB = 0. B is a matrix containing the state transition intensities, given by (Aj} and (1/,llj}. The mean return time for the link, TI, is defmed as the average of the individual mean return times of the states of Nib, weighted by their limiting probabilities and normalized: (5) For ease of implementation, this time window is expressed as a number of call arrivals. The window length Lj for class j is computed by multiplying the mean return time by the arrival rate, Lj = Aj T[, and rounding off to an integer. Although the window size varies with Aj, this variation is partly compensated by T[ decreasing with increasing Aj • 4.3 Prediction of Future Call Arrival Rates The prediction of future arrival call rates is naturally based on measures of recent arrival rates. In this work, the following representation of the history of the arrival process is used: for all classes j E J, exponentially weighted running averages hj = (hj), ... , hjM) of the inter-arrival times are computed on different time scales. These history vectors are computed using forgetting factors {a), ... ,aM } taking values in the interval (0, 1): hik) = a i[t/k) - t/k 1) 1 + (1 - a;)hik 1) , where fj(k) is the arrival time of the k-th call from class j. (6) In studies of time-series prediction, non-linear feed-forward NN s outperform linear predictors on time series with long memory [13]. We employ RBF NNs with symmetric Gaussian basis functions. The activations of the RBF units are normalized by division by the sum of activations, to produce a smooth output function. The locations and widths of the RBF units can be determined by inspection of the data sets, to cover the region of history vectors. The NN is trained with the average inter-arrival time as target. After every new call arrival, the prediction error €j(k) is computed: Lj Elk) = L I [ t(k + i) t(k + i-I)] - y/k). J i~ ' (7) Learning is performed on-line using the least mean squares rule, which means that the upd)lting must be delayed by Lj call arrivals. The predicted per-class arrival rates A/k) = y(k)-' are used to select a control policy on the arrival of a call request. Given the prediction horizon and the arrival rate predictor, ai' ... ,aM can be tuned by linear search to minimize the prediction error on sample traffic traces. 5 Numerical study The performance of the gain scheduled admission controller was evaluated on a simulated link with capacity C = 24 [units/s], that was offered calls from self-similar call arrival processes. For comparison, the simulations were repeated with three other link admission controllers: two TD-based controllers, one table-based and one NN based, and a controller using complete sharing, i.e. to accept a call if the free capacity on the link is sufficient. The NN based TD controller [7] uses RBF NNs (one per n EN), receiving (h" h2) as input. Each NN has 65 hidden units, factorized to 8 units per call class, plus a default activation unit. Its weights were initialized to favor acceptance of all feasible calls in all states. The table-based TD controller assumes Poisson call arrival processes. From this, it follows that the call number tuples n E N constitute Markovian states. Consequently, the value function table stores only one value per n. This controller was used for evaluation of the performance loss from incorrectly modelling self-similar call traffic by Poisson traffic. 5.1 Synthesis of Call Traffic Synthetic traffic traces were generated from a Gaussian fractional auto-regressive integrated moving average model, FARIMA (0, d, 0). This results in a statistically self-similar arrival process, where the Hurst parameter is easily tuned [7]. We generated traces containing arrival/departure pairs from two call classes, characterized by bandwidth requirements bi = 1 (narrow-band) and ~ = 6 (wide-band) [units/s] and call holding times with mean 1/,u1 = 1/,u2= 1 [s]. A Hurst parameter of 0.85 was used, and the call arrival rates were scaled to make the expected long-term arrival rates A, and A2 for the two classes fulfill b,A,/,u, + b).2/,u2 = 1.25 C. The ratio A,/A2 was varied from 0.4 to 2.0. 5.2 Gain Scheduling For simplicity, a constant prediction horizon was used throughout the simulations. This was computed according to section 4.2. By averaging the resulting prediction windows for A,/A2 = 0.4, 1.0 and 2.0, a window size L, = L2 = 6 was obtained. A A The table of policies to be used for gain scheduling was computed for predicted A, and A2 ranging from 0.5 to 15 with step size 0.5; in total 900 policies. The two rate-prediction NNs both had 9 hidden units. The NNs' weights were initialized to O. 5.3 Numerical results Both the TD learning controllers and the gain scheduling controller were allowed to adapt to the first 400 000 simulated call arrivals of the traffic traces. The throughput obtained by all four methods was measured on the subsequent 400000 call arrivals. o 1000 2000 3000 4000 0.5 1 1.5 2 2.5 3 3.5 4.0 call arrivals x 105 call arrivals (a) Initial weight evolution in neural predictor (b) Long-term weight evolution in neural predictor 11 9 1.5 2 2.5 3 3.5 4.0 x 105 call arrivals Throughput [units/s] 17.4 17.2 17.0 16.8 16.6 16.4 16.2 16.0 15.8 GSIRBF TDIRBF TDITBL CS 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 AdA2 (c) Weight evolution in NN based TD controller (d) Throughput versus arrival rate ratio Figure 1: Weight evolution for NN predictor (a, b); NN based TD-controller (c). Performance (d). Figure 1 (a, b) shows the evolution of the weights of the call arrival rate predictor for class 2, and figure 1 (c) displays nine weights of the RBF NN corresponding to the call number tuple (n!, n2) = (6,2), which is a candidate for intelligent blocking. These weights correspond to eight different class-2 center vectors, plus the default activation. The majority of the weights of the gain scheduling RBF NN seems to converge in a few thousand call arrivals, whereas the TD learning controller needs about tOO 000 call arrivals to converge. This is not surprising, since the RBF NNs of the TD learning controllers split up the set of training data, so that a single NN is updated much less frequently than a ratepredicting NN in the gain scheduling controller. Secondly, the TD learning NNs are trained on moving targets, due to the temporal-difference learning rule, stochastic action selection and a changing policy. A few of the weights of the gain scheduling NN change considerably even after long training. These weights correspond to RBF units that are activated by rare, large inputs. Figure t (d) evaluates performance in terms of throughput versus arrival rate ratio. Each data point is the averaged throughput for 10 traffic traces. Gain scheduling (GS/RBF) achieves the same throughput as TD learning with RBF NNs (TD/RBF), up to 1.3% compared to tabular TD learning (TDITBL), and up to 5.7% better than complete sharing (CS). The difference in throughput between TD learning and complete sharing is greatest for low arrival rate ratios, since the throughput increase by reserving bandwidth for highrate wideband calls is considerably higher than the loss of throughput from the blocked lowrate narrowband traffic. 6 Conclusion We have presented predictive gain scheduling, a technique for decomposing reinforcement learning problems. Link admission control, a sub-problem of network routing, was used to demonstrate the technique. By predicting near-future call arrival rates from one part of the full state descriptor, precomputed policies for Poisson call arrival processes (computed from the rest of the state descriptor) were selected. This increased the on-line convergence rate approximately 50 times, compared to a TD-based admission controller getting the full state descriptor as input. The decomposition did not result in any performance loss. The computational complexity of the controller using predictive gain scheduling may reach a computational bottleneck if the size of the state space is increased: the determination of optimal policies for Poisson traffic by policy iteration. This can be overcome by state aggregation [2], or by parametrization the relative value function combined with temporaldifference learning [10]. It is also possible to significantly reduce the number of relative value functions. In [14], we showed that linear interpolation of relative value functions distributed by an error-driven algorithm enables the use of less than 30 relative value functions without performance loss. Further, we have successfully employed gain scheduled link admission control as a building block of network routing [9], where the performance improvement compared to conventional methods is larger than for the link admission control problem. The use of gain scheduling to reduce the complexity of reinforcement learning problems is not limited to link admission control. In general, the technique should be applicable to problems where parts of the state descriptor can be used, directly or after preprocessing, to select among policies for instances of a simplified version of the original problem. References [1] Z. Dziong, ATM Network Resource Management, McGraw-Hill, 1997. [2] D.P. Bertsekas, Dynamic Programming and Optimal Control, Athena Scientific, Belmont, Mass., 1995. [3] V. Paxson and S. Floyd, "Wide-Area Traffic: The Failure of Poisson Modeling", IEEF/ACM Transactions on Networking, vol. 3, pp. 226-244, 1995. [4] W.E. Leland, M.S. Taqqu, W. Willinger and D.V. Wilson, "On the Self-Similar Nature of Ethemet Traffic (Extended Version)", IEEF/ACM Transactions on Networking, vol. 2, no. 1, pp. 1- 15, Feb. 1994. [5] A Feldman, AC. Gilbert, W. Willinger and T.G. Kurtz, "The Changing Nature of Network Traffic: Scaling Phenomena", Computer Communication Review, vol. 28, no. 2, pp. 5- 29, April 1998. [6] R.S. Sutton and AG. Barto, Reinforcement Learning: An Introduction, MIT Press, Cambridge, Mass., 1998. [7] J. Carlstrom and E. Nordstrom, "Reinforcement Learning for Control of Self-Similar Call Traffic in Broadband Networks", Teletraffic Engineering in a Competitive World - Proceedings of The 16th International Teletraffic Congress (ITC 16), pp. 571- 580, Elsevier Science B.V., 1999. [8] Z. Dziong and L. Mason,"Call Admission Control and Routing in Multi-service Loss Networks", IEEE Transactions on Communications, vol. 42, no. 2. pp. 2011- 2022, Feb. 1994. [9] J. Carlstrom and E. Nordstrom, "Gain Scheduled Routing in Multi-Service Networks", Technical Report 2000-009, Dept. of Information Technology, Uppsala University, Uppsala, Sweden, April 2000. [10] P. Marbach, O. Mihatsch and J.N. Tsitsiklis, "Call Admission Control and Routing in Integrated Service Networks Using Neuro-Dynarnic Programming", IEEE J. Sel. Areas ofComm, Feb. 2000. [11] H. Tong and T. Brown, "Adaptive Call Admission Control Under Quality of Service Constraints: A Reinforcement Learning Solution", IEEE Journal on Selected Areas in Communications, Feb. 2000. [12] K.J. Astrom and B. Wittenmark, Adaptive Control, 2nd ed., Addison-Wesley, 1995. [13] S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd ed., Macmillan College Publishing Co., Englewood Cliffs, NJ, 1999. [14] J. Carlstrom, "Efficient Approximation of Values in Gain Scheduled Routing", Technical Report 2000-010, Dept. of Information Technology, Uppsala University, Uppsala, Sweden, April 2000.
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Generalizable Singular Value Decomposition for Ill-posed Datasets Ulrik Kjerns Lars K. Hansen Department of Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby, Denmark uk, lkhansen@imm. dtu. dk Abstract Stephen C. Strother PET Imaging Service VA medical center Minneapolis steve@pet. med. va. gov We demonstrate that statistical analysis of ill-posed data sets is subject to a bias, which can be observed when projecting independent test set examples onto a basis defined by the training examples. Because the training examples in an ill-posed data set do not fully span the signal space the observed training set variances in each basis vector will be too high compared to the average variance of the test set projections onto the same basis vectors. On basis of this understanding we introduce the Generalizable Singular Value Decomposition (GenSVD) as a means to reduce this bias by re-estimation of the singular values obtained in a conventional Singular Value Decomposition, allowing for a generalization performance increase of a subsequent statistical model. We demonstrate that the algorithm succesfully corrects bias in a data set from a functional PET activation study of the human brain. 1 Ill-posed Data Sets An ill-posed data set has more dimensions in each example than there are examples. Such data sets occur in many fields of research typically in connection with image measurements. The associated statistical problem is that of extracting structure from the observed high-dimensional vectors in the presence of noise. The statistical analysis can be done either supervised (Le. modelling with target values: classification, regresssion) or unsupervised (modelling with no target values: clustering, PCA, ICA). In both types of analysis the ill-posedness may lead to immediate problems if one tries to apply conventional statistical methods of analysis, for example the empirical covariance matrix is prohibitively large and will be rank-deficient. A common approach is to use Singular Value Decomposition (SVD) or the analogue Principal Component Analysis (PCA) to reduce the dimensionality of the data. Let the N observed i-dimensional samples Xj, j = L .N, collected in the data matrix X = [Xl ... XN] of size I x N, I> N . The SVD-theorem states that such a matrix can be decomposed as (1) where U is a matrix of the same size as X with orthogonal basis vectors spanning the space of X, so that UTU = INxN. The square matrix A contains the singular values in the diagonal, A = diag( AI, ... , >w), which are ordered and positive Al ~ A2 ~ ... ~ AN ~ 0, and V is N x N and orthogonal V TV = IN. If there is a mean value significantly different from zero it may at times be advantageous to perform the above analysis on mean-subtracted data, i.e. X - X = U A V T where columns of X all contain the mean vector x = Lj xj/N. Each observation Xj can be expressed in coordinates in the basis defined by the vectors of U with no loss of information[Lautrup et al., 1995]. A change of basis is obtained by qj = U T Xj as the orthogonal basis rotation Q = [ql ... qN] = U T X = UTUAVT = AVT. (2) Since Q is only N x Nand N « I, Q is a compact representation of the data. Having now N examples of N dimension we have reduced the problem to a marginally illposed one. To further reduce the dimensionality, it is common to retain only a subset of the coordinates, e.g. the top P coordinates (P < N) and the supervised or unsupervised model can be formed in this smaller but now well-posed space. So far we have considered the procedure for modelling from a training set. Our hope is that the statistical description generalizes well to new examples proving that is is a good description of the generating process. The model should, in other words, be able to perform well on a new example, x*, and in the above framework this would mean the predictions based on q* = U T x* should generalize well. We will show in the following, that in general, the distribution of the test set projection q* is quite different from the statistics of the projections of the training examples qj. It has been noted in previous work [Hansen and Larsen, 1996, Roweis, 1998, Hansen et al., 1999] that PCA/SVD of ill-posed data does not by itself represent a probabilistic model where we can assign a likelihood to a new test data point, and procedures have been proposed which make this possible. In [Bishop, 1999] PCA has been considered in a Bayesian framework, but does not address the significant bias of the variance in training set projections in ill-posed data sets. In [Jackson, 1991] an asymptotic expression is given for the bias of eigen-values in a sample covariance matrix, but this expression is valid only in the well-posed case and is not applicable for ill-posed data. 1.1 Example Let the signal source be I-dimensional multivariate Gaussian distribution N(O,~) with a covariance matrix where the first K eigen-values equal u 2 and the last 1- K are zero, so that the covariance matrix has the decomposition ~=u2YDyT, D=diag(1, ... ,1,0, ... ,0), yTY=I (3) Our N samples of the distribution are collected in the matrix X = [Xij] with the SVD A = diag(Al, ... , AN) (4) and the representation ofthe N examples in the N basis vector coordinates defined by U is Q = [%] = U T X = A V T. The total variance per training example is ~ LX;j ~Tr(XTX) = ~Tr(VAUTUAVT) = ~Tr(VA2VT) i,j = ~ Tr(VVT A2) = ~ Tr(A2) = ~L A; i (5) Note that this variance is the same in the U-basis coordinates: 1 '" 2 N L...J qij = ~Tr(QTQ) = ~Tr(VA2VT) = ~ LA~ i,j i We can derive the expected value of this variance: (~ LX;) = (LxL) = (x? Xl) = Tr:E = a2K i ,j (6) (7) Now, consider a test example X* '" N(O,:E) with the projection q* = U T x* which will have the average total variance (Tr[(UT x*{ (UT x*)]) = Tr [(x*x* T)UUT] Tr[:EUUT] = Tr[DUUT] = a2min(N,K) (8) In summary, this means that the orthogonal basis U computed from the training set spans all the variance in the training set but fails to do so on the test examples when N < K, i.e. for ill-posed data. The training set variance is K / N a 2 on average per coordinate, compared to a 2 for the test examples. So which of the two variances is "correct" ? From a modelling point of view, the variance from the test example tells us the true story, so the training set variance should be regarded as biased. This suggests that the training set singular values should be corrected for this bias, in the above example by re-estimating the training set projections using Q = J N / K Q. In the more general case we do not know K, and the true covariance may have an arbitrary eigen-spectrum. The GenSVD algorithm below is a more general algorithm for correcting for the training set bias. 2 The GenSVD Algorithm The data matrix consists of N statistically independent samples X = [Xl ... XN ] so X is size I x N, and each column of X is assumed multivariate Gaussian, Xj '" N(O,:E) and is ill-posed with rank:E > N. With the SVD X = UoAoVaT, we now make the approximation that Uo contains an actual subset of the true eigen-vectors of :E (9) where we have collected the remaining (unspanned by X) eigen-vectors and values in UJ. and Ai , satisfying uluJ. = I and UJUJ. = 0. The unknown 'true' eigen-values corresponding to the observed eigen-vectors are collected in A = diag(Al, ... AN), which are the values we try to estimate in the following. It should be noted that a direct estimation of :E using f: = j;y X X T yields f: = j;yuoAoVaTVoAoUJ = j;yUoA~UJ, i.e., the nonzero eigen-vectors and values of f: is Uo and Ao. The distribution of test samples x* inside the space spanned by Uo is (10) The problem is that Uo and the examples Xj are not independent, so UJ Xj is biased, e.g. the SVD estimate -k A ~ of A 2 assigns all variance to lie within Uo. The GenSVD algorithm bypasses this problem by, for each example, computing a basis on all other examples, estimating the variances in A 2 in a leave-one-out manner. Consider (11) where we introduce the notation X_j for the matrix of all examples except the j'th, and this matrix is decomposed as X_j = B_jA_j lC;' The operation B_jB_; Xj projects the example onto the basis defined by the remaining examples, and back again, so it 'strips' off the part of signal space which is special for Xj which could be signal which does not generalize across examples. Since B_j and Xj are independent B-"J Xj has the same distribution as the projection of a test example x*, B_; x*. Thus, B_jB_; Xj and B_jB_; x* have the same distribution as well. Now, since span B_j=span X_j and span Uo=span [X_j Xj] we have that span B_j~span Uo so we see that Z j and U J B_jB-"J X* are identically distributed. This means that Zj has the covariance UJ B_jB-"J~B_jB_;Uo and using Eq. (9) and that ul B_j = 0 (since uluo = 0) we get (12) We note that this distribution is degenerate because the covariance is of rank N -l. For a sample Zj from the above distribution we have that UJ B_jB_;Uozj = UJ B_jB_;UoUJ B_jB_; Xj = UJ B_jB_; Xj = Zj (13) As a second approximation, assume that the observed Zj are independent so that we can write the likelihood of A ~ log [(27r)N/21(uJ B_J(B-"JUo)A2(UJ B_J(R;Uo)l l/2] +~ ~zJ (UJ B_J(B_;Uo)A -2(UJ B_j) (B-"JUo)zj j N ~ 2 1~ T 2 C + log A· + z· A - z· 2 J 2 J J j j (14) where we have used Eq. (13) and that the determinantl is approximated by IA21. This above expression is maximized when 5.~ = ~ ~ Z~j' (15) j A T A A A The GenSVD of X is then X = UoAV ,A = diag(Al' ... , AN). In practice, using Eq. (11) directly to compute an SVD of the matrix X_j for each example is computationally demanding. It is possible to compute Zj in a more efficient two-level procedure with the following algorithm: Compute UOAoVOT = svd(X) and Qo = [qj] = AoVOT lSince Zj is degenerate, we define the likelihood over the space where Zj occur, i.e. the determinant in Eq. 14 should be read as 'the product of non-zero eigenvalues'. foreach j = L.N Compute B_;A_; v..; = svd( Q.J Zj = B_;B-"J qj A2 1 2 '\ = Iii L:j Zij If the data has a mean value that we wish to remove prior to the SVD it is important that this is done within the GenSVD algorithm. Consider a centered matrix Xc = X - X where X contains the mean x replicated in all N columns. The signal space in Xc is now corrupted because each centered example will contain a component of all examples, which means the 'stripping' of signal components not spanned by other examples no longer works: B_; Xj is no longer distributed like B_; x*. This suggests the alternative algorithm for data with removal of mean component: Compute UOAoVOT = svd(X) and Qo = [qj] = AoVOT foreach j = L.N 1 '"" q-j = N-1 6j'¥j qj' T Compute B_;A_; v..; = svd(Q_; - Q.;) Zj = B_;B-"J (qj ii-j) A2 _ 1 2 Ai N -1 L:j Zij Finally, note that it is possible to leave out more than one example at a time if the data is independent only in block, i.e. let Q.k would be Qo with the k'th block left out. Example With PET Scans We compared the performance of GenSVD to conventional SVD on a functional [150] water PET activation study of the human brain. The study consisted of 18 subjects, who were scanned four times while tracing a star-shaped maze with a joy-stick with visual feedback, in total 72 scans of dimension '" 25000 spatial voxels. After the second scan, the visual feedback was mirrored, and the subject accomodated to and learned the new control environment during the last two scans. Scans were normalized by 1) dividing each scan by the average voxel value measured inside a brain mask and 2) for each scan subtracting the average scan for that subject thereby removing subject effects and 3) intra and inter-subject normalization and transformation using rigid body reorientation and affine linear transformations respectively. Voxels inside aforementioned brain mask were arranged in the data matrix with one scan per column. Figure 1 shows the results of an SVD decomposition compared to GenSVD. Each marker represents one scan and the glyphs indicate scan number out of the four (circle-square-star-triangle). The ellipses indicate the mean and covariances of the projections in each scan number. The 32 scans from eight subjects were used as a training set and 40 scans from the remaining 10 subjects for testing. The training set projections are filled markers, test-set projections onto the basis defined by the training set are open markers (i.e. we plot the first two columns of UoAo for SVD and of UoA for GenSVD). We see that there is a clear difference in variance in the train- and test-examples, which is corrected quite well by GenSVD. The lower plot in Figure 1 shows the singular values for the PET data set. We see that GenSVD estimates are much closer to the actual test projection standard deviations than the SVD singular values. 3 Conclusion We have demonstrated that projection of ill-posed data sets onto a basis defined by the same examples introduces a significant bias on the observed variance when comparing to projections of test examples onto the same basis. The GenSVD algorithm has been presented as a tool for correcting for this bias using a leave-one-out re-estimation scheme, and a computationally efficient implementation has been proposed. We have demonstrated that the method works well on an ill-posed real-world data set, were the distribution of the GenSVD-corrected training test set projections matched the distribution of the observed test set projections far better than the uncorrected training examples. This allows a generalization performance increase of a subsequent statistical model, in the case of both supervised and unsupervised models. Acknowledgments This work was supported partly by the Human Brain Project grant P20 MH57180, the Danish Research councils for the Natural and Technical Sciences through the Danish Computational Neural Network Center (CONNECT) and the Technology Center Through Highly Oriented Research (THOR). References [Bishop, 1999] Bishop, C. (1999). Bayesian pca. In Kearns, M. S., Soli a, S. A., and Cohn, D. A., editors, Advances in Neural Information Processing Systems, volume 11. The MIT Press. [Hansen et al., 1999] Hansen, L., Larsen, J. , Nielsen, F., Strother, S., Rostrup, E., Savoy, R., Lange, N., Sidtis, J., Svarer, C., and Paulson, O. (1999) . Generalizable patterns in neuroimaging: How many principal components? NeuroImage, 9:534- 544. [Hansen and Larsen, 1996] Hansen, L. K. and Larsen, J. (1996). Unsupervised learning and generalization. In Proceedings of IEEE International Conference on Neural Networks, pages 25- 30. [Jackson, 1991] Jackson, J . E. (1991). A User's Guide to Principal Components. Wiley Series on Probability and Statistics, John Wiley and Sons. [Lautrup et aI., 1995] Lautrup, B., Hansen, L. K., Law, I., M0rch, N., Svarer, C., and Strother, S. (1995). Massive weight sharing: A cure for extremely ill-posed problems. In Hermann, H. J ., Wolf, D. E., and Poppel, E. P., editors, Proceedings of Workshop on Supercomputing in Brain Research: Prom Tomography to Neural Networks: Prom tomography to neural networks, HLRZ, KFA Jillich, Germany, pages 137- 148. World Scientific. [Roweis, 1998] Roweis, S. (1998) . Em algorithms for pca and spca. In Jordan, M. I., Kearns, M. J., and Soli a, S. A., editors, Advances in Neural Information Processing Systems, volume 10. The MIT Press. 3.00 2.00 'E 11 1.00 0 "E 8 0 0.00 > (jJ -g 0 i;l - 1.00 (jJ - 2.00 - 3.00 - 4.00 1.50 1.00 " 11 ~ 0.50 8 o ~ 0.00 <D (!) "C 8 - 0.50 ell - 1.00 * * * .. - 3.00 - 2.00 * 1< Conventional SVD • * 1< • ~ . Oc .· . ~ " ~: j~.~ .Ii. .Ii. ..: .J>. .Ii. • - 1.00 0.00 1.00 First SVD component Generalizable SVD • '!' • .. • • 2.00 3.00 Solid: Train Open:Test o Trace scan 1 o Trace scan 2 * Mirror scan 1 £J. Mirror scan 2 4.00 - 1.50'--_~ __ ~ _______________ ---J c: o iii .~ - 2.00 - 1.50 - 1.00 - 0.50 0.00 0.50 1.00 1.50 2.00 First GenSVD component 2.00 r-~--------------------'---' SVD training set projection stdev -- GenSVD training set proj. stdev Test set projection stdev 1.50 \ \ ~ 1.00 {g ~ 0.50 0.00 '-----------------------'---' 5 10 Component 15 20 Figure 1: Projections of PET data in SVD and GenSVD. Each subject's four scans are indicated by: circle, square, star, triangle. Training set scans are marked with filled glyphs and test set with open glyphs. Solid and dotted Ellipses indicate test/train covariance per scan number. The third plot shows the standard deviations for the training and test set for SVD and GenSVD projections.
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Vicinal Risk Minimization Olivier Chapelle, Jason Weston* , Leon Bottou and Vladimir Vapnik AT&T Research Labs, 100 Schultz drive, Red Bank, NJ, USA * Barnhill BioInformatics.com, Savannah, GA, USA. {chapelle, weston,leonb, vlad}@research.att.com Abstract The Vicinal Risk Minimization principle establishes a bridge between generative models and methods derived from the Structural Risk Minimization Principle such as Support Vector Machines or Statistical Regularization. We explain how VRM provides a framework which integrates a number of existing algorithms, such as Parzen windows, Support Vector Machines, Ridge Regression, Constrained Logistic Classifiers and Tangent-Prop. We then show how the approach implies new algorithms for solving problems usually associated with generative models. New algorithms are described for dealing with pattern recognition problems with very different pattern distributions and dealing with unlabeled data. Preliminary empirical results are presented. 1 Introduction Structural Risk Minimisation (SRM) in a learning system can be achieved using constraints on the parameter vectors, using regularization terms in the cost function, or using Support Vector Machines (SVM). All these principles have lead to well established learning algorithms. It is often said, however, that some problems are best addressed by generative models. The first problem is of missing data. We may for instance have a few labeled patterns and a large number of unlabeled patterns. Intuition suggests that these unlabeled patterns carry useful information. The second problem is of discriminating classes with very different pattern distributions. This situation arises naturally in anomaly detection systems. This also occurs often in recognition systems that reject invalid patterns by defining a garbage class for grouping all ambiguous or unrecognizable cases. Although there are successful non-generative approaches (Schuurmans and Southey, 2000) (Drucker, Wu and Vapnik, 1999), the generative framework is undeniably appealing. Recent results (Jaakkola, Meila and Jebara, 2000) even define generative models that contain SVM as special cases. This paper discusses the Vicinal Risk Minimization (VRM) principle, summarily introduced in (Vapnik, 1999). This principle was independently hinted at by Tong and Koller (Tong and Koller, 2000) with a useful generative interpretation. In particular, they proved that SVM are a limiting case of their Restricted Bayesian Classifiers. We extend Tong's and Koller's result by showing that VRM subsumes several well known techniques such as Ridge Regression (Hoerl and Kennard, 1970), Constrained Logistic Classifier, or Tangent Prop (Simard et aI., 1992). We then go on to show how VRM naturally leads to simple algorithms that can deal with problems for which one would have formally considered purely generative models. We provide algorithms and preliminary empirical results for dealing with unlabeled data or recognizing classes with very different pattern distributions. 2 Vicinal Risk Minimization The learning problem can be formulated as the search of the function f E F that minimizes the expectation of a given loss £(f(x), y). R(f) = f £(f(x), y) dP(x, y) (1) In the classification framework, y takes values ±1 and £(f(x) , y) is a step function such as 1 - Sign(yf(x)), whereas in the regression framework, y is a real number and commonly £(f(x), y) is the mean squared error (f(x) _ y)2. The expectation (1) cannot be computed since the distribution P(x, y) is unknown. However, given a training set {(Xi, Yi) h <i<n, it is common to minimize instead the empirical ri~: -1 n Remp(f) = - L £(f(Xi)' Yi) n i=l Empirical Risk Minimization (ERM) is therefore equivalent to minimizing the expectation of the loss function with respect to an empirical distribution Pemp(x,y) formed by assembling delta functions located on each example: 1 n dPemp(x, y) = - LOx, (X)Oy, (y) n i=l (2) It is quite natural to consider improved density estimates by replacing the delta functions ox, (x) by some estimate of the density in the vicinity of the point Xi, PXi (X). 1 n dPest(x, y) = - L dPx, (x)Oy'(y) n i=l (3) We can define in this way the vicinal risk of a function as: Rvic(f) = f £(f(x),y) dPest(x,y) = ~ t f £(f(x), Yi)dPXi (x) (4) ~=1 The Vicinal Risk Minimization principle consists of estimating argmin!EFR(f) by the function which minimizes the vicinal risk (4). In general, one can construct the VRM functional using any estimate dPest (x, y) of the density dP(x, y), instead of restricting our choices to pointwise kernel estimates. Spherical gaussian kernel functions Nu(x - Xi) are otherwise an obvious choice for the local density estimate dPXi (x). The corresponding density estimate dPest is a Parzen windows estimate. The parameter u controls the scale of the density estimate. The extreme case u = 0 leads to the estimation of the density by delta functions and therefore leads to ERM. This must be distinguished from the case u -t 0 because the limit is taken after the minimization of the integral, leading to different results as shown in the next section. The theoretical analysis of ERM (Vapnik, 1999) shows that the crucial factor is the capacity of the class F offunctions. Large classes entail the risk of overfitting, whereas small classes entail the risk of underfitting. Two factors however are responsible for generalization of VRM, namely the quality of the estimate dPest and the size of the class F of functions. If dPest is a poor approximation to P then VRM can still perform well if F has suitably small capacity. ERM indeed uses a very naive estimate of dP and yet can provide good results. On the other hand, if F is not chosen with suitably small capacity then VRM can still perform well if the estimate dPest is a good approximation to dP. One can even take the set of all possible functions (whose capacity is obviously infinite) and still find a good solution if the estimate dPest is close enough to dP with an adequate metric. For example, if dPest is a Parzen window density estimate, then the Vicinal Risk minimizer is the Parzen window classifier. This latter property contrasts nicely with the ERM principle whose results strongly depend on the choice of the class of functions. Although we do not have a full theoretical understanding of VRM at this time, we expect considerable differences in the theoretical analysis of ERM and VRM. 3 Special Cases We now discuss the relationship of VRM to existing methods. There are obvious links between VRM and Parzen windows or Nearest Neighbour when the set of functions F is unconstrained. Furthermore, many existing algorithms can be viewed as special cases of VRM for different choices of F and dPest . a) VRM Regression and Ridge Regression Consider the case of VRM for regression with spherical Parzen windows (using gaussian kernel) with standard deviation u and with a family F of linear functions fw ,b(X) = W . x + b. We can write the vicinal risk as: Rvic(f) The resulting expression is the empirical risk augmented by a regularization term. The particular cost function above is known as the Ridge Regression cost function (Hoed and Kennard, 1970). This result can be extended to the case of non linear functions f by performing a Taylor expansion of f(Xi + £). The corresponding regularization term then combines successive derivatives offunction f. Useful mathematical arguments can be found in (Leen, 1995). b) VRM and Invariant Learning Generating synthetic examples is a simple way to incorporate selected invariances in a learning system. For instance, we can augment a optical character recognition database by applying applying translations or rotations to the initial examples. In the limit, this is equivalent to replacing each initial example by a distribution whose shape represents the desired invariances. This formulation naturally leads to a special case of VRM in which the local density estimates PXi (x) are elongated in the direction of invariance. Tangent-Prop (Simard et aI., 1992) is a more sophisticated way to incorporate invariances by adding an adequate regularization term to the cost function. Tangent-Prop has been formally proved to be equivalent to generating synthetic examples with infinitesimal deformations (Leen, 1995). This analysis makes Tangent-Prop a special case ofVRM. The local density estimate PXi is simply formed by Gaussian kernels with a covariance matrix whose eigenvectors describe the tangent direction to the invariant manifold. The eigenvalues then represent the respective strengths of the selected invariances. The tangent covariance matrix used in the SVM context by (Scholkopf et aI., 1998) specifies invariances globally. It can also been seen as a special case of VRM. c) VRM Classifier and Constrained Logistic Classifier Consider the case of VRM for classification with spherical Parzen windows with standard deviation 0' and with a family F of linear functions fw,b(X) = W . x + b. We can assume without loss of generality that JJwJJ = 1. We can write the vicinal risk as: RVic(w,b) 1 n f :;;: L -Yi Sign(b + w . x) dPXi (x) i=l = 1 n f :;;: L -Yi Sign(b + W· Xi + W· e:) dNu(e:) ,=1 We can decompose e: = WEw + e:~ where WEw represents its component parallel to wand e:~ represents its orthogonal component. Since JJwJJ = 1, we have W • e: = Ew. After integrating over e: ~ we are left with the following expression: The latter integral can be seen as the convolution of the Gaussian Nu (x) with the step function Sign(x), which is a sigmoid shaped function with asymptotes at ±1. Using notation rp(x) = 2 erf(x) - 1, we can write: 1 n (w. Xi + b) RVic(W, b) = :;;: ?= -Yi rp 0' ,=1 By rescaling wand b by a factor 1/0', we can write the following equivalent formulation of the VRM: Arg Min - L Yi rp(w· Xi + b) { In :i': constra:til~wJJ = 1/0' (5) Except for the minor shape difference between sigmoid functions, the above formulation describes a Logistic Classifier with a constraint on the weights. This formulation is also very close to using a single artificial neuron with a sigmoid transfer function and a weight decay. The above proof illustrates a general identity. Transforming the empirical probability estimate (2) by convolving it with a kernel function is equivalent to transforming the loss function £(f (x), y) by convolving it with the same kernel function. This is summarized in the following equality, where * represents the convolution operator. f £(f(x),y) [NuO * dPemp(',y)] (x) = f [£(f(.),y) *NuO] (x) dPemp(x,y) d) VRM Classifier and SVM (Tong and Koller, 2000) Consider again the case of VRM for classification with spherical Parzen windows with standard deviation 0' and with a family F of linear functions fw,b(X) = W . x + b. The resulting algorithm is in fact a Restricted Bayesian Classifier (Tong and Koller, 2000). Assuming that the examples are separable, Tong and Koller have shown that the resulting decision boundary tends towards the hard margin SVM decision boundary when a tends towards zero. The proof is based on the following observation: when a ~ 0, the vicinal risk (4) is dominated by the terms corresponding to the examples whose distance to the decision boundary is minimal. These examples in fact are the support vectors. On the other hand, choosing a > ° generates a decision boundary which depends on all the examples. The contribution of each example decreases exponentially when its distance to the decision boundary increases. This is only slightly different from a soft margin SVM whose boundary relies on support vectors that can be more distant than those selected by hard margin SVM. The difference here is just in the cost functions (sigmoid compared to linear loss). e) SVM and Constrained Logistic Classifiers The two previous paragraphs show that the same particular case of VRM is (a) equivalent to a Logistic Classifier with a constraint on the weights, and (b) tends towards the SVM classifier when a ~ ° and when the examples are separable. As a consequence, we can state that the Logistic Classifier decision boundary tends towards the SVM decision boundary when we relax the constraint on the weights. In practice we can find the SVM solution with a Logistic Classifier by simply using an iterative weight update algorithm such as gradient descent, choosing small initial weights, and letting the norm of the weights grow slowly while the iterative algorithm is running. Although this algorithm is not exact, it is fast and efficient. This is in fact similar to what is usually done with back-propagation neural networks (LeCun et aI., 1998). The same algorithm can be used for the VRM. In that context early stopping is similar to choosing the optimal a using cross-validation. 4 New Algorithms and Results 4.1 Adaptive Kernel Widths It is known in density estimation theory that the quality of the density estimate can be improved using variable kernel widths (Breiman, Meisel and Purcell, 1977). In regions of the space where there is little data, it is safer to have a smooth estimate of the density, whereas in the regions of the space there is more data one wants to be as accurate as possible via sharper kernel estimates. The VRM principle can take advantage of these improved density estimates for other problem domains. We consider here the following density estimate: 1 dPest(x, y) = - L 8Yi (y) NUi (x - Xi) dx n . ~ where the specific kernel width ai for each training example Xi is computed from the training set. a) Wisconsin Breast Cancer We made a first test of the method on the Wisconsin breast cancer dataset l which contains 589 examples on 30 dimensions. We compared VRM using the set of linear classifiers with various underlying density estimates. The minimization was achieved using gradient descent on the vicinal risk. All hyperparameters were determined using cross-validation. The following table reports results averaged on 100 runs. 1 h up:1 /horn. first. gmd .de/ ..... raetschl data/breast -cancer. SoftSVM VRM VRM Training Set HardSVM Beste Best fixed U Adaptive Ui 10 11.3% 11.1% 10.8% 9.6% 20 8.3% 7.5% 6.9% 6.6% 40 6.3% 5.5% 5.2% 4.8% 80 5.4% 4.0% 3.9% 3.7% The adaptive kernel width Ui were computed by multiplying a global factor by the average distance of the five closest training examples. The best global factor is determined by crossvalidation. These results suggest that VRM with adaptive kernel widths can outperform state of the art classifiers on small training sets. b) MNIST "I" versus other digits A second test was performed using the MNIST handwritten digits2• We considered the sub-problem of recognizing the ones versus all other digits. The testing set contains 10000 digits (5000 ones and 5000 non-ones). Two training set sizes were considered with 250 or 500 ones and an equal number of non-ones. Computations were achieved using the algorithm suggested in section (3.e). We simply trained a single linear unit with a sigmoid transfer function using stochastic gradient updates. This is appropriate for implementing an approximate VRM with a single kernel width. Adaptive kernel widths are implemented by simply changing the slope of the sigmoid for each example. For each example Xi, the kernel width Ui is computed from the training set using the 5/1000th quantile of the distances of all other examples to example Xi. The sigmoid slopes are then computed by renormalizing the Ui in order to make their mean equal to 1. Early stopping was achieved with cross-validation. Training Set HardSVM VRM VRM Fixed slope Adaptive slope 250+250 3.34% 2.79% 2.54% 500+500 3.11% 2.47% 2.27% 1000+1000 2.94% 2.08% 1.96% The statistical signifiance of these results can be asserted with very high probability by comparing the list of errors performed by each system (Bottou and Vapnik, 1992). Again these results suggest that VRM with adaptive kernel widths can be very useful with small training sets. 4.2 Unlabeled Data In some applications unlabeled data is in abundance whereas labeled data is not. The use of unlabeled data falls into the framework of VRM by simply making the same vicinal loss for unlabeled points. Given m unlabeled points xi, ... , x:n, one obtains the following formulation: 1 n f 1 m f Rvic(f) =;;: L l(f(X),Yi)dPXi(x) + m L l(f(x),f(xn)dPx;(x). i=l i=l To give an example of the usefulness of our approach consider the following example. Two normal distributions on the real line N( -1.6,1) and N(1.6, 1) model the patterns of two classes with equal probability; 20 labeled points and 100 unlabeled points are drawn. The following table compares the true generalization error of VRM with gaussian kernels and linear functions. Results are averaged over 100 runs. Two different kernel widths UL and Uu were used for kernels associated with labeled or unlabeled examples. Best kernel widths were obtained by cross-validation. We also studied the case UL -+ 0 in order to provide a result equivalent to a plain SVM. 2http://www.research.att.com/ ... yannlocr/index.html aL -+ 0 Best aL Best au Best au Labeled 6.5% 5.0% Labeled+Unlabeled 5.6% 4.3% Note that when both aL and au tend to zero, this algorithm reverts to a transduction algorithm due to Vapnik which was previously solved by the more difficult optimization procedure of integer programming (Bennet and Demiriz, 1999). 5 Conclusion In conclusion, the Vicinal Risk Minimization VRM principle provides a useful bridge between generative models and SRM methods such as SVM or Statistic Regularization. Several well known algorithms are in fact special cases of VRM. The VRM principle also suggests new algorithms. In this paper we proposed algorithms for dealing with unlabeled data and recognizing classes with very different pattern distributions, obtaining initial promising results. We hope that this approach can lead to further understanding of existing methods and also to suggest new ones. References Bennet, K. and Demiriz, A. (1999). Semi-supervised support vector machines. In Advances in Neural Information Processing Systems 11, pages 368-374. MIT Press. Bottou, L. and Vapnik, V. N. (1992). Local learning algorithms, appendix on confidence intervals. Neural Computation, 4(6):888- 900. Breiman, L., Meisel, W., and Purcell, E. (1977). Variable kernel estimates of multivariate densities. Technometrics, 19:135- 144. Drucker, H., Wu, D., and Vapnik, V. (1999). Support vector machines for spam categorization. Neural Networks, 10:1048- 1054. Hoed, A. and Kennard, R. W. (1970). Ridge regression: Biased estimation for non orthogonal problems. Technometrics, 12(1):55--67. Jaakkola, T., Meila, M., and Jebara, T. (2000). Maximum entropy discrimination. In Advances in Neural Information Processing Systems 12. MIT Press. LeCun, Y., Bottou, L., Orr, G., and Muller, K. (1998). Efficient backprop. In Orr, G. and K., M., editors, Neural Networks: Tricks of the Trade. Springer. Leen, T. K. (1995). Invariance and regularization in learning. In Advances in Neural Infonnation Processing Systems 7. MIT Press. Scholkopf, B., Simard, P., Smola, A., Vapnik, V. (1998). Prior knowledge in support vector kernels. In Advances in Neural Information Processing Systems 10. MIT Press. Schuurmans, D. and Southey, F. (2000). An adaptive regularization criterion for supervised learning. In Proceedings of the Seventeenth International Conference on Machine Learning (ICML2000). Simard, P., Victorri, B., Le Cun, Y., and Denker, J. (1992). Tangent prop: a formalism for specifying selected invariances in adaptive networks. In Advances in Neural Information Processing Systems 4, Denver, CO. Morgan Kaufman. Tong, S. and Koller, D. (2000). Restricted bayes optimal classifiers. Proceedings of the 17th National Conference on Artificial Intelligence (AAAI). Vapnik, V. (1999). The Nature of Statistical Learning Theory (Second Edition). Springer Verlag, New York.
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Means. Correlations and Bounds M.A.R. Leisink and H.J. Kappen Department of Biophysics University of Nijmegen, Geert Grooteplein 21 NL 6525 EZ Nijmegen, The Netherlands {martijn,bert}@mbfys.kun.nl Abstract The partition function for a Boltzmann machine can be bounded from above and below. We can use this to bound the means and the correlations. For networks with small weights, the values of these statistics can be restricted to non-trivial regions (i.e. a subset of [-1 , 1]). Experimental results show that reasonable bounding occurs for weight sizes where mean field expansions generally give good results. 1 Introduction Over the last decade, bounding techniques have become a popular tool to deal with graphical models that are too complex for exact computation. A nice property of bounds is that they give at least some information you can rely on. For instance, one may find that a correlation is definitely between 0.4 and 0.6. An ordinary approximation might be more accurate, but in practical situations there is absolutely no warranty for that. The best known bound is probably the mean field bound, which has been described for Boltzmann machines in [1] and later for sigmoid belief networks in [2]. Apart from its bounding properties, mean field theory is a commonly used approximation technique as well. Recently this first order bound was extended to a third order approximation for Boltzmann machines and sigmoid belief networks in [3] and [4], where it was shown that this particular third order expansion is still a bound. In 1996 an upper bound for Boltzmann machines was described in [5] and [6]. In the same articles the authors derive an upper bound for a special case of sigmoid belief networks: the two-layered networks. In this article we will focus solely on Boltzmann machines, but an extension to sigmoid belief networks is quite straightforward. This article is organized as follows: In section 2 we start with the general theory about bounding techniques. Later in that section the upper and lower bound are briefly described. For a full explanation we refer to the articles mentioned before. The section is concluded by explaining how these bounds on the partition function can be used to bound means and correlations. In section 3 results are shown for fully connected Boltzmann machines, where size of weights and thresholds as well as network size are varied. In section 4 we present our conclusions and outline possible extensions. 2 Theory There exists a general method to create a class of polynomials of a certain order, which all bound a function of interest, fo(x). Such a class of order 2n can be found if the 2n-th order derivative of fo(x), written as hn(x), can be bounded by a constant. When this constant is zero, the class is actually of order 2n-1. It turns out that this class is parameterized by n free parameters. Suppose we have a function b2k for some integer k which bounds the function 12k from below (an upper bound can be written as a lower bound by using the negative of both functions). Thus (1) Now construct the primitive functions 12k -1 and b2k -1 such that 12k - 1 (p) = b2k- 1(p) for a free to choose value for p. This constraint can always be achieved by adding an appropriate constant to the primitive function b2k- 1 . It is easy to prove that { 12k -1 (x) :S b2k -1 (x) 12k -1 (x) 2: b2k -1 (x) or in shorthand notation hk-1(x) § b2k- 1(X). for x < p for x 2: p (2) If we repeat this procedure and construct the primitive functions hk-2 and b2k- 2 such that hk-2(p) = b2k- 2(p) for the same p, one can verify that Vx hk-2(x) 2: b2k- 2(X) (3) Thus given a bound 12k (x) 2: b2k (x) we can construct a class of bounding functions for hk-2 parameterized by p. Since we assumed hn (x) can be bounded from below by a constant, we can apply the procedure n times and we finally find fa (x) 2: bo (x), where bo (x) is parameterized by n free parameters. This procedure can be found in more detail in [4]. 2.1 A third order lower bound for Boltzmann machines Boltzmann machines are stochastic neural networks with N binary valued neurons, Si, which are connected by symmetric weights Wij. Due to this symmetry the probability distribution is a Boltzmann-Gibbs distribution which is given by (see also [7]) p (siB, w) = ~ exp (~L. WijSiSj + L BiSi) = ~ exp (-E (s, B, w)) (4) 'J ' where the Bi are threshold values and Z (B , w) = L exp ( - E (s, B, w)) (5) all S is the normalization known as the partition function. This partition function is especially important, since statistical quantities such as means and correlations can be directly derived from it. For instance, the means can be computed as (sn) = LP (siB, w) Sn = L P (s, Sn =+l IB, w) - P (s, Sn = - l iB, w) all S all s/sn Z+ (B, w) - Z_ (B, w) Z (B, w) (6) where Z+ and Z_ are partition functions over a network with Sn clamped to +1 and -1 , respectively. This explains why the objective of almost any approximation method is the partition function given by equation 5. In [3] and [4] it is shown that the standard mean field lower bound can be obtained by applying the linear bound (7) to all exponentially many terms in the sum. Since J.l may depend on S, one can choose J.l (s) = J.li Si + J.lo , which leads to the standard mean field equations, where the J.li turn out to be the local fields. Moreover, the authors show that one can apply the procedure of 'upgrading bounds' (which is described briefly at the beginning of this section) to equation 7, which leads to the class of third order bounds for eX. This is achieved in the following way: 'r/X,V h(x) = eX 2': eV (1 + x - v) = b2(x) h(x)=ex'§ell-+ev ((1+J.l-v)(x-J.l)+~(x-J.l) 2) =bdx) (8) 'r/X,Il-,A fo(x) = eX 2': ell- { 1 + x - J.l + eA C ; >.. (x - J.l)2 + ~ (x - J.l)3) } = bo(x) with>" = v - J.l. In principle, this third order bound could be maximized with respect to all the free parameters, but here we follow the suggestion made in [4] to use a mean field optimization, which is much faster and generally almost as good as a full optimization. For more details we refer to [4]. 2.2 An upper bound An upper bound for Boltzmann machines has been described in [5] and [6]1. Basically, this method uses a quadratic upper bound on log cosh x, which can easily be obtained in the following way: h(x) = 1 - tanh2 x::; 1 = b2(x) h(x) = tanh x ~ x - J.l + tanhJ.l = bdx) (9) 1 2 fa (x) = log cosh x ::; "2 (x - J.l) + (x - J.l) tanh J.l + log cosh J.l = bo (x) Using this bound, one can derive Z (e , w) = L exp (~L WijSiSj + L eiSi) all s ij i = ~ 2exp (lOg cosh (L WniSi + en)) exp (~ .L WijSiSj + L eiSi) all sisn , 'J i'n ' i'n ::; L exp (~ L W~jSiSj + L e;Si + k) = ek . Z (e' , W') allsls n iji'n ii'n (10) INote: The articles referred to, use Si E {O, I} instead of the +1/-1 coding used here. where k is a constant and el and Wi are thresholds and weights in a reduced network given by I Wij = Wij + WniWnj e;j = ei + Wni (en J-Ln + tanhJ-Ln) (11) 1 2 1 2 k = "2 (en J-Ln + tanhJ-Ln) -"2 tanh J-Ln + log 2 cosh J-Ln Hence, equation 10 defines a recursive relation, where each step reduces the network by one neuron. Finally, after N steps, an upper bound on the partition function is found 2 . We did a crude minimization with respect to the free parameters J-L. A more sophisticated method can probably be found, but this is not the main objective of this article. 2.3 Bounding means and correlations The previous subsections showed very briefly how we can obtain a lower bound, ZL, and an upper bound, ZU , for any partition function. We can use this in combination with equation 6 to obtain a bound on the means: ZL _ ZU Zu _ ZL (sn)L = + X -::::; (sn)::::; + y - = (snt (12) where X = ZU if the nominator is positive and X = ZL otherwise. For Y it is the opposite. The difference, (sn)U - (sn)L, is called the bandwidth. Naively, we can compute the correlations similarly to the means using (13) where the partition function is computed for all combinations Sn Sm. Generally, however, this gives poor results, since we have to add four bounds together, which leads to a bandwidth which is about twice as large as for the means. We can circumvent this by computing the correlations using (14) where we allow the sum in the partition functions to be taken over Sn , but fix Sm either to Sn or its negative. Finally, the computation of the bounds (SnSm)L and (snsmt is analogue to equation 12. There exists an alternative way to bound the means and correlations. One can write ( ) _ Z+ - Z _ _ Z+/Z_ - 1 _ z - 1 - f ( ) Sn z Z+ + Z _ Z+/Z_ + 1 z + 1 with z = Z+/Z_ , which can be bounded by ZL Zu ----± < z < ----± Z~ Z~ (15) (16) Since f (z) is a monotonically increasing function of z, the bounds on (Sn) are given by applying this function to the left and right side of equation 16. The correlations can be bounded similarly. It is still unknown whether this algorithm would yield better results than the first one, which is explored in this article. 2The original articles show that it is not necessary to do all the N steps. However, since this is based on mixing approximation techniques with exact calculations, it is not used here as it would hide the real error the approximation makes. 13 ir==================i~----~----~ Exact Mean field lower bound 12.5 Upper bound Third order lower bound 12 11 10.5 10'--~·""'--: ' ' - ... ....... . _,_ ....... - ,'" o 0.2 0.4 a w 0.6 , , . ' 0.8 ,. , . , ,-,. ,. Figure 1: Comparison of 1) the mean field lower bound, 2) the upper bound and 3) the third order lower bound with the exact log partition function. The network was a fully connected Boltzmann machine with 14 neurons and (J'B = 0.2. The size of the weights is varied on the x-axis. Each point was averaged over 100 networks. 3 Results In all experiments we used fully connected Boltzmann machines of which the thresholds and weights both were drawn from a Gaussian with zero mean and standard deviation (J'B and (J'w/VN, respectively, where N is the network size. This is the so called sK-model (see also [8]). Generally speaking, the mean field approximation breaks down for (J'B = 0 and (J'w > 0.5, whereas it can be proven that any expansion based approximation is inaccurate when (J'w > 1 (which is the radius of convergence as in [9]). If (J'B #- 0 these maximum values are somewhat larger. In figure 1 we show the logarithm of the exact partition function , the first order or mean field bound, the upper bound (which is roughly quadratic) and the third order lower bound. The weight size is varied along the horizontal axis. One can see clearly that the mean field bound is not able to capture the quadratic form of the exact partition function for small weights due to its linear behaviour. The error made by the upper and third order lower bound is small enough to make non-trivial bounds on the means and correlations. An example of this bound is shown in figure 2 for the specific choice (J'B = (J'w = 0.4. For both the means and the correlations a histogram is plotted for the upper and lower bounds computed with equation 12. Both have an average bandwidth of 0.132, which is a clear subset of the whole possible interval of [-1 , 1]. In figure 3 the average bandwidth is shown for several values of (J'e and (J' w ' For bandwidths of 0.01,0.1 and 1 a line is drawn. We conclude that almost everywhere the bandwidth is non-trivially reduced and reaches practically useful values for (J'w less than 0.5. This is more or less equivalent to the region where the mean field approximation performs well. That approximation, however, gives no information on how close it actually is to the exact value, whereas the bounding method limits it to a definite region. Means 80 60 40 20 -0.2 0.2 Distance to exact Correlations 600,---------=----, ,------,6010 500 400 300 200 100 -0.2 -0.1 o Distance to exact 100 0.1 0.2 Figure 2: For the specific choice IJo = IJw = 0.4 thirty fully connected Boltzmann machines with 14 neurons were initialized and the bounds were computed. The two left panels show the distance between the lower bound and the exact means (left) and similarly for the upper bound (right). The right two panels show the distances of both bounds for the correlations. 0.8 1.5 1.5 0.6 o~ 0.4 0.5 0.5 0.2 00 Ow Figure 3: In the left panel the average bandwidth is colour coded for the means, where IJo and IJw are varied in ten steps along the axes. The right panel shows the same for the correlations. For each IJo , IJw thirty fully connected Boltzmann machines were initialized and the bounds on all the means and correlations were computed. For three specific bandwidths a line is drawn. 0.01 0.4 2 0.008 °e=0.3 °e=0.3 °e=0.3 0 =0.1 0.3 ° =0.3 1.5 ° =0.5 .r::; w w w ~0006 0.2 "0 @0.004 c:J 0.002 0.1 0.5 00 10 20 30 40 00 10 20 30 40 00 10 20 30 40 Network size Figure 4: For (Tw = 0.1, 0.3 and 0.5 the bandwidth for the correlations is shown versus the network size. (To = 0.3 in all cases, but the plots are nearly the same for other values. Please note the different scales for the y-axis. A similar graph for the means is not shown here, but it is roughly the same. The solid line is the average bandwidth over all correlations, whereas the dashed lines indicate the minimum and maximum bandwidth found. Unfortunately, the bounds have the unwanted property that the error scales badly with the size of the network. Although this makes the bounds unsuitable for very large networks, there is still a wide range of networks small enough to take advantage of the proposed method and still much too large to be treated exactly. The bandwidth versus network size is shown in figure 4 for three values of (T w' Obviously, the threshold of practical usefulness is reached earlier for larger weights. Finally, we remark that the computation time for the upper bound is (') (N4) and (') (N 3 ) for the mean field and third order lower bound. This is not shown here. 4 Conclusions In this article we combined two already existing bounds in such a way that not only the partition function of a Boltzmann machine is bounded from both sides, but also the means and correlations. This may seem superfluous, since there exist already several powerful approximation methods. Our method, however, can be used apart from any approximation technique and gives at least some information you can rely on. Although approximation techniques might do a good job on your data, you can never be sure about that. The method outlined in this paper ensures that the quantities of interest, the means and correlations, are restricted to a certain region. We have seen that , generally speaking, the results are useful for weight sizes where an ordinary mean field approximation performs well. This makes the method applicable to a large class of problems. Moreover, since many architectures are not fully connected, one can take advantage of that structure. At least for the upper bound it is shown already that this can improve computation speed and tightness. This would partially cancel the unwanted scaling with the network size. Finally, we would like to give some directions for further research. First of all, an extension to sigmoid belief networks can easily be done, since both a lower and an upper bound are already described. The upper bound, however, is only applicable to two layer networks. A more general upper bound can probably be found. Secondly one can obtain even better bounds (especially for larger weights) if the general constraint (17) is taken into account. This might even be extended to similar constraints, where three or more neurons are involved. Acknowledgelllents This research is supported by the Technology Foundation STW, applied science devision of NWO and the technology programme of the Ministry of Economic Affairs. References [1] C. Peterson and J. Anderson. A mean field theory learning algorithm for neural networks. Complex systems, 1:995- 1019, 1987. [2] S.K. Saul, T.S. Jaakkola, and M.l. Jordan. Mean field theory for sigmoid belief networks. Journal of Artificial Intelligence Research, 4:61- 76, 1996. [3] Martijn A.R. Leisink and Hilbert J. Kappen. A tighter bound for graphical models. In Todd K. Leen, Thomas G. Dietterich, and Volker Tresp, editors, Advances in Neural Information Processing Systems 13, pages 266- 272. MIT Press, 2001. [4] Martijn A.R. Leisink and Hilbert J. Kappen. A tighter bound for graphical models. Neural Computation, 13(9), 2001. To appear. [5] T. Jaakkola and M.l. Jordan. Recursive algorithms for approximating probabilities in graphical models. MIT Compo Cogn. Science Technical Report 9604, 1996. [6] Tommi S. Jaakkola and Michael 1. Jordan. Computing upper and lower bounds on likelihoods in intractable networks. In Proceedings of the Twelfth Annual Conference on Uncertainty in Artificial Intelligence (UAI- 96), pages 340- 348, San Francisco, CA, 1996. Morgan Kaufmann Publishers. [7] D. Ackley, G. Hinton, and T. Sejnowski. A learning algorithm for Boltzmann machines. Cognitive Science, 9:147-169, 1985. [8] D. Sherrington and S. Kirkpatrick. Solvable model of a spin-glass. Physical Review Letters, 35(26):1793-1796, 121975. [9] T. Plefka. Convergence condition of the TAP equation for the infinite-ranged ising spin glass model. J.Phys.A: Math. Gen., 15:1971-1978, 1981.
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Group Redundancy Measures Reveal Redundancy Reduction in the Auditory Pathway Gal Chechik Amir Globerson Naftali Tishby School of Computer Science and Engineering and The Interdisciplinary Center for Neural Computation Hebrew University of Jerusalem, Israel ggal@cs.huji.ac.il Michael J. Anderson Eric D. Young Department of Biomedical Engineering Johns Hopkins University, Baltimore, MD, USA Israel N elken Department of Physiology, Hadassah Medical School and The Interdisciplinary Center for Neural Computation Hebrew University of Jerusalem, Israel Abstract The way groups of auditory neurons interact to code acoustic information is investigated using an information theoretic approach. We develop measures of redundancy among groups of neurons, and apply them to the study of collaborative coding efficiency in two processing stations in the auditory pathway: the inferior colliculus (IC) and the primary auditory cortex (AI). Under two schemes for the coding of the acoustic content, acoustic segments coding and stimulus identity coding, we show differences both in information content and group redundancies between IC and AI neurons. These results provide for the first time a direct evidence for redundancy reduction along the ascending auditory pathway, as has been hypothesized for theoretical considerations [Barlow 1959,2001]. The redundancy effects under the single-spikes coding scheme are significant only for groups larger than ten cells, and cannot be revealed with the redundancy measures that use only pairs of cells. The results suggest that the auditory system transforms low level representations that contain redundancies due to the statistical structure of natural stimuli, into a representation in which cortical neurons extract rare and independent component of complex acoustic signals, that are useful for auditory scene analysis. 1 Introduction How do groups of sensory neurons interact to code information and how do these interactions change along the ascending sensory pathways? According to the a common view, sensory systems are composed of a series of processing stations, representing more and more complex aspects of sensory inputs. The changes in representations of stimuli along the sensory pathway reflect the information processing performed by the system. Several computational principles that govern these changes were suggested, such as information maximization and redundancy reduction [2, 3, 11]. In order to investigate such changes in practice, it is necessary to develop methods to quantify information content and redundancies among groups of neurons, and trace these measures along the sensory pathway. Interactions and high order correlations between neurons were mostly investigated within single brain areas on the level of pairs of cells (but also for larger groups of cells [9]) showing both synergistic and redundant interactions [8, 10, 21, 6, 7, 13]. The current study develops information theoretic redundancy measures for larger groups of neurons, focusing on the case of stimulus-conditioned independence. We then compare these measures in electro-physiological recordings from two auditory stations: the auditory mid-brain and the primary auditory cortex. 2 Redundancy measures for groups of neurons To investigate high order correlations and interactions within groups of neurons we start by defining information measures for groups of cells and then develop information redundancy measures for such groups. The properties of these measures are then further discussed for the specific case of stimulus-conditioned independence. Formally, the level of independence of two variables X and Y is commonly quantified by their mutual information (MI) [17,5]. This well known quantity, now widely used in analysis of neural data, is defined by J(X; Y) = DKL[P(X, Y)IIP(X)P(Y)] = ~p(x, y)log (:~~~~~)) (1) and measures how close the joint distribution P(X, Y) is to the factorization by the marginal distributions P(X)P(Y) (DKL is the Kullback Leiber divergence [5]). For larger groups of cells, an important generalized measure quantifies the information that several variables provide about each other. This multi information measure [18] is defined by (2) Similar to the mutual information case, the multi information measures how close the joint distribution is to the factorization by the marginals. It thus vanishes when variables are independent and is otherwise positive. We now turn to develop measures for group redundancies. Consider first the simple case of a pair of neurons (Xl, X 2 ) conveying information about the stimulus S. In this case, the redundancy-synergy index ([4, 7]) is defined by (3) Intuitively, RSpairs measures the amount of information on the stimulus S gained by observing the joint distribution of both Xl and X 2 , as compared with observing the two cells independently. In the extreme case where Xl = X 2 , the two cells are completely redundant and provide the same information about the stimulus, yielding RSpairs = I(Xl' X 2 ; S) - I(Xl ; S) - I(X2 ; S) = -I(Xl; S), which is always non-positive. On the other hand, positive RSpairs values testify for synergistic interaction between Xl and X 2 ([8, 7, 4]). For larger groups of neurons, several different measures of redundancy-synergy may be considered, that encompass different levels of interactions. For example, one can quantify the residual information obtained from a group of N neurons compared to all its N - 1 subgroups. As with inclusion-exclusion calculations this measure takes the form of a telescopic sum: RSNIN-l = I(XN; S) - L{XN-l} I(XN-\ S) + ... + (_l)N-l L{Xd I(Xi ; S), where {Xk} are all the subgroups of size k out of the N available neurons. Unfortunately, this measure involves 2N information terms, making its calculation infeasible even for moderate N values 1. A different RS measure quantifies the information embodied in the joint distribution of N neurons compared to that provided by N single independent neurons, and is defined by N RSNll = I(Xl ' ... , XN; S) - 2..: I(Xi ; S) (4) i=l Interestingly, this synergy-redundancy measure may be rewritten as the difference between two multi-information terms N I(Xl ' ... , XN; S) - 2..: I(Xi ; S) = (5) i = l N H(Xl' ... ,XN) - H(Xl' ... , XNIS) - 2..: H(Xi) - H(XiIS) = i=l I(Xl ; ... ; XNIS) - I(Xl ; ... ;XN) where H(X) = - L xP(x)log(P(x)) is the entropy of X 2 . We conclude that the index RSNll can be separated into two terms: one that is always non-negative, and measures the coding synergy, and the second which is always non-positive and quantifies the redundancy. These two terms correspond to two types of interactions between neurons: The first type are within-stimulus correlations (sometimes termed noise correlations) that emerge from functional connections between neurons and contribute to synergy. The second type are between stimulus correlations (or across stimulus correlations) that reflect the fact that the cells have similar responses per stimulus, and contribute to redundancy. Being interested in the latter type of correlations, we limit the discussion to the redundancy term -I(Xl; ... ; XN)' Formulating RSNll as in equation 5 proves highly useful when neural activities are independent given the stimulus P(XIS) = II~l P(XiIS). In this case, the first (synergy) term vanishes, thus limiting neural interactions to the redundant lOur results below suggest that some redundancy effects become significant only for groups larger than 10-15 cells. 2When comparing redundancy in different processing stations, one must consider the effects of the baseline information conveyed by single neurons. We thus use the normalized redundancy (compare with [15] p.315 and [4]) defined by !iSNll = RSNldI(Xl; ... ; X N; S) regime. More importantly, under the independence assumption we only have to estimate the marginal distributions P(XiIS = s) for each stimulus s instead of the full distribution P(XIS = s). It thus allows to estimate an exponentially smaller number of parameters, which in our case of small sample sizes, provides more accurate information estimates. This approximation makes it possible to investigate redundancy among considerably larger groups of neurons than the 2-3 neuron groups considered previously in the literature. How reasonable is the conditional-independence approximation ? It is a good approximation whenever neuronal activity is mostly determined by the presented stimulus and to a lesser extent by interactions with nearby neurons. A possible example is the high input regime of cortical neurons receiving thousands of inputs, where a single input has only a limited influence on the activity of the target cell. The experimental evidence in this regard is however mixed (see e.g.[9]). One should note however, that stimulus-conditioned independence is implicitly assumed in analysis of non-simultaneously recorded data. To summarize, the stimulus-conditioned independence assumption limits interactions to the redundant regime, but allows to compare the extent of redundancy among large groups of cells in different brain areas. 3 Experimental Methods To investigate redundancy in the auditory pathway, we analyze extracellular recordings from two brain areas of gas-anesthetized cats: 16 cells from the Inferior Colliculus (Ie) - the third processing station of the ascending auditory pathway - and 19 cells from the Primary Auditory Cortex (AI) - the fifth station. Neural activity was recorded non-simultaneously from a total of 6 different animals responding to a set of complex natural and modified stimuli. Because cortical auditory neurons respond differently to simple and complex stimuli [12, 1], we refrain from using artificial over-simplified acoustic stimuli but instead use a set of stimuli based on bird vocalizations which contains complex 'real-life' acoustic features. A representative example is shown in figure 1. Q) "0 . .e "1i E '" 20 40 60 80 time (milliseconds) 100 7 6 20 40 60 80 100 time (milliseconds) Figure 1: A representative stimulus containing a short bird vocalization recorded in a natural environment. The set of stimuli consisted of similar natural and modified recordings. A. Signal in time domain B. Signal in frequency domain. 4 Experimental Results In practice, in order to estimate the information conveyed by neural activity from limited data, one must assume a decoding procedure, such as focusing on a simple statistic of the spike trains that encompasses some of its informative properties. In this paper we consider two extreme cases: coding short acoustic segments with single spikes and coding the stimulus identity with spike counts in a long window. In addition, we estimated information and redundancy obtained with two other statistics. First, the latency of the first spike after stimulus onset, and secondly, a statistic which generalizes the counts statistics for a general renewal process [19]. These calculations yielded higher information content on average, but similar redundancies as presented below. Their detailed results will be reported elsewhere. 1.2 ~0 . 8 .$ :0 iO.6 0.4 Auditory Cortex (AI) o~========== 5 10 15 no of cells 0.15 ~ 0.1 '" c u C :::l U ~ 0.05 <ii c o t5 jg 0 Inlerior Colliculus (IC) -0.05 L--".--'--~---~--~--o 5 10 15 20 no of cells Figure 2: A. Information about stimulus frames as a function of number of cells. Information calculation was repeated for several subgroups of each size, and with several random seed initializations. The dark curve depicts the expected information provided by independent neurons (this expected curve is corrected for saturation effects [16] and is thus sub linear). The curved line depicts average information from joint distribution of sets of neurons Mean[J(Xl' ... Xk; S)]. All information estimations were corrected for small-samples bias by shuffling methods [14] . B. Fractional redundancy (difference of the mutual information from the expected baseline information divided by the baseline) as a function of number of neurons. 4.1 Coding acoustics with single spikes The current section focuses on the relation between single spikes and short windows of the acoustic stimuli shortly preceding them (which we denote as frames). As the set of possible frames is very large and no frame actually repeats itself, we must first pre-process the stimuli to reduce frames dimensionality. To this end, we first transformed the stimuli into the frequency domain (roughly approximating the cochlear transformation) and then extracted overlapping windows of 50 millisecond length, with 1 millisecond spacing. This set was clustered into 32 representatives, using a metric that groups together acoustic segments with the same spectro-temporal energy structure. This representation allowed us to estimate the joint distribution (under the stimulus-conditioned independence assumption) of cells' activity and stimuli, for groups of cells of different sizes. Figure 2A shows the mutual information between spikes and stimulus frames as a function of the number of cells for both AI and Ie neurons. Ie neurons convey high information but largely deviate from the information expected for independent neurons. On the other hand, AI neurons provide an order of magnitude less information than Ie cells but their information sums almost linearly, as expected from independent neurons. The difference between an information curve and its linear baseline measures the redundancy RSNII of equation 5. Figure 2B presents the normalized redundancy as a function of number of cells, showing that Ie cells are significantly more redundant than AI cells. Q) u c 0.6rr==--------,--------; D Primary Auditory Cortex A 1 _ Inferior Colliculus IC 0.5 ~'-"'=-c.:..::..::.:..:.:::.'--===::...:..::-----" 504 u u o 00.3 :§"' ~02 o 0. 0.1 o -0.5 -04 -0.3 -0.2 -0.1 pairwise redundancy -I(X;Y)/I(X;Y;S) o 0.6rr==--------,--------; D Primary Auditory Cortex A 1 _ Inferior Colliculus IC 0.5 ~'-"'=--"-"-':.:..:.:::.'--===::...:..::'---------" Q) u c 504 u u o 00.3 :§"' ~0.2 o 0. 0.1 -8.8 -0.6 -04 -0.2 0 triplets fractional redundancy -I(X;Y;Z)/I(X;Y;Z;S) Figure 3: Distribution of pairs (A.) and triplets (B.) normalized redundancies. AI cells (light bars) are significantly more independent than Ie cells (dark bars). Spike counts were collected over a window that maximizes mean single cells MI. Number of bins in counts-histogram was optimized separately for every cell. Information estimations were corrected for small-samples bias by shuffling methods [14]. 4.2 Coding stimuli by spike counts We now turn to investigate a second coding paradigm, and calculate the information conveyed by AI and Ie spike counts about the identity of the presented stimulus. To this end, we calculate a histogram of spike counts and estimate the counts' distribution as obtained from repeated presentations of the stimuli. The distribution of fractional redundancy in pairs of AI and Ie neurons is presented in figure 3A, and that of triplets in figure 3B 3 . As in the case of coding with single spikes, single AI cells convey on average less information about the stimulus. However, they are also more independent, thus making it possible to gain more information from groups of neurons. Ie neurons on the other hand, provide more information when considered separately but are more redundant. As in the case of coding acoustics with single spikes, single Ie cells provide more information than AI cells (data not shown) but this time AI cells convey half the information that Ie cells provide, while they convey ten times less information than Ie cells about acoustics. This suggests that AI cells poorly code the physical characteristics of the sound but convey information about its global properties. To illustrate the high information provided by both sets, we trained a neural network classifier that predicts the identity of the presented stimulus according to spike counts of a limited set of neurons. Figure 4 shows that both sets of neurons achieve considerable prediction accuracy, but Ie neurons obtain average accuracy of more than 90 percent already with five cells, while the average prediction accuracy using cortical neurons rises continuously 4. 3Unlike the binary case of single spikes, the limited amount of data prevents a robust estimation of information from spike counts for more than triplets of cells. 4The probability of accurate prediction is exponentially related to the input-output mutual information, via the relation Pcorrect = exp( -missing nats) yielding Mlnats = In(no. of stimuli) + In(Pcorrect). Classification thus provides lower bounds on information content. Figure 4. Prediction accuracy of stimulus identity as a function of number of Ie (upper curve) and AI (lower curve) cells used by the classifier. Error bars denote standard deviation across several subgroups of the same size. For each subgroup, a one-hidden layer neural network was trained separately for each stimulus using some stimulus presentations as a training set and the rest for testing. Performance reported is for the testing set. 5 Discussion >-" 0.95 [IS 0.9 :0 " al <:0.85 o "u ~ 0.8 Q. 0.75 I I I I I I • I I I I I I I I I I I I I I I I jI" I 0.7 '-------=---~--~---~--~ 5 10 15 20 number of cells We have developed information theoretic measures of redundancy among groups of neurons and applied them to investigate the collaborative coding efficiency in the auditory modality. Under two different coding paradigms, we show differences in both information content and group redundancies between Ie and cortical auditory neurons. Single Ie neurons carry more information about the presented stimulus, but are also more redundant. On the other hand, auditory cortical neurons carry less information but are more independent, thus allowing information to be summed almost linearly when considering groups of few tens of neurons. The results provide for the first time direct evidence for redundancy reduction along the ascending auditory pathway, as has been hypothesized by Barlow [2, 3]. The redundancy effects under the single-spikes coding paradigm are significant only for groups larger than ten cells, and cannot be revealed with the standard redundancy measures that use only pairs of cells. Our results suggest that transformations leading to redundancy reduction are not limited to low level sensory processing (aimed to reduce redundancy in input statistics) but are applied even at cortical sensory stations. We suggest that an essential experimental prerequisite to reveal these effects is the use of complex acoustic stimuli whose processing occurs at high level processing stations. The above findings are in agreement with the view that along the ascending sensory pathways, the number of neurons increase, their firing rates decrease, and neurons become tuned to more complex and independent features. Together, these suggest that the neural representation is mapped into a representation with higher effective dimensionality. Interestingly, recent advances in kernel-methods learning [20] have shown that nonlinear mapping into higher dimension and over-complete representations may be useful for learning of complex classifications. It is therefore possible that such mappings provide easier readout and more efficient learning in the brain. Acknowledgements This work supported in part by a Human Frontier Science Project (HFSP) grant RG 0133/1998 and by a grant from the Israeli Ministry of Science. References [1] O. Bar-Yosef and I. Nelken. Responses of neurons in cat primary auditory cortex to bird chirps: Effects of temporal and spectral context. J. Neuroscience, in press, 2001. [2] H.B. Barlow. Sensory mechanisms, the reduction of redundancy, and intelligence. In Mechanisation of thought processes, pages 535- 539. Her Majesty's stationary office, London, 1959. [3] H.B. Barlow. Redundancy reduction revisited. Network: Computation in neural systems, 12:241-253, 200l. [4] N. Brenner, S.P. Strong, R . Koberle, R. de Ruyter van Steveninck, and W. Bialek. Synergy in a neural code. Neural Computation, 13(7):1531, 2000. [5] T.M. Cover and J.A. Thomas. The elements of information theory. Plenum Press, New York, 1991. [6] Y. Dan, J.M. Alonso, W.M. Usrey, and R.C. Reid. Coding of visual information by precisely correlated spikes in the lateral geniculate nucleus. Nature Neuroscience, 1(6):501- 507, 1998. [7] I. Gat and N. Tishby. Synergy and redundancy among brain cells of behaving monkeys. In M.S. Kearns, S.A. Solla, and D.A.Cohn, editors, Advances in Neural Information Proceedings systems, volume 11, Cambridge, MA, 1999. MIT Press. [8] T.J. Gawne and B.J. Richmond. How independent are the messages carried by adjacent inferior temporal cortical neurons? J. Neurosci., 13(7):2758- 2771, 1993. [9] P.M. Gochin, M. Colombo, G. A. Dorfman, G.L. Gerstein, and C.G. Gross. Neural ensemble coding in inferior temporal cortex. J. Neurophysiol., 71:2325- 2337, 1994. [10] M. Meister. Multineural codes in retinal signaling. Proc. Natl. Acad. Sci., 93:609- 614, 1996. [11] J .P. Nadal, N. Brunei, and N. Parga. Nonlinear feedforward networks with stochastic outputs: infomax implies redundancy reduction. Network: Computation in neural systems, 9:207- 217, 1998. [12] I. Nelken, Y. Rotman, and O. Bar-Yosef. Specialization of the auditory system for the analysis of natural sounds. In J. Brugge and P.F. Poon, editors, Central Auditory Processing and Neural Modeling. Plenum, New York, 1997. [13] S. Nirenberg, S.M. Carcieri, A.L. Jacobs, and P.E. Latham. Retinal ganglion cells act largely as independent encoders. Nature, 411:698- 701, 200l. [14] LM. Optican, T.J. Gawne, B.J. Richmond, and P.J . Joseph. Unbiased measures of transmitted information and channel capacity from multivariate neuronal data. Bioi. Cyber, 65:305- 310, 1991. [15] E. T. Rolls and A. Treves. Neural Networks and Brain Function. Oxford Univ. Press, 1998. [16] I. Samengo. Independent neurons representing a fintie set of stimuli: dependence of the mutual information on the number of units sampled. Network: Comput. Neural Syst., 12:21- 31, 200l. [17] C.E. Shanon. A mathematical theory of communication. The Bell systems technical journal, 27:379- 423,623- 656, 1948. [18] M. Studenty and J. Vejnarova. The multiinformation function as a tool for measuring stochastic dependence. In M.I. Jordan, editor, Learning in Graphical Models, pages 261-297. Dordrecht: Kluwer, 1998. [19] C. van Vreeswijk. Information trasmission with renewal neurons. In J.M. Bower, editor, Computational Neuroscience: Trends in Research. Elsevier Press, 200l. [20] V.N. Vapnik. The nature of statistical learning theory. Springer-Verlag, Berlin, 1995. [21] DK. Warland, P. Reinagel, and M. Meister. Decoding visual information from a population of retinal ganglion cells. J. Neurophysiol., 78:2336- 2350, 1997.
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Global Coordination of Local Linear Models Sam Roweis , Lawrence K. Saul , and Geoffrey E. Hinton Department of Computer Science, University of Toronto Department of Computer and Information Science, University of Pennsylvania Abstract High dimensional data that lies on or near a low dimensional manifold can be described by a collection of local linear models. Such a description, however, does not provide a global parameterization of the manifold—arguably an important goal of unsupervised learning. In this paper, we show how to learn a collection of local linear models that solves this more difficult problem. Our local linear models are represented by a mixture of factor analyzers, and the “global coordination” of these models is achieved by adding a regularizing term to the standard maximum likelihood objective function. The regularizer breaks a degeneracy in the mixture model’s parameter space, favoring models whose internal coordinate systems are aligned in a consistent way. As a result, the internal coordinates change smoothly and continuously as one traverses a connected path on the manifold—even when the path crosses the domains of many different local models. The regularizer takes the form of a Kullback-Leibler divergence and illustrates an unexpected application of variational methods: not to perform approximate inference in intractable probabilistic models, but to learn more useful internal representations in tractable ones. 1 Manifold Learning Consider an ensemble of images, each of which contains a face against a neutral background. Each image can be represented by a point in the high dimensional vector space of pixel intensities. This representation, however, does not exploit the strong correlations between pixels of the same image, nor does it support many useful operations for reasoning about faces. If, for example, we select two images with faces in widely different locations and then average their pixel intensities, we do not obtain an image of a face at their average location. Images of faces lie on or near a low-dimensional, curved manifold, and we can represent them more usefully by the coordinates on this manifold than by pixel intensities. Using these “intrinsic coordinates”, the average of two faces is another face with the average of their locations, poses and expressions. To analyze and manipulate faces, it is helpful to imagine a “magic black box” with levers or dials corresponding to the intrinsic coordinates on this manifold. Given a setting of the levers and dials, the box generates an image of a face. Given an image of a face, the box deduces the appropriate setting of the levers and dials. In this paper, we describe a fairly general way to construct such a box automatically from an ensemble of high-dimensional vectors. We assume only that there exists an underlying manifold of low dimensionality and that the relationship between the raw data and the manifold coordinates is locally linear and smoothly varying. Thus our method applies not only to images of faces, but also to many other forms of highly distributed perceptual and scientific data (e.g., spectrograms of speech, robotic sensors, gene expression arrays, document collections). 2 Local Linear Models The global structure of perceptual manifolds (such as images of faces) tends to be highly nonlinear. Fortunately, despite their complicated global structure, we can usually characterize these manifolds as locally linear. Thus, to a good approximation, they can be represented by collections of simpler models, each of which describes a locally linear neighborhood[3, 6, 8]. For unsupervised learning tasks, a probabilistic model that nicely captures this intuition is a mixture of factor analyzers (MFA)[5]. The model is used to describe high dimensional data that lies on or near a lower dimensional manifold. MFAs parameterize a joint distribution over observed and hidden variables:
(1) where the observed variable, ! , represents the high dimensional data; the discrete hidden variables, " $# % &')()(*(+-, , indexes different neighborhoods on the manifold; and the continuous hidden variables, . / 0 !1 , represent low dimensional local coordinates. The model assumes that data is sampled from different neighborhoods on the manifold with prior probabilities 243 , and that within each neighborhood, the data’s local coordinates are normally distributed1 as: 256&87.:9 1;<
=?>A@ BDC % & *E F ( (2) Finally, the model assumes that the data’s high and low dimensional coordinates are related by linear processes parameterized by centers G , loading matrices H and noise levels I :
J &7 I 8 9K ;< =?>A@LBDC % & M C G C H : N E I 9
K M C G C H : N F ( (3) The marginal data distribution, O , is obtained by summing/integrating out the model’s discrete and continuous latent variables. The result is a mixture of Gaussian distributions with parameterized covariance matrices of the form: 2QP 3R - &872 H H E "S I ) 9
K ;< =)>T@ B C % &LM C G N E H H E S I 9
K M C G N:FU( (4) The learning problem for MFAs is to estimate the centers G , transformations H , and noise levels I of these linear processes, as well as the prior probabilities 3 of sampling data from different parts of the manifold. Parameter estimation in MFAs can be handled by an Expectation-Maximization (EM) algorithm[5] that attempts to maximize the logprobability, VXWY Z , averaged over training examples. Note that the parameter space of this model exhibits an invariance: taking H -[ H :\/ , where \ are ]_^`] orthogonal matrices ( \ \ E ba ), does not change the marginal distribution, 2 . The transformations H [ H \ correspond to arbitrary rotations and reflections of the local coordinates in each linear model. The objective function for the EM algorithm is unchanged by these transformations. Thus, maximum likelihood estimation in MFAs does not favor any particular alignment; instead, it produces models whose internal representations change unpredictably as one traverses connected paths on the manifold. Can we encourage models whose local coordinate systems are aligned in a consistent way? 3 Global Coordination Suppose the data lie near a smooth manifold with a locally flat (developable) structure. Then there exist a single set of “global coordinates” c which parametrize the manifold 1Although in principle each neighborhood could have a different prior on its local coordinates, without loss of generality we have made the standard assumption that d/egfih*j k?l is the same for all settings of k and absorbed the shape of each local Gaussian model into the matrices m2h . s,z g x global hidden variables coordinates data Figure 1: Graphical model for globally coordinated MFAs. Although global coordinates are unobserved, they affect the learning through a regularization term. After learning, inferences about the global variables are made by computing posterior distributions, d/e j .l . Likewise, data can easily be generated by sampling from the conditional distribution, d/eZj l . All these operations are particularly tractable due to the conditional independencies of the model. everywhere. Furthermore, to a good approximation, these global coordinates can be related to the local coordinates of different neighborhoods (in their region of validity) by linear2 transformations: c 6
S ( (5) What does it mean to say that the coordinates c ? provide a global parameterization of the manifold? Intuitively, if a data point belongs to overlapping neighborhoods, then the global coordinates computed from their local coordinate systems, given by eq. (5), should agree. We can formalize this “global coordination” of different local models by treating the coordinates c as unobserved variables and incorporating them into the probabilistic model: c
$ A c C C ) (6) (Here we posit a deterministic relationship between local and global coordinates, although it is possible to add noise to this mapping as well.) The globally coordinated MFA is represented by the graphical model in Fig. 1. We can appeal to its conditional independencies to make other useful inferences. In particular: c R: ] R R c
' (7) c ' P 6 ' c R: ?( (8) Now, if two or more mixture components—say, K and < —explain a data point with non-negligible probability, then the posterior distributions for the global coordinates of this data point, as induced by eq. (8), should be nearly identical: that is, c R: K c < . To enforce this criterion of agreement, we need to penalize models whose posterior distributions c ' given by eq. (8) are multimodal, since multiple modes only arise when different mixture components give rise to inconsistent global coordinates. While directly penalizing multimodality of c ' is difficult, a penalty which encourages consistency can be easily incorporated into the learning algorithm. We introduce a family of unimodal distributions over both c and , and encourage the true posteriors, c T ' , to be close to some member, c : ' , of this family. Developing this idea further, we introduce a new objective function for unsupervised learning in MFAs. The new objective function incorporates a regularizer to encourage the global consistency of local models: P VXWY ' C P c c T
i VXWY c :
i c T (9) The first term in this objective function computes the log-probability of the data. The second term computes a sum of Kullback-Leibler (KL) divergences; these are designed to 2Without loss of generality, the matrices h can be taken to be symmetric and positive-definite, by exploiting the polar factorization and absorbing reflection and rotation into the local coordinate systems. (In practice, though, it may be easier to optimize the objective function without constraining the matrices to be of this form.) In the experiments reported below, we have further restricted them to be diagonal. Together, then, the coordination matrices h and vectors
h account for an axis-aligned scaling and uniform translation between the global and local coordinate systems. penalize MFAs whose posterior distributions over global coordinates are not unimodal. The twin goals of density estimation and manifold learning in MFAs are pursued by attempting to balance these terms in the objective function. The factor controls the tradeoff between density modeling and global coordination: as [ only strict invariances (which do not affect likelihood) are exploited in order to achieve submodel agreement. In what follows we have set % arbitrarily; further optimization is possible. The most convenient way to parameterize the family of unimodal distributions is a factorized form involving a Gaussian density and a multinomial: c T '2 c ' T ' c ' c ' 6T '2 (10) Note that the distribution c T ' in eq. (10) factorizes over and c , implying that— according to this family of models—the global coordinate c is independent of the mixture component given the data point . Also, c is Gaussian, and thus unimodal. These are exactly the constraints we wish to impose on the posterior c :T i . At each iteration of learning, the means c , covariance matrices , and mixture weights are determined separately for each data point, so as to maximize the objective function in eq. (9): this amounts to computing the unimodal distributions, c :T , best matched to the true posterior distributions, c : ' . 4 Learning Algorithm Latent variable models are traditionally estimated by maximum likelihood or Bayesian methods whose objective functions do not reward the interpretability of their internal representations. Note how the goal of developing more useful internal representations has changed the learning problem in a fundamental way. Now we have additional “coordination” parameters–the offsets and weights –that must also be learned from examples. We also have auxiliary parameters for each data point—the means c , covariance matrices , and mixture weights —that determine the target distributions, c T ' . All these parameters, as well as the MFA model parameters #3 H G I , , must be chosen to “stitch together” the local coordinates systems in a smooth way and to learn internal representations easily coordinated by the local-to-global mapping in eq. (6). Optimization of the objective function in eq. (9) is reminiscent of so-called “variational” methods for approximate learning[7]. In these methods, an approximation to an exact (but intractable) posterior distribution is fitted by minimizing a KL divergence between the two distributions. The auxiliary parameters of the approximating distribution are known as variational parameters. Our objective function illustrates an unexpected application of such variational methods: not to perform approximate inference in intractable probabilistic models, but to learn more useful internal representations in tractable ones. We introduce the unimodal and factorized distributions c T ' to regularize the multimodal distributions c T . Penalizing the KL divergence between these distributions lifts a degeneracy in the model’s parameter space and favors local linear models that can be globally aligned. 4.1 Computing and optimizing the objective function Evaluating the objective function in eq. (9) requires a sum and integral over the latent variables of the model. These operations are simplified by rewriting the objective function as: P c c : ' M C V W Y c T
' S VXWY R c N ( (11) The factored form of the distributions c :T ' makes it straightforward to perform the required sums and integrals. The final result is a simple form in terms of entropies
and energies associated with the th data point: P O
C (12)
% & VXWY C V W Y S ] & V W Y &7. (13) % & c E c S % & E I 9
K C c E E H E I 9
K S % & M
N S % & VXWY I 8 S V W Y C VXWY 3R S S ] & VXWY 6&87.? (14) where we have introduced simplifying notation for the vector differences C G and c c C and the local precision matrices 9K a S H E I 9
K H 9
K . Iteratively maximizing the objective function by coordinate ascent now leads to a learning algorithm of the same general style as EM. 4.2 E-step Maximizing the objective function, eq. (9), with respect to the regularizing parameters # c R , (and subject to the constraint % ) leads to the fixed point equations: P 9K c P 9 ! #" %$ 9 &" $ ( (15) where " S E H E I 9
K . These equations can be solved by iteration with initialization 43 . Notice that and only need to be computed once before iterating the fixed point equations. The objective function is completely invariant to translation and rescaling of c and (since , and c appear only in the form c C E ). To remove this degeneracy, after solving the equations above we further constrain the global coordinates to have mean zero and unit variance in each direction. These constraints are enforced without changing the value of the objective function by simply translating the offsets ' and rescaling the diagonal matrices . 4.3 M-step The M-step consists of maximizing the objective function, eq. (9), with respect to the generative model parameters. Let us denote the updated parameter estimates by #)( 3R +* ,* G ( H ( I ( *, . Letting , the M-step updates for the first three of these are: ( 3 . * !/ P %$ $ * , 9
K P c * G 9K P Z R( (16) The remaining updates, to be performed in the order shown, are given in terms of updated difference vectors *
C * G 0* c c C * , the correlations 1 .* +* c E , and the variances 2 M S * c * c E N . ( H 1 2 9
K (17) 3 ( I 465 9
K P B 3 * C ( H 9
K * c 4 < 5 S 3 ( H 9K E ( H E 465?F (18) ( 9
K a S H E I 9
K H 9
K87 E * S H E ( I 9
K 1 +9 2 9K (19) At the optimum, the coordination weights satisfy an algebraic Riccati equation which can be solved by iterating the update shown above. (Such equations can also be solved by much more sophisticated methods well known in the engineering community. Most approaches involve inverting the previous value of which may be expensive for full matrices but is fast in our diagonal implementation.) Figure 2: Global coordination of local linear models. (left) A model trained using maximum likelihood, with the arrows indicating the direction of increase for each factor analyzer’s local coordinate system. (right) A coordinated model; arrows indicate the direction in the data space corresponding to increasing the global coordinate as inferred by the algorithm. The ellipses show the one standard deviation contour of the density of each analyzer. 5 Experiments We have tested our model on simple synthetic manifolds whose structure is known as well as on collections of images of handwritten digits and faces. Figure 2 illustrates the basic concept of coordination, as achieved by our learning rule. In the coordinated model, the global coordinate always points in the same direction along the data manifold, as defined by the composition of the transformations H and . In the model trained with maximum likelihood, the density is well captured but each local latent variable has a random orientation along the manifold. We also applied the algorithm to collections of images of handwritten digits and of faces. The representation of was an unprocessed vector of raw 8-bit grayscale pixel intensities for each image (of dimensionality 256 for the % ^ % digits and 560 for the & ^ & faces.) The MFAs had 64 local models and the global coordinates were two dimensional. After training, the coordinated MFAs had learned a smooth, continuous mapping from the plane to images of digits or of faces. This allows us both to infer a two-dimensional location given any image by computing c 2 and to generate new images from any point in the plane by computing c . (Precisely what we wanted from the magic box.) In general, both of these conditional distributions have the form of a mixture of Gaussians. Figure 3 shows the inferred global coordinates c (i.e. the means of the unimodal distributions c ) of the training points after the last iteration of training as well as examples of new images from the generative model, created by evaluating the mean of c along straight line paths in the global coordinate space. In the case of digits, it seems as though our models have captured tilt/shape and identity and represented them as the two axes of the c space; in the case of the faces the axes seem to capture pose and expression. (For the faces, the final c space was rotated by hand to align interpretable directions with the coordinate axes.) As with all EM algorithms, the coordinated MFA learning procedure is susceptible to local optima. Crucial to the success of our experiments is a good initialization, which was provided by the Locally Linear Embedding algorithm[9]. We clamped c equal to the embedding coordinate provided by LLE and to a small value and trained until convergence (typically 30-100 iterations). Then we proceeded with training using the full EM equations to update c , again until convergence (usually 5-10 more iterations). Note, however, that LLE and other embedding algorithms such as Isomap[10] are themselves unsupervised, so the overall procedure, including this initial phase, is still unsupervised. 6 Discussion Mixture models provide a simple way to approximate the density of high dimensional data that lies on or near a low dimensional manifold. However, their hidden representations do not make explicit the relationship between dissimilar data vectors. In this paper, we have shown how to learn global coordinates that can act as an encapsulating interface, so that other parts of a learning system do not need to interact with the individual components of a mixture. This should improve generalization as well as facilitate the propagation and exchange of information when these models are incorporated into a larger (perhaps Figure 3: Automatically constructed two dimensional global parameterizations of manifolds of digits and faces. Each plot shows the global coordinate space discovered by the unsupervised algorithm; points indicate the inferred means for each training item at the end of learning. The image stacks on the borders are not from the training set but are generated from the model itself and represent the mean of the predictive distribution d/eZj l at the corresponding open circles (sampled along the straight lines in the global space). The models provide both a two degree-of-freedom generator for complex images via d/eZj l as well as a pose/slant recognition system via d/e j .l . For the handwritten digits, the training set consisted of 1100 examples of the digit “2” (shown as crosses above) mixed with 1100 examples of “3”s (shown as triangles). The digits are from the NIST dataset, digitized at 16x16 pixels. For the faces, we used 2000 images of a single person with various poses and expressions taken from consecutive frames of a video digitized at 20x20 pixels. Brendan Frey kindly provided the face data. hierarchical) architecture for probabilistic reasoning. Two variants of our purely unsupervised proposal are possible. The first is to use an embedding algorithm (such as LLE or Isomap) not only as an initialization step but to provide clamped values for the global coordinates. While this supervised approach may work in practice, unsupervised coordination makes clear the objective function that is being optiFigure 4: A situation in which an un-coordinated mixture model–trained to do density estimation–cannot be “postcoordinated”. Noise has caused one of the local density models to orient orthogonal to the manifold. In globally coordinated learning, there is an additional pressure to align with neighbouring models which would force the local model to lie in the correct subspace. mized, which unifies the goals of manifold learning and density estimation. Another variant is to train an unsupervised mixture model (such as a MFA) using a traditional maximum likelihood objective function and then to “post-coordinate” its parameters by applying local reflections/rotations and translations to create global coordinates. As illustrated in figure 4, however, this two-step procedure can go awry because of noise in the original training set. When both density estimation and coordination are optimized simultaneously there is extra pressure for local experts to fit the global structure of the manifold. Our work can be viewed as a synthesis of two long lines of research in unsupervised learning. In the first are efforts at learning the global structure of nonlinear manifolds [1, 4, 9, 10]; in the second are efforts at developing probabilistic graphical models for reasoning under uncertainty[5, 6, 7]. Our work proposes to model the global coordinates on manifolds as latent variables, thus attempting to combine the representational advantages of both frameworks. It differs from embedding by providing a fully probabilistic model valid away from the training set, and from work in generative topographic mapping[2] by not requiring a uniform discretized gridding of the latent space. Moreover, by extending the usefulness of mixture models,it further develops an architecture that has already proved quite powerful and enormously popular in applications of statistical learning. Acknowledgements We thank Mike Revow for sharing his unpublished work (at the University of Toronto) on coordinating mixtures, and Zoubin Ghahramani, Peter Dayan, Jakob Verbeek and two anonymous reviewers for helpful comments and corrections. References [1] D. Beymer & T. Poggio. Image representations for visual learning. pringerScience 272 (1996). [2] C. Bishop, M. Svensen, and C. Williams. GTM: The generative topographic mapping. Neural Computation 10 (1998). [3] C. Bregler & S. Omohundro. Nonlinear image interpolation using manifold learning. Advances in Neural Information Processing Systems 7 (1995). [4] D. DeMers & G.W. Cottrell. Nonlinear dimensionality reduction. Advances in Neural Information Processing Systems 5 (1993). [5] Ghahramani, Z. and Hinton, G. The EM algorithm for mixtures of factor analyzers. University of Toronto Technical Report CRG-TR-96-1 (1996). [6] Hinton, G., Dayan, P., and Revow, M. Modeling the manifolds of images of handwritten digits. IEEE Transactions on Neural Networks 8 (1997). [7] M. Jordan, Z. Ghahramani, T. Jaakkola, and L. Saul. An introduction to variational methods for graphical models. Machine Learning 37(2) (1999). [8] N. Kambhatla and T. K. Leen. Dimension reduction by local principal component analysis. Neural Computation 9 (1997). [9] S. T. Roweis & L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science 290 (2000). [10] J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science 290 (2000).
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Correlation Codes in Neuronal Populations Maoz Shamir and Haim Sompolinsky Racah Institute of Physics and Center for Neural Computation, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Abstract Population codes often rely on the tuning of the mean responses to the stimulus parameters. However, this information can be greatly suppressed by long range correlations. Here we study the efficiency of coding information in the second order statistics of the population responses. We show that the Fisher Information of this system grows linearly with the size of the system. We propose a bilinear readout model for extracting information from correlation codes, and evaluate its performance in discrimination and estimation tasks. It is shown that the main source of information in this system is the stimulus dependence of the variances of the single neuron responses. 1 Introduction Experiments in the last years have shown that in many cortical areas, the fluctuations in the responses of neurons to external stimuli are significantly correlated [1, 2, 3, 4], raising important questions regarding the computational implications of neuronal correlations. Recent theoretical studies have addressed the issue of how neuronal correlations affect the efficiency of population coding [4, 5, 6]. It is often assumed that the information about stimuli is coded mainly in the mean neuronal responses, e.g., in the tuning of the mean firing rates, and that by averaging the tuned responses across large populations, an accurate estimate can be obtained despite the significant noise in the single neuron responses. Indeed, for uncorrelated neurons the Fisher Information of the population is extensive [7]; namely, it increases linearly with the number of neurons in the population. Furthermore, it has been shown that this extensive information can be extracted by relatively simple linear readout mechanisms [7, 8]. However, it was recently shown [6] that positive correlations which vary smoothly with space may drastically suppress the information in the mean responses. In particular, the Fisher Information of the system saturates to a finite value as the system size grows. This raises questions about the computational utility of neuronal population codes. Neuronal population responses can represent information in the higher order statistics of the responses [3], not only in their means. In this work, we study the accuracy of coding information in the second order statistics. We call such schemes correlation codes. Specifically, we assume that the neuronal responses obey multivariate Gaussian statistics governed by a stimulus-dependent correlation matrix. We ask whether the Fisher Information of such a system is extensive even in the presence of strong correlations in the neuronal noise. Secondly, we inquire how information in the second order statistics can be efficiently extracted. 2 Fisher Information of a Correlation Code Our model consists of a system of neurons that code a 2D angle , . Their stochastic response is given by a vector of activities
where
is the activity of the
-th neuron in the presence of a stimulus , and is distributed according to a multivariate Gaussian distribution "!$#&% ')( (+* ,.-0/ 21 (3* 04 (1) Here
is the mean activity of the
-th neuron and its dependence on is usually referred to as the tuning curve of the neuron; / is the correlation matrix; and is a normalization constant. Here we shall limit ourselves to the case of multiplicative modulation of the correlations. Specifically we use 5
76 8 9
09 6 :<; 5
=6 (2) ; 5
76 ; 5 ?>
( > 6 @
76)A ( @
=6 !B#%DC ( >
( > 6 E F (3) 9
8 9 ?>
( :G !$#&% CGHBI J K>
( LNM F (4) where and E are the correlation strength and correlation length respectively; L defines the tuning width of the correlations; and >
denotes the angle at which the variance of the
-th neuron, 9 M
, is maximal. An example is shown in Fig. 1. It is important to note that the variance adds a contribution to 5
76 which is larger than the contribution of the smooth part of the correlations. For reasons that will become clear below, we write, 5
76 :G 5PO
76 : A 5PQ
.@
=6 (5) where 5 O
=6 denotes the smooth part of the correlation matrix and 5 Q
the discontinuous diagonal part, which in the example of Eqs. (2)-(4) is 5PQ
G ( ,9 M
(6) A useful measure of the accuracy of a population code is the Fisher Information (FI). In the case of uncorrelated populations it is well known that FI increases linearly with system size [7], indicating that the accuracy of the population coding improves as the system size is increased. Furthermore, it has been shown that relatively simple, linear schemes can provide reasonable readout models for extracting the information in uncorrelated populations [8]. In the case of a correlated multivariate Gaussian distribution, FI is given as R R SUT.V2W A R:XZY0[0[ , where R SUT.V2W * .\ / : 1 * ,\ (7) R:XZY0[0[ N] ^ / 1 / 0.\_ M (8) where * \ and / \ denote derivatives of * and / with respect to , respectively. The form of these terms reveals that in general the correlations play two roles. First they control the efficiency of the information encoded in the mean activities * : (note the dependence of R SUT.V2W on 5 ). Secondly, / provides an additional source of information about the stimulus ( R&XZY0[0[ ). When the correlations are independent of the stimulus,
` 9
a bdc e , it was shown [6] that positive correlations, gf , with long correlation length, E ih , −180 −120 −60 0 60 120 180 ψ [deg] C(φ,ψ) φ=−120o φ=−60o φ=0o φ=60o φ=120o Figure 1: The stimulus-dependent correlation matrix, Eqs. (2)-(4), depicted as a function of two angles, 5 K> , where > >
( and >
( . Here, i , E and L . cause the saturation of FI to a finite limit at large . This implies that in the presence of such correlations, population averaging cannot overcome the noise even in large networks. This analysis however, [6], did not take into account stimulus-dependent correlations, which is the topic of the present work. Analyzing the dependence of RNXZY,[ [ , Eq. (8), we find it useful to write R XZY0[0[ R Q A R O (9) where R Q
C 5 Q
, \ 5 Q
F M (10) is FI of an uncorrelated population with stimulus-dependent variance which equals 5 Q
, and scales linearly with ; R O R:XZY0[0[ ( R Q . Evaluating these terms for the multiplicative model, Eq. (2), we find that R O is positive, so that R R Q . Furthermore, numerical evaluation of this term shows that R O saturates at large to a small finite value, so that for large R:XZY0[ [ R Q M
> C 9 \ ?> 9 ?> F M (11) as shown in Fig. 2. We thus conclude that R XZY0[0[ increases linearly with and is equal, for large , to the FI of variance coding namely to R of an independent population in which information is encoded in their activity variances. Since in our system the information is encoded in the second order statistics of the population responses, it is obvious that linear readouts are inadequate. This raises the question of whether there are relatively simple nonlinear readout models for such systems. In the next sections we will study bilinear readouts and show that they are useful models for extracting information from correlation codes. 3 A Bilinear Readout for Discrimination Tasks In a two-interval discrimination task the system is given two sets of neuronal activities M generated by two proximal stimuli and A @ and must infer which stimulus generated which activity. The Maximum-Likelihood (ML) discrimination yields the 0 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 N Jcorr [deg−2] 0 200 400 600 800 1000 0 0.5 1x 10 −3 N Js [deg−2] Figure 2: (a) Fisher Information, R XZY0[0[ , of the stimulus-dependent correlations, Eqs. (2)(4), as a function of the number of neurons in the system. In (b) we show the difference between the full FI and the contribution of the diagonal term, R O - as defined by Eq. (9). Here , E and L . Note the different scales in (a) and (b). probability of error given by \ , where d 1 M ` 1
M and the discriminability \ equals \ @ R (12) It has been previously shown that in the case of uncorrelated populations with mean coding, the optimal linear readouts achieves the Maximum-Likelihood discrimination performance in large N [7]. In order to isolate the properties of correlation coding we will assume that no information is coded in the average firing rates of the neurons, and take * hereafter. We suggest a bilinear readout as a simple generalization of the linear readout to correlation codes. In a discrimination task the bilinear readout makes a decision according to the sign of
=6
76
6 ( M
M 6 (13) where a A ( decision refers to A @ . Maximizing the signal-to-noise ratio of this rule, the optimal bilinear discriminator (OBD) matrix is given by
76 5 1 \
76 (14) Using the optimal weights to evaluate the discrimination error we obtain that in large the performance of the OBD saturates the ML performance, Eq. (12). Thus, since FI of this model increases linearly with the size of the system, the discriminability increases as . Since the correlation matrix / depends on the stimulus, , the OBD matrix, Eq. (14), will also be stimulus dependent. Thus, although the OBD is locally efficient, it cannot be used as such as a global efficient readout. 4 A Bilinear Readout for Angle Estimation 4.1 Optimal bilinear readout for estimation To study the global performance of bilinear readouts we investigate bilinear readouts which minimize the square error of estimating the angle averaged over the whole range of . For convenience we use complex notation for the encoded angle, and write as the estimator of `
. Let
76
=6
K 26 (15) where
76 are stimulus independent complex weights. We define the optimal bilinear estimator (OBE) as the set of weights that minimizes on average the quadratic estimation error of an unbiased estimator. This error is given by G @ M (
(16) where is the Lagrange multiplier of the constraint . In general, it is impossible to find a perfectly unbiased estimator for a continuously varied stimulus, using a finite number of weights. However, in the case of angle estimation, we can employ the underlying rotational symmetry to generate such an estimator. For this we use the symmetry of the correlation matrix, Eq. (2). In this case one can show that the Lagrange multipliers have the simple form of ) `
, and the OBE weight matrix is in the form of
76 ?>
( > 6 !B#%DC
>
A > 6 F (17) where ?> G ( > and ?> ) ( ?> A d . This form of a readout matrix, Eq. (17), guarantees that the estimator will be unbiased. Using these symmetry properties, K> can be written in the following form (for even ) ?> @
A M 1 W W HBI J ^ b ( > _ ( > " (18) Figure 3 (a) presents an example of the function K> . These numerical results (Fig. 3 (a)) also suggest that the function K> is mainly determined by a few harmonics plus a delta peak at > . Below we will use this fact to study simpler forms of bilinear readout. Further analysis of the OBE performance in the large limit yields the following asymptotic result @ : M 1 Q M 5 Q
?> `
M Q M 5 Q
?> 0 M ( c > , (19) Figure 3 (b) shows the numerical calculation of the OBE error (open circles) as a function of . The dashed line is the asymptotic behavior, given by Eq. (19). The dotted line is the Creamer-Rao bound. From the graph one can see that the estimation efficiency of this readout grows linearly with the size of the system, , but is lower than the bound. 4.2 Truncated bilinear readout Motivated by the simple structure of the optimal readout matrix observed in Fig. 3 (a), we studied a bilinear readout of the form of Eqs. (17) and (18) with K> which has a delta function peak at the origin plus a few harmonics. Restricting the number of harmonics to relatively small integers, we evaluated numerically the optimal values of the coefficients W for large systems. Surprisingly we found that for small and large , these coefficients approach a value which is independent of the specifics of the model and equals W ( B , yielding a bilinear weight matrix of the form
76 @
=6 ( W HBI J ^ b ( K>
( > 6 _ !B#%DC
>
A > 6 F (20) Figure 4 shows the numerical results for the squared average error of this readout for several values of ! and 8 . The results of Fig. 4 show that for a given the 0 100 200 300 400 0 0.05 0.1 N ∆θ−2 [deg−2] J (b) −2 0 2 0 φ w(φ) (a) Figure 3: (a) Profile of ?> , Eq. (17), for the OBE with . (b) Numerical evaluation of one over the squared estimation error, for the optimal bilinear readout in the multiplicative modulation model (open circles). The dashed line is the asymptotic behavior, given by Eq. (19). Here @ & M M , for the optimal bilinear readout in the multiplicative modulation model. The dotted line is the FI bound. In these simulations , E and L were used. inverse square error initially increases linearly with but saturates in the limit of large . However, the saturation size O V increases rapidly with . The precise form of O V depends on the specifics of the correlation model. For the exponentially decaying correlations assumed in Eq. (2), we find O V . Figure 4 shows that for this range of , and the deviations of the inverse square error from linearity are small. Thus, in the regime O V , @ : M is given by the asymptotic behavior, Eq. (19), shown by the dashed line. We thus conclude that the OBE (with unlimited ) will generate an inverse square estimation error which increases linearly with with a coefficient given by Eq. (19), and that this value can be achieved for reasonable values of by an approximate bilinear weight matrix, of the form of Eq. (20), with small . The asymptotic result, Eq. (19), is smaller than the optimal value given by the full FI, Eq. (11), see Fig. 4 (dotted line). In fact, it is equal to the error of an independent population with a variance which equals 5 Q
and a quadratic population vector readout of the form
M
`
(21) It is important to note that in the presence of correlations, the quadratic readout of Eq. (21) is very inefficient, yielding a finite error for large as shown in Fig. 4 (line marked ‘quadratic’). 5 Discussion To understand the reason for the simple form of the approximately optimal bilinear weight matrix, Eq. (20), we rewrite Eq. (15) with of Eq. (20) as
?`
(22)
60 @
76 ( W 1 `
W 1 26 (23) 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 N 1/ δθ2 deg−2 quadratic p=1 p=2 p=3 J Figure 4: Inverse square estimation error of the finite approximation for the OBE, Eq. (20). Solid curves from the bottom . The bottom curve is . The dashed line is the asymptotic behavior, given by Eq. (19). The FI bound is shown by the dotted line. For the simulations , E and L were used. Comparing this form with Eq. (21) it can be seen that our readout is in the form of a bilinear population vector in which the lowest Fourier modes of the response vector have been removed. Retaining only the high Fourier modes in the response profile suppresses the cross-correlations between the different components of the residual responses
because the underlying correlations have smooth spatial dependence, whose power is concentrated mostly in the low Fourier modes. On the other hand, the information contained in the variance is not removed because the variance contains a discontinuous spatial component, 5 Q
. In other words, the variance of a correlation profile which has only high Fourier modes can still preserve its slowly varying components. Thus, by projecting out the low Fourier modes of the spatial responses the spatial correlations are suppressed but the information in the response variance is retained. This interpretation of the bilinear readout implies that although all the elements of the correlation matrix depend on the stimulus, only the stimulus dependence of the diagonal elements is important. This important conclusion is borne out by our theoretical results concerning the performance of the system. As Eqs. (11) and (19) show, the asymptotic performance of both the full FI as well as that of the OBE are equivalent to those of an uncorrelated population with a stimulus dependent variance which equals 5 Q
. Although we have presented results here concerning a multiplicative model of correlations, we have studied other models of stimulus dependent correlations. These studies indicate that the above conclusions apply to a broad class of populations in which information is encoded in the second order statistics of the responses. Also, for the sake of clarity we have assumed here that the mean responses are untuned, * . Our studies have shown that adding tuned mean inputs does not modify the picture since the smoothly varying positive correlations greatly suppress the information embedded in the first order statistics. The relatively simple form of the readout Eq. (22) suggests that neuronal hardware may be able to extract efficiently information embedded in local populations of cells whose noisy responses are strongly correlated, provided that the variances of their responses are significantly tuned to the stimulus. This latter condition is not too restrictive, since tuning of variances of neuronal firing rates to stimulus and motor variables is quite common in the nervous system. Acknowledgments This work was partially supported by grants from the Israel-U.S.A. Binational Science Foundation and the Israeli Science Foundation. M.S. is supported by a scholarship from the Clore Foundation. References [1] E. Fetz, K. Yoyoma and W. Smith, Cerebral Cortex (Plenum Press, New York, 1991). [2] D. Lee, N.L. Port, W. Kruse and A.P. Georgopoulos, J. Neurosci. , 1161 (1998). [3] E.M. Maynard, N.G. Hatsopoulos, C.L. Ojakangas, B.D. Acuna, J.N. Sanes, R.A. Normann, and J.P. Donoghue, J. Neurosci. 19, 8083 (1999). [4] E. Zohary, M.N. Shadlen and W.T. Newsome, Nature , 140 (1994). [5] L.F. Abbott and P. Dayan, Neural Computation , 91 (1999). [6] H. Sompolinsky, H. Yoon, K. Kang and M. Shamir, Phys. Rev. E, , 051904 (2001); H. Yoon and H. Sompolinsky, Advances in Neural Information Processing Systems 11 (pp. 167). Kearns M.J, Solla S.A and Cohn D.A, Eds., (Cambridge, MA: MIT Press, 1999). [7] S. Seung and H. Sompolinsky, Proc. Natl. Acad. Sci. USA , 10794 (1993). [8] E. Salinas and L.F. Abbott, J. Comp. Neurosci. , 89 (1994).
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Grammatical Bigrams Mark A. Paskin Computer Science Division University of California, Berkeley Berkeley, CA 94720 paskin@cs.berkeley.edu Abstract Unsupervised learning algorithms have been derived for several statistical models of English grammar, but their computational complexity makes applying them to large data sets intractable. This paper presents a probabilistic model of English grammar that is much simpler than conventional models, but which admits an efficient EM training algorithm. The model is based upon grammatical bigrams, i.e., syntactic relationships between pairs of words. We present the results of experiments that quantify the representational adequacy of the grammatical bigram model, its ability to generalize from labelled data, and its ability to induce syntactic structure from large amounts of raw text. 1 Introduction One of the most significant challenges in learning grammars from raw text is keeping the computational complexity manageable. For example, the EM algorithm for the unsupervised training of Probabilistic Context-Free Grammars- known as the Inside-Outside algorithm- has been found in practice to be "computationally intractable for realistic problems" [1]. Unsupervised learning algorithms have been designed for other grammar models (e.g., [2, 3]). However, to the best of our knowledge, no large-scale experiments have been carried out to test the efficacy of these algorithms; the most likely reason is that their computational complexity, like that of the Inside-Outside algorithm, is impractical. One way to improve the complexity of inference and learning in statistical models is to introduce independence assumptions; however, doing so increases the model's bias. It is natural to wonder how a simpler grammar model (that can be trained efficiently from raw text) would compare with conventional models (which make fewer independence assumptions, but which must be trained from labelled data) . Such a model would be a useful tool in domains where partial accuracy is valuable and large amounts of unlabelled data are available (e.g., Information Retrieval, Information Extraction, etc.) . In this paper, we present a probabilistic model of syntax that is based upon grammatical bigrams, i.e., syntactic relationships between pairs of words. We show how this model results from introducing independence assumptions into more conventhe quick brown fox jumps over the lazy dog Figure 1: An example parse; arrows are drawn from head words to their dependents. The root word is jumps; brown is a predependent (adjunct) of fox; dog is a postdependent (complement) of over. tional models; as a result, grammatical bigram models can be trained efficiently from raw text using an O(n3 ) EM algorithm. We present the results of experiments that quantify the representational adequacy of the grammatical bigram model, its ability to generalize from labelled data, and its ability to induce syntactic structure from large amounts of raw text. 2 The Grammatical Bigram Model We first provide a brief introduction to the Dependency Grammar formalism used by the grammatical bigram model; then, we present the probability model and relate it to conventional models; finally, we sketch the EM algorithm for training the model. Details regarding the parsing and learning algorithms can be found in a companion technical report [4]. Dependency Grammar Formalism. 1 The primary unit of syntactic structure in dependency grammars is the dependency relationship, or link- a binary relation between a pair of words in the sentence. In each link, one word is designated the head, and the other is its dependent. (Typically, different types of dependency are distinguished, e.g, subject, complement, adjunct, etc.; in our simple model, no such distinction is made.) Dependents that precede their head are called pre dependents, and dependents that follow their heads are called postdependents. A dependency parse consists of a set of links that, when viewed as a directed graph over word tokens, form an ordered tree. This implies three important properties: 1. Every word except one (the root) is dependent to exactly one head. 2. The links are acyclic; no word is, through a sequence of links, dependent to itself. 3. When drawn as a graph above the sentence, no two dependency relations cross-a property known as projectivity or planarity. The planarity constraint ensures that a head word and its (direct or indirect) dependents form a contiguous subsequence of the sentence; this sequence is the head word's constituent. See Figure 1 for an example dependency parse. In order to formalize our dependency grammar model, we will view sentences as sequences of word tokens drawn from some set of word types. Let V = {tl' t2, ... , t M } be our vocabulary of M word types. A sentence with n words is therefore represented as a sequence S = (Wl, W2 , ... ,wn ), where each word token Wi is a variable that ranges over V. For 1 :S i ,j :S n, we use the notation (i,j) E L to express that Wj is a dependent of Wi in the parse L. IThe Dependency Grammar formalism described here (which is the same used in [5, 6]) is impoverished compared to the sophisticated models used in Linguistics; refer to [7] for a comprehensive treatment of English syntax in a dependency framework. Because it simplifies the structure of our model, we will make the following three assumptions about Sand L (without loss of generality): (1) the first word WI of S is a special symbol ROOT E V; (2) the root of L is WI; and, (3) WI has only one dependent. These assumptions are merely syntactic sugar: they allow us to treat all words in the true sentence (i.e., (W2, ... ,Wn )) as dependent to one word. (The true root of the sentence is the sole child of WI.) Probability Model. A probabilistic dependency grammar is a probability distribution P(S, L) where S = (WI,W2, .. . ,wn ) is a sentence, L is a parse of S, and the words W2, ... ,Wn are random variables ranging over V. Of course, S and L exist in high dimensional spaces; therefore, tractable representations of this distribution make use of independence assumptions. Conventional probabilistic dependency grammar models make use of what may be called the head word hypothesis: that a head word is the sole (or primary) determinant of how its constituent combines with other constituents. The head word hypothesis constitutes an independence assumption; it implies that the distribution can be safely factored into a product over constituents: n P(S,L) = II P((Wj: (i,j) E L) is the dependent sequencelwi is the head) i=1 For example, the probability of a particular sequence can be governed by a fixed set of probabilistic phrase-structure rules, as in [6]; alternatively, the predependent and postdependent subsequences can be modeled separately by Markov chains that are specific to the head word, as in [8]. Consider a much stronger independence assumption: that all the dependents of a head word are independent of one another and their relative order. This is clearly an approximation; in general, there will be strong correlations between the dependents of a head word. More importantly, this assumption prevents the model from representing important argument structure constraints. (For example: many words require dependents, e.g. prepositions; some verbs can have optional objects, whereas others require or forbid them.) However, this assumption relieves the parser of having to maintain internal state for each constituent it constructs, and therefore reduces the computational complexity of parsing and learning. We can express this independence assumption in the following way: first, we forego modeling the length of the sentence, n, since in parsing applications it is always known; then, we expand P(S, Lin) into P(S I L)P(L I n) and choose P(L I n) as uniform; finally, we select P(S I L) II P( Wj is a [pre/post]dependent I Wi is the head) (i,j)EL This distribution factors into a product of terms over syntactically related word pairs; therefore, we call this model the "grammatical bigram" model. The parameters of the model are <"(xy P(predependent is ty I head is tx ) 6. "(~ P(postdependent is ty I head is tx ) We can make the parameterization explicit by introducing the indicator variable wi, whose value is 1 if Wi = tx and a otherwise. Then we can express P(S I L) as P(S I L) (i,j)EL x=1 y=1 j<i (i,j)EL x=1 y=1 i<j Parsing. Parsing a sentence S consists of computing L* L:, argmaxP(L I S,n) = argmaxP(L, Sin) = argmaxP(S I L) L L L Yuret has shown that there are exponentially many parses of a sentence with n words [9], so exhaustive search for L * is intractable. Fortunately, our grammar model falls into the class of "Bilexical Grammars" , for which efficient parsing algorithms have been developed. Our parsing algorithm (described in the tech report [4]) is derived from Eisner's span-based chart-parsing algorithm [5], and can find L* in O(n3 ) time. Learning. Suppose we have a labelled data set where Sk = (Wl,k, W2,k,·· · , Wnk,k) and Lk is a parse over Sk. The maximum likelihood values for our parameters given the training data are where the indicator variable et is equal to 1 if (i,j) E Lk and 0 otherwise. As one would expect, the maximum-likelihood value of ,;; (resp. ,~ ) is simply the fraction of tx's predependents (resp. postdependents) that were ty. In the unsupervised acquisition problem, our data set has no parses; our approach is to treat the Lk as hidden variables and to employ the EM algorithm to learn (locally) optimal values of the parameters ,. As we have shown above, the et are sufficient statistics for our model; the companion tech report [4] gives an adaptation of the Inside-Outside algorithm which computes their conditional expectation in O(n3 ) time. This algorithm effectively examines every possible parse of every sentence in the training set and calculates the expected number of times each pair of words was related syntactically. 3 Evaluation This section presents three experiments that attempt to quantify the representational adequacy and learnability of grammatical bigram models. Corpora. Our experiments make use of two corpora; one is labelled with parses, and the other is not. The labelled corpus was generated automatically from the phrase-structure trees in the Wall Street Journal portion of the Penn Treebank-III [10].2 The resultant corpus, which we call C, consists of 49,207 sentences (1,037,374 word tokens). This corpus is split into two pieces: 90% of the sentences comprise corpus Ctrain (44,286 sentences, 934,659 word tokens), and the remaining 10% comprise Ctest (4,921 sentences, 102,715 word tokens). The unlabelled corpus consists of the 1987- 1992 Wall Street Journal articles in the TREC Text Research Collection Volumes 1 and 2. These articles were segmented on sentence boundaries using the technique of [11], and the sentences were postprocessed to have a format similar to corpus C. The resultant corpus consists of 3,347,516 sentences (66,777,856 word tokens). We will call this corpus U. 2This involved selecting a head word for each constituent, for which the head-word extraction heuristics described in [6] were employed. Additionally, punctuation was removed, all words were down-cased, and all numbers were mapped to a special <#> symbol. The model's vocabulary is the same for all experiments; it consists of the 10,000 most frequent word types in corpus U; this vocabulary covers 94.0% of word instances in corpus U and 93.9% of word instances in corpus L. Words encountered during testing and training that are outside the vocabulary are mapped to the <unk> type. Performance metric. The performance metric we report is the link precision of the grammatical bigram model: the fraction of links hypothesized by the model that are present in the test corpus Ltest. (In a scenario where the model is not required to output a complete parse, e.g., a shallow parsing task, we could similarly define a notion of link recall; but in our current setting, these metrics are identical.) Link precision is measured without regard for link orientation; this amounts to ignoring the model's choice of root, since this choice induces a directionality on all of the edges. Experiments. We report on the results of three experiments: I. Retention. This experiment represents a best-case scenario: the model is trained on corpora Ltrain and Ltest and then tested on Ltest. The model's link precision in this setting is 80.6%. II. Generalization. In this experiment, we measure the model's ability to generalize from labelled data. The model is trained on Ltrain and then tested on Ltest. The model's link precision in this setting is 61.8%. III. Induction. In this experiment, we measure the model's ability to induce grammatical structure from unlabelled data. The model is trained on U and then tested on Ltest . The model's link precision in this setting is 39.7%. Analysis. The results of Experiment I give some measure of the grammatical bigram model's representational adequacy. A model that memorizes every parse would perform perfectly in this setting, but the grammatical bigram model is only able to recover four out of every five links. To see why, we can examine an example parse. Figure 2 shows how the models trained in Experiments I, II, and III parse the same test sentence. In the top parse, syndrome is incorrectly selected as a postdependent of the first on token rather than the second. This error can be attributed directly to the grammatical bigram independence assumption: because argument structure is not modeled, there is no reason to prefer the correct parse, in which both on tokens have a single dependent, over the chosen parse, in which the first has two dependents and the second has none.3 Experiment II measures the generalization ability of the grammatical bigram model; in this setting, the model can recover three out of every five links. To see why the performance drops so drastically, we again turn to an example parse: the middle parse in Figure 2. Because the forces -+ on link was never observed in the training data, served has been made the head of both on tokens; ironically, this corrects the error made in the top parse because the planarity constraint rules out the incorrect link from the first on token to syndrome. Another error in the middle parse is a failure to select several as a predependent of forces; this error also arises because the combination never occurs in the training data. Thus, we can attribute this drop in performance to sparseness in the training data. We can compare the grammatical bigram model's parsing performance with the results reported by Eisner [8]. In that investigation, several different probability models are ascribed to the simple dependency grammar described above and 3 Although the model's parse of acquired immune deficiency syndrome agrees with the labelled corpus, this particular parse reflects a failure of the head-word extraction heuristics; acquired and immune should be predependents of deficiency, and deficiency should be a predependent of syndrome. r 1. 843 '.88' 3.2",-d fir hr-. n~ 14 . 383 I. <root> she has also served on several task forces on acquired immune deficiency syndrome 1. 528 r 9 . 630 1 '. 803 r,m 1.358 1 12 . 527 fir hn fO ~8~\ A , 14 . 264 II. <root> she has also served on several task forces on acquired immune deficiency syndrome 1. 990 0.913 4 . 124 - 1.709 (~ 13 . 585 k 0 .14 9 tI 1--' f' ~ ( III. <root> she has also served on several task forces on acquired immune deficiency syndrome Figure 2: The same test sentence, parsed by the models trained in each of the three experiments. Links are labelled with -log2 IXY I I:~1 IXY, the mutual information of the linked words; dotted edges are default attachments. are compared on a task similar to Experiment 11.4 Eisner reports that the bestperforming dependency grammar model (Model D) achieves a (direction-sensitive) link precision of 90.0%, and the Collins parser [6] achieves a (direction-sensitive) link precision of 92.6%. The superior performance of these models can be attributed to two factors: first, they include sophisticated models of argument structure; and second, they both make use of part-of-speech taggers, and can "back-off" to non-lexical distributions when statistics are not available. Finally, Experiment III shows that when trained on unlabelled data, the grammatical bigram model is able to recover two out of every five links. This performance is rather poor, and is only slightly better than chance; a model that chooses parses uniformly at random achieves 31.3% precision on L\est . To get an intuition for why this performance is so poor, we can examine the last parse, which was induced from unlabelled data. Because Wall Street Journal articles often report corporate news, the frequent co-occurrence of has -+ acquired has led to a parse consistent with the interpretation that the subject she suffers from AIDS, rather than serving on a task force to study it. We also see that a flat parse structure has been selected for acquired immune deficiency syndrome; this is because while this particular noun phrase occurs in the training data, its constituent nouns do not occur independently with any frequency, and so their relative co-occurrence frequencies cannot be assessed. 4 Discussion Future work. As one would expect, our experiments indicate that the parsing performance of the grammatical bigram model is not as good as that of state-ofthe-art parsers; however, its performance in Experiment II suggests that it may be useful in domains where partial accuracy is valuable and large amounts of unlabelled data are available. However, to realize that potential, the model must be improved so that its performance in Experiment III is closer to that of Experiment II. To that end, we can see two obvious avenues of improvement. The first involves increasing the model's capacity for generalization and preventing overfitting. The 4The labelled corpus used in that investigation is also based upon a transformed version of Treebank-III, but the head-word extraction heuristics were slightly different, and sentences with conjunctions were completely eliminated. However, the setup is sufficiently similar that we think the comparison we draw is informative. model presented in this paper is sensitive only to pairwise relationships to words; however, it could make good use of the fact that words can have similar syntactic behavior. We are currently investigating whether word clustering techniques can improve performance in supervised and unsupervised learning. Another way to improve the model is to directly address the primary source of parsing error: the lack of argument structure modeling. We are also investigating approximation algorithms that reintroduce argument structure constraints without making the computational complexity unmanageable. Related work. A recent proposal by Yuret presents a "lexical attraction" model with similarities to the grammatical bigram model [9]; however, unlike the present proposal, that model is trained using a heuristic algorithm. The grammatical bigram model also bears resemblance to several proposals to extend finite-state methods to model long-distance dependencies (e.g., [12, 13]), although these models are not based upon an underlying theory of syntax. References [1] K. Lari and S. J. Young. The estimation of stochastic context-free grammars using the Inside-Outside algorithm. Computer Speech and Language, 4:35- 56, 1990. [2] John Lafferty, Daniel Sleator, and Davy Temperley. Grammatical trigrams: A probabilistic model of link grammar. In Proceedings of the AAAI Conference on Probabilistic Approaches to Natural Language, October 1992. [3] Yves Schabes. Stochastic lexicalized tree-adjoining grammars. In Proceedings of the Fourteenth International Conference on Computational Linguistics, pages 426-432, Nantes, France, 1992. [4] Mark A. Paskin. Cubic-time parsing and learning algorithms for grammatical bigram models. Technical Report CSD-01-1148, University of California, Berkeley, 2001. [5] Jason Eisner. Bilexical grammars and their cubic-time parsing algorithms. In Harry Bunt and Anton Nijholt, editors, Advances in Probabilistic and Other Parsing Technologies, chapter 1. Kluwer Academic Publishers, October 2000. [6] Michael Collins. Head-driven Statistical Models for Natural Language Parsing. PhD thesis, University of Pennsylvania, Philadelphia, Pennsylvania, 1999. [7] Richard A. Hudson. English Word Grammar. B. Blackwell, Oxford, UK, 1990. [8] Jason M. Eisner. An empirical comparison of probability models for dependency grammars. Technical Report ICRS-96-11, CIS Department, University of Pennsylvania, 220 S. 33rd St. Philadelphia, PA 19104- 6389, 1996. [9] Deniz Yuret. Discovery of Linguistic Relations Using Lexical Attraction. PhD thesis, Massachusetts Institute of Technology, May 1998. [10] M. Marcus, B. Santorini, and M. Marcinkiewicz. Building a large annotated corpus of english: The penn treebank. Computational Linguistics, 19:313- 330, 1993. [11] Jeffrey C. Reynar and Adwait Ratnaparkhi. A maximum entropy approach to identifying sentence boundaries. In Proceedings of the Fifth Conference on Applied Natural Language Processing, Washington, D.C., March 31 - April 3 1997. [12] S. Della Pietra, V. Della Pietra, J. Gillett, J. Lafferty, H. Printz, and L. Ures. Inference and estimation of a long-range trigram model. In Proceedings of the Second International Colloquium on Grammatical Inference and Applications, number 862 in Lecture Notes in Artificial Intelligence, pages 78- 92. Springer-Verlag, 1994. [13] Ronald Rosenfeld. Adaptive Statistical Language Modeling: A Maximum Entropy Approach. PhD thesis, Carnegie Mellon University, 1994.
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The Steering Approach for Multi-Criteria Reinforcement Learning Shie Mannor and Nahum Shimkin Department of Electrical Engineering Technion, Haifa 32000, Israel {shie,shimkin}@{tx,ee}.technion.ac.il Abstract We consider the problem of learning to attain multiple goals in a dynamic environment, which is initially unknown. In addition, the environment may contain arbitrarily varying elements related to actions of other agents or to non-stationary moves of Nature. This problem is modelled as a stochastic (Markov) game between the learning agent and an arbitrary player, with a vector-valued reward function. The objective of the learning agent is to have its long-term average reward vector belong to a given target set. We devise an algorithm for achieving this task, which is based on the theory of approachability for stochastic games. This algorithm combines, in an appropriate way, a finite set of standard, scalar-reward learning algorithms. Sufficient conditions are given for the convergence of the learning algorithm to a general target set. The specialization of these results to the single-controller Markov decision problem are discussed as well. 1 Introduction This paper considers an on-line learning problem for Markov decision processes with vector-valued rewards. Each entry of the reward vector represents a scalar reward (or cost) function which is of interest. Focusing on the long-term average reward, we assume that the desired performance is specified through a given target set, to which the average reward vector should eventually belong. Accordingly, the specified goal of the decision maker is to ensure that the average reward vector will converge to the target set. Following terminology from game theory, we refer to such convergence of the reward vector as approaching the target set. A distinctive feature of our problem formulation is the possible incorporation of arbitrarily varying elements of the environment, which may account for the influence of other agents or non-stationary moves of Nature. These are collectively modelled as a second agent, whose actions may affect both the state transition and the obtained rewards. This agent is free to choose its actions according to any control policy, and no prior assumptions are made regarding its policy. This problem formulation is derived from the so-called theory of approachability that was introduced in [3] in the context of repeated matrix games with vector payoffs. Using a geometric viewpoint, it characterizes the sets in the reward space that a player can guarantee for himself for any possible policy of the other player, and provides appropriate policies for approaching these sets. Approachability theory has been extended to stochastic (Markov) games in [14], and the relevant results are briefly reviewed in Section 2. In this paper we add the learning aspect, and consider the problem of learning such approaching policies on-line, using Reinforcement Learning (RL) or similar algorithms. Approaching policies are generally required to be non-stationary. Their construction relies on a geometric viewpoint, whereby the average reward vector is “steered” in the direction of the target set by the use of direction-dependent (and possibly stationary) control policies. To motivate the steering viewpoint, consider the following one dimensional example of an automatic temperature controlling agent. The measured property is the temperature which should be in some prescribed range [T, T], the agent may activate a cooler or a heater at will. An obvious algorithm that achieves the prescribed temperature range is – when the average temperature is higher than T choose a “policy” that reduces it, namely activate the cooler; and if the average temperature is lower than T use the heater. See Figure 1(a) for an illustration. Note that this algorithm is robust and requires little knowledge about the characteristics of the processes, as would be required by a procedure that tunes the heater or cooler for continuous operation. A learning algorithm needs only determine which element to use at each of the two extreme regions. a b T T Temperature Heating policy Cooling policy Temperature Humidity Target Figure 1: (a) The single dimensional temperature example. If the temperature is higher than T the control is to cool, and if the temperature is lower than T the control is to heat. (b) The two dimensional temperature-humidity example. The learning directions are denoted by arrows, note that an infinite number of directions are to be considered. Consider next a more complex multi-objective version of this controlling agent. The controller’s objective is as before to have the temperature in a certain range. One can add other parameters such as the average humidity, frequency of switching between policies, average energy consumption and so on. This problem is naturally characterized as a multi-objective problem, in which the objective of the controller is to have the average reward in some target set. (Note that in this example, the temperature itself is apparently the object of interest rather than its long-term average. However, we can reformulate the temperature requirement as an average reward objective by measuring the fraction of times that the temperature is outside the target range, and require this fraction to be zero. For the purpose of illustration we shall proceed here with the original formulation). For example, suppose that the controller is also interested in the humidity. For the controlled environment of, say, a greenhouse, the allowed level of humidity depends on the average temperature. An illustrative target set is shown in Figure 1(b). A steering policy for the controller is not as simple anymore. In place of the two directions (left/right) of the one-dimensional case, we now face a continuum of possible directions, each associated with a possibly different steering policy. For the purpose of the proposed learning algorithm we shall require to consider only a finite number of steering policies. We will show that this can always be done, with negligible effect on the attainable performance. The analytical basis for this work relies on three elements: stochastic game models, which capture the Markovian system dynamics while allowing arbitrary variation in some elements of the environment; the theory of approachability for vector-valued dynamic games, which provides the basis for the steering approach; and RL algorithms for (scalar) average reward problems. For the sake of brevity, we do not detail the mathematical models and proofs and concentrate on concepts. Reinforcement Learning (RL) has emerged in the last decade as a unifying discipline for learning and adaptive control. Comprehensive overviews may be found in [2, 7]. RL for average reward Markov Decision Processes (MDPs) was suggested in [13, 10] and later analyzed in [1]. Several methods exist for average reward RL, including Q-learning [1] the E3 algorithm [8], actor-critic schemes [2] and more. The paper is organized as follows: In Section 2 we describe the stochastic game setup, recall approachability theory, and mention a key theorem that allows to consider only a finite number of directions for approaching a set. Section 3 describes the proposed multi-criteria RL algorithm and outlines its convergence proof. We also briefly discuss learning in multi-criteria single controller environments, as this case is a special case of the more general game model. An illustrative example is briefly described in Section 4 and concluding remarks are drawn in Section 5. 2 Multi-Criteria Stochastic Games In this section we present the multi-criteria stochastic game model. We recall some known results from approachability theory for stochastic games with vector-valued reward, and state a key theorem which decomposes the problem of approaching a target set into a finite number of scalar control problems. We consider a two-person average reward stochastic game model, with a vector-valued reward function. We refer to the players as P1 (the learning agent) and P2 (the arbitrary adversary). The game is defined by: the state space S; the sets of actions for P1 and P2, respectively, in each state s, A and B; the state transition kernel, P = (P(s′|s, a, b)); a vector-valued reward function m : S × A × B →IRk. The reward itself is allowed to be random, in which case it is assumed to have a bounded second moment. At each time epoch n ≥0, both players observe the current state sn, and then P1 and P2 simultaneously choose actions an and bn, respectively. As a result P1 receives the reward vector mn = m(sn, an, bn) and the next state is determined according to the transition probability P(·|sn, an, bn). More generally, we allow the actual reward mn to be random, in which case m(sn, an, bn) denotes its mean and a bounded second moment is assumed. We further assume that both players observe the previous rewards and actions (however, in some of the learning algorithms below, the assumption that P1 observes P2’s action may be relaxed). A policy π ∈Π for P1 is a mapping which assigns to each possible observed history a mixed action in ∆(A), namely a probability vector over P1’s action set A. A policy σ ∈Σ for P2 is defined similarly. A policy of either player is called stationary if the mixed action it prescribes depends only on the current state sn. Let ˆmn denote the average reward by time n: ˆmn △= 1 n Pn−1 t=0 mt. The following recurrence assumption will be imposed. Let state s∗denote a specific reference state to which a return is guaranteed. We define the hitting time of state s∗as: τ △= min{n > 0 : sn = s∗}. Assumption 1 (Recurrence) There exist a state s∗∈S and a finite constant N such that Es πσ(τ 2) < N for all π ∈Π, σ ∈Σ and s ∈S, where Es πσ is the expectation operator when starting from state s0 = s and using policies π and σ for P1 and P2, respectively. If the game is finite then this assumption is satisfied if state s∗is accessible from all other states under any pair of stationary deterministic policies [14]. We note that the recurrence assumption may be relaxed in a similar manner to [11]. Let u be a unit vector in the reward space IRk. We often consider the projected game in direction u as the zero-sum stochastic game with same dynamic as above, and scalar rewards rn := mn · u. Here “·” stands for the standard inner product in IRk. Denote this game by Γs(u), where s is the initial state. The scalar stochastic game Γs(u), has a value, denoted vΓs(u), if vΓs(u) = sup π inf σ lim inf n→∞Es πσ( ˆmn · u) = inf σ sup π lim sup n→∞Es πσ( ˆmn · u) . For finite games, the value exists [12]. Furthermore, under Assumption 1 the value is independent of the initial state and can be achieved in stationary policies [6]. We henceforth simply write vΓ(u) for this value. We next consider the task of approaching a given target set in the reward space, and introduce approaching policies for the case where the game parameters are fully known to P1. Let T ⊂IRk denote the target set. In the following, d is the Euclidean distance in IRk, and P s π,σ is the probability measure induced by the policies π and σ, with initial state s. Definition 2.1 The set T ⊂IRk is approachable (from initial state s) if there exists a Tapproaching policy π∗of P1 such that d( ˆmn, T) →0 P s π∗,σ-a.s., for every σ ∈Σ at a uniform rate over Σ. The policy π∗in that definition will be called an approaching policy for P1. A set is approachable if it is approachable from all states. Noting that approaching a set and its closure are the same, we shall henceforth suppose that the set T is closed. We recall the basic results from [14] regarding approachability for known stochastic games, which generalize Blackwell’s conditions for repeated matrix games. Let φ(π, σ) △= Es∗ π,σ(Pτ−1 t=0 mt) Es∗ π,σ(τ) (1) denote the average per-cycle reward vector, which is the expected total reward over the cycle that starts and ends in the reference state, divided by the expected duration of that cycle. For any x ̸∈T, denote by Cx a closest point in T to x, and let ux be the unit vector in the direction of Cx −x, which points from x to the goal set T, see Figure 2 for an illustration. Theorem 2.1 [14] Let Assumption 1 hold. Assume that for every point x ̸∈T there exists a policy π(x) such that: (φ(π(x), σ) −Cx) · ux ≥0 , ∀σ ∈Σ . (2) Then T is approachable by P1. An approaching policy is: If sn = s∗and ˆmn ̸∈T, play π( ˆmn) until the next visit to state s∗; otherwise, play arbitrarily. Figure 2: An illustration of approachability. π(x) brings P1 to the other side of the hyperplane perpendicular to the segment between Cx and x. Geometrically, the condition in (2) means that P1 can ensure, irrespectively of P2’s policy, that the average per-cycle reward will be on the other side (relative to x) of the hyperplane which is perpendicular to the line segment that points from x to Cx. We shall refer to the direction ux as the steering direction from point x, and to the policy π(x) as the steering policy from x. The approaching policy uses the following rule: between successive visits to the reference state, a fixed (possibly stationary) policy is used. When in the reference state, the current average reward vector ˆmn is inspected. If this vector is not in T, then the steering policy that satisfies (2) with x = ˆmn is selected for the next cycle. Consequently, the average reward is “steered” towards T, and eventually converges to it. Recalling the definition of the projected game in direction u and its value vΓ(u), the condition in (2) may be equivalently stated as vΓ(ux) ≥Cx · ux. Furthermore, the policy π(x) can always be chosen as the stationary policy which is optimal for P1 in the game Γ(ux). In particular, the steering policy π(x) needs to depend only on the corresponding steering direction ux. It can be shown that for convex target sets, the condition of the last theorem turns out to be both sufficient and necessary. Standard approachability results, as outlined above, require to consider an infinite number of steering directions whenever the reward in non-scalar. The corresponding set of steering policies may turn out to be infinite as well. For the purpose of our learning scheme, we shall require an approaching policy which relies on a finite set of steering directions and policies. The following results show that this can indeed be done, possibly requiring to slightly expand the target set. In the following, let M be an upper bound on the magnitude of the expected one-stage reward vector, so that ||m(s, a, b)|| < M for all (s, a, b) (|| · || denote the Euclidean norm). We say that a set of vectors (u1, . . . , uJ) is an ϵ-cover of the unit ball if for every vector in the unit ball u there exists a vector ui such that ||ui −u|| ≤ϵ . Theorem 2.2 Let Assumption 1 hold and suppose that the target set T ⊂IRk satisfies condition (2). Fix ϵ > 0. Let {u1, . . . , uJ} be an ϵ/M cover of the unit ball. Suppose that πi is an optimal strategy in the scalar game Γ(ui) (1 ≤i ≤J). Then the following policy approaches T ϵ, the ϵexpansion of T: If sn = s∗and ˆmn ̸∈T ϵ, then choose j so that u ˆmn is closest to uj (in Euclidean norm) and play πj until the next visit to state s∗; otherwise, play arbitrarily. Proof: (Outline) The basic observation is that if two directions, u and ui are close, then vΓ(u) and vΓ(ui) are close. Consequently, by playing a strategy which is optimal in Γ(ui) results in a play which is almost optimal in Γ(u). Finally we can apply Blackwell’s Theorem (2.1) for the expansion of T, by noticing that a “good enough” strategy is played in every direction. Remark: It follows immediately from the last theorem that the set T itself (rather than its ϵexpansion) is approachable with a finite number of steering directions if T −ϵ, the ϵ shrinkage of T, satisfies (2). Equivalently, T is required to satisfy (2) with the 0 on the right-hand-side replaced by ϵ > 0. 3 The Multi-Criteria Reinforcement Learning Algorithm In this section we introduce and prove the convergence of the MCRL (Multi-Criteria Reinforcement Learning) algorithm. We consider the controlled Markov model of Section 2, but here we assume that P1, the learning agent, does not know the model parameters, namely the state transition probabilities and reward functions. A policy of P1 that does not rely on knowledge of these parameters will be referred to as a learning policy. P1’s task is to approach a given target set T, namely to ensure convergence of the average reward vector to this set irrespective of P2’s actions. The proposed learning algorithm relies on the construction of the previous section of approaching policies with a finite number of steering directions. The main idea is to apply a (scalar) learning algorithm for each of the projected games Γ(uj) corresponding to these directions. Recall that each such game is a standard zero-sum stochastic game with average reward. The required learning algorithm for game Γ(u) should secure an average reward that is not less than the value vΓ(u) of that game. Consider a zero-sum stochastic game, with reward function r(s, a, b), average reward ˆrn and value v. Assume for simplicity that the initial state is fixed. We say that a learning policy π of P1 is ϵ-optimal in this game if, for any policy σ of P2, the average reward satisfies lim inf n→∞ˆrn ≥v −ϵ Pπσ a.s. , where Pπσ is the probability measure induced by the algorithm π, P2’s policy σ and the game dynamics. Note that P1 may be unable to learn a min-max policy as P2 may play an inferior policy and refrain from playing certain actions, thereby keeping some parts of the game unobserved. Remark: RL for average reward zero-sum stochastic games can be devised in a similar manner to average reward Markov decision processes. For example, a Q-learning based algorithm which combines the ideas of [9] with those of [1] can be devised. An additional assumption that is needed for the analysis is that all actions of both players are used infinitely often. A different type of a scalar algorithm that overcomes this problem is [4]. The algorithm there is similar to the E3 algorithm [8] which is based on explicit exploration-exploitation tradeoffand estimation of the game reward and transition structure. We now describe the MCRL algorithm that nearly approaches any target set T that satisfies (2). The parameters of the algorithm are ϵ and M. ϵ is the approximation level and M is a known bound on the norm of the expected reward per step. The goal of the algorithm is to approach T ϵ, the ϵ expansion of T. There are J learning algorithms that are run in parallel, denoted by π1, . . . πJ. The MCRL is described in Figure 3 and is given here as a meta-algorithm (the scalar RL algorithms πi are not specified). When arriving to s∗, the decision maker checks if the average reward vector is outside the set T ϵ. In that case, he switches to an appropriate policy that is intended to “steer” the average reward vector towards the target set. The steering policy (πj) is chosen according to closest direction (uj) to the actual direction needed according to the problem geometry. Recall that each πj is actually a learning policy with respect to a scalar reward function. In general, when πj is not played, its learning pauses and the process history during that time is ignored. Note however that some “off-policy” algorithms (such as Q-learning) can learn the optimal policy even while playing a different policy. In that case a more efficient version of the MCRL is suggested, in which learning is performed by all learning policies πj continuously and concurrently. 0. Let u1, . . . uJ be an ϵ/2M cover of the unit ball. Initialize J different ϵ/2-optimal scalar algorithms, π1, . . . , πJ. 1. If s0 ̸= s∗play arbitrarily until sn = s∗. 2. (sn = s∗) If ˆmn ∈T ϵ goto step 1. Else let i = arg min1≤i≤J ||ui −u ˆmn||2. 3. While sn ̸= s∗play according to πi, the reward πi receives is ui · mn. 4. When sn = s∗goto step 2. Figure 3: The MCRL algorithm Theorem 3.1 Suppose that Assumption 1 holds and the MCRL algorithm is used with ϵ-optimal scalar learning algorithms. If the target set T satisfies (2), then T ϵ is approached using MCRL. Proof: (Outline) If a direction is played infinitely often, then eventually the learned strategy in this direction is nearly optimal. If a direction is not played infinitely often it has a negligible effect on the long term average reward vector. Since the learning algorithms are nearly optimal, then any policy πj that is played infinitely often, eventually attains a (scalar) average reward of vΓ(uj)−ϵ/2. One can apply Theorem 2.2 for the set T ϵ/2 to verify that the overall policy is an approaching policy for the target set. Note that for convex target sets the algorithm is consistent in the sense that if the set is approachable then the algorithm attains it. Remark: Multi-criteria Markov Decision Process (MDP) models may be regarded as a special case of the stochastic game model that was considered so far, with P2 eliminated from the problem. The MCRL meta-algorithm of the previous section remains the same for MDPs. Its constituent scalar learning algorithms are now learning algorithms for the optimal polices in average-reward MDPs. These are generally simpler than for the game problem. Examples of optimal or ϵ-optimal algorithms are Q-Learning with persistent exploration [2], Actor-critic schemes [2], an appropriate version of the E3 algorithm [8] and others. In the absence of an adversary, the problem of approaching a set becomes much simpler. Moreover, it can be shown that if a set is approachable then it may be approached using a stationary (possibly randomized) policy. Indeed, any point in feasible set of state-action frequencies may be achieved by such a stationary policy [5]. Thus, alternative learning schemes may be applicable to this problem. Another observation is that all steering policies learned and used within the MCRL may now be deterministic stationary policies, which simplifies the implementation of this algorithm. 4 Example Recall the humidity-temperature example from the introduction. Suppose that the system is modelled in such a way that P1 chooses a temperature-humidity curve. Then Nature (modelled as P2) chooses the exact location on the temperature-humidity curve. In Figure 4(a) we show three different temperature-humidity curves, that can be determined by P1 (each defined by a certain strategy of P1 - f0, f1, f2). We implemented MCRL algorithm with nine directions. In each direction a version of Littman’s Q-learning ([9]), adapted for average cost games, was used. A sample path of the average reward generated by the MCRL algorithm is shown in Figure 4(b). The sample path started at ’S’ and finished at ’E’. For this specific run, an even smaller number of directions would have sufficed (up and right). It can be seen that the learning algorithm pushes towards the target set so that the path is mostly on the edge of the target set. Note that in this example a small number of directions was quite enough for approaching the target set. a b 0.5 1 1.5 2 0.5 1 1.5 2 Temperature Humidity f0 f1 f2 Problem dynamics for different strategies 0.8 1 1.2 1.4 1.6 1.8 2 0.8 1 1.2 1.4 1.6 1.8 2 S E Temperature Humidity A sample path of average reward Figure 4: (a) Greenhouse problem dynamics. (b) A sample path from ’S’ to ’E’ 5 Conclusion We have presented a learning algorithm that approaches a prescribed target set in multi-dimensional performance space, provided this set satisfies a certain sufficient condition. Our approach essentially relies on the theory of approachability for stochastic games, which is based on the idea of steering the average reward vector towards the target set. We provided a key result stating that a set can be approached to a given precision using only a finite number of steering policies, which may be learned on-line. An interesting observation regarding the proposed learning algorithm is that the learned optimal polices in each direction are essentially independent of the target set T. Thus, the target set need not be fixed in advance and may be modified on-line without requiring a new learning process. This may be especially useful for constrained MDPs. Of further interest is the question of reduction of the number of steering directions used in the algorithm. In some cases, especially when the requirements embodied by the target set T are not stringent, this number may be quite small compared to the worst-case estimate used above. A possible refinement of the algorithm is to eliminate directions that are not required. The scaling of he algorithm with the dimension of the reward space is exponential. The problem is that as the dimension increases, exponentially many directions are needed to cover the unit ball. While in general this is necessary, it might happen that considerably less directions are needed. Conditions and algorithms that use much less than exponential number of directions are under current study. Acknowledgement This research was supported by the fund for the promotion of research at the Technion. References [1] J. Abounadi, D. Bertsekas, and V. Borkar. Learning algorithms for markov decision processes with average cost. LIDS-P 2434, Lab. for Info. and Decision Systems, MIT, October 1998. [2] A.G. Barto and R.S. Sutton. Reinforcement Learning. MIT Press, 1998. [3] D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific J. Math., 6(1):1–8, 1956. [4] R.I. Brafman and M. Tennenholtz. A near optimal polynomial time algorithm for learning in certain classes of stochastic games. Artificial Intelligence, 121(1-2):31–47, April 2000. [5] C. Derman. Finite state Markovian decision processes. Academic Press, 1970. [6] J. Filar and K. Vrieze. Competitive Markov Decision Processes. Springer Verlag, 1996. [7] L.P. Kaelbling, M. Littman, and A.W. Moore. Reinforcement learning - a survey. Journal of Artificial Intelligence Research, (4):237–285, May 1996. [8] M. Kearns and S. Singh. Near-optimal reinforcement learning in polynomial time. In Proc. of the 15th Int. Conf. on Machine Learning, pages 260–268. Morgan Kaufmann, 1998. [9] M.L. Littman. Markov games as a framework for multi-agent reinforcement learning. In Morgan Kaufman, editor, Eleventh International Conference on Machine Learning, pages 157–163, 1994. [10] S. Mahadevan. Average reward reinforcement learning: Foundations, algorithms, and empirical results. Machine Learning, 22(1):159–196, 1996. [11] S. Mannor and N. Shimkin. The empirical bayes envelope approach to regret minimization in stochastic games. Technical report EE- 1262, Faculty of Electrical Engineering, Technion, Israel, October 2000. [12] J.F. Mertens and A. Neyman. Stochastic games. International Journal of Game Theory, 10(2):53–66, 1981. [13] A. Schwartz. A reinforcement learning method for maximizing undiscounted rewards. In Proceedings of the Tenth International Conference on Machine Learning, pages 298–305. Morgan Kaufmann, 1993. [14] N. Shimkin and A. Shwartz. Guaranteed performance regions in markovian systems with competing decision makers. IEEE Trans. on Automatic Control, 38(1):84–95, January 1993.
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Stabilizing Value Function with the Xin Wang Department of Computer Science Oregon State University Corvallis, OR, 97331 wangxi@cs. orst.edu Thomas G Dietterich Department of Computer Science Oregon State University Corvallis, OR, 97331 tgd@cs. orst. edu Abstract We address the problem of non-convergence of online reinforcement learning algorithms (e.g., Q learning and SARSA(A)) by adopting an incremental-batch approach that separates the exploration process from the function fitting process. Our BFBP (Batch Fit to Best Paths) algorithm alternates between an exploration phase (during which trajectories are generated to try to find fragments of the optimal policy) and a function fitting phase (during which a function approximator is fit to the best known paths from start states to terminal states). An advantage of this approach is that batch value-function fitting is a global process, which allows it to address the tradeoffs in function approximation that cannot be handled by local, online algorithms. This approach was pioneered by Boyan and Moore with their GROWSUPPORT and ROUT algorithms. We show how to improve upon their work by applying a better exploration process and by enriching the function fitting procedure to incorporate Bellman error and advantage error measures into the objective function. The results show improved performance on several benchmark problems. 1 Introduction Function approximation is essential for applying value-function-based reinforcement learning (RL) algorithms to solve large Markov decision problems (MDPs). However, online RL algorithms such as SARSA(A) have been shown experimentally to have difficulty converging when applied with function approximators. Theoretical analysis has not been able to prove convergence, even in the case-of linear function approximators. (See Gordon (2001), however, for a non-divergence result.) The heart of the problem is that the approximate values of different states (e.g., 81 and 82) are coupled through the parameters of the function approximator. The optimal policy at state 81 may require increasing a parameter, while the optimal policy at state 82 may require decreasing it. As a result, algorithms based on local parameter updates tend to oscillate or even to diverge. To avoid this problem, a more global approach is called for-an approach that can consider Sl and S2 simultaneously and find a solution that works well in both states. One approach is to formulate the reinforcement learning problem as a global search through a space of parameterized policies as in the policy gradient algorithms (Williams, 1992; Sutton, McAllester, Singh, & Mansour, 2000; Konda & Tsitsiklis, 2000; Baxter & Bartlett, 2000). This avoids the oscillation problem, but the resulting algorithms are slow and only converge to local optima. We pursue an alternative approach that formulates the function approximation problem as a global supervised learning problem. This approach, pioneered by Boyan and Moore in their GROWSUPPORT (1995) and ROUT (1996) algorithms, separates the reinforcement learning problem into two subproblems: the exploration problem (finding a good partial value function) and the representation problem (representing and generalizing that value function). These algorithms alternate between two phases. During the exploration phase, a support set of points is constructed whose optimal values are known within some tolerance. In the function fitting phase, a function approximator V is fit to the support set. In this paper, we describe two ways of improving upon GROWSUPPORT and ROUT. First, we replace the support set with the set of states that lie along the best paths found during exploration. Second, we employ a combined error function that includes terms for the supervised error, the Bellman error, and the advantage error (Baird, 1995) into the function fitting process. The resulting BFBP (Batch Fit to Best Paths) method gives significantly better performance on resource-constrained scheduling problems as well as on the mountain car toy benchmark problem. 2 GrowSupport, ROUT, and BFBP Consider a deterministic, episodic MDP. Let s' == a(s) denote the state s' that results from performing a in s and r(a, s) denote the one-step reward. Both GROWSUPPORT and ROUT build a support set S == {(Si' V(Si))} of states whose optimal values V (s) are known with reasonable accuracy. Both algorithms initialize S with a set of terminal states (with V(s) == 0). In each iteration, a function approximator V is fit to S to minimize :Ei[V(Si) - V(Si)]2. Then, an exploration process attempts to identify new points to include in S. In GROWSUPPORT, a sample of points X is initially drawn from the state space. In each iteration, after fitting V, GROWSUPPORT computes a new estimate V(s) for each state sEX according to V(s) == maxa r(s, a) + V(a(s)), where V(a(s)) is computed by executing the greedy policy with respect to V starting in a(s). If V(a(s)) is within c of V(a(s)), for all actions a, then (s, V(s)) is added to S. ROUT employs a different procedure suitable for stochastic MDPs. Let P(s'ls, a) be the probability that action a in state s results in state s' and R(s'ls, a) be the expected one-step reward. During the exploration phase, ROUT generates a trajectory from the start state to a terminal state and then searches for a state s along that trajectory such that (i) V(s) is not a good approximation to the backedup value V(s) == maxa :Est P(s'ls, a)[R(s'ls, a) + V(s')], and (ii) for every state s along a set of rollout trajectories starting at s', V(s) is within c of the backed-up value maxa :Est P(s'ls, a)[R(s'ls, a) +V(s')]. If such a state is found, then (s, V(s)) is added to S. Both GROWSUPPORT and ROUT rely on the function approximator to generalize well at the boundaries of the support set. A new state s can only be added to S if V has generalized to all of s's successor states. H this occurs consistently, then eventually the support set will expand to include all of the starting states of the MDP, at which point a satisfactory policy has been found. However, if this "boundary generalization" does not occur, then no new points will be added to S, and both GROWSUPPORT and ROUT. terminate without a solution. Unfortunately, most regression methods have high bias and variance near the boundaries of their training data, so failures of boundary generalization are common. These observations led us to develop the BFBP algorithm. In BFBP, the exploration process maintains a data structure S that stores the best known path from the start state to a terminal state and a "tree" of one-step departures from this best path (Le., states that can be reached by executing an action in some state on the best path). At each state Si E S, the data structure stores the action at executed in that state (to reach the next state in the path), the one-step reward ri, and the estimated value V(Si). S also stores each action a_ that causes a departure from the best path along with the resulting state S_, reward r_ and estimated value V(s_). We will denote by B the subset of S that constitutes the best path. The estimated values V are computed as folloV1S. For states S'i E B, V(Si) is computed 'by summing the immediate rewards rj for all steps j 2: i along B. For the one-step departure states s_, V(s_) is computed from an exploration trial in which the greedy policy was followed starting in state s_. fuitially, S is empty, so a random trajectory is generated from the start state So to a terminal state, and it becomes the initial best known path. fu subsequent iterations, a state Si E B is chosen at random, and an action a' 1= at is chosen and executed to produce state s' and reward r'. Then the greedy policy (with respect to the current V) is executed until a terminal state is reached. The rewards along this new path are summed to produce V(s'). If V(s') +r' > V(Si), then the best path is revised as follows. The new best action in state Si becomes a l with estimated value V(s') +r'. This improved value is then propagated backwards to update the V estimates for all ancestor states in B. The old best action at in state Si becomes an inferior action a_ with result state s_. Finally all descendants of s_ along the old best path are deleted. This method of investigating one-step departures from the best path is inspired by Harvey and Ginsberg's (1995) limited discrepancy search (LDS) algorithm. In each exploration phase, K one-step departure paths are explored. After the exploration phase, the value function approximation V is recomputed with the goal of minimizing a combined error function: J(V) == AsL (V(s) - V(S))2 + AbL (V(s) - [r(s, a*) + V(a*(s))])2 + sES sEB Aa L L ([r(s,a-) +V(a-(s))] - [r(s,a*) + V(a*(s))]):. The three terms of this objective function are referred to as the supervised, Bellman, and advantage terms. Their relative importance is controlled by the coefficients As, Ab' and Au. The supervised term is the usual squared error between the V(s) values stored in S and the fitted values V(s). The Bellman term is the squared error between the fitted value and the backed-up value of the next state on the best path. And the advantage term penalizes any case where the backed-up value of an inferior action a_ is larger than the backed-up value of the best action a*. The notation (u)+ == u if u 2: 0 and 0 otherwise. TheoreIll 1 Let M be a deterministic MDP such that (aJ there are only a finite number of starting states, (bJ there are only· a finite set of actions executable in each state, and (c) all policies reach a terminal state. Then BFBP applied to M converges. Proof: The LDS exploration process is monotonic, since the data structure S is only updated if a new best path is found. The conditions of the theorem imply that there are only a finite number of possible paths that·can be explored from the starting states to the terminal states. Hence, the data structure S will eventually converge. Consequently, the value function V fit to S will also converge. Q.E.D. The theorem requires that the MDP contain no cycles. There are cycles in our jobshop scheduling problems, but we eliminate them by remembering all states visited along the current trajectory and barring any action that would return to a previously visited state. Note also that the theorem applies to MDPs with continuous state spaces provided the action space and the start states are finite. Unfortunately, BFBP does not necessarily converge to an optimal policy. This is because LDS exploration can get stuck in a local optimum such that all one step departures using the V-greedy policy produce trajectories that do not improve over the current best path. Hence, although BFBP resembles policy iteration, it does not have the same optimality guarantees,. because policy iteration evaluates the current greedy policy in all states in the state space. Theoretically, we could prove convergence to the optimal policy under modified conditions. If we replace LDS exploration with €-greedy exploration, then exploration will converge to the optimal paths with probability 1. When trained on those paths, if the function approximator fits a sufficiently accurate V, then BFBS will converge optimally. hI our experiments, however, we have found that €-greedy gives no improvement over LDS, whereas LDS exploration provides more complete coverage of one-step departures from the current best path, and these are used in J(V). 3 Experimental Evaluation We have studied five domains: Grid World and Puddle World (Boyan & Moore, 1995), Mountain Car (Sutton, 1996), and resource-constrained scheduling problems ART-1 and ART-2 (Zhang & Dietterich, 1995). For the first three domains, following Boyan and Moore, we compare BFBP with GROWSUPPORT. For the final domain, it is difficult to draw a sample of states X from the state space to initialize GROWSUPPORT. Hence, we compare against ROUT instead. As mentioned above, we detected and removed cycles from the scheduling domain (since ROUT requires this). We retained the cycles in the first three problems. On mountain car, we also applied SARSA(A) with the CMAC function approximator developed by Sutton (1996). We experimented with two function approximators: regression trees (RT) and locally-weighted linear regression (LWLR). Our regression trees employ linear separating planes at the internal nodes and linear surfaces at the leaf nodes. The trees are grown top-down in the usual fashion. To determine the splitting plane at a node, we choose a state Si at random from S, choose one of its inferior children S_, and construct the plane that is the perpendicular bisector of these two points. The splitting plane is evaluated by fitting the resulting child nodes to the data (as leaf nodes) and computing the value of J (V). A number C of parent-child pairs (Si' S - ) are generated and evaluated, and the best one is retained to be the splitting plane. This process is then repeated recursively until a node contains fewer than M data points~ The linear surfaces at the leaves are trained by gradient descent to minimize J(V). The gradient descent terminates after 100 steps or earlier if J becomes very small. In our experiments, we tried all combinations of the following parameters and report the best results: (a) 11 learning rates (from 0.00001 to 0.1), (b) M == 1, Table 1: Comparison of results on three toy domains. Problem Domain Algorithms Optimal Policyfj Best Policy Length Grid World GROWSUPPORT Yes 39 BFBP Yes 39 Puddle World G ROWSUPPORT Yes 39 BFBP Yes 39 Mountain Car SARSA(A) No 103 GROWSUPPORT No 93 BFBP Yes 88 Table 2: Results of ROUT and BFBP on scheduling problem ART-I-TRNOO I Performance I ROUT (RT) I ROUT (LWLR) I BFBP (RT) I Best policy explored I 1.75 I 1.55 I 1.50 I Best final learned policy I 1.8625 I 1.8125 I 1.55 10, 20, 50, 100, 1000, (c) C == 5, 10, 20, 50, 100, and (d) K == 50, 100, 150, 200. For locally-weighted linear regression, we replicated the methods of B'oyan and Moore. To compute V(s), a linear regression is performed using all points Si E S weighted by their distance to S according to the kernel exp -(Ilsi - sII 2/a2 ). We experimented with all combinations of the following parameters and report the best results: (a) 29 values (from 0.01 to 1000.0) of the tolerance E that controls the addition of new points to S, and (b) 39 values (from 0.01 to 1000.0) for a. We execute ROUT and GROWSUPPORT to termination. We execute BFBP for 100 iterations, but it converges much earlier: 36 iterations for the grid world, 3 for puddle world, 10 for mountain car, and 5 for the job-shop scheduling problems. Table 1 compares the results of the algorithms on the toy domains with parameters for each method tuned to give the best results and with As == 1 and Ab == Aa == o. In all cases, BFBP matches or beats the other methods. In Mountain Car, in particular, we were pleased that BFBP discovered the optimal policy very quickly. Table 2 compares the results of ROUT and BFBP on job-shop scheduling problem TRNOO from problem set ART-1 (again with As == 1 and Ab == Aa == 0). For ROUT, results with both LWLR and RT are shown. LWLR gives better results for ROUT. We conjecture that this is because ROUT needs a value function approximator that is conservative near the boundary of the training data, whereas BFBP does not. We report both the best policy found during the iterations and the final policy at convergence. Figure 1 plots the r,esults for ROUT (LWLR) against BFBP (RT) for eight additional scheduling problems from ART-I. The figure ofmerit is RDF, which is a normalized measure of schedule length (small values are preferred). BFBP's learned policy out-performs ROUT's in every case. The experiments above all employed only the supervised term in the error function J. These experiments demonstrate that LDS exploration gives better training sets than the support set methods of GROWSUPPORT and ROUT. Now we turn to the question of whether the Bellman and advantage terms can provide improved results. For the grid world and puddle world tasks, the supervised term already gives optimal performance, so we focus on the mountain car and job-shop scheduling problems. Table 3 summarizes the results for BFBP on the mountain car problem. All parameter settings, except for the last, succeed in finding the optimal policy. To get best policy explored + y=xbest finalleamed policy x Xx + x x 2.4 2.2 G:' Q es <l) 1.8 § § ~ 1.6 0.0... ff P=l 1.4 1.2 1 1 1.2 + + 1.4 1.6 1.8 ROUT performance (RDF) 2.2 2.4 Figure 1: Performance of Rout vs. BFBP over 8 job shop scheduling problems Table 3: Fraction of parameter settings that give optimal performance for BFBP on the mountain car problem .As .Ab .Aa # settings As Ab Aa # settings 0.0 0.0 1.0 2/1311 0.0 1.0 0.0 1/1297 1.0 0.0 0.0 52/1280 1.0 0.0 10.0 184/1291 1.0 10.0 0.0 163/1295 1.0 0.0 100.0 133/1286 1.0 100.0 0.0 4/939 1.0 1000.0 0.0 0/1299 a sense of the robustness of the method, we also report the fraction of parameter settings that gave the optimal policy. The number of parameter settings tested (the denominator) should be the same for all combinations of A values. Nonetheless, for reasons unrelated to the parameter settings, some combinations failed to be executed by our distributed process scheduler. The best settings combine As == 1 with either Ab == 10 or Aa == 10. However, if we employ either the Bellman or the advantage term alone, the results are poor. Hence, it appears that the supervised term is very important for good performance, but that the advantage and Bellman terms can improve performance substantially .and reduce the sensitivity of BFBP to the settings of the other parameters. Table 4 shows the performance of BFBP on ART-I-TRNOO. The best performance (at convergence) is obtained with As == Aa == 1 and Ab == O. As with mountain car, these experiments show that the supervised term is the most important, but that it gives even better results when combined with the advantage term. All of the above experiments compare performance on single problems. We also tested the ability of BFBP to generalize to similar problems following the formulation of (Zhang & Dietterich, 1995). Figure 2 compares the performance of neural networks and regression trees as function approximators for BFBP. Both were trained on job shop scheduling problem set ART-2. Twenty of the problems in ART-2 were used for training, 20 for cross-validation, and 50 for testing. Eleven different values for As, Ab' Aa and eight different values for the learning rate were tried, with the best values selected according to the cross-validation set. Figure 2 shows that BFBP is significantly better than the baseline performance (with RDF Table 4: Performance ofBFBP on ART-1-TENOO for different settings of the .A parameters. The ('perform;' column gives the best RDF in any iteratIon and the RDF at convergence. .A8 .Ab .Aa perform. .A8 .Ab .Aa perform. .A8 .Ab .Aa perform. 0 0 1 1.50/1.75 0 1 0 1.50/1.775 1 0 0 1.50/1.55 0 1 1 1.50/1.775 0 1 10 1.50/1.825 0 1 100 1.50/1.65 0 10 1 1.50/1.775 0 100 1 1.50/1.738 1 1 a 1.50/1.563 1 0 1 1.50/1.488 1 0 10 1.463/1.525 1 0 100 1.50/1.588 1 1 1 1.525/1.55 1 1 10 1.50/1.588 1 1 100 1.50/1.675 1.8 .------,----.-------,.---...,----,------, 1.75 BFBP neural net ----*---. BFBP regression tree --- -G---1.7 _________________________RP_E _ 1.65 LL 1.6 o0: ~ 1.55 ~ ~ 1.45 1.4'¥._?4<.,_. ,'"'::-..... ,c ._••_._._.-!n-._,._. . ...._._.,.._._._. ._,._. . ._., ._.._._. ,,_._. ._.,_T._..O_L__. . 1.35 30 25 20 15 LOS iteration 10 1.3 L..--__---l--__--'--__--L -'---__---'-__-----' o Figure 2: BFBP on ART-2 using neural nets and regression trees. "RDF" is a hand-coded heuristic, "TDL" is Zhang's TD(.A) neural network. as a search heuristic) and that its performance is comparable to TD(A) with neural networks (Zhang & Dietterich, 1995). Figure 3 shows that for ART-2, using parent/inferior-child pair splits gives better results than using axis-parallel splits. 4 Conclusions This paper has shown that the exploration strategies underlying GROWSUPPORT and ROUT can be improved by simply remembering and training on the best paths found between start and terminal states. Furthermore, the paper proved that the BFBP method converges for arbitrary function approximators, which is a result that has not yet been demonstrated for online methods such as SARSA(A). In addition, we have shown that the performance of our BFBP algorithm can be further improved (and made more robust) by incorporating a penalty for violations of the Bellman equation or a penalty for preferring inferior actions (an advantage error). Taken together, these results show that incremental-batch value function approximation can be a reliable, convergent method for solving deterministic reinforcement learning problems. The key to the success of the method is the ability to separate the exploration process from the function approximation process and to make the exploration process convergent. This insight should also be applicable to stochastic episodic MDPs. 1.9 ,-------.-----,------,----..,-------,-----, axis-parallel ....*... parent/inferior-child .. ··11 •••. 30 25 20 15 LDS iteration 10 1.3 L-__--l-'---__---'-L-__---L.__----I o Figure 3: Axis parallel splits versus parent/inferior-child pair splits on ART-2 Acknowledgments The authors gratefully acknowledge the support of AFOSR under contract F4962098-1-0375, and the NSF under grants IRl-9626584, I1S-0083292, 1TR-5710001197, and EIA-9818414. We thank Valentina Zubek for her careful reading of the paper. References Baird, L. C. (1995). Residual algorithms: Reinforcement learning with function approximation. In ICML-95, 30-37, San Francisco, CA. Morgan Kaufmann. Baxter, J., & Bartlett, P. L. (2000). Reinforcement learning in POMDP's via direct gradient ascent. In ICML-2000, 41-48. Morgan Kaufmann, San Francisco, CA. Boyan, J. A., & Moore, A. W. (1995). Generalization in reinforcement learning: Safely approximating the value function. In NIPS-7, 369-376. The MIT Press, Cambridge. Boyan, J. A., & Moore, A. W. (1996). Learning evaluation functions for large acyclic domains. In ICML-96, 63-70. Morgan Kaufmann, San Francisco, CA. Gordon, G. J. (2001). Reinforcement learning with function approximation converge to a region. In NIPS-13, 1040-1046. The MIT Press. Harvey, W. D., & Ginsberg, L. P. (1995). Limited discrepancy search. In IJCAI-95, 825-830. Morgan Kaufmann. Konda, V. R., & Tsitsiklis, J. N. (2000). Policy gradient methods for reinforcement learning with function approximation. In NIPS-12, 1008-1014 Cambridge, MA. MIT Press. Moll, R., Barto, A. G., Perkins, T. J., & Sutton, R. S. (1999). Learning instanceindependent value functions to enhance local search. In NIPS-ll, 1017-1023. Sutton, R. S., McAllester, D., Singh, S., & Mansour, Y. (2000). Policy gradient methods for reinforcement learning with function approximation. In NIPS-12, 1057-1063. Sutton, R. S. (1996). Generalization in reinforcement learning: Successful examples using sparse coarse coding. In NIPS-8, 1038-1044. The MIT Press, Cambridge. Williams, R. J. (1992). Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8,229. -.. Zhang, W., & Dietterich, T. G. (1995). A reinforcement learning approach to job-shop scheduling. In IJCAI-95, 1114-1120. Morgan Kaufmann, San Francisco, CA.
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Tree-based reparameterization for approximate inference on loopy graphs Martin J. Wainwright, Tommi Jaakkola, and Alan S. Will sky Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, MA 02139 mjwain@mit.edu tommi@ai.mit.edu willsky@mit.edu Abstract We develop a tree-based reparameterization framework that provides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles. It includes belief propagation (BP), which can be reformulated as a very local form of reparameterization. More generally, we consider algorithms that perform exact computations over spanning trees of the full graph. On the practical side, we find that such tree reparameterization (TRP) algorithms have convergence properties superior to BP. The reparameterization perspective also provides a number of theoretical insights into approximate inference, including a new characterization of fixed points; and an invariance intrinsic to TRP /BP. These two properties enable us to analyze and bound the error between the TRP /BP approximations and the actual marginals. While our results arise naturally from the TRP perspective, most of them apply in an algorithm-independent manner to any local minimum of the Bethe free energy. Our results also have natural extensions to more structured approximations [e.g., 1, 2]. 1 Introduction Given a graphical model, one important problem is the computation of marginal distributions of variables at each node. Although highly efficient algorithms exist for this task on trees, exact solutions are prohibitively complex for more general graphs of any substantial size. This difficulty motivates the use of approximate inference algorithms, of which one of the best-known and most widely studied is belief propagation [3], also known as the sum-product algorithm in coding [e.g., 4]. Recent work has yielded some insight into belief propagation (BP). Several researchers [e.g., 5, 6] have analyzed the single loop case, where BP can be reformulated as a matrix powering method. For Gaussian processes on arbitrary graphs, two groups [7, 8] have shown that the means are exact when BP converges. For graphs corresponding to turbo codes, Richardson [9] established the existence of fixed points, and gave conditions for their stability. More recently, Yedidia et al. [1] showed that BP corresponds to constrained minimization of the Bethe free energy, and proposed extensions based on Kikuchi expansions [10]. Related extensions to BP were proposed in [2]. The paper [1] has inspired other researchers [e.g., 11, 12] to develop more sophisticated algorithms for minimizing the Bethe free energy. These advances notwithstanding, much remains to be understood about the behavior of BP. The framework of this paper provides a new conceptual view of various algorithms for approximate inference, including BP. The basic idea is to seek a reparameterization of the distribution that yields factors which correspond, either exactly or approximately, to the desired marginal distributions. If the graph is acyclic (i.e., a tree) , then there exists a unique reparameterization specified by exact marginal distributions over cliques. For a graph with cycles, we consider the idea of iteratively reparameterizing different parts of the distribution, each corresponding to an acyclic subgraph. As we will show, BP can be interpreted in exactly this manner, in which each reparameterization takes place over a pair of neighboring nodes. One of the consequences of this interpretation is a more storage-efficient "message-free" implementation of BP. More significantly, this interpretation leads to more general updates in which reparameterization is performed over arbitrary acyclic subgraphs, which we refer to as tree-based reparameterization (TRP) algorithms. At a low level, the more global TRP updates can be viewed as a tree-based schedule for message-passing. Indeed, a practical contribution of this paper is to demonstrate that TRP updates tend to have better convergence properties than local BP updates. At a more abstract level, the reparameterization perspective provides valuable conceptual insight, including a simple tree-consistency characterization of fixed points, as well as an invariance intrinsic to TRP /BP. These properties allow us to derive an exact expression for the error between the TRP /BP approximations and the actual marginals. Based on this exact expression, we derive computable bounds on the error. Most of these results, though they emerge very naturally in the TRP framework, apply in an algorithm-independent manner to any constrained local minimum of the Bethe free energy, whether obtained by TRP /BP or an alternative method [e.g., 11, 12]. More details of our work can be found in [13, 14]. 1.1 Basic notation An undirected graph Q = (V, £) consists of a set of nodes or vertices V = {l, ... ,N} that are joined by a set of edges £. Lying at each node s E V is a discrete random variable Xs E {a, ... ,m - I}. The underlying sample space X N is the set of all N vectors x = {xs I S E V} over m symbols, so that IXNI = m N . We focus on stochastic processes that are Markov with respect to Q, so that the Hammersley-Clifford theorem [ e.g., 3] guarantees that the distribution factorizes as p(x) ex: [lcEe 'l/Jc(xc) where 'l/Jc(xc) is a compatibility function depending only on the subvector Xc = {xs I SEC} of nodes in a particular clique C. Note that each individual node forms a singleton clique, so that some of the factors 'l/Jc may involve functions of each individual variable. As a consequence, if we have independent measurements Ys of Xs at some (or all) of the nodes, then Bayes' rule implies that the effect of including these measurements i.e., the transformation from the prior distribution p(x) to the conditional distribution p(x I y) is simply to modify the singleton factors. As a result, throughout this paper, we suppress explicit mention of measurements, since the problem of computing marginals for either p(x) or p(x I y) are of identical structure and complexity. The analysis of this paper is restricted to graphs with singleton ('l/Js) and pairwise ('l/Jst} cliques. However, it is straightforward to extend reparameterization to larger cliques, as in cluster variational methods [e.g., 10]. 1.2 Exact tree inference as reparameterization Algorithms for optimal inference on trees have appeared in the literature of various fields [e.g., 4, 3]. One important consequence of the junction tree representation [15] is that any exact algorithm for optimal inference on trees actually computes marginal distributions for pairs (s, t) of neighboring nodes. In doing so, it produces an alternative factorization p(x) = TI sEV Ps TI(s,t)E£ Pst/(PsPt) where Ps and Pst are the single-node and pairwise marginals respectively. This {Ps, Pst} representation can be deduced from a more general factorization result on junction trees [e.g. 15]. Thus, exact inference on trees can be viewed as computing a reparameterized factorization of the distribution p(x) that explicitly exposes the local marginal distributions. 2 Tree-based reparameterization for graphs with cycles The basic idea of a TRP algorithm is to perform successive reparameterization updates on trees embedded within the original graph. Although such updates are applicable to arbitrary acyclic substructures, here we focus on a set T 1 , ... , TL of embedded spanning trees. To describe TRP updates, let T be a pseudomarginal probability vector consisting of single-node marginals Ts(xs) for 8 E V; and pairwise joint distributions Tst (x s, Xt) for edges (s, t) E [. Aside from positivity and normalization (Lx Ts = 1; L x x Tst = 1) constraints, a given vecs s , t tor T is arbitraryl, and gives rises to a parameterization of the distribution as p(x; T) ex: TIsEV Ts TI(S,t)E£ Tst/ {(Lx. Tst)(L Xt Tst)}, where the dependence of Ts and Tst on x is omitted for notational simplicity. Ultimately, we shall seek vectors T that are consistent i.e., that belong to <C = {T I Lx. Tst = Tt \;/ (8, t) E [}. In the context of TRP, such consistent vectors represent approximations to the exact marginals of the distribution defined by the graph with cycles. We shall express TRP as a sequence of functional updates Tn I-t T n+1 , where superscript n denotes iteration number. We initialize at TO via T~t = Ii 'l/Js'I/Jt'I/Jst and T~ = Ii 'l/Js TItEN(S) [L X t 'l/Jst'I/Jt], where Ii denotes a normalization factor; and N(8) is the set of neighbors of node 8. At iteration n, we choose some spanning tree Ti(n) with edge set [i(n), and factor the distribution p(x; Tn) into a product of two terms ex: (la) ex: (lb) corresponding, respectively, to terms in the spanning tree; and residual terms over edges in [/ [i(n) removed to form Ti(n). We then perform a reparameterization update on pi(n) (x; Tn) explicitly: pi(n) (x'; Tn) for all (s,t) E [i(n) (2) x, s.t( x ~ ,x;)=(x. ,xtl with a similar update for the single-node marginals {Ts I s E V}. These marginal computations can be performed efficiently by any exact tree algorithm applied to Ti(n). Elements of T n+1 corresponding to terms in ri(n) (x; Tn) are left unchanged lIn general, T need not be the actual marginals for any distribution. (i.e., Ts~+l = Tst for all (8, t) E E /Ei(n)) . The only restriction placed on the spanning tree set T1, ... ,TL is that each edge (8, t) E E belong to at least one spanning tree. For practical reasons, it is desirable to choose a set of spanning trees that leads to rapid mixing throughout the graph. A natural choice for the spanning tree index i(n) is the cyclic ordering, in which i(n) == n(modL) + 1. 2.1 BP as local reparameterization Interestingly, BP can be reformulated in a "message-free" manner as a sequence of local rather than global reparameterization operations. This message-free version of BP directly updates approximate marginals, Ts and Tst, with initial values determined from the initial messages M~t and the original compatibility functions of the graphical model as T~ = Ii 'l/Js ITuEN(S) M~s for all 8 E V and T~t = Ii 'l/Jst'l/Js'l/Jt ITuEN(s)/t M~s ITuEN(t) /s M~t for all (8, t) E E, where Ii denotes a normalization factor. At iteration n, these quantities are updated according to the following recursions: (3a) T;'t (3b) The reparameterization form of BP decomposes the graph into a set of two-node trees (one for each edge (8, t)); performs exact inference on such tree via equation (3b); and merges the marginals from each tree via equation (3a). It can be shown by induction [see 13] that this simple reparameterization algorithm is equivalent to the message-passing version of BP. 2.2 Practical advantages of TRP updates Since a single TRP update suffices to transmit information globally throughout the graph, it might be expected to have better convergence properties than the purely local BP updates. Indeed, this has proven to be the case in various experiments that we have performed on two graphs (a single loop of 15 nodes, and a 7 x 7 grid). We find that TRP tends to converge 2 to 3 times faster than BP on average (rescaled for equivalent computational cost); more importantly, TRP will converge for many problems where BP fails [13]. Further research needs to address the optimal choice of trees (not necessarily spanning) in implementing TRP. 3 Theoretical results The TRP perspective leads to a number of theoretical insights into approximate inference, including a new characterization of fixed points, an invariance property, and error analysis. 3.1 Analysis of TRP updates Our analysis of TRP updates uses a cost function that is an approximation to the Kullback-Leibler divergence between p(x; T) and p(x; U) namely, the quantity Xs Given an arbitrary U E C, we show that successive iterates {Tn} of TRP updates satisfy the following "Pythagorean" identity: G(U ;T n) = G(U ;T n+l ) + G(T n+1; T n) (4) which can be used to show that TRP fixed points T * satisfy the necessary conditions to be local minima of G subject to the constraint T * E C. The cost function G, though distinct from the Bethe free energy [1], coincides with it on the constraint set C, thereby allowing us to establish the equivalence of TRP and BP fixed points. 3.2 Characterization of fixed points From the reparameterization perspective arises an intuitive characterization of any TRP /BP fixed point T *. Shown in Figure l(a) is a distribution on a graph with T1: T2~ T3~ T1: T2~ T3~ T~T; T4; T; T; T5: T; T ~ TtT; T2*T; T; T ~ (a) Fixed point on full graph (b) Tree consistency condition. Figure 1. Illustration of fixed point consistency condition. (a) Fixed point T * = {T;, T;t} on the full graph with cycles. (b) Illustration of consistency condition on an embedded tree. The quantities {T;, T;t } must be exact marginal probabilities for any tree embedded within the full graph. cycles, parameterized according to the fixed point T * = {Ts*t, T;}. The consistency condition implies that if edges are removed from the full graph to form a spanning tree, as shown in panel (b), then the quantities Ts*t and Ts* correspond to exact marginal distributions over the tree. This statement holds for any acyclic substructure embedded within the full graph with cycles not just the spanning trees Tl , ... ,TL used to implement TRP. Thus, algorithms such as TRP /BP attempt to reparameterize a distribution on a graph with cycles so that it is consistent with respect to each embedded tree. It is remarkable that the existence of such a parameterization (though obvious for trees) should hold for a positive distribution on an arbitrary graph. Also noteworthy is the parallel to the characterization of max-product2 fixed points obtained by Freeman and Weiss [16]. Finally, it can be shown [13, 14] that this characterization, though it emerged very naturally from the TRP perspective, applies more generally to any constrained local minimum of the Bethe free energy, whether obtained by TRP /BP, or an alternative technique [e.g., 11, 12]. 2Max-product is a related but different algorithm for computing approximate MAP assignments in graphs with cycles. 3.3 Invariance of the distribution A fundamental property of TRP updates is that they leave invariant the full distribution on the graph with cycles. This invariance follows from the decomposition of equation (1): in particular, the distribution pi(n) (x; Tn) is left invariant by reparameterization; and TRP does not change terms in ri(n) (x; Tn). As a consequence, the overall distribution remains invariant i.e., p(x; Tn) == p(x; TO) for all n. By continuity of the map T f-7 p(x; T), it follows that any fixed point T* of the algorithm also satisfies p(x; T*) == p(x; TO). This fixed point invariance is also an algorithmindependent result in particular, all constrained local minima of the Bethe free energy, regardless of how they are obtained, are invariant in this manner [13, 14]. This invariance has a number of important consequences. For example, it places severe restrictions on cases (other than trees) in which TRP /BP can be exact; see [14] for examples. In application to the linear-Gaussian problem, it leads to an elementary proof of a known result [7, 8] namely, the means must be exact if the BP updates converge. 3.4 Error analysis Lastly, we can analyze the error arising from any TRP /BP fixed point T* on an arbitrary graph. Of interest are the exact single-node marginals Ps of the original distribution p(x; TO) defined by the graph with cycles, which by invariance are equivalent to those of p(x; T*). Now the quantities Ts* have two distinct interpretations: (a) as the TRP /BP approximations to the actual single-node marginals on the full graph; and (b) as the exact marginals on any embedded tree (as in Figure 1). This implies that the approximations T; are related to the actual marginals Ps on the full graph by a relatively simple perturbation namely, removing edges from the full graph to reveal an embedded tree. From this observation, we can derive the following exact expression for the difference between the actual marginal PS;j and the TRP /BP approximation3 T;j: [{ ri(X;T*)} .J lEpi (x;T* ) Z(T*) - 1 J(xs = J) (5) where i E {1, ... ,L} is an arbitrary spanning tree index; pi and ri are defined in equation (1a) and (1b) respectively; Z(T*) is the partition function of p(x; T*); J(xs = j) is an indicator function for Xs to take the value j; and lEpi (x;T * ) denotes expectation using the distribution pi(x; T*). Unfortunately, while the tree distribution pi (x; T*) is tractable, the argument of the expectation includes all terms r i (x; T*) removed from the original graph to form spanning tree Ti. Moreover, computing the partition function Z (T*) is intractable. These difficulties motivate the development of bounds on the error. In [14], we use convexity arguments to derive a particular set of bounds on the approximation error. Such error bounds, in turn, can be used to compute upper and lower bounds on the actual marginals Ps;l. Figure 2 illustrates the TRP /BP approximation, as well as these bounds on the actual marginals for a binary process on a 3 x 3 grid under two conditions. Note that the tightness of the bounds is closely related to approximation accuracy. Although it is unlikely that these bounds will remain quantitatively useful for general problems on large graphs, they may still yield useful qualitative information. 3The notation T;;j denotes the /h element of the vector T; . 0.9 0.8 0.7 :;:::'0.6 " :5-"b.5 e "- o. 0.2 0.1 Bounds on single node marginals °1~~--~--~4~~5---6~~~~~~ Node number (a) Weak potentials 0.9 0.8 Bounds on single node marginals 4 5 6 Node number (b) Strong mixed potentials Figure 2. Behavior of bounds on 3 x 3 grid. Plotted are the actual marginals P s;l versus the TRP approximations T;'l> as well as upper and lower bounds on the actual marginals. (a) For weak potentials, TRP /BP approximation is excellent; bounds on exact marginals are tight. (b) For strong mixed potentials, approximation is poor. Bounds are looser, and for certain nodes, the TRP /BP approximation lies above the upper bounds on the actual marginal P8 ;1 . Much of the analysis of this paper -- including reparameterization, invariance, and error analysis -- can be extended [see 14] to more structured approximation algorithms [e.g., 1, 2]. Figure 3 illustrates the use of bounds in assessing when to use a more structured approximation. For strong attractive potentials on the 3 x 3 grid, the TRP /BP approximation in panel (a) is very poor, as reflected by relatively loose bounds on the actual marginals. In contrast, the Kikuchi approximation in (b) is excellent, as revealed by the tightness of the bounds. 4 Discussion The TRP framework of this paper provides a new view of approximate inference; and makes both practical and conceptual contributions. On the practical side, we find that more global TRP updates tend to have better convergence properties than local BP updates. The freedom in tree choice leads to open problems of a graphtheoretic nature: e.g., how to choose trees so as to guarantee convergence, or to optimize the rate of convergence? Among the conceptual insights provided by the reparameterization perspective are a new characterization of fixed points; an intrinsic invariance; and analysis of the approximation error. Importantly, most of these results apply to any constrained local minimum of the Bethe free energy, and have natural extensions [see 14] to more structured approximations [e.g., 1, 2]. Acknowledgments This work partially funded by ODDR&E MURI Grant DAAD19-00-1-0466; by ONR Grant N00014-00-1-0089; and by AFOSR Grant F49620-00-1-0362; MJW also supported by NSERC 1967 fellowship. References [1] J. Yedidia, W. T. Freeman, and Y. Weiss. Generalized belief propagation. In NIPS 13, pages 689- 695. MIT Press, 2001. Bounds on single node marginals Bounds on single node marginals - - - -0 - - - -0- - e - - - M M 0.8 _ 0 - - - - 0 - - - -0- - - - €l - - -;o::V " " :5-"b. :5-"b.5 £> £> a.. 0.4 a.. 0.4 0.3 0.2r r -+-:Ac-,-tu--:al----. 0.1 -+- TAP I BP - 0 · Bounds °1~~==~~-4~~5~~-~-~~ Node number (a) TRP /BP 0.3 :~ II =-:= ~~r~~lured approx. 1 ~ r l=-e~B=o= un=ds~==~~~~-~-~-~ °1 4 5 Node number (b) Kikuchi Figure 3. When to use a more structured approximation? (a) For strong attractive potentials on the 3 x 3 grid, BP approximation is poor, as reflected by loose bounds on the actual marginal. (b) Kikuchi approximation [1] for same problem is excellent; corresponding bounds are tight. [2] T. P. Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, MIT Media Lab, 2001. [3] J. Pearl. Probabilistic reasoning in intelligent systems. Morgan Kaufman, San Mateo, 1988. [4] F. Kschischang and B. Frey. Iterative decoding of compound codes by probability propagation in graphical models. IEEE Sel. Areas Comm., 16(2):219- 230, February 1998. [5] J. B. Anderson and S. M. Hladnik. Tailbiting map decoders. IEEE Sel. Areas Comm., 16:297- 302, February 1998. [6] Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural Computation, 12:1-41, 2000. [7] Y. Weiss and W. T. Freeman. Correctness of belief propagation in Gaussian graphical models of arbitrary topology. In NIPS 12, pages 673- 679. MIT Press, 2000. [8] P. Rusmevichientong and B. Van Roy. An analysis of turbo decoding with Gaussian densities. In NIPS 12, pages 575- 581. MIT Press, 2000. [9] T. Richardson. The geometry of turbo-decoding dynamics. IEEE Trans. Info. Theory, 46(1):9- 23, January 2000. [10] R. Kikuchi. The theory of cooperative phenomena. Physical Review, 81:988- 1003, 1951. [11] M. Welling and Y. Teh. Belief optimization: A stable alternative to loopy belief propagation. In Uncertainty in Artificial Intelligence, July 2001. [12] A. Yuille. A double-loop algorithm to minimize the Bethe and Kikuchi free energies. Neural Computation, To appear, 2001. [13] M. J . Wainwright, T. Jaakkola, and A. S. Willsky. Tree-based reparameterization for approximate estimation on graphs with cycles. LIDS Tech. report P-2510: available at http://ssg.rnit.edu/group/rnjyain/rnjyain.shtrnl, May 2001. [14] M. Wainwright. Stochastic processes on graphs with cycles: geometric and variational approaches. PhD thesis, MIT, Laboratory for Information and Decision Systems, January 2002. [15] S. L. Lauritzen. Graphical models. Oxford University Press, Oxford, 1996. [16] W. Freeman and Y. Weiss. On the optimality of solutions of the max-product belief propagation algorithm in arbitrary graphs. IEEE Trans. Info. Theory, 47:736- 744, 2001.
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Cobot: A Social Reinforcement Learning Agent Charles Lee Isbell, Jr. Christian R. Shelton AT&T Labs-Research Stanford University Michael Kearns Satinder Singh Peter Stone University of Pennsylvania Syntek Capital AT&T Labs-Research Abstract We report on the use of reinforcement learning with Cobot, a software agent residing in the well-known online community LambdaMOO. Our initial work on Cobot (Isbell et al.2000) provided him with the ability to collect social statistics and report them to users. Here we describe an application of RL allowing Cobot to take proactive actions in this complex social environment, and adapt behavior from multiple sources of human reward. After 5 months of training, and 3171 reward and punishment events from 254 different LambdaMOO users, Cobot learned nontrivial preferences for a number of users, modifing his behavior based on his current state. Here we describe LambdaMOO and the state and action spaces of Cobot, and report the statistical results of the learning experiment. 1 Introduction While most applications of reinforcement learning (RL) to date have been to problems of control, game playing and optimization (Sutton and Barto1998), there has been a recent handful of applicationsto human-computer interaction. Such applications present a number of interesting challenges to RL methodology(such as data sparsity and inevitable violations of the Markov property). These previous studies focus on systems that encounter human users one at a time, such as spoken dialogue systems (Singh et al.2000). In this paper, we report on an RL-based agent for LambdaMOO, a complex, open-ended, multi-user chat environment, populated by a community of human users with rich and often enduring social relationships. Our long-term goal is to build an agent who can learn to perform useful, interesting and entertaining actions in LambdaMOO on the basis of user feedback. While this is a deliberately ambitious and underspecified goal, we describe here our implementation, the empirical experiences of our agent so far, and some of the lessons we have learned about this challenging domain. In previous work (Isbell et al.2000), we developed the software agent Cobot, who interacted in various ways with LambdaMOO users. Cobot had two primary functions. First, Cobot gathered “social statistics” (e.g. how frequently and in what ways users interacted with one another), and provided summaries of these statistics as a service. Second, Cobot had rudimentary chatting abilities based on the application of information retrieval methods to large documents. The original Cobot was entirely reactive , in that he never initiated interaction with human users, but would only respond to their actions. As we documented in our earlier paper, Cobot proved tremendously popular with LambdaMOO users, setting the stage for our current efforts. We modified Cobot to allow him to take certain actions (such as proposing conversation topics, introducing users, or engaging in common word play routines) under his own initiative. The hope is to build an agent that will eventually take unprompted actions that are meaningful, useful or amusing to users. Rather than hand-code complex rules specifying Here we mean “responding only to human-invoked interaction”, rather than “non-deliberative”. Characters in LambdaMOO have gender. Cobot’s description to users indicates that he is male. when each action is appropriate (rules that would be inaccurate and quickly become stale), we wanted Cobot to learn the individual and communal preferences of users. Thus, we provided a mechanism for users to reward or punish Cobot, and programmed Cobot to use RL algorithms to alter his behavior on the basis of this feedback. The application of RL (or any machine learning methodology) to such an environment presents a number of interesting domain-specific challenges, including: Choice of an appropriate state space. To learn how to act in a social environment such as LambdaMOO, Cobot must represent the salient features. These should include social information such as which users are present, how experienced they are in LambdaMOO, how frequently they interact with one another, and so on. Multiple reward sources. Cobot lives in an environment with multiple, often conflicting sources of reward from different human users. How to integrate these sources reasonably is a nontrivial empirical question. Inconsistency and drift of user rewards and desires. Individual users may be inconsistent in the rewards they provide (even when they implicitly have a fixed set of preferences), and their preferences may change over time (for example, due to becoming bored or irritated with an action). Even when their rewards are consistent, there can be great temporal variation in their reward pattern. Variability in user understanding. There is great variation in users’ understanding of Cobot’s functionality, and the effects of their rewards and punishments. Data sparsity. Training data is scarce for many reasons, including user fickleness, and the need to prevent Cobot from generating too much spam in the environment. Irreproducibility of experiments. As LambdaMOO is a globally distributed community of human users, it is virtually impossible to replicate experiments taking place there. We do not have any simple answers (nor do we believe that simple answers exist), but here we provide a case study of our choices and findings. Our primary findings are: Inappropriateness of average reward. We found that the average reward that Cobot received over time, the standard measure of successfor RL experiments, is an inadequate and perhaps even inappropriate metric of performance in the LambdaMOO domain. Reasons include that user preferences are not stationary, but drift as users become habituated or bored with Cobot’s behavior; and the tendency for satisfied users to stop providing Cobot with any feedback, positive or negative. Despite the inadequacy of average reward, we are still able to establish several measures by which Cobot’s RL succeeds, discussed below. A small set of dedicated “parents”. While many users provided only a moderate or small amount of RL training (rewards and punishments) to Cobot, a handful of users did invest significant time in training him. Some parents have strong opinions. While many of the users that trained Cobot did not exhibit clear preferences for any of his actions over the others, some users clearly and consistently rewarded and punished particular actions over the others. Cobot learns matching policies. For those users who exhibited clear preferences through their rewards and punishments, Cobot successfully learned corresponding policies of behavior. Cobot responds to his dedicated parents. For those users who invested the most training time in Cobot, the observed distribution of his actions is significantly altered by their presence. Some preferences depend on state. Although some users for whom we have sufficient data seem to have preferences that do not depend upon the social state features we constructed for the RL, others do in fact appear to change their preferences depending upon prevailing social conditions. The outline for the rest of the paper is as follows. In Section 2, we give brief background on LambdaMOO. In Section 3, we describe our earlier (non-RL) work on Cobot. Section 4 provides some brief background on RL. In Sections 5, 6 and 7 we describe our implementation of Cobot’s RL action space, reward mechanisms and state features, respectively. Our primary findings are presented in Section 8, and Section 9 offers conclusions. 2 LambdaMOO LambdaMOO, founded in 1990 by Pavel Curtis at Xerox PARC, is the oldest continuously operating MUD, a class of online worlds with roots in text-based multiplayer role-playing games. MUDs (multi-user dungeons) differ from most chat and gaming systems in their use of a persistent representation of a virtual world, often created by the participants, who are represented as characters of their own choosing. LambdaMOO appears as a series of interconnected rooms, populated by users and objects who may move between them. Each room provides a shared chat channel, and typically has an elaborate text description that imbues it with its own “look and feel.” In addition to speech, users express themselves via a large collection of verbs, allowing a rich set of simulated actions, and the expression of emotional states: (1) Buster is overwhelmed by all these deadlines. (2) Buster begins to slowly tear his hair out, one strand at a time. (3) HFh comforts Buster. (4) HFh [to Buster]: Remember, the mighty oak was once a nut like you. (5) Buster [to HFh]: Right, but his personal growth was assured. Thanks anyway, though. (6) Buster feels better now. Lines (1) and (2) are initiated by verb commands by user Buster, expressing his emotional state, while lines (3) and (4) are examples of verbs and speech acts, respectively, by HFh. Lines (5) and (6) are speech and verb acts by Buster. Though there are many standard verbs, such as the use of the verb comfort in line (3) above, the variety is essentially unlimited, as players have the ability to create their own verbs. The rooms and objects in LambdaMOO are created by users themselves, who devise descriptions, and control access by other users. Users can also create objects with verbs that can be invoked by other players. As last count, the database contains 118,154 objects, including 4836 active user accounts. LambdaMOO’s long existence and its user-created nature combine to give it one of the strongest senses of virtual community in the on-line world. Many users have interacted extensively with each other over many years, and users are widely acknowledged for their contribution of interesting objects. LambdaMOO is an attractive environment for experiments in AI (Foner1997; Mauldin1994), including learning. The population is generally curious and technically savvy, and users are interested in automated objects meant to display some form of intelligence. 3 Cobot Cobot is a software agent residing in LambdaMOO. Like a human user, he connects via telnet, and from the point of view of the LambdaMOO server, is a user with all the rights and responsibilities implied. Once actually connected, Cobot wanders into the Living Room, where he spends most of his time. The Living Room is a central public place, frequented both by many regulars, and by users new to LambdaMOO. There are several permanent objects in the Living Room, including a couch with various features and a cuckoo clock. The Living Room usually has between five and twenty users, and is perpetually busy. Over a year, Cobot noted over 2.5 million separate events (about one event every eleven seconds) Previously, we implemented a variety of functionality on Cobot centering around gathering and reporting social statistics. Cobot notes who takes what actions, and on whom. Cobot can answer queries about these statistics, and describe the similarities and differences between users. He also has a rudimentary chatting ability based on the application of information retrieval methods to large documents. He can also search the web to answer specific questions posed to him. A more complete description of Cobot’s abilities, and his early experiences as a social agent in LambdaMOO, can be found in (Isbell et al.2000). Our focus here is to make Cobot proactive—i.e., let him take actions under his own initiative—in a way that is useful, interesting, or pleasing to LambdaMOO users. It is impossible to program rules anticipating when any given action is appropriate in such a complex and dynamic environment, so we applied reinforcement learning to adapt directly from user feedback. We emphasize that Cobot’s original reactive functionality remained on during the RL experiment. Cobot’s persona is largely due to this original functionality, and we felt it was most interesting, and even necessary, to add RL work in this context. Null Action Choose to remain silent for this time period. Topic Change (4) Introduce a conversationaltopic. Cobot declares that he wants to discuss sports or politics, or he utters a sentence from either the sports section or political section of the Boston Globe. Roll Call (2) Initiate a “roll call,” a common word play routine in LambdaMOO. For example, someone who is tired of Monica Lewinsky may emote “TIRED OF LEWINSKY ROLL CALL.” Sympathetic users agree with the roll call. Cobot takes a recent utterance, and extracts either a single noun, or a verb phrase. Social Commentary Make a comment describing the current social state of the Living Room, such as “It sure is quiet” or “Everyone here is friendly.” These statements are based on Cobot’s statistics from recent activity. Several different utterances possible, but they are treated as a single action for RL purposes. Introductions Introduce two users who have not yet interacted in front of Cobot. Table 1: The 9 RL actions available to Cobot. 4 RL Background In RL, problems of decision-making by agents interacting with uncertain environments are usually modeled as Markov decision processes (MDPs). In the MDP framework, at each time step the agent senses the state of the environment, and chooses and executes an action from the set of actions available to it in that state. The agent’s action (and perhaps other uncontrolled external events) cause a stochastic change in the state of the environment. The agent receives a (possibly zero) scalar reward from the environment. The agent’s goal is to choose actions so as to maximize the expected sum of rewards over some time horizon. An optimal policy is a mapping from states to actions that achieves the agent’s goal. Many RL algorithms have been developed for learning good approximations to an optimal policy from the agent’s experience in its environment. At a high level, most algorithms use this experience to learn value functions (or -values) that map state-action pairs to the maximal expected sum of reward that can be achieved starting from that state-action pair. The learned value function is used to choose actions stochastically, so that in each state, actions with higher value are chosen with higher probability. In addition, many RL algorithms use some form of function approximation (parametric representations of complex value functions) both to map state-action features to their values and to map states to distributions over actions (i.e., the policy). See (Sutton and Barto1998) for an extensive introduction to RL. In the next sections, we describe the Cobot’s actions, our choice of state features, and how we dealt with multiple sources of reward. The particular RL algorithm we use is a variant of (Sutton et al.1999)’s policy gradient algorithm. Its details are beyond the scope of this paper; however, see (Shelton2000) for details. One aspect of our RL algorithm that is relevant to understanding our results is that we use a linear function approximator to store our policy. In other words, for each state feature, we maintain a vector of real-valued weights indexed by the possible actions. A positive weight for some action means that the feature increases the probability of taking that action, while a negative weight decreases the probability. The weight’s magnitude determines the strength of this contribution. 5 Cobot’s RL Actions To have any hope of learning to behave in a way interesting to LambdaMOO users, Cobot’s actions must “make sense” to them, fit in with the social chat-based environment, and minimize the risk of causing irritation. Conversation, word play, and emoting routines are among the most common activity in LambdaMOO, so we designed a set of actions along these lines, as detailed in Table 1. Many of these actions extract an utterance from the recent conversations, or from a continually changing external source, such as the online Boston Globe. Thus a single action may cause an infinite variety of behavior by Cobot. At set time intervals (only every few minutes on average, to minimize spam), Cobot selects an action to perform from this set according to a distribution determined by the Q-values in his current state. Any rewards or punishments received before the next RL action are attributed to the current action, and used to update Cobot’s value functions. It is worth remembering that Cobot has two different categories of action: those actions taken proactively as a result of the RL, and those actions taken in response to a user’s action towards Cobot. Some users are certainly aware of the distinction and can easily determine which actions fall into which category, but other users may occasionally reward or punish Cobot in response to a reactive action. Such “erroneous” rewards and punishments act as a source of noise in the training process. 6 The RL Reward Function Cobot learns to behave directly from the feedback of LambdaMOO users, any of whom can reward or punish him. There are both explicit and implicit feedback mechanisms. We implemented explicit reward and punish verbs on Cobot that LambdaMOO users can invoke at any time. These verbs give a numerical (positive and negative, respectively) training signal to Cobot that is the basis of the RL. The signal is attributed as immediate feedback for the current state and RL action, and “backed up” to previous states and actions in accordance with the standard RL algorithms. There are several standard LambdaMOO verbs that are commonly used to express, sometimes playfully, approval or disapproval. Examples of the former include the verb hug, and of the latter the verb spank. In the interest of allowing the RL process to integrate naturally with the LambdaMOO environment, we chose to accept a number of such verbs as implicit reward and punishment signals for Cobot; however, such implicit feedback is numerically weaker than the feedback generated by the explicit mechanisms. One fundamental design choice is whether to learn a single value function for the entire community, or to learn separate value functions for each user based on individual feedback, combining the value functions of those present to determine how to act at each moment. We opted for the latter for three primary reasons. First, it was clear that for learning to have any hope of success, ths system must represent who is present at any given moment—different users simply have different personalities and preferences. We felt that representing which users are present as additional state features would throw away valuable domain information, as the RL would have to discover on its own the primacy of user identity. Having separate reward functions for each user is thus a way of asserting the importance of identity to the learning process. Second, despite the extremely limited number of training examples available in this domain ( per month), learning must be quick and significant. Without a clear sense that their training has some impact on Cobot’s behavior, users will quickly lose interest in providing feedback. A known challenge for RL is the “curse of dimensionality,” (i.e. the size of the state space increases exponentially with the number of state features). By avoiding the need to represent the presence or absence of roughly 250 users, we are able to maintain a fairly small state space and so speed up learning. Third, we (correctly) anticipated the fact that certain users would provide an inordinate amount of training to Cobot, and we did not want the overall policy followed by Cobot to be dominated by the preferences of these individuals. By learning separate policies for each user, and then combining these policies among those users present, we can limit the impact any single user can have on Cobot’s actions. 7 Cobot’s RL State Features The decision to maintain and learn separate value functions for each user means that we can maintain separate state spaces as well, in the hopes of simplifying states and speeding learning. Cobot can be viewed as running a large number of separate RL processes in parallel, with each process having a different state space. The state space for a user contains a number of features containing statistics about that particular user. LambdaMOO is a social environment, and Cobot is learning to take social actions, so we felt that his state features should contain information allowing him to gauge social activity and relationships. Table 2 provides a description of the state features used for RL by Cobot for each user. Even though we have simplified the state space by partitioning by user, the state space for a single user remains sufficiently complex to preclude standard table-based representation of value functions (also, each user’s state space is effectively infinite, as there are real-valued state features). Thus, linear function approximation is used for each user’s policy. Cobot’s RL actions are then chosen according to a mixture of the policies of the users present. We refer the reader to (Shelton2000) for more details on the method by which policies are learned and combined. Social Summary Vector A vector of four numbers: the rate at which the user is producing events; the rate at which events are being produced that are directed at the user; the percentage of the other users present who are among this user’s ten most frequently interacted-with users (“playmates”); and the percentage of the other users present for whom this user is among their top ten playmates. Mood Vector A vector measuring the recent use of eight groups of common verbs (e.g., one group includes verbs grin and smile). Verbs were grouped according to how well their usage was correlated. Rates Vector A vector measuring the rate at which events are produced by those present. Current Room The room where Cobot currently resides. Roll Call Vector Indicates if Cobot’s currently saved roll call text has been used before, if someone has done a roll call since the last time Cobot did, and if there has been a roll call since the last time Cobot grabbed new text. Bias Each user has one feature that is always “on”; that is, this bias is always set to a value of 1. Intuitively, it is the feature indicating the user’s “presence.” Table 2: State space of Cobot. Each user has his own state space and value function; the table thus describes the state space maintained for a generic user. 8 Experimental Procedure and Findings Cobot has been present in LambdaMOO more or less continuously since September, 1999. The RL version of Cobot debuted May 10, 2000. Again, Cobot’s various reactive functionality was left intact for the duration of the RL experiment. Cobot is a working system with real human users, and we wanted to perform the RL experiment in this context. Upon launching the RL functionality publicly in the Living Room, Cobot logged all RL-related data (states visited, actions taken, rewards received from each user, parameters of the value functions, etc.) from May 10 until October 10, 2000. During this time, 63123 RL actions were taken (in addition, of course, to many more reactive non-RL actions), and 3171 reward and punishment events were received from 254 different users. The findings we now summarize are based on these extensive logs: Inappropriatenessof average reward. The most standard and obvious sign of successfulRL would be an increase in the average reward over time. Instead, as shown in Figure 1a, the average cumulative reward received by Cobot actually goes down. However, rather than indicating that users are becoming more dissatisfied as Cobot learns, the decay in reward reveals some peculiarities of human feedback in such an open-ended environment. There are at least two difficulties with average cumulative reward in an environment of human users. The first is that humans are fickle, and their tastes and preferences may drift over time. Indeed, our experiences as users, and with the original reactive functionality of Cobot, suggest that novelty is highly valued in LambdaMOO. Thus a feature that is popular and exciting to users when it is introduced may eventually become an irritant (there are many examples of this phenomenon). In RL terminology, we do not have a fixed, consistent reward function, and thus we are always learning a moving target. While difficult to quantify in such a complex environment, this phenomenon is sufficiently prevalent in LambdaMOO to cast serious doubts on the use of average cumulative reward as the primary measure of performance. The second and related difficulty is that even when users do maintain relatively fixed preferences, they tend to give Cobot less feedback of either type (reward or punishment) as he manages to learn their preferences accurately. Simply put, once Cobot seems to be behaving as they wish, users feel no need to continually provide reward for his “correct” actions or to punish him for the occasional “mistake.” This reward pattern is in contrast to typical RL applications, where there is an automated and indefatigable reward source. Strong empirical evidence for this second phenomenon is provided by User M and User S. These two users were among Cobot’s most dedicated trainers, each had strong preferences for certain actions, and Cobot learned to strongly modify his behavior in their presence to match their preferences. Nevertheless, both users tended to provide less frequent feedback to Cobot as the experiment progressed, as shown in Figure 1a. We conclude that there are serious conceptual difficulties with the use of average cumulative reward in such a human-centric application of RL, and that alternative measures must be investigated, which we do below. A small set of dedicated “parents.” Among the 254 users who gave at least one reward or punishment event to Cobot, 218 gave 20 or fewer, while 15 gave 50 or more. Thus, we found that while many users exhibited a passing interest in training Cobot, there was a small group that was willing to invest nontrivial time and effort in teaching Cobot their preferences. In particular, User M and User S, generated 594 and 69 rewards and punishments events, respectively. By “event”, we simply mean an RL action that received some feedback. The actual absolute User O Roll Call. User O appears to especially dislike roll call actions when there have been repeated roll calls and/or Cobot is repeating the same roll calls. Rates. The overall rate of events being generating has slightly more relevance than that of the rate of events being generated just by User O. User B Social Summary. User B is effected by the presence of his friends. Not shown here are other Social Summary features (deviating about 6 degrees). It appears that User B is more likely to ignore Cobot when he is with many friends. User C Roll Call. User C appears to have strong preferences about Cobot’s behavior when a “roll call party” is in progress (i.e., everyone is generating roll calls). User P Room. User P would follow Cobot to his home, where he is generally alone, and has trained him there. He appears to have different preferences for Cobot under those circumstances. Table 3: Relevant features for users with non-uniform policies. Several of our top users had some features that deviated from their bias feature. The second column indicates the number of degrees between the weight vectors for those features and the weight vectors for the bias feature. We have only included features that deviated by more than 10 degrees. For the users above the double line, we have included only features whose weights had a length greater than 0.2. Each of these users had bias weights of length greater than 1. For those below the line, we have included only features with a length greater than 0.1 (these all had bias weights of length much less than 1). Some parents have strong opinions. For the vast majority of users who participated in the RL training of Cobot, the policy learned was quite close to the uniform distribution. Quantification of this statement is somewhat complex, since policies are dependent on state. However, we observed that for most users the learned policy’s dependence on state was weak, and the resulting distribution near uniform (though there are interesting and notable exceptions, as we shall see below). This result is perhaps to be expected: most users provided too little feedback for Cobot to detect strong preferences, and may not have been exhibiting strong and consistent preferences in the feedback they did provide. However, there was again a small group of users for whom a highly non-uniform policy was learned. In particular, for Users M and S mentioned above, the resulting policies were relatively independent of state and their entropies were 0.03 and 1.93, respectively. (The entropy of the uniform distribution over the actions is 2.2.) Several other users also exhibited less dramatic but still non-uniform distributions. User M seemed to have a strong preference for roll call actions, with the learned policy selecting these with probability 0.99, while User S preferred social commentary actions, with the learned policy giving them probability 0.38. (Each action in the uniform distribution is given weight 1/9 = 0.11.) Cobot learns matching policies. In Figure 1b, we demonstrate that the policy learned by Cobot for User M does in fact reflect the empirical pattern of rewards received over time. Similar results obtain for User S, not shown here. Thus, repeated feedback given to Cobot for a non-uniform set of preferences clearly pays off in a corresponding policy. Cobot responds to his dedicated parents. The policies learned by Cobot for users can have strong impact on the empirical distribution of actions he actually ends up taking in LambdaMOO. For User M, we find that his presence causes a significant shift towards his preferences. In other words, Cobot does his best to “please” these dedicated trainers whenever they arrive in the Living Room, and returns to a more uniform policy upon their departure. Some preferences depend on state. Finally, we show that the policies learned by Cobot sometimes depend upon the features Cobot maintains in his state. We use two facts about the RL weights (described in Section 4) maintained by Cobot to determine which features are relevant for a given user. First, we note that by construction, the RL weights learned for the bias feature described in Table 2 representthe user’s preferences independentof state (since this feature is always on whenever the user is present). Second, we note that because we initialized all weights to 0, only features with non-zero weights will contribute to the policy that Cobot uses. Thus, we can determine that a feature is relevant for a user if that feature’s weight vector is far from that user’s bias feature weight vector, and from the all-zero vector. For our purposes, we have used (1) the normalized inner product (the cosine of the angle between two vectors) as a measure of a feature’s distance from the bias feature, and (2) a feature’s weight vector length to determine if it is away from zero. Thesemeasures show that for most users, Cobot learned a state-independent policy (e.g., User M prefers roll calls); however, as we can see in Table 3, Cobot has learned a policy for some users that depends upon state. numerical reward received may be larger or smaller than 1 at any time, as implicit rewards provide fractional reward, and the user may repeatedly reward or punish an action, with the feedback being summed. For example, the total absolute value of rewards and punishments provided by User M was 607.63 over 594 feedback events. 0 1 2 3 4 5 6 x 10 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Average Cumulative Reward per Timestep Time Reward reward all users abs reward all users reward user M abs reward user M reward user S abs reward user S 1 2 3 4 5 6 7 8 9 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Rewards / Policy / Empirical Distribution Comparision for User M action change rewards policy empirical Figure 1: a) Average Cumulative Reward Over Time. b) Rewards received, policy learned, and effect on actions for User M. Figure a) shows that average cumulative reward decreases over time, for both total and absolute reward; however, Figure b shows that proper learning is taking place. For each of the RL actions, three quantities are shown. The blue bars (left) show the average reward given by User M for each action (the average reward given by User M across all actions has been subtracted off to indicate relative preferences). The yellow bars (middle) show the policy learned by Cobot for User M (the probability assigned to each action in the uniform distribution (1/9) has been subtracted off). The red bars (right) show the empirical frequency with which each action was taken in the presence of User M (minus the empirical frequency with which that action was taken by Cobot over all time steps). These bars indicate the extent to which the presence of User M biases Cobot’s behavior towards M’s preferences. We see that the policy learned by Cobot for User M aligns nicely with the preferences expressed by M and that Cobot’s behavior shifts strongly towards the learned policy for User M wheneverM is present. To go beyond a qualitative visual analysis, we have defined a metric that measures the extent to which two rankings of actions agree, while taking into account that some actions are extremely close in the each ranking. The details are beyond the scope of the paper, but the agreement between the action rankings shown here are in near-perfect agreement by this measure. Similar results obtain for User S. 9 Conclusions We have reported on our efforts to apply reinforcement learning in a complex human online social environment where many of the standard assumptions (stationary rewards, Markovian behavior, appropriateness of average reward) are clearly violated. We feel that the results obtained with Cobot so far are compelling, and offer promise for the application of RL in such open-ended social settings. Cobot continues to take RL actions and receive rewards and punishments from LambdaMOO users, and we plan to continue and embellish this work as part of our overall efforts on Cobot. References Foner, L. (1997). Entertaining Agents: a Sociological Case Study. In Proceedings of the First International Conference on Autonomous Agents. Isbell, C. L., Kearns, M., Kormann, D., Singh, S., and Stone, P. (2000). Cobot in LambdaMOO: A Social Statistics Agent. To appear in Proceedings of AAAI-2000. Mauldin, M. (1994). Chatterbots, TinyMUDs, and the Turing Test: Entering the Loebner Prize Competition. In Proceedings of the Twelfth National Conference on Artificial Intelligence. Shelton, C. R. (2000). Balancing Multiple Sources of Reward in Reinforcement Learning. Submitted for publication in Neural Information Processing Systems-2000. Singh, S., Kearns, M., Littman, D., and Walker, M. (2000). Empirical Evaluation of a Reinforcement Learning Dialogue System. To appear in Proceedings of AAAI-2000. Sutton, R. S. and Barto, A. G. (1998). Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA. Sutton, R. S., McAllester, D., Singh, S., and Mansour, Y. (1999). 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Semi-Supervised MarginBoost F. d'Alche-Buc LIP6,UMR CNRS 7606, Universite P. et M. Curie 75252 Paris Cedex, France florence. dAlche@lip6.fr Yves Grandvalet Heudiasyc, UMR CNRS 6599, Universite de Technologie de Compiegne, BP 20.529, 60205 Compiegne cedex, France Yves. Grandvalet@hds.utc.fr Christophe Ambroise Heudiasyc, UMR CNRS 6599, Universite de Technologie de Compiegne, BP 20.529, 60205 Compiegne cedex, France Christophe A mbroise@hds.utc.fr Abstract In many discrimination problems a large amount of data is available but only a few of them are labeled. This provides a strong motivation to improve or develop methods for semi-supervised learning. In this paper, boosting is generalized to this task within the optimization framework of MarginBoost . We extend the margin definition to unlabeled data and develop the gradient descent algorithm that corresponds to the resulting margin cost function. This meta-learning scheme can be applied to any base classifier able to benefit from unlabeled data. We propose here to apply it to mixture models trained with an Expectation-Maximization algorithm. Promising results are presented on benchmarks with different rates of labeled data. 1 Introduction In semi-supervised classification tasks, a concept is to be learnt using both labeled and unlabeled examples. Such problems arise frequently in data-mining where the cost of the labeling process can be prohibitive because it requires human help as in video-indexing, text-categorization [12] and medical diagnosis. While some works proposed different methods [16] to learn mixture models [12], [1], SVM [3], cotrained machines [5] to solve this task, no extension has been developed so far for ensemble methods such as boosting [7, 6]. Boosting consists in building sequentially a linear combination of base classifiers that focus on the difficult examples. For AdaBoost and extensions such as MarginBoost [10], this stage-wise procedure corresponds to a gradient descent of a cost functional based on a decreasing function of the margin, in the space of linear combinations of base classifiers. We propose to generalize boosting to semi-supervised learning within the framework of optimization. We extend the margin notion to unlabeled data, derive the corresponding criterion to be maximized, and propose the resulting algorithm called Semi-Supervised MarginBoost (SSMBoost). This new method enhances our previous work [9] based on a direct plug-in extension of AdaBoost in the sense that all the ingredients of the gradient algorithm such as the gradient direction and the stopping rule are defined from the expression of the new cost function. Moreover, while the algorithm has been tested using the mixtures of models [1], 55MBoost is designed to combine any base classifiers that deals with both labeled and unlabeled data. The paper begins with a brief presentation of MarginBoost (section 2). Then, in section 3, the 55MBoost algorithm is presented. Experimental results are discussed in section 5 and we conclude in section 6. 2 Boosting with MarginBoost Boosting [7, 6, 15] aims at improving the performance of any weak "base classifier" by linear combination. We focus here on normalized ensemble classifiers gt E LinCH) whose normalized1 coefficients are noted aT = I ~: I and each base classifier with outputs in [-1, 1] is hT E 1{: t gt(x) = L aThT(x) (1) T=l Different contributions [13, 14],[8], [10] have described boosting within an optimization scheme, considering that it carries out a gradient descent in the space of linear combinations of base functions. We have chosen the MarginBoost algorithm, a variant of a more general algorithm called Any Boost [10], that generalizes AdaBoost and formally justifies the interpretation in terms of margin. If S is the training sample {(Xi,Yi) ,i = l..l}, MarginBoost, described in Fig. 1, minimizes the cost functional C defined for any scalar decreasing function c of the margin p : I C(gt) = L c(p(gt(Xi), Yi))) (2) i=l Instead of taking exactly ht+l = - \1C(gt) which does not ensure that the resulting function gt+! belongs to Lin(1{), ht+! is chosen such as the inner product2 - < \1C(gt), ht+l > is maximal. The equivalent weighted cost function to be maximized can thus be expressed as : JF = L Wt(i)Yiht+! (Xi) iES 3 Generalizing MarginBoost to semi-supervised classification 3.1 Margin Extension (3) For labeled data, the margin measures the quality of the classifier output. When no label is observed, the usual margin cannot be calculated and has to be estimated. A first estimation could be derived from the expected margin EypL(gt(X) , y). We can use the output of the classifier (gt(x) + 1)/2 as an estimate of the posterior probability P(Y = +llx). This leads to the following margin pi; which depends on the input and is linked with the response of the classifier: lOr> 0 and L1 norm is used for normalization: IOrl = L~=l Or 2< f, 9 >= LiE S f(X;)g(Xi) Let wo(i) = l/l, i = 1, ... ,l. Let go(x) = 0 For t = 1 ... T (do the gradient descent): 1. Learn a gradient direction htH E 1i with a high value of J{ = L,iEswt(i)YihtH(Xi) 2. Apply the stopping rule: if J{ ::::: L,iES Wt(i)Yigt(Xi) then return gt else go on. 3. Choose a step-length for the obtained direction by a line-search or by fixing it as a constant f 4 Add the new direction to obtain 9 = (l a t I9t+a ttlhtt') . HI lattl l 5. Fix the weight distribution: Wt 1 = c'(p(9ttl(Xi),Yi)) + 2: jE S c'(p(9ttl(Xj),Yj)) Figure 1: MarginBoost algorithm (with L1 normalization of the combination coefficients) Another way of defining the extended margin is to use directly the maximum a posteriori estimate of the true margin. This MAP estimate depends on the sign of the classifier output and provides the following margin definition pC; : (5) 3.2 Semi-Supervised MarginBoost : generalization of marginBoost to deal with unlabeled data The generalization of the margin can be used to define an appropriate cost functional for the semi-supervised learning task. Considering that the training sample S is now divided into two disjoint subsets L for labeled data and U for unlabeled data, the cost falls into two parts involving PL = P and PU: (6) iEL iEU The maximization of - < \lC(gt), htH > is equivalent to optimize the new quantity JtS that falls now into two terms J{ = Jf + J? The first term one can be directly obtained from equation (3) : Jf = LWt(i).YihtH(Xi) (7) iEL The second term, J?, can be expressed as following: (8) with the weight distribution Wt now defined as : { c'(pL(9t(Xi),Yi)) ( .) _ IWt l Wt z c'(PU(9t(Xi))) IWt l if i E L .. with IWt I = 2:= Wt (i) If z E U iES (9) This expression of JP comes directly from differential calculus and the chosen inner product: ( )() { YiC'(Pd9t(Xi),Yi)) if x = Xi and i E L 'VC gt Xi = c'(p (g (x.))) apU(9t(Xi)) if x = x, and i E U U t t a9t(Xi) 0 (10) Implementation of 55MBoost with margins pI[; and Pu requires their derivatives. Let us notice that the "signed margin", pus, is not derivable at point O. However, according to the results of convex analysis (see for instance [2]), it is possible to define the "sub derivative' of Pus since it is a continuous and convex function. The value of the sub derivative corresponds here to the average value of the right and left derivatives. apUS(gt(Xi)) = {sign(g(Xi)) agt (Xi) 0 if X :f": 0 if x = 0 And, for the "squared margin" Pu9 , we have: apu 9 (gt(Xi)) = 2g(Xi) agt(Xi) (11) (12) This completes the set of ingredients that must be incorporated into the algorithm of Fig. 1 to obtain 55MBoost. 4 Base Classifier The base classifier should be able to make use of the unlabeled data provided by the boosting algorithm. Mixture models are well suited for this purpose, as shown by their extensive use in clustering. Hierarchical mixtures provide flexible discrimination tools, where each conditional distribution f(xlY = k) is modelled by a mixture of components [4]. At the high level, the distribution is described by K f(x; if» = 2:= Pk!k (x; Ok) , (13) k=l where K is the number of classes, Pk are the mixing proportions, Ok the conditional distribution parameters, and if> denotes all parameters {Pk; 0df=l. The high-level description can also be expressed as a low-level mixture of components, as shown here for binary classification: Kl K2 f(x;if» = 2:= PkJkl(X;Okl) + 2:= Pk2!k2(X;Ok2) (14) With this setting, the EM algorithm is used to maximize the log-likelihood with respect to if> considering the incomplete data is {Xi, Yi}~= l and the missing data is the component label Cik, k = 1, ... , K 1 + K2 [11]. An original implementation of EM based on the concept of possible labels [1] is considered here. It is well adapted to hierarchical mixtures, where the class label Y provides a subset of possible components. When Y = 1 the first Kl modes are possible, when Y = -1 the last K2 modes are possible, and when an example is unlabeled, all modes are possible. A binary vector Zi E {0,1}(Kl+K2) indicates the components from which feature vector Xi may have been generated, in agreement with the assumed mixture model and the (absence of) label Yi. Assuming that the training sample {Xi, Zi }i=l is i.i.d, the weighted log-likelihood is given by I L(<I>;{Xi,zdi=l = LWt(i) log (j(Xi,zi;<I») , (15) i=l where Wt(i) are provided by boosting at step t. L is maximized using the following EM algorithm: E-Step Compute the expectation of L( <I>; {Xi, zdi=l) conditionally to {Xi, zdi=l and the current value of <I> (denoted <I>q): with Uik I Kl+K2 L L Wt(i)Uik log (Pk!k(Xi; Ok)) i=l k=l ZikPk!k(Xi; Ok) L£ ZUP£!£(Xi; O£) M-Step Maximize Q(<I>I<I>q) with respect to <I>. (16) Assuming that each mode k follows a normal distribution with mean ILk' and covariance ~k ' <I>q+l = {ILk+! ; ~k+!;Pk+l}f~iK2 is given by: (17) (18) 5 Experimental results Tests of the algorithm are performed on three benchmarks of the boosting literature: twonorm and ringnorm [6] and banana [13]. Information about these datasets and the results obtained in discrimination are available at www.first.gmd.de/-raetsch/ 10 different samples were used for each experiment. We first study the behavior of 55MBoost according the evolution of the test error with increasing rates of unlabeled data (table 1). We consider five different settings where 0%, 50%, 75%, 90% and 95% of labels are missing. 55MB is tested for the margins P~ and Pu with c(x) = exp( -x). It is compared to mixture models and AdaBoost. 55MBoost and AdaBoost are trained identically, the only difference being that AdaBoost is not provided with missing labels. Both algorithms are run for T = 100 boosting steps, without special care of overfitting. The base classifier (called here base(EM)) is a hierarchical mixture model with an arbitrary choice of 4 modes per class but the algorithm (which may be stalled in local minima) is restarted 100 times from different initial solutions, and the best final solution (regarding training error rate) is selected. We report mean error rates together with the lower and upper quartiles in table 1. For sake of space, we did not display the results obtained without missing labels: in this case, AdaBoost and 55MBoost behave nearly identically and better than EM only for Banana. For rates of unlabeled data inferior to 95%, 55MBoost beats slightly AdaBoost for Ringnorm and Twonorm (except for 75%) but is not able to do as well as Table 1: Mean error rates (in %) and interquartiles obtained with 4 different percentages of unlabeled data for mixture models base(EM), AdaBoost and 55MBoost. Ringnorm 50% 75% 90% 95% base(EM) 2.1 [ 1.7, 2.1] 4.3[ 1.9, 5.7] 9.5 [ 2.7,12.0] 23.7[14.5,27.0] AdaBoost 1.8[ 1.6, 2.0] 3.1[ 1.9, 4.1] 11.5[ 4.2,12.1] 28.7[11.5,37.6] 55MBoost pS 1. 7[ 1.5, 1.8] 2.0 [ 1.5, 2.4] 3.7[ 2.1, 4.8] 6.9[ 5.6,10.7] 55MBoost pg 1. 7[ 1.6, 1.8] 2.O[ 1.4, 2.5] 4.5 [ 2.2, 3.6] 8.1 [ 4.2, 9.0] Twonorm 50% 75% 90% 95% base(EM) 3.2[ 2.7, 3.1] 6.5[ 3.0, 9.0] 20.6[10.3,22.5] 24.8[18.3,31.9] AdaBoost 3.2[ 2.9, 3.2] 3.2[ 3.0, 3.5] 11.0[ 5.2,14.2] 38.9[29.4,50.0] 55MBoost pS 2.7[ 2.5, 2.9] 3.4[ 2.8, 4.3] 10.1 [ 5.8,13.6] 20.4[11.9,32.3] 55MBoost pg 2.7[ 2.5, 2.8] 3.4[ 2.8, 4.2] 11.0[ 5.6,16.2] 21.1 [12.5,30.8] Banana 50% 75% 90% 95% base(EM) 18.2[16.7,18.6] 21.8[18.0,25.0] 26.1[20.7,29.8] 31.7[23.8,35.8] AdaBoost 12.6[11.7,13.1] 15.2 [13.0,16.8] 22.1 [18.0,24.3] 37.5 [32.2,42.2] 55MBoost pS 13.3 [12.7,14.3] 17.0[15.3,17.8] 22.2[18.0,28.0] 28.3 [20.2,35.2] 55MBoost pg 13.3[12.8,14.2] 16.9[15.6,17.8] 22.8[18.3,29.3] 28.6[21.5,34.2] AdaBoost on Banana data. One possible explanation is that the discrimination frontiers involved in the banana problem are so complex that the labels really bring crucial informations and thus adding unlabeled data does not help in such a case. Nevertheless, at rate 95% which is the most realistic situation, the margin Pu obtains the minimal error rate for each of the three problems. It shows that it is worth boosting and using unlabeled data. As there is no great difference between the two proposed margins, we conducted further experiments using only the Pu' Second, in order to study the relation between the presence of noise in the dataset and the ability of 55MBoost to enhance generalization performance, we draw in Fig. 2, the test errors obtained for problems with different values of Bayes error when varying the rate of labeled examples. We see that even for difficult tasks (very noisy problems), the degradation in performance for large subsets of unlabeled data is still low. This reflects some consistency in the behavior of our algorithm. Third, we test the sensibility of 55MBoost to overfitting. Overfitting can usually be avoided by techniques such as early stopping, softenizing of the margin ([13], [14]) or using an adequate margin function such as 1 - tanh(p) instead of exp( -p) [10]. Here we keep using c = exp and ran 55MBoost with a maximal number of step T = 1000 with 95% of unlabeled data. Of course, this does not correspond to a realistic use of boosting in practice but it allows to check if the algorithm behaves consistently in terms of gradient steps number. It is remarkable that no overfitting is observed and in the Twonorm case (see Fig. 3), the test error still decreases ! We also observe that the standard error deviation is reduced at the end of the process. For the banana problem (see Fig. 3 b.), we observe a stabilization near the step t = 100. A massive presence of unlabeled data implies thus a regularizing effect. 50 40 20 10 Bayes error:;;; 2.3% Bayes error:;;; 15.7% Bayes error:;;; 3 1.2% °0L---~,7 0 --~207---~~7----4~0--~5~0--~6=0--~7=0----8=0----9=0~~ '00 Rate of missing labels (%) Figure 2: Consistency of the 55MBoost behavior: evolution of test error versus the missing labels rate with respect to various Bayes error (twonorm ). 70 60' , , I \ "' \ " Mean (Error Test) +/- 1 std Mean (Error test) -'-",---.- - --/----oL-~ __ ~ __ ~ __ _L __ _L __ ~ __ ~ __ ~ __ L_~ o ~ ~ _ ~ ~ ~ ~ ~ ~ _ Steptofgradientdescent(boosting process} 70 60 ~ 50 i \!) 0: 40" , \ 10 I ~ Mean of Error Test +/- std Mean of Error test ~_~~ __ -r~/_ ~ ~",~~,.~. '-. -.I~" ~~-I °OL-~'OO~-2~OO~~3~OO--~400~~500~-=~~~7~OO~~8=OO--~~~~'~ Step t of gradient descent Figure 3: Evolution of Test error with respect to maximal number T of iterations with 95% of missing labels (Two norm and Banana). 6 Conclusion MarginBoost algorithm has been extended to deal with both labeled and unlabeled data. Results obtained on three classical benchmarks of boosting litterature show that it is worth using additional information conveyed by the patterns alone. No overfitting was observed during processing 55MBoost on the benchmarks when 95% of the labels are missing: this should mean that the unlabeled data should playa regularizing role in the ensemble classifier during the boosting process. After applying this method to a large real dataset such as those of text-categorization, our future works on this theme will concern the use of the extended margin cost function on the base classifiers itself like multilayered perceptrons or decision trees. Another approach could also be conducted from the more general framework of AnyBoost that optimize any differential cost function. References [1] C. Ambroise and G. Govaert. EM algorithm for partially known labels. In IFCS 2000, july 2000. [2] J.-P. Aubin. L 'analyse non lineaire et ses applications d l'economie. Masson, 1984. [3] K P. Bennett and A. Demiriz. Semi-supervised support vector machines. In D. Cohn, M. Kearns, and S. Solla, editors, Advances in Neural Information Processing Systems, pages 368-374. MIT Press, 1999. [4] C.M. Bishop and M.E. Tipping. A hierarchical latent variable model for data vizualization. IEEE PAMI, 20:281- 293, 1998. [5] A. Blum and Tom Mitchell. Combining labeled and unlabeled data with co-training. In Proceedings of the 1998 Conference on Computational Learning Theory, July 1998. [6] L. Breiman. Prediction games and arcing algorithms. Technical Report 504, Statistics Department, University of California at Berkeley, 1997. [7] Y. Freund and R. E. Schapire. Experiments with a new boosting algorithm. In Machine Learning: Proceedings of the Thirteenth International Conference, pages 148- 156. Morgan Kauffman, 1996. [8] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of boosting. The Annals of Statistics, 28(2):337- 407, 2000. [9] Y. Grandvalet, F. d'Alche Buc, and C. Ambroise. Boosting mixture models for semisupervised learning. In ICANN 2001, august 200l. [10] L. Mason, J. Baxter, P. L. Bartlett, and M. Frean. Functional gradient techniques for combining hypotheses. In Advances in Large Margin Classifiers. MIT, 2000. [11] G.J. McLachlan and T. Krishnan. The EM algorithm and extensions. Wiley, 1997. [12] K Nigam, A. K McCallum, S. Thrun, and T. Mitchell. Text classification from labeled and unlabeled documents using EM. Machine learning, 39(2/3):135- 167, 2000. [13] G. Riitsch, T. Onoda, and K-R. Muller. Soft margins for AdaBoost. Technical report, Department of Computer Science, Royal Holloway, London, 1998. [14] G. Riitsch, T. Onoda, and K-R. Muller. Soft margins for AdaBoost. Machine Learning, 42(3):287- 320, 200l. [15] R. E. Schapire, Y. Freund, P. Bartlett, and W. S. Lee. Boosting the margin: A new explanation for the effectiveness of voting methods. Th e Annals of Statistics, 26(5):1651- 1686, 1998. [16] Matthias Seeger. Learning with labeled and unlabeled data,www.citeseer.nj.nec.com/seegerOllearning.html.
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ALGONQUIN - Learning dynamic noise models from noisy speech for robust speech recognition Brendan J. Freyl, Trausti T. Kristjanssonl , Li Deng2 , Alex Acero2 1 Probabilistic and Statistical Inference Group, University of Toronto http://www.psi.toronto.edu 2 Speech Technology Group, Microsoft Research Abstract A challenging, unsolved problem in the speech recognition community is recognizing speech signals that are corrupted by loud, highly nonstationary noise. One approach to noisy speech recognition is to automatically remove the noise from the cepstrum sequence before feeding it in to a clean speech recognizer. In previous work published in Eurospeech, we showed how a probability model trained on clean speech and a separate probability model trained on noise could be combined for the purpose of estimating the noisefree speech from the noisy speech. We showed how an iterative 2nd order vector Taylor series approximation could be used for probabilistic inference in this model. In many circumstances, it is not possible to obtain examples of noise without speech. Noise statistics may change significantly during an utterance, so that speechfree frames are not sufficient for estimating the noise model. In this paper, we show how the noise model can be learned even when the data contains speech. In particular, the noise model can be learned from the test utterance and then used to de noise the test utterance. The approximate inference technique is used as an approximate E step in a generalized EM algorithm that learns the parameters of the noise model from a test utterance. For both Wall Street Journal data with added noise samples and the Aurora benchmark, we show that the new noise adaptive technique performs as well as or significantly better than the non-adaptive algorithm, without the need for a separate training set of noise examples. 1 Introduction Two main approaches to robust speech recognition include "recognizer domain approaches" (Varga and Moore 1990; Gales and Young 1996), where the acoustic recognition model is modified or retrained to recognize noisy, distorted speech, and "feature domain approaches" (Boll 1979; Deng et al. 2000; Attias et al. 2001; Frey et al. 2001), where the features of noisy, distorted speech are first denoised and then fed into a speech recognition system whose acoustic recognition model is trained on clean speech. One advantage of the feature domain approach over the recognizer domain approach is that the speech modeling part of the denoising model can have much lower complexity than the full acoustic recognition model. This can lead to a much faster overall system, since the denoising process uses probabilistic inference in a much smaller model. Also, since the complexity of the denoising model is much lower than the complexity of the recognizer, the denoising model can be adapted to new environments more easily, or a variety of denoising models can be stored and applied as needed. We model the log-spectra of clean speech, noise, and channel impulse response function using mixtures of Gaussians. (In contrast, Attias et al. (2001) model autoregressive coefficients.) The relationship between these log-spectra and the log-spectrum of the noisy speech is nonlinear, leading to a posterior distribution over the clean speech that is a mixture of non-Gaussian distributions. We show how a variational technique that makes use of an iterative 2nd order vector Taylor series approximation can be used to infer the clean speech and compute sufficient statistics for a generalized EM algorithm that can learn the noise model from noisy speech. Our method, called ALGONQUIN, improves on previous work using the vector Taylor series approximation (Moreno 1996) by modeling the variance of the noise and channel instead of using point estimates, by modeling the noise and channel as a mixture mixture model instead of a single component model, by iterating Laplace's method to track the clean speech instead of applying it once at the model centers, by accounting for the error in the nonlinear relationship between the log-spectra, and by learning the noise model from noisy speech. 2 ALGONQUIN's Probability Model For clarity, we present a version of ALGONQUIN that treats frames of log-spectra independently. The extension of the version presented here to HMM models of speech, noise and channel distortion is analogous to the extension of a mixture of Gaussians to an HMM with Gaussian outputs. Following (Moreno 1996), we derive an approximate relationship between the log spectra of the clean speech, noise, channel and noisy speech. Assuming additive noise and linear channel distortion, the windowed FFT Y(j) for a particular frame (25 ms duration, spaced at 10 ms intervals) of noisy speech is related to the FFTs of the channel H(j), clean speech 5(j) and additive noise N(j) by Y(j) = H(j)5(j) + N(j). (1) We use a mel-frequency scale, in which case this relationship is only approximate. However, it is quite accurate if the channel frequency response is roughly constant across each mel-frequency filter band. For brevity, we will assume H(j) = 1 in the remainder of this paper. Assuming there is no channel distortion simplifies the description of the algorithm. To see how channel distortion can be accounted for in a nonadaptive way, see (Frey et al. 2001). The technique described in this paper for adapting the noise model can be extended to adapting the channel model. Assuming H(j) = 1, the energy spectrum is obtained as follows: IY(j)12 = Y(j)*Y(j) = 5(j)* 5(j) + N(j)* N(j) + 2Re(N(j)* 5(j)) = 15(j)12 + IN(j)12 + 2Re(N(j)* 5(j)) , where "*,, denotes complex conjugate. If the phase of the noise and the speech are uncorrelated, the last term in the above expression is small and we can approximate the energy spectrum as follows: IYUW ~ ISUW + INUW· (2) Although we could model these spectra directly, they are constrained to be nonnegative. To make density modeling easier, we model the log-spectrum instead. An additional benefit to this approach is that channel distortion is an additive effect in the log-spectrum domain. Letting y be the vector containing the log-spectrum log IY(:W, and similarly for s and n , we can rewrite (2) as exp(y) ~ exp(s) + exp(n) = exp(s) 0 (1 + exp(n - s)), where the expO function operates in an element-wise fashion on its vector argument and the "0" symbol indicates element-wise product. Taking the logarithm, we obtain a function gO that is an approximate mapping of sand n to y (see (Moreno 1996) for more details): y ~ g([~]) = s + In(l + exp(n - s)). (4) "T" indicates matrix transpose and InO and expO operate on the individual elements of their vector arguments. Assuming the errors in the above approximation are Gaussian, the observation likelihood is p(yls,n) =N(y;g([~]),W), (5) where W is the diagonal covariance matrix of the errors. A more precise approximation to the observation likelihood can be obtained by writing W as a function of s and n , but we assume W is constant for clarity. Using a prior p(s, n), the goal of de noising is to infer the log-spectrum of the clean speech s, given the log-spectrum ofthe noisy speech y. The minimum squared error estimate of sis s = Is sp(sly), where p(sly) ex InP(yls, n)p(s, n). This inference is made difficult by the fact that the nonlinearity g([s n]T) in (5) makes the posterior non-Gaussian even if the prior is Gaussian. In the next section, we show how an iterative variational method that uses a 2nd order vector Taylor series approximation can be used for approximate inference and learning. We assume that a priori the speech and noise are independent p(s, n) = p(s)p(n) and we model each using a separate mixture of Gaussians. cS = 1, ... , NS is the class index for the clean speech and en = 1, ... ,Nn is the class index for the noise. The mixing proportions and Gaussian components are parameterized as follows: p(s) = LP(cS)p(slcS), p(CS) =7r~s , p(slcS) =N(s;JL~ s ,~~ s ), C S We assume the covariance matrices ~~ s and ~~n are diagonal. Combining (5) and (6), the joint distribution over the noisy speech, clean speech class, clean speech vector, noise class and noise vector is p(y, s, cs, n , en) = N(y; g([~]), w)7r~s N(s; JL~ s , ~~s )7r~N(n; JL~n , ~~n). (7) Under this joint distribution, the posterior p(s, nly) is a mixture of non-Gaussian distributions. In fact, for a given speech class and noise class, the posterior p(s, nics, en, y) may have multiple modes. So, exact computation of s is intractable and we use an approximation. 3 Approximating the Posterior For the current frame of noisy speech y, ALGONQUIN approximates the posterior using a simpler, parameterized distribution, q: p(s,cS, n,cnly) ~ q(s,cS,n,cn). The "variational parameters" of q are adjusted to make this approximation accurate, and then q is used as a surrogate for the true posterior when computing § and learning the noise model (c.f. (Jordan et al. 1998)). For each cS and en, we approximate p(s, nics, en, y) by a Gaussian, (9) where 1J~'en and 1J~'en are the approximate posterior means of the speech and noise for classes cS and en, and <P ~~en, <P~.r;,n and <P~::'en specify the covariance matrix for the speech and noise for classes cS and en. Since rows of vectors in (4) do not interact and since the likelihood covariance matrix q, and the prior covariance matrices ~ ~. and ~~n are diagonal, the matrices <P~~ en, <P~.r;,n and <P~::'en are diagonal. The posterior mixing proportions for classes cS and en are q( cS , en) = Pc' en. The approximate posterior is given by q(s,n,cs,cn) = q(s,nlcs,cn)q(cS, en). The goal of variational inference is to minimize the relative entropy (KullbackLeibler divergence) between q and p: K "''''11 ( S n) q(s,n ,cS,cn) = ~ ~ q s, n , c ,c In ( S n I ). c' en s n P s, c , n , c y This is a particularly good choice for a cost function, because, since lnp(y) doesn't depend on the variational parameters, minimizing K is equivalent to maximizing () K "''''11 ( S n) p(s,cS,n,cn,y) F = lnp y = ~ ~ q s, n , c ,c In ( S n) , e' en s n q s, n, c ,c which is a lower bound on the log-probability of the data. So, variational inference can be used as a generalized E step (Neal and Hinton 1998) in an algorithm that alternatively maximizes a lower bound on lnp(y) with respect to the variational parameters and the noise model parameters, as described in the next section. Variational inference begins by optimizing the means and variances in (9) for each CS and en. Initially, we set the posterior means and variances to the prior means and variances. F does not have a simple form in these variational parameters. So, at each iteration, we make a 2nd order vector Taylor series approximation of the likelihood, centered at the current variational parameters, and maximize the resulting approximation to F. The updates are where g' 0 is a matrix of derivatives whose rows correspond to the noisy speech y and whose columns correspond to the clean speech and noise [s n]. The inverse posterior covariance matrix is the sum of the inverse prior covariance matrix and the inverse likelihood covariance matrix, modified by the Jacobian g' 0 for the mapping from s and n to y The posterior means are moved towards the prior means and toward values that match the observation y. These two effects are weighted by the inverse prior covariance matrix and the inverse likelihood covariance matrix. After iterating the above updates (in our experiments, 3 to 5 times) for each eS and en, the posterior mixing proportions that maximize :F are computed: where A is a normalizing constant that is computed so that L e.en Pe'en = 1. The minimum squared error estimate of the clean speech, s, is We apply this algorithm on a frame-by-frame basis, until all frames in the test utterance have been denoised. 4 Speed Since elements of s, nand y that are in different rows do not interact in (4), the above matrix algebra reduces to efficient scalar algebra. For 256 speech components, 4 noise components and 3 iterations of inference, our unoptimized C code takes 60 ms to denoise each frame. We are confident that this time can be reduced by an order of magnitude using standard implementation tricks. 5 Adapting the Noise Model Using Noisy Speech The version of ALGONQUIN described above requires that a mixture model of the noise be trained on noise samples, before the log-spectrum of the noisy speech can be denoised. Here, we describe how the iterative inference technique can be used as the E step in a generalized EM algorithm for learning the noise model from noisy speech. For a set of frames y(1), . .. , yeT) in a noisy test utterance, we construct a total bound :F = L:F(t) :::; Llnp(y(t)). t t The generalized EM algorithm alternates between updating one set of variational (t) n(t) t £ h f tIT d ... T· h parameters Pe.en, 11 e'en, e c. or eac rame = , ... , ,an maximizIng.r WIt respect to the noise model parameters 7r~n, J.t~n and ~~n. Since:F:::; Ltlnp(y(t)), this procedure maximizes a lower bound on the log-probability of the data. The use of the vector Taylor series approximations leads to an algorithm that maximizes an approximation to a lower bound on the log-probability of the data. Restaurant Street Airport Station Average 20 dB 2.12 2.96 1.82 1.73 2.16 15 dB 3.87 4.78 2.27 3.24 3.54 10 dB 9.18 10.73 5.49 6.48 7.97 5 dB 20.51 13.52 14.97 15.18 18.54 o dB 47.04 45.68 36.00 37.24 41.49 -5dB 78.69 72.34 69.04 67.26 71.83 Average 16.54 17.53 12.11 12.77 14.74 Table 1: Word error rates (in percent) on set B of the Aurora test set, for the adaptive version of ALGONQUIN with 4 noise componentsset. Setting the derivatives of :F with respect to the noise model parameters to zero, we obtain the following M step updates: ~n ('"' '"' (t) (opnn(t) +d· (( n (t) n)( n (t) n )T))) / ('"' '"' (t) ) en +--~ ~ Pe. en e' en lag 11e' en -#-ten 11e' en -#-ten ~ ~ Pe. en . t c B t c B The variational parameters can be updated multiple times before updating the model parameters, or the variational parameters can updated only once before updating the model parameters. The latter approach may converge more quickly in some situations. 6 Experimental Results After training a 256-component speech model on clean speech, we used the adaptive version of ALGONQUIN to denoise noisy test utterances on two tasks: the publically available Aurora limited vocabulary speech recognition task (http://www.etsi.org/technicalactiv/dsr.htm); the Wall Street Journal (WSJ) large vocabulary speech recognition task, with Microsoft's Whisper speech recognition system. We obtained results on all 48 test sets from partitions A and B of the Aurora database. Each set contains 24,000 sentences that have been corrupted from one of 4 different noise types and one of 6 different signal to noise ratios. Table 1 gives the error rates for the adaptive version of ALGONQUIN, with 4 noise components. These error rates are superior to error rates obtained by our spectral subtraction technique for (Deng et al. 2000), and highly competitive with other results on the Aurora task. Table 2 compares the performances of the adaptive version of ALGONQUIN and the non-adaptive version. For the non-adaptive version, 20 non-speech frames are used to estimate the noise model. For the adaptive version, the parameters are initialized using 20 non-speech frames and then 3 iterations of generalized EM are used to learn the noise model. The average error rate over all noise types and SNRs for set B of Aurora drops from 17.65% to 15.19% when the noise adaptive algorithm is used to update the noise model. This is a relative gain of 13.94%. When 4 components are used there is a further gain of 2.5%. The Wall Street Journal test set consists of 167 sentences spoken by female speakers. The Microsoft Whisper recognizer with a 5,000 word vocabulary was used to recognize these sentences. Table 2 shows that the adaptive version of algonquin WER WER Reduction WER Reduction 20 frames 1 comp in WER 4 comps in WER Aurora, Set A 18.10% 15.91% 12.10% 15.62% 13.70% Aurora, Set B 17.65% 15.19% 13.94% 14.74% 16.49% WSJ, XD14, 10dB 30.00% 21.8% 27.33% 21.50% 28.33% WSJ, XD10, 10dB 21.80% 20.6% 5.50'70 20.6% 5.50 '70 Table 2: Word error rates (WER) and percentage reduction in WER for the Aurora test data and the Wall Street Journal test data, without scaling. performs better than the non-adaptive version, especially on noise type "XD14", which consists of the highly-nonstationary sound of a jet engine shutting down. For noise type "XD1O", which is stationary noise, we observe a gain, but we do not see any further gain for multiple noise components. 7 Conclusions A far as variational methods go, ALGONQUIN is a fast technique for denoising logspectrum or cepstrum speech feature vectors. ALGONQUIN improves on previous work using the vector Taylor series approximation, by using multiple component speech and noise models, and it uses an iterative variational method to produce accurate posterior distributions for speech and noise. By employing a generalized EM method, ALGONQUIN can estimate a noise model from noisy speech data. Our results show that the noise adaptive ALGONQUIN algorithm can obtain better results than the non-adaptive version. This is especially important for nonstationary noise, where the non-adaptive algorithm relies on an estimate of the noise based on a subset of the frames, but the adaptive algorithm uses all the frames of the utterance, even those that contain speech. A different approach to denoising speech features is to learn time-domain models. Attias et al. (2001) report results on a non-adaptive time-domain technique. Our results cannot be directly compared with theirs, since our results are for unscaled data. Eventually, the two approaches should be thoroughly compared. References Attias, H., Platt, J . C., Acero, A., and Deng, L. 2001. Speech denoising and dereverberation using probabilistic models. In Advances in Neural Information Processing Systems 13. MIT Press, Cambridge MA. Boll, S. 1979. Suppression of acoustic noise in speech using spectral subtraction. IEEE Transactions on Acoustics, Speech and Signal Processing, 27:114- 120. Deng, L., Acero, A., Plumpe, M., and Huang, X. D. 2000. Large-vocabulary speech recognition under adverse acoustic environments. In Proceedings of the International Conference on Spoken Language Processing, pages 806- 809. Frey, B. J., Deng, L. , Acero, A., and Krist jansson, T. 2001. ALGONQUIN: Iterating Laplace's method to remove multiple types of acoustic distortion for robust speech recognition. In Proceedings of Eurospeech 2001. Gales, M. J. F. and Young, S. J . 1996. Robust continuous speech recognition using parallel model combination. IEEE Speech and Audio Processing, 4(5):352- 359. Jordan, M. 1., Ghahramani, Z., Jaakkola, T. S., and Saul, L. K. 1998. An introduction to variational methods for graphical models. In Jordan, M. 1., editor, Learning in Graphical Models. Kluwer Academic Publishers, Norwell MA. Moreno, P. 1996. Speech Recognition in Noisy Environments. Carnegie Mellon University, Pittsburgh PA. Doctoral dissertation. Neal, R. M. and Hinton, G. E. 1998. A view of the EM algorithm that justifies incremental, sparse, and other variants. In Jordan, M. 1., editor, Learning in Graphical Models, pages 355- 368. Kluwer Academic Publishers, Norwell MA. Varga, A. P. and Moore, R. K. 1990. Hidden Markov model decomposition of speech and noise. In Proceedings of the International Conference on Acoustics, Speech and Signal Processing, pages 845- 848. IEEE Press.
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Adaptive N earest Neighbor Classification using Support Vector Machines Carlotta Domeniconi, Dimitrios Gunopulos Dept. of Computer Science, University of California, Riverside, CA 92521 { carlotta, dg} @cs.ucr.edu Abstract The nearest neighbor technique is a simple and appealing method to address classification problems. It relies on the assumption of locally constant class conditional probabilities. This assumption becomes invalid in high dimensions with a finite number of examples due to the curse of dimensionality. We propose a technique that computes a locally flexible metric by means of Support Vector Machines (SVMs). The maximum margin boundary found by the SVM is used to determine the most discriminant direction over the query's neighborhood. Such direction provides a local weighting scheme for input features. We present experimental evidence of classification performance improvement over the SVM algorithm alone and over a variety of adaptive learning schemes, by using both simulated and real data sets. 1 Introduction In a classification problem, we are given J classes and l training observations. The training observations consist of n feature measurements x = (Xl,'" ,Xn)T E ~n and the known class labels j = 1, ... , J. The goal is to predict the class label of a given query q. The K nearest neighbor classification method [4, 13, 16] is a simple and appealing approach to this problem: it finds the K nearest neighbors of q in the training set, and then predicts the class label of q as the most frequent one occurring in the K neighbors. It has been shown [5, 8] that the one nearest neighbor rule has asymptotic error rate that is at most twice the Bayes error rate, independent of the distance metric used. The nearest neighbor rule becomes less appealing with finite training samples, however. This is due to the curse of dimensionality [2]. Severe bias can be introduced in the nearest neighbor rule in a high dimensional input feature space with finite samples. As such, the choice of a distance measure becomes crucial in determining the outcome of nearest neighbor classification. The commonly used Euclidean distance implies that the input space is isotropic, which is often invalid and generally undesirable in many practical applications. Several techniques [9, 10, 7] have been proposed to try to minimize bias in high dimensions by using locally adaptive mechanisms. The "lazy learning" approach used by these methods, while appealing in many ways, requires a considerable amount of on-line computation, which makes it difficult for such techniques to scale up to large data sets. The feature weighting scheme they introduce, in fact, is query based and is applied on-line when the test point is presented to the "lazy learner" . In this paper we propose a locally adaptive metric classification method which, although still founded on a query based weighting mechanism, computes off-line the information relevant to define local weights. Our technique uses support vector machines (SVMs) as a guidance for the process of defining a local flexible metric. SVMs have been successfully used as a classification tool in a variety of areas [11, 3, 14], and the maximum margin boundary they provide has been proved to be optimal in a structural risk minimization sense. The solid theoretical foundations that have inspired SVMs convey desirable computational and learning theoretic properties to the SVM's learning algorithm, and therefore SVMs are a natural choice for seeking local discriminant directions between classes. The solution provided by SVMs allows to determine locations in input space where class conditional probabilities are likely to be not constant, and guides the extraction of local information in such areas. This process produces highly stretched neighborhoods along boundary directions when the query is close to the boundary. As a result, the class conditional probabilities tend to be constant in the modified neighborhoods, whereby better classification performance can be achieved. The amount of elongation-constriction decays as the query moves further from the boundary vicinity. 2 Feature Weighting SVMs classify patterns according to the sign(f(x)), where f(x) L:~=l (XiYiK(Xi, x) - b, K(x,y) = cpT(x). cp(y) (kernel junction), and cp: 3(n -+ 3(N is a mapping of the input vectors into a higher dimensional feature space. Here we assume Xi E 3(n, i = I, . . . ,l, and Yi E {-I,I}. Clearly, in the general case of a non-linear feature mapping cp, the SVM classifier gives a non-linear boundary f(x) = 0 in input space. The gradient vector lld = "Vdj, computed at any point d of the level curve f(x) = 0, gives the perpendicular direction to the decision boundary in input space at d. As such, the vector lld identifies the orientation in input space on which the projected training data are well separated, locally over d's neighborhood. Therefore, the orientation given by lld, and any orientation close to it, is highly informative for the classification task at hand, and we can use such information to define a local measure of feature relevance. Let q be a query point whose class label we want to predict. Suppose q is close to the boundary, which is where class conditional probabilities become locally non uniform, and therefore estimation of local feature relevance becomes crucial. Let d be the closest point to q on the boundary f(x) = 0: d = argminp Ilq - pll, subject to the constraint f(p) = O. Then we know that the gradient lld identifies a direction along which data points between classes are well separated. As a consequence, the subspace spanned by the orientation lld, locally at q, is likely to contain points having the same class label as q . Therefore, when applying a nearest neighbor rule at q, we desire to stay close to q along the lld direction, because that is where it is likely to find points similar to q in terms of class posterior probabilities. Distances should be constricted (large weight) along lld and along directions close to it. The farther we move from the lld direction, the less discriminant the correspondent orientation becomes. This means that class labels are likely not to change along those orientations, and distances should be elongated (small weight), thus including in q's neighborhood points which are likely to be similar to q in terms of the class conditional probabilities. Formally, we can measure how close a direction t is to lld by considering the dot product lla ·t. In particular, by denoting with Uj the unit vector along input feature j, for j = 1, . .. , n, we can define a measure of relevance for feature j, locally at q (and therefore at d), as Rj(q) == Iu] . lldl = Ind,j l, where lld = (nd,l,'" ,nd,n)T. The measure of feature relevance, as a weighting scheme, can then be given by the following exponential weighting scheme: Wj(q) = exp(ARj(q))1 2::7=1 exp(ARi(q)), where A is a parameter that can be chosen to maximize (minimize) the influence of R j on Wj' When A = 0 we have Wj = lin, thereby ignoring any difference between the Rj's. On the other hand, when A is large a change in Rj will be exponentially reflected in Wj' The exponential weighting scheme conveys stability to the method by preventing neighborhoods to extend infinitely in any direction. This is achieved by avoiding zero weights, which would instead be allowed by linear or quadratic weightings. Thus, the exponential weighting scheme can be used as weights associated with features for weighted distance computation D(x,y) = )2::7=1 Wi(Xi - Yi)2. These weights enable the neighborhood to elongate less important feature dimensions, and, at the same time, to constrict the most influential ones. Note that the technique is query-based because weightings depend on the query. 3 Local Flexible Metric Classification based on SVMs To estimate the orientation of local boundaries, we move from the query point along the input axes at distances proportional to a given small step (whose initial value can be arbitrarily small, and doubled at each iteration till the boundary is crossed). We stop as soon as the boundary is crossed along an input axis i, i.e. when a point Pi is reached that satisfies the condition sign(f(q)) x sign(f(pi)) = -1. Given Pi, we can get arbitrarily close to the boundary by moving at (arbitrarily) small steps along the segment that joins Pi to q. Let us denote with di the intercepted point on the boundary along direction i. We then approximate lld with the gradient vector lldi = \7 di f, computed at di. We desire that the parameter A in the exponential weighting scheme increases as the distance of q from the boundary decreases. By using the knowledge that support vectors are mostly located around the boundary surface, we can estimate how close a query point q is to the boundary by computing its distance from the closest non bounded support vector: Bq = minsi Ilq - si ll, where the minimum is taken over the non bounded (0 < D:i < C) support vectors Si. Following the same principle, in [1] the spatial resolution around the boundary is increased by enlarging volume elements locally in neighborhoods of support vectors. Then, we can achieve our goal by setting A = D - B q , where D is a constant input parameter of the algorithm. In our experiments we set D equal to the approximated average distance between the training points Xk and the boundary: D = t 2::xk {minsi Ilxk - sill}. If A becomes negative it is set to zero. By doing so the value of A nicely adapts to each query point according to its location with respect to the boundary. The closer q is to the decision boundary, the higher the effect of the Rj's values will be on distances computation. We observe that this principled guideline for setting the parameters of our technique takes advantage of the sparseness representation of the solution provided by the SVM. In fact, for each query point q, in order to compute Bq we only need to consider the support vectors, whose number is typically small compared to the Input: Decision boundary f(x) = a produced by a SVM; query point q and parameter K. 1. Compute the approximated closest point d i to q on the boundary; 2. Compute the gradient vector ndi = \l dJ; 3. Set feature relevance values Rj(q) = Indi,jl for j = 1, . . . ,n; 4. Estimate the distance of q from the boundary as: Bq = minsi Ilq - sill; 5. Set A = D - B q , where D = t EXk {minsi Ilxk - sill}; 6. Set Wj(q) = exp(ARj(q))/ E~=l exp(ARi(q)), for j 1, ... ,n; 7. Use the resulting w for K-nearest neighbor classification at the query point q. Figure 1: The LFM-SVM algorithm total number of training examples. Furthermore, the computation of D's value is carried out once and off-line. The resulting local flexible metric technique based on SVMs (LFM-SVM) is summarized in Figure 1. The algorithm has only one adjustable tuning parameter, namely the number K of neighbors in the final nearest neighbor rule. This parameter is common to all nearest neighbor classification techniques. 4 Experimental Results In the following we compare several classification methods using both simulated and real data. We compare the following classification approaches: (1) LFM-SVM algorithm described in Figure 1. SV Mlight [12] with radial basis kernels is used to build the SVM classifier; (2) RBF-SVM classifier with radial basis kernels. We used SV Mlight [12], and set the value of"( in K(Xi' x) = e-rllxi-xI12 equal to the optimal one determined via cross-validation. Also the value of C for the soft-margin classifier is optimized via cross-validation. The output of this classifier is the input of LFM-SVM; (3) ADAMENN-adaptive metric nearest neighbor technique [7]. It uses the Chi-squared distance in order to estimate to which extent each dimension can be relied on to predict class posterior probabilities; (4) Machete [9]. It is a recursive partitioning procedure, in which the input variable used for splitting at each step is the one that maximizes the estimated local relevance. Such relevance is measured in terms of the improvement in squared prediction error each feature is capable to provide; (5) Scythe [9]. It is a generalization of the machete algorithm, in which the input variables influence each split in proportion to their estimated local relevance; (6) DANN-discriminant adaptive nearest neighbor classification [10]. It is an adaptive nearest neighbor classification method based on linear discriminant analysis. It computes a distance metric as a product of properly weighted within and between sum of squares matrices; (7) Simple K-NN method using the Euclidean distance measure; (8) C4.5 decision tree method [15]. In all the experiments, the features are first normalized over the training data to have zero mean and unit variance, and the test data features are normalized using the corresponding training mean and variance. Procedural parameters (including K) for each method were determined empirically through cross-validation. 4.1 Experiments on Simulated Data For all simulated data, 10 independent training samples of size 200 were generated. For each of these, an additional independent test sample consisting of 200 observations was generated. These test data were classified by each competing method using the respective training data set. Error rates computed over all 2,000 such classifications are reported in Table 1. The Problems. (1) Multi-Gaussians. The data set consists of n = 2 input features, l = 200 training data, and J = 2 classes. Each class contains two spherical bivariate normal subclasses, having standard deviation 1. The mean vectors for one class are (-3/4, -3) and (3/4,3); whereas for the other class are (3, -3) and (-3,3). For each class, data are evenly drawn from each of the two normal subclasses. The first column of Table 1 shows the results for this problem. The standard deviations are: 0.17, 0.01, 0.01, 0.01, 0.01 0.01, 0.01 and 1.50, respectively. (2) Noisy-Gaussians. The data for this problem are generated as in the previous example, but augmented with four predictors having independent standard Gaussian distributions. They serve as noise. Results are shown in the second column of Table 1. The standard deviations are: 0.18, 0.01, 0.02, 0.01, 0.01, 0.01, 0.01 and 1.60, respectively. Results. Table 1 shows that all methods have similar performances for the MultiGaussians problem, with C4.5 being the worst performer. When the noisy predictors are added to the problem (NoisyGaussians), we observe different levels of deterioration in performance among the eight methods. LFM-SVM shows the most robust behavior in presence of noise. K-NN is instead the worst performer. In Figure 2 we plot the performances of LFM-SVM and RBF-SVM as a function of an increasing number of noisy features (for the same MultiGaussians problem). The standard deviations for RBF -SVM (in order of increasing number of noisy features) are: 0.01, 0.01, 0.03, 0.03, 0.03 and 0.03. The standard deviations for LFM-SVM are: 0.17,0.18,0.2,0.3,0.3 and 0.3. The LFM-SVM technique shows a considerable improvement over RBF -SVM as the amount of noise increases. Table 1: Average classification error rates for simulated and real data. MultiGauss NoisyGauss Iris Sonar Liver Vote Breast OQ Pima LFM-SVM 3.3 3.4 4.0 11.0 28.1 2.6 3.0 3.5 19.3 RBF-SVM 3.3 4.1 4.0 12.0 26.1 3.0 3.1 3.4 21.3 ADAMENN 3.4 4.1 3.0 9.1 30.7 3.0 3.2 3.1 20.4 Machete 3.4 4.3 5.0 21.2 27.5 3.4 3.5 7.4 20.4 Scythe 3.4 4.8 4.0 16.3 27.5 3.4 2.7 5.0 20.0 DANN 3.7 4.7 6.0 1.1 30.1 3.0 2.2 4.0 22.2 K-NN 3.3 7.0 6.0 12.5 32.5 7.8 2.7 5.4 24.2 C4.5 5.0 5.1 8.0 23.1 38.3 3.4 4.1 9.2 23.8 4.2 Experiments on Real Data In our experiments we used seven different real data sets. They are all taken from DCI Machine Learning Repository at http://www.cs.uci.edu/,,,-,mlearn/ MLRepository.html. For a description of the data sets see [6]. For the Iris, Sonar, Liver and Vote data we perform leave-one-out cross-validation to measure performance, since the number of available data is limited for these data sets. For the 36'--'--'---r--'--~--'--'--~--.--'--~ 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 ~ ~~=='P'LFM-SVM --+-RBF-SVM ---)(--O L-~--~--~~--~--~~--~--~~--~ o 10 12 14 16 18 20 22 Number of Noisy Variables Figure 2: Average Error Rates of LFM-SVM and RBF-SVM as a function of an increasing number of noisy predictors. i J. T I I I - 1~ • • -~ ""'!"" :E :E z " i z z 3 z 1j z z ~ > OJ "" ;l :z :E ~ Q '" "' "" "" " Q ..J "" Figure 3: Performance distributions for real data. Breast, OQ-Ietter and Pima data we randomly generated five independent training sets of size 200. For each of these, an additional independent test sample consisting of 200 observations was generated. Table 1 (columns 3-9) shows the cross-validated error rates for the eight methods under consideration on the seven real data. The standard deviation values are as follows. Breast data: 0.2, 0.2, 0.2, 0.2, 0.2, 0.9, 0.9 and 0.9, respectively. OQ data: 0.2, 0.2, 0.2, 0.3, 0.2, 1.1, 1.5 and 2.1, respectively. Pima data: 0.4, 0.4, 0.4, 0.4, 0.4, 2.4, 2.1 and 0.7, respectively. Results. Table 1 shows that LFM-SVM achieves the best performance in 2/7 of the real data sets; in one case it shows the second best performance, and in the remaining four its error rate is still quite close to the best one. Following Friedman [9], we capture robustness by computing the ratio bm of the error rate em of method m and the smallest error rate over all methods being compared in a particular example: bm = emf minl~k~8 ek· Figure 3 plots the distribution of bm for each method over the seven real data sets. The dark area represents the lower and upper quartiles of the distribution that are separated by the median. The outer vertical lines show the entire range of values for the distribution. The spread of the error distribution for LFM-SVM is narrow and close to one. The results clearly demonstrate that LFM-SVM (and ADAMENN) obtained the most robust performance over the data sets. The poor performance of the machete and C4.5 methods might be due to the greedy strategy they employ. Such recursive peeling strategy removes at each step a subset of data points permanently from further consideration. As a result, changes in an early split, due to any variability in parameter estimates, can have a significant impact on later splits, thereby producing different terminal regions. This makes predictions highly sensitive to the sampling fluctuations associated with the random nature of the process that produces the traning data, thus leading to high variance predictions. The scythe algorithm, by relaxing the winner-take-all splitting strategy of the machete algorithm, mitigates the greedy nature of the approach, and thereby achieves better performance. In [10], the authors show that the metric employed by the DANN algorithm approximates the weighted Chi-squared distance, given that class densities are Gaussian and have the same covariance matrix. As a consequence, we may expect a degradation in performance when the data do not follow Gaussian distributions and are corrupted by noise, which is likely the case in real scenarios like the ones tested here. We observe that the sparse solution given by SVMs provides LFM-SVM with principled guidelines to efficiently set the input parameters. This is an important advantage over ADAMENN, which has six tunable input parameters. Furthermore, LFM-SVM speeds up the classification process since it applies the nearest neighbor rule only once, whereas ADAMENN applies it at each point within a region centered at the query. We also observe that the construction of the SVM for LFM-SVM is carried out off-line only once, and there exist algorithmic and computational results which make SVM training practical also for large-scale problems [12]. The LFM-SVM offers performance improvements over the RBF-SVM algorithm alone, for both the (noisy) simulated and real data sets. The reason for such performance gain may rely on the effect of our local weighting scheme on the separability of classes, and therefore on the margin, as shown in [6]. Assigning large weights to input features close to the gradient direction, locally in neighborhoods of support vectors, corresponds to increase the spatial resolution along those orientations, and therefore to improve the separability of classes. As a consequence, better classification results can be achieved as demonstrated in our experiments. 5 Related Work In [1], Amari and Wu improve support vector machine classifiers by modifying kernel functions. A primary kernel is first used to obtain support vectors. The kernel is then modified in a data dependent way by using the support vectors: the factor that drives the transformation has larger values at positions close to support vectors. The modified kernel enlarges the spatial resolution around the boundary so that the separability of classes is increased. The resulting transformation depends on the distance of data points from the support vectors, and it is therefore a local transformation, but is independent of the boundary's orientation in input space. Likewise, our transformation metric depends, through the factor A, on the distance of the query point from the support vectors. Moreover, since we weight features, our metric is directional, and depends on the orientation of local boundaries in input space. This dependence is driven by our measure of feature relevance, which has the effect of increasing the spatial resolution along discriminant directions around the boundary. 6 Conclusions We have described a locally adaptive metric classification method and demonstrated its efficacy through experimental results. The proposed technique offers performance improvements over the SVM alone, and has the potential of scaling up to large data sets. It speeds up, in fact, the classification process by computing offline the information relevant to define local weights, and by applying the nearest neighbor rule only once. Acknowledgments This research has been supported by the National Science Foundation under grants NSF CAREER Award 9984729 and NSF IIS-9907477, by the US Department of Defense, and a research award from AT&T. References [1] S. Amari and S. Wu, "Improving support vector machine classifiers by modifying kernel functions", Neural Networks, 12, pp. 783-789, 1999. [2] R.E. Bellman, Adaptive Control Processes. Princeton Univ. Press, 1961. [3] M. Brown, W. Grundy, D. Lin, N. Cristianini, C. Sugnet, T. Furey, M. Ares, and D. Haussler, "Knowledge-based analysis of microarray gene expressions data using support vector machines", Tech. Report, University of California in Santa Cruz, 1999. [4] W.S. Cleveland and S.J. Devlin, "Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting", J. Amer. Statist. Assoc. 83, 596-610, 1988 [5] T.M. Cover and P.E. Hart, "Nearest Neighbor Pattern Classification", IEEE Trans. on Information Theory, pp. 21-27, 1967. [6] C. Domeniconi and D. Gunopulos, "Adaptive Nearest Neighbor Classification using Support Vector Machines", Tech. Report UCR-CSE-01-04, Dept. of Computer Science, University of California, Riverside, June 200l. [7] C. Domeniconi, J. Peng, and D. Gunopulos, "An Adaptive Metric Machine for Pattern Classification", Advances in Neural Information Processing Systems, 2000. [8] R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis. John Wiley & Sons, Inc., 1973. [9] J.H. Friedman "Flexible Metric Nearest Neighbor Classification", Tech. Report, Dept. of Statistics, Stanford University, 1994. [10] T. Hastie and R. Tibshirani, "Discriminant Adaptive Nearest Neighbor Classification", IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 18, No.6, pp. 607-615, 1996. [11] T. Joachims, "Text categorization with support vector machines", Pmc. of European Conference on Machine Learning, 1998. [12] T. Joachims, "Making large-scale SVM learning practical" Advances in Kernel Methods - Support Vector Learning, B. Sch6lkopf and C. Burger and A. Smola (ed.), MITPress, 1999. http://www-ai.cs.uni-dortmund.de/thorsten/svm_light.html [13] D.G. Lowe, "Similarity Metric Learning for a Variable-Kernel Classifier", Neural Computation 7(1):72-85, 1995. [14] E. Osuna, R. Freund, and F. Girosi, "Training support vector machines: An application to face detection", Pmc. of Computer Vision and Pattern Recognition, 1997. [15] J.R. Quinlan, C4.5: Programs for Machine Learning. Morgan-Kaufmann Publishers, Inc., 1993. [16] C.J. Stone, Nonparametric regression and its applications (with discussion). Ann. Statist. 5, 595, 1977.
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Probabilistic Inference of Hand Motion from Neural Activity in Motor Cortex Y. Gao M. J. Black E. Bienenstock S. Shoham J. P. Donoghue Division of Applied Mathematics, Brown University, Providence, RI 02912 Dept. of Computer Science, Brown University, Box 1910, Providence, RI 02912 Princeton University, Dept. of Molecular Biology Princeton, NJ, 08544 Dept. of Neuroscience, Brown University, Providence, RI 02912 gao@cfm.brown.edu, black@cs.brown.edu, elie@dam.brown.edu, sshoham@princeton.com, john donoghue@brown.edu Abstract Statistical learning and probabilistic inference techniques are used to infer the hand position of a subject from multi-electrode recordings of neural activity in motor cortex. First, an array of electrodes provides training data of neural firing conditioned on hand kinematics. We learn a nonparametric representation of this firing activity using a Bayesian model and rigorously compare it with previous models using cross-validation. Second, we infer a posterior probability distribution over hand motion conditioned on a sequence of neural test data using Bayesian inference. The learned firing models of multiple cells are used to define a nonGaussian likelihood term which is combined with a prior probability for the kinematics. A particle filtering method is used to represent, update, and propagate the posterior distribution over time. The approach is compared with traditional linear filtering methods; the results suggest that it may be appropriate for neural prosthetic applications. 1 Introduction This paper explores the use of statistical learning methods and probabilistic inference techniques for modeling the relationship between the motion of a monkey’s arm and neural activity in motor cortex. Our goals are threefold: (i) to investigate the nature of encoding in motor cortex, (ii) to characterize the probabilistic relationship between arm kinematics (hand position or velocity) and activity of a simultaneously recorded neural population, and (iii) to optimally reconstruct (decode) hand trajectory from population activity to smoothly control a prosthetic robot arm (cf [14]). A multi-electrode array (Figure 1) is used to simultaneously record the activity of 24 neurons in the arm area of primary motor cortex (MI) in awake, behaving, macaque monkeys. This activity is recorded while the monkeys manually track a smoothly and randomly mov
C. ! !" # $ % & ' ( ) ) * + ,.-0/1 2 3 4 4 4 4 5 564 White Matter Connector Acrylic Bone Silicone Figure 1: Multi-electrode array. A. 10X10 matrix of electrodes. Separation 400 7 m (size 4X4 mm). B. Location of array in the MI arm area. C. Illustration of implanted array (courtesy N. Hatsopoulos). ing visual target on a computer monitor [12]. Statistical learning methods are used to derive Bayesian estimates of the conditional probability of firing for each cell given the kinematic variables (we consider only hand velocity here). Specifically, we use non-parametric models of the conditional firing, learned using regularization (smoothing) techniques with cross-validation. Our results suggest that the cells encode information about the position and velocity of the hand in space. Moreover, the non-parametric models provide a better explanation of the data than previous parametric models [6, 10] and provide new insight into neural coding in MI. Decoding involves the inference of the hand motion from the firing rate of the cells. In particular, we represent the posterior probability of the entire hand trajectory conditioned on the observed sequence of neural activity (spike trains). The nature of this activity results in ambiguities and a non-Gaussian posterior probability distribution. Consequently, we represent the posterior non-parametrically using a discrete set of samples [8]. This distribution is predicted and updated in non-overlapping 50 ms time intervals using a Bayesian estimation method called particle filtering [8]. Experiments with real and synthetic data suggest that this approach provides probabilistically sound estimates of kinematics and allows the probabilistic combination of information from multiple neurons, the use of priors, and the rigorous evaluation of models and results. 2 Methods: Neural Recording The design of the experiment and data collection is described in detail in [12]. Summarizing, a ten-by-ten array of electrodes is implanted in the primary motor cortex (MI) of a Macaque monkey (Figure 1) [7, 9, 12]. Neural activity in motor cortex has been shown to be related to the movement kinematics of the animal’s arm and, in particular, to the direction of hand motion [3, 6]. Previous behavioral tasks have involved reaching in one of a fixed number of directions [3, 6, 14]. To model the relationship between continuous, smooth, hand motion and neural activity, we use a more complex scenario where the monkey performs a continuous tracking task in which the hand is moved on a 2D tablet while holding a low-friction manipulandum that controls the motion of a feedback dot viewed on a computer monitor (Figure 2a) [12]. The monkey receives a reward upon completion of a successful trial in which the manipulandum is moved to keep the feedback dot within a pre-specified distance of the target. The path of the target is chosen to be a smooth random walk that effectively samples the space of hand positions and velocities: measured hand positions and velocities have a roughly Gaussian distribution (Figure 2b and c) [12]. Neural activity is amplified, waveforms are thresholded, and spike sorting is performed off-line to isolate the activity of individual cells [9]. Recordings from 24 motor cortical cells are measured simultaneously with hand kinematics. Monitor Tablet Manipulandum Trajectory Target 0 5 10 15 20 25 0 2 4 6 8 10 12 14 16 a b c Figure 2: Smooth tracking task. (a) The target moves with a smooth random walk. Distribution of the position (b) and velocity (c) of the hand. Color coding indicates the frequency with which different parts of the space are visited. (b) Position: horizontal and vertical axes represent and position of the hand. (c) Velocity: the horizontal axis represents direction, , and the vertical axis represents speed, . 0 0.5 1 1.5 2 2.5 3
cell 3 cell 16 cell 19 Figure 3: Observed mean conditional firing rates in 50 ms intervals for three cells given hand velocity. The horizontal axis represents the direction of movement, , in radians (“wrapping” around from to ). The vertical axis represents speed, , and ranges from 0 cm/s to 12 cm/s. Color ranges from dark blue (no measurement) to red (approximately 3 spikes). 3 Modeling Neural Activity Figure 3 shows the measured mean firing rate within 50 ms time intervals for three cells conditioned on the subject’s hand velocity. We view the neural firing activity in Figure 3 as a stochastic and sparse realization of some underlying model that relates neural firing to hand motion. Similar plots are obtained as a function of hand position. Each plot can be thought of as a type of “tuning function” [12] that characterizes the response of the cell conditioned on hand velocity. In previous work, authors have considered a variety of models of this data including a cosine tuning function [6] and a modified cosine function [10]. Here we explore a non-parametric model of the underling activity and, adopting a Bayesian formulation, seek a maximum a posterior (MAP) estimate of a cell’s conditional firing. Adopting a Markov Random Field (MRF) assumption [4], let the velocity space, ! "!#%$ , be discretized on a &(')' *+&(')' grid. Let g be the array of true (unobserved) conditional neural firing and , be the corresponding observed mean firing. We seek the posterior probability -/. g 01,123 5476 .98:-/.9; 6<0>=?6@2A4CBDFE/G -/. =?6<0H=?6?IJ2K2 (1) −3 −2 −1 0 1 2 3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 −3 −2 −1 0 1 2 3 −12 −10 −8 −6 −4 −2 0 a b Figure 4: Prior probability of firing variation ( = ). (a) Probability of firing variation computed from training data (blue). Proposed robust prior model (red) plotted for ' . (b) Logarithm of the distributions shown to provide detail. where 8 is a normalizing constant independent of g, ; 6 and = 6 are the observed and true mean firing at velocity respectively, = 6 I represents the firing rate for the th neighboring velocity of , and the neighbors are taken to be the four nearest velocities ( ). The first term on the right hand side represents the likelihood of observing a particular firing rate ; 6 given that the true rate is =6 . Here we compare two generative models of the neural spiking process within 50 ms; a Poisson model, , and a Gaussian model,
: . ; 0>= 2 & ; =
.9; 0H= 2 & ? . ; =A2 ! The second term is a spatial prior probability that encodes our expectations about = , the variation of neural activity in velocity space. The MRF prior states that the firing, =?6 , at velocity depends only on the firing at neighboring velocities. We consider two possible prior models for the distribution of = : Gaussian and “robust”. A Gaussian prior corresponds to an assumption that the firing rate varies smoothly. A robust prior assumes a heavy-tailed distribution of the spatial variation (see Figure 4), = , (derivatives of the firing rate in the and directions) and implies piecewise smooth data. The two spatial priors are -#" . = 2 %$ . '& = 2
. = 2 & ? . = 2( )! The various models (cosine, a modified cosine (Moran and Schwartz [10]), Gaussian+Gaussian, and Poisson+Robust) are fit to the training data as shown in Figure 5. G In the case of the Gaussian+Gaussian and Poisson+Robust models, the optimal value of the parameter is computed for each cell using cross validation. During cross-validation, each time 10 trials out of 180 are left out for testing and the models are fit with the remaining training data. We then compute the log likelihood of the test data given the model. This provides a measure of how well the model captures the statistical variation in the training set and is used for quantitative comparison. The whole procedure is repeated 18 times for different test data sets. The solution to the Gaussian+Gaussian model can be computed in closed form but for the Poisson+Robust model no closed form solution for g exists and an optimal Bayesian estimate could be achieved with simulated annealing [4]. Instead, we derive an approximate * By “Gaussian+Gaussian” we mean both the likelihood and prior terms are Gaussian whereas “Poisson+Robust”implies a Poisson likelihood and robust spatial prior. 0.2 0.4 0.6 0.8 1 1.2 0.5
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0.3 0.5 0.7 0.6 0.8 1 0.5 0.6 0.7 0.8 Cosine Moran&Schwartz (M&S) Gaussian+Gaussian Poisson+Robust cell 3 cell 16 cell 19 Figure 5: Estimated firing rate for cells in Figure 3 using different models. Method: Log Likelihood Ratio p-value G+G over Cosine 24.9181 7.6294e-06 G+G over M&S 15.8333 0.0047 P+R over Cosine 50.0685 7.6294e-06 P+R over M&S 32.2218 7.6294e-06 Table 1: Numerical comparison; log likelihood ratio of pairs of models and the significance level given by Wilcoxon signed rank test (Splus, MathSoft Inc., WA). solution for g in (1) by minimizing the negative logarithm of the distribution using standard regularization techniques [1, 13] with missing data, the learned prior model, and a Poisson likelihood term [11]. Simple gradient descent [1] with deterministic annealing provides a reasonable solution. Note that the negative logarithm of the prior term can be approximated by the robust statistical error function . =A2 = . & . =A2 12 which has been used extensively in machine vision and image processing for smoothing data with discontinuities [1, 5]. Figure 5 shows the various estimates of the receptive fields. Observe that the pattern of firing is not Gaussian. Moreover, some cells appear to be tuned to motion direction, , and not to speed, , resulting in vertically elongated patterns of firing. Other cells (e.g. cell 19) appear to be tuned to particular directions and speeds; this type of activity is not well fit by the parametric models. Table 1 shows a quantitative comparison using cross-validation. The log likelihood ratio (LLR) is used to compare each pair of models: LLR(model 1, model 2) = log( (observed firing 0 model 1)/Pr(observed firing 0 model 2)). The positive values in Table 1 indicate that the non-parametric models do a better job of explaining new data than the parametric models with the Poisson+Robust fit providing the best description of the data. This P+R model implies that the conditional firing rate is well described by regions of smooth activity with relatively sharp discontinuities between them. The non-parametric models reduce the strong bias of the parametric models with a slight increase in variance and hence achieve a lower total error. 4 Temporal Inference Given neural measurements our goal is to infer the motion of the hand over time. Related approaches have exploited simple linear filtering methods which do not provide a probabilistic interpretation of the data that can facilitate analysis and support the principled combination of multiple sources of information. Related probabilistic approaches have exploited Kalman filtering [2]. We note here however, that the learned models of neural activity are not-Gaussian and the dynamics of the hand motion may be non-linear. Furthermore with a small number of cells, our interpretation of the neural data may be ambiguous and the posterior probability of the kinematic variables, given the neural activity, may be best modeled by a non-Gaussian, multi-modal, distribution. To cope with these issues in a sound probabilistic framework we exploit a non-parametric approach that uses factored sampling to discretely approximate the posterior distribution, and particle filtering to propagate and update this distribution over time [8]. Let the state of the system be s 7 ! ?# at time . Let D be the mean firing rate of cell at time where the mean firing rate is estimated within non-overlapping 50 ms temporal windows. Also, let c G # represent the firing rate of all cells at time . Similarly let D represent the sequence of these firing rates for cell up to time and let C G # represent the firing of all cells up to time . We assume that the temporal dynamics of the states, s , form a Markov chain for which the state at time depends only on the state at the previous time instant: -/. s 0 S G 2 . s 0 s G 2 where S s F s
(# denotes the state history. We also assume that given s , the current observation c and the previous observations C G are independent. Using Bayes rule and the above assumptions, the probability of observing the state at time given the history of firing can be written as . s H0 C J2 8 . c 0 s "2 -/. s H0 C G 2 (2) where 8 is a normalizing term that insures that the distribution integrates to one. The likelihood term -/. c 0 s 2 DFE/G . D 0 s 2 assumes conditional independence of the individual cells where the likelihood for the firing rate of an individual cell is taken to be a Poisson distribution with the mean firing rate for the speed and velocity given by s determined by the conditional firing models learned in the previous section. Plotting this likelihood term for a range of states reveals that its structure is highly non-Gaussian with multiple peaks. The temporal prior term, . s 0 C G 2 can be written as -/. s H0 C G 2 -/. s H0 s G 2 -/. s G 0 C G 2 s G (3) where . s H0 s G 2 embodies the temporal dynamics of the hand velocity which are assumed to be constant with Gaussian noise; that is, a diffusion process. Note, . s G 0 C G 2 is the posterior distribution over the state space at time & . The posterior, . s 0 C J2 , is represented with a discrete, weighted set, of ')'' random samples which are propagated in time using a standard particle filter (see [8] for details). Unlike previous applications of particle filtering, the likelihood of firing for an individual cell in 125 126 127 128 129 130 131 132 133 134 135 -10 -5 0 5 10 trial No. 8, Vx in cm/s, blue:true, red:reconstruction time in second 125 126 127 128 129 130 131 132 133 134 -10 -5 0 5 10 Vy in cm/s 125 126 127 128 129 130 131 132 133 134 135 -10 -5 0 5 10 trial No. 8, Vx in cm/s, blue:true, red:reconstruction time in second 125 126 127 128 129 130 131 132 133 134 135 -10 -5 0 5 10 Vy in cm/s a b Figure 6: Tracking results using 1008 synthetic cells showing horizontal velocity, , (top) and vertical velocity, , (bottom). Blue indicates true velocity of hand. (a) Bayesian estimate using particle filtering. Red curve shows expected value of the posterior. The error is ' for and ' ' for . (b) Linear filtering method shown in red; ' for and ' ' for . 50 ms provides very little information. For the posterior to be meaningful we must combine evidence from multiple cells. Our experiments indicate that the responses from our 24 cells are insufficient for this task. To demonstrate the feasibility of the particle filtering method, we synthesized approximately 1000 cells by taking the learned models of the 24 cells and translating them along the axis to generate a more complete covering of the velocity space. Note that the assumption of such a set of cells in MI is quite reasonable give the sampling of cells we have observed in multiple monkeys. From the set of synthetic cells we then generate a synthetic spike train by taking a known sequence of hand velocities and stochastically generating spikes using the learned conditional firing models with a Poisson generative model. Particle filtering is used to estimate the posterior distribution over hand velocities given the synthetic neural data. The expected value of the horizontal and vertical velocity is displayed in Figure 6a. For comparison, a standard linear filtering method [6, 14] was trained on the synthetic data from 50 ms intervals. The resulting prediction is shown in Figure 6b. Linear filtering works well over longer time windows which introduce lag. The Bayesian analysis provides a probabilistic framework for sound causal estimates over short time intervals. We are currently experimenting with modified particle filtering schemes in which linear filtering methods provide a proposal distribution and importance sampling is used to construct a valid posterior distribution. We are also comparing these results with those of various Kalman filters. 5 Conclusions We have described a Bayesian model for neural activity in MI that relates this activity to actions in the world. Quantitative comparison with previous models of MI activity indicate that the non-parametric models computed using regularization more accurately describe the neural activity. In particular, the robust spatial prior term suggests that neural firing in MI is not a smooth function of velocity but rather exhibits discontinuities between regions of high and low activity. We have also described the Bayesian decoding of hand motion from firing activity using a particle filter. Initial results suggest that measurements from several hundred cells may be required for accurate estimates of hand velocity. The application of particle filtering to this problem has many advantages as it allows complex, non-Gaussian, likelihood models that may incorporate non-linear temporal properties of neural firing (e.g. refractory period). Unlike previous linear filtering methods this Bayesian approach provides probabilistically sound, causal, estimates in short time windows of 50ms. Current work is exploring correlations between cells [7] and the relationship between the neural activity and other kinematic variables [12]. Acknowledgments. This work was supported by the Keck Foundation and by the National Institutes of Health under grants #R01 NS25074 and #N01-NS-9-2322 and by the National Science Foundation ITR Program award #0113679. We are very grateful to M. Serruya, M. Fellows, L. Paninski, and N. Hatsopoulos who provided the neural data and valuable insight. References [1] M. Black and A. Rangarajan. On the unification of line processes, outlier rejection, and robust statistics with applications in early vision. IJCV, 19(1):57–92, 1996. [2] E. Brown, L. Frank, D. Tang, M. Quirk, and M. Wilson. A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells. J. Neuroscience, 18(18):7411–7425, 1998. [3] Q-G. Fu, D. Flament, J. Coltz, and T. Ebner. Temporal encoding of movement kinematics in the discharge of primate primary motor and premotor neurons. J. of Neurophysiology, 73(2):836– 854, 1995. [4] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions and Bayesian restoration of images. PAMI, 6(6):721–741, November 1984. [5] S. Geman and D. McClure. Statistical methods for tomographic image reconstruction. Bulletin of the Int. Stat. Inst., LII-4:5–21, 1987. [6] A. Georgopoulos, A. Schwartz, and R. Kettner. Neuronal population coding of movement direction. Science, 233:1416–1419, 1986. [7] N. Hatsopoulos, C. Ojakangas, L. Paninski, and J. Donoghue. Information about movement direction obtained from synchronous activity of motor cortical neurons. Proc. Nat. Academy of Sciences, 95:15706–15711, 1998. [8] M. Isard and A. Blake. Condensation – conditional density propagation for visual tracking. IJCV, 29(1): 5–28, 1998. [9] E. Maynard, N. Hatsopoulos, C. Ojakangas, B. Acuna, J. Sanes, R. Normann, and J. Donoghue. Neuronal interaction improve cortical population coding of movement direction. J. of Neuroscience, 19(18):8083–8093, 1999. [10] D. Moran and A. Schwartz. Motor cortical representation of speed and direction during reaching. J. Neurophysiol, 82:2676-2692, 1999. [11] R. Nowak and E. Kolaczyk. A statistical multiscale framework for Poisson inverse problems. IEEE Inf. Theory, 46(5):1811–1825, 2000. [12] L. Paninski, M. Fellows, N. Hatsopoulos, and J. Donoghue. Temporal tuning properties for hand position and velocity in motor cortical neurons. submitted, J. Neurophysiology, 2001. [13] D. Terzopoulos. Regularization of inverse visual problems involving discontinuities. PAMI, 8(4):413–424, 1986. [14] J. Wessberg, C. Stambaugh, J. Kralik, P. Beck, M. Laubach, J. Chapin, J. Kim, S. Biggs, M. Srinivasan, and M. Nicolelis. Real-time prediction of hand trajectory by ensembles of cortical neurons in primates. Nature, 408:361–365, 2000.
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Fast, large-scale transformation-invariant clustering Brendan J. Frey Machine Learning Group University of Toronto www.psi.toronto.edu/∼frey Nebojsa Jojic Vision Technology Group Microsoft Research www.ifp.uiuc.edu/∼jojic Abstract In previous work on “transformed mixtures of Gaussians” and “transformed hidden Markov models”, we showed how the EM algorithm in a discrete latent variable model can be used to jointly normalize data (e.g., center images, pitch-normalize spectrograms) and learn a mixture model of the normalized data. The only input to the algorithm is the data, a list of possible transformations, and the number of clusters to find. The main criticism of this work was that the exhaustive computation of the posterior probabilities over transformations would make scaling up to large feature vectors and large sets of transformations intractable. Here, we describe how a tremendous speed-up is acheived through the use of a variational technique for decoupling transformations, and a fast Fourier transform method for computing posterior probabilities. For N ×N images, learning C clusters under N rotations, N scales, N x-translations and N y-translations takes only (C + 2 log N)N 2 scalar operations per iteration. In contrast, the original algorithm takes CN 6 operations to account for these transformations. We give results on learning a 4-component mixture model from a video sequence with frames of size 320×240. The model accounts for 360 rotations and 76,800 translations. Each iteration of EM takes only 10 seconds per frame in MATLAB, which is over 5 million times faster than the original algorithm. 1 Introduction The task of clustering raw data such as video frames and speech spectrograms is often obfuscated by the presence of random, but well-understood transformations in the data. Examples of these transformations include object motion and camera motion in video sequences and pitch modulation in spectrograms. The machine learning community has proposed a variety of sophisticated techniques for pattern analysis and pattern classification, but these techniques have mostly assumed the data is already normalized (e.g., the patterns are centered in the images) or nearly normalized. Linear approximations to the transformation manifold have been used to significantly improve the performance of feedforward discriminative classifiers such as nearest neighbors and multilayer perceptrons (Simard, LeCun and Denker 1993). Linear generative models (factor analyzers, mixtures of factor analyzers) have also been modified using linear approximations to the transformation manifold to build in some degree of transformation invariance (Hinton, Dayan and Revow 1997). A multi-resolution approach can be used to extend the usefulness of linear approximations (Vasconcelos and Lippman 1998), but this approach is susceptable to local minima – e.g. a pie may be confused for a face at low resolution. For significant levels of transformation, linear approximations are far from exact and better results can be obtained by explicitly considering transformed versions of the input. This approach has been used to design “convolutional neural networks” that are invariant to translations of parts of the input (LeCun et al. 1998). In previous work on “transformed mixtures of Gaussians” (Frey and Jojic 2001) and “transformed hidden Markov models” (Jojic et al. 2000), we showed how the EM algorithm in a discrete latent variable model can be used to jointly normalize data (e.g., center video frames, pitch-normalize spectrograms) and learn a mixture model of the normalized data. We found “that the algorithm is reasonably fast (it learns in minutes or hours) and very effective at transformation-invariant density modeling.” Those results were for 44 × 28 images, but realistic applications such as home video summarization require near-real-time processing of medium-quality video at resolutions near 320 × 240. In this paper, we show how a variational technique and a fast Fourier method for computing posterior probabilities can be used to achieve this goal. 2 Background In (Frey and Jojic 2001), we introduced a single discrete variable that enumerates a discrete set of possible transformations that can occur in the input. Here, we break the transformation into a sequence of transformations. Tk is the random variable for the transformation matrix at step k. So, if Tk is the set of possible transformation matrices corresponding to the type of transformation at step k (e.g., image rotation), Tk ∈Tk. The generative model is shown in Fig. 1a and consists of picking a class c, drawing a vector of image pixel intensities z0 from a Gaussian, picking the first transformation matrix T1 from Tk, applying this transformation to z0 and adding Gaussian noise to obtain z1, and repeating this process until the last transformation matrix TK is drawn from TK and is applied to zK−1 to obtain the observed data zK. The joint distribution is p(c, z0, T1, z1, . . . , TK, zK) = p(c)p(z0|c) K Y k=1 p(Tk)p(zk|zk−1, Tk). (1) The probability of class c ∈{1, . . . , C} is parameterized by p(c) = πc and the untransformed latent image has conditional density p(z0|c) = N(z0; µc, Φc), (2) where N() is the normal distribution, µc is the mean image for class c and Φc is the diagonal noise covariance matrix for class c. Notice that the noise modeled by Φc gets transformed, so Φc can model noise sources that depend on the transformations, such as background clutter and object deformations in images. (c) TK z0 z1 T1 c (b) z z0 z1 T1 TK c (a) K Figure 1: (a) The Bayesian network for a generative model that draws an image z0 from class c, applies a randomly drawn transformation matrix T1 of type 1 (e.g., image rotation) to obtain z1, and so on, until a randomly drawn transformation matrix TK of type K (e.g., image translation) is applied to obtain the observed image zK. (b) The Bayesian network for a factorized variational approximation to the posterior distribution, given zK. (c) When an image is measured on a discrete, radial 2-D grid, a scale and rotation correspond to a shift in the radial and angular coordinates. The probability of transformation matrix Tk at step k is p(Tk) = λk,Tk. (In our experiments, we often fix this to be uniform.) At each step, we assume a small amount of noise with diagonal covariance matrix Ψ is added to the image, so p(zk|zK−1, Tk) = N(zk; Tkzk−1, Ψ). (3) Tk operates on zk−1 to produce a transformed image. In fact, Tk can be viewed as a permutation matrix that rearranges the pixels in zk−1. Usually, we assume Ψ = ψI and in our experiments we often set ψ to a constant, small value, such as 0.01. In (2001), an exact EM algorithm for learning this model is described. The sufficient statistics for πc, µc and Φc are computed by averaging the derivatives of ln(πcN(z0; µc, Φc)) over the posterior distribution, p(c, z0|zK) = X T1 · · · X TK p(z0|c, T1, . . . , TK, zK)p(c, T1, . . . , TK|zK). (4) Since z0, . . . , zK are jointly Gaussian given c and T1, . . . , TK, p(z0|c, T1, . . . , TK, zK) is Gaussian and its mean and covariance are computed using linear algebra. Also, p(c, T1, . . . , TK|zK) is computed using linear algebra. The problem with this direct approach is that the number of scalar operations in (4) is very large for large feature vectors and large sets of transformations. For N × N images, learning C clusters under N rotations, N scales, N x-translations and N y-translations leads to N 4 terms in the summation. Since there are N 2 pixels, each term is computed using N 2 scalar operations. So, each iteration of EM takes CN 6 scalar operations per training case. For 10 classes and images of size 256 × 256, the direct approach takes 2.8 × 1015 scalar operations per image for each iteration of EM. We now describe how a variational technique for decoupling transformations, and a fast Fourier transform method for computing posterior probabilities can reduce the above number to (C + 2 log N)N 2 scalar operations. For 10 classes and images of size 256 × 256, the new method takes 2, 752, 512 scalar operations per image for each iteration of EM. 3 Factorized variational technique To simplify the computation of the required posterior in (4), we use a variational approximation (Jordan et al. 1998). As shown in Fig. 1b, our variational approximation is a completely factorized approximation to the true posterior: p(c, z0, T1, z1, . . . , TK|zK) ≈q(c, z0, T1, z1, . . . , TK) = q(c)q(z0) K−1 Y k=1 q(Tk)q(zk) q(TK). (5) The q-distributions are parameterized and these variational parameters are varied to make the approximation a good one. p(c, z0|zK) ≈q(c)q(zK), so the sufficient statistics can be readily determined from q(c) and q(zK). The variational parameters are q(c) = ρc, q(Tk) = γk,Tk, q(zk) = N(zk; ηk, Ωk). The generalized EM algorithm (Neal and Hinton 1998) maximizes a lower bound on the log-likelihood of the observed image zK: B = X Z q(c, z0, T1, z1, . . . , TK) ln p(c, z0, T1, z1, . . . , TK, zK) q(c, z0, T1, z1, . . . , TK) ≤ln p(zK). (6) In the E step, the variational parameters are adjusted to maximize B and in the M step, the model parameters are adjusted to maximize B. Assuming constant noise, Ψ = ψI, the derivatives of B with respect to the variational parameters produce the following E-step updates: Ω0 ← X c ρcΦ−1 c + ψ−1I −1 η0 ←Ω0 X c ρcΦ−1 c µc + ψ−1 X T1 γ1,T1T−1 1 η1 (7) ρc ←πc exp −1 2tr(Ω0Φ−1 c ) −1 2(η0 −µc)′Φ−1 c (η0 −µc) Ωk ←1 2ψI ηk ←1 2 X Tk γk,TkTkηk−1 + X Tk+1 γk+1,Tk+1T−1 k+1ηk+1 (8) γk,Tk ←λk,Tk exp −1 2tr(Ωkψ−1) −1 2ψ−1(ηk −Tkηk−1)′(ηk −Tkηk−1) . (9) Each time the ρc’s are updated, they should be normalized and similarly for the γk,Tk’s. One or more iterations of the above updates are applied for each training case and the variational parameters are stored for use in the M-step, and as the initial conditions for the next E-step. The derivatives of B with respect to the model parameters produce the following M-step updates: πc ←⟨ρc⟩ µc ←⟨ρcη0⟩ Φc ←⟨ρc(Ω0 + diag((η0 −µc)(η0 −µc)′)⟩, (10) where ⟨⟩indicates an average over the training set. This factorized variational inference technique is quite greedy, since at each step, the method approximates the posterior with one Gaussian. So, the method works best for a small number of steps (2 in our experiments). 4 Inference using fast Fourier transforms The M-step updates described above take very few computations, but the E-step updates can be computationally burdensome. The dominant culprits are the computation of the distance of the form dT = (g −Th)′(g −Th) (11) in (9), for all possible transformations T, and the computation of the form X T γTTh (12) in (7) and (8). Since the variational approximation is more accurate when the transformations are broken into fewer steps, it is a good idea to pack as many transformations into each step as possible. In our experiments, x-y translations are applied in one step, and rotations are applied in another step. However, the number of possible x-y translations in a 320 × 240 image is 76,800. So, 76,800 dT’s must be computed and the computation of each dT uses a vector norm of size 76,800. It turns out that if the data is defined on a coordinate system where the effect of a transformation is a shift, the above quantities can be computed very quickly using fast Fourier transforms (FFTs). For images measured on rectangular grids, an x-y translation corresponds to a shift in the coordinate system. For images measured on a radial grid, such as the one shown in Fig. 1c, a scale and rotation corresponds to a shift in the coordinate system (Wolberg and Zokai 2000). When updating the variational parameters, it is straightforward to convert them to the appropriate coordinate system, apply the FFT method and convert them back. We now use a very different notation to describe the FFT method. The image is measured on a discrete grid and x is the x-y coordinate of a pixel in the image (x is a 2-vector). The images g and h in (11) and (12) are written as functions of x: g(x), h(x). In this representation, T is an integer 2-vector, corresponding to a shift in x. So, (11) becomes d(T) = X x (g(x) −h(x + T))2 = X x (g(x)2 −2g(x)h(x + T) + h(x + T)2) (13) and (12) becomes X T γ(T)h(x + T). (14) The common form is the correlation f(T) = X x g(x)h(x + T), (15) For an N × N grid, computing the correlation directly for all T takes N 4 scalar operations. The FFT can be used to compute the correlation in N 2 log N time. The FFTs G(ω) and H(ω) of g and h are computed in N 2 log N time. Then, the FFT F(ω) of f is computed in N 2 as follows, F(ω) = G(ω)∗H(ω), (16) where “∗” denotes complex conjugate. Then the inverse FFT f(T) of F(ω) is computed in N 2 log N time. Using this method, the posterior and sufficient statistics for all N 2 shifts in an N × N grid can be computed in N 2 log N time. Using this method along with the variational technique, C classes, N scales, N rotations, N x-translations and N y-translations can be accounted for using (C + 2 log N)N 2 scalar operations. 5 Results In order to compare our new learning algorithm with the previously published result, we repeated the experiment on clustering head poses in 200 44x28 frames. We achieved essentially the same result, but in only 10 seconds as opposed to 40 minutes that the original algorithm needed to compete the task. Both algorithms were implemented in Matlab. It should be noted that the original algorithm tested only for 9 vertical and 9 horizontal shifts (81 combinations), while the new algorithm dealt with all 1232 possible discrete shifts. This makes the new algorithm 600 times faster on low resolution data. The speed-up is even more drastic at higher resolutions, and when rotations and scales are added, since the complexity of the original algorithm is CN 6, where C is the number of classes and N is the number of pixels. The speed-up promised in the abstract is based on our computations, but obviously we were not able to run the original algorithm on full 320x240 resolution data. To illustrate that the fast variational technique presented here can be efficiently used to learn data means in the presence of scale change, significant rotations and translations in the data, we captured 10 seconds of a video at 320x240 resolution and trained a two-stage transformation-invariant where the first stage modeled rotations and scales as shifts in the log-polar coordinate system and the second stage modeled all possible shifts as described above. In Fig. 2 we show the results of training an ordinary Gaussian model, shift-invariant model and finally the scale, rotation and shift invariant model on the sequence. We also show three frames from the sequence stabilized using the variational inference. 6 Conclusions We describes how a tremendous speed-up in training transformation-invariant generative model can be achieved through the use of a variational technique for decoupling transformations, and a fast Fourier transform method for computing posterior probabilities. For N × N images, learning C clusters under N rotations, N scales, N x-translations and N y-translations takes only (C +2 log N)N 2 scalar operations per iteration. In contrast, the original algorithm takes CN 6 operations to account for these transformations. In this way we were able to reduce the computation to only seconds per frame for the images of 320x240 resolution using a simple Matlab implementation. This opens the door for generative models of pixel intensities in video to be efficiently used for transformation-invariant video summary and search. As opposed to most techniques used in computer vision today, the generative modeling approach provides the likelihood model useful for search or retrieval, automatic clustering of the data and the extensibility through adding new hidden variables. The model described here could potentially be useful for other high-dimensional data, such as audio. References Dempster, A. P., Laird, N. M., and Rubin, D. B. 1977. Maximum likelihood from incomplete data via the EM algorithm. Proceedings of the Royal Statistical Society, B-39:1–38. Frey, B. J. and Jojic, N. 2001. Transformation invariant clustering and dimensionality reduction. IEEE Transactions on Pattern Analysis and Machine Intelligence. To appear. Available at http://www.cs.utoronto.ca/∼frey. Figure 2: Learning a rotation, scale and translation invariant model on 320x240 video Hinton, G. E., Dayan, P., and Revow, M. 1997. Modeling the manifolds of images of handwritten digits. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8:65–74. Jojic, N., Petrovic, N., Frey, B. J., and Huang, T. S. 2000. Transformed hidden markov models: Estimating mixture models of images and inferring spatial transformations in video sequences. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., and Saul, L. K. 1998. An introduction to variational methods for graphical models. In Jordan, M. I., editor, Learning in Graphical Models. Kluwer Academic Publishers, Norwell MA. LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. 1998. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324. Neal, R. M. and Hinton, G. E. 1998. A view of the EM algorithm that justifies incremental, sparse, and other variants. In Jordan, M. I., editor, Learning in Graphical Models, pages 355–368. Kluwer Academic Publishers, Norwell MA. Simard, P. Y., LeCun, Y., and Denker, J. 1993. Efficient pattern recognition using a new transformation distance. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5. Morgan Kaufmann, San Mateo CA. Vasconcelos, N. and Lippman, A. 1998. Multiresolution tangent distance for affineinvariant classification. In Jordan, M. I., Kearns, M. I., and Solla, S. A., editors, Advances in Neural Information Processing Systems 10. MIT Press, Cambridge MA. Wolberg, G. and Zokai, S. 2000. Robust image registration using log-polar transform. In Proceedings IEEE Intl. Conference on Image Processing, Vancouver, Canada.
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A Rotation and Translation Invariant Discrete Saliency Network Lance R. Williams Dept. of Computer Science Univ. of New Mexico Albuquerque, NM 87131 John W. Zweck Dept. of CS and EE Univ. of Maryland Baltimore County Baltimore, MD 21250 Abstract We describe a neural network which enhances and completes salient closed contours. Our work is different from all previous work in three important ways. First, like the input provided to V1 by LGN, the input to our computation is isotropic. That is, the input is composed of spots not edges. Second, our network computes a well defined function of the input based on a distribution of closed contours characterized by a random process. Third, even though our computation is implemented in a discrete network, its output is invariant to continuous rotations and translations of the input pattern. 1 Introduction There is a long history of research on neural networks inspired by the structure of visual cortex whose functions have been described as contour completion, saliency enhancement, orientation sharpening, or segmentation[6, 7, 8, 9, 12]. A similiar network has been proposed as a model of visual hallucinations[1]. In this paper, we describe a neural network which enhances and completes salient closed contours. Our work is different from all previous work in three important ways. First, like the input provided to V1 by LGN, the input to our computation is isotropic. That is, the input is composed of spots not edges. Second, our network computes a well defined function of the input based on a distribution of closed contours characterized by a random process. Third, even though our computation is implemented in a discrete network, its output is invariant to continuous rotations and translations of the input pattern. There are two important properties which a computation must possess if it is to be invariant to rotations and translations, i.e., Euclidean invariant. First, the input, the output, and all intermediate representations must be Euclidean invariant. Second, all transformations of these representations must also be Euclidean invariant. The models described in [6, 7, 8, 9, 12] are not Euclidean invariant, first and foremost, because their input representations are not Euclidean invariant. That is, not all rotations and translations of the input can be represented equally well. This problem is often skirted by researchers by choosing input patterns which match particular choices of sampling rate and phase. For example, Li [7] used only six samples in orientation (including ) and Heitger and von der Heydt[5] only twelve (including , and ). Li’s first test pattern was a dashed line of orientation, , while Heitger and von der Heydt used a Kanizsa Triangle with sides of , , and orientation. There is no reason to believe that the experimental results they showed would be similiar if the input patterns were rotated by as little as . To our knowledge, no researcher in this area has ever commented on this problem before. 2 A continuum formulation of the saliency problem The following section reviews the continuum formulation of the contour completion and saliency problem as described in Williams and Thornber[11]. 2.1 Shape distribution Mumford[3] observed that the probability distribution of object boundary shapes could be modeled by a Fokker-Planck equation of the following form:
"! (1) where $#&% $' )( +* is the probability that a particle is located at position, % #,"'-* , and is moving in direction, , at time, . This partial differential equation can be viewed as a set of independent advection equations in and (the first and second terms) coupled in the dimension by the diffusion equation (the third term). The advection equations translate probability mass in direction, , with unit speed, while the diffusion term models the Brownian motion in direction, with diffusion parameter, . The combined effect of these three terms is that particles tend to travel in straight lines, but over time they drift to the left or right by an amount proportional to . Finally, the effect of the fourth term is that particles decay over time, with a half-life given by the decay constant, . 2.2 The propagators The Green’s function, . #&% /' )( 10324% 5 '76 ( 98:* , gives the probability that a particle observed at position, % 5 , and direction, 6 , at time, 8 , will later be observed at position, % , and direction, , at time, 0 . It is the solution, $#&% /' )( 0 * , of the Fokker-Planck initial value problem with initial value, $#&% "' )( 98;* =< #&% % 5 * < # 6>* where < is the Dirac delta function. The Green’s function is used to define two propagators. The long-time propagator: ? 8 #&% /' 2@% 5 '76>* BADC 8FE /GH#I+* . #&% /' )( @24% 5 '16 ( * (2) gives the probability that #&% /' * and #&% 5 '16* are distinct edges from the boundary of a single object. 1 The short-time propagator: ? 0 #:% $' 2@% 5 '16* BA C 8 E KJ GL#,+*NM . #&% /' )( @24% 5 '76 ( * (3) gives the probability that #&% $' * and #:% 5 '16>* are from the boundary of a single object but are really the same edge. In both of these propagators, GH#O!P* is a cut-off function with GL# * and Q RTSIU C GL#,+* : GL#,+* 0 WV X Y[Z7Y W\I] V S ^ `_"a&bca ! (4) The cut-off function is characterized by three parameters, _ , ] , and d . The parameter, _ , specifies where the cut-off is and ] specifies how hard hard it is. The parameter, d , is the scale of the edge detection process. 1We assume that the probability that two edges are the same depends only on the distance between them, and that e/fOg,h ikj h l g m@ne/fpo9m for particles travelling at unit speed. 2.3 Eigenfunctions The integral linear operator, #+! * , combines three sources of information: 1) the probability that two edges belong to the same object; 2) the probability that the two edges are distinct; and 3) the probability that the two edges exist. It is defined as follows: #&% /' 24% 5 '76>* #&% >* ? 8#&% /' 2@% 5 '76>* #&% 5 * (5) where the input bias function, #&% >* , gives the probability that an edge exists at % . As described in Williams and Thornber[11], the right and left eigenfunctions, #+! * and #O!P* , of #O!P* with largest positive real eigenvalue, , play a central role in the computation of saliency: #&% /' * AWATA
E % 5 E 6 #&% $' 24% 5 '76>* #:% 5 '16>* (6) #&% $' * AWATA E % 5 E 6 #:% 5 '16>* #&% 5 '16 24% "' * ! (7) Because #+! * is invariant under a transformation which reverses the order and direction of its arguments: #&% /' 24% 5 '76>* #&% 5 '76 24% "' * (8) the right and left eigenfunctions are related as follows: #:% $' * #:% $' * ! (9) 2.4 Stochastic completion field The magnitude of the stochastic completion field, #&% 5 '16* , equals the probability that a closed contour satisfying a subset of the constraints exists at #&% 5 '76>* . It is the sum of three terms: #&% 5 '76>* 8#&% 5 '16>* 8 #&% 5 '76>* 8#&% 5 '76>* 0 #:% 5 '16* 0 #&% 5 '76>* 8 #&% 5 '76>* AWATA E % E #&% $' * #&% "' * (10) where #&% 5 '76>* is a source field, and #&% 5 '76>* is a sink field: #:% 5 '16>* ATATA E % E ? #&% 5 '16 2@% "' * #&% >* #&% $' * (11) #:% 5 '16>* ATATA E % E #&% /' * #&% >* ? #:% "' 2@% 5 '16*
! (12) The purpose of writing #&% 5 '16* in this way is to remove the contribution, 0 #&% 5 '76>* 0 #&% 5 '16* , of closed contours at scales smaller than d which would otherwise dominate the completion field. Given the above expression for the completion field, it is clear that the key problem is computing the eigenfunction, #O!P* , of #O!P* with largest positive real eigenvalue. To accomplish this, we can use the well known power method (see [4]). In this case, the power method involves repeated application of the linear operator, #O!P* , to the function, #O!P* , followed by normalization: 0 #&% "' * ATATA E % 5 E 6 #&% $' 2 % 5 '76>* #&% 5 '76>* AWA A AWATA E % E E % 5 E 6 #&% $' 2 % 5 '76>* #&% 5 '76>* ! (13) In the limit, as gets very large, 0 #&% /' * converges to the eigenfunction of #+! * , with largest positive real eigenvalue. We observe that the above computation can be considered a continuous state, discrete time, recurrent neural network. 3 A discrete implementation of the continuum formulation The continuous functions comprising the state of the computation are represented as weighted sums of a finite set of shiftable-twistable basis functions. The weights form the coefficient vectors for the functions. The computation we describe is biologically plausible in the sense that all transformations of state are effected by linear transformations (or other vector parallel operations) on the coefficient vectors. 3.1 Shiftable-twistable bases The input and output of the above computation are functions defined on the continuous space, 0 , of positions in the plane, , and directions in the circle, 0 . For such computations, the important symmetry is determined by those transformations,
, of 0 , which perform a shift in by % 8 , followed by a twist in 0 through an angle, 8 . A twist through an angle, 8 , consists of two parts: (1) a rotation,
, of and (2) a translation in 0 , both by 8 . The symmetry,
, which is called a shift-twist transformation, is given by the formula,
#&% $' * #
#&% % 8:*>' k 8:* ! (14) A visual computation, , on 0 is called shift-twist invariant if, for all #&% 8' 8:* 0 , a shift-twist of the input by #&% 8 ' 8 * produces an identical shift-twist of the output. This property can be depicted in the following commutative diagram: #&% "' * #&% /' *
#
#&% % 8 *c' k 8 * #
#&% % 8 *c' k 8 * where #O!P* is the input, #+! * , is the output, is the computation, and ! "# is the shifttwist transformation. Correspondingly, we define a shiftable-twistable basis2 of functions on 0 to be a set of functions on 0 with the property that whenever a function, $ #&% $' * , is in their span, then so is $ #
#&% /' *+* , for every choice of #&% 8 ' 8 * in 0 . As such, the notion of a shiftable-twistable basis on %& 0 generalizes that of a shiftablesteerable basis on [2, 10]. Shiftable-twistable bases can be constructed as follows. Let ' #&% $' * be a function on 0 which is periodic (with period ( ) in both spatial variables, % . In analogy with the definition of a shiftable-steerable function on , we say that ' is shiftable-twistable on ) 0 if there are integers, * and + , and interpolation functions, ,- . #&% 8 ' 8 * , such that for each #:% 8 ' 8 */ 0 , the shift-twist of ' by #:% 8 ' 8 * is a linear combination of a finite number of basic shift-twists of ' by amounts # % 0 d ' d
* , i.e., if ' # 1 2
#&% /' *+* 43 . ,5 . #&% 8' 8;* ' # 6 . ^ ^ " #&% "' *+*
! (15) Here d (879* is the basic shift amount and d
7:+ is the basic twist amount. The sum in equation (15) is taken over all pairs of integers, % 0 # 0 ' 0<; * , in the range, >= 0 ' 0 ;? * , and all integers, , in the range, >= ? + . The Gaussian-Fourier basis is the product of a shiftable-steerable basis of Gaussians in % and a Fourier series basis in . For the experiments in this paper, the standard deviation of the Gaussian basis function, @ #&% * 0 ^&A<BDC C FE ^ , equals the basic shift amount, d . We regard @ #&% * as a periodic function of period, ( , which is chosen to be much larger than d , so that @ # (87 ' (87 * and its derivatives are essentially zero. For each frequency, G , and shift amount, d (where * (H74d is an integer), we define the Gaussian-Fourier basis functions, ' . I , by ' . I #&% $' * @ #&% % 0 d * AKJ IL
! (16) Zweck and Williams[13] showed that the Gaussian-Fourier basis is shiftable-twistable. 2We use this terminology even though the basis functions need not be linearly independent. 3.2 Power method update formula Suppose that #&% $' * can be represented in the Gaussian-Fourier basis as #&% /' * 3 . I . I ' . I #&% /' *c! (17) The vector, , with components, . I , will be called the coefficient vector of #&% /' * . In the next two sections, we demonstrate how the following integral linear transform: 0 #:% $' * A AWA
E % 5 E 6 ? 8 #:% $' 2@% 5 '16>* #&% 5 * #:% 5 '16>* (18) (i.e., the basic step in the power method) can be implemented as a discrete linear transform in a Gaussian-Fourier shiftable-twistable basis: 0 ! (19) 3.3 The propagation operator P In practice, we do not explicitly represent the matrix, . Instead we compute the necessary matrix-vector product using the advection-diffusion-decayoperator in the Gaussian-Fourier shiftable-twistable basis, , described in detail in Zweck and Williams[13]: 0 Q R U C (20) where 8
8 and where: 0 GH# d +*
0 (21)
0 # *
! (22) In the shiftable-twistable basis, the advection operator, , is a discrete convolution: 3 . I , B . B I # d +* . I (23) with the following kernel: , . # d +* 0 X A X 8 , . # d J - ' ) M * "# * E (24) where the , . are sinc functions. Let ! be the number of Fourier series frequencies, G , used in the shiftable-twistable basis, and let d 7"! . The diffusion-decay operator, , is a diagonal matrix: 0 . I A B ^ S E$# # A B J I ^
# * A J I ^
:* . I (25) where % ^ S ^
. 3.4 The bias operator In the continuum, the bias operator effects a multiplication of the function, #:% >* , by the input bias function, #:% * . Our aim is to identify an equivalent linear operator in the shiftabletwistable basis. Suppose that both and are represented in a Gaussian basis, @ . #:% >* . Their product is: #&% >* #&% * 3 . . @ . #&% >*'& 3 @ #&% * 3 . 3 . @ . #&% >* @ #&% >* ! (26) Now, the product of two Gaussian basis functions, @ . and @ , is a Gaussian of smaller variance which cannot be represented in the Gaussian basis, @ . . Because #&% >* #:% >* is a linear combination of the products of pairs of Gaussian basis functions, it cannot be represented in the Gaussian basis either. However, we observe that the convolution of #:% * #:% >* and a Gaussian, #&% * J #:% >* #&% >*9M , where #:% >* 0 ^ X A B E ^ , can be represented in the Gaussian basis. It follows that there exists a matrix, , such that: #&% * J #:% * #&% *NM 3 . J M . @ . #&% *c! (27) The formula for the matrix, , is derived by first completing the square in the exponent of the product of two Gaussians to obtain: @ #&% d % 0 * @ #&% d % * @ # #:% ^ # % 0 % *+** @ # ^ # % 0 % **c! (28) This product is then convolved with to obtain a function, $ #&% >* , which is a shift of the Gaussian basis function, @ #:% * . Finally we use the shiftability formula: @ #&% % 8;* 3 . ,5 . #&% 8[* @ . #&% * (29) where , . are the interpolation functions, @ . #&% * equals @ #:% d % 0 * , and d (879* is the shift amount, to express $ #:% * in the Gaussian basis. The result is: . 3 J J $# 2 2 % $ % 2 2 7 * , . # d # % @ % * 7 *c! (30) 4 Experimental results In our experiments the Gaussian-Fourier basis consisted of *
translates (in each spatial dimension) of a Gaussian (of period, ( ! ), and !
harmonic signals in the orientation dimension. The standard deviation of the Gaussian was set equal to the shift amount, d (879* . For illustration purposes, all functions were rendered at a resolution of 4 4 . The diffusion parameter, , equaled ! , and the decay constant, , equaled ! . The time step, d , used to solve the Fokker-Planck equation in the basis equaled d 7 . The parameters for the cut-off function used to eliminate self-loops were _` and ] ; . In the first experiment, the input bias function, #&% >* , consisted of twenty randomly positioned spots and twenty spots on the boundary of an avocado. The positions of the spots are real valued, i.e., they do not lie on the grid of basis functions. See Fig. 1 (left). The stochastic completion field computed using 32 iterations of the power method is shown in Fig. 1 (right). In the second experiment, the input bias function from the first experiment was rotated by and translated by half the distance between the centers of adjacent basis functions, # #&% J ^ ' ^ M *+* . See Fig. 2 (left). The stochastic completion field is identical (up to rotation and translation) to the one computed in the first experiment. This demonstrates the Euclidean invariance of the computation. See Fig. 2 (right). The estimate of the largest positive real eigenvalue, , as a function of , the power method iteration is shown in Fig. 3. 5 Conclusion We described a neural network which enhances and completes salient closed contours. Even though the computation is implemented in a discrete network, its output is invariant under continuous rotations and translations of the input pattern. References [1] Cowan, J.D., Neurodynamics and Brain Mechanisms, Cognition, Computation and Consciousness, Ito, M., Miyashita, Y. and Rolls, E., (Eds.), Oxford UP, 1997. Figure 1: Left: The input bias function, #:% * . Twenty randomly positioned spots were added to twenty spots on the boundary of an avocado. The positions are real valued, i.e., they do not lie on the grid of basis functions. Right: The stochastic completion field, A #&% 5 '76>* E 6 , computed using
basis functions. Figure 2: Left: The input bias function from Fig. 1, rotated by and translated by half the distance between the centers of adjacent basis functions, # #&% J ^ ' ^ M /*+* . Right: The stochastic completion field, is identical (up to rotation and translation) to the one shown in Fig. 1. This demonstrates the Euclidean invariance of the computation. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 5 10 15 20 25 30 35 Figure 3: The estimate of the largest positive real eigenvalue, , as a function of , the power method iteration. Both the final value and all intermediate values are identical in the rotated and non-rotated cases. [2] Freeman, W., and Adelson, E., The Design and Use of Steerable Filters, IEEE Transactions on Pattern Analysis and Machine Intelligence 13 (9), pp.891-906, 1991. [3] Mumford, D., Elastica and Computer Vision, Algebraic Geometry and Its Applications, Chandrajit Bajaj (ed.), Springer-Verlag, New York, 1994. [4] Golub, G.H. and C.F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins Univ. Press, 1996. [5] Heitger, R. and von der Heydt, R., A Computational Model of Neural Contour Processing, Figure-ground and Illusory Contours, Proc. of 4th Intl. Conf. on Computer Vision, Berlin, Germany, 1993. [6] Iverson, L., Toward Discrete Geometric Models for Early Vision, Ph.D. dissertation, McGill University, 1993. [7] Li, Z., A Neural Model of Contour Integration in Primary Visual Cortex, Neural Computation 10(4), pp. 903-940, 1998. [8] Parent, P., and Zucker, S.W., Trace Inference, Curvature Consistency and Curve Detection, IEEE Transactions on Pattern Analysis and Machine Intelligence 11, pp. 823-889, 1989. [9] Shashua, A. and Ullman, S., Structural Saliency: The Detection of Globally Salient Structures Using a Locally Connected Network, 2nd Intl. Conf. on Computer Vision, Clearwater, FL, pp. 321-327, 1988. [10] Simoncelli, E., Freeman, W., Adelson E. and Heeger, D., Shiftable Multiscale Transforms, IEEE Trans. Information Theory 38(2), pp. 587-607, 1992. [11] Williams, L.R., and Thornber, K.K., Orientation, Scale, and Discontinuity as Emergent Properties of Illusory Contour Shape, Neural Computation 13(8), pp. 16831711, 2001. [12] Yen, S. and Finkel, L., Salient Contour Extraction by Temporal Binding in a Cortically-Based Network, Neural Information Processing Systems 9, Denver, CO, 1996. [13] Zweck, J., and Williams, L., Euclidean Group Invariant Computation of Stochastic Completion Fields Using Shiftable-Twistable Functions, Proc. European Conf. on Computer Vision (ECCV ’00), Dublin, Ireland, 2000.
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A theory of neural integration in the head-direction system Richard H.R. Hahnloser , Xiaohui Xie and H. Sebastian Seung Howard Hughes Medical Institute Dept. of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 rhahnloser|xhxie|seung @mit.edu Abstract Integration in the head-direction system is a computation by which horizontal angular head velocity signals from the vestibular nuclei are integrated to yield a neural representation of head direction. In the thalamus, the postsubiculum and the mammillary nuclei, the head-direction representation has the form of a place code: neurons have a preferred head direction in which their firing is maximal [Blair and Sharp, 1995, Blair et al., 1998, ?]. Integration is a difficult computation, given that head-velocities can vary over a large range. Previous models of the head-direction system relied on the assumption that the integration is achieved in a firing-rate-based attractor network with a ring structure. In order to correctly integrate head-velocity signals during high-speed head rotations, very fast synaptic dynamics had to be assumed. Here we address the question whether integration in the head-direction system is possible with slow synapses, for example excitatory NMDA and inhibitory GABA(B) type synapses. For neural networks with such slow synapses, rate-based dynamics are a good approximation of spiking neurons [Ermentrout, 1994]. We find that correct integration during high-speed head rotations imposes strong constraints on possible network architectures. 1 Introduction Several network models have been designed to emulate the properties of head-direction neurons (HDNs) [Zhang, 1996, Redish et al., 1996, Goodridge and Touretzky, 2000]. The model by Zhang reproduces persistent activity during stationary head positions. Persistent neural activity is generated in a ring-attractor network with symmetric excitatory and inhibitory synaptic connections. Independently, he and Redish et al. showed that integration is possible by adding asymmetrical connections to the attractor network. They assumed that the strength of these asymmetrical connections is modulated by head-velocity. When the rat moves its head to the right, the asymmetrical connections induce a rightward shift of the activity in the attractor network. A more plausible model without multiplicative modulation of connections has been studied recently by Goodridge and Touretzky. There, the head-velocity input has a modulatory influence on firing rates of intermittent neurons rather than on connection strengths. The intermittent neurons are divided into two groups that make spatially offset connections, one group to the right, the other to the left. The different types of neurons in the Goodridge and Touretzky model have firing properties that are comparable to neurons in the various nuclei of the head-direction system. What all these previous models have in common is that the integration is performed in an inherent double-ring network with very fast synapses (less than ms for [Goodridge and Touretzky, 2000]). The connections made by one ring are responsible for rightward turns and the connections made by the other ring are responsible for leftward turns. In order to derive a network theory of integration valid for fast and slow synapses, here we solve a simple double-ring network in the linear and in the saturated regimes. An important property of the head-direction system is that the integration be linear over a large range of head-velocities. We are interested in finding those type of synaptic connections that yield a large linear range and pose our findings as predictions on optimal network architectures. Although our network is conceptually simpler than previous models, we show that using two simple read-out methods, averaging and extracting the maximum, it is possible to approximate head-velocity independent tuning curves as observed in the Postsubiculum (PoS) and anticipatory responses in the anterior dorsal thalamus (ADN). 2 Definition of the model We assume that the number of neurons in the double-ring network is large and write its dynamics as a continuous neural field
(1)
(2) where ! #" $&% (') '+* $), ('& .-/ 102.354 '76 398: ! " $ , ('& ' -/ ; $ % ('& * 102.3 4 6 398:=< " > 0 : @?BADC E
> denotes a rectification nonlinearity. and are the firing rates of neurons in the left and right ring, respectively. The quantities D and F represent synaptic activations (amount of neurotransmitter release caused by the firing rates and ). is a synaptic time constant. The vestibular inputs 3G4 ' 6 3 and 354H 6 3 are purely excitatory, ' 354JI 6 3 IK354 . For simplicity, we assume that 6 3 is proportional to angular headvelocity. The synaptic connection profiles $ % between neurons on the same ring and $ , between neurons on different rings are given by: $)% LNM4OJM QPSRT $), UWVB4L V XPGRT S< (3) M4 , M , VB4 and V define the intra and inter-ring connection strengths. * is the intra-ring connection offset and the inter-ring offset. 3 Integration When the animal is not moving, the vestibular inputs to the two rings are equal, 6 3O E . In this case, within a certain range of synaptic connections, steady bumps of activities appear on the two rings. When the head of the animal rotates, the activity bumps travel at a velocity determined by 6 3 . For perfect integration, should be proportional to 6 3 over the full range of possible head-velocities. This is a difficult computational problem, in particular for slow synapses. 4 Small head-velocity approximation When the head is not rotating ( 6 3 E ), the two stationary bumps of synaptic activation are of the form " PGRT (') 4 ' 0 : and L " PSR T (') 4 ' 0 : (4) where 4 is the current head direction and is the offset between the two bumps. How to calculate , and is shown in the Appendix. The half width of these bumps is given by WA PPSRT
S< (5) When the angular head velocity is small ( 6 3
3S4 ), we linearize the dynamics around the stationary solution Eq. (4), see Appendix. We find that G (' " ' PGRT ('& 4 ' ' ' 0 : (6) F (' " PGRT ('& 4 ' ; 'J 10 : (7) where the velocity is given by 6 3 M XT 1* " M4 ' VB4F ' 'J M4 ' VB4 T 0 (8) and M PGRT * ' V ' M T * (9) T PGRT ' ! T < (10) Equation (8) is the desired result, relating the velocity of the two bumps to the differential vestibular input 6 3 . In Fig. 1 we show simulation results using slow synapses ( " E ms). The integration is linear over almost the entire range of head-velocities (up to more than # EE $%
!'&%( ) when V M , i.e., when the amplitudes of inter-ring and intra-ring connections are equal. We point out that the condition V NM cannot directly be deduced from the above formulas, some empirical tuning (for example V 4 E ) was necessary to achieve this large range of linearity (large both in 6 3 and ). When the bumps move, their amplitudes tend to decrease. Fig. 1d shows the peak firing rates of neurons in the two rings as a function of vestibular input. As can be seen, the firing rates are a linear function of vestibular input, in agreement with equations 17 and 18 of the Appendix. However, a linear firing-rate modulation by head velocity is not universal, for some parameters we have seen asymmetrically head-velocity tuning, with a preference for small head velocities (not shown). a. b. −1 −0.5 0 0.5 1 −800 −600 −400 −200 0 200 400 600 800 ∆ b/b0 v (degrees/sec) Simulation Theory −1 −0.5 0 0.5 1 −800 −600 −400 −200 0 200 400 600 800 ∆ b/b0 v (degrees/sec) Simulation Theory c. d. −1 −0.5 0 0.5 1 −600 −400 −200 0 200 400 600 ∆ b/b0 v (degrees/sec) Simulation Theory −1 −0.5 0 0.5 1 50 55 60 65 70 75 ∆ b/b0 Firing rate (Hz) Left Right Figure 1: Velocity of activity bumps as a function of vestibular input 6 3
3G4 . a. Sublinear integration. V M , * E $ , VB4 E . b. Supralinear integration. V M , * $ , V 4 E . c. Linear (perfect) integration. V M , * $ , V 4 E . d. Head-velocity dependent modulation of firing rates (on the right and on the left ring). Same parameters as in c. " E ms. * " $ , and " $ . 5 Saturating velocity When 6 3 is very large, at some point, the left ring becomes inactive. Because inactivating the left ring means that the push-pull competition between the two rings is minimized, we are able to determine the saturating velocity of the double-ring network. The saturating velocity is given by the on-ring connections $ % . Define $ M4 @M XPGRT ('+* M4 @M XPGRT * PSR T 9A * T $&% ' 9A * $ % where $)% M4 M XPGRT * PGRT . Now, let be the steady solution of a ring network with symmetric connections $&% . By differentiating, it follows that '
is the solution of a ring network with connections $ . Hence, the saturating velocity is given by 9A * < (11) Notice that a traveling solution may not always exist if one ring is inactive (this is the case when there are no intra-ring excitatory connections). However, even without a traveling solution, equation (11) remains valid. In Figs. 1a and b, the saturating velocity is indicated by the horizontal dotted lines, in Fig. 1a we find " E $%
!'&%( and in Fig. 1b E $'
'&'( . 6 ADN and POs neurons Goodridge and Touretzky’s integrator model was designed to emulate details of neuronal tuning as observed in the different areas of the head-direction system. Wondering whether the simple double ring studied here can also reproduce multiple tuning curves, we analyze simple read-out methods of the firing rates and . What we find is that two readout methods can indeed approximate response behavior resembling that of ADN and POs neurons. ADN neurons: By reading out firing rates using a maximum operation, ?BADC ! , anticipatory head-direction tuning arises due to the fact that there is an activity offset between the two rings, equation (13). When the head turns to the right, the activity on the right ring is larger than on the left ring and so the tuning of is biased to the right. Similarly, for left turns, is biased to the left. Thus, the activity offset between the two rings leads to an anticipation time for ADN neurons, see Figure 2. Because, by assumption is head-velocity independent, it follows that is inversely proportional to head-velocity (assuming perfect integration),
. In other words, the anticipation time tends to be smaller for fast head rotations and larger for slow head rotations. POs neurons: By reading out the double ring activity as an average,
D , neurons in POs do not have any anticipation time: because averaging is a symmetric operation, all information about the direction of head rotations is lost. 0 90 180 270 360 Head−direction (degs) Firing Rate Left ring Right ring Max Average Left turn Right turn Figure 2: Snapshots of the activities on the two rings (top). Reading out the activities by averaging and by a maximum operation (bottom). 7 Discussion Here we discuss how the various connection parameters contribute to the double-ring network to function as an integrator. In particular we discuss how parameters have to be tuned in order to yield an integration that is large in 6 3 and in . : By assumption the synaptic time constant is large. has the simplest effect of all parameters on the integrator properties. According to equation (8), scales the range of . Notice that if were small, a large range of could be trivially achieved. The art here is to achieve this with large . * : The connection offset * between neurons receiving similar vestibular input is the sole parameter besides determing the saturating head-velocity, beyond which integration is impossible. According to equation (11), the saturating velocity is large if * is close to E $ (we want the saturating velocity to be large). In other words, for good integration, excitatory connections should be strongest (or inhibitory connections weakest) for neuron pairs with preferred head-directions differing by close to E $ . - : The connection offset between neurons receiving different vestibular input determines the anticipation time of thalamic neurons. If is large, then , the activity offset in equation (13) is large. And, because is proportional to (assuming perfect integration), we conclude that should preferentially be large (close to E $ ) if is to be large. Notice that by equation (8), the range of is not affected by - . VB4 and V : The inter-ring connections should be mainly excitatory, which implies that VB4 should not be too negative ( V=4 E was found to be optimal). The intuitive reason is the following. We want the integration to be as linear in 6 3 as possible, which means that we want our linear expansions (6) and (7) to deviate as little as possible from (4). Hence, the differential gain between the two rings should be small, which is the case when the two rings excite each other. The interring excitation makes sure, even for large values of 6 3 , that there are comparable activity levels on the two rings. This is one of the main points of this study. M4 and M : The intra-ring connections should be mainly inhibitory, which implies that M4 should be strongly negative. The reason for this is that inhibition is necessary to result in proper and stable integration. Since inhibition cannot come from the inter-ring connections, it has to come from M 4 . Notice also that according to equation (15), M cannot be much larger than V . If this were the case, the persistent activity in the no head-movement case would become unstable. For linear integration we have found that the condition V M is necessary; small deviations from this condition cause the integrator to become sub- or supralinear. 8 Conclusion We have presented a theory for integration in the head-direction system with slow synapses. We have found that in order to achieve a large range of linear integration, there should be strong excitatory connections between neurons with dissimilar head-velocity tuning and inhibitory connections between neurons with similar head-velocity tuning (see the discussion). Similar to models of the occulomotor integrator [Seung, 1996], we have found that linear integration can only be achieved by precise tuning of synaptic weights (for example V NM ). Appendix To study the traveling pulse solution with velocity , it is convenient to go into a moving coordinate frame by the change of variables B' . The stationary solution in the moving frame reads ' and ' W (12) Set E . In order to find the fixed points of equation (12), we use the ansatz (4) and equate the coefficients of the 3 Fourier modes T , PGRT and the -independent mode. This leads to A PT M QT 1*
V ' (13) 3 4 ' M4L.VB4F 4 ' PGRT (14) M XPGRT * V ' M T 1* 8 (15) where the functions D4 and are given by 4 U D " T '& PGRT 10 D " ' T 0 < The above set of equations fully characterize the solution for E . Eq. (13) determines the offset between the two rings, eq. (15) determines the threshold , eq. (14) the amplitude and eq. (5) the bias . When the vestibular input 6 3 is small, we assume that the perturbed solution around and takes the form: PGRT (') 4 ' PGRT ') 4F ' S< We linearize the dynamics (12) (to first order in ) and equate the Fourier coefficients. This leads to M T * ! " T
' T 0 (16) where ' and ' . We determine and by solving the linearized dynamics of the differential mode
' ' PSR T ' 4F ' . Comparing once more the Fourier coefficients leads to 6 3 " ' ' T 0 (17) 6 3 " ' ' T 0 (18) where M 4 ' V 4
. By substituting and into Eq. (16), we find equation (8). References [Blair et al., 1998] Blair, H., Cho, J., and Sharp, P. (1998). Role of the lateral mammillary nucleus in the rat head direction circuit: A combined single unit recording and lesion study. Neuron, 21:1387–1397. [Blair and Sharp, 1995] Blair, H. and Sharp, P. (1995). Anticipatory head diirection signals in anterior thalamus: evidence for a thalamocortical circuit that integrates angular head motion to compute head direction. The Journal of Neuroscience, 15(9):6260–6270. [Ermentrout, 1994] Ermentrout, B. (1994). Reduction of conductance-based models with slow synapses to neural nets. Neural Computation, 6:679–695. [Goodridge and Touretzky, 2000] Goodridge, J. and Touretzky, D. (2000). Modeling attractor deformation in the rodent head-direction system. The Journal of Neurophysiology, 83:3402–3410. [Redish et al., 1996] Redish, A., Elga, A. N., and Touretzky, D. (1996). A coupled attractor model of the rodent head direction system. Network: Computation in Neural Systems, 7:671–685. [Seung, 1996] Seung, H. S. (1996). How the brain keeps the eyes still. Proc. Natl. Acad. Sci. USA, 93:13339–13344. [Zhang, 1996] Zhang, K. (1996). Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: A theory. J. Neurosci., 16(6):2112–2126.
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(Not) Bounding the True Error John Langford Department of Computer Science Carnegie-Mellon University Pittsburgh, PA 15213 jcl+@cs.cmu.edu Rich Caruana Department of Computer Science Cornell University Ithaca, NY 14853 caruana@cs.cornell.edu Abstract We present a new approach to bounding the true error rate of a continuous valued classifier based upon PAC-Bayes bounds. The method first constructs a distribution over classifiers by determining how sensitive each parameter in the model is to noise. The true error rate of the stochastic classifier found with the sensitivity analysis can then be tightly bounded using a PAC-Bayes bound. In this paper we demonstrate the method on artificial neural networks with results of a order of magnitude improvement vs. the best deterministic neural net bounds. 1 Introduction In machine learning it is important to know the true error rate a classifier will achieve on future test cases. Estimating this error rate can be suprisingly difficult. For example, all known bounds on the true error rate of artificial neural networks tend to be extremely loose and often result in the meaningless bound of “always err” (error rate = 1.0). In this paper, we do not bound the true error rate of a neural network. Instead, we bound the true error rate of a distribution over neural networks which we create by analysing one neural network. (Hence, the title.) This approach proves to be much more fruitful than trying to bound the true error rate of an individual network. The best current approaches [1][2] often require , , or more examples before producing a nontrivial bound on the true error rate. We produce nontrivial bounds on the true error rate of a stochastic neural network with less than examples. A stochastic neural network is a neural network where each weight
is perturbed by a gaussian with variance every time it is evaluated. Our approach uses the PAC-Bayes bound [5]. The approach can be thought of as a redivision of the work between the experimenter and the theoretician: we make the experimenter work harder so that the theoretician’s true error bound becomes much tighter. This “extra work” on the part of the experimenter is significant, but tractable, and the resulting bounds are much tighter. An alternative viewpoint is that the classification problem is finding a hypothesis with a low upper bound on the future error rate. We present a post-processing phase for neural networks which results in a classifier with a much lower upper bound on the future error rate. The post-processing can be used with any artificial neural net trained with any optimization method; it does not require the learning procedure be modified, re-run, or even that the threshold function be differentiable. In fact, this post-processing step can easily be adapted to other learning algorithms. David MacKay [4] has done significant work to make approximate Bayesian learning tractable with a neural network. Our work here is complimentary rather than competitive. We exhibit a technique which will likely give nontrivial true error rate bounds for Bayesian neural networks regardless of approximation or prior modeling errors. Verification of this statement is work in progress. The post-processing step finds a “large” distribution over classifiers, which has a small average empirical error rate. Given the average empirical error rate, it is straightforward to apply the PAC-Bayes bound in order to find a bound on the average true error rate. We find this large distribution over classifiers by performing a simple noise sensitivy analysis on the learned model. The noise model allows us to generate a distribution of classifiers with a known, small, average empirical error rate. In this paper we refer to the distribution of neural nets that results from this noise analysis as a stochastic neural net model. Why do we expect the PAC-Bayes bound to be a significant improvement over standard covering number and VC bound approaches? There exist learning problems for which the difference between the lower bound and the PAC-Bayes upper bound are tight up to where is the number of training examples. This is superior to the guarantees which can be made for typical covering number bounds where the gap is, at best, known up to an (asymptotic) constant. The guarantee that PAC-Bayes bounds are sometimes quite tight encourages us to apply them here. The next sections will: 1. Describe the bounds we will compare. 2. Describe our algorithm for constructing a distribution over neural networks. 3. Present experimental results. 2 Theoretical setup We will work in the standard supervised batch learning setting. This setting starts with the assumption that all examples are drawn from some fixed (unknown) distribution, , over (input, output) pairs,
. The output is drawn from the space and the input space is arbitrary. The goal of machine learning is to use a sample set of pairs to find a classifier, , which maps the input space to the output space and has a small true error, ! #"%$&
(' ) . Since the distribution is unknown, the true error rate is not observable. However, we can observe the empirical error rate, * +,- . #"/
0' ) ) 1 32 54 1
6' ) . Now that the basic quantities of interest are defined, we will first present a modern neural network bound, then specialize the PAC-Bayes bound to a stochastic neural network. A stochastic neural network is simply a neural network where each weight in the neural network is drawn from some distribution whenever it is used. We will describe our technique for constructing the distribution of the stochastic neural network. 2.1 Neural Network bound We will compare a specialization of the best current neural network true error rate bound [2] with our approach. The neural network bound is described in terms of the following parameters: 1. A margin, 798:7 . 2. An arbitrary function (unrelated to the neural network sigmoid function) ; defined by ;< =
) if
:7 , ;< =
) if
?> , and linear in between. 3. @ , an upper bound on the sum of the magnitude of the weights in the A th layer of the neural network 4. B , a Lipschitz constant which holds for the A th layer of the neural network. A Lipschitz constant is a bound on the magnitude of the derivative. 5. C , the size of the input space. With these parameters defined, we get the following bound. Theorem 2.1 (2 layer feed-forward Neural Network true error bound) D" $ EGF (HJI K L>9M5NPO QSR T8UWVYX[Z where R 8 ) 1 [2 ; Q
Q 1 B 1 B @ 1 @ Proof: Given in [2]. The theorem is actually only given up to a universal constant. “ ” might be the right choice, but this is just an educated guess. The neural network true error bound above is (perhaps) the tightest known bound for general feed-forward neural networks and so it is the natural bound to compare with. This 2 layer feed-forward bound is not easily applied in a tight manner because we can’t calculate a priori what our weight bound @ should be. This can be patched up using the principle of structural risk minimization. In particular, we can state the bound for @ 1 )! #" where $ is some non-negative integer and > is a constant. If the $ th bound holds with probability % '& " , then all bounds will hold simultaneously with probability Z , since ( ) " 4 1 $ )+* , Applying this approach to the values of both @ 1 and @ , we get the following theorem: Theorem 2.2 (2 layer feed-forward Neural Network true error bound) #" $ E F HJI K L>3M NO Q ".R T8 /$0 V X9Z where R 8<1$ 0 ) 1 2 ; Q
Q 1 B 1 B " 2354 7689 :; </= Proof: Apply the union bound to all possible values of $ and 0 as discussed above. In practice, we will use ) 2 ) > and report the value of the tightest applicable bound for all $.0 . 2.2 Stochastic Neural Network bound Our approach will start with a simple refinement [3] of the original PAC-Bayes bound [5]. We will first specialize this bound to stochastic neural networks and then show that the use of this bound in conjunction with a post-processing algorithm results in a much tighter true error rate upper bound. First, we will need to define some parameters of the theorem. 1. ? is a distribution over the hypotheses which can be found in an example dependent manner. 2. @ is a distribution over the hypotheses which is chosen a priori—without dependence on the examples. 3. BA )+CD E AD is the true error rate of the stochastic hypothesis which, in any evaluation, draws a hypothesis from ? , and outputs
. 4. * BA )FCG HE A * +, is the average empirical error rate of the same stochastic hypothesis. Now, we are ready to state the theorem. Theorem 2.3 (PAC-Bayes Relative Entropy Bound) For all priors, @ , #" $ E F ? K KL * A +,JII HA LK KL M?NIOI @ QP N & V X Z where KL /?NIOI @ )SR UT P NWV O X Y J C is the Kullback-Leibler divergence between the distributions ? and @ and KL * ZA IOI A is the KL divergence between a coin of bias * HA and a coin of bias [A . Proof: Given in [3]. We need to specialize this theorem for application to a stochastic neural network with a choice of the “prior”. Our “prior” will be zero on all neural net structures other than the one we train and a multidimensional isotropic gaussian on the values of the weights in our neural network. The multidimensional gaussian will have a mean of and a variance in each dimension of R . This choice is made for convenience and happens to work. The optimal value of R is unknown and dependent on the learning problem so we will wish to parameterize it in an example dependent manner. We can do this using the same trick as for the original neural net bound. Use a sequence of bounds where R ) " for and some constants and $ a nonnegative number. For the $ th bound set Z % & " . Now, the union bound will imply that all bounds hold simultaneously with probability at least Z . Now, assuming that our “posterior” ? is also defined by a multidimensional gaussian with the mean and variance in each dimension defined by and , we can specialize to the following corollary: Corollary 2.4 (Stochastic Neural Network bound) Let 0 be the number of weights in a neural net, be the A th weight and be the variance of the A th weight. Then, we have #" $ F ? K KL * A +,JII HA K9M5NPO " 2 4 1 P N 9
9 1 P N " & X Z (1) Proof: Analytic calculation of the KL divergence between two multidimensional Gaussians and the union bound applied for each value of $ . We will choose ) > and -) > as reasonable default values. One more step is necessary in order to apply this bound. The essential difficulty is evaluting * A . This quantity is observable although calculating it to high precision is difficult. We will avoid the need for a direct evaluation by a monte carlo evaluation and a bound on the tail of the monte carlo evaluation. Let * A +, #" A /
' ) be the observed rate of failure of a random hypotheses drawn according to ? and applied to a random training example. Then, the following simple bound holds: Theorem 2.5 (Sample Convergence Bound) For all distributions, ? , for all sample sets , #" A E KL * A IOI* HA +, K P N & VX[Z where is the number of evaluations of the stochastic hypothesis. Proof: This is simply an application of the Chernoff bound for the tail of a Binomial where a “head” occurs when an error is observed and the bias is * A +, . In order to calculate a bound on the expected true error rate, we will first bound the expected empirical error rate * ZA +, with confidence & then bound the expected true error rate A with confidence & , using our bound on * A +, . Since the total probability of failure is only & & ) Z our bound will hold with probability Z . In practice, we will use ) evaluations of the empirical error rate of the stochastic neural network. 2.3 Distribution Construction algorithm One critical step is missing in the description: How do we calculate the multidimensional gaussian, ? ? The variance of the posterior gaussian needs to be dependent on each weight in order to achieve a tight bound since we want any “meaningless” weights to not contribute significantly to the overall sample complexity. We use a simple greedy algorithm to find the appropriate variance in each dimension. 1. Train a neural net on the examples 2. For every weight, , search for the variance, , which reduces the empirical accuracy of the stochastic neural network by some fixed target percentage (we use ) while holding all other weights fixed. 0.01 0.1 1 10 100 10000 100000 error pattern presentations SNN bound NN bound SNN Train error NN Train error SNN Test error NN Test error 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10000 100000 error pattern presentations Figure 1: Plot of measured errors and error bounds for the neural network (NN) and the stochastic neural network (SNN) on the synthetic problem. The training set has 100 cases and the reduction in empirical error is 5%. Note that a true error bound of “100” (visible in the graph on the left) implies that at least more examples are required in order to make a nonvacuous bound. The graph on the right expands the vertical scale by excluding the poor true error bound that has error above 100. The curves for NN and SNN are qualitatively similar on the train and test sets. As expected, the SNN consistently performs 5% worse than the NN on the train set (easier to see in the graph on the right). Surprisingly, the SNN performs worse than the NN by less than 5% on the test sets. Both NN and SNN exhibit overfitting after about 6000-12000 pattern presentations (600-1200 epochs). The shape of the SNN bound roughly mimics the shape of the empirically measured true error (this is more visible in the graph on the right) and thus might be useful for indicating where the net begins overfitting. 3. The stochastic neural network defined by
will generally have a too-large empirical error. Therefore, we calculate a global multiplier such that the stochastic neural network defined by
decreases the empirical accuracy by only the same (absolute error rate) used in Step 2. 4. Then, we evaluate the empirical error rate of the resulting stochastic neural net by repeatedly drawing samples from the stochastic neural network. In the work reported here we use samples. 3 Experimental Results How well can we bound the true error rate of a stochastic neural network? The answer is much better than we can bound the true error rate of a neural network. We use two datasets to empirically evaluate the quality of the new bound. The first is a synthetic dataset which has 25 input dimensions and one output dimension. Most of these dimensions are useless—simply random numbers drawn from a !" # $ Gaussian. One of the 25 input dimensions is dependent on the label. First, the label % is drawn uniformly from & # , then the special dimension is drawn from a !%' # $ Gaussian. Note that this learning problem can not be solved perfectly because some examples will be drawn from the tails where the gaussians overlap. The “ideal” neural net to use in solving this synthetic problem is a single node perceptron. We will instead use a 2 layer neural net with 2 hidden nodes using the sigmoid transfer function. This overly complex neural net will result in the potential for significant overfitting which makes the bound prediction problem interesting. It is also somewhat more “realistic” if the neural net structure does not exactly match the learning problem. The second dataset is the ADULT problem from the UCI Machine Learning Repository. We use a 2 layer neural net with 2 hidden units for this problem as well because preliminary experiments showed that nets this small can overfit the ADULT dataset if the training sample is small. To keep things challenging, we use just (*)+ examples in our experiments. As 0.01 0.1 1 10 100 10000 100000 error pattern presentations SNN bound NN bound SNN Train error NN Train error SNN Test error NN Test error 0 0.1 0.2 0.3 0.4 0.5 0.6 10000 100000 error pattern presentations Figure 2: Plot of measured errors and error bounds for the neural network (NN) and the stochastic neural network (SNN) on the UCI ADULT dataset. These graphs show the results obtained using a 1% reduction in empirical error instead of the 5% reduction used in Figure 1. The training sample size for this problem is 200 cases. NN and SNN exhibit overfitting after approximately 12000 pattern presentations (600 epochs). As in Figure 1, a true error bound of “100” implies that at least more examples are required in order to make a nonvacuous bound. The graph on the right expands the vertical scale by excluding the poor true error bound. we will see in Figures 1 and 2, constructing a nonvacuous bound for a continuous hypothesis space with only ) examples is quite difficult. The conventional bounds are hopelessly loose. Figure 1 shows the results for the synthetic problem. For this problem we use 100 training cases and a 5% reduction in empirical error. The results for the ADULT problem are presented in Figure 2. For this problem we use 200 training cases and a 1% reduction in empirical error. Experiments performed on these problems using somewhat smaller and larger training samples yielded similar results. The choice of reduction in empirical error is somewhat arbitrary. We see qualitatively similar results if we switch to a 1% reduction for the synthetic problem and a 5% reduction for the ADULT problem. There are several things worth noting about the results in the two figures. 1. The SNN upper bounds are 2-3 orders of magnitude lower than the NN upper bounds. While not as tight as might be desired, the SNN upper bounds are orders of magnitude better and are not vacuous. 2. The SNNs perform somewhat better than expected. In particular, on the synthetic problem the SNN true error rate is at most worse than the true error rate of the NN (true error rates are estimated using large test sets). This is suprising considering that we fixed the difference in empirical error rates at for the synthetic problem. Similarly, on the ADULT problem we observe that the true error rates between the SNN and NN typically is only about 0.5%, about half of the target difference of 1%. This is good because it suggests that we do not lose as much accuracy as might be expected when creating the SNN. 3. On both test problems, the shape of the SNN bound is somewhat similar to the shape of the true error rate. In particular, the local minima in the SNN bound occur roughly where the local minima in the true error rates occur. The SNN bound may weakly predict the overfitting points of the SNN and NN nets. The comparison between the neural network bound and the stochastic neural network bound is not quite “fair” due to the form of the bound. In particular, the stochastic neural network bound can never return a value greater than “always err”. This implies that when the bound is near the value of “ ”, it is difficult to judge how rapidly extra examples will improve the stochastic neural network bound. We can judge the sample complexity of the stochastic bound by plotting the value of the numerator in equation 1. Figure 3 plots the complexity versus the number of pattern presentations in training. In this figure, we 1 10 100 10000 100000 Complexity pattern presentations Complexity Figure 3: We plot the “complexity” of the stochastic network model (numerator of equation 1) vs. training epoch. Note that the complexity increases with more training as expected and stays below , implying nonvacuous bounds on a training set of size . observe the expected result: the “complexity” (numerator of equation 1) increases with more training and is significantly less than the number of examples (100). The stochastic bound is a radical improvement on the neural network bound but it is not yet a perfectly tight bound. Given that we do not have a perfectly tight bound, one important consideration arises: does the minimum of the stochastic bound predict the minimum of the true error rate (as predicted by a large holdout dataset). In particular, can we use the stochastic bound to determine when we should cease training? The stochastic bound depends upon (1) the complexity which increases with training time and (2) the training error which decreases with training time. This dependence results in a minima which occurs at approximately 12000 pattern presentations for both of our test problems. The point of minimal true error (for the stochastic and deterministic neural networks) occurs at approximately 6000 pattern presentations for the synthetic problem, and at about 18000 pattern presentations for the ADULT problem, indicating that the stochastic bound weakly predicts the point of minimum error. The neural network bound has no such minimum. Is the choice of 1-5% increased empirical error optimal? In general, the “optimal” choice of the extra error rate depends upon the learning problem. Since the stochastic neural network bound (corollary 2.4) holds for all multidimensional gaussian distributions, we are free to optimize the choice of distribution in anyway we desire. Figure 4 shows the resulting bound for different choices of posterior ? . The bound has a minimum at > extra error indicating that our initial choices of > and > are in the right ballpark, and > may be unnecessarily large. Larger differences in empirical error rate such as > are easier to obtain reliably with fewer samples from the stochastic neural net, but we have not had difficulty using as few as 100 samples from the SNN with as small as a 1% increase in empirical error. Also note that the complexity always decreases with increasing entropy in the distribution of our stochastic neural net. The existence of a minimum in Figure 4 is the “right” behaviour: the increased empirical error rate is significant in the calculation of the true error bound. 4 Conclusion We have applied a PAC-Bayes bound for the true error rate of a stochastic neural network. The stochastic neural network bound results in a radically tighter ( orders of mag0.1 1 10 100 0 0.02 0.04 0.06 0.08 0.1 true error bound or complexity extra training error Stochastic NN bound Complexity Figure 4: Plot of the stochastic neural net (SNN) bound for “posterior” distributions chosen according to the extra empirical error they introduce. nitude) bound on the true error rate of a classifier while increasing the empirical and true error rates only a small amount. Although, the stochastic neural net bound is not completely tight, it is not vacuous with just examples and the minima of the bound weakly predicts the point where overtraining occurs. The results with two datasets (one synthetic and one from UCI) are extremely promising—the bounds are orders of magnitude better. Our next step will be to test the method on more datasets using a greater variety of net architectures to insure that the bounds remain tight. In addition, there remain many opportunities for improving the application of the bound. For example, it is possible that shifting the weights when finding a maximum acceptable variance will result in a tighter bound. Also, we have not taken into account symmetries within the network which would allow for a tighter bound calculation. References [1] Peter Bartlett, “The Sample Complexity of Pattern Classification with Neural Networks: The Size of the Weights is More Important than the Size of the Network”, IEEE Transactions on Information Theory, Vol. 44, No. 2, March 1998. [2] V. Koltchinskii and D. Panchenko, “Empirical Margin Distributions and Bounding the Generalization Error of Combined Classifiers”, preprint, http://citeseer.nj.nec.com/386416.html [3] John Langford and Matthias Seeger, “Bounds for Averaging Classifiers.” CMU tech report, 2001. [4] David MacKay, “Probable Networks and Plausible Predictions - A Review of Practical Bayesian Methods for Supervised Neural Networks”, ?? [5] David McAllester, “Some PAC-Bayes bounds”, COLT 1999.
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Novel iteration schemes for the Cluster Variation Method Hilbert J. Kappen Department of Biophysics Nijmegen University Nijmegen, the Netherlands bert©mbfys.kun.nl Wim Wiegerinck Department of Biophysics Nijmegen University Nijmegen, the Netherlands wimw©mbfys.kun.nl Abstract The Cluster Variation method is a class of approximation methods containing the Bethe and Kikuchi approximations as special cases. We derive two novel iteration schemes for the Cluster Variation Method. One is a fixed point iteration scheme which gives a significant improvement over loopy BP, mean field and TAP methods on directed graphical models. The other is a gradient based method, that is guaranteed to converge and is shown to give useful results on random graphs with mild frustration. We conclude that the methods are of significant practical value for large inference problems. 1 Introduction Belief Propagation (BP) is a message passing scheme, which is known to yield exact inference in tree structured graphical models [1]. It has been noted by several authors that Belief Propagation can can also give impressive results for graphs that are not trees [2]. The Cluster Variation Method (CVM), is a method that has been developed in the physics community for approximate inference in the Ising model [3]. The CVM approximates the joint probability distribution by a number of (overlapping) marginal distributions (clusters). The quality of the approximation is determined by the size and number of clusters. When the clusters consist of only two variables, the method is known as the Bethe approximation. Recently, the method has been introduced by Yedidia et a1.[4] into the machine learning community, showing that in the Bethe approximation, the CVM solution coincides with the fixed points of the belief propagation algorithm. For clusters consisting of more than two variables, [4] present a message passing scheme called generalized belief propagation (GBP). This approximation to the free energy is often referred to as the Kikuchi approximation. They show, that GBP gives a significant improvement over the Bethe approximation for a small two dimensional Ising lattice with random couplings. However, for larger latices, both GBP and BP fail to converge [4, 5]. In [5] the CCCP method is proposed, which is a double loop iteration algorithm that is guaranteed to converge for the general CVM problem. Intuitively, the method consists of iteration a sequence of convex subproblem (outer loop) each of which is solved using a fixed point iteration method (inner loop). In this sense, the method is similar to the UPS algorithm of [6] which identifies trees as subproblems. In this paper, we propose two algorithms, one is a fixed point iteration procedure, the other a gradient based method. We show that the fixed point iteration method gives very fast convergence and accurate results for some classical directed graphical models. However, for more challenging cases the fixed point method does not converge and the gradient based approach, which is guaranteed to converge, is preferable. 2 The Cluster Variation Method In this section, we briefly present the cluster variation method. For a more complete treatment see for instance [7]. Let x = (Xl, ... ,xn ) be a set of variables, where each Xi can take a finite number of values. Consider a probability distribution on X of the form ( ) __ 1_ -H(x) PH X Z(H)e Z = 2:= e-H(x) x It is well known, that PH can be obtained as the minimum of the free energy, which is a functional over probability distributions of the following form: FH(P) = (H) + (logp) , (1) where the expectation value is taken with respect to the distribution p, i.e. (H) = L xP(x)H(x). When one minimizes FH(P) with respect to P under the constraint of normalization L xP(X) = 1, one obtains PH. Computing marginals of PH such as PH(Xi) or PH(Xi, Xj) involves sums over all states, which is intractable for large n. Therefore, one needs tractable approximations to PH. The cluster variation method replaces the probability distribution PH(X) by a large number of (possibly overlapping) probability distributions, each describing a sub set (cluster) of variables. Due to the one-to-one correspondence between a probability distribution and the minima of a free energy we can define approximate probability distributions by constructing approximate free energies and computing their minimum. This is achieved by approximating Eq. 1 in terms of the cluster probabilities. The solution is obtained by minimizing this approximate free energy subject to normalization and consistency constraints. Define clusters as subsets of distinct variables: Xa = (XiI' ... ,Xik), with 1 ~ i j ~ n. Consider the set of clusters P that describe the interactions in H and write H as a sum of these interactions: H(x) = 2:= Hl(xoJ a EP We now define a set of clusters B, that will determine our approximation in the cluster variation method. For each cluster a E B, we introduce a probability distribution Pa(xa) which jointly must approximate p(x). B should at least contain the interactions in p(x) in the following way: Va E P => 30:' E B,a c a'. In addition, we demand that no two clusters in B contain each other: a, a' E B => a rt a', a' rt a. The minimal choice for B is to chose clusters from P itself. The maximal choice for B is the cliques obtained when constructing the junction tree[8]. In this case, the clusters in B form a tree structure and the CVM method is exact. Define a set of clusters M that consist of any intersection of a number of clusters of B: M = {,BI,B = nkak, ak E B}, and define U = BuM. Once U is given, we define numbers a/3 recursively by the Moebius formula 1 = L ao;, V (3 E U o;EU,o;"J/3 In particular, this shows that ao; = 1 for 0: E B. The Moebius formula allows us to rewrite (H) in terms of the cluster probabilities (H) = Lao; LPo;(xo;)Ho;(xo;), (2) o;EU x" with Ho;(xo;) = L./3EP,/3co; Hh(X/3) . Since interactions Hh may appear in more than one Ho;, the constants ao; ensure that double counting is compensated for. Whereas (H) can be written exactly in terms of Po;, this is not the case for the entropy term in Eq. 1. The approach is to decompose the entropy of a cluster 0: in terms of 'connected entropies' in the following way: 1 (3) x" /3Co; where the sum over (3 contains all sub clusters of 0:. Such a decomposition can be made for any cluster. In particular it can be made for the 'cluster' consisting of all variables, so that we obtain S = - LP(x) logp(x) = L Sh· (4) x /3 The cluster variation method approximates the total entropy by restricting this latter sum to only clusters in U and re-expressing Sh in terms of So;, using the Moebius formula and the definition Eq. 3. (5) /3EU /3EU 0;"J/3 o;EU Since So; is a function of Po; (Eq. 3) , we have expressed the entropy in terms of cluster probabilities Po; . The quality of this approximation is illustrated in Fig. 1 for the SK model. Note, that both the Bethe and Kikuchi approximation strongly deteriorate around J = 1, which is where the spin-glass phase starts. For J < 1, the Kikuchi approximation is superior to the Bethe approximation. Note, however, that this figure only illustrates the quality of the truncations in Eq. 5 assuming that the exact marginals are known. It does not say anything about the accuracy of the approximate marginals using the approximate free energy. Substituting Eqs. 2 and 5 into the free energy Eq. 1 we obtain the approximate free energy of the Cluster Variation method. This free energy must be minimized subject to normalization constraints L.x" Po; (x o; ) = 1 and consistency constraints Po;(X/3) = P/3(X/3), 0:,(3 E U,(3 C 0:. (6) with Po; (X/3) = L.x Po; (xo;). "\f3 IThis decomposition is similar to writing a correlation in terms of means and covariance. For instance when a = (i), S(i) = SIi) is the usual mean field entropy and S(ij) = Sli) + SIj) + Slij) defines the two node correction Slij)" 12 10 8 >-a. e 6 c lJ.J 4 "2 """0 0.5 1.5 2 J Figure 1: Exact and approximate entropies for the fully connected Boltzmann-Gibbs distribution on n = 10 variables with random couplings (SK model) as a function of mean coupling strength. Couplings Wij are chosen from a normal Gaussian distribution with mean zero and standard deviation J /..;n. External fields ()i are chosen from a normal Gaussian distribution with mean zero and standard deviation 0.1. The exact entropy is computed from Eq. 4. The Bethe and Kikuchi approximations are computed using the approximate entropy expression Eq. 5 with exact marginals and by choosing B as the set of all pairs and all triplets, respectively. The set of consistency constraints can be significantly reduced because some constraints imply others. Let 0:,0:', .. . denote clusters in Band fJ, fJ', ... denote clusters in M. • If fJ c fJ' Co: and Pa(x/3') = P/3' (x/3') and Pa(x/3 ) = P/3(x/3), then P/3' (x/3) = P/3 (X/3)' This means that constraints between clusters in M can be removed . • If fJ c fJ' c 0:, 0:' and Pa(x/3') = Pa' (x/3') and p,,,(x/3) = P/3 (x/3 ), then Pa,(x/3) = P/3 (x/3)' This means that some constraints between clusters in B and M can be removed. We denote the remaining necessary constraints by 0: ---t fJ. Adding Lagrange multipliers for the constraints we obtain the Cluster Variation free energy: aEU x " - L Aa (LPa(Xa) - 1) - L L L Aa/3 (X/3) (Pa(x/3 ) - P/3 (x/3)) aEU x " /3EM a-+/3 X f3 (7) 3 Iterating Lagrange multipliers By setting 88Fc(vm), a E U equal to zero, one can express the cluster probabilities in PO! X o: terms of the Lagrange multipliers: ; exp (-Ha(Xa) + L )..a(3 (X(3)) a (3f-a (8) ; exp (-H(3 (X(3 ) - a1 L )..a(3 (X(3 )) (3 (3 a-t(3 (9) The remaining task is to solve for the Lagrange multipliers such that all constraints (Eq. 6) are satisfied. We present two ways to do this. When one substitutes Eqs. 8-9 into the constraint Eqs. 6 one obtains a system of coupled non-linear equations. In Yedidia et al.[4] a message passing algorithm was proposed to find a solution to this problem. Here, we will present an alternative method, that solves directly in terms of the Lagrange multipliers. 3.1 Fixed point iteration Consider the constraints Eq. 6 for some fixed cluster fJ and all clusters a -+ fJ and define B(3 = {a E Bla -+ fJ }· We wish to solve for all constraints a -+ fJ, with a E B(3 by adjusting )..a(3, a E B(3. This is a sub-problem with IB(3 I IX(3 I equations and an equal number of unknowns, where IB(3 1 is the number of elements of B(3 and IX(3 1 is the number of values that x(3 can take. The probability distribution P(3 (Eq. 9) depends only on these Lagrange multipliers. Pa (Eq. 8) depends also on other Lagrange multipliers. However, we consider only its dependence on )..a(3 , a E B(3 , and consider all other Lagrange multipliers as fixed. Thus, (10) with Pa independent of )..a(3, a E B(3 . Substituting, Eqs. 9 and 10 into Eq. 6, we obtain a set of linear equations for )..a(3(x(3 ) which we can solve in closed form: )..a(3 (X(3 ) = alB IH(3 (X(3 ) - L AaadogPa l (X(3 ) a(3 + (3 a' with 1 Aaal = /jaa l --c=--:a(3 + IB(3 1 We update the probabilities with the new values of the Lagrange multipliers using Eqs. 9 and 10. We repeat the above procedure for all fJ E M until convergence. 3.2 Gradient descent We define an auxiliary cost function C = L LP(3 (X(3 ) log P(3 ((X(3 )) = L Ca(3 a(3 Xf3 Pa x(3 a(3 (11) that is zero when all constraints are satisfied and positive otherwise and minimize this cost function with respect to the Lagrange multipliers )..a(3 (X(3 ). The gradient of C is given by: 8C Pf3(Xf3 ) ""' (Pf3(Xf3 ) ) ""' (PI () ()) --- ~ log ( ) - Calf3 ~ af3' xf3 - Pa xf3 af3 a/-tf3 Pa' Xf3 13' +--a with 4 Numerical results 4.1 Directed Graphical models We show the performance of the fixed point iteration procedure on several 'real world' directed graphical models. In figure 2a, we plot the exact single node marginals against the approximate marginals for the Asia problem [8]. Clusters in B are defined according to the conditional probability tables. Convergence was reached in 6 iterations using fixed point iteration. Maximal error on the marginals is 0.0033. For comparison, we computed the mean field and TAP approximations, as previously introduced by [9]. Although TAP is significantly better than MF, it is far worse than the CVM method. This is not surprising, since both the MF and TAP approximation are based on single node approximation, whereas the CVM method uses potentials up to size 3. In figure 2b, we plot the exact single node marginals against the approximate CVM marginals for the alarm network [10]. The structure and CPTs were downloaded from www.cs.huji.ac.il;-nir. Clusters in B are defined according to the conditional probability tables and maximally contain 5 variables. Convergence was reached in 15 iterations using fixed point iteration. Maximal error on the marginals is 0.029. Ordinary loopy BP gives an error in the marginals of approximately 0.25 [2]. Mean field and TAP methods did not give reproducible results on this problem. 0.5,------------.--f'l---+ 0.4 (fj co c ·~0 .3 C1l E x e 0.2 c. c. <t: 0.1 + -F® Exact marginals x x (a) Asia problem (n = 8). 0.8 (fj co .a, 0.6 Co E :2 0.4 > () 0.2 ,... .' .. ' .' .' OL-----------~ o 0.5 Exact marginals (b) Alarm problem (n = 37). Figure 2: Comparison of single node marginals on two real world problems. Finally, we tested the cluster variation method on randomly generated directed graphical models. Each node is randomly connected to k parents. The entries of the probability tables are randomly generated between zero and one. Due to the large number of loops in the graph, the exact method requires exponential time in the maximum clique size, which can be seen from Table 1 to scale approximately linear with the network size. Therefore exact computation is only feasible for small graphs (up to size n = 40 in this case). For the CVM, clusters in B are defined according to the conditional probability tables. Therefore, maximal cluster size is k + 1. On these more challenging cases, the fixed point iteration method does not converge. The results shown are obtained with conjugate gradient descent on the auxiliary cost function Eq. 11. The results are shown in Table 1. n Iter IGI Potential error Margin error G 10 16 8 0.018 0.004 9.7e-ll 20 189 12 0.019 0.029 2.4e-4 30 157 16 0.033 0.130 2.1e-3 40 148 21 0.048 0.144 3.6e-3 50 132 26 4.5e-3 Table 1: Comparison of CYM method for large directed graphical models. Each node is connected to k = 5 parents. IGI is the tree width of the triangulated graph required for the exact computation. Iter is the number of conjugate gradient descent iterations of the CYM method. Potential error and margin error are the maximum absolute distance (MAD) in any of the cluster probabilities and single variable marginals computed with CYM, respectively. G is given by Eq. 11 after termination of CYM. 4.2 Markov networks We compare the Bethe and Kikuchi approximations for the SK model with n = 5 neurons as defined in Fig. 1. We expect that for small J the CVM approximation gives accurate results and deteriorates for larger J. We compare the Bethe approximation, where we define clusters for all pairs of nodes and a Kikuchi approximation where we define clusters for all sub sets of three nodes. The results are given in Table 2. We see that for the Bethe approximation, the results of the fixed point iteration method (FPI) and the gradient based approach agree. For the Kikuchi approximation the fixed point iteration method does not converge and results are omitted. As expected, the Kikuchi approximation gives more accurate results than the Bethe approximation for small J. 5 Conclusion We have presented two iteration schemes for finding the minimum of the constraint problem Eq. 7. One method is a fixed point iteration method that is equivalent to belief propagation for pairwise interactions. This method is very fast and gives very accurate results for 'not too complex' graphical models, such as real world directed graphical models and frustrated Boltzmann distributions in the Bethe approximation. However, for more complex graphs such as random directed graphs or more complex approximations, such as the Kikuchi approximation, the fixed point iteration method does not converge. Empirically, it is found that smoothing may somewhat help, but certainly does not solve this problem. For these more complex problems we propose to minimize an auxiliary cost function using a gradient Bethe Kikuchi FPI gradient gradient J Iter Error Iter Error Iter Error 0.25 7 0.000161 7 0.000548 120 0.000012 0.50 9 0.001297 11 0.001263 221 0.000355 0.75 13 0.004325 14 0.004392 86 0.021176 1.00 17 0.009765 15 0.009827 49 0.036882 1.50 38 0.027217 16 0.027323 150 0.059977 2.00 75 0.049955 20 0.049831 137 0.088481 Table 2: Comparison of Bethe and Kikuchi approximation for Boltzmann distributions. Iter is the number of iterations needed. Error is the MAD of single variable marginals. based method. Clearly, this approach is guaranteed to converge. Empirically, we have found no problems with local minima. However, we have found that obtaining solution with C close to zero may require many iterations. Acknowledgments This research was supported in part by the Dutch Technology Foundation (STW). I would like to thank Taylan Cemgil for providing his Matlab graphical models toolkit and Sebino Stramaglia (Bari, Italy) for useful discussions. References [1] J. Pearl. Probabilistic reasoning in intelligent systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco, California, 1988. [2] Kevin P. Murphy, Yair Weiss, and Michael I. Jordan. Loopy belief propagation for approximate inference: An empirical study. In Proceedings of Uncertainty in AI, pages 467- 475, 1999. [3] R. Kikuchi. Physical Review, 81:988, 1951. [4] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Generalized belief propagation. In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13 (Proceedings of the 2000 Conference), 2001. In press. [5] A.L. Yuille and A. Rangarajan. The convex-concave principle. In T.G. Dieterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14, 2002. In press. [6] Y. Teh and M. Welling. The unified propagation and scaling algorithm. In T.G. Dieterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14, 2002. In press. [7] H.J. Kappen. The cluster variation method for approximate reasoning in medical diagnosis. In G. Nardulli and S. Stramaglia, editors, Modeling Bio-medical signals. World-Scientific, 2002. In press. [8] S.L. Lauritzen and D.J. Spiegelhalter. Local computations with probabilties on graphical structures and their application to expert systems. J. Royal Statistical society B, 50:154- 227, 1988. [9] H.J. Kappen and W.A.J.J. Wiegerinck. Second order approximations for probability models. In Todd Leen, Tom Dietterich, Rich Caruana, and Virginia de Sa, editors, Advances in Neural Information Processing Systems 13, pages 238- 244. MIT Press, 2001. [10] 1. Beinlich, G. Suermondt, R. Chaves, and G. Cooper. The alarm monitoring system: A case study with two probabilistic inference techniques for belief networks. In 2'nd European Conference on AI in Medicin e, 1989.
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Infinite Mixtures of Gaussian Process Experts Carl Edward Rasmussen and Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR, England edward,zoubin@gatsby.ucl.ac.uk http://www.gatsby.ucl.ac.uk Abstract We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using an input-dependent adaptation of the Dirichlet Process, we implement a gating network for an infinite number of Experts. Inference in this model may be done efficiently using a Markov Chain relying on Gibbs sampling. The model allows the effective covariance function to vary with the inputs, and may handle large datasets – thus potentially overcoming two of the biggest hurdles with GP models. Simulations show the viability of this approach. 1 Introduction Gaussian Processes [Williams & Rasmussen, 1996] have proven to be a powerful tool for regression. They combine the flexibility of being able to model arbitrary smooth functions if given enough data, with the simplicity of a Bayesian specification that only requires inference over a small number of readily interpretable hyperparameters – such as the length scales by which the function varies along different dimensions, the contributions of signal and noise to the variance in the data, etc. However, GPs suffer from two important limitations. First, because inference requires inversion of an covariance matrix where is the number of training data points, they are computationally impractical for large datasets. Second, the covariance function is commonly assumed to be stationary, limiting the modeling flexibility. For example, if the noise variance is different in different parts of the input space, or if the function has a discontinuity, a stationary covariance function will not be adequate. Goldberg et al [1998] discussed the case of input dependent noise variance. Several recent attempts have been aimed at approximate inference in GP models [Williams & Seeger 2001, Smola & Bartlett 2001]. These methods are based on selecting a projection of the covariance matrix onto a smaller subspace (e.g. a subset of the data points) reducing the overall computational complexity. There have also been attempts at deriving more complex covariance functions [Gibbs 1997] although it can be difficult to decide a priori on a covariance function of sufficient complexity which guarantees positive definiteness. In this paper we will simultaneously address both the problem of computational complexity and the deficiencies in covariance functions using a divide and conquer strategy inspired by the Mixture of Experts (ME) architecture [Jacobs et al, 1991]. In this model the input space is (probabilistically) divided by a gating network into regions within which specific separate experts make predictions. Using GP models as experts we gain the double advantage that computation for each expert is cubic only in the number of data point in its region, rather than in the entire dataset, and that each GP-expert may learn different characteristics of the function (such as lengths scales, noise variances, etc). Of course, as in the ME, the learning of the experts and the gating network are intimately coupled. Unfortunately, it may be (practically and statistically) difficult to infer the appropriate number of experts for a particular dataset. In the current paper we sidestep this difficult problem by using an infinite number of experts and employing a gating network related to the Dirichlet Process, to specify a spatially varying Dirichlet Process. An infinite number of experts may also in many cases be more faithful to our prior expectations about complex real-word datasets. Integrating over the posterior distribution for the parameters is carried out using a Markov Chain Monte Carlo approach. Tresp [2001] presented an alternative approach to mixtures of GPs. In his approach both the experts and the gating network were implemented with GPs; the gating network being a softmax of GPs. Our new model avoids several limitations of the previous approach, which are covered in depth in the discussion. 2 Infinite GP mixtures The traditional ME likelihood does not apply when the experts are non-parametric. This is because in a normal ME model the data is assumed to be iid given the model parameters:
"!
# where and are inputs and outputs (boldface denotes vectors), are the parameters of expert , ! are the parameters of the gating network and are the discrete indicator variables assigning data points to experts. This iid assumption is contrary to GP models which solely model the dependencies in the joint distribution (given the hyperparameters). There is a joint distribution corresponding to every possible assignment of data points to experts; therefore the likelihood is a sum over (exponentially many) assignments: $
%&' () *
+,(- "!
&/. 01324 65 )78924 65 )7*
;:<,(- "!
#= (1) Given the configuration (>?,@ 4=A=A=4 $<BC
, the distribution factors into the product, over experts, of the joint Gaussian distribution of all data points assigned to each expert. Whereas the original ME formulation used expectations of assignment variables called responsibilities, this is inadequate for inference in the mixture of GP experts. Consequently, we directly represent the indicators, , and Gibbs sample for them to capture their dependencies. In Gibbs sampling we need the posterior conditional distribution for each indicator given all the remaining indicators and the data: , D (FE $ G $ C "!
H% I J "(FE $
D (FE $ "!
# where (CE denotes all indicators except number K . We defer discussion of the second term defining the gating network to the next section. As discussed, the first term being the likelihood given the indicators factors into independent terms for each expert. For Gibbs sampling we therefore need the probability of output under GP number : 924L 5MON K $PL C7* <2Q CL 5 PL )7
#= For a GP model, this conditional density is the well known Gaussian [Williams & Rasmussen, 1996]: E $
#
E @ 6E $
E @
(2) where the covariance matrix depends on the parameters . Thus, for the GP expert, we compute the above conditional density by simply evaluating the GP on the data assigned to it. Although this equation looks computationally expensive, we can keep track of the inverse covariance matrices and reuse them for consecutive Gibbs updates by performing rank one updates (since Gibbs sampling changes at most one indicator at a time). We are free to choose any valid covariance function for the experts. In our simulations we employed the following Gaussian covariance function:
G "! # %$I $ 0 & $
('()* $+-, @ . K K0/
(3) with hyperparameters 1 controlling the signal variance, @ controlling the noise variance, and ) $ controlling the length scale or (inverse) relevance of the 2 -th dimension of in relation to predicting ; . is the Kronecker delta function (i.e. . K K /
! if K K / , o.w. 0). 3 The Gating network The gating network assigns probability to different experts based entirely on the input. We will derive a gating network based on the Dirichlet Process which can be defined as the limit of a Dirichlet distribution when the number of classes tends to infinity. The standard Dirichlet Process is not input dependent, but we will modify it to serve as a gating mechanism. We start from a symmetric Dirichlet distribution on proportions: 43 @ A= = = 365C 76
- 8:9&;9=<?>A@& BQC7 '%D
G E 476
E 47 'FD
5 G3HJI 5 E @ where 7 is the (positive) concentration parameter. It can be shown [Rasmussen, 2000] that the conditional probability of a single indicator when integrating over the 3 variables and letting D tend to infinity is given by: components where E K MLGN : 1 ( E 76
6E K O ! , 7 all other components combined: 1 N & for all K / N K (CE 76
7 O ! , 7 (4) where GE 4K ( QP &0R S . , 8
) is the occupation number of expert excluding observation K , and is the total number of data points. This shows that the probabilities are proportional to the occupation numbers. To make the gating network input dependent, we will simply employ a local estimate 1 for this occupation number using a kernel classifier: E 4K T !
P R S JUWV
. , 8
P R S FUWV &
(5) where the delta function selects data points assigned to class , and U is the kernel function parametrized by ! . As an example we use a Gaussian kernel function: UOV
GX1 Y! # $ $ $
' ! $ + (6) 1this local estimate won’t generally be an integer, but this doesn’t have any adverse consequences parameterized by length scales ! $ for each dimension. These length scales allow dimensions of space to be more or less relevant to the gating network classification. We Gibbs sample from the indicator variables by multiplying the input-dependent Dirichlet process prior eq. (4) and (5) with the GP conditional density eq. (2). Gibbs sampling in an infinite model requires that the indicator variables can take on values that no other indicator variable has already taken, thereby creating new experts. We use the auxiliary variable approach of Neal [1998] (algorithm 8 in that paper). In this approach hyperparameters for new experts are sampled from their prior and the likelihood is evaluated based on these. This requires finding the likelihood of a Gaussian process with no data. Fortunately, for the covariance function eq. (3) this likelihood is Gaussian with zero mean and variance , @ . If all data points are assigned to a single GP, the likelihood calculation will still be cubic in the number of data points (per Gibbs sweep over all indicators). We can reduce the computational complexity by introducing the constraint that no GP expert can have more than max data points assigned to it. This is easily implemented2 by modifying the conditionals in the Gibbs sampler. The hyperparameter 7 controls the prior probability of assigning a data point to a new expert, and therefore influences the total number of experts used to model the data. As in Rasmussen [2000], we give a vague inverse gamma prior to 7 , and sample from its posterior using Adaptive Rejection Sampling (ARS) [Gilks & Wild, 1992]. Allowing 7 to vary gives the model more freedom to infer the number of GPs to use for a particular dataset. Finally we need to do inference for the parameters of the gating function. Given a set of indicator variables one could use standard methods from kernel classification to optimize the kernel widths in different directions. These methods typically optimize the leave-oneout pseudo-likelihood (ie the product of the conditionals), since computing the likelihood in a model defined purely from conditional distributions as in eq. (4), (5) & (6) is generally difficult (and as pointed out in the discussion section there may not even be a single likelihood). In our model we multiply the pseudo-likelihood by a (vague) prior and sample from the resulting pseudo-posterior. 4 The Algorithm The individual GP experts are given a stationary Gaussian covariance function, with a single length scale per dimension, a signal variance and a noise variance, i.e. , # (where is the dimension of the input) hyperparameters per expert, eq. (3). The signal and noise variances are given inverse gamma priors with hyper-hypers and (separately for the two variances). This serves to couple the hyperparameters between experts, and allows the priors on and *@ (which are used when evaluating auxiliary classes) to adapt. Finally we give vague independent log normal priors to the lenght scale paramters ) and ! . The algorithm for learning an infinite mixture of GP experts consists of the following steps: 1. Initialize indicator variables to a single value (or a few values if individual GPs are to be kept small for computational reasons). 2. Do a Gibbs sampling sweep over all indicators. 3. Do Hybrid Monte Carlo (HMC) [Duane et al, 1987] for hyperparameters of the GP covariance function,
@ ) $ , for each expert in turn. We used 10 leapfrog iterations with a stepsize small enough that rejections were rare. 4. Optimize the hyper-hypers, & , for each of the variance parameters. 5. Sample the Dirichlet process concentration parameter, 7 using ARS. 2We simply set the conditional probability of joining a class which has been deemed full to zero. 0 10 20 30 40 50 60 −150 −100 −50 0 50 100 Time (ms) Acceleration (g) iMGPE stationary GP 0 10 20 30 40 50 60 −150 −100 −50 0 50 100 Time (ms) Acceleration (g) Figure 1: The left hand plot shows the motorcycle impact data (133 points) together with the median of the model’s predictive distribution, and for comparison the mean of a stationary covariance GP model (with optimized hyperparameters). On the right hand plot we show ! NJN samples from the posterior distribution for the iMGPE of the (noise free) function evaluated intervals of 1 ms. We have jittered the points in the plot along the time dimension by adding uniform N = # ms noise, so that the density can be seen more easily. Also, the # std error ( ) confidence interval for the (noise free) function predicted by a stationary GP is plotted (thin lines). 6. Sample the gating kernel widths, ! ; we use the Metropolis method to sample from the pseudo-posterior with a Gaussian proposal fit at the current ! 3 7. Repeat from 2 until the Markov chain has adequately sampled the posterior. 5 Simulations on a simple real-world data set To illustrate our algorithm, we used the motorcycle dataset, fig. 1, discussed in Silverman [1985]. This dataset is obviously non-stationary and has input-dependent noise. We noticed that the raw data is discretized into bins of size ! = # g; accordingly we cut off the prior for the noise variance at ' ! # . The model is able to capture the general shape of the function and also the input-dependent nature of the noise (fig. 1). This can be seen from the right hand plot in fig. 1, where the uncertainty of the function is very low for ! N
owing to a small inferred noise level in this region. For comparison, the predictions from a stationary GP has been superimposed in fig. 1. Whereas the medians of the predictive distributions agree to a large extent (left hand plot), we see a huge difference in the predictive distributions (right hand). The homoscedastic GP cannot capture the very tight distribution for ! N ms offered by iMGPE. Also for large LJN ms, the iMGPE model predicts with fairly high probability that the signal could be very close to zero. Note that the predictive distribution of the function is multimodal, for example, around time 35 ms. Multimodal predictive distributions could in principle be obtained from an ordinary GP by integrating over hyperparameters, however, in a mixture of GP’s model they can arise naturally. The predictive distribution of the function appears to have significant mass around N g which seems somewhat artifactual. We explicitly did not normalize or center the data, which has a large range in output. The 3The Gaussian fit uses the derivative and Hessian of the log posterior wrt the log length scales. Since this is an asymmetric proposal the acceptance probabilities must be modified accordingly. This scheme has the advantage of containing no tunable parameters; however when the dimension is large, it may be computationally more efficient to use HMC, to avoid calculation of the Hessian. 0 10 20 30 40 50 60 70 80 90 100 20 40 60 80 100 120 20 40 60 80 100 120 5 10 15 20 25 30 0 2 4 6 8 10 12 14 number of occupied experts frequency Figure 2: The left hand plot shows the number of times, out of 100 samples, that the indicator variables for two data points were equal. The data have been sorted from left-toright according to the value of the time variable (since the data is not equally spaced in time the axis of this matrix cannot be aligned with the plot in fig.1). The right hand plot shows a histogram over the 100 samples of the number of GP experts used to model the data. Gaussian processes had zero mean a priori, which coupled with the concentration of data around zero may explain the posterior mass at zero. It would be more natural to treat the GP means as separate hyperparameters controlled by a hyper-hyperparameter (centered at zero) and do inference on them, rather than fix them all at 0. Although the scatter of data from the predictive distribution for iMGPE looks somewhat messy with multimodality etc, it is important to note that it assigns high density to regions that seem probable. The motorcycle data appears to have roughly three regions: a flat low-noise region, followed by a curved region, and a flat high noise region. This intuition is bourne out by the model. We can see this in two ways. Fig 2. (left) shows the number of times two data points were assigned to the same expert. A clearly defined block captures the initial flat region and a few other points that lie near the N g line; the middle block captures the curved region, with a more gradual transition to the last flat region. A histogram of the number of GP experts used shows that the posterior distribution of number of needed GPs has a broad peak between and ! N , where less than 3 occupied experts is very unlikely, and above ! N becoming progressively less likely. Note that it never uses just a single GP to model the data which accords with the intuition that a single stationary covariance function would be inadequate. We should point out that the model is not trying to do model selection between finite GP mixtures, but rather always assumes that there are infinitely many available, most of which contribute with small mass4 to a diffuse density in the background. In figure 3 we assessed the convergence rate of the Markov Chain by plotting the autocorrelation function for several parameters. We conclude that the mixing time is around 100 iterations5. Consequently, we run the chain for a total of !J! NJN%N iterations, discarding the initial ! NJN%N (burn-in) and keeping every ! N%N ’th after that. The total computation time was around 1 hour (1 GHz Pentium). The right hand panel of figure 3 shows the distribution of the gating function kernel width 4The total mass of the non-represented experts is , where the posterior for in this experiment is peaked between and
(see figure 3, bottom right panel), corresponding to about of the total mass 5the sum of the auto-correlation coefficients from to is an estimate of the mixing time 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 time lag in iterations auto correlation coefficient log number of occupied experts log gating kernel width log Dirichlet concentration −1 −0.5 0 0.5 0 5 10 log (base 10) gating function kernel width frequency −0.5 0 0.5 1 0 5 10 log (base 10) Dirichlet process concentration frequency Figure 3: The left hand plot shows the auto-correlation for various parameters of the model based on !%! NJN%N iterations. The right hand plots show the distribution of the (log) kernel width ! and (log) Dirichlet concentration parameter 7 , based on ! N%N samples from the posterior. ! and the concentration parameter of the Dirichlet process. The posterior 6 kernel width ! lies between N = ! and ; comparing to the scale of the inputs these are quite short distances, corresponding to rapid transitions between experts (as opposed to lengthy intervals with multiple active experts). This corresponds well with our visual impression of the data. 6 Discussion and Conclusions We now return to Tresp [2000]. There are four ways in which the infinite mixture of GP experts differs from, and we believe, improves upon the model presented by Tresp. First, in his model, although a gating network divides up the input space, each GP expert predicts on the basis of all of the data. Data that was not assigned to a GP expert can therefore spill over into predictions of a GP, which will lead to bias near region boundaries especially for experts with long length scales. Second, if there are experts, Tresp’s model has GPs (the experts, noise models, and separate gating functions) each of which requires computations over the entire dataset resulting in 4
computations. In our model since the experts divide up the data points, if there are experts equally dividing the data an iteration takes '
computations (each of Gibbs updates requires a rank-one computation '
for each of the experts and the Hybrid Monte Carlo takes times '
). Even for modest (e.g. ! N ) this can be a significant saving. Inference for the gating length scale parameters is
if the full Hessian is used, but can be reduced to
for a diagonal approximation, or Hybrid Monte Carlo if the input dimension is large. Third, by going to the Dirichlet process infinite limit, we allow the model to infer the number of components required to capture the data. Finally, in our model the GP hyperparameters are not fixed but are instead inferred from the data. We have defined the gating network prior implicitly in terms of the conditional distribution of an indicator variable given all the other indicator variables. Specifically, the distribution of this indicator variable is an input-dependent Dirichlet process with counts given by local estimates of the data density in each class eq. (5). We have not been able to prove that these conditional distributions are always consistent with a single joint distribution over 6for comparison the (vague) prior on the kernel width is log normal with of the mass between and , corresponding to very short (sub sample) distances upto distances comparable to the entire input range the indicators. If indeed there does not exist a single consistent joint distribution the Gibbs sampler may converge to different distributions depending on the order of sampling. We are encouraged by the preliminary results obtained on the motorcycle data. Future work should also include empirical comparisons with other state-of-the-art regression methods on multidimensional benchmark datasets. We have argued here that single iterations of the MCMC inference are computationally tractable even for large data sets, experiments will show whether mixing is sufficiently rapid to allow practical application. We hope that the extra flexibility of the effective covariance function will turn out to improve performance. Also, the automatic choice of the number of experts may make the model advantageous for practical modeling tasks. Finally, we wish to come back to the modeling philosophy which underlies this paper. The computational problem in doing inference and prediction using Gaussian Processes arises out of the unrealistic assumption that a single covariance function captures the behavior of the data over its entire range. This leads to a cumbersome matrix inversion over the entire data set. Instead we find that by making a more realistic assumption, that the data can be modeled by an infinite mixture of local Gaussian processes, the computational problem also decomposes into smaller matrix inversions. References Gibbs, M. N. (1997). Bayesian Gaussian Processes for Regression and Classification. PhD thesis. University of Cambridge. Goldberg, P. W., Williams, C. K. I., & Bishop C. M. (1998). Regression with Inputdependent Noise, NIPS 10. Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D. (1987). Hybrid Monte Carlo, Physics letters B, vol. 55, pp. 2774–2777. Gilks, W. R. & Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics 41, 337–348. Jacobs, R. A., Jordan, M. I., Nowlan, S. J. & Hinton, G. E. (1991). Adaptive mixture of local experts. Neural Computation, vol 3, pp 79–87. Neal, R. M. (1998). Markov chain sampling methods for Dirichlet process mixture models. Technical Report 4915, Department of Statistics, University of Toronto. http://www.cs.toronto.edu/ radford/mixmc.abstract.html. Rasmussen, C. E. (2000). The Infinite Gaussian Mixture Model, NIPS 12, S.A. Solla, T.K. Leen and K.-R. M¨uller (eds.), pp. 554–560, MIT Press. Silverman, B. W. (1985). Some aspects of the spline smoothing approach to non-parametric regression curve fitting. J. Royal Stat. Society. Ser. B, vol. 47, pp. 1–52. Smola A. J. and Bartlett, P. (2001). Sparse Greedy Gaussian Process Regression, NIPS 13. Tresp V. (2001). Mixtures of Gaussian Process, NIPS 13. Williams, C. K. I. and Seeger, M. (2001). Using the Nystr¨om Method to Speed Up Kernel Machines, NIPS 13. Williams, C. K. I. and C. E. Rasmussen (1996). Gaussian Processes for Regression, in D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (editors), NIPS 8, MIT Press.
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Relative Density Nets: A New Way to Combine Backpropagation with HMM's Andrew D. Brown Department of Computer Science University of Toronto Toronto, Canada M5S 3G4 andy@cs.utoronto.ca Abstract Geoffrey E. Hinton Gatsby Unit, UCL London, UK WCIN 3AR hinton@gatsby.ucl.ac.uk Logistic units in the first hidden layer of a feedforward neural network compute the relative probability of a data point under two Gaussians. This leads us to consider substituting other density models. We present an architecture for performing discriminative learning of Hidden Markov Models using a network of many small HMM's. Experiments on speech data show it to be superior to the standard method of discriminatively training HMM's. 1 Introduction A standard way of performing classification using a generative model is to divide the training cases into their respective classes and then train a set of class conditional models. This unsupervised approach to classification is appealing for two reasons. It is possible to reduce overfitting, because the model learns the class-conditional input densities P(xlc) rather than the input-conditional class probabilities P(clx). Also, provided that the model density is a good match to the underlying data density then the decision provided by a probabilistic model is Bayes optimal. The problem with this unsupervised approach to using probabilistic models for classification is that, for reasons of computational efficiency and analytical convenience, very simple generative models are typically used and the optimality of the procedure no longer holds. For this reason it is usually advantageous to train a classifier discriminatively. In this paper we will look specifically at the problem of learning HMM's for classifying speech sequences. It is an application area where the assumption that the HMM is the correct generative model for the data is inaccurate and discriminative methods of training have been successful. The first section will give an overview of current methods of discriminatively training HMM classifiers. We will then introduce a new type of multi-layer backpropagation network which takes better advantage of the HMM's for discrimination. Finally, we present some simulations comparing the two methods. 19 ' S1 c1=" 1 1 1 [tn] [tn] [tn] HMM's \V Sequence Figure 1: An Alphanet with one HMM per class. Each computes a score for the sequence and this feeds into a softmax output layer. 2 Alphanets and Discriminative Learning The unsupervised way of using an HMM for classifying a collection of sequences is to use the Baum-Welch algorithm [1] to fit one HMM per class. Then new sequences are classified by computing the probability of a sequence under each model and assigning it to the one with the highest probability. Speech recognition is one of the commonest applications of HMM's, but unfortunately an HMM is a poor model of the speech production process. For this reason speech researchers have looked at the possibility of improving the performance of an HMM classifier by using information from negative examples examples drawn from classes other than the one which the HMM was meant to model. One way of doing this is to compute the mutual information between the class label and the data under the HMM density, and maximize that objective function [2]. It was later shown that this procedure could be viewed as a type of neural network (see Figure 1) in which the inputs to the network are the log-probability scores C(Xl:TIH) of the sequence under hidden Markov model H [3]. In such a model there is one HMM per class, and the output is a softmax non-linearity: (1) Training this model by maximizing the log probability of correct classification leads to a classifier which will perform better than an equivalent HMM model trained solely in a unsupervised manner. Such an architecture has been termed an "AIphanet" because it may be implemented as a recurrent neural network which mimics the forward pass of the forward-backward algorithm.l 3 Backpropagation Networks as Density Comparators A multi-layer feedforward network is usually thought of as a flexible non-linear regression model, but if it uses the logistic function non-linearity in the hidden layer, there is an interesting interpretation of the operation performed by each hidden unit. Given a mixture of two Gaussians where we know the component priors P(9) and the component densities P(xl9) then the posterior probability that Gaussian, 90, generated an observation x, is a logistic function whose argument is the negative log-odds of the two classes [4] . This can clearly be seen by rearranging lThe results of the forward pass are the probabilities of the hidden states conditioned on the past observations, or "alphas" in standard HMM terminology. the expression for the posterior: P(Qolx) P(xI9o)P(Qo) P(xI9o)P(Qo) + P(xI9dP (Qd 1 1 + exp {-log P(x IQo) - log P(Qo) } P(x lQd P(Ql) If the class conditional densities in question are multivariate Gaussians P(xI9k) = 121f~1-~ exp {-~(x - Pk)T ~-l(X - Pk)} (2) (3) with equal covariance matrices, ~ , then the posterior class probability may be written in this familiar form: where, w b 1 P(Qo Ix) = -l-+-e-xp-{-=---(:-x=Tw-+-b---:-) (4) (5) (6) Thus, the multi-layer perceptron can be viewed as computing pairwise posteriors between Gaussians in the input space, and then combining these in the output layer to compute a decision. 4 A New Kind of Discriminative Net This view of a feedforward network suggests variations in which other kinds of density models are used in place of Gaussians in the input space. In particular, instead of performing pairwise comparisons between Gaussians, the units in the first hidden layer can perform pairwise comparisons between the densities of an input sequence under M different HMM's. For a given sequence the log-probability of a sequence under each HMM is computed and the difference in log-probability is used as input to the logistic hidden unit.2 This is equivalent to computing the posterior responsibilities of a mixture of two HMM's with equal prior probabilities. In order to maximally leverage the information captured by the HMM's we use (~) hidden units so that all possible pairs are included. The output of a hidden unit h is given by (7) where we have used (mn) as an index over the set, (~) , of all unordered pairs of the HMM's. The results of this hidden layer computation are then combined using a fully connected layer of free weights, W, and finally passed through a soft max function to make the final decision. ak = L W(m ,n)kh(mn) (mn) E (~) (8) (9) 2We take the time averaged log-probability so that the scale of the inputs is independent of the length of the sequence. Density Comparator Units Figure 2: A multi-layer density net with HMM's in the input layer. The hidden layer units perform all pairwise comparisons between the HMM's. where we have used u(·) as shorthand for the logistic function, and Pk is the value of the kth output unit. The resulting architecture is shown in figure 2. Because each unit in the hidden layer takes as input the difference in log-probability of two HMM's, this can be thought of as a fixed layer of weights connecting each hidden unit to a pair of HMM's with weights of ±l. In contrast to the Alphanet, which allocates one HMM to model each class, this network does not require a one-to-one alignment between models and classes and it gets maximum discriminative benefit from the HMM's by comparing all pairs. Another benefit of this architecture is that it allows us to use more HMM's than there are classes. The unsupervised approach to training HMM classifiers is problematic because it depends on the assumption that a single HMM is a good model of the data and, in the case of speech, this is a poor assumption. Training the classifier discriminatively alleviated this drawback and the multi-layer classifier goes even further in this direction by allowing many HMM's to be used to learn the decision boundaries between the classes. The intuition here is that many small HMM's can be a far more efficient way to characterize sequences than one big HMM. When many small HMM's cooperate to generate sequences, the mutual information between different parts of generated sequences scales linearly with the number of HMM's and only logarithmically with the number of hidden nodes in each HMM [5]. 5 Derivative Updates for a Relative Density Network The learning algorithm for an RDN is just the backpropagation algorithm applied to the network architecture as defined in equations 7,8 and 9. The output layer is a distribution over class memberships of data point Xl:T, and this is parameterized as a softmax function. We minimize the cross-entropy loss function: K f = 2: tk logpk (10) k = l where Pk is the value of the kth output unit and tk is an indicator variable which is equal to 1 if k is the true class. Taking derivatives of this expression with respect to the inputs of the output units yields of -=tk-Pk (11) oak o£ Oak -,---- = (tk - Pk)h(mn) OW(mn) ,k oak OW(mn),k O£ (12) The derivative of the output of the (mn)th hidden unit with respect to the output of ith HMM, £i, is oh(mn) ~ = U(£m - £n)(l - U(£m - £n))(bim - bin) (13) where (bim - bin) is an indicator which equals +1 if i = m, -1 if i = n and zero otherwise. This derivative can be chained with the the derivatives backpropagated from the output to the hidden layer. For the final step of the backpropagation procedure we need the derivative of the log-likelihood of each HMM with respect to its parameters. In the experiments we use HMM's with a single, axis-aligned, Gaussian output density per state. We use the following notation for the parameters: • A: aij is the transition probability from state i to state j • II: 7ri is the initial state prior • f./,i: mean vector for state i • Vi: vector of variances for state i • 1-l: set of HMM parameters {A, II, f./" v} We also use the variable St to represent the state of the HMM at time t. We make use of the property of all latent variable density models that the derivative of the log-likelihood is equal to the expected derivative of the joint log-likelihood under the posterior distribution. For an HMM this means that: O£(Xl:TI1-l) '" 0 o1-li = ~ P(Sl:Tlxl:T' 1-l) o1-li log P(Xl:T' Sl:TI1-l) Sl:T The joint likelihood of an HMM is: (logP(Xl:T' Sl:TI1-l)) = T L(b81 ,i)log 7ri + LL(b8"jb8,_1 ,i)log aij + t=2 i,j (14) ~ ~(b8" i) [-~ ~IOgVi 'd - ~ ~(Xt 'd - f./,i,d) 2 /Vi,d] + canst (15) where (-) denotes expectations under the posterior distribution and (b8 , ,i) and (b 8 , ,jb8'_1 ,i) are the expected state occupancies and transitions under this distribution. All the necessary expectations are computed by the forward backward algorithm. We could take derivatives with respect to this functional directly, but that would require doing constrained gradient descent on the probabilities and the variances. Instead, we reparameterize the model using a softmax basis for probability vectors and an exponential basis for the variance parameters. This choice of basis allows us to do unconstrained optimization in the new basis. The new parameters are defined as follows: . _ exp(e;; ») . _ exp(e ; ~») . _ (v) a'J - 2: (e (a» ) , 7r, - 2: (e ( ~»)' V"d - exp(Oi,d ) JI exp 1JI if exp i This results in the following derivatives: O£(Xl:T 11-l) oO(a) 'J T L [(b8 , ,jb8'_1 ,i) - (b 8'_1 ,i)aij ] t = 2 (16) 8£(Xl:T 11£) (8S1 ,i) 1fi (17) 80(7r) • 8£(Xl:T 11£) T l)8st ,i)(Xt,d - f..li,d)/Vi,d (18) 8f..li,d t= l 8£(Xl:T 11£) 1 T 80(v) 2"l)8st ,i) [(Xt,d - f..li,d)2/Vi,d - IJ (19) .,d t=l When chained with the error signal backpropagated from the output, these derivatives give us the direction in which to move the parameters of each HMM in order to increase the log probability of the correct classification of the sequence. 6 Experiments To evaluate the relative merits of the RDN, we compared it against an Alphanet on a speaker identification task. The data was taken from the CSLU 'Speaker Recognition' corpus. It consisted of 12 speakers uttering phrases consisting of 6 different sequences of connected digits recorded multiple times (48) over the course of 12 recording sessions. The data was pre-emphasized and Fourier transformed in 32ms frames at a frame rate of lOms. It was then filtered using 24 bandpass, mel-frequency scaled filters. The log magnitude filter response was then used as the feature vector for the HMM's. This pre-processing reduced the data dimensionality while retaining its spectral structure. While mel-cepstral coefficients are typically recommended for use with axis-aligned Gaussians, they destroy the spectral structure of the data, and we would like to allow for the possibility that of the many HMM's some of them will specialize on particular sub-bands of the frequency domain. They can do this by treating the variance as a measure of the importance of a particular frequency band using large variances for unimportant bands, and small ones for bands to which they pay particular attention. We compared the RDN with an Alphanet and three other models which were implemented as controls. The first of these was a network with a similar architecture to the RDN (as shown in figure 2), except that instead of fixed connections of ±1, the hidden units have a set of adaptable weights to all M of the HMM's. We refer to this network as a comparative density net (CDN). A second control experiment used an architecture similar to a CDN without the hidden layer, i.e. there is a single layer of adaptable weights directly connecting the HMM's with the softmax output units. We label this architecture a CDN-l. The CDN-l differs from the Alphanet in that each softmax output unit has adaptable connections to the HMM's and we can vary the number of HMM's, whereas the Alphanet has just one HMM per class directly connected to each softmax output unit. Finally, we implemented a version of a network similar to an Alphanet, but using a mixture of Gaussians as the input density model. The point of this comparison was to see if the HMM actually achieves a benefit from modelling the temporal aspects of the speaker recognition task. In each experiment an RDN constructed out of a set of, M, 4-state HMM's was compared to the four other networks all matched to have the same number of free parameters, except for the MoGnet. In the case of the MoGnet, we used the same number of Gaussian mixture models as HMM's in the Alphanet, each with the same number of hidden states. Thus, it has fewer parameters, because it is lacking the transition probabilities of the HMM. We ran the experiment four times with a) ~ b) ~ 0.95 ~ e 0.95 e = E=:l 0.9 ~ 0.9 0.85 0.85 0.8 0.8 B 0.75 8 0 0.75 0.7 0.7 EJ 0.65 0.65 0.6 0.6 0.55 0.55 RDN Alphanet MaGnet CDN CDN-1 RDN Alphanet MaGnet CDN CDN-1 Architecture Architecture C) $ d) ~ ~ ~ D e 0.9 ~ ~ * 8 *0.8 a: a: c gO.7 U 0 ~ ~O.8 ~ ~0.6 ·in ·in gj B gj Ci CiO.5 8 0.4 0.3 0.6 RDN Alphanet MeG net CDN CDN-1 RDN Alphanet MeGnet CDN CDN-1 Architecture Architecture Figure 3: Results of the experiments for an RDN with (a) 12, (b) 16, (c) 20 and (d) 24 HMM's. values of M of 12, 16, 20 and 24. For the Alphanet and MoGnet we varied the number of states in the HMM's and the Gaussian mixtures, respectively. For the CDN model we used the same number of 4-state HMM's as the RDN and varied the number of units in the hidden layer of the network. Since the CDN-1 network has no hidden units, we used the same number of HMM's as the RDN and varied the number of states in the HMM. The experiments were repeated 10 times with different training-test set splits. All the models were trained using 90 iterations of a conjugate gradient optimization procedure [6] . 7 Results The boxplot in figure 3 shows the results of the classification performance on the 10 runs in each of the 4 experiments. Comparing the Alphanet and the RDN we see that the RDN consistently outperforms the Alphanet. In all four experiments the difference in their performance under a paired t-test was significant at the level p < 0.01. This indicates that given a classification network with a fixed number of parameters, there is an advantage to using many small HMM's and using all the pairwise information about an observed sequence, as opposed to using a network with a single large HMM per class. In the third experiment involving the MoGnet we see that its performance is comparable to that of the Alphanet. This suggests that the HMM's ability to model the temporal structure of the data is not really necessary for the speaker classification task as we have set it Up.3 Nevertheless, the performance of both the Alphanet and 3If we had done text-dependent speaker identification, instead of multiple digit phrases the MoGnet is less than the RDN. Unfortunately the CDN and CDN-l networks perform much worse than we expected. While we expected these models to perform similarly to the RDN, it seems that the optimization procedure takes much longer with these models. This is probably because the small initial weights from the HMM's to the next layer severely attenuate the backpropagated error derivatives that are used to train the HMM's. As a result the CDN networks do not converge properly in the time allowed. 8 Conclusions We have introduced relative density networks, and shown that this method of discriminatively learning many small density models in place of a single density model per class has benefits in classification performance. In addition, there may be a small speed benefit to using many smaller HMM's compared to a few big ones. Computing the probability of a sequence under an HMM is order O(TK2 ), where T is the length of the sequence and K is the number of hidden states in the network. Thus, smaller HMM's can be evaluated faster. However, this is somewhat counterbalanced by the quadratic growth in the size of the hidden layer as M increases. Acknowledgments We would like to thank John Bridle, Chris Williams, Radford Neal, Sam Roweis, Zoubin Ghahramani, and the anonymous reviewers for helpful comments. References [1] L. E. Baum, T. Petrie, G. Soules, and N. Weiss, "A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains," The Annals of Mathematical Statistics, vol. 41, no. 1, pp. 164-171, 1970. [2] 1. R. Bahl, P. F. Brown, P. V. de Souza, and R. 1. Mercer, "Maximum mutual information of hidden Markov model parameters for speech recognition," in Proceeding of the IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 49- 53, 1986. [3] J. Bridle, "Training stochastic model recognition algorithms as networks can lead to maximum mutual information estimation of parameters," in Advances in Neural Information Processing Systems (D. Touretzky, ed.), vol. 2, (San Mateo, CA), pp. 211- 217, Morgan Kaufmann, 1990. [4] M. I. Jordan, "Why the logistic function? A tutorial discussion on probabilities and neural networks," Tech. Rep. Computational Cognitive Science, Technical Report 9503, Massachusetts Institute of Technology, August 1995. [5] A. D. Brown and G. E. Hinton, "Products of hidden Markov models," in Proceedings of Artificial Intelligence and Statistics 2001 (T. Jaakkola and T. Richardson, eds.), pp. 3- 11, Morgan Kaufmann, 2001. [6] C. E. Rasmussen, Evaluation of Gaussian Processes and other Methods for NonLinear Regression. PhD thesis, University of Toronto, 1996. Matlab conjugate gradient code available from http ://www .gatsby.ucl.ac.uk/~edward/code/. then this might have made a difference.
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Neural Implementation of Bayesian Inference in Population Codes Si Wu Computer Science Department Sheffield University, UK Shun-ichi Amari Lab. for Mathematic Neuroscience, RIKEN Brain Science Institute, JAPAN Abstract This study investigates a population decoding paradigm, in which the estimation of stimulus in the previous step is used as prior knowledge for consecutive decoding. We analyze the decoding accuracy of such a Bayesian decoder (Maximum a Posteriori Estimate), and show that it can be implemented by a biologically plausible recurrent network, where the prior knowledge of stimulus is conveyed by the change in recurrent interactions as a result of Hebbian learning. 1 Introduction Information in the brain is not processed by a single neuron, but rather by a population of them. Such a coding strategy is called population coding. It is conceivable that population coding has advantage of being robust to the fluctuation in a single neuron's activity. However, people argue that population coding may have other computationally desirable properties. One such property is to provide a framework for encoding complex objects by using basis functions [1]. This is inspired by the recent progresses in nonlinear function approximation, such as, sparse coding, overcomplete representation and kernel regression. These methods are efficient and show some interesting neuron-like behaviors [2,3]. It is reasonable to think that similar strategies are used in the brain under the support of population codes. However, to confirm this idea, a general suspicion has to be clarified: can the brain perform such complex statistic inference? An important work towards the answer of this question was done by Pouget and co-authors [4,5]. They show that Maximum Likelihood (ML) Inference, which is usually thought to be complex, can be implemented by a biologically plausible recurrent network using the idea of line attractor. ML is a special case of Bayesian inference when the stimulus is (or assumed to be) uniformly distributed. In case there is prior knowledge on the stimulus distribution, Maximum a Posteriori (MAP) Estimate has better performance. Zhang et al. has successfully applied MAP for reconstructing the rat position in a maze from the activity of hippocampal place cells [6]. In their method, the prior knowledge is the rat's position in the previous time step, which restricts the variability of rat's position in the current step under the continuity constraint. It turns out that MAP has a much better performance than other decoding methods, and overcomes the inefficiency of ML when information is not sufficient (when the rat stops running). This result implies that MAP may be used by the nervous system. So far, in the literature MAP has been mainly studied as a mathematic tool for reconstructing data, though its potential neural implementation was pointed out by [1,6]. In the present study, we will firmly show how to implement MAP in a biologic way. The same kind of recurrent network for achieving ML is used [4,5]. The decoding process consists of two steps. In the first step when there is no prior knowledge of the stimulus, the network implements ML. Its estimation is subsequently used to form the prior distribution of stimulus for consecutive decoding, which we assume is a Gaussian function with the mean value being the estimation. It turns out that this prior knowledge can be naturally conveyed by the change in the recurrent interactions according to the Hebbian learning rule. This is an interesting finding and suggests a new role of Hebbian learning. In the second step, with the changed interactions, the network implements MAP. The decoding accuracy of MAP and the optimal form of Gaussian prior are also analyzed in this paper. 2 MAP in Population Codes Let us consider a standard population coding paradigm. There are N neurons coding for a stimulus x. The population activity is denoted by r = {rd. Here ri is the response of the ith neuron, which is given by (1) where fi(X) is the tuning function and fi is a random noise. The encoding process of a population code is specified by the conditional probability Q(rlx) (i.e., the noise model). The decoding is to infer the value of x from the observed r. We consider a general Bayesian inference in a population code, which estimates the stimulus by maximizing a log posterior distribution, In P(xlr) , i.e., argmaxx In P(xlr) , argmaxx InP(rlx) + InP(x), (2) where P(rlx) is the likelihood function. It can be equal to or different from the real encoding model Q(rlx) , depending on the available information of the encoding process [7]. P(x) is the distribution of x, representing the prior knowledge. This method is also called Maximum a Posteriori (MAP). When the distribution of x is, or is assumed to be (when there is no prior knowledge) uniform, MAP is equivalent to ML. MAP could be used in the information processing of the brain in several occasions. Let us consider the following scenario: a stimulus is decoded in multiple steps. This happens when the same stimulus is presented through multiple steps, or during a single presentation, neural signals are sampled many times. In both cases, the brain successively gains a rough estimation of the stimulus in each step decoding, which can serve to be the prior knowledge when further decoding is concerned. It is therefore natural to use MAP in this situation. Experiencing slightly different stimuli in consecutive steps as studied in [6], or more generally, stimulus slowly changes with time (multiple-step diagram is a discreted approximation), is a similar scenario. For simplicity, we only consider that stimulus is unchanged in the present study. 2.1 The Performance of MAP Let us analyze the performance of MAP. Some notations are introduced first. Denote Xt a particular estimation of the stimulus in the tth step, and 0; the corresponding variance. The prior distribution of x in the t + lth step is assumed to be a Gaussian with the mean value X"~ i.e., P(xIXt) = _1_ exp-CX-Xt)2 /2r;, .,J2irTt (3) where the parameter Tt reflects the estimator's confidence on xt, whose optimal value will be calculated later. The posterior distribution of x in the t + lth step is given by P( I ) = P(rlx)P(xlxt) xr P(r) , and the solution of MAP is obtained by solving \7 In P(Xt+1 Ir) \7lnP(rlxt+l) - (Xt+l-Xt)/T;, O. (4) (5) We calculate the decoding accuracies iteratively. In the first step decoding, since there is no prior knowledge on x, ML is used, whose decoding accuracy is known to be [7] 02«\7lnP(rlx))2> 1 - < -\7\7lnP(rlx) >2' (6) where the bracket < . > denotes averaging over Q(rlx). Note that, to get the above result, we have considered that ML is asymptotically or quasi-asymptotically (when an unfaithful model is used) efficient [7]. This includes the cases when neural responses are independent, weakly correlated, uniformly correlated, correlated with strength proportional to firing rate (multiplicative correlation), or the fluctuation in neural responses are sufficiently small. In other strong correlation cases, ML is proved to be non-Fisherian, i.e, its decoding error satisfies a Cauchy type of distribution with variance diverging. Decoding accuracy can no longer be quantified by variance in such situations (for details, please refer to [8]). Now come to calculate the decoding error in the second step. Suppose X2 is close enough to x. By expanding \7lnP(rlx2) at x in eq.(5), we obtain \7lnP(rlx) + \7\7lnP(rlx)(x2 - x) - (X2 - X1)/T; = O. (7) The random variable Xl can be decomposed as Xl = x + f1, where f1 is a random number satisfying Gaussian distribution of zero mean and variance Oi. By using the notation of f1, we have A \7lnP(rlx)+fdTf X2 -x = l/T; - \7\7lnP(rlx)' (8) For the correlation cases considered in the present study (i.e, those ensure ML asymptotically or quasi-asymptotically efficient), - \7\7 In P(rlx) can be approximated as a (positive) constant according to the law of large numbers [7,8]. Therefore, we can define a constant variable a = T;(-\7\7lnP(rlx)), (9) and a random variable R = \71nP(rlx) - \7\71n P(rlx) . Obviously R satisfies the Gaussian distribution of zero mean and variance 0I. By using the notations 0: and R, we get o:R+fl X2- X = --1+0: whose variance is calculated to be (10) (11) (12) Since (1 + 0:2)/(1 + 0:)2 ::::: 1 holds for any positive 0:, the decoding accuracy in the second step is always improved. It is not difficult to check that its minimum value is 0 2 - !02 2 2 1> when 0: = 1, or, the optimal value of Tl is 1 Tl=----,.......,.....,.. - \7\71n P(rlx) (13) (14) When a faithful model is used, -\7\71nQ(rlx) is the Fisher information. Tl hence equals to the variance of decoding error. This is understandable. Following the same procedure, it can be proved that the optimal decoding accuracy in the tth step is 0; = tOI when the width of Gaussian prior being Tl = tTl. It is interesting to see that the above multiple decoding procedure, when the optimal values of Tt are used, achieves the same decoding accuracy as a one-step ML by using all N x t signals. This is the best for any estimator to achieve. However, the multiple decoding is not a trivial replacement of one-step ML, and has many advantages. One of them is to save memory, considering that only N signals and the value of previous estimation are stored in each step. Moreover, if a slowly changing stimulus is concerned, the multiple decoding outperforms one-step ML for the balance between adaptation and memory. These properties are valuable when information is processed in the brain. 3 Network Implementation of MAP In this section, we investigate how to implement MAP by a recurrent network. A two-step decoding is studied. Without loss of generality, we consider N ---+ 00 and do calculation in the continuous limit. The network we consider is a fully connected one-dimensional homogeneous neural field, in which c denotes the position coordinate, i.e., the neurons' preferred stimuli. The tuning function of the neuron with preferred stimulus c is fc(x) = _1_ exp-(c- x)2/2a2 . (15) "fiifa For simplicity, we consider an encoding process in which the fluctuations in neurons' responses are independent Gaussian noises (more general correlated cases can be handled similarly), that is, Q(rlx) = ~ exp- ~ j(Tc - f c (x))2 dC, (16) where p is the neuron density and Z is the normalization factor. A faithful model is used in both steps decoding, i.e., P(rlx) = Q(rlx) (again, generalization to more general cases of P(rlx) -::/:- Q(rlx) is straightforward.). For the above model setting, the solution of ML in the first step is calculated to be Xl = argmaxx J rc!e(x)de, where the condition J J;(x)de = const has been used. The solution of MAP in the second step is X2 = argmaxx J rc!e(x)de - (x - xd2/ 2Tf. (17) (18) Compared with eq.(17), eq.(18) has one more term corresponding to the contribution of prior distribution. Now come to the study of using a recurrent network to realize eqs.(17) and (18). Following the idea of Pouget et al. [4,5], the following network dynamics is constructed. Let Ue denote the (average) internal state of neuron at e, and We,e' the recurrent connection weights from neurons at e to those at e'. The dynamics of neural excitation is governed by dUe J 0 ' dt = -Ue + We,e' e, de + Ie, (19) where U; o e = ----;;-=--=1 + f..LJU;de (20) is the activity of neurons at e and Ie is the external input arriving at e. The recurrent interactions are chosen to be W - exp-(e-e')2/2a2 c,c' , (21) which ensures that when there is no external input (Ie = 0), the network is neutrally stable on line attractor, 'r:/z, (22) where the parameter D is constant and can be determined easily. Note that the line attractor has the same shape as the tuning function. This is crucial, which allows the network perform template-matching by using the tuning function, being as same as ML and MAP. When a sufficiently small input Ie is added, the network is no longer neutrally stable on the line attractor. It can be proved that the steady state of the network has approximately the same shape as eq.(22) (the deviation is of the 2nd order of the magnitude of Ie.), whereas, its steady position on the line attractor (i.e., the network estimation) is determined by maximizing the overlap between Ie and Oe(Z) [4,9]. Thus, if Ie = ere in the first step1, where e is a sufficiently small number, the network estimation is given by 21 = argmaxz J reOe(z)de, (23) ------------lConsider an instant input, triggering the network to be initially at Oe(t = 0) = r e, as used in [5], has the same result. which has the same value as the solution of ML (see eq.(I7)). We say that the network implements ML. To implement MAP in the second step, it is critical to identify a neural mechanism which can 'transmit' the prior knowledge obtained in the first step to the second one. We find that this is naturally done by Hebbian learning. After the first step decoding, the recurrent interaction changes a small amount according to the Hebbian rule, whose new value is (24) where TJ is a small positive number representing the Hebbian learning rate, and Oe(,2d is the neuron activity in the first step. With the new recurrent interactions, the net input from other neurons to the one at c is calculated to be J We,el Oel dc' J We,el Oel dc' +TJOe(,2d J Oe/(zdOe,dc', (25) where 1/ is a small constant. To get the last approximation, the following facts have been used: 1) The initial state of neuron in the second step is at Oe(Z1 ), 2) The neuron activity Oe during the second step is between Oe(zd and Oe(Z2 ), where Z2 is the position of the steady state; 3) (Z1 - z2 )2/2a2 « 1, considering that neurons are widely tuned as seen in data (a is large) and consecutive estimations are close enough. These factors ensures the approximation, J Oe/ (zdOe,dc' :=;:j const to be good enough. Substituting eq.(25) in (19), we see that the network dynamics in the second step, when compared with the first one, is in effect to modify the input Ie to be I~ = €(re + AOc(zd), where A is a constant and can be determined easily. Thus, the network estimation in the second step is determined by maximizing the overlap between I~ and Oc(z), which gives Z2 = argmaxz J rcOc(z)dc + A J Oe(zdOe(z)dc. (26) The first term in the right handside is known to achieve ML. Let us see the contribution of the second one, which can be transformed to J Oe(zdOc(z)dc = Bexp-CZI-Z)2/4a2, :=;:j -B(z - zd 2 /4a2 + terms not on z, (27) where B is a constant. Again, in the above calculation, (Z1 - z)2/4a2 « 1 is used for the same argument discussed above. Compare eqs.(I8) and (27), we see that the second term plays the same role as the prior knowledge in MAP. Thus, the network indeed implements MAP. The value of A (or the Hebbian learning rate) can be adjusted accordingly to match the optimal choice of Tf . The above result is confirmed by the simulation experiment (Table.I) , which was done with 101 neurons uniformly distributed in the region [-3,3] and the true stimulus being at O. It shows that the estimation of the network agrees well with MAP. Table 1: Comparing the decoding accuracies of the network and MAP with different values of a (the corresponding values of T[ and A are adjusted.). The parameters are a = 1, f-t = 0.5 and (J2 = 0.01. The data is obtained after 100 trials. 4 Conclusion and Discussion In summary we have investigated how to implement MAP by using a biologically plausible recurrent network. A two-step decoding paradigm is studied. In the first step when there is no prior knowledge, the network implements ML, whose estimation is subsequently used to form the prior distribution of stimulus for consecutive decoding. In the second step, the network implements MAP. Line attractor and Hebbian learning are two critical elements to implement MAP. The former enables the network to do template-matching by using the tuning function, being as same as ML and MAP. The latter provides a mechanism that conveys the prior knowledge obtained from the first step to the second one. Though the results in this paper may quantitatively depend on the formulation of the models, it is reasonable to believe that they are qualitatively true, as both Hebbian learning and line attractor are biologically plausible. Line attractor comes from the translation invariance of network interactions, and has been shown to be involved in several neural computations [10-12]. We expect that the essential idea of Bayesian inference of utilizing previous knowledge for successive decoding is used in the information processing of the brain. We also analyzed the decoding accuracy of MAP in a population code and the optimal form of Gaussian prior. In the present study, stimulus is kept to be fixed during consecutive decodings. A generalization to the case when stimulus slowly changes over time is straightforward. References [1] A. Pouget, P. Dayan & R. Zemel. Nature Reviews Neuroscience, 1, 125-132, 2000. [2] B. Olshausen & D. Field. Nature, 381, 607-609, 1996. [3] T. Poggio & F. Girosi. Neural Computation, 10, 1445-1454, 1998. [4] A. Pouget & K. Zhang. NIPS, 9, 1997. [5] S. Deneve, P. E. Latham & A. Pouget. Nature Neuroscience, 2, 740-745, 1999. [6] K. Zhang, 1. Ginzburg, B. McNaughton & T. Sejnowski. J. Neurophysiol., 79, 10171044, 1998. [7] S. Wu, H. Nakahara & S. Amari. Neural Computation, 13, 775-798, 200l. [8] S. Wu, S. Amari & H. Nakahara. CNS*Ol (to appear). [9] S. Wu, S. Amari & H. Nakahara. Neural Computation (in press). [10] S. Amari. Biological Cybernetics, 27, 77-87, 1977. [11] K. Zhang. J. Neurosci., 16, 2112-2126, 1996. [12] H. Seung. Proc. Natl. Acad. Sci. USA , 93, 13339-13344, 1996.
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TAP Gibbs Free Energy, Belief Propagation and Sparsity Lehel Csat´o and Manfred Opper Neural Computing Research Group School of Engineering and Applied Science Aston University, Birmingham B4 7ET, UK. [csatol,opperm]@aston.ac.uk Ole Winther Center for Biological Sequence Analysis, BioCentrum Technical University of Denmark, B208, 2800 Lyngby, Denmark. winther@cbs.dtu.dk Abstract The adaptive TAP Gibbs free energy for a general densely connected probabilistic model with quadratic interactions and arbritary single site constraints is derived. We show how a specific sequential minimization of the free energy leads to a generalization of Minka’s expectation propagation. Lastly, we derive a sparse representation version of the sequential algorithm. The usefulness of the approach is demonstrated on classification and density estimation with Gaussian processes and on an independent component analysis problem. 1 Introduction There is an increasing interest in methods for approximate inference in probabilistic (graphical) models. Such approximations may usually be grouped in three classes. In the first case we approximate self-consistency relations for marginal probabilities by a set of nonlinear equations. Mean field (MF) approximations and their advanced extensions belong to this group. However, it is not clear in general, how to solve these equations efficiently. This latter problem is of central concern to the second class, the Message passing algorithms, like Bayesian online approaches (for references, see e.g. [1]) and belief propagation (BP) which dynamically update approximations to conditional probabilities. Finally, approximations based on Free Energies allow us to derive marginal moments by minimising entropic loss measures. This method introduces new possibilities for algorithms and also gives approximations for the log-likelihood of observed data. The variational method is the most prominent member of this group. One can gain important insight into an approximation, when it can be derived by different approaches. Recently, the fixed points of the BP algorithm were identified as the stable minima of the Bethe Free Energy, an insight which led to improved approximation schemes [2]. While BP is good and efficient on sparse tree-like structures, one may look for an approximation that works well in the opposite limit of densely connected graphs where individual dependencies are weak but their overall effect cannot be neglected. A interesting candidate is the adaptive TAP (ADATAP) approach introduced in [3] as a set of self-consistency relations. Recently, a message passing algorithm of Minka (termed expectation propagation) [1] was found to solve the ADATAP equations efficiently for models with Gaussian Process (GP) priors. The goal of this paper is three-fold. We will add a further derivation of ADATAP using an approximate free energy. A sequential algorithm for minimising the free energy generalises Minka’s result. Finally, we discuss how a sparse representation of ADATAP can be achieved for GP models, thereby extending previous sparse on-line approximations to the batch case [4]. We will specialize to probabilistic models on undirected graphs with nodes that are of the type
!" $#% (1) The set of &" ’s encodes the dependencies between the random variables ')(+*,*+*,( . , whereas the factorising term /01 32 (called likelihood in the following) usually encodes observed data at sites and also incorporates all local constraints of the (the range, discreteness, etc). Hence, depending on these constraints, maybe discrete or continuous. Eq. (1) is a sufficiently rich and interesting class of models containing Boltzmann machines, models with Gaussian process priors [3], probabilistic independent component analysis [5] as well as Bayes belief networks and probabilistic neural networks (when the space of variables is augmented by auxiliary integration variables). 2 ADATAP approach from Gibbs Free Energy We use the minimization of an approximation to a Gibbs Free Energy 4 in order to rederive the ADATAP approximation. The Gibbs Free Energy provides a method for computing marginal moments of as well as of 57698 within the same approach. It is defined by a constrained relative entropy minimization which is, for the present problem defined as 4 ;: (< >=@? 8 ACBEDGF IH ( JLKM/0N A >: ( M0O+N A <QP 5G698 ( (2) where the brackets denote expectations with respect to the distribution H and M O N A is shorthand for a vector with elements M O N A . Finally, DRF H ( J * TS@UVHW 698 A X9Y[Z \E] X9YEZ . Since at the total minimum of 4 (with respect to its arguments) the minimizer in (2) is just H^ _ , we conclude that =`? 8badc ef4 I: (g< h 5i698 and the desired marginal moments of are M/0N ( M O Ngj >kElgm!=@? 8 adc e 4 I:V . We will search for an approximation to 4 which is based on splitting 4 n 4po Lqsr 4 , where 4to is the Gibbs free energy for a factorising model that is obtained from (1) by setting all " >u . Previous attempts [6, 7] were based on a truncation of the power series expansion of r 4 with respect to the &v at second order. While this truncation leads to the correct TAP equations for the large w limit of the so-called SK-model in statistical physics, its general significance is unclear. In fact, it will not be exact for a simple model with Gaussian likelihood. To make our approximation exact for such a case, we define (generalizing an idea of [8]) for an arbitrary Gaussian likelihood yx , r 4 x ;: (< * 4 $z;: (< 5p4po {z ;: (g< . The main reason for this definition is the fact that r 4 x ;: (< is independent of the actual Gaussian likelihood x chosen to compute 4 ! This result depends crucially on the moment constraints in (2). Changes in a Gaussian likelihood can always be absorbed within the Lagrange-multipliers for the constraints. We use this universal form r 4 x to define the ADATAP approximation as 4 \ 4po qGr 4 x , which by construction is exact for any Gaussian likelihood . Introducing appropriate Lagrange multipliers and , we get r 4 x ;: (< >=Wk c 57698 x ( q : R5 < R5 6 8
5 O (3) with x ( S U 5 ' O O q ;y !v * Finally, setting o o ( o SnU o 5 ' O o O , we have 4 o =Wk "! c !$# 5 6 8 o o ( o q : o 5 < o&% * (4) 3 Sequential Algorithm The expression of 4' \ in terms of moments ;: (< and Lagrange parameters ( and o ( o suggests that we may find local minima of 4 \ by iteratively alternating between updates of moments and Lagrange multipliers. Of special interest is the following sequential algorithm, which is a generalization of Minka’s EP [1] for Gaussian process classification to an arbritary model of the type eq. (1). We choose a site and define the updates by using the saddle points of 4 with respect to the moments and Lagrange multipliers in the following sequential order (where ( is a diagonal matrix with elements ): )+* , c , 4 fu . "/ 10 ( 532 54 ' 6 v &
5 O / 0 ( 532 4 ' 6 )78, c 9 , 4 fu:. o / 5 5 7 , 9 , 4 78; , & o / 5 5 ' 9 , 4 7<; , ) * ! , c ! , 4 u . / ) * ! , 698 o &
/ 5 ) ! , 698 o )78, c 9 , 4 fu:. "/ 5 o 5 7 , 9 , 4 78; , & =/ 5 o 5 ' 9 , 4 7<; , . The algorithm proceeds then by choosing a new site. The computation of (T5>2 4 ' can be performed efficiently using the Sherman-Woodbury formula because only one element is changed in each update. 3.1 Cavity interpretation At the fixed point, we may take $? , XA@&Z B ! , o 5 ' O o O as the ADATAP approximation to the true marginal distribution of [3]. The sequential approach may thus be considered as a belief propagation algorithm for ADATAP. Although is usually not Gaussian, we can also derive the moments : and < from the Gaussian distribution corresponding to x . This auxiliary Gaussian model x has a likelihood &x 8C 5 ' O O q and provides us also with an additional approximation to the matrix of covariances via D ( 532 4 ' . This is useful when the coupling matrix 2 must be adapted to a set of observations by maximum likelihood II. We will give an example of this for independent component analysis below. It is important to understand the role of o and o within the “cavity” approach to the TAP equations. Defining E &v , it is easy to show that o M E NGF and o M E O NHF 5 M E N O F where the brackets denote an expectation with respect to the distribution of all remaining variables x C 2 c &x + when node is deleted from the graph. This statistics of E corresponds to the empty ”cavity” at site . The marginal distribution as computed by ADATAP is equivalent to the approximation that the cavity distribution is Gaussian. 4 Examples 4.1 Models with Gaussian Process Priors For this class of models, we assume that the graph is embedded in , where the vector is the restriction of a Gaussian process (random field)
0 with , to a set of training inputs via
. is the posterior distribution corresponding to a local likelihood model, when we set 2 5 4 ' and the matrix is obtained from a positive definite covariance kernel as D v D o ( . The diagonal element D 4 ' 9 is included in the likelihood term. Our ADATAP approximation can be extended from the finite set of inputs to the entire space by extending the auxiliary Gaussian distribution x with its likelihoods &x to a Gaussian process with mean M
0 N and posterior covariance kernel D ( which approximates the posterior process. A calculation similar to [4] leads to the representation M
0 N D o ( (5) D ( D o ( q c D o ( D D o ( (6) Algorithms for the update of ’s and D ’s will usually suffer from time consuming matrix multiplications when w is large. This common problem for GP models can be overcome by a sparsity approximation which extends previous on-line approaches [4] to the batch ADATAP approach. The idea is to replace the current version x of the approximate Gaussian with a further approximation x for which both the the corresponding as well as D are nonzero only, when the nodes and belong to a smaller subset of nodes called ”basis vectors” (BV) of size [4]. For fixed BV set, the parameters of x are determined by minimizing the relative entropy DGF x ( x . This yields and ( ! ( with the #"Gw projection matrix Q 4 ' $&% (' . Here is the kernel matrix between BVs and and ' the kernel matrix between BVs and all nodes. The new distribution x can be written in the form (1) with a likelihood that contains only BVs x $)% j $&% 5 +* $&% +, O * (7) Eq. (7) can be used to compute the sparse approximation within the sequential algorithm. We will only give a brief discussion here. In order to recompute the appropriate ”cavity” parameters o and o when a new node is chosen by the algorithm, one removes a ”pseudovariable” $&% from the likelihood and recomputes the statistics of the remaining ones. When is in the BV set, then simply $&% $&% and the computation reduces to the previous one. We will demonstrate the significance of this approach for two examples. 4.2 Independent Component Analysis We consider a measured signal -/. which is assumed to be an instantaneous linear mixing of sources corrupted with additive white Gaussian noise 0 that is, -/. 21i . q 03. ( (8) where 1 is a (time independent) mixing matrix and the noise vector is assumed to be without temporal correlations having time independent covariance matrix . We thus have the following likelihood for parameters and sources at time W . K 1 ( ( . j 4 ; 4 ; X
4 YZ X
4 YIZ * (9) and for all times W K 1 ( ( 0i 2 . W . K 1 ( ( . . The aim of independent component analysis is to recover the unknown quantities: the sources , the mixing matrix 1 and the noise covariance from the observed data using the assumption of statistical independence of the sources W . 2 W . . Following [5], we estimate the mixing matrix 1 and the noise covariance , by an MLII procedure, i.e. by maximizing the Likelihood W K 1 ( j S`U W K 1 ( ( 0yW . The corresponding estimates are 1 . -. M . N . M . . N 4 ' and ' M 5 1i 5 170 N * These estimates require averages over the posterior of which has again the structure of the model eq. (1). They can be obtained efficiently using our sequential belief propagation algorithm in an iterative EM fashion, where the E-step amounts to estimating M . N and M/ . . N with fixed 1 and and the M-step consists of updating 1 and . 5 Simulations 5.1 Classification with GPs This problem has been studied before [9, 4] using a sequential, sparse algorithm, based on a single sweep through the data only. Within the ADATAP approach we are able to perform multiple sweeps in order to achieve a self-consistent solution. The outputs are binary ! * 5 ( , and the likelihood is based on the probit model W ! K
"bg $#jl&%0' j ' ( O*) S + 4-, U 5 . ; O * where 'G !
"b/.0 o and 0 o measures the noise level. The predictive distribution for a new test input " is # l/%, ! M
"b N . .1032! with 0 O 2 40 O o q D . " ( "b , which is easily rewritten in terms of the parameters ’s and D ’s according to eqs. (5). We used the USPS dataset1 of gray-scale handwritten digit images of size 5 " 5 with 6 7 training patterns and u!u86 test patterns. For the kernel we choose the RBF kernel D o " ( " :98; 5=< " 5 " < O . 0 O ; g where is the dimension of the inputs ( >5 in this case), and 9 ; and 0 ; are parameters. In the simulations we used 6 uEu!u random training examples. We performed simulations for different sizes of the BV set and compared multiple iterations with a single sweep through the dataset. The results are displayed in Fig. 1. The lines show the average results of > runs where the task was to classify the digits into fours/non-fours. Our results show that, in contrast to the online learning, the fluctuations caused by the order of presentation are diminished (marked with bars on the figure). 5.2 Density estimation with GPs Bayesian non-parametric models for density estimation can be defined [10] by parametrising densities ? as ? "K
@ ; X 2 Z A @ ; X 2 ZB 2 and using a Gaussian process prior over the space of functions
. Observing w data points C D" ' (,*+*,*+( " - , we can express the predictive distribution (again, E denotes the expectation over the GP prior) as FGIH-J KMLON P Q=R S FGIH-J TULWV X YZ[ FGIH Y J TUL\]N P Q ^`_badc egfhh V R S TUijGIHLkV X YZ[ TUilGIH Y L-monqpsrt ;vuxwy{z*w \ | a}c e fh Q p h V R p S TUilGIHL~V X YZ[ TUilGIH Y L \ 1Available from http://www.kernel-machines.org/data/ 50 150 250 350 450 550 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 #BV Test error % 1 iterations 4 iterations USPS: 4 <−> non 4 Figure 1: Results for classification for different BV sizes (x-axis) and multiple sweeps through the data. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 2: The GP estimation (continuous line) of a mixture of Gaussians (dotted line) using u BVs. In the last expression, we have introduced an expectation over a new, effective Gaussian obtained by multiplying the old prior and the term 4 A @ ; X 2 ZB 2 and normalizing by . We assume that for sufficiently large w the integral over can be performed by Laplace’s method, leaving us with an approximate predictor of the form ? "K C C M
O "bgN , where the brackets denote posterior expectation for a GP model with a kernel that is a solution to the integral equation D " ( ! D o " ( ! 5 S U D o " ( D ( ! . The likelihood of the fields *
" at the observation points is O 4 ; X ; Z , , @ ; . For any fixed , we can apply the sparse ADATAP algorithm to this problem. After convergence of this inner loop, a new value of must be determined from (following a Laplace argument) M
O " N until global convergence is achieved. To give a simplified toy example, we choose a kernel D o " ( ! which reproduces itself after convolution. Hence, the dependence is scaled out and we work with and normalised at the end. We used a periodic kernel for data in u ( given by D o " ( ! 5 o " 5 ! q ? 8 1 o " 5 ! g " 5 ! * D o has constant Fourier coefficients up to a cutoff frequency o ( o 5 in our simulations). For the experiment we are using artificial data from a mixture of two Gaussians (dotted line in Fig. 2). We apply the sparse algorithm with multiple sweeps through the data. The sparsity also avoids the numerical problems caused by a possible close to singular Gram matrix. For the experiments, the size of the BV set was not limited a priori, and a similar criterion as in [4] was chosen in order to decide whether a data point should be included in the BV set or not. As a result, for > uEu training data, only u were retained in the BV set. (continuous line in Fig. 2). 5.3 Independent Component Analysis We have tested the sequential algorithm on an ICA problem for local feature extraction in hand written digits, i.e. extracting the different stroke styles [5] . We assumed positive components of 1 (enforced by Lagrange multipliers) and a positive prior W . ` . 5 . (10) As in [5] we used 500 handwritten ’3’s which are assumed to be generated by 25 hidden images. We compared a traditional parallel update algorithm with the sequential belief propagation algorithm. Both algorithms have computational complexity w . We find that the sequential algorithm needs only on average 7 sweeps through the sites to reach the desired accuracy whereas the parallel one fails to reach the desired accuracy in 100 sweeps using a somewhat larger number of flops. The adaptive TAP method using the sequential belief propagation approach is also not more computationally expensive than the linear response method used in [5]. 6 Conclusion and Outlook An obvious future direction for the ADATAP approach is the investigation of other minimization algorithms as an alternative to the EP approach outlined before. Also an extension of the sparse approximation to other non-GP models will be interesting. A highly important but difficult problem is the assessment of the accuracy of the approximation. Acknowledgments M. Opper is grateful to Lars Kai Hansen for suggesting the non-parametric density model. O. Winther thanks Pedro Højen-Sørensen for the use of his Matlab code. The work is supported by EPSRC grant no. GR/M81601 and by the Danish Research Councils through Center for Biological Sequence Analysis. References [1] T.P. Minka. Expectation propagation for approximate Bayesian inference. PhD thesis, Dep. of Electrical Eng. and Comp. Sci.; MIT, 2000. [2] J. S. Yedidia, W. T. Freeman and Y. Weiss, Generalized Belief Propagation, to appear in Advances in Neural Information Processing Systems (NIPS’2000), MIT Press (2001). [3] M. Opper and O. Winther, Tractable approximations for probabilistic models: The adaptive TAP approach, Phys. Rev. Lett. 86, 3695 (2001). [4] L. Csat´o and M. Opper. Sparse Gaussian Processes. Neural Computation accepted (2001). [5] P.A.d.F.R. Højen-Sørensen, O. Winther, and L. K. Hansen, Mean Field Approaches to Independent Component Analysis, Neural Computation accepted (2001). Available from http://www.cbs.dtu.dk/winther/ [6] T. Plefka, Convergence condition of the TAP equations for the infinite-ranged Ising spin glass model, J. Phys. A 15, 1971 (1982). [7] T. Tanaka, Mean-Field Theory of Boltzmann Machine Learning, Phys. Rev. E 58, 2302(1998). [8] G. Parisi and M. Potters, Mean-Field Equations for Spin Models with Orthogonal Interaction Matrices, J. Phys. A (Math. Gen.) 28, 5267 (1995). [9] L. Csat´o, E. Fokou´e, M. Opper, B. Schottky, and O. Winther. Efficient approaches to Gaussian process classification. In Advances in Neural Information Processing Systems, volume 12, (2000). [10] D.M. Schmidt. Continuous probability distributions from finite data. arXiv:physics/9808005 (1998)
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Quantizing Density Estimators Peter Meinicke Neuroinformatics Group University of Bielefeld Bielefeld, Germany pmeinick@techfak.uni-bielefeld.de Helge Ritter Neuroinformatics Group University of Bielefeld Bielefeld, Germany helge@techfak.uni-bielefeld.de Abstract We suggest a nonparametric framework for unsupervised learning of projection models in terms of density estimation on quantized sample spaces. The objective is not to optimally reconstruct the data but instead the quantizer is chosen to optimally reconstruct the density of the data. For the resulting quantizing density estimator (QDE) we present a general method for parameter estimation and model selection. We show how projection sets which correspond to traditional unsupervised methods like vector quantization or PCA appear in the new framework. For a principal component quantizer we present results on synthetic and realworld data, which show that the QDE can improve the generalization of the kernel density estimator although its estimate is based on significantly lower-dimensional projection indices of the data. 1 Introduction Unsupervised learning is essentially concerned with finding alternative representations for unlabeled data. These alternative representations usually reflect some important properties of the underlying distribution and usually they try to exploit some redundancy in the data. In that way many unsupervised methods aim at a complexity-reduced representation of the data, like the most common approaches, namely vector quantization (VQ) and principal component analysis (PCA). Both approaches can be viewed as specific kinds of quantization, which is a basic mechanism of complexity reduction. The objective of our approach to unsupervised learning is to achieve a suitable quantization of the data space which allows for an optimal reconstruction of the underlying density from a finite sample. In that way we consider unsupervised learning as density estimation on a quantized sample space and the resulting estimator will be referred to as quantizing density estimator (QDE). The construction of a QDE first requires to specify a suitable class of parametrized quantization functions and then to select from this set a certain function with good generalization properties. While the first point is common to unsupervised learning, the latter point is addressed in a density estimation framework where we tackle the model selection problem in a data-driven and nonparametric way. It is often overlooked that modern Bayesian approaches to unsupervised learning and model selection are almost always based on some strong assumptions about the data distribution. Unfortunately these assumptions usually cannot be inferred from human knowledge about the data domain and therefore the model building process is usually driven by computational considerations. Although our approach can be interpreted in terms of a generative model of the data, in contrast to most other generative models (see [10] for an overview), the present approach is nonparametric, since no specific assumptions about the functional form of the data distribution have to be made. In that way our approach compares well with other quantization methods, like principal curves and surfaces [4, 13, 6], which only have to make rather general assumptions about the underlying distribution. The QDE approach can utilize these methods as specific quantization techniques and shows a practical way how to further automatize the construction of unsupervised learning machines. 2 Quantization by Density Estimation We will now explain how the QDE may be derived from a generalization of the kernel density estimator (KDE), one of the most popular methods for nonparametric density estimation [12, 11]. If we construct a kernel density estimator on the basis of a quantized sample, we have the following estimator
(1) where
"! is a sample from the target distribution, $#% #& denotes the kernel function and ('*)+-,/.*0213)+ with parameter vector is a given quantization or projection function which maps a point to a parametrized subset . 0 of the sample space 4 576 98 : 4 ;<8 :=>7?A@ B>CEDGF HJILKNM POQ6*R S M (2) Thereby the projection index 8T:UV#W associates a data point with its nearest neighbour in the projection set . 0 which is parametrized according to . 0 X'AQ76 R Y ;ZR\[P]^1_)a` ! (3) where ] is the set of all possible projection indices which are realizations of the deterministic latent variable R . For a fixed non-zero kernel bandwidth the parameters of the quantization function may be determined by nonparametric maximum likelihood (ML) estimation, as will be introduced in the next section. For an intuitive motivation of the QDE, one may ask from a data compression perspective whether it is necessary to store all the sample data
! for the realization of the kernel density estimator or if it is possible to first reduce the data by some suitable quantization method and then construct the estimator from the more parsimonious complexity-reduced data set. Clearly, we would prefer a quantizer which does not decrease the performance of the estimator on unseen and unquantized data. To get an idea of how to select a suitable quantization function let us consider an example from a 1D data space. In one dimension a natural projection set can be specified by a set of b quantization levels on the real line, i.e. . 0 dc cfe ! . For a fixed kernel bandwidth, we can now perform maximum likelihood estimation of the level coordinates. In that way we obtain a maximum likelihood estimator of the form agh5 e =i gY c (4) with i 3j k 'mlno?A@ BCEDpFZqnj gdrfO c qsj ! j counting the number of data points which are quantized to level l . In this case, it remains the question how to choose the number of quantization levels. From a different starting point the authors in [3] proposed the same functional form of a nonparametric ML density estimator with respect to Gaussian kernels of equal width centered on variable positions. As with the traditional Gaussian KDE (fixed kernel centers on data points), for consistency of the estimator the bandwidth has to be decreased as the sample size increases. In [3] the authors reported that for a fixed non-zero bandwidth, MLestimation of the kernel centers always resulted in a smaller number of actually distinct centers, i.e. several kernels coincided to maximize the likelihood. Therefore the resulting estimator had the form of (4) where b corresponds to the number of distinct centers with i counting the number of kernels coinciding at c . The optimum number of effective quantization levels for a given bandwidth therefore arises as an automatic byproduct of ML estimation. Finally one has to choose an appropriate kernel width which implicitly determines the complexity of the quantizer. The bandwidth selection problem has been tackled in the domain of kernel density estimation for some time and many approaches have been proposed (see e.g. [5] for an overview), among which the cross-validation methods are most common. In the next section we will adopt the method of likelihood cross-validation to find a practical answer to the bandwidth selection problem. 3 General Learning Scheme By applying the method of sieves as proposed in [3], for a fixed non-zero bandwidth we can estimate the parameters of the quantization function via maximization of the log-likelihood
BZ9a w.r.t. to . For consistency of the resulting density estimator the bandwidth has to be decreased as the sample size increases, since asymptotically the estimator must converge to a mixture of delta functions centered on the data points. Thus, for decreasing bandwidth, the quantization function of the QDE must converge to the identity function, i.e. the QDE must converge to the kernel density estimator. For a fixed bandwidth, maximization of the likelihood can be achieved by applying the EMalgorithm [2] which provides a convenient optimization scheme, especially for Gaussian kernels. The EM-scheme requires to iterate the following two steps E-Step: r Yr
q ;=q (5) M-Step: ?A@ B5C ? 0
r r B ; r Y (6) for a sequence s i with suitable initial parameter vector and sufficient convergence at . Thereby r denotes the posterior probability that data point l has been “generated” by mixture component k with density ; r Y . For further insight one may realize that the M-Step requires to solve a constrained optimization problem by searching for C ? 0
r r B 6*R "! # R R
(7) subject to R 7?A@ B CEDpF HJILK M OX6 R Y M (8) In general this optimization problem can only be solved by iterative techniques. Therefore it may be convenient not to maximize but only to increase the log-likelihood at the M-Step which then corresponds to an application of the generalized EM-algorithm. Without (8) unconstrained maximization according to (7) yields another class of interesting learning schemes which for reasons of space will not be considered in this paper. For Gaussian kernels and an Euclidean metric for the projection, in the limiting case of a vanishing bandwidth, EM-optimization of the QDE parameters corresponds to minimization of the following error or risk
CEDpF HJILK M OX6 R Y M Minimization of such error functions corresponds to traditional approaches to unsupervised learning of projection models which can be viewed as a special case of QDE-based learning. 3.1 Bandwidth Selection It is easy to see that the kernel bandwidth cannot be determined by ML-estimation since maximization of the likelihood would drive the bandwidth towards zero. For selection of the kernel bandwidth, we therefore apply the method of likelihood cross-validation (see e.g. [12]), which can be realized by a slight extension of the above EM-scheme. With the leave-one-out QDE O r ;Yr 4 (9) the idea is to maximize
BZ with respect to the kernel bandwidth. For a Gaussian kernel with bandwidth an appropriate EM scheme requires the following MStep update rule
r r M OX r Y M (10) The posterior probabilities r are easily derived from a leave-one-out version of (5). In an overall optimization scheme one may now alter the estimation of and or alternatively one may estimate both by likelihood cross-validation. 4 Projection Sets in Multidimensions By the specification of a certain class of quantization functions we can incorporate domain knowledge into the density estimation process, in order to improve generalization. Thereby the idea is to reduce the variance of the density estimator by reducing the variation of the quantized training set. The price is an increase of the bias which requires a careful selection of the set of admissible quantization functions. Then the QDE offers the chance to find a better bias/variance trade-off then with the “non-quantizing” KDE. We will now show how to utilize existing methods for unsupervised learning within the current density estimation framework. Because many unsupervised methods can be stated in terms of finding optimal projection sets, it is straightforward to apply the corresponding types of quantization functions within the current framework. Thus in the following we shall consider specific parametrizations of the general projection set (3) which correspond to traditional unsupervised learning methods. 4.1 Vector Quantization Vector quantization (VQ) is a standard technique among unsupervised methods and it is easily incorporated into the current density estimation framework by straightforward generalization of the one-dimensional quantizer in section 2 to the multi-dimensional case. Again with a fixed kernel bandwidth ML estimation yields a certain number of b distinct (“effective”) quantization levels, similar to maximum entropy clustering [9, 1]. The projection set of a vector quantizer can be parametrized according to a general basis function representation [7] 6 4 > ;N[ ! (11) with -dimensional vector of basis functions $#& containing discrete delta functions, i.e. rL s5 rq for component k . The QDE on the basis of a vector quantizer can be expected to generalize well if some cluster structure is present within the data. In multi-dimensional spaces the data are often concentrated in certain regions which allows for a sparse representation by some reference vectors well-positioned in those regions. An alternative approach has been proposed in [14] where the application of the support vector formalism to density estimation results in a sparse representation of the data distribution. 4.2 Principal Component Analysis A linear affine parametrization of the projection set yields candidate functions of the form 6 R 4 5
Rs RX[N)a` (12) with . The PCA approach reflects our knowledge that in most high-dimensional data spaces, the data are concentrated around some manifold of lower dimensionality. To exploit this structure PCA divides the sample space into two subspaces which are quantized in different ways: within the “inner” subspace spanned by the directions of the projection manifold we have no quantization at all; within the orthogonal “outer” subspace the data are quantized to a single level. With a Gaussian kernel with fixed bandwidth the constrained optimization problem at the M-Step takes a convenient form which facilitates further analysis of the learning algorithm. From (7) and (8) it follows that one has to maximize the following objective function sN> const. O
r r M O r O L5O M (13) where matrix has orthogonal columns which span the subspace of the projection manifold. From the consideration of the corresponding stationarity conditions one finds that the sample mean Q
is an estimator of the shift vector . Maximization of (13) with respect to then requires to maximize the following trace tr ! O" O #a$ S tr &%' (14) with symmetric matrices !
r r # r r # ;"
r r # r
r (15) Thus (14) is maximized if contains all eigenvectors of % , associated with positive eigenvalues, i.e. with ( ( (*) being the eigenvalues of % we have the optimal subspace dimensionality j +( ' ( , sal ! j (16) which complements a recent result about parametric dimensionality estimation with respect to a -factor model with isotropic Gaussian noise [8]. For the QDE, the two limiting cases of zero and infinite bandwidth, are of particular interest. With the positive definite sample covariance matrix
O O one can show DpC % O DpC % (17) Thus for sufficiently large bandwidth % becomes negative definite, which implies a zero subspace dimensionality estimator , i.e. all data are quantized to the sample mean. For sufficiently small bandwidth % becomes positive definite implying , i.e. no quantization takes place. 4.3 Independent Component Analysis The PCA method provides a rather coarse quantization scheme since it only decides between one-level and no quantization for each subspace dimension. A natural refinement would therefore be to allow for a certain number of effective quantization levels for each component. Such an approach may be viewed as a nonparametric variant of independent component analysis (ICA). The idea is to quantize each coordinate axis separately, which yields a multi-dimensional quantization grid according to 6 R Y > + #s R\[ ! + (18) with [ )+ , [ )
and $#& as in (11). Thereby the components of contain the quantization levels of the l -th coordinate axis with direction . Further, it makes sense to normalize the direction vectors according to M M . There are strong similarities with a parametric ICA model which has been suggested in [10], where source densities have been mixtures of delta functions and additive noise has been isotropic Gaussian. Other unsupervised learning methods which correspond to different projection sets, like principal curves or multilayer perceptrons (see [7] for an overview) can as well be incorporated into the QDE framework and will be considered elsewhere. 5 Experiments In the following experiments we investigated the PCA based QDE with Gaussian kernel and compared the generalization performance with that of the “non-quantizing” KDE. All parameters, including the bandwidth of the KDE, were estimated by likelihood crossvalidation. In the first experiment we sampled 100 points from a stretched and rotated uniform distribution with support on a h rectangle. In this case the QDE extracted a one-dimensional “unquantized” subspace. Generalization performance was measured by the average log-likelihood on an independent 1000-point test set. With an automatically selected 1D subspace (compression ratio
) the PCA-QDE could improve the performance of the KDE from to . Thus, the PCA-QDE could successfully exploit the elongated structure of the distribution. The estimated density functions are depicted in figure 1, where grey-values are proportional to
on a grid. From the images one can see, that the QDE better captures the global structure of the distribution while the KDE is more sensitive to local variations in the data. In a second experiment we trained PCA-QDEs with -dimensional real-world data ( images) which had been derived from the MNIST database of handwritten digits (http://www.research.att.com/ yann/ocr/mnist/). For each digit class a -point training set and a -point test set were used to compare the PCA-QDE with Figure 1: Left: stretched uniform distribution in 2D with white points indicating 100 data points used for estimation; middle: Estimated density using the PCA-QDE; right: kernel density estimate. the KDE, with results shown in table 1. Again the PCA-QDE improved the generalization performance of the KDE although the QDE decided to remove about 40 “redundant” dimensions per digit class. Table 1: Results on -dimensional digit data for different digit classes ’0’...’9’ (first row); second row: difference between average log-likelihoods of (PCA-)QDE and KDE on test set; third row: optimal subspace dimensionality of QDE Digit: 0 1 2 3 4 5 6 7 8 9 : 1.87 0.66 1.02 1.38 1.58 1.54 1.44 0.64 1.53 1.33 : 22 29 26 24 24 25 24 27 21 25 6 Conclusion The QDE offers a nonparametric approach to unsupervised learning of quantization functions which can be viewed as a generalization of the kernel density estimator. While the KDE is directly constructed from the given data set the QDE first creates a quantized representation of the data. Unlike traditional quantization methods which minimize the associated reconstruction error of the data points, the QDE adjusts the quantizer to optimize an estimate of the data density. This feature allows for a convenient model selection procedure, since the complexity of the quantizer can be controlled by the kernel bandwidth, which in turn can be selected in a data-driven way. For a practical realization we outlined EM-schemes for parameter estimation and bandwidth selection. As an illustration, we discussed examples with different projection sets which correspond to VQ, PCA and ICA methods. We presented experiments which demonstrate that the bias imposed by the quantization can lead to an improved generalization as compared to the “non-quantizing” KDE. This suggests that QDEs offer a promising approach to unsupervised learning that allows to control bias without the usually rather strong distributional assumptions of the Bayesian approach. Acknowledgement This work was funded by the Deutsche Forschungsgemeinschaft within the project SFB 360. References [1] J. M. Buhmann and N. Tishby. Empirical risk approximation: A statistical learning theory of data clustering. In C. M. Bishop, editor, Neural Networks and Machine Learning, pages 57–68. Springer, Berlin Heidelberg New York, 1998. [2] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society Series B, 39:1–38, 1977. [3] Stuart Geman and Chii-Ruey Hwang. Nonparametric maximum likelihood estimation by the method of sieves. The Annals of Statistics, 10(2):401–414, 1982. [4] T. Hastie and W. Stuetzle. Principal curves. Journal of the American Statistical Association, 84:502–516, 1989. [5] M. C. Jones, J. S. Marron, and S. J. Sheather. A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association, 91(433):401– 407, 1996. [6] B. K´egl, A. Krzyzak, T. Linder, and K. Zeger. Learning and design of principal curves. IEEE Transaction on Pattern Analysis and Machine Intelligence, 22(3):281– 297, 2000. [7] Peter Meinicke. Unsupervised Learning in a Generalized Regression Framework. PhD thesis, Universitaet Bielefeld, 2000. http://archiv.ub.unibielefeld.de/disshabi/2000/0033/. [8] Peter Meinicke and Helge Ritter. Resolution-based complexity control for Gaussian mixture models. Neural Computation, 13(2):453–475, 2001. [9] K. Rose, E. Gurewitz, and G. C. Fox. Statistical mechanics and phase transitions in clustering. Physical Review Letters, 65(8):945–948, 1990. [10] Sam Roweis and Zoubin Ghahramani. A unifying review of linear Gaussian models. Neural Computation, 11(2):305–345, 1999. [11] D. W. Scott. Multivariate Density Estimation. Wiley, 1992. [12] B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman and Hall, London and New York, 1986. [13] Alex J. Smola, Robert C. Williamson, Sebastian Mika, and Bernhard Sch¨olkopf. Regularized principal manifolds. In Proc. 4th European Conference on Computational Learning Theory, volume 1572, pages 214–229. Springer-Verlag, 1999. [14] Vladimir N. Vapnik and Sayan Mukherjee. Support vector method for multivariate density estimation. In S. A. Solla, T. K. Leen, and K.-R. M¨uller, editors, Advances in Neural Information Processing Systems, volume 12, pages 659–665. The MIT Press, 2000.
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The Noisy Euclidean Traveling Salesman Problem and Learning Mikio L. Braun, Joachim M. Buhmann braunm@cs.uni-bonn.de, jb@cs.uni-bonn.de Institute for Computer Science, Dept. III, University of Bonn R6merstraBe 164, 53117 Bonn, Germany Abstract We consider noisy Euclidean traveling salesman problems in the plane, which are random combinatorial problems with underlying structure. Gibbs sampling is used to compute average trajectories, which estimate the underlying structure common to all instances. This procedure requires identifying the exact relationship between permutations and tours. In a learning setting, the average trajectory is used as a model to construct solutions to new instances sampled from the same source. Experimental results show that the average trajectory can in fact estimate the underlying structure and that overfitting effects occur if the trajectory adapts too closely to a single instance. 1 Introduction The approach in combinatorial optimization is traditionally single-instance and worst-case-oriented. An algorithm is tested against the worst possible single instance. In reality, algorithms are often applied to a large number of related instances, the average-case performance being the measurement of interest. This constitutes a completely different problem: given a set of similar instances, construct solutions which are good on average. We call this kind of problem multiple-instances and average-case-oriented. Since the instances share some information, it might be expected that this problem is simpler than solving all instances separately, even for NP-hard problems. We will study the following example of a multiple-instance average-case problem, which is built from the Euclidean travelings salesman problem (TSP) in the plane. Consider a salesman who makes weekly trips. At the beginning of each week, the salesman has a new set of appointments for the week, for which he has to plan the shortest round-trip. The location of the appointments will not be completely random, because there are certain areas which have a higher probability of containing an appointment, for example cities or business districts within cities. Instead of solving the planning problem each week from scratch, a clever salesman will try to exploit the underlying density and have a rough trip pre-planned, which he will only adapt from week to week. An idealizing formulization of this setting is as follows. Fix the number of appointments n E N. Let Xl, ... , Xn E ]R2 and (J E 114. Then, the locations of the appointments for each week are given as samples from the normally distributed random vectors (i E {1, ... ,n}) (1) The random vector (Xl, ... ,Xn ) will be called a scenario, sampled appointment locations (sampled) instance. The task consists in finding the permutation 7r E Sn which minimizes 7r I-t d7r(n)1f(l) + L~:ll d1f(i)1f(iH) , where dij := IIXi - Xj112' and Sn being the set of all bijective functions on the set {1, ... , n}. Typical examples are depicted in figure l(a)- (c). It turns out that the multiple-instance average-case setting is related to learning theory, especially to the theory of cost-based unsupervised learning. This relationship becomes clear if one considers the performance measure of interest. The algorithm takes a set of instances It, ... ,In as input and outputs a number of solutions Sl,···, Sn· It is then measured by the average performance (l/n) L~=l C(Sk, h), where C(s , I) denotes the cost of solution s on instance I. We now modify the performance measure as follows. Given a finite number of instances It, ... ,In, the algorithm has to construct a solution s' on a newly sampled instance I'. The performance is then measured by the expected cost E (C (s' ,I')). This can be interpreted as a learning task. The instances 11 , ... ,In are then the training data, E(C(s', I')) is the analogue of the expected risk or cost, and the set of solutions is identified with the hypothesis class in learning theory. In this paper, the setting presented in the previous paragraph is studied with the further restriction that only one training instance is present. From this training instance, an average solution is constructed, represented by a closed curve in the plane. This average trajectory is supposed to capture the essential structure of the underlying probability density, similar to the centroids in K-means clustering. Then, the average trajectory is used as a seed for a simple heuristic which constructs solutions on newly drawn instances. The average trajectories are computed by geometrically averaging tours which are drawn by a Gibbs sampler at finite temperature. This will be discussed in detail in sections 2 and 3. It turns out that the temperature acts as a scale or smoothing parameter. A few comments concerning the selection of this parameter are given in section 6. The technical content of our approach is reminiscent of the "elastic net" -approaches of Durbin and Willshaw (see [2], [5]), but differs in many points. It is based on a completely different algorithmic approach using Gibbs sampling and a general technique for averaging tours. Our algorithm has polynomial complexity per Monte Carlo step and convergence is guaranteed by the usual bounds for Markov Chain Monte Carlo simulation and Gibbs sampling. Furthermore, the goal is not to provide a heuristic for computing the best solution, but to extract the relevant statistics of the Gibbs distribution at finite temperatures to generate the average trajectory, which will be used to compute solutions on future instances. 2 The Metropolis algorithm The Metropolis algorithm is a well-known algorithm which simulates a homogeneous Markov chain whose distribution converges to the Gibbs distribution. We assume that the reader is familiar with the concepts, we give here only a brief sketch of the relevant results and refer to [6], [3] for further details. Let M be a finite set and f: M -+ lit The Gibbs distribution at temperature T E Il4 is given by (m E M) 9T(m) := exp( - f(m)/T~ . Lm/EM exp( - f(m )/T) (2) The Metropolis algorithm works as follows. We start with any element m E M and set Xl +- m. For i ~ 2, apply a random local update m':= ¢(Xi). Then set with probability min {I, exp( -(f(Xi) - f(m'))/T)} else (3) This scheme converges to the Gibbs distribution if certain conditions on ¢ are met. Furthermore, a L2-law of large numbers holds: For h: M --t ]R, ~ L:~=l h(Xk ) --t L:mEM gT(m)h(m) in L2. For TSP, M = Sn and ¢ is the Lin-Kernighan two-change [4], which consists in choosing two indexes i, j at random and reversing the path between the appointments i and j. Note that the Lin-Kernighan two-change and its generalizations for neighborhood search are powerful heuristic in itself. 3 Averaging Tours Our goal is to compute the average trajectory, which should grasp the underlying structure common to all instances, with respect to the Gibbs measure at non-zero temperature T . The Metropolis algorithm produces a sequence of permutations 7rl, 7r2, ... with P{ 7rn = .} --t gT(.) for n --t 00. Since permutations cannot be added, we cannot simply compute the empirical means of 7rn. Instead, we map permutations to their corresponding trajectories. Definition 1 (trajectory) The trajectory of 7r E Sn given n points Xl, ... ,Xn is a mapping r( 7r): {I, ... , n} --t ]R2 defined by r( 7r) (i) := X1C(i). The set of all trajectories (for all sets of n points) is denoted by Tn (this is the set of all mappings T {I, ... , n} --t ]R2 ). Addition of trajectories and multiplication with scalars can be defined pointwise. Then it is technically possible to compute t L:~=l r(7rk). Unfortunately, this does not yield the desired results, since the relation between permutations and tours is not one-to-one. For example, the permutation obtained by starting the tour at a different city still corresponds to the same tour. We therefore need to define the addition of trajectories in a way which is independent of the choice of permutation (and therefore trajectory) to represent the tour. We will study the relationship between tours and permutations first in some detail, since we feel that the concepts introduced here might be generally useful for analyzing combinatorial optimization problems. Definition 2 (tour and length of a tour) Let G = (V, E) be a complete (undirected) graph with V = {I, ... ,n} and E = (~). A subset tEE is called a tour iff It I = n, for every v E V, there exist exactly two el, e2 E t such that v E el and v E e2, and (V, t) is connected. Given a symmetric matrix (dij ) of distances, the length of a tour t is defined by C(t) := L:{i,j} Et dij . The tour corresponding to a permutation 7r E Sn is given by n-l t(7r) :={ {7r(I), 7r(n)}} U U {{7r(i),7r(i + I)}}. (4) i=l If t(7r) = t for a permutation 7r and a tour t, we say that 7r represents t. We call two permutations 7r, 7r' equivalent, if they represent the same tour and write 7r ,...., 7r'. Let [7r] denote the equivalence class of 7r as usual. Note that the length of a permutation is fully determined by its equivalence class. Therefore, ,...., describes the intrinsic symmetries of the TSP formulated as an optimization problem on Sn, denoted by TSP(Sn). We have to define the addition EB of trajectories such that the sum is independent of the representation. This means that for two tours h, t2 such that h is represented by 'lf1, 'If~ and t2 by 'lf2, 'If~ it holds that f('lf1) EB f('lf2) ~ f('lfD EB f('If~). The idea will be to normalize both summands before addition. We will first study the exact representation symmetry of TSP(Sn) ' The TSP(Sn) symmetry group Algebraically speaking, Sn is a group with concatenation of functions as multiplication, so we can characterize the equivalence classes of ~ by studying the set of operations on a permutation which map to the same equivalent class. We define a group action of Sn on itself by right translation ('If, 9 E Sn): " . " : Sn x Sn -+ Sn, g. 'If:= 'lfg- 1. (5) Note that any permutation in Sn can be mapped to another by an appropriate group action (namely 'If -+ 'If' by ('If,-l'lf) . 'If.), such that the group action of Sn on itself suffices to study the equivalence classes of ~. For certain 9 E Sn, it holds that t(g· 'If) = t('If). We want to determine the maximal set H t of elements which keeps t invariant. It even holds that H t is a subgroup of Sn: The identity is trivially in H t . Let g, h be t-invariant, then t((gh-1) . 'If) = t(g ·(h- 1 . 'If)) = t(h- 1 . 'If) = t(h ·(h- 1 . 'If) = t( 'If). H t will be called the symmetry group of TSP(Sn) and it follows that ['If] = H t · 'If :={h · 'If I hE Hd. The shift u and reversal (2 are defined by (i E {I, ... , n} ) (.) . __ {i + 1 i < n, u z. 1 . , z = n (2(i) :=n + 1- i, (6) and set H :=((2, u), the group generated by u and (2. It holds that (this result is an easy consequence of (2(2 = id{l, ... ,n}, (2U = u-1(2 and un = id{l, ... ,n}) H = {uk IkE {I, ... , n}} U {(2uk IkE {l, ... ,n}}. (7) The fundamental result is Theorem 1 Let t be the mapping which sends permutations to tours as defined in (4). Then, H t = H , where H t is the set of all t-invariant permutations and H is defined in (7). Proof: It is obvious that H ~ H t . Now, let h- 1 E H t . We are going to prove that t-invariant permutations are completely defined by their values on 1 and 2. Let hE H t and k:= h(l) . Then, h(2) = u(k) or h(2) = u - 1(k), because otherwise, h would give rise to a link {{'If(h(1),'If(h(2»}} 1. t('If) . For the same reason, h(3) must be mapped to u±2(k). Since h must be bijective, h(3) =I- h(l) , so that the sign of the exponent must be the same as for h(2). In general, h(i) = u±(i- 1l(k). Now note that for i,k E {l , ... ,n}, ui(k) = uk(i) and therefore, { u k- 1 if h(i) = ui-1 (k) h= (2un-k ifh(i)=u-i+1(k)' D Adding trajectories We can now define equivalence for trajectories. First define a group action of Sn on Tn analogously to (5): the action of h E Ht on "( E Tn is given by h · "( := "( 0 h- 1 . Furthermore, we say that "( ~ 1}, if Ht · "( = Ht ·1}. Our approach is motivated geometrically. We measure distances between trajectories as follows. Let d: ]R2 x ]R2 -+ Il4 be a metric. Then define h, 1} E Tn) dh,1}):= 2::=1 dh(k),1}(k). (8) Before adding two trajectories we will first choose equivalent representations "(', 1}' which minimize d( "(' , 1}'). Because of the results presented so far, searching through all equivalent trajectories is computationally tractable. Note that for h E H t , it holds that d( h . ,,(, h . rJ) = db, rJ) as h only reorders the summands. It follows that it suffices to change the representations only for one argument, since d(h· ,,(, i· rJ) = db, h- 1i· rJ)· So the time complexity of one addition reduces to 2n computation of distances which involve n subtractions each. The normalizing action is defined by b, rJ E Tn) n, 1J := argmin d( ,,(, n . rJ)· n EH t Assuming that the normalizing action is unique1 , we can prove (9) Theorem 2 Let ,,(, rJ be two trajectories, and n, 1J the unique normalizing action as defined in (9). Then, the operation "( EB rJ := "( + n, 1J . rJ (10) is representation invariant. Proof: Let "(I = g. ,,(, rJl = h· rJ for g, h E Ht . We claim that n,I1J1 = gn' 1Jh-1. The normalizing action is defined by n,I1J1 = argmin db/, nl . rJl) = argmin d(g . ,,(, nih· rJ) = argmin db , g-l nlh· rJ), n l EHt n l EH t n l EH t (11) by inserting g-l parallelly before both arguments in the last step. Since the normalizing action is unique, it follows that for the n l realizing the minimum it holds that g-ln l h = n, 1J and therefore n l = n, I1J1 = gn' 1Jh-1. Now, consider the sum which proves the representation independence. 0 The sum of more than two trajectories can be defined by normalizing everything with respect to the first summand, so that empirical sums t EB~=l f(?ri) are now well-defined. 4 Inferring Solutions on New Instances We transfer a trajectory to a new set of appointments Xl, .. . ,Xn by computing the relaxed tour using the following finite-horizon adaption technique: First of all, passing times ti for all appointments are computed. We extend the domain of a trajectory "( from {I, ... , n} to the interval [1, n + 1) by linear interpolation. Then we define ti such that "((ti) is the earliest point with minimal distance between appointment Xi and the trajectory. The passing times can be calculated easily by simple geometric considerations. The permutation which sorts (ti)~l is the relaxed solution of"( to (Xi) . In a post-processing step, self-intersections are removed first. Then, segments of length w are optimized by exhaustive search. Let ?r be the relaxed solution. The path from ?rei) to ?r(i + w + 2) (index addition is modulo n) is replaced by the best alternative through the appointments ?r(i + 1), ... , ?r(i + w + 1). Iterate for all i E {I, . . . , n} until there is no further improvement. Since this procedure has time complexity w!n, it can only be done efficiently for small w. lOtherwise, perturb the locations of the appointments by infinitesimal changes. 5 Experiments For experiments, we used the following set-up: We took the 11.111-norm to determine the normalizing action. Typical sample-sizes for the Markov chain Monte Carlo integration were 1000 with 100 steps in between to decouple consecutive samples. Scenarios were modeled after eq. (1), where the Xi were chosen to form simple geometric shapes. Average trajectories for different temperatures are plotted in figures l(a)- (c). As the temperature decreases, the average trajectory converges to the trajectory of a single locally optimal tour. The graphs demonstrate that the temperature T acts as a smoothing parameter. To estimate the expected risk of an average trajectory, the post-processed relaxed (PPR) solutions were averaged over 100 new instances (see figure l(d)-(g)) in order to estimate the expected costs. The costs of the best solutions are good approximations, within 5% of the average minimum as determined by careful simulated annealing. An interesting effect occurs: the expected costs have their minimum at non-zero temperature. The corresponding trajectories are plotted in figure l(e),(f). They recover the structure of the scenario. In other words, average trajectories computed at temperatures which are too low, start to overfit to noise present only in the instance for which they were computed. So computation of the global optimum of a noisy combinatorial optimization problem might not be the right strategy, because the solutions might not reflect the underlying structure. Averaging over many suboptimal solutions provides much better statistics. 6 Selection of the Temperature The question remains how to select the optimal temperature. This problem is essentially the same as determining the correct model complexity in learning theory, and therefore no fully satisfying answer is readily available. The problem is nevertheless suited for the application of the heuristic provided by the empirical risk approximation (ERA) framework [1], which will be briefly sketched here. The main idea of ERA is to coarse-grain the set of hypotheses M by treating hypotheses as equivalent which are only slightly different. Hypotheses whose £1 mutual distance (defined in a similar fashion as (8)) is smaller than the parameter "( E Il4 are considered statistically equivalent. Selecting a subset of solutions such that £l-spheres of radius "( cover M results in the coarse-grained hypothesis set M,. VC-type large deviation bounds depending on the size of the coarse-grained hypothesis class can now be derived: ( n(c - "()2 ) p{ C2 (m"! ) min C2 (m) > 2c} :::; 21M"!1 sup exp ( )' mEM mEM., am + c c "( (13) am depending on the distribution. The bound weighs two competing effects. On the one hand, increasing "( introduces a systematic bias in the estimation. On the other hand, decreasing "( increases the cardinality of the hypothesis class. Given a confidence J > 0, the probability of being worse than c > 0 on a second instance and "( are linked. So an optimal coarsening "( can be determined. ERA then advocates to either sample from the ,,(-sphere around the empirical minimizer or average over these solutions. Now it is well known, that the Gibbs sampler is concentrated on solutions whose costs are below a certain threshold. Therefore, the ERA is suited for our approach. In the relating equation the log cardinality of the approximation set occurs, which is usually interpreted as micro canonical entropy. This relates back to statistical physics, the starting point of our whole approach. Now interpreting "( as energy, we can compute the stop temperature from the optimal T Using the well-known relation from statistical physics ~ee:t:~:: = T - 1 , we can derive a lower bound on the optimal temperature depending on variance estimates of the specific scenario given. 7 Conclusion In reality, optimization algorithms are often applied to many similar instances. We pointed out that this can be interpreted as a learning problem. The underlying structure of similar instances should be extracted and used in order reduce the computational complexity for computing solutions to related instances. Starting with the noisy Euclidean TSP, the construction of average tours is studied in this paper, which involves determining the exact relationship between permutation and tours, and identifying the intrinsic symmetries of the TSP. We hope that this technique might prove to be useful for other applications in the field of averaging over solutions of combinatorial problems. The average trajectories are able to capture the underlying structure common to all instances. A heuristic for constructing solutions on new instances is proposed. An empirical study of these procedures is conducted with results satisfying our expectations. In terms of learning theory, overfitting effects can be observed. This phenomenon points at a deep connection between combinatorial optimization problems with noise and learning theory, which might be bidirectional. On the one hand, we believe that noisy (in contrast to random) combinatorial optimization problems are dominant in reality. Robust algorithms could be built by first estimating the undistorted structure and then using this structure as a guideline for constructing solutions for single instances. On the other hand, hardness of efficient optimization might be linked to the inability to extract meaningful structure. These connections, which are subject of further studies, link statistical complexity to computational complexity. Acknowledgments The authors would like to thank Naftali Tishby, Scott Kirkpatrick and Michael Clausen for their helpful comments and discussions. References [1] J. M. Buhmann and M. Held. Model selection in clustering by uniform convergence bounds. Advances in Neural Information Processing Systems, 12:216- 222, 1999. [2] R. Durbin and D. Willshaw. An analogue approach to the travelling salesman problem using an elastic net method. Nature, 326:689- 691, 1987. [3] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchio Optimisation by simulated annealing. Science, 220:671- 680, 1983. [4] S. Lin and B. Kernighan. An effective heuristic algorithm for the traveling salesman problem. Operations Research, 21:498- 516, 1973. [5] P.D. Simic. Statistical mechanics as the underlying theory of "elastic" and "neural" optimizations. Network, 1:89-103, 1990. [6] G. Winkler. Image Analysis, Random fields and Dynamic Monte Carlo Methods, volume 27 of Application of Mathematics. Springer, Heidelberg, 1995. i 17.7 &: 17.6 " j 17.5 f 17.4 <"Il 17.3 ~ 11.5 ~ &: 11 .45 " j 11.4 ~ ~ 11.35 -sigma2 = O.03 temperatureT (d) -si ma = O.025 temperatureT (f) T.,...,:0.15OO:Xl Lenglt.: 5.179571 o 0 o o T.,...,: 0.212759 Lenglt.:6.295844 o o e o o o.I H CD o n 5O"",11I>1S20_025 1 510 0_7654.2.()_742680 _231390 .057211.(l.Q1597 0 . 2 1 479 0.83224 0 .5833a1~ g Figure 1: (a) Average trajectories at different temperatures for n = 100 appointments on a circle with a 2 = 0.03. (b) Average trajectories at different temperatures, for multiple Gaussian sources, n = 50 and a2 = 0.025. (c) The same for an instance with structure on two levels. (d) Average tour length of the post-processed relaxed (PPR) solutions for the circle instance plotted in (a). The PPR width was w = 5. The average fits to noise in the data if the temperature is too low, leading to overfitting phenomena. Note that the average best solution is :s: 16.5. (e) The average trajectory with the smallest average length of its PPR solutions in (d). (f) Average tour length as in (d). The average best solution is :s: 10.80. (g) Lowest temperature trajectory with small average PPR solution length in (f).
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The Concave-Convex Procedure (CCCP) A. L. Yuille and Anand Rangarajan * Smith-Kettlewell Eye Research Institute, 2318 Fillmore Street, San Francisco, CA 94115, USA. Tel. (415) 345-2144. Fax. (415) 345-8455. Email yuille@ski.org * Prof. Anand Rangarajan. Dept. of CISE, Univ. of Florida Room 301, CSE Building Gainesville, FL 32611-6120 Phone: (352) 392 1507 Fax: (352) 392 1220 e-mail: anand@cise.ufl.edu Abstract We introduce the Concave-Convex procedure (CCCP) which constructs discrete time iterative dynamical systems which are guaranteed to monotonically decrease global optimization/energy functions. It can be applied to (almost) any optimization problem and many existing algorithms can be interpreted in terms of CCCP. In particular, we prove relationships to some applications of Legendre transform techniques. We then illustrate CCCP by applications to Potts models, linear assignment, EM algorithms, and Generalized Iterative Scaling (GIS). CCCP can be used both as a new way to understand existing optimization algorithms and as a procedure for generating new algorithms. 1 Introduction There is a lot of interest in designing discrete time dynamical systems for inference and learning (see, for example, [10], [3], [7], [13]). This paper describes a simple geometrical Concave-Convex procedure (CCCP) for constructing discrete time dynamical systems which can be guaranteed to decrease almost any global optimization/energy function (see technical conditions in section (2)). We prove that there is a relationship between CCCP and optimization techniques based on introducing auxiliary variables using Legendre transforms. We distinguish between Legendre min-max and Legendre minimization. In the former, see [6], the introduction of auxiliary variables converts the problem to a min-max problem where the goal is to find a saddle point. By contrast, in Legendre minimization, see [8], the problem remains a minimization one (and so it becomes easier to analyze convergence). CCCP relates to Legendre minimization only and gives a geometrical perspective which complements the algebraic manipulations presented in [8]. CCCP can be used both as a new way to understand existing optimization algorithms and as a procedure for generating new algorithms. We illustrate this by giving examples from Potts models, EM, linear assignment, and Generalized Iterative Scaling. Recently, CCCP has also been used to construct algorithms to minimize the Bethe/Kikuchi free energy [13]. We introduce CCCP in section (2) and relate it to Legendre transforms in section (3). Then we give examples in section (4). 2 The Concave-Convex Procedure (CCCP) The key results of CCCP are summarized by Theorems 1,2, and 3. Theorem 1 shows that any function, subject to weak conditions, can be expressed as the sum of a convex and concave part (this decomposition is not unique). This implies that CCCP can be applied to (almost) any optimization problem. Theorem 1. Let E(x) be an energy function with bounded Hessian [J2 E(x)/8x8x. Then we can always decompose it into the sum of a convex function and a concave function. Proof. Select any convex function F(x) with positive definite Hessian with eigenvalues bounded below by f > o. Then there exists a positive constant A such that the Hessian of E(x) + AF(x) is positive definite and hence E(x) + AF(x) is convex. Hence we can express E(x) as the sum of a convex part, E(x) + AF(x) , and a concave part -AF(x). Figure 1: Decomposing a function into convex and concave parts. The original function (Left Panel) can be expressed as the sum of a convex function (Centre Panel) and a concave function (Right Panel). (Figure courtesy of James M. Coughlan). Our main result is given by Theorem 2 which defines the CCCP procedure and proves that it converges to a minimum or saddle point of the energy. Theorem 2. Consider an energy function E(x) (bounded below) of form E(x) = Evex (x) + E cave (x) where Evex (x), E cave (x) are convex and concave functions of x respectively. Then the discrete iterative CCCP algorithm ;zt f-7 ;zt+1 given by: -t+l _ -t \1Evex (x ) - -\1Ecave (x ), (1) is guaranteed to monotonically decrease the energy E(x) as a function of time and hence to converge to a minimum or saddle point of E(x). Proof. The convexity and concavity of Evex (.) and Ecave (.) means that Evex (X2) 2: Evex (xd + (X2 -xd· ~ Evex (xd and Ecave (X4) :S Ecave (X3) + (X4 -X3)· ~ Ecave (X3 ), for all X1 ,X2,X3,X4. Now set Xl = xt+l,X2 = xt,X3 = xt,X4 = xt+1. Using the algorithm definition (i.e. ~Ev ex (xt+1) = -~Ecave (xt)) we find that Evex (xt+1) + Ecave (xt+1) :S Evex (xt) + Ecave (xt), which proves the claim. We can get a graphical illustration of this algorithm by the reformulation shown in figure (2) (suggested by James M. Coughlan). Think of decomposing the energy function E(x) into E1(x) - E2(x) where both E1(x) and E2(x) are convex. (This is equivalent to decomposing E(x) into a a convex term E1(x) plus a concave term -E2(X)) . The algorithm proceeds by matching points on the two terms which have the same tangents. For an input Xo we calculate the gradient ~ E2 (xo) and find the point Xl such that ~ E1 (xd = ~ E2 (xo). We next determine the point X2 such that ~E1(X2) = ~E2 (X1)' and repeat. 7~------~--------~------, 60 50 40 30 20 o 10 O L---~=-~O-=~~~~----~ 10 XO Figure 2: A CCCP algorithm illustrated for Convex minus Convex. We want to minimize the function in the Left Panel. We decompose it (Right Panel) into a convex part (top curve) minus a convex term (bottom curve). The algorithm iterates by matching points on the two curves which have the same tangent vectors, see text for more details. The algorithm rapidly converges to the solution at x = 5.0. We can extend Theorem 2 to allow for linear constraints on the variables X, for example Li et Xi = aM where the {en, {aM} are constants. This follows directly because properties such as convexity and concavity are preserved when linear constraints are imposed. We can change to new coordinates defined on the hyperplane defined by the linear constraints. Then we apply Theorem 1 in this coordinate system. Observe that Theorem 2 defines the update as an implicit function of xt+1. In many cases, as we will show, it is possible to solve for xt+1 directly. In other cases we may need an algorithm, or inner loop, to determine xt+1 from ~Evex (xt+1). In these cases we will need the following theorem where we re-express CCCP in terms of minimizing a time sequence of convex update energy functions Et+1 (xt+1) to obtain the updates xt+1 (i.e. at the tth iteration of CCCP we need to minimize the energy Et+1 (xt+1 )). We include linear constraints in Theorem 3. Theorem 3. Let E(x) = Evex (x) + Ecave(x) where X is required to satisfy the linear constraints Li et Xi = aM, where the {et}, { aM} are constants. Then the update rule for xt+1 can be formulated as minimizing a time sequence of convex update energy functions Et+1 (;rt+1): where the lagrange parameters P'J1} impose linear comnstraints. Proof. Direct calculation. (2) The convexity of EH1 (;rt+1) implies that there is a unique minimum corresponding to ;rt+1. This means that if an inner loop is needed to calculate ;rt+1 then we can use standard techniques such as conjugate gradient descent (or even CCCP). 3 Legendre Transformations The Legendre transform can be used to reformulate optimization problems by introducing auxiliary variables [6]. The idea is that some of the formulations may be more effective (and computationally cheaper) than others. We will concentrate on Legendre minimization, see [7] and [8], instead of Legendre min-max emphasized in [6]. An advantage of Legendre minimization is that mathematical convergence proofs can be given. (For example, [8] proved convergence results for the algorithm implemented in [7].) In Theorem 4 we show that Legendre minimization algorithms are equivalent to CCCP. The CCCP viewpoint emphasizes the geometry of the approach and complements the algebraic manipulations given in [8]. (Moreover, our results of the previous section show the generality of CCCP while, by contrast, the Legendre transform methods have been applied only on a case by case basis). Definition 1. Let F(x) be a convex function. For each value y let F*(ff) = minx{F(x) +y·x.}. Then F*(Y) is concave and is the Legendre transform of F(x). Moreover, F (x) = maxy{ F* (y) - y. x} . Property 1. F(.) and F*(.) are related by a:; (fJ) = {~~} - 1(_Y), -~~(x) = {a{y* } -1 (x). (By { a{y* } -1 (x) we mean the value y such that a{y* (y) = x.) Theorem 4. Let E1 (x) = f(x) + g(x) and E2(x, Y) = f(x) + x· Y + h(i/), where f(.), h(.) are convex functions and g(.) is concave. Then applying CCCP to E1 (x) is equivalent to minimizing E2 (x, Y) with respect to x and y alternatively (for suitable choices of g(.) and h(.). Proof. We can write E1(X) = f(x) +miny{g*(Y) +x·y} where g*(.) is the Legendre transform of g(.) (identify g(.) with F*( .) and g*(.) with F(.) in definition 1). Thus minimizing E1 (x) with respect to x is equivalent to minimizing E1 (x, Y) = f(x) + x . y + g* (Y) with respect to x and y. (Alternatively, we can set g* (Y) = h(Y) in the expression for E2(x,i/) and obtain a cost function E2(x) = f(x) + g(x).) Alternatively minimization over x and y gives: (i) of/ax = y to determine Xt+1 in terms of Yt, and (ii) ag* / ay = x to determine Yt in terms of Xt which, by Property 1 of the Legendre transform is equivalent to setting y = -ag / ax. Combining these two stages gives CCCP: af (_) ag (_) ax Xt+1 = - ax Xt . 4 Examples of CCCP We now illustrate CCCP by giving four examples: (i) discrete time dynamical systems for the mean field Potts model, (ii) an EM algorithm for the elastic net, (iii) a discrete (Sinkhorn) algorithm for solving the linear assignment problem, and (iv) the Generalized Iterative Scaling (GIS) algorithm for parameter estimation. Example 1. Discrete Time Dynamical Systems for the Mean Field Potts Model. These attempt to minimize discrete energy functions of form E[V] = 2:i,j,a,b Tijab Via V)b + 2:ia Bia Via, where the {Via} take discrete values {a, I} with linear constraints 2:i Via = 1, Va. Discussion. Mean field algorithms minimize a continuous effective energy Eett [S; T] to obtain a minimum of the discrete energy E[V] in the limit as T f-7 a. The {Sial are continuous variables in the range [0,1] and correspond to (approximate) estimates of the mean states of the {Via}. As described in [12}, to ensure that the minima of E[V] and E ett [S; T] all coincide (as T f-7 0) it is sufficient that Tijab be negative definite. Moreover, this can be attained by adding a term -K 2:ia Vi! to E[V] (for sufficiently large K) without altering the structure of the minima of E[V] . Hence, without loss of generality we can consider 2:i,j,a,b Tijab Via V)b to be a concave function. We impose the linear constraints by adding a Lagrange multiplier term 2:a Pa {2:i Via - I} to the energy where the {Pa} are the Lagrange multipliers. The effective energy becomes: i,j,a ,b ia ia a We can then incorporate the Lagrange multiplier term into the convex part. This gives: Evex [S] = T2:ia SialogSia + 2:aPa{2:iSia -I} and Ecave[S] = 2:i jab TijabSiaSjb + 2:ia BiaSia · Taking derivatives yields: &g Evex [S] = TI~~Sia + Pa and &t E cave [S] = 2 2:j,b TijabSjb + Bia· Applying eeep by setting &:s::~ (StH) = - &:5;:e (st) gives T{l + log Sia (t + I)} + Pa = -2 2:j,b TijabSjb(t)Bia· We solve for the Lagrange multipliers {Pal by imposing the constraints 2:i Sia(t + 1) = 1, Va. This gives a discrete update rule: (-1/T){2 2:. b TijabSjb(t)+Oia} e J, Sia (t + 1) = 2: ' . 2:c e( -1/T){2 j,b TijcbSjb(tl+Oi c} (4) Algorithms of this type were derived in [lO}, [3} using different design principles. Our second example relates to the ubiquitous EM algorithm. In general EM and CCCP give different algorithms but in some cases they are identical. The EM algorithm seeks to estimate a variable f* = argmaxt log 2:{I} P(f, l), where {f}, {l} are variables that depend on the specific problem formulation. It was shown in [4] that this is equivalent to minimizing the following effective energy with respect to the variables f and P(l): Eett [!, P(l)] = - ~ 2:1 P(l) log P(f, l) + ~ 2:{I} P(l) log P(l). To apply CCCP to an effective energy like this we need either: (a) to decompose Eett [!, P(l)] into convex and concave functions of f, P(l), or (b) to eliminate either variable and obtain a convex concave decomposition in the remaining variable (d. Theorem 4). We illustrate (b) for the elastic net [2]. (See Yuille and Rangarajan, in preparation, for an illustration of (a)). Example 2. The elastic net attempts to solve the Travelling Salesman Problem (TSP) by finding the shortest tour through a set of cities at positions {Xi}' The elastic net is represented by a set of nodes at positions {Ya} with variables {Sial that determine the correspondence between the cities and the nodes of the net. Let Eel I [S, 171 be the effective energy for the elastic net, then the {y} variables can be eliminated and the resulting Es[S] can be minimized using GGGP. (Note that the standard elastic net only enforces the second set of linear constraints). Discussion. The elastic net energy function can be expressed as [11]: ia a,b i,a where we impose the conditions L:a Sia = 1, V i and L:i Sia = 1, V a. The EM algorithm can be applied to estimate the {Ya}. Alternatively we can solve for the {Ya} variables to obtain Yb = L:i a PabSiaXi where {Pab } = {Jab + 2')'Aab} -1. We substitute this back into Eell [S, 171 to get a new energy Es[S] given by: (6) i ,j,a,b i,a Once again this is a sum of a concave and a convex part (the first term is concave because of the minus sign and the fact that {Pba } and Xi . Xj are both positive semidefinite.) We can now apply GGGP and obtain the standard EM algorithm for this problem. (See Yuille and Rangarajan, in preparation, for more details). Our final example is a discrete iterative algorithm to solve the linear assignment problem. This algorithm was reported by Kosowsky and Yuille in [5] where it was also shown to correspond to the well-known Sinkhorn algorithm [9]. We now show that both Kosowsky and Yuille's linear assignment algorithm, and hence Sinkhorn's algorithm are examples of CCCP (after a change of variables). Example 3. The linear assignment problem seeks to find the permutation matrix {TIia} which minimizes the energy E[m = L:ia TIia Aia, where {Aia} is a set of assignment values. As shown in [5} this is equivalent to minimizing the (convex) Ep[P] energy given by Ep[P] = L:aPa + ~ L:i log L:a e-,B(Aia+Pa) , where the solution is given by TI;a = e-,B(Aia+Pa) / L:b e-,B(Aib+Pb) rounded off to the nearest integer (for sufficiently large fJ). The iterative algorithm to minimize Ep[P] (which can be re-expressed as Sinkhorn's algorithm, see [5}) is of form: (7) and can be re-expressed as GGGP. Discussion. By performing the change of coordinates fJPa = - log r a V a (for r a > 0, Va) we can re-express the Ep[P] energy as: (8) Observe that the first term of Er[r] is convex and the second term is concave (this can be verified by calculating the Hessian). Applying CCCP gives the update rule: 1 e-,BAia rt+l = 2:= 2::: e-,BAibrt' a i b b (9) which corresponds to equation (7). Example 4. The Generalized Iterative Scaling (GIS) Algorithm [ll for estimating parameters in parallel. Discussion. The GIS algorithm is designed to estimate the parameter X of a distribution P(x : X) = eX.¢(x) IZ[X] so that 2:::x P(x; X)¢(x) = h, where h are observation data (with components indexed by j.t). It is assumed that ¢fJ,(x) ::::: 0, V j.t,x, hfJ, ::::: 0, V j.t, and 2:::fJ, ¢fJ, (x) = 1, V x and 2:::fJ, hfJ, = 1. (All estimation problems of this type can be transformed into this form [lj). Darroch and Ratcliff [ll prove that the following GIS algorithm is guaranteed to converge to value X* that minimizes the (convex) cost function E(X) = log Z[X]-X.h and hence satisfies 2:::x P(x; X*)¢(x) = h. The GIS algorithms is given by: Xt+! = Xt - log ht + log h, (10) where ht = 2:::x P(x; Xt)¢(x) {evaluate log h componentwise: (log h)fJ, = log hf),') To show that GIS can be reformulated as CCCP, we introduce a new variable iJ = eX (componentwise). We reformulate the problem in terms of minimizing the cost function E,B [iJ] = log Z[log(iJ)] - h . (log iJ). A straightforward calculation shows that -h . (log iJ) is a convex function of iJ with first derivative being -hi iJ (where the division is componentwise). The first derivative of log Z[log(iJ)] is (II iJ) 2:::x ¢(x)P(x: log ,8) (evaluated componentwise). To show that log Z[log(iJ)] is concave requires computing its Hessian and applying the Cauchy-Schwarz inequality using the fact that 2:::fJ, ¢fJ,(x) = 1, V x and that ¢fJ,(x) ::::: 0, V j.t,x. We can therefore apply CCCP to E,B [iJ] which yields l/iJH1 = l/iJt x Ilh x ht (componentwise), which is GIS (by taking logs and using log ,8 = X). 5 Conclusion CCCP is a general principle which can be used to construct discrete time iterative dynamical systems for almost any energy minimization problem. It gives a geometric perspective on Legendre minimization (though not on Legendre min-max). We have illustrated that several existing discrete time iterative algorithms can be reinterpreted in terms of CCCP (see Yuille and Rangarajan, in preparation, for other examples). Therefore CCCP gives a novel way ofthinking about and classifying existing algorithms. Moreover, CCCP can also be used to construct novel algorithms. See, for example, recent work [13] where CCCP was used to construct a double loop algorithm to minimize the Bethe/Kikuchi free energy (which are generalizations of the mean field free energy). There are interesting connections between our results and those known to mathematicians. After this work was completed we found that a result similar to Theorem 2 had appeared in an unpublished technical report by D. Geman. There also are similarities to the work of Hoang Tuy who has shown that any arbitrary closed set is the projection of a difference of two convex sets in a space with one more dimension. (See http://www.mai.liu.se/Opt/MPS/News/tuy.html). Acknowledgements We thank James Coughlan and Yair Weiss for helpful conversations. Max Welling gave useful feedback on this manuscript. We thank the National Institute of Health (NEI) for grant number R01-EY 12691-01. References [1] J.N. Darroch and D. Ratcliff. "Generalized Iterative Scaling for Log-Linear Models". The Annals of Mathematical Statistics. Vol. 43. No.5, pp 1470-1480. 1972. [2] R. Durbin, R. Szeliski and A.L. Yuille." An Analysis of an Elastic net Approach to the Traveling Salesman Problem". Neural Computation. 1, pp 348-358. 1989. [3] LM. Elfadel "Convex potentials and their conjugates in analog mean-field optimization". Neural Computation. Volume 7. Number 5. pp. 1079-1104. 1995. [4] R. Hathaway. "Another Interpretation of the EM Algorithm for Mixture Distributions". Statistics and Probability Letters. Vol. 4, pp 53-56. 1986. [5] J. Kosowsky and A.L. Yuille. "The Invisible Hand Algorithm: Solving the Assignment Problem with Statistical Physics". Neural Networks. , Vol. 7, No.3, pp 477-490. 1994. [6] E. Mjolsness and C. Garrett. "Algebraic Transformations of Objective Functions". Neural Networks. Vol. 3, pp 651-669. [7] A. Rangarajan, S. Gold, and E. Mjolsness. "A Novel Optimizing Network Architecture with Applications" . Neural Computation, 8(5), pp 1041-1060. 1996. [8] A. Rangarajan, A.L. Yuille, S. Gold. and E. Mjolsness." A Convergence Proof for the Softassign Quadratic assignment Problem". In Proceedings of NIPS'96. Denver. Colorado. 1996. [9] R. Sinkhorn. "A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices". Ann. Math. Statist .. 35, pp 876-879. 1964. [10] F.R. Waugh and R.M. Westervelt. "Analog neural networks with local competition: L Dynamics and stability". Physical Review E, 47(6), pp 4524-4536. 1993. [11] A.L. Yuille. "Generalized Deformable Models, Statistical Physics and Matching Problems," Neural Computation, 2 pp 1-24. 1990. [12] A.L. Yuille and J.J. Kosowsky. "Statistical Physics Algorithms that Converge." Neural Computation. 6, pp 341-356. 1994. [13] A.L. Yuille. "A Double-Loop Algorithm to Minimize the Bethe and Kikuchi Free Energies" . Neural Computation. In press. 2002.
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Active Learning in the Drug Discovery Process Manfred K. Warmuth , Gunnar R¨atsch , Michael Mathieson , Jun Liao , Christian Lemmen Computer Science Dep., Univ. of Calif. at Santa Cruz FHG FIRST, Kekul´estr. 7, Berlin, Germany DuPont Pharmaceuticals,150 California St. San Francisco. manfred,mathiesm,liaojun @cse.ucsc.edu, Gunnar.Raetsch@anu.edu.au, clemmen@biosolveit.de Abstract We investigate the following data mining problem from Computational Chemistry: From a large data set of compounds, find those that bind to a target molecule in as few iterations of biological testing as possible. In each iteration a comparatively small batch of compounds is screened for binding to the target. We apply active learning techniques for selecting the successive batches. One selection strategy picks unlabeled examples closest to the maximum margin hyperplane. Another produces many weight vectors by running perceptrons over multiple permutations of the data. Each weight vector votes with its prediction and we pick the unlabeled examples for which the prediction is most evenly split between and . For a third selection strategy note that each unlabeled example bisects the version space of consistent weight vectors. We estimate the volume on both sides of the split by bouncing a billiard through the version space and select unlabeled examples that cause the most even split of the version space. We demonstrate that on two data sets provided by DuPont Pharmaceuticals that all three selection strategies perform comparably well and are much better than selecting random batches for testing. 1 Introduction Two of the most important goals in Computational Drug Design are to find active compounds in large databases quickly and (usually along the way) to obtain an interpretable model for what makes a specific subset of compounds active. Activity is typically defined All but last author received partial support from NSF grant CCR 9821087 Current address: Austrialian National University, Canberra, Austrialia. Partially supported by DFG (JA 379/9-1, MU 987/1-1) and travel grants from EU (Neurocolt II). Current address: BioSolveIT GmbH, An der Ziegelei 75, Sankt Augustin, Germany as binding to a target molecule. Most of the time an iterative approach to the problem is employed. That is in each iteration a batch of unlabeled compounds is screened against the target using some sort of biological assay[MGST97]. The desired goal is that many active hits show up in the assays of the selected batches. From the Machine Learning point of view all examples (compounds) are initially unlabeled. In each iteration the learner selects a batch of un-labeled examples for being labeled as positive (active) or negative (inactive). In Machine Learning this type of problem has been called “query learning” [Ang88] “selective sampling” [CAL90] or “active learning” [TK00]. A Round0 data set contains 1,316 chemically diverse examples, only 39 of which are positive. A second Round1 data set has 634 examples with 150 positives. 1This data set is preselected on the basis of medicinal chemistry intuition. Note that our classification problem is fundamentally asymmetric in that the data sets have typically many more negative examples and the Chemists are more interested in the positive hits because these compounds might lead to new drugs. What makes this problem challenging is that each compound is described by a vector of 139,351 binary shape features. The vectors are sparse (on the average 1378 features are set per Round0 compound and 7613 per Round1 compound). We are working with retrospective data sets for which we know all the labels. However, we simulate the real-life situation by initially hiding all labels and only giving to the algorithm the labels for the requested batches of examples (virtual screening). The long-term goal of this type of research is to provide a computer program to the Chemists which will do the following interactive job: At any point new unlabeled examples may be added. Whenever a test is completed, the labels are given to the program. Whenever a new test needs to be set up, the Chemist asks the program to suggest a batch of unlabeled compounds. The suggested batch might be “edited” and augmented using the invaluable knowledge and intuition of the medicinal Chemist. The hope is that the computer assisted approach allows for mining larger data sets more quickly. Note that compounds are often generated with virtual Combinatorial Chemistry. Even though compound descriptors can be computed, the compounds have not been synthesized yet. In other words it is comparatively easy to generate lots of unlabeled data. In our case the Round0 data set consists of compounds from Vendor catalog and corporate collecFigure 1: Three types of compounds/points: are active, are inactive and are yet unlabeled. The Maximum Margin Hyperplane is used as the internal classifier. tions. Much more design effort went into the harder Round1 data set. Our initial results are very encouraging. Our selection strategies do much better than choosing random batches indicating that the long-term goal outlined above may be feasible. Thus from the Machine Learning point of view we have a fixed set of points in
that are either unlabeled or labeled positive or negative. (See Figure 1). The binary descriptors of the compounds are rather “complete” and the data is always linearly separable. Thus we concentrate on simple linear classifiers in this paper.2 We analyzed a large number of different ways to produce hyperplanes and combine hyperplanes. In the next section we describe different selection strategies on the basis of these hyperplanes in detail and provide an experimental comparison. Finally in Section 3 we give some theoretical justification for why the strategies are so effective. 1Data provided by DuPont Pharmaceuticals. 2On the current data sets kernels did not improve the results (not shown). 2 Different Selection Criteria and their Performance A selection algorithm is specified in three parts: a batch size, an initialization and a selection strategy. In practice it is not cost effective to test single examples at a time. We always chose 5% of the total data set as our batch size, which matches reasonably with typical experimental constraints. The initial batches are chosen at random until at least one positive and one negative example are found. Typically this is achieved with the first batch. All further batches are chosen using the selection strategy. As we mentioned in the introduction, all our selection strategies are based on linear classifiers of the data labeled so far. All examples are normalized to unit-length and we consider homogeneous hyperplanes where the normal direction is again unitlength. A plane predicts with sign
on the example/compound . Once we specify how the weight vector is found then the next batch is found by selecting the unlabeled examples closest to this hyperplane. The simplest way to obtain such a weight vector is to run a perceptron over the labeled data until it produces a consistent weight vector (Perc). Our second selection strategy (called SVM) uses the maximum margin hyperplane [BGV92] produced by a Support Vector Machine. When using the perceptron to predict for example handwritten characters, it has been shown that “voting” the predictions of many hyperplanes improves the predictive performance [FS98]. So we always start from the weight vector zero and do multiple passes over the data until the perceptron is consistent. After processing each example we store the weight vector. We remember all weight vectors for each pass3 and do this for 100 random permutations of the labeled examples. Each weight vector gets one vote. The prediction on an example is positive if the total vote is larger than zero and we select the unlabeled examples whose total vote is closest to zero4. We call this selection strategy VoPerc. The dot product is commutative. So when then the point lies on the positive side of the hyperplane . In a dual view the point lies on the positive side of the hyperplane (Recall all instances and weight vectors have unit-length). A weight vector that is consistent with all -labeled examples
must lie on the -side of the plane for all . The set of all consistent weight vectors is called the version space which is a section of the unit hypersphere bounded by the planes corresponding to the labeled examples. An unlabeled hyperplane bisects the version space. For our third selection strategy (VolEst) a billiard is bounced 1000 times inside the version space and the fraction ! of bounce points on the positive side of " is computed. The prediction for # is positive if ! is larger than half and the strategy selects unlabeled points whose fraction is closest to half. In Figure 2 (left) we plot the true positives and false positives w.r.t. the whole data set for Perc and VoPerc showing that VoPerc performs slightly better. Also VoPerc has lower variance (Figure 2 (right)). Figure 3 (left) shows the averaged true positives and false positives of VoPerc, SVM, and VolEst. We note that all three perform similarly. We also plotted ROC curves after each batch has been added (not shown). These plots also show that all three strategies are comparable. The three strategies VoPerc, SVM, and VolEst all perform much better than the corresponding strategies where the selection criterion is to select random unlabeled examples instead of using a “closest” criterion. For example we show in Figure 4 that SVM is significantly better than SVM-Rand. Surprisingly the improvement is larger on the easier Round0 data set. The reason is that the Round0 has a smaller fraction of positive examples (3%). Recall 3Surprisingly with some smart bookkeeping this can all be done with essentially no computational overhead. [FS98] 4Instead of voting the predictions of all weight vectors one can also average all the weight vectors after normalizing them and select unlabeled examples closest to the resulting single weight vector. This way of averaging leads to slightly worse results (not shown). 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected number of examples Perc true pos Perc false pos VoPerc true pos VoPerc false pos 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 fraction of examples selected standard deviation Perc true pos Perc false pos VoPerc true pos VoPerc false pos Figure 2: (left) Average (over 10 runs) of true positives and false positives on the entire Round1 data set after each 5% batch for Perc and VoPerc. (right) Standard deviation over 10 runs. 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected number of examples VoPerc true pos VoPerc false pos SVM true pos SVM false pos VolEst true pos VolEst false pos 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected total number of hits 5% batch size 1 example batch size Figure 3: (left) Average (over 10 runs) of true and false positives on entire Round1 data set after each 5% batch for VoPerc, SVM, and VolEst. (right) Comparison of 5% batch size and 1 example batch size for VoPerc on Round1 data. that the Round1 data was preselected by the Chemists for actives and the fraction was raised to about 25%. This suggest that our methods are particularly suitable when few positive examples are hidden in a large set of negative examples. The simple strategy SVM of choosing unlabeled examples closest to the maximum margin hyperplane has been investigated by other authors (in [CCS00] for character recognition and in [TK00] for text categorization). The labeled points that are closest to the hyperplane are called the support vectors because if all other points are removed then the maximum margin hyperplane remains unchanged. In Figure 5 we visualize the location of the points in relation to the center of the hyperplane. We show the location of the points projected onto the normal direction of the hyperplane. For each 5% batch the location of the points is scattered onto a thin stripe. The hyperplane crosses the stripe in the middle. In the left plot the distances are scaled so that the support vectors are at distance 1. In the right plot the geometric distance to the hyperplane is plotted. Recall that we pick unlabeled points closest to the hyperplane (center of the stripe). As soon as the “window” between the support vectors is cleaned most positive examples have been found (compare with the SVM curves given in Figure 3 (left)). Also shrinking the width of the geometric window corresponds to improved generalization. So far our three selection strategies VoPerc, SVM and VolEst have shown similar performance. The question is whether the performance criterion considered so far is suitable for the drug design application. Here the goal is to label/verify many positive compounds quickly. We therefore think that the total number of positives (hits) among all examples tested so far is the best performance criterion. Note that the total number of hits of the random selection strategy grows linearly with the number of batches (In each random batch 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 40 fraction of examples selected number of examples random true pos random false pos closest true pos closest false pos 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected number of examples random true pos random false pos closest true pos closest false pos Figure 4: Comparisons of SVM using random batch selection and closest batch selection. (left) Round0 data. (right) Round1 data. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 normalized distance to hyperplane fraction of examples selected −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 geometric distance to hyperplane fraction of examples selected Figure 5: (left) Scatter plot of the distance of examples to the maximum margin hyperplane normalized so support vectors are at 1. (right) Scatter plot of the geometric distance of examples to the hyperplane. Each stripe shows location of a random sub-sample of points (Round1 data) after an additional 5% batch has been labeled by SVM. Selected examples are black x, unselected positives are red plus, unselected negatives are blue square. we expect 5% hits). In contrast the total number of hits of VoPerc, SVM and VolEst is 5% in the first batch (since it is random) but much faster thereafter (See Figure 6). VoPerc performs the best. Since the positive examples are much more valuable in our application, we also changed the selection strategy SVM to selecting unlabeled examples of largest positive distance 5 to the maximum margin hyperplane (SVM ) rather than smallest distance . Correspondingly VoPerc picks the unlabeled example with the highest vote and VolEst picks the unlabeled example with the largest fraction ! . The total hit plots of the resulting modified strategies SVM , VoPerc and VolEst are improved ( see Figure 7 versus Figure 6 ). However the generalization plots of the modified strategies (i.e. curves like Figure 3(left)) are slightly worse for the new versions. Thus in some sense the original strategies are better at ”exploration” (giving better generalization on the entire data set) while the modified strategies are better at ”exploitation” (higher number of total hits). We show this trade-off in Figure 8 for SVM and SVM . The same trade-off occurs for the VoPerc
and VolEst
pairs of strategies(not shown). Finally we investigate the effect of batch size on performance. For simplicity we only show total hit plots for VoPerc( Figure 3 (right) ). Note that for our data a batch size of 5% (31 examples for Round1) is performing not much worse than the experimentally unrealistic batch size of only 1 example. Only when the results for batch size 1 are much better than 5In Figure 5 this means we are selecting from right to left 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 40 fraction of examples selected total number of hits VoPerc SVM VolEst 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected total number of hits VoPerc SVM VolEst Figure 6: Total hit performance on Round0 (left) and Round1 (right) data of , VoPerc and VolEst with 5% batch size. 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 40 fraction of examples selected total number of hits VoPerc+ SVM+ VolEst+ 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected total number of hits VoPerc+ SVM+ VolEst+ Figure 7: Total hit performance on Round0 (left) and Round1 (right) data of , VoPerc and VolEst with 5% batch size. the results for larger batch sizes, more sophisticated selection strategies are worth exploring that pick say a batch that is “close” and at the same time “diverse”. At this point our data sets are still small enough that we were able to precompute all dot products (the kernel matrix). After this preprocessing, one pass of a perceptron is at most
, where is the number of labeled examples and the number of mistakes. Finding the maximum margin hyperplane is estimated at
time. For the computation of VolEst we need to spend
per bounce of the billiard. In our implementations we used SVM Light [Joa99] and the billiard algorithm of [Ruj97, RM00, HGC99]. If we have the internal hypothesis of the algorithm then for applying the selection criterion we need to evaluate the hypothesis for each unlabeled point. This cost is proportional to the number of support vectors for the SVM-based methods and proportional to the number of mistakes for the perceptron-based methods. In the case of VolEst we again need
time per bounce, where is the number of labeled points. Overall VolEst was clearly the slowest. For much larger data sets VoPerc seems to be the simplest and the most adaptable. 3 Theoretical Justifications As we see in Figure 5(right) the geometric margin of the support vectors (half the width of the window) is shrinking as more examples are labeled. Thus the following goal is reasonable for designing selection strategies: pick unlabeled examples that cause the margin to shrink the most. The simplest such strategy is to pick examples closest to the maximum margin hyperplane since these example are expected to change the maximum margin 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected total number of hits SVM+ SVM 0 0.2 0.4 0.6 0.8 1 0 50 100 150 fraction of examples selected number of examples SVM+ true pos SVM true pos SVM+ false pos SVM false pos Figure 8: Exploitation versus Exploration: (left) Total hit performance and (right) True and False positives performance (right) of SVM and on Round 1 data hyperplane the most [TK00, CCS00]. An alternative goal is to reduce the volume of the version space. This volume is a rough measure of the remaining uncertainty in the data. Recall that both the weight vectors and instances have unit length. Thus is the distance of the point to the plane as well as (in the dual view) the distance of the point to the plane . The maximum margin hyperplane is the point in version space with the largest sphere that is completely contained in the version space [Ruj97, RM00]. After labeling only one side of the plane remains. So if passes close to the point then about half of the largest sphere is eliminated from the version space. So this is a second justification for selecting unlabeled examples closest to the maximum margin hyperplane. Our selection strategy VolEst starts from any point inside the version space and then bounces a billiard 1000 times. The billiard is almost always ergodic (See discussion in [Ruj97]). Thus the fraction ! of bounces on the positive side of an unlabeled hyperplane is an estimate of the fraction of volume on the positive side of . Since it is unknown how will be labeled, the best example are those that split the version space in half. Thus in VolEst we select unlabeled points for which ! is closest to half. The thinking underlying our strategy VolEst is most closely related to the Committee Machine where random concepts in the version space are asked to vote on the next random example and the label of that example is requested only if the vote is close to an even split [SOS92]. We tried to improve our estimate of the volume by replacing ! by the fraction of the total trajectory located on the positive side of # . On our two data sets this did not improve the performance (not shown). We also averaged the 1000 bounce points. The resulting weight vector (an approximation to the center of mass of the version space) approximates the so called Bayes point [Ruj97] which has the following property: Any unlabeled hyperplane passing through the Bayes point cuts the version space roughly 6 in half. We thus tested a selection strategy which picks unlabeled points closest to the estimated center of mass. This strategy was again indistinguishable from the other two strategies based on bouncing the billiard. We have no rigorous justification for the variants of our algorithms. 4 Conclusion We showed how the active learning paradigm ideally fits the drug design cycle. After some deliberations we concluded that the total number of positive examples (hits) among the tested examples is the best performance criterion for the drug design application. We found 6Even in dimension two there is no point that does this exactly [Ruj97]. that a number of different selection strategies with comparable performance. The variants that select the unlabeled examples with the highest score (i.e. the variants) perform better. Overall the selection strategies based on the Voted Perceptron were the most versatile and showed slightly better performance. References [Ang88] D. Angluin. Queries and concept learning. Machine Learning, 2:319–342, 1988. [BGV92] B.E. Boser, I.M. Guyon, and V.N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory, pages 144–152, 1992. [CAL90] D. Cohn, L. Atlas, and R. Ladner. Training connectionist networks with queries and selective sampling. Advances in Neural Information Processing Systems, 2:566–573, 1990. [CCS00] C. Campbell, N. Cristianini, and A. Smola. Query learning with large margin classifiers. In Proceedings of ICML2000, page 8, Stanford, CA, 2000. [FS98] Y. Freund and R. Schapire. Large margin classification using the perceptron algorithm. In Proc. 11th Annu. Conf. on Comput. Learning Theory. ACM Press, New York, NY, July 1998. [HGC99] Ralf Herbrich, Thore Graepel, and Colin Campbell. Bayes point machines: Estimating the bayes point in kernel space. In Proceedings of IJCAI Workshop Support Vector Machines, pages 23–27, 1999. [Joa99] T. Joachims. Making large–scale SVM learning practical. In B. Sch¨olkopf, C.J.C. Burges, and A.J. Smola, editors, Advances in Kernel Methods — Support Vector Learning, pages 169–184, Cambridge, MA, 1999. MIT Press. [MGST97] P. Myers, J. Greene, J. Saunders, and S. Teig. Rapid, reliable drug discovery. Today’s Chemist at Work, 6:46–53, 1997. [RM00] P. Ruj´an and M. Marchand. Computing the bayes kernel classifier. In Advances in Large Margin Classifiers, volume 12, pages 329–348. MIT Press, 2000. [Ruj97] P. Ruj´an. Playing billiard in version space. Neural Computation, 9:99–122, 1997. [SOS92] H. Seung, M. Opper, and H. Sompolinsky. Query by committee. In Proceedings of the Fifth Workshop on Computational Learning Theory, pages 287– 294, 1992. [TK00] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. In Proceedings of the Seventeenth International Conference on Machine Learning, San Francisco, CA, 2000. Morgan Kaufmann.
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Intransitive Likelihood-Ratio Classifiers Jeff Bilmes and Gang Ji Department of Electrical Engineering University of Washington Seattle, WA 98195-2500 bilmes,gji @ee.washington.edu Marina Meil˘a Department of Statistics University of Washington Seattle, WA 98195-4322 mmp@stat.washington.edu Abstract In this work, we introduce an information-theoreticbased correction term to the likelihood ratio classification method for multiple classes. Under certain conditions, the term is sufficient for optimally correcting the difference between the true and estimated likelihood ratio, and we analyze this in the Gaussian case. We find that the new correction term significantly improves the classification results when tested on medium vocabulary speech recognition tasks. Moreover, the addition of this term makes the class comparisons analogous to an intransitive game and we therefore use several tournament-like strategies to deal with this issue. We find that further small improvements are obtained by using an appropriate tournament. Lastly, we find that intransitivity appears to be a good measure of classification confidence. 1 Introduction An important aspect of decision theory is multi-way pattern classification whereby one must determine the class for a given data vector that minimizes the overall risk: argmin
where is the loss in choosing when the true class is . This decision rule is provably optimal for the given loss function [3]. For the 0/1-loss functions, it is optimal to simply use the posterior probability to determine the optimal class argmax This procedure may equivalently be specified using a tournament style game-playing strategy. In this case, there is an implicit class ordering "!#!$!# %& , and a class-pair ( ' and ( ) scoring function for an unknown sample : )+*-, *., 0/21 *., such that *-, 43#576 8 '9 :; 8 ( is the log-likelihood ratio and 1 *., 43$56 '<=:> ( is the log prior odds. The strategy proceeds by evaluating )8<?< @ which if positive is followed by ) ? A and otherwise by ) A @ . This continues until a “winner” is found. Of course, the order of the classes does not matter, as the same winner is found for all permutations. In any event, this style of classification can be seen as a transitive game [5] between players who correspond to the individual classes. In this work we extend the likelihood-ratio based classification with a term, based on the Kullback-Leibler divergence [2], that expresses the inherent posterior confusability between the underlying likelihoods being compared for a given pair of players. We find that by including this term, the results of a classification system significantly improve, without changing or increasing the quantity of the estimated free model parameters. We also show how, under certain assumptions, the term can be seen as an optimal correction between the estimated model likelihood ratio and the true likelihood ratio, and gain further intuition by examining the case when the likelihoods 8 '9 are Gaussians. Furthermore, we observe that the new strategy leads to an intransitive game [5], and we investigate several strategies for playing such games. This results in further (but small) improvements. Finally, we consider the instance of intransitivity as a confidence measure, and investigate an iterative approach to further improve the correction term. Section 2 first motivates and defines our approach, and shows the conditions under which it is optimal. Section 2.1 then reports experimental results which show significant improvements where the likelihoods are hidden Markov models trained on speech data. Section 3 then recasts the procedure as intransitive games, and evaluates a variety of game playing strategies yielding further (small) error reductions. Section 3.1 attempts to better understand our results via empirical analysis, and evaluates additional classification strategies. Section 4 explores an iterative strategy for improving our technique, and finally Section 5 concludes and discusses future work. 2 Extended Likelihood-Ratio-based Classification The Kullback-Leibler (KL) divergence[2], an asymmetric measure of the distance between two probability densities, is defined as follows: 83$56 where and are probability densities over the same sample space. The KL-divergence is also called the average (under ) information for discrimination in favor of over . For our purposes, we are interested in KL-divergence between class-conditional likelihoods 8 '9 where ' is the class number: ' (
3$56 8 '< 8 ( 8 '<7 One intuitive way of viewing ' ( is as follows: if ' ( is small, then samples of class ' are more likely to be erroneously classified as class ( than when ' ( is large. Comparing ' ( and ( >'< should tell us which of ' and ( is more likely to have its samples mis-classified by the other model. Therefore, the difference ' ( ( '9 , when positive, indicates that samples of class ( are more likely to be mis-classified as class ' than samples of class ' are to be mis-classified as class ( (and vice-versa when the difference is negative). In other words, ' “steals” from ( more than ( steals from ' when the difference is positive, thereby suggesting that class ( should receive aid in this case. This difference can be viewed as a form of posterior (i.e., based on the data) “bias” indicating which class should receive favor over the other.1 We can adjust *., (the loglikelihood ratio) with this posterior bias, to obtain a new function comparing classes ' and ( as follows: ) *-, *-, / 1 *-, *., ! 1Note that this is not the normal notion of statistical bias as in where is an estimate of model parameters. where *-, ' ( ( >'< The likelihood ratio is adjusted in favor of ( when *-, is positive, and in favor of ' when *., is negative. We then use ) *-, , and when it is positive, choose class ' . The above intuition does not explain why such a correction factor should be used, since using *-, along with 1 *-, is already optimal. In practice, however, we do not have access to the true likelihood ratios but instead to an approximation that has been estimated from training data. Let the variable *.,
3$56 8 '<=:> ( be the true log-likelihood ratio, and *., 3#576 8 '9 : 8 ( be the model-based log ratio. Furthermore, let ' (
3$56 8 '< 8 ( 8 '<7 be the modified KL-divergence between the class conditional models, measured modulo the true distribution '< , and let *., ' ( ( >'< . Finally, let 1 *., (resp. 1 *., ) be the true (resp. estimated) log prior odds. Our (usable) scoring function becomes: )+*., *-, 0/ 1 *-, *-, ! (1) which has an intuitive explanation similar to the above. There are certain conditions under which the above approach is theoretically justifiable. Let us assume for now a two-class problem where ' and ( are the two classes, so '9;/ ( . A sufficient condition for the estimated quantities above to yield optimal performance is for /
1 / 1 for all .2 Since this is not the case in practice, an ' ( -dependent constant term may be added correcting for any differences as best as possible. This yields / 1 / 1 / . We can define an -dependent cost function
/21 1 which, when minimized, yields
/ 1
1 stating that the optimal under this cost function is just the mean of the difference of the remaining terms. Note that
'< ' ( ( ( '< and
'< ' ( ( ( '9 . Several additional assumptions lead to Equation 1. First, let us assume that the prior probabilities are equal (so '< ! ) and that the estimated and true priors are negligibly different (i.e., 1 1 ). Secondly, if we assume that
, this implies that '< :; ( ( >'<=: ' ( which means that ( >'< ' ( under equal priors. While KL-divergence is not symmetric in general, we can see that if this holds (or is approximately true for a given problem) then the remaining correction is exactly yielding *-, in Equation 1. To gain further insight, we can examine the case when the likelihoods are Gaussian univariate distributions, with means * , and variances * , . In this case, *., * , , * / * , ! , *#"$ (2) It is easy to see that for * , the value of *-, is zero for any * , . By computing the derivative %'&)(+* %', @ ( we can show that *-, is monotonically increasing with * . Hence, *., is positive iff *. , and therefore it penalizes the distribution (class) with higher variance. 2Note that we have dropped the / argument for notational simplicity. VOCAB SIZE WER *-, WER )+*-, 75 2.33584 1.91561 150 3.31072 2.89833 300 5.22513 4.51365 600 7.39268 6.18517 Table 1: Word error rates (WER) for likelihood ratio *., and augmented likelihood ratio )+*-, based classification for various numbers of classes (VOCAB SIZE). Similar relations hold for multivariate Gaussians with means * , and variances * , . *-, * , , * / * , , * * , (3) The above is zero when the two covariance matrices are equal. This implies that for Gaussians with equal covariance matrices, ( $ '< '$ ( and our correction term is optimal. This is the same as the condition for Fisher’s linear discriminant analysis (LDA). Moreover, in the case * , with , we have that *-, for and *-, for
which again implies that *., penalizes the class that has larger covariance. 2.1 Results We tried this method (assuming that 1 *-, 1 *., ) on a medium vocabulary speech recognition task. In our case the likelihood functions '< are hidden Markov model (HMM) scores3. The task we chose is NYNEX PHONEBOOK[4], an isolated word speech corpus. Details of the experimental setup, training/test sets, and model topologies, are described in [1]4. In general, there are a number of ways to compute *-, . These include 1) analytically, using estimated model parameters (possible, for example, with Gaussian densities), 2) computing the KL-divergences on training data using a law-of-large-numbers-like average of likelihood ratios and using training-data estimated model parameters, 3) doing the same as 2 but using test data where hypothesized answers come from a first pass *., -based classification, and 4) Monte-Carlo methods where again the same procedure as 2 is used, but the data is sampled from the training-data estimated distributions. For HMMs, method 1 above is not possible. Also, the data set we used (PHONEBOOK) uses different classes for the training and test sets. In other words, the training and test vocabularies are different. During training, phone models are constructed that are pieced together for the test vocabularies. Therefore, method 2 above is also not possible for this data. Either method 3 or 4 can be used in our case, and we used method 3 in all our experiments. Of course, using the true test labels in method 3 would be the ideal measure of the degree of confusion between models, but these are of course not available (see Figure 2, however, showing the results of a cheating experiment). Therefore, we use the hypothesized labels from a first stage to compute *-, . The procedure thus is as follows: 1) obtain '< using maximum likelihood EM training, 2) classify the test set using only *-, and record the error rate, 3) using the hypothesized class labels (answers with errors) to step 2, compute *., , 4) re-classify the test set using the score ) *-, *-, *., and record the new error rate. ) *-, is used if either one of '# ( 3Using 4 state per phone, 12 Gaussian mixtures per state HMMs, totaling 200k free model parameters for the system. 4Note, however, that error results here are reported on the development set, i.e., PHONEBOOK lists a,b,c,d o,y VOCAB *-, RAND1 RAND500 RAND1000 WORLD CUP 75 2.33584 1.87198 1.82047 1.91467 2.12777 150 3.31072 2.88505 2.71881 2.72809 2.79516 300 5.22513 4.41428 4.34608 4.28930 3.81583 600 7.39268 6.15828 6.13085 5.91440 5.93883 Table 2: The WER under different tournament strategies or ( $ '< is below a threshold (i.e., when a likely confusion exists), otherwise *-, is used for classification. Table 1 shows the result of this experiment. The first column shows the vocabulary size of the system (identical to the number of classes)5. The second column shows the word error rate (WER) using just *-, , and the third column shows WER using ) *., . As can be seen, the WER decreases significantly with this approach. Note also that no additional free parameters are used to obtain these improvements. 3 Playing Games We may view either *-, or ) *., as providing a score of class ' over ( — when positive, class ' wins, and when negative, class ( wins. In general, the classification procedure may be viewed as a tournament-style game, where for a given sample , different classes correspond to different players. Players pair together and play each other, and the winner goes on to play another match with a different player. The strategy leading to table 1 required a particular class presentation order — in that case the order was just the numeric ordering of the arbitrarily assigned integer classes (corresponding to words in this case). Of course when *-, alone is used, the order of the comparisons do not matter, leading to a transitive game [5] (the order of player pairings do not change the final winner). The quantity ) *-, , however, is not guaranteed to be transitive, and when used in a tournament it results in what is called an intransitive game[5]. This means, for example, that might win over who might win over who then might win over . Games may be depicted as directed graphs, where an edge between two players point towards the winner. In an intransitive game, the graph contains directed cycles. There has been very little research on intransitive game strategies — there are in fact a number of philosophical issues relating to if such games are valid or truly exist. Nevertheless, we derived a number of tournament strategies for playing such intransitive games and evaluated their performance in the following. Broadly, there are two tournament types that we considered. Given a particular ordering of the classes "!#!$!$= % , we define a sequential tournament when plays , the winner plays , the winner plays and so on. We also define a tree-based tournament when plays , plays , and so on. The tree-based tournament is then applied recursively on the resulting : winners until a final winner is found. Based on the above, we investigated several intransitive game playing strategies. For RAND1, we just choose a single random tournament order in a sequential tournament. For RAND500, we run 500 sequential tournaments, each one with a different random order. The ultimate winner is taken to be the player who wins the most tournaments. The third strategy plays 1000 rather than 500 tournaments. The final strategy is inspired by worldcup soccer tournaments: given a randomly generated permutation, the class sequence is 5The 75-word case is an average result of 8 experiments, the 150-word case is an average of 4 cases, and the 300-word case is an average of 2 cases. There are 7291 separate test samples in the 600-word case, and on average about 911 samples per 75-word test case. vocabulary var max var max 75 1.0047 0.0071 2.7662 1.0285 0.0759 3.8230 150 1.0061 0.0126 3.6539 1.0118 0.0263 3.8724 300 1.0241 0.0551 4.0918 1.0170 0.0380 3.9072 600 1.0319 0.0770 5.0460 1.0533 0.1482 5.5796 Table 3: The statistics of winners. Columns 2-4: 500 random tournaments, Columns 5-7: 1000 random tournaments. separated into 8 groups. We pick the winner of each group using a sequential tournament (the “regionals”). Then a tree-based tournament is used on the group winners. Table 1 compares these different strategies. As can be seen, the results get slightly better (particularly with a larger number of classes) as the number of tournaments increases. Finally, the single word cup strategy does surprisingly well for the larger class sizes. Note that the improvements are statistically significant over the baseline (0.002 using a difference of proportions significance test) and the improvements are more dramatic for increasing vocabulary size. Furthermore, the it appears that the larger vocabulary sizes benefit more from the larger number (1000 rather than 500) of random tournaments. 1 2 3 4 5 6 0 10 20 30 40 50 60 length of cycle probability of error (%) 0 1 2 3 4 5 0 10 20 30 40 50 60 70 number of cycles detected probability of error (%) Figure 1: 75-word vocabulary case. Left: probability of error given that there exists a cycle of at least the given length (a cycle length of one means no cycle found). Right:probability of error given that at least the given number of cycles exist. 3.1 Empirical Analysis In order to better understand our results, this section analyzes the 500 and 1000 random tournament strategies described above. Each set of random tournaments produces a set of winners which may be described by a histogram. The entropy of that histogram describes its spread, and the number of typical winners is approximately . This is of course relative to each sample so we may look at the average ( ), variance, and maximum of this number (the minimum is 1.0 in every case). This is given in Table 3 for the 500 and 1000 cases. The table indicates that there is typically only one winner since is approximately 1 and the variances are small. This shows further that the winner is typically not in a cycle, as the existence of a directed cycle in the tournament graph would probably lead to different winners for each random tournament. The relationship between properties of cycles and WER is explored below. When the tournament is intransitive (and therefore the graph possess a cycle), our second analyses shows that the probability of error tends to increase. This is shown in Figure 1 showing that the error probability increases both as the detected cycle length and the numvocabulary *., skip WER #cycles(%) break WER #cycles(%) 75 2.33584 1.90237 13.89 1.90223 9.34 150 3.31072 2.76814 19.6625 2.67814 16.83 300 5.22513 4.46296 22.38 4.46296 21.34 600 7.39268 6.50117 31.96 6.50117 31.53 Table 4: WER results using two strategies (skip and break) that utilize information about cycles in the tournament graphs, compared to baseline *., . The and columns show the number of cycles detected relative to the number of samples in each case. ber of detected cycles increases. 6 This property suggests that the existence of intransitivity could be used as a confidence measure, or could be used to try to reduce errors. As an attempt at the latter, we evaluated two very simple heuristics that try to eliminate cycles as detected during classification. In the first method (skip), we run a sequential tournament (using a random class ordering) until either a clear winner is found (a transitive game), or a cycle is detected. If a cycle is detected, we select two players not in the cycle, effectively jumping out of the cycle, and continue playing until the end of the class ordering. If winner cannot be determined (because there are too few players remaining), we backoff and use *-, to select the winner. In a second method (break), if a cycle is detected, we eliminate the class having the smallest likelihood from that cycle, and then continue playing as before. Neither method detects all the cycles in the graph (their number can be exponentially large). As can be seen, the WER results still provide significant improvements over the baseline, but are no better than earlier results. Because the tournament strategy is coupled with cycle detection, the cycles detected are different in each case (the second method detecting fewer cycles presumably because the eliminated class is in multiple cycles). In any case, it is apparent that further work is needed to investigate the relationship between the existence and properties of cycles and methods to utilize this information. 4 Iterative Determination of KL-divergence In all of our experiments so far, KL-divergence is calculated according to the initial hypothesized answers. We would expect that using the true answers to determine the KLdivergence would improve our results further. The top horizontal lines in Figure 2 shows the original baseline results, and the bottom lines show the results using the true answers (a cheating experiment) to determine the KL-divergence. As can be seen, the improvement is significant thereby confirming that using *-, can significantly improve classification performance. Note also that the relative improvement stays about constant with increasing vocabulary size. This further indicates that an iterative strategy for determining KL-divergence might further improve our results. In this case, *-, is used to determine the answers to compute the first set of KL-divergences used in )
*-, . This is then used to compute a new set of answers which then is used to compute a new scores )
*-, and so on. The remaining plots in Figure 2 show the results of this strategy for the 500 and 1000 random trials case (i.e., the answers used to compute the KL-divergences in each case are obtained from the previous set of random tournaments using the histogram peak procedure described earlier). Rather surprisingly, the results show that iterating in this fashion does not influence the results in 6Note that this shows a lower bound on the number of cycles detected. This is saying that if we find, for example, four or more cycles then the chance of error is high. 0 2 4 6 8 10 1.5 2 2.5 number of iterations word error rate (%) 75 classes 0 2 4 6 8 10 2 2.5 3 3.5 number of iterations word error rate (%) 150 classes 0 2 4 6 8 10 4 4.2 4.4 4.6 4.8 5 5.2 5.4 number of iterations word error rate (%) 300 classes 0 2 4 6 8 10 5.5 6 6.5 7 7.5 number of iterations word error rate (%) 600 classes baseline cheating 500 trials 1000 trials Figure 2: Baseline using likelihood ratio (top lines), cheating results using correct answers for KL-divergence (bottom lines), and the iterative determination of KL-distance using hypothesized answers from previous iteration (middle lines). any appreciable way — the WERs seem to decrease only slightly from their initial drop. It is the case, however, that as the number of random tournaments increases, the results become closer to the ideal as the vocabulary size increases. We are currently studying further such iterative procedures for recomputing the KL-divergences. 5 Discussion and Conclusion We have introduced a correction term to the likelihood ratio classification method that is justified by the difference between the estimated and true class conditional probabilities 8 '<> '< . The correction term *-, is an estimate of the classification bias that would optimally compensate for these differences. The presence of *-, makes the class comparisons intransitive and we introduce several tournament-like strategies to compensate. While the introduction of *., consistently improves the classification results, further improvements are obtained by the selection of the comparison strategy. Further details and results of our methods will appear in forthcoming publications and technical reports. References [1] J. Bilmes. Natural Statistic Models for Automatic Speech Recognition. PhD thesis, U.C. Berkeley, Dept. of EECS, CS Division, 1999. [2] T.M. Cover and J.A. Thomas. Elements of Information Theory. Wiley, 1991. [3] R.O. Duda, P.E. Hart, and D.G. Stork. Pattern Classification. John Wiley and Sons, Inc., 2000. [4] J. Pitrelli, C. Fong, S.H. Wong, J.R. Spitz, and H.C. Lueng. PhoneBook: A phonetically-rich isolated-word telephone-speech database. In Proc. IEEE Intl. Conf. on Acoustics, Speech, and Signal Processing, 1995. [5] P.D. Straffin. Game Theory and Strategy. The Mathematical ASsociation of America, 1993.
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Transform-invariant image decomposition with similarity templates Chris Stauffer, Erik Miller, and Kinh Tieu MIT Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 {stauffer,emiller,tieu}@ai.mit.edu Abstract Recent work has shown impressive transform-invariant modeling and clustering for sets of images of objects with similar appearance. We seek to expand these capabilities to sets of images of an object class that show considerable variation across individual instances (e.g. pedestrian images) using a representation based on pixel-wise similarities, similarity templates. Because of its invariance to the colors of particular components of an object, this representation enables detection of instances of an object class and enables alignment of those instances. Further, this model implicitly represents the regions of color regularity in the class-specific image set enabling a decomposition of that object class into component regions. 1 Introduction Images of a class of objects are often not effectively characterized by a Gaussian distribution or even a mixture of Gaussians. In particular, we are interested in modeling classes of objects that are characterized by similarities and differences between image pixels rather than by the values of those pixels. For instance, images of pedestrians (at a certain scale and pose) can be characterized by a few regions of regularity (RORs) such as shirt, pants, background, and head, that have fixed properties such as constant color or constant texture within the region, but tend to be different from each other. The particular color (or texture) of those regions is largely irrelevant. We shall refer to sets of images that fit this general description as images characterized by regions of regularity, or ICRORs. Jojic and Frey [1] and others [2] have investigated transform-invariant modeling and clustering for images of a particular object (e.g., an individual’s face). Their method can simultaneously converge on a model and align the data to that model. This method has shown positive results for many types of objects that are effectively modeled by a Gaussian or a mixture of Gaussians. Their work with transformed component analysis (TCA) shows promise for handling considerable variation within the images resulting from lighting or slight misalignments. However, because these models rely on an image set with a fixed mean or mixture of means, they are not directly applicable to ICRORs. We would also like to address transform-invariant modeling, but use a model which is invariant to the particular color of component regions. One simple way to achieve this is to use edge templates to model local differences in image color. In contrast, we have chosen to model global similarities in color using a similarity template (ST). While representations of pixel similarity have previously been exploited for segmentation of single images [3, 4], we have chosen to use them for aggregate modeling of image sets. Similarity templates enable alignment of image sets and decomposition of images into class-specific pixel regions. We note also that registration of two ICRORs can be accomplished by minimizing the mutual information between corresponding pixels [5]. But, there is no obvious way of extending this method to large sets of images without a combinatorial explosion. Section 2 briefly introduces similarity templates. We investigate their uses for modeling and detection. Section 3 discusses dataset alignment. Section 4 covers their application to decomposing a class-specific set of images into component regions. Future avenues of research and conclusions are discussed Section 5. 2 Similarity templates This section begins with a brief explanation of the similarity template followed by the mechanics of computing and comparing similarity templates. A similarity template S for an N-pixel image is an NxN matrix. The element Si,j represents the probability that pixel locations pi and pj would result from choosing a region and drawing (iid) two samples (pixel locations) from it. More formally, Si,j = X r p(r)p(pi|r)p(pj|r), (1) where p(r) is the probability of choosing region r and p(pi|r) is the probability of choosing pixel location pi from region r. 2.1 The “ideal” similarity template Consider sampling pixel pairs as described above from an N-pixel image of a particular object (e.g., a pedestrian) segmented by an oracle into disjoint regions (e.g., shirt, pants, head, feet, background). Assuming each region is equally likely to be sampled and that the pixels in the region are selected with uniform probability, then Si,j = ( 1 R)( 1 Sr )2 if ri = rj 0 otherwise, (2) where R is the number of regions, Sr is the number of pixels in region r, and ri is the region label of pi. If two pixels are from the same region, the corresponding value is the product of the probability 1 R of choosing a particular region and the probability ( 1 Sr )2 of drawing that pixel pair. This can be interpreted as a block diagonal co-occurrence matrix of sampled pixel pairs. In this ideal case, two images of different pedestrians with the same body size and shape would result in the same similarity template regardless of the colors of their clothes, since the ST is a function only of the segmentation. An ST of an image without a pedestrian would exhibit different statistics. Note that even the ST of an image of a blank wall (segmented as a single region) would be different because pixels that are in different regions under the ideal pedestrian ST would be in the same region. Unfortunately, images do not typically come with labeled regions, and so computation of a similarity template is impossible. However, in this paper, we take advantage of the observation that properties within a region, such as color, are often approximately constant. Using this observation, we can approximate true similarity templates from unsegmented images. 2.2 Computing similarity templates For the purposes of this paper, our model for similarity is based solely on color. Since there is a correlation between color similarity and two pixels being in the same region, we approximate the corresponding value ˜Si,j with a measure of color similarity: ˜Si,j = 1 NZi exp −||Ii −Ij||2 σ2 i , (3) where Ii and Ij are pixel color values, σ2 i is a parameter that adjusts the color similarity measure as a function of the pixel color distribution in the image, and Zi is the sum of the ith row. This normalization is required because large regions have a disproportionate effect on the ST estimate. The choice of σ2 i had little effect on the resulting ST. If each latent region had a constant but unique color and the regions were of equal size, then as σ2 i approaches zero this process reconstructs the “ideal” similarity template defined in Equation 1. Although region colors are neither constant nor unique, this approximation has proven to work well in practice. It is possible to add a spatial prior based on the relative pixel location to model the fact that similarities tend to local, but we will rely on the statistics of the images in our data set to determine whether (and to what extent) this is the case. Also, it may be possible to achieve better results using a more complex color model (e.g., hsv with full covariance) or broadening the measure of similarity to include other modalities (e.g., texture, motion, depth, etc.). Figure 1 shows two views of the same similarity template. The first view represents each pixel’s similarity to every other pixel. The second view contains a sub-image for each pixel which highlights the pixels that are most likely produced by the same region. Pixels in the shirt tend to highlight the entire shirt and the pants (to a lesser amount). Pixels in the background tend to be very dissimilar to all pixels in the foreground. 2.3 Aggregate similarity templates (AST) We assume each estimated ST is a noisy measurement of the true underlying joint distribution. Hence we compute an aggregate similarity template (AST) as the mean ¯S of the ST estimates over an entire class-specific set of K images: ¯Si,j = 1 K K X k=1 ˜Sk i,j. (4) For this quantity to be meaningful, the RORs must be in at least partial correspondence across the training set. Note that this is a less restrictive assumption than assuming edges of regions are in correspondence across an image set, since regions have greater support. Being the mean of a set of probability distributions, the AST is also a valid joint probability distribution. (a) (b) Figure 1: (a) The NxN aggregate similarity template for pedestrian data set. (b) An alternate view of (a). This view is a width2xheight2 version of (a). Each subimage represents the row of the original AST that corresponds to that pixel. Each sub-image highlights the pixels that are most similar to the pixel it represents. 2.4 Comparing similarity templates To compare an estimated similarity template ˜S to an aggregate similarity template ¯S we evaluate their dot product1: s( ¯S, ˜S) = X i X j ¯Si,j ˜Si,j. (5) We are currently investigating other measures for comparison. By thresholding the ratio of the dot product of a particular image patch under and AST trained on pedestrian image patches versus an AST trained on random image patches, we can determine whether a person is present in the image. In previous work [6], we have illustrated encouraging detection performance. 3 Data set alignment In this paper, we investigate a more difficult problem: alignment of a set of images. To explore this problem, we created a set of 128x64 images of simulated pedestrians. These pedestrians were generated by creating four independently-colored regions corresponding to shirts, pants, head, and background. Each region was given a random color. The RGB components were chosen from a uniform distribution [0, 1]. Then, independent Gaussian noise was added to each pixel (σ = .1). Finally the images were translated uniformly up to 25% of the size of the object. Figure 2 shows examples of these images. 1In our experimentation KL-divergence, typically used to compare estimates of distributions, proved less robust. Figure 2: A set of randomly generated “pedestrian” images used in alignment experimetns. Using the congealing procedure of Miller et al. [2], we iteratively estimated the latent variables (translations) that maximized the probability of the image STs to the AST and re-estimated the AST. We were able to align the images to within .5 pixels on average. 4 Decomposing the similarity template This section explains how to derive a factorized representation from the AST that will be useful for recognition of particular instances of a class and for further refinement of detection. This representation is also useful in approximating the template to avoid the O(N 2) storage requirements. An AST represents the similarity of pixels within an image across an entire classspecific data set. Pairwise statistics have been used for segmentation previously [3]. Recently, work centered on factoring joint distributions has gained increasing attention [7, 8, 9, 10]. Rather than estimating two sets of marginals (conditioned on a latent variable) that explain co-occurrence data (e.g. word-document pairs), we seek a single set of marginals conditioned on a latent variable (the ROR) that explain our co-occurrence data (pixel position pairs). Hence, it is a density factorization in which the two conditional factors are identical (Equation 1). We refer to this as symmetric factorization of a joint density. Also, rather than treating pixel brightness (darkness, redness, blueness, or hue) as a value to be reconstructed in the decomposition, we chose to represent pixel similarity. In contrast to simply treating images as additive mixtures of basis functions [9], our decomposition will get the same results on a database of images of digits written in black on white paper or in white on a black board and color images introduce no difficulties for our methods. We would like to estimate the factors from Equation 1 that best reconstruct our measured AST, ¯S. Let ˆS be the estimate of ¯S constructed from these factors. Given the number of regions R, it is possible to estimate the priors for each region p(r) and the probability of each region producing each pixel p(pi|r). The error function we minimize is the KL-divergence between the empirically measured ¯S and our parameterized estimate ˆS, E = X i X j ¯Si,j log ¯Si,j ˆSi,j ! (6) as in [8]. Because our model ¯S is symmetric, this case can be updated with only two rules: pnew(pi|r) ∝p(pi|r) X pj p(r)p(pj|r) ˆS(pi, pj) ¯S(pi, pj), and (7) pnew(r) ∝p(r) X pi X pj p(pj|r)p(pi|r) ˆS(pi, pj) ¯S(pi, pj). (8) 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 Figure 3: The similarity template and the corresponding automatically generated binary decomposition of the images in the pedestrian data set. The root node represents every pixel in the image. The first branch splits foreground vs. background pixels. Other nodes correspond to shirt, legs, head, and background regions. The more underlying regions we allow our model, the closer our estimate will approximate the true joint distribution. These region models tend to represent parts of the object class. p(pi|r) will tend to have high probabilities for a set of pixels belonging to the same region. We take advantage of the fact that aligned pedestrian images are symmetric about the vertical axis by adding a “reflected” aggregate similarity template to the aggregate similarity template. The resulting representation provides a compact approximation of the AST (O(RN) rather than O(N 2)). Rather than performing a straight R-way decomposition of the AST to obtain R pixel region models, we extracted a hierarchical segmentation in the form of a binary tree. Given the initial region-conditioned marginals p(pi|r0) and p(pi|r1), each pixel was assigned to the region with higher likelihood. This was iteratively applied to the ASTs defined for each sub-region. Region priors were set to 0.5 and not adapted in order to encourage a balanced cut. The probabilistic segmentation can be employed to accumulate robust estimates of statistics of the region. For instance, the mean pixel value can be calculated as a weighted mean where the pixels are weighted by p(pi|r). 4.1 Decomposing pedestrians Because the data collected at our lab showed limited variability in lighting, background composition, and clothing, we used the MIT CBCL pedestrian data set which contains images of 924 unique, roughly aligned pedestrians in a wide variety of environments to estimate the AST. Figure 3 shows the resulting hierarchical segmentation for the pedestrian AST. Since this intuitive representation was derived automatically with absolutely no knowledge about pedestrians, we hope other classes of objects can be similarly decomposed into RORs. In our experience, a color histogram of all the pixels within a pedestrian is not useful for recognition and was almost useless for data mining applications. Here we propose a class-conditional color model. It determines a color model over each region that our algorithm has determined contain similar color information within this class of objects. This allows us to obtain robust estimates of color in the regions Figure 4: Results of automatic clustering on three components: shirt, pants, and the background. Each shows the feature, the most unusual examples of that region, followed by the 12 most likely examples for the eight prototypical colors of that region. of regularity. Further, as a result of our probabilistic segmentation, the values of p(pi|r) indicate which pixels are most regular in a region which enables us to weight the contribution of each pixel to the color model. For the case of pedestrian-conditional color models, the regions roughly correspond to shirt color, pant color, feet color, head color, and some background color regions. The colors in a region of a single image can be modeled by color histograms, Gaussians, or mixtures of Gaussians. These region models can be clustered across images to determine a density of shirt colors, pant colors, and other region colors within a particular environment. This enables not only an efficient factored color component codebook, but anomaly detection based on particular regions and higher order models of co-occurrences between particular types of regions. To illustrate the effectiveness of our representation we chose the simplest model for the colors in each region–a single Gaussian in RGB space. The mean and variance of each Gaussian was computed by weighting the pixels represented by the corresponding node by p(pi|r). This biases the estimate towards the “most similar” pixels in the region (e.g., the center of the shirt or the center of the legs). This allows us to represent the colors of each pedestrian image with 31 means and variances corresponding to the (2treeheight −1) nodes. We investigated unsupervised clustering on components of the conditional color model. We fit a mixture of eight Gaussians to the 924 color means for each region. Figure 4 shows the 12 pedestrians with the highest probability under each of the eight models and the 12 most unusual pedestrians with respect to that region for three of the nodes of the tree: shirt color, pant color, and color of the background. Red, white, blue, and black shirts represent a significant portion of the database. Blue jeans are also very common in the Boston area (where the CBCL database was collected). Indoor scenes tended to be very dark, and cement is much more common than grass. 5 Conclusions While this representation shows promise, it is not ideal for many problems. First, it is expensive in both memory and computation. Here, we are only using a simple measure of pairwise similarity–color similarity. In the future, similarity templates could be applied to different modalities including texture similarity, depth similarity, or motion similarity. While computationally intensive, we believe that similarity templates can provide a unified approach to the extraction of possible class-specific targets from an image database, alignment of the candidate images, and precomputation of meaningful features of that class. For the case of pedestrians, it could detect potential pedestrians in a database, align them, derive a model of pedestrians, and extract the parameters for each pedestrian. Once the features are computed, query and retrieval can be done efficiently. We have introduced a new image representation based on pixel-wise similarity. We have shown its application in both alignment and decomposition of pedestrian images. References [1] Jojic, N. and B. J. Frey. “Topographic transformation as a discrete latent variable.” In NIPS 12, S. A. Solla, T. K. Leen and K.-R. Muller (eds), MIT Press, Cambridge, MA. [2] Miller, E., N. Matsakis, and P. Viola, (2000) ”Learning from One Example Through Shared Densities on Transforms.” CVPR2000, Vol. 1, pp. 464-471. [3] Shi, J. and J. Malik. “Normalized Cuts and Image Segmentation,” In CVPR San juan, Puerto Rico, June 1997. [4] Boykov, Y., O. Veksler and R. Zabih. Fast Approximate Energy Minimization via Graph Cuts, In ICCV (99), September 1999. [5] Viola, P. Alignment by Maximization of Mutual Information. MIT Artificial Intelligence Lab, Ph.D. Thesis AI-TR #1548, June, 1995. [6] Stauffer, C. and W.E.L. Grimson. “Similarity templates for detection and recognition,” submitted to CVPR (2001). [7] Pereira, F.C., N. Tishby, and L. Lee. “Distributional clustering of English words.” In 30th Annual Meeting of the Association for Computational Linguistics, Columbus, Ohio, pages 183–190, 1993. [8] Thomas Hofmann, “Probabilistic Latent Semantic Analysis,” UAI (99), Morgan Kaufmann Publishers, Inc., San Francisco, 1999. [9] Lee, D. D. and H. S. Seung. “Learning the parts of objects by non-negative matrix factorization.” Nature 401, 788-791 (1999). [10] Stauffer, C.. ”Automatic hierarchical classification using time-based co-occurrences.” CVPR (1999), Fort Colins, CO, (June 1999).
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Activity Driven Adaptive Stochastic Resonance Gregor Wenning and Klaus Oberrnayer Department of Electrical Engineering and Computer Science Technical University of Berlin Franklinstr. 28/29, 10587 Berlin {grewe, oby}@cs.tu-berlin.de Abstract Cortical neurons might be considered as threshold elements integrating in parallel many excitatory and inhibitory inputs. Due to the apparent variability of cortical spike trains this yields a strongly fluctuating membrane potential, such that threshold crossings are highly irregular. Here we study how a neuron could maximize its sensitivity w.r.t. a relatively small subset of excitatory input. Weak signals embedded in fluctuations is the natural realm of stochastic resonance. The neuron's response is described in a hazard-function approximation applied to an Ornstein-Uhlenbeck process. We analytically derive an optimality criterium and give a learning rule for the adjustment of the membrane fluctuations, such that the sensitivity is maximal exploiting stochastic resonance. We show that adaptation depends only on quantities that could easily be estimated locally (in space and time) by the neuron. The main results are compared with simulations of a biophysically more realistic neuron model. 1 Introduction Energetical considerations [1] and measurements [2] suggest, that sub-threshold inputs, i.e. inputs which on their own are not capable of driving a neuron, play an important role in information processing. This implies that measures must be taken, such that the relevant information which is contained in the inputs is amplified in order to be transmitted. One way to increase the sensitivity of a threshold device is the addition of noise. This phenomenon is called stochastic resonance (see [3] for a review) , and has already been investigated and experimentally demonstrated in the context of neural systems (e.g. [3, 4]). The optimal noise level, however, depends on the distribution of the input signals, hence neurons must adapt their internal noise levels when the statistics of the input is changing. Here we derive and explore an activity dependent learning rule which is intuitive and which only depends on quantities (input and output rates) which a neuron could - in principle - estimate. The paper is structured as follows. In section 2 we describe the neuron model and we introduce the membrane potential dynamics in its hazard function approximation. In section 3 we characterize stochastic resonance in this model system and we calculate the optimal noise level as a function of the input and output rates. In section 4 we introduce an activity dependent learning rule for optimally adjusting the internal noise level, demonstrate its usefulness by applying it to the Ornstein-Uhlenbeck neuron and relate the phenomenon of stochastic resonance to its experimentally accessible signature: the adaptation of the neuron's transfer function. Section 5 contains a comparison to the results from a biophysically more realistic neuron model. Section 6, finally, concludes with a brief discussion. 2 The abstract Neuron Model Figure 1 a) shows the basic model setup. A leaky integrate-and-fire neuron receives .", a) "0 {5 0.9 b) " '-< 0.8 -5 > O/l 0.7 c ~;=".~ >8 rateAo .~ train with rate As 8 ;:: 0.6 " 0.5 ~ E 0.4 /'~ 0.3 IWn I .S "-' 0.2 0 2 N balanced Poisson .0 0.1 spike trains with rates ;:: 00 0. 0.2 0.4 0.6 0.8 As average membrane potential Figure 1: a)The basic model setup. For explanation see text. b) A family of Arrhenius type hazard functions for different noise levels. 1 corresponds to the threshold e and values below 1 are subthreshold . a "signal" input, which we assume to be a Poisson distributed spike train with a rate As. The rate As is low enough, so that the membrane potential V of the neuron remains sub-threshold and no output spikes are generated. For the following we assume that the information the input and output of the neuron convey is coded by its input and output rates As and Ao only. Sensitivity is then increased by adding 2N balanced excitatory and inhibitory "noise" inputs (N inputs each) with rates An and Poisson distributed spikes. Balanced inputs [5, 6] were chosen, because they do not affect the average membrane potential and allow to separate the effect of decreasing the distance of the neuron's operating point to the threshold potential from the effect of increasing the variance of the noise. Signal and noise inputs are coupled to the neuron via synaptic weights Ws and Wn for the signal and noise inputs. The threshold of the neuron is denoted bye. Without loss of generality the membrane time-constant, the neuron's resting potential, and the neuron's threshold are set to one, zero, and one, respectively. If the total rate 2N An of incoming spikes on the "noise" channel is large and the individual coupling constants Wn are small, the dynamics of the membrane potential can be approximated by an Ornstein-Uhlenbeck process, dV = - V dt + J.l dt + (J" dW, (1) where drift J.l and variance (J" are given by J.l = wsAs and (J"2 = w1As + 2NwYvAN, and where dW describes a Gaussian noise process with mean zero and variance one [8]. Spike initiation is included by inserting an absorbing boundary with reset. Equation (1) can be solved analytically for special cases [8], but here we opt for a more versatile approximation (cf. [7]). In this approximation, the probability of crossing the threshold, which is proportional to the instantaneous output rate of the neuron, is described by an effective transfer function. In [7] several transfer functions were compared in their performance, from which we choose an Arrheniustype function , Ao(t) = c exp{ _ (e ~(t))2}, cr (2) where e - x(t) is the distance in voltage between the noise free trajectory of the membrane potential x(t) and the threshold e, x(t) is calculated from eq. (1) without its diffusion term. Note that x(t) is a function of As, c is a constant. Figure 1 b) shows a family of Arrhenius type transfer functions for different noise levels cr. 3 Stochastic Resonance in an Ornstein-Uhlenbeck Neuron Several measures can be used to quantify the impact of noise on the quality of signal transmission through threshold devices. A natural choice is the mutual information [9] between the distributions p( As) and p( Ao) of input and output rates, which we will discuss in section 4, see also figure 3f. In order to keep the analysis and the derivation of the learning rule simple, however, we first consider a scenario, in which a neuron should distinguish between two sub-threshold input rates As and As + ~s. Optimal distinguishability is achieved if the difference ~o of the corresponding output rates is maximal, i.e. if ~ o = /(As + ~ s) - /(As) = max , (3) where / is the transfer function given by eq. (2). Obviously there is a close connection between these two measures, because increasing both of them leads to an increase in the entropy of p( Ao) . Fig. 2 shows plots of the difference ~ o of output rates vs. the level of noise, cr, for 0.16 ~s 0.4 ~s 0.14 0.35 0.12 AS= 7 :5 0.1 :5 0 0 <:::] 0.08 <:::] 0.06 0.04 0.02 0.05 00 50 100 50 100 0 2 [per cent] [per cent] Figure 2: ~ o vs. cr2 for two different base rates As = 2 (left) and 7 (right) and 10 different values of ~ s = 0.01 , 0.02, ... , 0.1. cr2 is given in per cent of the maximum cr2 = 2N W;An. The arrows above the x-axis indicate the position of the maximum according to eq. (3), the arrowheads below the x-axis indicate the optimal value computed using eq. (5) (67% and 25%). Parameters were: N = la, An = 7, Ws = 0.1 , and Wn E [0, 0.1]. different rates As and different values of ~ s . All curves show a clear maximum at a particular noise level. The optimal noise level increases with decreasing the input rate As, but is roughly independent of the difference ~ s as long as ~ s is small. Therefore, one optimal noise level holds even if a neuron has to distinguish several sub-threshold input rates - as long as these rates are clustered around a given base rate As. The optimal noise level for constant As (stationary states) is given by the condition d d(j2 (f(A s + ~ s) - f(As)) = 0 , (4) where f is given by eq. (2). Equation (4) can be evaluated in the limit of small values of ~ s using a Taylor expansion up to the second order. We obtain (j;pt = 2(1 - ws As)2 (5) if the main part of the variance of the membrane potential is a result of the balanced . "f 2 '" 2N 2 , (f (1)) S' 2 (1 - W, A, )2 (2) (5) mput , l.e. 1 (j '" WNAN c . eq. . mce (jopt - log(Ao/C) , eq. , eq. is equivalent to 1 + 2 log( Ao (A; ;0"2)) = O. This shows that the optimal noise level depends either only on As or on Ao (As; (j2), both are quantities which are locally available at the cell. 4 Adaptive Stochastic Resonance We now consider the case, that a neuron needs to adapt its internal noise level because the base input rate As changes. A simple learning rule which converges to the optimal noise level is given by (j2 ~(j2 = f log( - 2-) , (6) (j opt where the learning parameter f determines the time-scale of adaptation. Inserting the corresponding expressions for the actual and the optimal variance we obtain a learning rule for the weights W n , I ( 2NAnw; ) ~wn = -f og ( )2 . 2 1 - ws As (7) Note, that equivalent learning rules (in the sense of eq. (6)) can be formulated for the number N of the noise inputs and for their rates An as well. The r.h.s. of eqs. (6) and (7) depend only on quantities which are locally available at the neuron. Fig. 3ab shows the stochastic adaptation of the noise level, using eq. (7) , to randomly distributed As which are clustered around a base rate. Fig. 3c-f shows an application ofthe learning rule, eq. (7) to an Ornstein-Uhlenbeck neuron whose noise level needs to adapt to three different base input rates. T he figure shows the base input rate As (Fig. 3a). In fig. 3b the adaptation of Wn according to eq. (7) is shown (solid line), for comparison the Wn which maximizes eq. (3) is also displayed (dashed dotted line). Mutual information was calculated between a distribution of randomly chosen input rates which are clustered around the base rate As. The Wn that maximizes mutual Information between input and output rates is displayed in fig. 3d (dashed line). Fig. 3e shows the ratio ~ o / ~ s computed by using eq. (3) and the Wn calculated with eq. (8) (dashed dotted line) and the same ratio for the quadratic approximation. Fig. 3f shows the mutual information between the input and output rates as a function of the changing w n . I [n~[ :~--/ 0 • ~I 0 500 1000 1500 2000 2500 3000 0.15 o:i,ek ' I I,rJ b) 110 w n 0.1 I1S 0.05 0 500 1000 1500 2000 2500 3000 O'h • ri" 00 W 500 1000 1500 2000 2500 3000 n 0.1 d) .. 10 ~ a) 0 0 500 1000 1500 2000 2500 3000 AS As ':1 C) • I • I I 5 0 00 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 time [update steps] time [update steps 1 Figure 3: a) Input rates As are evenly distributed around a base rate with width 0.5, in each time step one As is presented. b) Application of the learning rule eq. (7) to the rates shown in a). Adaptation of the noise level to three different input base rates As. c) The three base rates As. d) Wn as a function of time according to eq. (7) (solid line) , the optimal Wn that maximizes eq. (3) (dashed dotted line) and the optimal Wn that maximizes the mutual information between the input and output rates (dashed). T he optimal values of Wn as the quadratic approximation, eq. (5) yield are indicated by the black arrows. e ) The ratio b.. o / b.. s computed from eq. (3) (dashed dotted line) and the quadratic approximation (solid line) . f) Mutual information between input and output rates as a function of base rate and changing synaptic coupling constant W n . For calculating the mutual information the input rates were chosen randomly from the interval [As - 0.25, As + 0.25] in each time step. Parameters as in fig . 2. T he figure shows, that the learning rule, eq. (7) in the quadratic approximation leads to values for () which are near-optimal, and that optimizing the difference of output rates leads to results similar to the optimization of the mutual information. 5 Conductance based Model Neuron To check if and how the results from the abstract model carryover to a biophysically mode realistic one we explore a modified Hodgkin-Huxley point neuron with an additional A-Current (a slow potassium current) as in [11] . T he dynamics of the membrane potential V is described by the following equation C~~ - gL(V(t) - EL) - !iNam~ h(t)(V - ENa) - !iKn(t)4(V - EK) - !iAa~ b(t)(V - EK) + l syn + lapp, (8) the parameters can be found in the appendix. All parameters are kept fixed througha) 'N 10 tS ~ peak o ~ 5 conductances ~ a 80,---------------------------------, II~ 70 b) "} 60 ?::l c 50 ::I 8 40 ~ 30 .~ .5 20 <t> U 10 ~ 00'----0 ,"" 2 =:::...0-.4""""'-=--,L---~--,~ ,2---" 1 .4 ~ 00 2 4 6 B 10 'i3 noiselevel in multiples of peak conductances Figure 4: a) Transfer function for the conductance based model neuron with additional balanced input, a = 1, 2, 3, 4 b ) Demonstration of SR for the conductance based model neuron. The plot shows the resonance for two different base currents lapp = 0.7 and lapp = 0.2 and a E [0, 10]. ~ -I ~ & '0 -~ E = .j 7 a) E3 5 43 :2 1 90 ,-------------------------------, -B gp ao ~ ~ 70 --&., ~ 60 b 50 ~ 40 1G _~30 ~ 20 :;:: -~ 10 0) 0 0 0.5 ~----------~~------~~~1 °0~----------0~.5~--~------~ I drift [n~] s"U-othresl""1<:>ld p<:>te:n.t:ial (8 = 1 ) Figure 5: a) Optimal noise-level as a function ofthe base current in the conductance based model. b) Optimal noise-level as a function of the noise-free membrane potential in the abstract model. out all shown data. As balanced input we choose an excitatory Poisson spike train with rate Ane = 1750 Hz and an inhibitory spike train with rate Ani = 750 Hz . These spike trains are coupled to the neuron via synapses resulting in a synaptic current as in [12] ls yn = ge(V(t) - Ee) + gi(V(t) - Ei)). (9) Every time a spike arrives at the synapse the conductance is increased by its peak conductance ge i and decreases afterwards exponentially like exp{ - _t_, } . The corI T e, t responding parameters are ge = a * 0.02 * gL , gi = a * 0.0615 * gL. The common factor a is varied in the simulations and adjusts the height of the peak conductances, gL is the leak conductance given above. Excitatory and inhibitory input are called balanced if the impact of a spike-train at threshold is the same for excitation and inhibition TegeAne(Ee - B) = - TigiAni(Ei - B) (10) with Te i = ~ fooo ge i(t)dt . Note that the factor a does cancel in eq . (10). I ge,t J! , Fig. 4a displays transfer functions in the conductance based setting with balanced input. A family of functions with varying peak conductances for the balanced input is drawn. ~ 100 r-----~----~----~----~----~-----. rJJ d) ~ 50 ~ °OL'~--~--~----~--~~--~--~ '-J 50 100 150 200 250 300 ,-..., i':k"-: - ..... :~ : '" 0 50 100 150 200 250 300 1~-:o 50 100 150 200 250 300 Figure 6: Adaptive SR in the conductance based model. a) Currents drawn from a uniform distribution of width 0.2 nA centered around base currents of 3, 8, 1 nA respectively. b) Optimal noise-level in terms of a. Optimality refers to a semi-linear fit to the data of fig. 5a. c) adapting the peak conductances using a in a learning rule like eg. (8). d) Difference in spike count, for base currents I ± 0.1 nA and using a as specified in c) . For studying SR in the conductance based framework , we apply the same paradigm as in the abstract model. Given a certain average membrane potential, which is adjusted via injecting a current I (in nA), we calculate the difference in the output rate given a certain difference in the average membrane potential (mediated via the injected current) I ± t:.I. A demonstration of stochastic resonance in the conductance based neuron can be seen in fig. 4b. In fig. 5a the optimal noise-level, in terms of multiples a of the peak conductances, is plotted versus all currents that yield a sub-threshold membrane voltage. For comparison we give the corresponding relationship for the abstract model in fig. 5b. Fig. 6 shows the performance of the conductance based model using a learning rule like eg. (7). Since we do not have an analytically derived expression for (J opt in the conductance based case, the relation (Jopt (I) , necessary for using eg. (7), corresponds to a semi-linear fit to the (a opt , I) relation in fig. 5a. 6 Conclusion and future directions In our contribution we have shown, that a simple and activity driven learning rule can be given for the adaptation of the optimal noise level in a stochastic resonance setting. The results from the abstract framework are compared with results from a conductance based model neuron. A biological plausible mechanism for implementing adaptive stochastic resonance in conductance based neurons is currently under investigation. Acknowledgments Supported by: Wellcome Trust (061113/Z/00) Appendix: Paramet ers for the conductance based model n euron somatic conductances/ion-channel properties: em = 1.0 :':2 ,gL = 0.05 ~ ,gNa = 100 ~,gJ( = 40 ~,gA = 20 ~,EL = -65 mV,ENa = 55 mV, EJ( - 80 mV, TA = 20 ms, synaptic coupling: Ee = 0 mV, Ei = -80 mV, Te = 5 ms, Ti = 10 ms, spike initiation: dh = ~ dn = ngo - n <jQ = ~ dt Th ' dt Tn' dt T A ' mCO = <>m<>-ti3m' O:m = -O.l(V + 30)/(exp( -O.l(V + 30)) - 1), f3m = 4exp( -(V + 55)/18), hoo = <>h":i3h' O:h = 0.07exp(-(V + 44)/20), f3h = l/(exp(-O. l(V + 14)) + 1) , n co = <>n+i3n ' O:n = -O.Ol(V + 34)/(exp( -O.l(V + 34)) -1) , f3n = 0.125exp( -(V + 44)/80) a oo = l/(exp( -(V + 50)/20) + 1) , boo = l/(exp((V + 80)/6) + 1), Th = in/(O:h + f3h) , Tn = in/(O:n + f3n), in = 0.1 References [1] S. B. Laughlin, RR de Ruyter van Steveninck and J.C. Anderson, The metabolic cost of neural information, Nature Neuroscience, 1(1), 1998, p.36-41 [2] J. S. Anderson, I. Lampl, D. C. Gillespie and D. Ferster, The Contribution of Noise to Contrast Invariance of Orientation Tuning in Cat Visual Cortex, Science, 290, 2000, p.1968-1972 [3] L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, Stochastic Resonance Reviews of modern Physics, 70(1) , 1998, p.223-287 [4] D. F. Russel, L. A. Wilkens and F. Moss, Use of behavioral stochastic resonance by paddle fish for feeding, Nature, 402, 1999, p.291-294 [5] M. N. Shadlen, and W. T. Newsome, The Variable Discharge of Cortical Neurons: Implications for Connectivity, Computation, and Information Coding, The Journal of Neuroscience, 18(10), 1998, p.3870-3896 [6] M.V. Tsodyks and T. Sejnowski, Rapid state switching in balanced cortical network models, Network: Computation in Neural Systems, 6, 1995, p.I11-124 [7] H. E. Plesser and W. Gerstner, Noise in Integrate-and-Fire Neurons: From Stochastic Input to Escape Rates, Neural Computation, 12, 2000, p.367-384 [8] H. C. Tuckwell, Introduction to theoretical neurobiology: volume 2 nonlinear and stochastic theories, Cambridge University Press, 1998 [9] T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley Series in Telecommunications, 1991, 2nd edition [10] A.R Bulsara, and A. Zador, Threshold detection of wide band signals: a noise-induced maximum in the mutual information., PRE, 54(3), 1996, R2185-2188 [11] O. Shriki, D. Hansel and H. Sompolinsky, Modeling neuronal networks in cortex by rate models using the current-frequency response properties of cortical cells, Soc. Neurosci. Abstr., 24, p.143, 1998 [12] E. Salinas and T .J. Sejnowski, Impa ct of Correlated Synaptic Input on Output Firing Rate and Variability in Simple Neuronal Models J. Neurosci. 20, 2000, p.6193-6209
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Reinforcement Learning Memory Bram Bakker Dept. of Psychology, Leiden University / IDSIA P.O. Box 9555; 2300 RB, Leiden; The Netherlands bbakker@fsw.leidenuniv.nl Abstract This paper presents reinforcement learning with a Long ShortTerm Memory recurrent neural network: RL-LSTM. Model-free RL-LSTM using Advantage(,x) learning and directed exploration can solve non-Markovian tasks with long-term dependencies between relevant events. This is demonstrated in a T-maze task, as well as in a difficult variation of the pole balancing task. 1 Introduction Reinforcement learning (RL) is a way of learning how to behave based on delayed reward signals [12]. Among the more important challenges for RL are tasks where part of the state of the environment is hidden from the agent. Such tasks are called non-Markovian tasks or Partially Observable Markov Decision Processes. Many real world tasks have this problem of hidden state. For instance, in a navigation task different positions in the environment may look the same, but one and the same action may lead to different next states or rewards. Thus, hidden state makes RL more realistic. However, it also makes it more difficult, because now the agent not only needs to learn the mapping from environmental states to actions, for optimal performance it usually needs to determine which environmental state it is in as well. Long-term dependencies. Most approaches to solving non-Markovian RL tasks have problems if there are long-term dependencies between relevant events. An example of a long-term dependency problem is a maze navigation task where the only way to distinguish between two T-junctions that look identical is to remember an observation or action a long time before either T-junction. Such a case prese~ts obvious problems for fixed size history window approaches [6], which attempt toresolve the hidden state by making the chosen action depend not only on the current observation, but also on a fixed number of the most recent observations and actions. If the relevant piece of information to be remembered falls outside the history window, the agent cannot use it. McCallum's variable history window [8] has, in principle, the capacity to represent long-term dependencies. However, the system starts with zero history and increases the depth of the history window step by step. This makes learning long-term dependencies difficult, especially when there are no short-term dependencies to build on. Other approaches to non-Markovian tasks are based on learning Finite State Automata [2], recurrent neural networks (RNNs) [10, 11, 6], or on learning to set memory bits [9]. Unlike history window approaches, they do not have to represent (possibly long) entire histories, but can in principle extract and represent just the relevant information for an arbitrary amount of time. However, learning to do that has proven difficult. The difficulty lies in discovering the correlation between a piece of information and the moment at which this information becomes relevant at a later time, given the distracting observations and actions between them. This difficulty can be viewed as an instance of the general problem of learning long-term dependencies in timeseries data. This paper uses one particular solution to this problem that has worked well in supervised timeseries learning tasks: Long ShortTerm Memory (LSTM) [5, 3]. In this paper an LSTM recurrent neural network is used in conjunction with model-free RL, in the same spirit as the model-free RNN approaches of [10,6]. The next section describes LSTM. Section 3 presents LSTM's combination with reinforcement learning in a system called RL-LSTM. Section 4 contains simulation results on non-Markovian RL tasks with long-term dependencies. Section 5, finally, presents the general conclusions. 2 LSTM LSTM is a recently proposed recurrent neural network architecture, originally designed for supervised timeseries learning [5, 3]. It is based on an analysis of the problems that conventional recurrent neural network learning algorithms, e.g. backpropagation through time (BPTT) and real-time recurrent learning (RTRL), have when learning timeseries with long-term dependencies. These problems boil down to the problem that errors propagated back in time tend to either vanish or blow up (see [5]). Memory cells. LSTM's solution to this problem is to enforce constant error flow in a number of specialized units, called Constant Error Carrousels (CECs). This actually corresponds to these CECs having linear activation functions which do not decay over time. In order to prevent the CECs from filling up with useless information from the timeseries, access to them is regulated using other specialized, multiplicative units, called input gates. Like the CECs, the input. gates receive input from the timeseries and the other units in the network, and they learn to open and close access to the CECs at appropriate moments. Access from the activations of the CECs to the output units (and possibly other units) of the network is regulated using multiplicative output gates. Similar to the input gates, the output gates learn when the time is right to send the information stored in the CECs to the output side of the network. A recent addition is forget gates [3], which learn to reset the activation of the CECs when the information stored in the CECs is no longer useful. The combination of a CEC with its associated input, output, and forget gate is called a memory cell. See figure 1b for a schematic of a memory cell. It is also possible for multiple CECs to be combined with only one input, output, and forget gate, in a so-called memory block. Activation updates. More formally, the network's activations at each timestep t are computed as follows. A standard hidden unit's activation yh, output unit activation yk, input gate activation yin, output gate activation y0'Ut, and forget gate activation yep is computed in the following standard way: (1) m where Wim is the weight of the connection from unit m to unit i. In this paper, Ii is the standard logistic sigmoid function for all units except output units, for which it is the identity function. The CEC activation Be"!, or the "state" of memory cell v :J cell output a. ,/' ,/' b. cell input memory cell Figure 1: a. The general LSTM architecture used in this paper. Arrows indicate unidirectional, fully connected weights. The network's output units directly code for the Advantages of different actions. b. One memory cell. in Illell10ry block j, is COITlputed as follows: (2) m where 9 is a logistic sigmoid function scaled to the range [-2,2]' and sc~(o) == o. 3 The memory cell's output ycj is calculated by ycj (t) == youtj (t)h(sc~ (t)) (3) 3 where h is a logistic sigmoid function scaled to the range [-1, IJ. Learning. At some or all timesteps of the timeseries, the output units of the network may make prediction' errors. Errors are propagated just one step back in time through all units other than the CEes, including the gates. However, errors are backpropagated through the CECs for an indefinite amount of time, using an efficient variation of RTRL [5, 3J. Weight updates are done at every timestep, which fits in nicely with the philosophy of online RL. The learning algorithm is adapted slightly for RL, as explained in the next section. 3 RL-LSTM RNNs, such as LSTM, can be applied to RL tasks in various ways. One way is to let the RNN learn a model of the environment, which learns to predict observations and rewards, and in this way learns to infer the environmental state at each point [6, IIJ. LSTM's architecture would allow the predictions to depend on information from long ago. The model-based system could then learn the mapping from (inferred) environmental states to actions as in the Markovian case, using standard techniques such as Q-learning [6, 2J, or by backpropagating through the frozen model to the controller [IIJ. An alternative, model-free approach, and the one used here, is to use the RNN to directly approximate the value function of a reinforcement learning algorithm [10, 6]. The state of the environment is approximated by the current observation, which is the input to the network, together with the recurrent activations in the network, which represent the agent's history. One possible advantage of such a model-free approach over a model-based approach is that the system may learn to only resolve hidden state insofar as that is useful for obtaining higher rewards, rather than waste time and resources in trying to predict features of the environment that are irrelevant for obtaining rewards [6, 8]. Advantage learning. In this paper, the RL-LSTM network approximates the value function of Advantage learning [4], which was designed as, an improvement on Q-Iearning for continuous-time RL. In continuous-time RL, values of adjacent states, and therefore optimal Q-values of different actions in a given state, typically differ by only small amounts, which can easily get lost in noise. Advantage learning remedies this problem by artificially decreasing the values of suboptimal actions in each state. Here Advantage learning is used for both continuous-time and discrete-time RL. Note that the same problem of small differences between values of adjacent states applies to any RL problem with long paths to rewards. And to demonstrate RLLSTM's potential to bridge long time lags, we need to consider such RL problems. In general, Advantage learning may be more suitable for non-Markovian tasks than Q-Iearning, because it seems less sensitive to getting the value estimations exactly right. The LSTM network's output units directly code for the Advantage values of different actions. Figure 1a shows the general network architecture used in this paper. As in Q-learning with a function approximator, the temporal difference error ETD(t), derived from the equivalent of the Bellman equation for Advantage learning [4], is taken as the function approximator's prediction error at timestep t: ETD(t) = V(s(t)) + r(t) +iV(S(t+ 1)) - V(s(t)) _ A(s(t),a(t)) (4) ~ where A(s, a) is the Advantage value of action a in state s, r is the immediate reward, and V(s) == maxa A(s, a) is the value of the state s. , is a discount factor in the range [0,1], and "" is a constant scaling the difference between values of optimal and suboptimal actions. Output units associated with other actions than the executed one do not receive error signals. Eligibility traces. In this work, Advantage learning is extended with eligibility traces, which have often been found to improve learning in RL, especially in non-Markovian domains [7]. This yields Advantage(A) learning, and the necessary computations turn out virtually the same as in Q(A)-learning [1]. It requires the storage of one eligibility trace eim per weight Wim. A weight update corresponds to ) TD) ) ayK(t) ) Wim(t+ 1 == Wim(t) +aE (t)eim(t, where eim(t == , Aeim(t-1) +-8--. (5 Wim K indicates the output unit associated with the executed action, a is a learning rate parameter, and Ais a parameter determining how fast the eligibility trace decays. eim(O) == 0, and eim(t - 1) is set to 0 if an exploratory action is taken. Exploration. Non-Markovian RL requires extra attention to the issue of exploration [2, 8]. Undirected exploration attempts to tryout actions in the same way in each environmental state. However, in non-Markovian tasks, th~ agent initially does not know which environmental state it is in. Part of the exploration must be aimed at discovering the environmental state structure. Furthermore, in many cases, the non-Markovian environment will provide unambiguous observations indicating the state in some parts, while providing ambiguous observations (hidden state) in other parts. In general, we want more exploration in the ambiguous parts. This paper employs a directed exploration technique based on these ideas. A separate multilayer feedforward neural network, with the same input as the LSTM network (representing the current observation) and one output unit yV, is trained concurrently with the LSTM network. It is trained, using standard backpropagation, to predict the absolute value of the current temporal difference error E TD (t) as defined byeq. 4, plus its own discounted prediction at the next timestep: Yd(t) == IETD (t)1 + (3yV(t + 1) (6) where Y'd(t) is the desired value for output yV(t), and (3 is a discount parameter in the range [0,1]. This amounts to attempting to identify which observations are G Figure 2: Long-term dependency T-maze with length of corridor N == 10. At the starting position S the agent's observation indicates where the goal position G is in this episode. "problematic" , in the sense that they are associated with large errors in the current value estimation (the first term), or precede situations with large such errors (the second term). yV(t) is linearly scaled and used as the temperature of a Boltzmann action selection rule [12]. The net result is much exploration when, for the current observation, differences between estimated Advantage values are small (the standard effect of Boltzmann exploration), or when there is much "uncertainty" about current Advantage values or Advantage values in the near future (the effect of this directed exploration scheme). This exploration technique has obvious similarities with the statistically more rigorous technique of Interval Estimation (see [12]), as well as with certain model-based approaches where exploration is greater when there is more uncertainty in the predictions of a model [11]. 4 Test problems Long-term dependency T-maze. The first test problem is a non-Markovian grid-based T-maze (see figure 2). It was designed to test RL-LSTM's capability to bridge long time lags, without confounding the results by making the control task difficult in other ways. The agent has four possible actions: move North, East, South, or West. The agent must learn to move from the starting position at the beginning of the corridor to the T-junction. There it must move either North or South to a changing goal position, which it cannot see. However, the location of the goal depends on a "road sign" the agent has seen at the starting position. If the agent takes the correct action at the T-junction, it receives a reward of 4. If it takes the wrong action, it receives a reward of -.1. In both cases, the episode ends and a new episode starts, with the new goal position set randomly either North or South. During the episode, the agent receives a reward of -.1 when it stands still. At the starting position, the observation is either 011 or 110, in the corridor the observation is 101, and at the T-junction the observation is 010. The length of the corridor N was systematically varied from 5 to 70. In each condition, 10 runs were performed. If the agent takes only optimal ac~ions to the T-junction, it must remember the observation from the starting position for N timesteps to determine the optimal action at the T-junction. Note that the agent is not aided by experiences in whiGh there are shorter time lag dependencies. In fact, the opposite is true~ Initially, it takes many more actions until even the T-junction is reached, and the experienced history is very variable from episode to episode. The agent must first learn to reliably move to the T-junction. Once this is accomplished, the agent will begin to experience more or less consistent and shortest possible histories of observations and actions, from which it can learn to extract the relevant piece of information. The directed exploration mechanism is crucial in this regard: it learns to set exploration low in the corridor and high at the T-junction. The LSTM network had 3 input units, 12 standard hidden units, 3 memory cells, and a == .0002. The following parameter values were used in all conditions: { == .98, ,\ == .8, flJ == .1. An empirical comparison was made with two alternative systems that have been used in non-Markovian tasks. The long-term dependency nature of the 5 10 15 20 25 30 40 50 60 70 N: lenQth of corridor en c::o ]1.5 o -* 1 ~ (]) OJ ~ 0.5 ~ 70 60 G---E) LSTM + - -+ Elman-BPTT ~ooooooX Memory bits ,, \ ,, \ \ "" 5 10 15 20 25 30 40 50 N: lenQth of corridor - 8 10 Figure 3: Results in noise-free T-maze task. Left: Number of successful runs (out of 10) as a function of N, length of the corridor. Right: Average number of timesteps until success as a function of N. 5 10 15 20 25 30 40 50 60 70 N: lenQth of corridor 70 \ \ \ "\ IG---E) LSTM , + - -+ Elman-BPTT \ \ ~ 00000 oX Memory bits ,, ~ ..+, \, \, "" 5 10 15 20 25 30 40 50 N: lenQth of corridor en 10 \ e 2 8 ~ 4 .0E ~ 2 Figure 4: Results in noisy T-maze task. Left: Number of successful runs (out of 10) as a function of N, length of the corridor. Right: Average number of timesteps until success as a function of N. task virtually rules out history window approaches. Instead, two alternative systems were used that, like LSTM, are capable in principle of representing information for arbitrary long time lags. In the first alternative, the LSTM network was replaced by an Elman-style Simple Recurrent Network, trained using BPTT [6]. Note that the unfolding of the RNN necessary for BPTT means that this is no longer truly online RL. The Elman network had 16 hidden units and 16 context units, and a == .001. The second alternative is a table-based system extended with memory bits that are part of the observation, and that the controller can switch on and off [9]. Because the task requires the agent to remember just one bit of information, this system had one memory bit, and a == .01. In order to determine the specific contribution of LSTM to performance, in both alternatives all elements of the overall system except LSTM (i.e. Advantage(A) learning, directed exploration) were left unchanged. A run was considered a success if the agent learned to take the correct action at the T-junction in over 80% of cases, using its stochastic action selection mechanism. In practice, this corresponds to 100% correct action choices at the T-junction using greedy action selection, as well as optimal or near-optimal action choices leading to the T-junction. Figure 3 shows the number of successful runs (out of 10) as a function of the length of the corridor N, for each of the three methods. It also shows the average number of timesteps needed to reach success. It is apparent that RL-LSTM is able to deal with much longer time lags than the two alternatives. RLLSTM has perfect performance up to N == 50, after which performance gradually decreases. In those cases where the alternatives also reach success, RL-LSTM also learns faster. The reason why the memory bits system performs worst is probably that, in contrast with the other two, it does not explicitly compute the gradient of performance with respect to past events. This should make credit assignment less directed and therefore less effective. The Elman-BPTT system does compute such a gradient, but in contrast to LSTM, the gradient information tends to vanish quickly with longer time lags (as explained in section 2). T-maze with noise. It is one thing to learn long-term dependencies in a noise-free task, it is quite another thing to do so in the presence of severe noise. To investigate this, a very noisy variation of the T-mazetask described above was designed. Now the observation in the corridor is aOb, where a and b are independent, uniformly distributed random values in the range [0, 1], generate online. All other aspects of the task remain the same as above. Both the LSTM and the Elman-BPTT system were also left unchanged. To allow for a fair comparison, the table-based memory bit system's observation was computed using Michie and Chambers's BOXES state aggregation mechanism (see [12]), partitioning each input dimension into three equal regions. Figure 4 shows the results. The memory bit system suffers most from the noise. This is not very surprising because a table-based system, even if augmented with BOXES state aggregation, does not give very sophisticated generalization. The two RNN approaches are hardly affected by the severe noise in the observations. Most importantly, RL-LSTM again significantly outperforms the others, both in terms of the maximum time lag it can deal with, and in terms of the number of timesteps needed to learn the task. Multi-mode pole balancing. The third test problem is less artificial than the T-mazes and has more complicated dynamics. It consists of a difficult variation of the classical pole balancing task. In the pole balancing task, an agent must balance an inherently unstable pole, hinged to the top of a wheeled cart that travels along a track, by applying left and right forces to the cart. Even in the Markovian version, the task requires fairly precise control to solve it. The version used in this experiment is made more difficult by two sources of hidden state. First, as in [6], the agent cannot observe the state information corresponding to the cart velocity and pole angular velocity. It has to learn to approximate this (continuous) information using its recurrent connections in order to solve the task. Second, the agent must learn to operate in two different modes. In mode A, action 1 is left push and action 2 is right push. In mode B, this is reversed: action 1 is right push and action 2 is left push. Modes are randomly set at the beginning of each episode. The information which mode the agent is operating in is provided to the agent only for the first second of the episode. After that, the corresponding input unit is set to zero and the agent must remember which mode it is in. Obviously, failing to remember the mode leads to very poor performance. The only reward signal is -1 if the pole falls past ±12° or if the cart hits either end of the track. Note that the agent must learn to remember the (discrete) mode information for an infinite amount of time if it is to learn to balance the pole indefinitely. This rules out history window approaches altogether. However, in contrast with the T-mazes, the system now has the benefit of starting with relatively short time lags. The LSTM network had 2 output units, 14 standard hidden units, and 6 memory cells. It has 3 input units: one each for cart position and pole angle; and one for the mode of operation, set to zero after one second of simulated time (50 timesteps). ry == .95, A == .6, fiJ == .2, a == .002. In this problem, directed exploration was not necessary, because in contrast to the T-mazes, imperfect policies lead to many different experiences with reward signals, and there is hidden state everywhere in the environment. For a continuous problem like this, a table-based memory bit system is not suited very well, so a comparison was only made with the ElmanBPTT system, which had 16 hidden and context units and a == .002. The Elman-BPTT system never reached satisfactory solutions in 10 runs. It only learned to balance the pole for the first 50 timesteps, when the mode information is available, thus failing to learn the long-term dependency. However, RL-LSTM learned optimal performance in 2 out of 10 runs (after an average of 6,250,000 timesteps of learning). After learning, these two agents were able to balance the pole indefinitely in both modes of operation. In the other 8 runs, the agents still learned to balance the pole in both modes for hundreds or even thousands of timesteps (after an average of 8,095,000 timesteps of learning), thus showing that the mode information was remembered for long time lags. In most cases, such an agent learns optimal performance for one mode, while achieving good but suboptimal performance in the other. 5 Conclusions The results presented in this paper suggest that reinforcement learning with Long Short-Term ~v1emory (RL-LSTI\,f) is a promising approach to solving non-:r-v1arkovi&t~ RL tasks with long-term dependencies. This was demonstrated in a T-maze task with minimal time lag dependencies of up to 70 timesteps, as well as in a nonMarkovian version of pole balancing where optimal performance requires remembering information indefinitely. RL-LSTM's main power is derived from LSTM's property of constant error flow, but for good performance in RL tasks, the combination with Advantage(A) learning and directed exploration was crucial. Acknowledgments The author wishes to thank Edwin de Jong, Michiel de Jong, Gwendid van der Voort van der Kleij, Patrick Hudson, Felix Gers, and Jiirgen Schmidhuber for valuable comments. References [1] B. Bakker. Reinforcement learning with LSTM in non-Markovian tasks with longterm dependencies. Technical report, Dept. of Psychology, Leiden University, 2001. [2] L. Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In Proc. of the 10th National Conf. on AI AAAI Press, 1992. [3] F. Gers, J. Schmidhuber, and F. Cummins. Learning to forget: Continual prediction with LSTM. Neural Computation, 12 (10):2451-2471, 2000. [4] M. E. Harmon and L. C. Baird. Multi-player residual advantage learning with general function approximation. Technical report, Wright-Patterson Air Force Base, 1996. [5] S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural Computation, 9 (8):1735-1780, 1997. [6] L.-J. Lin and T. Mitchell. Reinforcement learning with hidden states. In Proc. of the 2nd Int. Conf. on Simulation of Adaptive Behavior. MIT Press, 1993. [7] J. Loch and S. Singh. Using eligibility traces to find the best memoryless policy in Partially Observable Markov Decision Processes. In Proc. of ICML'98, 1998. [8] R. A. McCallum. Learning to use selective attention and short-term memory in sequential tasks. In Proc. 4th Int. Conf. on Simulation of Adaptive Behavior, 1996. [9] L. Peshkin, N. Meuleau, and L. P. Kaelbling. Learning policies with external memory. In Proc. of the 16th Int. Conf. on Machine Learning, 1999. [10] J. Schmidhuber. Networks adjusting networks. In Proc. of Distributed Adaptive Neural Information Processing, St. Augustin, 1990. [11] J. Schmidhuber. Curious model-building control systems. In Proc. of IJCNN'91, volume 2, pages 1458-1463, Singapore, 1991. [12] R. S. Sutton and A. G. Barto. Reinforcement learning: An introduction. MIT Press, Cambridge; MA, 1998.
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ADynamic HMM for On-line Segmentation of Sequential Data Jens Kohlmorgen* Fraunhofer FIRST.IDA Kekulestr. 7 12489 Berlin, Germany jek@first·fraunhofer.de Steven Lemm Fraunhofer FIRST.IDA Kekulestr. 7 12489 Berlin, Germany lemm@first·fraunhofer.de Abstract We propose a novel method for the analysis of sequential data that exhibits an inherent mode switching. In particular, the data might be a non-stationary time series from a dynamical system that switches between multiple operating modes. Unlike other approaches, our method processes the data incrementally and without any training of internal parameters. We use an HMM with a dynamically changing number of states and an on-line variant of the Viterbi algorithm that performs an unsupervised segmentation and classification of the data on-the-fly, i.e. the method is able to process incoming data in real-time. The main idea of the approach is to track and segment changes of the probability density of the data in a sliding window on the incoming data stream. The usefulness of the algorithm is demonstrated by an application to a switching dynamical system. 1 Introduction Abrupt changes can occur in many different real-world systems like, for example, in speech, in climatological or industrial processes, in financial markets, and also in physiological signals (EEG/MEG). Methods for the analysis of time-varying dynamical systems are therefore an important issue in many application areas. In [12], we introduced the annealed competition of experts method for time series from nonlinear switching dynamics, related approaches were presented, e.g., in [2, 6, 9, 14]. For a brief review of some of these models see [5], a good introduction is given in [3]. We here present a different approach in two respects. First, the segmentation does not depend on the predictability of the system. Instead, we merely estimate the density distribution of the data and track its changes. This is particularly an improvement for systems where data is hard to predict, like, for example, EEG recordings [7] or financial data. Second, it is an on-line method. An incoming data stream is processed incrementally while keeping the computational effort limited by a fixed • http://www.first.fraunhofer.de/..-.jek upper bound, i.e. the algorithm is able to perpetually segment and classify data streams with a fixed amount of memory and CPU resources. It is even possible to continuously monitor measured data in real-time, as long as the sampling rate is not too high.l The main reason for achieving a high on-line processing speed is the fact that the method, in contrast to the approaches above, does not involve any training, i.e. iterative adaptation of parameters. Instead, it optimizes the segmentation on-the-fly by means of dynamic programming [1], which thereby results in an automatic correction or fine-tuning of previously estimated segmentation bounds. 2 The segmentation algorithm We consider the problem of continuously segmenting a data stream on-line and simultaneously labeling the segments. The data stream is supposed to have a sequential or temporal structure as follows: it is supposed to consist of consecutive blocks of data in such a way that the data points in each block originate from the same underlying distribution. The segmentation task is to be performed in an unsupervised fashion, i.e. without any a-priori given labels or segmentation bounds. 2.1 Using pdfs as features for segmentation Consider Yl, Y2 , Y3, ... , with Yt E Rn, an incoming data stream to be analyzed. The sequence might have already passed a pre-processing step like filtering or subsampling, as long as this can be done on-the-fly in case of an on-line scenario. As a first step of further processing, it might then be useful to exploit an idea from dynamical systems theory and embed the data into a higher-dimensional space, which aims to reconstruct the state space of the underlying system, Xt = (Yt,Yt-n'" ,Yt-(m-l)r)' (1) The parameter m is called the embedding dimension and T is called the delay parameter of the embedding. The dimension of the vectors Xt thus is d = m n. The idea behind embedding is that the measured data might be a potentially non-linear projection of the systems state or phase space. In any case, an embedding in a higher-dimensional space might help to resolve structure in the data, a property which is exploited, e.g., in scatter plots. After the embedding step one might perform a sub-sampling of the embedded data in order to reduce the amount of data for real-time processing.2 Next, we want to track the density distribution of the embedded data and therefore estimate the probability density function (pdf) in a sliding window of length W. We use a standard density estimator with multivariate Gaussian kernels [4] for this purpose, centered on the data points3 in the window { ~ }W-l Xt-w w=o, () 1 ~l 1 ( (x - Xt_w)2) Pt x = W ~ (27fa2 )d/2 exp 2a2 . (2) The kernel width a is a smoothing parameter and its value is important to obtain a good representation of the underlying distribution. We propose to choose a proportional to the mean distance of each Xt to its first d nearest neighbors, averaged over a sample set {xt}. 1 In our reported application we can process data at 1000 Hz (450 Hz including display) on a 1.33 GHz PC in MATLAB/C under Linux, which we expect is sufficient for a large number of applications. 2In that case, our further notation of time indices would refer to the subsampled data. 3We use if to denote a specific vector-valued point and x to denote a vector-valued variable. 2.2 Similarity of two pdfs Once we have sampled enough data points to compute the first pdf according to eq. (2), we can compute a new pdf with each new incoming data point. In order to quantify the difference between two such functions, f and g, we use the squared L2-Norm, also called integrated squared error (ISE), d(f,g) = J(f - g)2 dx, which can be calculated analytically if f and 9 are mixtures of Gaussians as in our case of pdfs estimated from data windows, (3) 2.3 The HMM in the off-line case Before we can discuss the on-line variant, it is necessary to introduce the HMM and the respective off-line algorithm first. For a given a data sequence, {X'dT=l' we can obtain the corresponding sequence of pdfs {Pt(X)}tES, S = {W, ... , T}, according to eq. (2). We now construct a hidden Markov model (HMM) where each of these pdfs is represented by a state s E S, with S being the set of states in the HMM. For each state s, we define a continuous observation probability distribution, ( ( ) I ) 1 ( d(Ps(X),Pt(x))) PPt X s-~ exp 22 ' V 21f <; <; (4) for observing a pdf Pt(x) in state s. Next, the initial state distribution {1fsLES of the HMM is given by the uniform distribution, 1fs = liN, with N = lSI being the number of states. Thus, each state is a-priori equally probable. The HMM transition matrix, A = (PijkjES, determines each probability to switch from a state Si to a state Sj. Our aim is to find a representation of the given sequence of pdfs in terms of a sequence of a small number of representative pdfs, that we call prototypes, which moreover exhibits only a small number of prototype changes. We therefore define A in such a way that transitions to the same state are k times more likely than transitions to any of the other states, _ { k+~-l Pij 1 k+N- l ;ifi=J ;ifi-j.J (5) This completes the definition of our HMM. Note that this HMM has only two free parameters, k and <;. The well-known Viterbi algorithm [13] can now be applied to the above HMM in order to compute the optimal - i.e. the most likely - state sequence of prototype pdfs that might have generated the given sequence of pdfs. This state sequence represents the segmentation we are aiming at. We can compute the most likely state sequence more efficiently if we compute it in terms of costs, c = -log(p), instead of probabilities p, i.e. instead of computing the maximum of the likelihood function L, we compute the minimum of the cost function, -log(L), which yields the optimal state sequence as well. In this way we can replace products by sums and avoid numerical problems [13]. In addition to that, we can further simplify the computation for the special case of our particular HMM architecture, which finally results in the following algorithm: For each time step, t = w, ... , T, we compute for all S E S the cost cs(t) of the optimal state sequence from W to t, subject to the constraint that it ends in state S at time t. We call these constrained optimal sequences c-paths and the unconstrained optimum 0* -path. The iteration can be formulated as follows, with ds,t being a short hand for d(ps(x)'pt(x)) and bs,s denoting the Kronecker delta function: Initialization, Vs E S: Cs(W) := ds,w, (6) Induction, Vs E S: cs(t) := ds t + min { cs(t - 1) + C (1- bs s)}, for t = W + 1, ... , T, (7) , sES ' Termination: 0* := min { cs(T) } . sES (8) The regularization constant C, which is given by C = 2C;2 10g(k) and thus subsumes our two free HMM parameters, can be interpreted as transition cost for switching to a new state in the path.4 The optimal prototype sequence with minimal costs 0* for the complete series of pdfs, which is determined in the last step, is obtained by logging and updating the c-paths for all states s during the iteration and finally choosing the one with minimal costs according to eq. (8). 2.4 The on-line algorithm In order to turn the above segmentation algorithm into an on-line algorithm, we must restrict the incremental update in eq. (7), such that it only uses pdfs (and therewith states) from past data points. We neglect at this stage that memory and CPU resources are limited. Suppose that we have already processed data up to T - 1. When a new data point YT arrives at time T, we can generate a new embedded vector XT (once we have sampled enough initial data points for the embedding), we have a new pdf pT(X) (once we have sampled enough embedded vectors Xt for the first pdf window), and thus we have given a new HMM state. We can also readily compute the distances between the new pdf and all the previous pdfs, dT,t, t < T, according to eq. (3). A similarly simple and straightforward update of the costs, the c-paths and the optimal state sequence is only possible, however, if we neglect to consider potential c-paths that would have contained the new pdf as a prototype in previous segments. In that case we can simply reuse the c-paths from T - 1. The on-line update at time T for these restricted paths, that we henceforth denote with a tilde, can be performed as follows: For T = W, we have cw(W) := o*(W) := dw,w = O. For T > W: 1. Compute the cost cT(T - 1) for the new state s = T at time T - 1: For t = w, ... , T - 1, compute { 0 ift=W CT(t) :=dT,t+ min{cT(t-1); o*(t-1)+C}: else (9) and update o*(t) := CT(t), if CT(t) < o*(t). (10) Here we use all previous optimal segmentations o*(t), so we don't need to keep the complete matrix (cs(t))S,tES and repeatedly compute the minimum 4We developed an algorithm that computes an appropriate value for the hyperparameter C from a sample set {it}. Due to the limited space we will present that algorithm in a forthcoming publication [8]. over all states. However, we must store and update the history of optimal segmentations 8* (t). 2. Update from T - 1 to T and compute cs(T) for all states s E S obtained so far, and also get 8*(T): For s = W, ... , T , compute cs(T) := ds,T + min {cs(T - 1); 8*(T - 1) + C} (11) and finally get the cost of the optimal path 8* (T) := min {cs(T)} . sES (12) As for the off-line case, the above algorithm only shows the update equations for the costs of the C- and 8* -paths. The associated state sequences must be logged simultaneously during the computation. Note that this can be done by just storing the sequence of switching points for each path. Moreover, we do not need to keep the full matrix (cs(t))s,tES for the update, the most recent column is sufficient. So far we have presented the incremental version of the segmentation algorithm. This algorithm still needs an amount of memory and CPU time that is increasing with each new data point. In order to limit both resources to a fixed amount, we must remove old pdfs, i.e. old HMM states, at some point. We propose to do this by discarding all states with time indices smaller or equal to s each time the path associated with cs(T) in eq. (11) exhibits a switch back from a more recent state/pdf to the currently considered state s as a result of the min-operation in eq. (11). In the above algorithm this can simply be done by setting W := s + 1 in that case, which also allows us to discard the corresponding old cs(T)- and 8* (t)-paths, for all s::::: sand t < s. In addition, the "if t = W" initialization clause in eq. (9) must be ignored after the first such cut and the 8* (W - I)-path must therefore still be kept to compute the else-part also for t = W now. Moreover, we do not have CT(W -1) and we therefore assume min {CT(W - 1); 8*(W - 1) + C} = 8*(W - 1) + C (in eq. (9)). The explanation for this is as follows: A switch back in eq. (11) indicates that a new data distribution is established, such that the c-path that ends in a pdf state s from an old distribution routes its path through one of the more recent states that represent the new distribution, which means that this has lower costs despite of the incurred additional transition. Vice versa, a newly obtained pdf is unlikely to properly represent the previous mode then, which justifies our above assumption about CT (W -1). The effect of the proposed cut-off strategy is that we discard paths that end in pdfs from old modes but still allow to find the optimal pdf prototype within the current segment. Cut-off conditions occur shortly after mode changes in the data and cause the removal of HMM states with pdfs from old modes. However, if no mode change takes place in the incoming data sequence, no states will be discarded. We therefore still need to set a fixed upper limit", for the number of candidate paths/pdfs that are simultaneously under consideration if we only have limited resources available. When this limit is reached because no switches are detected, we must successively discard the oldest path/pdf stored, which finally might result in choosing a suboptimal prototype for that segment however. Ultimately, a continuous discarding even enforces a change of prototypes after 2", time steps if no switching is induced by the data until then. The buffer size", should therefore be as large as possible. In any case, the buffer overflow condition can be recorded along with the segmentation, which allows us to identify such artificial switchings. 2.5 The labeling algorithm A labeling algorithm is required to identify segments that represent the same underlying distribution and thus have similar pdf prototypes. The labeling algorithm generates labels for the segments and assigns identical labels to segments that are similar in this respect. To this end, we propose a relatively simple on-line clustering scheme for the prototypes, since we expect the prototypes obtained from the same underlying distribution to be already well-separated from the other prototypes as a result of the segmentation algorithm. We assign a new label to a segment if the distance of its associated prototype to all preceding prototypes exceeds a certain threshold e, and we assign the existing label of the closest preceding prototype otherwise. This can be written as l(R) = { ne.wlabel,. if min1:'Sr<R {d(Pt(r) (x), Pt(R) (x))} > e (13) 1 (mdexmml:'Sr<R {d(Pt(r) (x), Pt(R) (x))} ), else; with the initialization l(l) = newlabel. Here, r = 1, ... , R, denotes the enumeration of the segments obtained so far, and t(·) denotes the mapping to the index of the corresponding pdf prototype. Note that the segmentation algorithm might replace a number of recent pdf prototypes (and also recent segmentation bounds) during the on-line processing in order to optimize the segmentation each time new data is presented. Therefore, a relabeling of all segments that have changed is necessary in each update step of the labeler. As for the hyperparameters (J and C, we developed an algorithm that computes a suitable value for e from a sample set {X'd. We refer to our forthcoming publication [8]. 3 Application We illustrate our approach by an application to a time series from switching dynamics based on the Mackey-Glass delay differential equation, dx(t) = -O.lx(t) 0.2x(t - td) . dt + 1 + x( t - td)l° (14) Eq. (14) describes a high-dimensional chaotic system that was originally introduced as a model of blood cell regulation [10]. In our example, four stationary operating modes, A, B, C, and D, are established by using different delays, td = 17, 23, 30, and 35, respectively. The dynamics operates stationary in one mode for a certain number of time steps, which is chosen at random between 200 and 300 (referring to sub-sampled data with a step size 6. = 6). It then randomly switches to one of the other modes with uniform probability. This procedure is repeated 15 times, it thus generates a switching chaotic time series with 15 stationary segments. We then added a relatively large amount of "measurement" noise to the series: zero-mean Gaussian noise with a standard deviation of 30% of the standard deviation of the original series. The on-line segmentation algorithm was then applied to the noisy data, i.e. processing was performed on-line although the full data set was already available in this case. The scalar time series was embedded on-the-fly by using m = 6 and T = 1 (on the sub-sampled data) and we used a pdf window of size W = 50. The algorithm processed 457 data points per second on a 1.33 GHz PC in MATLAB/C under Linux, including the display of the ongoing segmentation, where one can observe the re-adaptation of past segmentation bounds and labels when new data becomes available. actual modes mode D modeC mode B mode A labels 1 2 3 4 3 561 3 3 6 2 2 bounds on-line segmentation xl!) Figure 1: Segmentation of a switching Mackey-Glass time series with noise (bottom) that operates in four different modes (top). The on-line segmentation algorithm (middle), which receives the data points one by one, but not the mode information, yields correct segmentation bounds almost everywhere. The on-line labeler, however, assigns more labels (6) than desired (4), presumably due to the fact that the segments are very short and noisy. The final segmentation is shown in Fig. 1. Surprisingly, the bounds of the segments are almost perfectly recovered from the very noisy data set. The only two exceptions are the third segment from the right, which is noticeably shorter than the original mode, and the segment in the middle, which is split in two by the algorithm. This split actually makes sense if one compares it with the data: there is a visible change in the signal characteristics at that point (t ~ 1500) even though the delay parameter was not modified there. This change is recorded by the algorithm since it operates in an unsupervised way. The on-line labeling algorithm correctly assigns single labels to modes A, B, and C, but it assigns three labels (4, 5, and 6) to the segments of mode D, the most chaotic one. This is probably due to the small sample sizes (of the segments), in combination with the large amount of noise in the data. 4 Discussion We presented an on-line method for the unsupervised segmentation and identification of sequential data, in particular from non-stationary switching dynamics. It is based on an HMM where the number of states varies dynamically as an effect of the way the incoming data is processed. In contrast to other approaches, it processes the data on-line and potentially even in real-time without training of any parameters. The method provides and updates a segmentation each time a new data point arrives. In effect, past segmentation bounds and labels are automatically re-adapted when new incoming data points are processed. The number of prototypes and labels that identify the segments is not fixed but determined by the algorithm. We expect useful applications of this method in fields where complex non-stationary dynamics plays an important role, like, e.g., in physiology (EEG, MEG), climatology, in industrial applications, or in finance. References [1] Bellman, R. E. (1957). Dynamic Programming, Princeton University Press, Princeton, N J. [2] Bengio, Y, Frasconi, P. (1995). An Input Output HMM Architecture. In: Advances in Neural Information Processing Systems 7 (eds. Tesauro, Touretzky, Leen), Morgan Kaufmann, 427- 434. [3] Bengio, Y (1999). Markovian Models for Sequential Data. Neural Computing Surveys, http://www.icsi.berkeley.edu/~jagota/NCS, 2:129-162. [4] Bishop, C. M. (1995). Neural Networks for Pattern Recognition, Oxford Univ. Press, NY. [5] Husmeier, D. (2000). Learning Non-Stationary Conditional Probability Distributions. Neural Networks 13, 287- 290. [6] Kehagias, A., Petridis, V. (1997). Time Series Segmentation using Predictive Modular Neural Networks. Neural Computation 9, 1691- 1710. [7] Kohlmorgen, J., Miiller, K.-R., Rittweger, J., Pawelzik, K. (2000). Identification of Nonstationary Dynamics in Physiological Recordings, Bioi Cybern 83(1),73- 84. [8] Kohlmorgen, J., Lemm, S., to appear. [9] Liehr, S., Pawelzik, K., Kohlmorgen, J., Miiller, K.-R. (1999). Hidden Markov Mixtures of Experts with an Application to EEG Recordings from Sleep. Theo Biosci 118, 246- 260. [10] Mackey, M., Glass, 1. (1977). Oscillation and Chaos in a Physiological Control System. Science 197, 287. [11] Packard, N. H., Crutchfield J. P., Farmer, J . D., Shaw, R. S. (1980). Geometry from a Time Series. Phys Rev Letters 45, 712- 716. [12] Pawelzik, K., Kohlmorgen, J., Miiller, K.-R. (1996). Annealed Competition of Experts for a Segmentation and Classification of Switching Dynamics. Neural Computation 8(2), 340- 356. [13] Rabiner, L. R. (1989). A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition, Proceedings of the IEEE 77(2), 257- 286. [14] Ramamurti, V., Ghosh, J. (1999). Structurally Adaptive Modular Networks for Non-Stationary Environments. IEEE Tr. Neural Networks 10(1), 152- 160.
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The Intelligent Surfer: Probabilistic Combination of Link and Content Information in PageRank Matthew Richardson Pedro Domingos Department of Computer Science and Engineering University of Washington Box 352350 Seattle, WA 98195-2350, USA {mattr, pedrod}@cs.washington.edu Abstract The PageRank algorithm, used in the Google search engine, greatly improves the results of Web search by taking into account the link structure of the Web. PageRank assigns to a page a score proportional to the number of times a random surfer would visit that page, if it surfed indefinitely from page to page, following all outlinks from a page with equal probability. We propose to improve PageRank by using a more intelligent surfer, one that is guided by a probabilistic model of the relevance of a page to a query. Efficient execution of our algorithm at query time is made possible by precomputing at crawl time (and thus once for all queries) the necessary terms. Experiments on two large subsets of the Web indicate that our algorithm significantly outperforms PageRank in the (human-rated) quality of the pages returned, while remaining efficient enough to be used in today’s large search engines. 1 Introduction Traditional information retrieval techniques can give poor results on the Web, with its vast scale and highly variable content quality. Recently, however, it was found that Web search results can be much improved by using the information contained in the link structure between pages. The two best-known algorithms which do this are HITS [1] and PageRank [2]. The latter is used in the highly successful Google search engine [3]. The heuristic underlying both of these approaches is that pages with many inlinks are more likely to be of high quality than pages with few inlinks, given that the author of a page will presumably include in it links to pages that s/he believes are of high quality. Given a query (set of words or other query terms), HITS invokes a traditional search engine to obtain a set of pages relevant to it, expands this set with its inlinks and outlinks, and then attempts to find two types of pages, hubs (pages that point to many pages of high quality) and authorities (pages of high quality). Because this computation is carried out at query time, it is not feasible for today’s search engines, which need to handle tens of millions of queries per day. In contrast, PageRank computes a single measure of quality for a page at crawl time. This meas ure is then combined with a traditional information retrieval score at query time. Compared with HITS, this has the advantage of much greater efficiency, but the disadvantage that the PageRank score of a page ignores whether or not the page is relevant to the query at hand. Traditional information retrieval measures like TFIDF [4] rate a document highly if the query terms occur frequently in it. PageRank rates a page highly if it is at the center of a large sub-web (i.e., if many pages point to it, many other pages point to those, etc.). Intuitively, however, the best pages should be those that are at the center of a large sub-web relevant to the query. If one issues a query containing the word jaguar, then pages containing the word jaguar that are also pointed to by many other pages containing jaguar are more likely to be good choices than pages that contain jaguar but have no inlinks from pages containing it. This paper proposes a search algorithm that formalizes this intuition while, like PageRank, doing most of its computations at crawl time. The PageRank score of a page can be viewed as the rate at which a surfer would visit that page, if it surfed the Web indefinitely, blindly jumping from page to page. Our algorithm does something closer to what a human surfer would do, jumping preferentially to pages containing the query terms. A problem common to both PageRank and HITS is topic drift. Because they give the same weight to all edges, the pages with the most inlinks in the network being considered (either at crawl or query time) tend to dominate, whether or not they are the most relevant to the query. Chakrabarti et al. [5] and Bharat and Henzinger [6] propose heuristic methods for differentially weighting links. Our algorithm can be viewed as a more principled approach to the same problem. It can also be viewed as an analog for PageRank of Cohn and Hofmann’s [7] variation of HITS. Rafiei and Mendelzon's [8] algorithm, which biases PageRank towards pages containing a specific word, is a predecessor of our work. Haveliwala [9] proposes applying an optimized version of PageRank to the subset of pages containing the query terms, and suggests that users do this on their own machines. We first describe PageRank. We then introduce our query-dependent, contentsensitive version of PageRank, and demonstrate how it can be implemented efficiently. Finally, we present and discuss experimental results. 2 PageRank : The Random Surfer Imagine a web surfer who jumps from web page to web page, choosing with uniform probability which link to follow at each step. In order to reduce the effect of deadends or endless cycles the surfer will occasionally jump to a random page with some small probability β, or when on a page with no out-links. To reformulate this in graph terms, consider the web as a directed graph, where nodes represent web pages, and edges between nodes represent links between web pages. Let W be the set of nodes, N=|W|, Fi be the set of pages page i links to, and Bi be the set pages which link to page i. For pages which have no outlinks we add a link to all pages in the graph1. In this way, rank which is lost due to pages with no outlinks is redistributed uniformly to all pages. If averaged over a sufficient number of steps, the probability the surfer is on page j at some point in time is given by the formula: ∑ ∈ + − = j B i iF i P N j P ) ( ) 1( ) ( β β (1) 1 For each page s with no outlinks, we set Fs={all N nodes}, and for all other nodes augment Bi with s. (Bi ∪ {s}) The PageRank score for node j is defined as this probability: PR(j)=P(j). Because equation (1) is recursive, it must be iteratively evaluated until P(j) converges. Typically, the initial distribution for P(j) is uniform. PageRank is equivalent to the primary eigenvector of the transition matrix Z: M N Z NxN β β + − = 1 ) 1( ,with = otherwise 0 to from edge an is there if 1 j i F M i ji (2) One iteration of equation (1) is equivalent to computing xt+1=Zxt, where xj t=P(j) at iteration t. After convergence, we have xT+1=xT, or xT=ZxT, which means xT is an eigenvector of Z. Furthermore, since the columns of Z are normalized, x has an eigenvalue of 1. 3 Directed Surfer Model We propose a more intelligent surfer, who probabilistically hops from page to page, depending on the content of the pages and the query terms the surfer is looking for. The resulting probability distribution over pages is: ∑ ∈ → + ′ − = j B i q q q q j i P i P j P j P ) ( ) ( ) ( ) 1( ) ( β β (3) where Pq(i→j) is the probability that the surfer transitions to page j given that he is on page i and is searching for the query q. Pq’(j) specifies where the surfer chooses to jump when not following links. Pq(j) is the resulting probability distribution over pages and corresponds to the query-dependent PageRank score (QD-PageRankq(j) ≡ Pq(j)). As with PageRank, QD-PageRank is determined by iterative evaluation of equation 3 from some initial distribution, and is equivalent to the primary eigenvector of the transition matrix Zq, where ∑ ∈ → + ′ − = j ji B i q q q j i P j P Z ) ( ) ( ) 1( β β . Although Pq(i→j) and Pq’(j) are arbitrary distributions, we will focus on the case where both probability distributions are derived from Rq(j), a measure of relevance of page j to query q: ∑ ∈ = ′ W k q q q k R j R j P ) ( ) ( ) ( ∑ ∈ = → iF k q q q k R j R j i P ) ( ) ( ) ( (4) In other words, when choosing among multiple out-links from a page, the directed surfer tends to follow those which lead to pages whose content has been deemed relevant to the query (according to Rq). Similarly to PageRank, when a page’s outlinks all have zero relevance, or has no outlinks, we add links from that page to all other pages in the network. On such a page, the surfer thus chooses a new page to jump to according to the distribution Pq’(j). When given a multiple-term query, Q={q1,q2,…}, the surfer selects a q according to some probability distribution, P(q) and uses that term to guide its behavior (according to equation 3) for a large number of steps1. It then selects another term according to the distribution to determine its behavior, and so on. The resulting distribution over visited web pages is QD-PageRankQ and is given by 1 However many steps are needed to reach convergence of equation 3. ∑ ∈ = ≡ − Q q q Q j P q P j P j ) ( ) ( ) ( ) ( PageRank QD Q (5) For standard PageRank, the PageRank vector is equivalent to the primary eigenvector of the matrix Z. The vector of single-term QD-PageRankq is again equivalent to the primary eigenvector of the matrix Zq. An interesting question that arises is whether the QD-PageRankQ vector is equivalent to the primary eigenvector of a matrix ∑ ∈ = Q q q Q q P Z Z ) ( (corresponding to the combination performed by equation 5). In fact, this is not the case. Instead, the primary eigenvector of ZQ corresponds to the QD-PageRank obtained by a random surfer who, at each step, selects a new query according to the distribution P(q). However, QD-PageRankQ is approximately equal to the PageRank that results from this single-step surfer, for the following reason. Let xq be the L2-normalized primary eigenvector for matrix Zq (note element j of xq is QD-PageRankq(j)), thus satisfying xi=Tixi. Since xq is the primary eigenvector for Zq, we have [10]: r q q q Q r q x Z x Z ≥ ∈ ∀ : , . Thus, to a first degree of approximation, q q Q r r q x Z x Z κ ≈ ∑ ∈ . Suppose P(q)=1/|Q|. Consider ∑ ∈ = Q q q Q q P x x ) ( (see equation 5). Then ( ) Q Q q q Q q q q Q q Q r r q Q q q Q q q Q Q n Q Q Q Q x x x Z x Z x Z x Z κ κ κ ∑ ∑ ∑ ∑ ∑ ∑ ∈ ∈ ∈ ∈ ∈ ∈ = = ≈ = = 1 1 1 and thus xQ is approximately an eigenvector for ZQ. Since xQ is equivalent to QDPageRankQ, and ZQ describes the behavior of the single-step surfer, QD-PageRankQ is approximately the same PageRank that would be obtained by using the single-step surfer. The approximation has the least error when the individual random surfers defined by Zq are very similar, or are very dissimilar. The choice of relevance function Rq(j) is arbitrary. In the simplest case, Rq(j)=R is independent of the query term and the document, and QD-PageRank reduces to PageRank. One simple content-dependent function could be Rq(j)=1 if the term q appears on page j, and 0 otherwise. Much more complex functions could be used, such as the well-known TFIDF information retrieval metric, a score obtained by latent semantic indexing, or any heuristic measure using text size, positioning, etc…. It is important to note that most current text ranking functions could be easily incorporated into the directed surfer model. 4 Scalability The difficulty with calculating a query dependent PageRank is that a search engine cannot perform the computation, which can take hours, at query time, when it is expected to return results in seconds (or less). We surmount this problem by precomputing the individual term rankings QD-PageRankq, and combining them at query time according to equation 5. We show that the computation and storage requirements for QD-PageRankq for hundreds of thousands of words is only approximately 100-200 times that of a single query independent PageRank. Let W={q1, q2, …, qm} be the set of words in our lexicon. That is, we assume all search queries contain terms in W, or we are willing to use plain PageRank for those terms not in W. Let dq be the number of documents which contain the term q. Then ∑ ∈ = W q q d S is the number of unique document-term pairs. 4.1 Disk Storage For each term q, we must store the results of the computation. We add the minor restriction that a search query will only return documents containing all of the terms1. Thus, when merging QD-PageRankq’s, we need only to know the QD-PageRankq for documents that contain the term. Each QD-PageRankq is a vector of dq values. Thus, the space required to store all of the PageRanks is S, a factor of S/N times the query independent PageRank alone (recall N is the number of web pages). Further, note that the storage space is still considerably less than that required for the search engine’s reverse index, which must store information about all document-term pairs, as opposed to our need to store information about every unique document term pair. 4.2 Time Requirements If Rq(j)=0 for some document j, the directed surfer will never arrive at that page. In this case, we know QD-PageRankq(j)=0, and thus when calculating QD-PageRankq, we need only consider the subset of nodes for which Rq(j)>0. We add the reasonable constraint that Rq(j)=0 if term q does not appear in document j, which is common for most information retrieval relevance metrics, such as TFIDF. The computation for term q then only needs to consider dq documents. Because it is proportional to the number of documents in the graph, the computation of QD-PageRankq for all q in W will require O(S) time, a factor of S/N times the computation of the query independent PageRank alone. Furthermore, we have noticed in our experiments that the computation converges in fewer iterations on these smaller sub-graphs, empirically reducing the computational requirements to 0.75*S/N. Additional speedup may be derived from the fact that for most words, the sub-graph will completely fit in memory, unlike PageRank which (for any large corpus) must repeatedly read the graph structure from disk during computation. 4.3 Empirical Scalability The fraction S/N is critical to determining the scalability of QD-PageRank. If every document contained vastly different words, S/N would be proportional to the number of search terms, m. However, this is not the case. Instead, there are a very few words that are found in almost every document, and many words which are found in very few documents2; in both cases the contribution to S is small. In our database of 1.7 million pages (see section 5), we let W be the set of all unique words, and removed the 100 most common words3. This results in |W|=2.3 million words, and the ratio S/N was found to be 165. We expect that this ratio will remain relatively constant even for much larger sets of web pages. This means QDPageRank requires approximately 165 times the storage space and 124 times the computation time to allow for arbitrary queries over any of the 2.3 million words (which is still less storage space than is required by the search engine’s reverse index alone). 1 Google has this “feature” as well. See http://www.google.com/technology/whyuse.html. 2 This is because the distribution of words in text tends to follow an inverse power law [11]. We also verified experimentally that the same holds true for the distribution of the number of documents a word is found in. 3 It is common to remove “stop” words such as the, is, etc., as they do not affect the search. 5 Results We give results on two data sets: educrawl, and WebBase. Educrawl is a crawl of the web, restricted to .edu domains. The crawler was seeded with the first 18 results of a search for “University” on Google (www.google.com). Links containing “?” or “cgibin” were ignored, and links were only followed if they ended with “.html”. The crawl contains 1.76 million pages over 32,000 different domains. WebBase is the first 15 million pages of the Stanford WebBase repository [12], which contains over 120 million pages. For both datasets, HTML tags were removed before processing. We calculated QD-PageRank as described above, using Rq(j) = the fraction of words equal to q in page j, and P(q)=1/|Q|. We compare our algorithm to the standard PageRank algorithm. For content ranking, we used the same Rq(j) function as for QDPageRank, but, similarly to TFIDF, weighted the contribution of each search term by the log of its inverse document frequency. As there is nothing published about merging PageRank and content rank into one list, the approach we follow is to normalize the two scores and add them. This implicitly assumes that PageRank and content rank are equally important. This resulted in poor PageRank performance, which we found was because the distribution of PageRanks is much more skewed than the distribution of content ranks; normalizing the vectors resulted in PageRank primarily determining the final ranking. To correct this problem, we scaled each vector to have the same average value in its top ten terms before adding the two vectors. This drastically improved PageRank. For educrawl, we requested a single word and two double word search queries from each of three volunteers, resulting in a total of nine queries. For each query, we randomly mixed the top 10 results from standard PageRank with the top 10 results from QD-PageRank, and gave them to four volunteers, who were asked to rate each search result as a 0 (not relevant), 1 (somewhat relevant, not very good), or 2 (good search result) based on the contents of the page it pointed to. In Table 1, we present the final rating for each method, per query. This rating was obtained by first summing the ratings for the ten pages from each method for each volunteer, and then averaging the individual ratings. A similar experiment for WebBase is given in Table 2. For WebBase, we randomly selected the queries from Bharat and Henzinger [6]. The four volunteers for the WebBase evaluation were independent from the four for the educrawl evaluation, and none knew how the pages they were asked to rate were obtained. Table 1: Results on educrawl Table 2: Results on WebBase Query QD-PR PR Query QD-PR PR chinese association 10.75 6.50 alcoholism 11.50 11.88 computer labs 9.50 13.25 architecture 8.45 2.93 financial aid 8.00 12.38 bicycling 8.45 6.88 intramural 16.5 10.25 rock climbing 8.43 5.75 maternity 12.5 6.75 shakespeare 11.53 5.03 president office 5.00 11.38 stamp collecting 9.13 10.68 sororities 13.75 7.38 vintage car 13.15 8.68 student housing 14.13 10.75 Thailand tourism 16.90 9.75 visitor visa 19.25 12.50 Zen Buddhism 8.63 10.38 Average 12.15 10.13 Average 10.68 7.99 QD-PageRank performs better than PageRank, accomplishing a relative improvement in relevance of 20% on educrawl and 34% on WebBase. The results are statistically significant (p<.03 for educrawl and p<.001 for WebBase using a two-tailed paired ttest, one sample per person per query). Averaging over queries, every volunteer found QD-PageRank to be an improvement over PageRank, though not all differences were statistically significant. One item to note is that the results on multiple word queries are not as positive as the results on single word queries. As discussed in section 3, the combination of single word QD-PageRanks to calculate the QD-PageRank for a multiple word query is only an approximation, made for practical reasons. This approximation is worse when the words are highly dependent. Further, some queries, such as “financial aid” have a different intended meaning as a phrase than simply the two words “financial” and “aid”. For queries such as these, the words are highly dependent. We could partially overcome this difficulty by adding the most common phrases to the lexicon, thus treating them the same as single words. 6 Conclusions In this paper, we introduced a model that probabilistically combines page content and link structure in the form of an intelligent random surfer. The model can accommodate essentially any query relevance function in use today, and produces higherquality results than PageRank, while having time and storage requirements that are within reason for today’s large scale search engines. Acknowledgments We would like to thank Gary Wesley and Taher Haveliwala for their help with WebBase, Frank McSherry for eigen-help, and our experiment volunteers for their time. This work was partially supported by NSF CAREER and IBM Faculty awards to the second author. References [1] J. M. Kleinberg (1998). Authoritative sources in a hyperlinked environment. Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms. [2] L. Page, S. Brin, R. Motwani, and T. Winograd (1998). The PageRank citation ranking: Bringing order to the web. Technical report, Stanford University, Stanford, CA. [3] S. Brin and L. Page (1998). The anatomy of a large-scale hypertextual Web search engine. Proceedings of the Seventh International World Wide Web Conference. [4] G. Salton and M. J. McGill (1983). Introduction to Modern Information Retrieval. McGraw-Hill, New York, NY. [5] S. Chakrabarti, B. Dom, D. Gibson, J. Kleinberg, P. Raghavan, and S. Rajagopalan (1998). Automatic resource compilation by analyzing hyperlink structure and associated text. Proceedings of the Seventh International World Wide Web Conference. [6] K. Bharat and M. R. Henzinger (1998). Improved algorithms for topic distillation in a hyperlinked environment. Proceedings of the Twenty-First Annual International ACM SIGIR Conference on Research and Development in Information Retrieval. [7] D. Cohn and T. Hofmann (2001). The missing link - a probabilistic model of document content and hypertext connectivity. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13. MIT Press, Cambridge, MA. [8] D. Rafiei and A. Mendelzon (2000). What is this page known for? Computing web page reputations. Proceedings of the Ninth International World Wide Web Conference. [9] T. Haveliwala (1999). Efficient computation of PageRank. Technical report, Stanford University, Stanford, CA. [10] G. H. Golub and C. F. Van Loan (1996). Matrix Computations. Johns Hopkins University Press, Baltimore, MD, third edition. [11] G. K. Zipf (1949). Human Behavior and the Principle of Least Effort. Addison-Wesley, Cambridge, MA. [12] J. Hirai, S. Raghaven, H. Garcia-Molina, A. Paepcke (1999). WebBase: a repository of web pages. Proceedings of the Ninth World Wide Web Conference.
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Incremental A S. Koenig and M. Likhachev Georgia Institute of Technology College of Computing Atlanta, GA 30312-0280 skoenig, mlikhach @cc.gatech.edu Abstract Incremental search techniques find optimal solutions to series of similar search tasks much faster than is possible by solving each search task from scratch. While researchers have developed incremental versions of uninformed search methods, we develop an incremental version of A*. The first search of Lifelong Planning A* is the same as that of A* but all subsequent searches are much faster because it reuses those parts of the previous search tree that are identical to the new search tree. We then present experimental results that demonstrate the advantages of Lifelong Planning A* for simple route planning tasks. 1 Overview Artificial intelligence has investigated knowledge-based search techniques that allow one to solve search tasks in large domains. Most of the research on these methods has studied how to solve one-shot search problems. However, search is often a repetitive process, where one needs to solve a series of similar search tasks, for example, because the actual situation turns out to be slightly different from the one initially assumed or because the situation changes over time. An example for route planning tasks are changing traffic conditions. Thus, one needs to replan for the new situation, for example if one always wants to display the least time-consuming route from the airport to the conference center on a web page. In these situations, most search methods replan from scratch, that is, solve the search problems independently. Incremental search techniques share with case-based planning, plan adaptation, repair-based planning, and learning search-control knowledge the property that they find solutions to series of similar search tasks much faster than is possible by solving each search task from scratch. Incremental search techniques, however, differ from the other techniques in that the quality of their solutions is guaranteed to be as good as the quality of the solutions obtained by replanning from scratch. Although incremental search methods are not widely known in artificial intelligence and control, different researchers have developed incremental search versions of uninformed search methods in the algorithms literature. An overview can be found in [FMSN00]. We, on the other hand, develop an incremental version of A*, thus combining ideas from the algorithms literature and the artificial intelligence literature. We call the algorithm Lifelong Planning A* (LPA*), in analogy to “lifelong learning” [Thr98], because it reuses We thank Anthony Stentz for his support. The Intelligent Decision-Making Group is partly supported by NSF awards under contracts IIS9984827, IIS-0098807, and ITR/AP-0113881. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the sponsoring organizations and agencies or the U.S. government. information from previous searches. LPA* uses heuristics to focus the search and always finds a shortest path for the current edge costs. The first search of LPA* is the same as that of A* but all subsequent searches are much faster. LPA* produces at least the search tree that A* builds. However, it achieves a substantial speedup over A* because it reuses those parts of the previous search tree that are identical to the new search tree. 2 The Route Planning Task Lifelong Planning A* (LPA*) solves the following search task: It applies to finite graph search problems on known graphs whose edge costs can increase or decrease over time. denotes the finite set of vertices of the graph.
denotes the set of successors of vertex . Similarly,
denotes the set of predecessors of vertex . !"# %$'& denotes the cost of moving from vertex to vertex "() *+, . LPA* always determines a shortest path from a given start vertex ,-/.10324.56 to a given goal vertex 879 0;: , knowing both the topology of the graph and the current edge costs. We use <>=, to denote the start distance of vertex ?@ , that is, the length of a shortest path from -/.102A. to . To motivate and test LPA*, we use a special case of these search tasks that is easy to visualize. We apply LPA* to navigation problems in known eight-connected gridworlds with cells whose traversability can change over time. They are either traversable (with cost one) or untraversable. LPA* always determines a shortest path between two given cells of the gridworld, knowing both the topology of the gridworld and which cells are currently blocked. This is a special case of the graph search problems on eight-connectedgrids whose edge costs are either one or infinity. As an approximation of the distance between two cells, we use the maximum of the absolute differences of their x and y coordinates. This results in consistent heuristics that are for eight-connected grids what Manhattan distances are for four-connected grids. 3 Reusing Information from Previous Searches The graph search problems can be solved with traditional graph-search methods, such as breadth-first search, if they update the shortest path every time some edge costs change. They typically do not take advantage of information from previous searches. The following example, however, shows that this can be advantageous. Consider the gridworlds of size B 5CED F shown in Figure 1. The original gridworld is shown on top and the changed gridworld is shown at the bottom. The traversability of only a few cells has changed. In particular, three blocked cells became traversable (namely, B3, C5, and D2) and three traversable cells became blocked (namely, A1, A4, D3). Thus, two percent of the cells changed their status but the obstacle density remained the same. The figure shows the shortest paths in both cases, breaking ties towards the north. Note that we assume that one can squeeze through diagonal obstacles. (This is just an artifact of how we generated the underlying graphs from the mazes.) The shortest path changed since one cell on the original shortest path became blocked. Once the start distances of all cells are known, one can easily trace back a shortest path from the start cell to the goal cell by always greedily decreasing the start distance, starting at the goal cell. This is similar to how A* traces the shortest path back from 79 03: to -/.10324. using the search tree it has constructed. Thus, we only need to determine the start distances. The start distances are shown in each traversable cell of the original and changed gridworlds. Those cells whose start distances in the changed gridworld have changed from the corresponding ones in the original gridworld are shaded gray. There are two different ways of decreasing the search effort for determining the start distances for the changed gridworld. First, some start distances have not changed and thus need not get recomputed. This is what DynamicSWSF-FP [RR96] does. (DynamicSWSFOriginal Eight-Connected Gridworld 1 1 2 2 2 3 3 3 3 3 4 4 4 4 3 5 4 5 4 5 5 5 3 5 6 4 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 5 5 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 12 11 11 11 11 11 11 11 11 12 12 12 12 12 13 14 11 11 13 13 14 14 14 14 14 14 12 12 15 15 15 15 15 15 15 15 13 13 16 16 16 16 16 16 14 14 14 14 15 16 15 15 15 15 16 16 16 16 1 2 3 4 5 6 7 8 9 10 10 11 2 8 1 3 2 4 5 6 7 11 9 10 sstart sgoal 12 8 A B C D 1 2 3 4 5 Changed Eight-Connected Gridworld 1 1 2 2 2 3 3 3 3 3 3 2 4 4 4 4 4 4 5 5 5 5 5 5 5 3 5 5 6 6 4 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 5 5 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 12 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 13 13 14 14 14 13 13 14 14 14 14 14 14 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 17 17 17 17 17 18 18 18 18 18 18 19 19 19 19 1 7 8 11 1 7 8 sstart sgoal 2 3 4 5 6 9 10 11 12 Figure 1: Simple Gridworld FP, as originally stated, searches from the goal vertex to the start vertex and thus maintains estimates of the goal distances rather than the start distances. It is a simple matter of restating it to search from the start vertex to the goal vertex. Furthermore, DynamicSWSFFP, as originally stated, recomputes all goal distances that have changed. To avoid biasing our experimental results in favor of LPA*, we changed the termination condition of DynamicSWSF-FP so that it stops immediately after it is sure that it has found a shortest path.) Second, heuristic knowledge, in form of approximations of the goal distances, can be used to focus the search and determine that some start distances need not get computed at all. This is what A* [Pea85] does. We demonstrate that the two ways of decreasing the search effort are orthogonal by developing LPA* that combines both of them and thus is able to replan faster than either DynamicSWSF-FP or A*. Figure 2 shows in gray those cells whose start distances each of the four algorithms recomputes. (To be precise: it shows in gray the cells that each of the four algorithms expands.) During the search in the original gridworld, DynamicSWSF-FP computes the same start distances as breadth-first search during the first search and LPA* computes the same start distances as A*. During the search in the changed gridworld, however, both incremental search (DynamicSWSF-FP) and heuristic search (A*) individually decrease the number of start distances that need to get recomputed compared to breadth-first search, and together (LPA*) decrease the number even more. 4 Lifelong Planning A* Lifelong Planning A* (LPA*) is an incremental version of A* that uses heuristics to control its search. As for A*, the heuristics approximate the goal distances of the vertices . They need to be consistent, that is, satisfy 79 03: and $E, ; " " for all vertices 5 and " 5 + with 79 0;: . LPA* maintains an estimate < of the start distance <+= of each vertex . These values directly correspond to the g-values of an A* search. They are carried forward from search to search. LPA* also maintains a second kind of estimate of the start distances. The rhs-values are one-step lookahead values based on the g-values and thus potentially better informed Original Eight-Connected Gridworld sstart sstart sstart sstart uninformed search heuristic search breadth-first search A* DynamicSWSF-FP (with early termination) Lifelong Planning A* incremental search complete search sgoal sgoal sgoal sgoal Changed Eight-Connected Gridworld uninformed search heuristic search breadth-first search A* DynamicSWSF-FP (with early termination) Lifelong Planning A* incremental search complete search sstart sstart sstart sgoal sstart sgoal sgoal sgoal Figure 2: Performance of Search Methods in the Simple Gridworld than the g-values. They always satisfy the following relationship:
if "!$#%'&)( * ,+ .-%/021435/6 , otherwise. (1) A vertex is called locally consistent iff its g-value equals its rhs-value. This is similar to satisfying the Bellman equation for undiscounted deterministic sequential decision problems. Thus, this concept is important because the g-values of all vertices equal their start distances iff all vertices are locally consistent. However, LPA* does not make every vertex locally consistent. Instead, it uses the heuristics to focus the search and update only the g-values that are relevant for computing a shortest path from ,-/.10324. to 739 0;: . LPA* maintains a priority queue 7 that always contains exactly the locally inconsistent vertices. These are the vertices whose g-values LPA* potentially needs to update to make them locally consistent. The keys of the vertices in the priority queue correspond to the f-values used by A*, and LPA* always expands the vertex in the priority queue with the smallest key, similar to A* that always expands the vertex in the priority queue with the smallest f-value. By expanding a vertex, we mean executing 10-16 (numbers in brackets refer to line numbers in Figure 3). The key 8 of vertex is a vector with two components: The pseudocode uses the following functions to manage the priority queue: U.TopKey * + returns the smallest priority of all vertices in priority queue . (If is empty, then U.TopKey * + returns .) U.Pop * + deletes the vertex with the smallest priority in priority queue and returns the vertex. U.Insert * + inserts vertex into priority queue with priority . Finally, U.Remove * ,+ removes vertex from priority queue . procedure CalculateKey * ,+
01 return * * ,+ * +.+* ,+ * * ,+ " * ,++ ; procedure Initialize * +
02 ;
03 for all ! * ,+ * ,+ ;
04 * "!$#&%'(#+ ) ;
05 U.Insert * !$#&%'*# * !#%+'*# + ) + ; procedure UpdateVertex *-,5+
06 if *,/. !$#&%'(# + *,+ ! 1032 '"4$56-798 * * +;: * ,5+.+ ;
07 if *,! + U.Remove *-,5+ ;
08 if * *-,5+<. *-,5+.+ U.Insert *,= CalculateKey *-,5+.+ ; procedure ComputeShortestPath * +
09 while * U.TopKey * +?> @ CalculateKey * "A"B % C + OR * "A"B %C +<. * *A"B % C ++
10 , U.Pop * + ;
11 if * *-,5+ED *-,5+.+
12 *-,5+ *-,5+ ;
13 for all ! F,3:$: *-,5+ UpdateVertex * ,+ ;
14 else
15 *-,5+ ;
16 for all ! F,3:$: *-,5+3G
, UpdateVertex * ,+ ; procedure Main * +
17 Initialize * + ;
18 forever
19 ComputeShortestPath * + ;
20 Wait for changes in edge costs;
21 for all directed edges *-,F H+ with changed edge costs
22 Update the edge cost : *-,F H+ ;
23 UpdateVertex *H+ ; Figure 3: Lifelong Planning A*. I KJ IEL M I=N $O"6 (2) where 8QP SRUT&V < A 4 and 8?W XRUT&V < A 4 1 . Keys are compared according to a lexicographic ordering. For example, a key 8( is smaller than or equal to a key 8 "/ , denoted by 8( ZY $ 8 " , iff either 8 P 8 " P or ( 8 P 8 " P and 8[W $ 8 " W ). 8QP, corresponds directly to the f-values \ <+= used by A* because both the g-values and rhs-values of LPA* correspond to the g-values of A* and the h-values of LPA* correspond to the h-values of A*. 8 W corresponds to the g-values of A*. LPA* expands vertices in the order of increasing k P -values and vertices with equal k P -values in order of increasing k W -values. This is similar to A* that expands vertices in the order of increasing f-values (since the heuristics are consistent) and vertices with equal f-values that are on the same branch of the search tree in order of increasing g-values (since it grows the search tree). A locally inconsistent vertex is called overconsistent iff < ^] . When LPA* expands a locally overconsistent vertex 12-13 , then + <+=, because vertex has the smallest key among all locally inconsistent vertices. <+=, implies that 8 `_ \ baA< = *c and thus LPA* expands overconsistent vertices in the same order as A*. During the expansion of vertex , LPA* sets the g-value of vertex to its rhsvalue and thus its start distance 12 , which is the desired value and also makes the vertex locally consistent. Its g-value then no longer changes until LPA* terminates. A locally inconsistent vertex is called underconsistent iff < . When LPA* expands a locally underconsistent vertex 15-16 , then it simply sets the g-value of the vertex to infinity 15 . This makes the vertex either locally consistent or locally overconsistent. If the expanded vertex was locally overconsistent, then the change of its g-value can affect the local consistency of its successors 13 . Similarly, if the expanded vertex was locally underconsistent, then it and its successors can be affected 16 . LPA* therefore updates rhs-values of these vertices, checks their local consistency, and adds them to or removes them from the priority queue accordingly. LPA* expands vertices until 739 0;: is locally consistent and the key of the vertex to expand next is no smaller than the key of 79 0;: . This is similar to A* that expands vertices until it expands 879 03: at which point in time the g-value of 739 0;: equals its start distance and the f-value of the vertex to expand next is no smaller than the f-value of 79 0;: . It turns out that LPA* expands a vertex at most twice, namely at most once when it is underconsistent and at most once when it is overconsistent. Thus, ComputeShortestPath returns after a number of vertex expansions that is at most twice the number of vertices. If < 879 0;: & after the search, then there is no finite-cost path from .1032A. to 879 0;: . Otherwise, one can trace back a shortest path from -/.10324. to 79 0;: by always moving from the current vertex , starting at 79 03: , to any predecessor " that minimizes < "# " ; until -/.10324. is reached (ties can be broken arbitrarily), similar to what A* can do if it does not use backpointers. The resulting version of LPA* is shown in Figure 3. The main function Main() first calls Initialize() to initialize the search problem 17 . Initialize() sets the initial g-values of all vertices to infinity and sets their rhs-values according to Equation 1 03-04 . Thus, initially -/.10324. is the only locally inconsistent vertex and is inserted into the otherwise empty priority queue with a key calculated according to Equation 2 05 . This initialization guarantees that the first call to ComputeShortestPath() performs exactly an A* search, that is, expands exactly the same vertices as A* in exactly the same order, provided that A* breaks ties between vertices with the same f-values suitably. Notice that, in an actual implementation, Initialize() only needs to initialize a vertex when it encounters it during the search and thus does not need to initialize all vertices up front. This is important because the number of vertices can be large and only a few of them might be reached during the search. LPA* then waits for changes in edge costs 20 . If some edge costs have changed, it calls UpdateVertex() 23 to update the rhs-values and keys of the vertices potentially affected by the changed edge costs as well as their membership in the priority queue if they become locally consistent or inconsistent, and finally recalculates a shortest path 19 . 5 Optimizations of Lifelong Planning A* There are several simple ways of optimizing LPA* without changing its overall operation. The resulting version of LPA* is shown in Figure 4. First, a vertex sometimes gets removed from the priority queue and then immediately reinserted with a different key. For example, a vertex can get removed on line 07 and then be reentered on line 08 . In this case, it is often more efficient to leave the vertex in the priority queue, update its key, and only change its position in the priority queue. Second, when UpdateVertex on line 13 computes the rhs-value for a successor of an overconsistent vertex it is unnecessary to take the minimum over all of its respective predecessors. It is sufficient to compute the rhs-value as the minimum of its old rhs-value and the sum of the new g-value of the overconsistent vertex and the cost of moving from the overconsistent vertex to the successor. The reason is that only the g-value of the overconsistent vertex has changed. Since it decreased, it can only decrease the rhs-values of the successor. Third, when UpdateVertex on line 16 computes the rhs-value for a successor of an underconsistent vertex, the only g-value that has changed is the g-value of the underconsistent vertex. Since it increased, the rhs-value of the successor can only get affected if its old rhs-value was based on the old g-value of the underconsistent vertex. This can be used to decide whether the successor needs to get updated and its rhs-value needs to get recomputed 21’ . Fourth, the second and third optimization concerned the computations of the rhs-values of the successors after the g-value of a vertex has changed. Similar optimizations can be made for the computation of the rhs-value of a vertex after the cost of one of its incoming edges has changed. 6 Analytical and Experimental Results We can prove the correctness of ComputeShortestPath(). The pseudocode uses the following functions to manage the priority queue: U.Top * + returns a vertex with the smallest priority of all vertices in priority queue . U.TopKey * + returns the smallest priority of all vertices in priority queue . (If is empty, then U.TopKey * + returns 1 .) U.Insert * + inserts vertex into priority queue with priority . U.Update * $ + changes the priority of vertex in priority queue to . (It does nothing if the current priority of vertex already equals .) Finally, U.Remove * ,+ removes vertex from priority queue . procedure CalculateKey * ,+
01’ return * * ,+ * +.+ * ,+ * * ,+ * ,+.+ ; procedure Initialize * +
02’ ;
03’ for all ! * + * ,+ ;
04’ * !$#%+'*# + ) ;
05’ U.Insert * !$#%+'*# * !#%+'*# + ) + ; procedure UpdateVertex *-,5+
06’ if ( *,+<. *-,5+ AND , ! + U.Update *-,F CalculateKey *,+.+ ;
07’ else if * *-,5+ . *,+ AND , ! + U.Insert *-,= CalculateKey *-,5+.+ ;
08’ else if * *-,5+ *,+ AND , ! + U.Remove *-,+ ; procedure ComputeShortestPath * +
09’ while * U.TopKey * + > @ CalculateKey * A"B % C + OR * A"B % C + . * A"B % C +.+
10’ , U.Top * + ;
11’ if * *-,5+ D *-,5+.+
12’ *-,5+ *,+ ;
13’ U.Remove *-,5+ ;
14’ for all !Z =,9:(:*-,5+
15’ if * . !$#%+'*# + * ,+ * * ,+ 1 *,+/:*-,F ",+.+ ;
16’ UpdateVertex * ,+ ;
17’ else
18’ B C5 *-,+ ;
19’ *-,5+ ;
20’ for all !Z =,9:(:*-,5+G
,
21’ if * * ,+ B C5 : *-,F ",+ OR ,+
22’ if * . !#%+'*# + * ,+ ! 032 '*4(56 !$8 * * + :* ,+.+ ;
23’ UpdateVertex * ,+ ; procedure Main * +
24’ Initialize * + ;
25’ forever
26’ ComputeShortestPath * + ;
27’ Wait for changes in edge costs;
28’ for all directed edges *-,F H+ with changed edge costs
29’ : B C5 :*-,F H+ ;
30’ Update the edge cost :*-,= 1H+ ;
31’ if ( :*B C5 D :*-,= 1H+ )
32’ if *-H . !$#&%'*# + *-H + * *-H+ *-,5+ :*-,= 1H+.+ ;
33’ else if * *-H+ *-,5+ ;:"B C5 +
34’ if *-H . !$#&%'*# + *-H + ! 1032 '"4$5 68 * * + ;:* 1H+.+ ;
35’ UpdateVertex *-H + ; Figure 4: Lifelong Planning A* (optimized version) Theorem 1 ComputeShortestPath() terminates and one can then trace back a shortest path from ' to ' by always moving from the current vertex , starting at ' , to any predecessor / that minimizes -% / 21435 / 6 until is reached (ties can be broken arbitrarily). (The proofs can be found in [LK01].) We now compare breadth-first search, A*, DynamicSWSF-FP, and the optimized version of LPA* experimentally. (We use DynamicSWSF-FP with the same optimizations that we developed for LPA*, to avoid biasing our experimental results in favor of LPA*.) The priority queues of all four algorithms were implemented as binary heaps. Since all algorithms determine the same paths (if they break ties suitably), we need to compare their total search time until a shortest path has been found. Since the actual runtimes are implementation-dependent, we instead use three measures that all correspond to common operations performed by the algorithms and thus heavily influence their runtimes: the total number of vertex expansions (that is, updates of the g-values, similar to backup operations of dynamic programming for sequential decision problems), the total number of vertex accesses (for example, to read or change their values), and the total number of heap percolates (exchanges of a parent and child in the heap). Note that we count two vertex expansions, not just one vertex expansion, if LPA* expands the same vertex twice, to avoid biasing our experimental results in favor of LPA*. All of our experiments use fifty eight-connected gridworlds that have size
C
and an obstacle density of 40 percent. The start cell is at coordinates (34, 20) and the goal cell is at coordinates (5, 20), where the upper leftmost cell is at coordinates (0, 0). For each gridworld, the initial obstacle configuration was generated randomly. Then, it was changed 500 times in a row, each time by making eight randomly chosen blocked cells traversable and eight randomly chosen traversable cells blocked. Thus, each time one percent of the cells changed their status but the obstacle density remained the same. After each of the 500 changes, the algorithms recomputed a shortest path from the start cell to the goal cell. For each of the four algorithms and each of the three performance measures, the following table reports the mean of the performance measure for the 500 changes: both its average over the fifty mazes and its 95-percent confidence interval over the fifty mazes (assuming a normal distribution with unknown variance). The table confirms the observations made in Section 3: LPA* outperforms the other three search methods according to all three performance measures. uninformed search heuristic search complete search breadth-first search A* ve = 1331.7 4.4 ve = 284.0 5.9 va = 26207.2 84.0 va = 6177.3 129.3 hp = 5985.3 19.7 hp = 1697.3 39.9 incremental search DynamicSWSF-FP Lifelong Planning A* ve = 173.0 4.9 ve = 25.6 2.0 va = 5697.4 167.0 va = 1235.9 75.0 hp = 956.2 26.6 hp = 240.1 16.9 We have also applied LPA* successfully to more complex planning tasks, including the kind of route planning tasks that Focussed Dynamic A* [Ste95] applies to. The results will be reported separately. References [FMSN00] D. Frigioni, A. Marchetti-Spaccamela, and U. Nanni. Fully dynamic algorithms for maintaining shortest paths trees. Journal of Algorithms, 34(2):251– 281, 2000. [LK01] M. Likhachev and S. Koenig. Lifelong Planning A* and Dynamic A* Lite: The proofs. Technical report, College of Computing, Georgia Institute of Technology, Atlanta (Georgia), 2001. [Pea85] J. Pearl. Heuristics: Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, 1985. [RR96] G. Ramalingam and T. Reps. An incremental algorithm for a generalization of the shortest-path problem. Journal of Algorithms, 21:267–305, 1996. [Ste95] A. Stentz. The focussed D* algorithm for real-time replanning. In Proceedings of the International Joint Conference on Artificial Intelligence, pages 1652– 1659, 1995. [Thr98] Sebastian Thrun. Lifelong learning algorithms. In S. Thrun and L. Pratt, editors, Learning To Learn. Kluwer Academic Publishers, 1998.
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Grouping and dimensionality reduction by locally linear embedding Marzia Polito Division of Physics, Mathematics and Astronomy California Institute of Technology Pasadena, CA, 91125 polito@caltech.edu Pietro Perona Division of Engeneering and Applied Mathematics California Institute of Technology Pasadena, CA, 91125 perona@caltech.edu Abstract Locally Linear Embedding (LLE) is an elegant nonlinear dimensionality-reduction technique recently introduced by Roweis and Saul [2]. It fails when the data is divided into separate groups. We study a variant of LLE that can simultaneously group the data and calculate local embedding of each group. An estimate for the upper bound on the intrinsic dimension of the data set is obtained automatically. 1 Introduction Consider a collection of N data points Xi E ]RD. Suppose that, while the dimension D is large, we have independent information suggesting that the data are distributed on a manifold of dimension d < < D. In many circumstances it is beneficial to calculate the coordinates Yi E ]Rd of the data on the lower-dimensional manifold, both because the shape of the manifold may yield some insight in the process that produced the data, and because it is cheaper to store and manipulate the data when it is embedded in fewer dimensions. How can we compute such coordinates? Principal component analysis (PCA) is a classical technique which works well when the data lie close to a flat manifold [1]. Elegant methods for dealing with data that is distributed on curved manifolds have been recently proposed [3, 2]. We study one of them, Locally Linear Embedding (LLE) [2], by Roweis and Saul. While LLE is not designed to handle data that are disconnected, i.e. separated into groups, we show that a simple variation of the method will handle this situation correctly. Furthermore, both the number of groups and the upper bound on the intrinsic dimension of the data may be estimated automatically, rather than being given a-priori. 2 Locally linear embedding The key insight inspiring LLE is that, while the data may not lie close to a globally linear manifold, it may be approximately locally linear, and in this case each point may be approximated as a linear combination of its nearest neighbors. The coefficients of this linear combination carries the vital information for constructing a lower-dimensional linear embedding. More explicitly: consider a data set {Xd i=l...,N E ]RD. The local linear structure can be easily encoded in a sparse N by N matrix W, proceeding as follows. The first step is to choose a criterion to determine the neighbors of each point. Roweis and Saul chose an integer number K and pick, for every point, the K points nearest to it. For each point Xi then, they determine the linear combination of its neighbors which best approximates the point itself. The coefficients of such linear combinations are computed by minimizing the quadratic cost function: N f(W) = L IXi - L WijXj 12 (1) j=1 while enforcing the constraints Wij = 0 if Xj is not a neighbor of Xi , and L:.f=1 Wij = 1 for every i; these constraints ensure that the approximation of Xi ~ Xi = L:.f=1 WijXj lies in the affine subspace generated by the K nearest neighbors of Xi, and that the solution W is translation-invariant. This least square problem may be solved in closed form [2]. The next step consists of calculating a set {Yih=1, ... ,N of points in ]Rd, reproducing as faithfully as possible the local linear structure encoded in W. This is done minimizing a cost function N N <I>(Y) = L IYi - L Wij Yjl2 (2) i=1 j=1 To ensure the uniqueness of the solution two constraint are imposed: translation invariance by placing the center of gravity of the data in the origin, i.e. L:i Yi = 0, and normalized unit covariance of the Yi's, i.e. tt L:~1 Yi Q9 Yi = I. Roweis and Saul prove that <I>(Y) = tr(yT MY), where M is defined as M = (I - wf (I - W). The minimum of the function <I>(Y) for the d-th dimensional representation is then obtained with the following recipe. Given d, consider the d + 1 eigenvectors associated to the d + 1 smallest eigenvalues of the matrix M. Then discard the very first one. The rows of the matrix Y whose columns are given by such d eigenvectors give the desired solution. The first eigenvector is discarded because it is a vector composed of all ones, with 0 as eigenvalue. As we shall see, this is true when the data set is 'connected'. 2.1 Disjoint components In LLE every data point has a set of K neighbors. This allows us to partition of the whole data set X into K -connected components, corresponding to the intuitive visual notion of different 'groups' in the data set. We say that a partition X = UiUi is finer than a partition X = Uj 10 if every Ui is contained in some 10. The partition in K -connected components is the finest : •............................ 10 20 30 40 50 60 70 60 90 100 -020"'---------:::---;:;;-----O;---;;;------;';c------:::--~ Figure 1: (Top-left) 2D data Xi distributed along a curve (the index i increases from left to right for convenience). (Top-right) Coordinates Yi of the same points calculated by LLE with K = 10 and d = 1. The x axis represents the index i and the y axis represents Yi. This is a good parametrization which recognizes the intrinsically I-dimensional structure of the data. (Bottom-left) As above, the data is now disconnected, i.e. points in different groups do not share neighbors. (Bottom-right) One-dimensional LLE calculated on the data (different symbols used for points belonging to the different groups). Notice that the Yi's are not a good representation of the data any longer since they are constant within each group. partition of the data set such that if two points have at least one neighbor in common, or one is a neighbor of the other, then they belong to the same component. Note that for any two points in the same component, we can find an ordered sequence of points having them as endpoints, such that two consecutive points have at least one neighbor in common. A set is K -connected if it contains only one K-connected component. Consider data that is not K -connected, then LLE does not compute a good parametrization, as illustrated in Figure 1. 2.2 Choice of d. How is d chosen? The LLE method [2] is based on the assumption that d is known. What if we do not know it in advance? If we overestimate d it then LLE behaves pathologically. Let us consider a straight line, drawn in 1R3 . Figure 2 shows what happens if d is chosen equal to 1 and to 2. When the choice is 2 (right) then LLE 'makes up' information and generates a somewhat arbitrary 2D curve. As an effect of the covariance constraint, the representation curves the line, the Figure 2: Coordinates Yi calculated for data Xi distributed along a straight line in ]RD = ]R3 when the dimension d is chosen as d = 1 (Left), and d = 2 (Right). The index i is indicated along the x axis (Left) and along the 2D curve (Right). curvature can be very high, and even locally we possibly completely lose the linear structure. The problem is, we chosed the wrong target dimension. The onedimensional LLE works in fact perfectly (see Figure 2, left). PCA provides a principled way of estimating the intrinsic dimensionality of the data: it corresponds to the number of large singular values of the covariance matrix of the data. Is such an estimate possible with LLE as well? 3 Dimensionality detection: the size of the eigenvalues In the example of Figure 2 the two dimensional representation of the data (d = 2) is clearly the 'wrong' one, since the data lie in a one-dimensional linear subspace. In this case the unit covariance constraint in minimizing the function <I>(Y) is not compatible with the linear structure. How could one have obtained the correct estimate of d? The answer is that d + 1 should be less or equal to the number of eigenvalues of M that are close to zero. Proposition 1. Assume that the data Xi E ]RD is K -connected and that it is locally fiat, i.e. there exists a corresponding set Yi E ]Rd for some d > 0 such that Yi = L:j Wij}j (zero-error approximation), the set {Yi} has rank d, and has the origin as center of gravity: L:~ 1 Yi = o. Call z the number of zero eigenvalues of the matrix M. Then d < z. Proof. By construction the N vector composed of all 1 's is a zero-eigenvector of M. Moreover, since the Yi are such that the addends of <I> have zero error, then the matrix Y , which by hypothesis has rank d, is in the kernel of I - W and hence in the kernel of M. Due to the center of gravity constraint, all the columns of Y are orthogonal to the all 1 's vector. Hence M has at least d + 1 zero eigenvalues. D Therefore, in order to estimate d, one may count the number z of zero eigenvalues of M and choose any d < z. Within this range, smaller values of d will yield more compact representations, while larger values of d will yield more expressive ones, i.e. ones that are most faithful to the original data. What happens in non-ideal conditions, i.e. when the data are not exactly locally fiat, and when one has to contend with numerical noise? The appendix provides an argument showing that the statement in the proposition is robust with respect to ,,' ,,' ,,' ,,' ,,' ,,' ,,' ,,' ,,' ,,' ,,' ,,' 10 '0 10" 10 " 2nd eigen value 10" 2nd cigcnvuluc 10" 10 " 10" lst eigenva]ue 10" lst ci'cnv· "' 0 0 ., " Figure 3: (Left) Eigenvalues for the straight-line data Xi used for Figure 2. (Right) Eigenvalues for the curve data shown in the top-left panel of Figure 1. In both cases the two last eigenvalue are orders of magnitude smaller than the other eigenvalues, indicating a maximal dimension d = 1 for the data. noise, i.e. numerical errors and small deviations from the ideal locally flat data will result in small deviations from the ideal zero-value of the first d + 1 eigenvalues, where d is used here for the 'intrinsic' dimension of the data. This is illustrated in Figure 3. In Figure 4 we describe the successful application of the dimensionality detection method on a data set of synthetically generated grayscale images. 4 LLE and grouping In the first example (2.1) we pointed out the limits of LLE when applied to multiple components of data. It appears then that a grouping procedure should always preceed LLE. The data would be first split into its component groups, each one of which should be then analyzed with LLE. A deeper analysis of the algorithm though, suggests that grouping and LLE could actually be performed at the same time. Proposition 2. Suppose the data set {Xdi=l, ... ,N E ll~P is partitioned into m Kconnected components. Then there exists an m-dimensional eigenspace of M with zero eigenvalue which admits a basis {vih=l, ... ,m where the Vi have entries that are either '1' or '0'. More precisely: each Vi corresponds to one of the groups of the data and takes value Vi ,j = 1 for j in the group, Vi ,j = 0 for j not in the group. Proof. Without loss of generality, assume that the indexing of the data X i is such that the weight matrix W , and consequent ely the matrix M, are block-diagonal with m blocks, each block corresponding to one of the groups of data. This is achieved by a permutation of indices, which will not effect any further step of our algorithm. As a direct consequence of the row normalization of W, each block of M has exactly one eigenvector composed of all ones, with eigenvalue O. Therefore, there is an m-dimensional eigenspace with eigenvalue 0, and there exist a basis of it, each vector of which has value 1 on a certain component, 0 otherwise. D Therefore one may count the number of connected components by computing the eigenvectors of M corresponding to eigenvalue 0, and counting the number m of those vectors Vi whose components take few discrete values (see Figure 6). Each index i may be assigned to a group by clustering based on the value of Vl, ... , Vm . Figure 4: (Left) A sample from a data set of N=1000, 40 by 40 grayscale images, each one thought as a point in a 1600 dimensional vector space. In each image, a slightly blurred line separates a dark from a bright portion. The orientation of the line and its distance from the center of the image are variable. (Middle) The non-zero eigenvalues of M. LLE is performed with K=20. The 2nd and 3rd smallest eigenvalues are of smaller size than the others, giving an upper bound of 2 on the intrinsic dimension of the data set. (Right) The 2-dimensional LLE representation. The polar coordinates, after rescaling, are the distance of the dividing line from the center and its orientation. ". Figure 5: The data set is analogous to the one used above (N =1000, 40 by 40 grayscale images, LLE performed with K=20). The orientation of the line dividing the dark from the bright portion is now only allowed to vary in two disjoint intervals. (Middle) The non-zero eigenvalues of M. (Left and Right) The 3rd and 5th (resp. 4th and 6th) eigenvectors of M are used for the LLE representation of the first (resp. the second) K-component. ,,' ,,' ,,' ,,' ,,' 10'0 4th. 5th and 6th eigenvalues 10 " 10 " 10 " 1st. 2nd and 3rd eigenvalues Figure 6: (Left) The last six eigenvectors of M for the broken parabola of Figure 1 shown, top to bottom, in reverse order of magnitude of the corresponding eigenvalue. The x axis is associated to the index i. (Right) The eigenvalues of the same (log scale). Notice that the last six are practically zero. The eigenvectors corresponding to the three last eigenvalues have discrete values indicating that the data is split in three groups. There are z=6 zero-eigenvalues indicating that the dimension of the data is d:::; z/m - 1 = 1. In the Appendix (A) we show that such a process is robust with respect to numerical noise. It is also robust to small perturbations of the block-diagonal structure of M (see Figure 7). This makes the use of LLE for grouping purposes convenient. Should the K-connected components be completely separated, the partition would be easily obtained via a more efficient graph-search algorithm. The proof is carried out for ordered indices as in Fig. 3 but it is invariant under index permutation. The analysis of Proposition 1 may be extended to the dimension of each of the m groups according to Proposition 2. Therefore, in the ideal case, we will find z zero-eigenvalues of M which, together with the number m obtained by counting the discrete-valued eigenvectors may be used to estimate the maximal d using z ~ m(d + 1). This behavior may be observed experimentally, see Figures 6 and 5. 5 Conclusions We have examined two difficulties of the Locally Linear Embedding method [2] and shown that, in a neighborhood of ideal conditions, they may be solved by a careful exam of eigenvectors of the matrix M that are associated to very small eigenvalues. More specifically: the number of groups in which the data is partitioned corresponds to the number of discrete-valued eigenvectors, while the maximal dimension d of the low-dimensional embedding may be obtained by dividing the number of small eigenvalues by m and subtracting 1. Both the groups and the low-dimensional embedding coordinates may be computed from the components of such eigenvectors. Our algorithms have mainly been tested on synthetically generated data. Further investigation on real data sets is necessary in order to validate our theoretical results. w' ,----~-~-~-~--~______, w' w' w' w' w' w' 10 '· 3rd and 4th eigenvalues . 10 " 10 " 1st and 2ndeige!\values 10'" o~--=-----::------=-------=------,=--~ Figure 7: (Left) 2D Data Xi distributed along a broken parabola. Nevertheless, for K=14, the components are not completely K-disconnected (a different symbol is used for the neighbors of the leftmost point on the rightmost component). (Right) The set of eigenvalues for M. A set of two almost-zero eigenvalues and a set of two of small size are visible. References [1] C. Bishop, Neural Networks for Pattern Recognition, Oxford Univ. Press, (1995). [2] S. T. Roweis, L.K.Saul, Science, 290, p. 2323-2326, (2000). [3] J. Tenenbaum, V. de Silva, J. Langford, Science, 290, p. 2319-2323, (2000). A Appendix In Proposition 2 of Section 4 we proved that during the LLE procedure we can automatically detect the number of K -connected components, in case there is no noise. Similarly, in Proposition 1 of Section 3 we proved that under ideal conditions (no noise, locally flat data), we can determine an estimate for the intrinsic dimension of the data. Our next goal is to establish a certain robustness of these results in the case there is numerical noise, or the components are not completely separated, or the data is not exactly locally flat. In general, suppose we have a non degenerate matrix A, and an orthonormal basis of eigenvectors VI, ... , Vm , with eigenvalues AI , ... Am. As a consequence of a small perturbation of the matrix into A + dA, we will have eigenvectors Vi + dVi with eigenvalues Ai + dAi' The unitary norm constraint makes sure that dVi is orthogonal to Vi and could be therefore written as dVi = L:k#i O'.ikVk. Using again the orthonormality, one can derive expressions for the perturbations of Ai and Vi : dAi O'.ij (Ai - Aj) < vi,dAvi > < Vj,dAVi > . This shows that if the perturbation dA has order E, then the perturbations dA and O'.ij are also of order E. Notice that we are not interested in perturbations O'.ij within the eigenspace of eigenvalue 0, but rather those orthogonal to it, and therefore Ai =j:. Aj.
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Incorporating Invariances in Nonlinear Support Vector Machines Olivier Chapelle olivier.chapelle@lip6.fr LIP6, Paris, France Biowulf Technologies Bernhard Scholkopf bernhard.schoelkopf@tuebingen.mpg.de Max-Planck-Institute, Tiibingen, Germany Biowulf Technologies Abstract The choice of an SVM kernel corresponds to the choice of a representation of the data in a feature space and, to improve performance, it should therefore incorporate prior knowledge such as known transformation invariances. We propose a technique which extends earlier work and aims at incorporating invariances in nonlinear kernels. We show on a digit recognition task that the proposed approach is superior to the Virtual Support Vector method, which previously had been the method of choice. 1 Introduction In some classification tasks, an a priori knowledge is known about the invariances related to the task. For instance, in image classification, we know that the label of a given image should not change after a small translation or rotation. More generally, we assume we know a local transformation Lt depending on a parameter t (for instance, a vertical translation of t pixels) such that any point x should be considered equivalent to LtX, the transformed point. Ideally, the output of the learned function should be constant when its inputs are transformed by the desired invariance. It has been shown [1] that one can not find a non-trivial kernel which is globally invariant. For this reason, we consider here local invariances and for this purpose we associate at each training point Xi a tangent vector dXi, dXi = lim - (LtXi Xi) = LtXi 1 81 t--+o t 8t t=o In practice dXi can be either computed by finite difference or by differentiation. Note that generally one can consider more than one invariance transformation. A common way of introducing invariances in a learning system is to add the perturbed examples LtXi in the training set [7]. Those points are often called virtual examples. In the SVM framework, when applied only to the SVs, it leads to the Virtual Support Vector (VSV) method [10]. An alternative to this is to modify directly the cost function in order to take into account the tangent vectors. This has been successfully applied to neural networks [13] and linear Support Vector Machines [11]. The aim of the present work is to extend these methods to the case of nonlinear SVMs which will be achieved mainly by using the kernel peA trick [12]. The paper is organized as follows. After introducing the basics of Support Vector Machines in section 2, we recall the method proposed in [11] to train invariant linear SVMs (section 3). In section 4, we show how to extend it to the nonlinear case and finally experimental results are provided in section 5. 2 Support Vector Learning We introduce some standard notations for SVMs; for a complete description, see [15]. Let {(Xi, Yi) h<i<n be a set of training examples, Xi E IRd , belonging to classes labeled by Yi E {-1,1}. In kernel methods, we map these vectors into a feature space using a kernel function K(Xi' Xj) that defines an inner product in this feature space. The decision function given by an SVM is the maximal margin hyperplane in this space, g(X) = sign(f(x)), where f(x) = (~a?YiK(Xi'X) + b) . (1) The coefficients a? are obtained by maximizing the functional n 1 n W(o:) = Lai -"2 L aiajYiyjK(Xi,Xj) (2) i=l i,j=l under the constraints L:~= 1 aiYi = 0 and ai ~ O. This formulation of the SVM optimization problem is called the hard margin formulation since no training errors are allowed. In the rest of the paper, we will not consider the so called soft-margin SVM algorithm [4], where training errors are allowed. 3 Invariances for Linear SVMs For linear SVMs, one wants to find a hyperplane whose normal vector w is as orthogonal as possible to the tangent vectors. This can be easily understood from the equality f(Xi + dXi) - f(Xi) = w . dXi' For this purpose, it has been suggested [11] to minimize the functional n (1 - ')')w2 + ')' L(w, dXi)2 (3) i=l subject to the constraints Yi(W . Xi + b) ~ 1. The parameter,), trades off between normal SVM training (')' = 0) and full enforcement of the orthogonality between the hyperplane and the invariance directions (')' ---+ 1). Let us introduce c, ~ ((1-0)[ +0 ~dx'dxi) 'i', (4) the square root of the regularized covariance matrix of the tangent vectors. It was shown in [11] that training a linear invariant SVM, i.e. minimizing (3), is equivalent to a standard SVM training after the following linear transformation of the input space C- 1 X --+ , x. This method led to significant improvements in linear SVMs, and to small improvements when used as a linear preprocessing step in nonlinear SVMs. The latter, however, was a hybrid system with unclear theoretical foundations. In the next section we show how to deal with the nonlinear case in a principled way. 4 Extension to the nonlinear case In the nonlinear case, the data are first mapped into a high-dimensional feature space where a linear decision boundary is computed. To extend directly the previous analysis to the nonlinear case, one would need to compute the matrix C, in feature space, ( ) 1~ C, = (1 - '"Y)I + '"Y ~ dlJ> (Xi) dlJ> (Xi) T (5) and the new kernel function K(x, y) = C~ llJ>(x) . C~ llJ>(y) = lJ>(x) T C~ 21J>(y) (6) However, due to the high dimension of the feature space, it is impossible to do it directly. We propose two different ways for overcoming this difficulty. 4.1 Decomposition of the tangent Gram matrix In order to be able to compute the new kernel (6) , we propose to diagonalize the matrix C, (eq 5) using a similar approach as the kernel PCA trick [12]. In that article, they showed how it was possible to diagonalize the feature space covariance matrix by computing the eigendecomposition of the Gram matrix of those points. Presently, instead of having a set of training points {1J>(Xi)}, we have a set of tangent vectors {dlJ> (Xi)} and a tangent covariance matrix (the right term of the sum in (5)) Let us introduce the Gram matrix Kt of the tangent vectors: Kij = dlJ>(Xi)· dlJ>(xj) K(Xi +dXi, Xj +dxj) - K(Xi +dXi, Xj) - K(Xi' Xj +dxj) + K(Xi' Xj) (7) d T02K(Xi,Xj)d . (8) xi ~ ~ X J UXiUXj This matrix Kt can be computed either by finite differences (equation 7) or with the analytical derivative expression given by equation (8). Note that for a linear kernel, K(x,y) = x T y, and (8) reads Kfj = dxi dXj, which is a standard dot product between the tangent vectors. Writing the eigendecomposition of Kt as Kt = U AUT, and using the kernel PCA tools [12], one can show after some algebra (details in [2]) that the new kernel matrix reads K(x,y) 1 n1( 1 1) --Kx y + --I '"Y (,) ~ Ap '"Y Ap + 1 '"Y 1 '"Y (~ U. d T OK(Xi' X)) (~U . d T OK(Xi' y)) ~ 'p x, ~ ~ 'p x, ~ ~1 U~ ~1 U~ 4.2 The kernel PCA map A drawback of the previous approach appears when one wants to deal with multiple invariances (i.e. more than one tangent vector per training point). Indeed, it requires to diagonalize the matrix Kt (cf eq 7), whose size is equal to the number of different tangent vectors. For this reason, we propose an alternative method. The idea is to use directly the so-called kernel peA map, first introduced in [12] and extended in [14]. This map is based on the fact that even in a high dimensional feature space 1i, a training set {Xl , .. . , x n } of size n when mapped to this feature space spans a subspace E C 1i whose dimension is at most n . More precisely, if (VI"'" Vn ) E En is an orthonormal basis of E with each Vi being a principal axis of {<I>(xd, ... , <I> (xn )} , the kernel peA map 'i/J : X -+ ~n is defined coordinatewise as 'i/Jp (x) = <I>(x) . v P' 1:S p:S n. Each principal direction has a linear expansion on the training points {<I>(Xi)} and the coefficients of this expansion are obtained using kernel peA [12]. Writing the eigendecompostion of K as K = U AUT, with U an orthonormal matrix and A a diagonal one, it turns out that the the kernel peA map reads 'i/J(x) = A -1/2UTk(x), where k (x) = (K(x, Xl)"'" K(x, xn)) T . (9) Note that by definition, for all i and j , <I>(Xi) and <I>(Xj) lie in E and thus K(Xi' Xj) = <I>(Xi) . <I>(Xj) = 'i/J(Xi) . 'i/J(Xj). This reflects the fact that if we retain all principal components, kernel peA is just a basis transform in E, leaving the dot product of training points invariant. As a consequence, training a nonlinear SVM on {Xl , ... , xn} is equivalent to training a linear SVM on {'i/J(xd, . . . ,'i/J(xn)} and thus, thanks to the nonlinear mapping 'i/J, we can work directly in the linear space E and use exactly the technique described for invariant linear SVMs (section 3). However the invariance directions d<I>(Xi) do not necessarily belong to E. By projecting them onto E, some information might be lost. The hope is that this approximation will give a similar decision function to the exact one obtained in section 4.l. Finally, the proposed algorithm consists in training an invariant linear SVM as described in section 3 with training set {'i/J(XI) , ... ,'i/J(xn)} and with invariance directions {d'i/J(XI) , ... , d'i/J(xn)}, where d'i/J(Xi) = 'i/J(Xi + dXi ) - 'i/J(Xi), which can be expressed from equation (9) as 4.3 Comparisons with the VSV method One might wonder what is the difference between enforcing an invariance and just adding the virtual examples LtXi in the training set. Indeed the two approaches are related and some equivalence can be shown [6]. So why not just add virtual examples? This is the idea of the Virtual Support Vector (VSV) method [10]. The reason is the following: if a training point Xi is far from the margin, adding the virtual example LtXi will not change the decision boundary since neither of the points can become a support vector. Hence adding virtual examples in the SVM framework enforces invariance only around the decision boundary, which, as an aside, is the main reason why the virtual SV method only adds virtual examples generated from points that were support vectors in the earlier iteration. One might argue that the points which are far from the decision boundary do not provide any information anyway. On the other hand, there is some merit in not only keeping the output label invariant under the transformation Lt, but also the real-valued output. This can be justified by seeing the distance of a given point to the margin as an indication of its class-conditional probability [8]. It appears reasonable that an invariance transformation should not affect this probability too much. 5 Experiments In our experiments, we compared a standard SVM with several methods taking into account invariances: standard SVM with virtual examples (cf. the VSV method [10]) [VSV], invariant SVM as described in section 4.1 [ISVM] and invariant hyperplane in kernel peA coordinates as described in section 4.2 [ IHKPcA ]. The hybrid method described in [11] (see end of section 3) did not perform better than the VSV method and is not included in our experiments for this reason. Note that in the following experiments, each tangent vector d<I>(Xi) has been normalized by the average length vI: Ild<I>(xi)W/n in order to be scale independent. 5.1 Toy problem The toy problem we considered is the following: the training data has been generated uniformly from [-1, 1]2. The true decision boundary is a circle centered at the origin: f(x) = sign(x2 - 0.7). The a priori knowledge we want to encode in this toy problem is local invariance under rotations. Therefore, the output of the decision function on a given training point Xi and on its image R(Xi,C:) obtained by a small rotation should be as similar as possible. To each training point, we associate a tangent vector dXi which is actually orthogonal to Xi. A training set of 30 points was generated and the experiments were repeated 100 times. A Gaussian kernel K(x,y) = exp (_ II X2~~ 1I 2) was chosen. The results are summarized in figure 1. Adding virtual examples (VSV method) is already very useful since it made the test error decrease from 6.25% to 3.87% (with the best choice of a). But the use of ISVM or IHKPcA yields even better performance. On this toy problem, the more the invariances are enforced b -+ 1), the better the performances are (see right side of figure 1), reaching a test error of 1.11%. When comparing log a = 1.4 and log a = 0 (right side of of figure 1), one notices that the decrease in the test error does not have the same speed. This is actually the dual of the phenomenon observed on the left side of this figure: for a same value of gamma, the test error tends to increase, when a is larger. This analysis suggests that 'Y needs to be adapted in function of a. This can be done automatically by the gradient descent technique described in [3]. 0.14 0.12 0.1 0 .08 O.02 '----_~ , ------: -0~ .5--~ 0 --0~ .5,------~------'c-" . 5 Log sigma 0.12 0.06 0.04 0.02 log sigma=-O.8 log sigma=O - 10 si ma=1,4 %'------~-~-~ 6 -~8--,~ 0 -~ , 2~ - Log (1-gamma) Figure 1: Left: test error for different learning algorithms plotted against the width of a RBF kernel and "( fixed to 0.9. Right: test error of IHKPcA across "( and for different values of (5. The test errors are averaged over the 100 splits and the error bars correspond to the standard deviation of the means. 5.2 Handwritten digit recognition As a real world experiment, we tried to incorporate invariances for a handwritten digit recognition task. The USPS dataset have been used extensively in the past for this purpose, especially in the SVM community. It consists of 7291 training and 2007 test examples. According to [9], the best performance has been obtained for a polynomial kernel of degree 3, and all the results described in this section were performed using this kernel. The local transformations we considered are translations (horizontal and vertical). All the tangent vectors have been computed by a finite difference between the original digit and its I-pixel translated. We split the training set into 23 subsets of 317 training examples after a random permutation of the training and test set. Also we concentrated on a binary classification problem, namely separating digits a to 4 against 5 to 9. The gain in performance should also be valid for the multiclass case. Figure 2 compares ISVM, IHKPcA and VSV for different values of "(. From those figures, it can be seen that the difference between ISVM (the original method) and IHKPcA (the approximation) is much larger than in the toy example. The difference to the toy example is probably due to the input dimensionality. In 2 dimensions, with an RBF kernel, the 30 examples of the toy problem "almost span" the whole feature space, whereas with 256 dimensions, this is no longer the case. What is noteworthy in these experiments is that our proposed method is much better than the standard VSV. As explained in section 4.3, the reason for this might be that invariance is enforced around all training points and not only around support vectors. Note that what we call VSV here is a standard SVM with a double size training set containing the original data points and their translates. The horizontal invariance yields larger improvements than the vertical one. One of the reason might be that the digits in the USPS database are already centered vertically. 0.068 0.066 0.064 0.062 0.06 0.058 0.056 0.054 0 0.5 1.5 2 2.5 -Log (1-gamma) IHKPCA ISVM - VSV 3.5 Vertical translation (to the top) 0.068 0.066 0.064 0.062 0.06 0.058 0.056 0.054 0 0.5 1.5 2 2.5 -Log (1-gamma) IHKPCA ISVM - VSV 3.5 Horizontal translation (to the right) Figure 2: Comparison of ISVM, IHKPcA and VSV on the USPS dataset. The left of the plot ("( = 0) corresponds to standard SVM whereas the right part of the plot h -+ 1) means that a lot of emphasis is put on the enforcement of the constraints. The test errors are averaged over the 23 splits and the error bars correspond to the standard deviation of the means. 6 Conclusion We have extended a method for constructing invariant hyperplanes to the nonlinear case. We have shown results that are superior to the virtual SV method. The latter has recently broken the record on the NIST database which is the "gold standard" of handwritten digit benchmarks [5], therefore it appears promising to also try the new system on that task. For this propose, a large scale version of this method needs to be derived. The first idea we tried is to compute the kernel PCA map using only a subset of the training points. Encouraging results have been obtained on the lO-class USPS database (with the whole training set), but other methods are also currently under study. References [1] C. J. C. Burges. Geometry and invariance in kernel based methods. In B. Sch6lkopf, C. J . C. Burges, and A. J . Smola, editors, Advances in Kernel Methods Support Vector Learning. MIT Press, 1999. [2] O. Chapelle and B. Sch6lkopf. Incorporating invariances in nonlinear Support Vector Machines, 2001. Availabe at: www-connex.lip6.frrchapelle. [3] O. Chapelle, V. Vapnik, O. Bousquet, and S. Mukherjee. Choosing multiple parameters for support vector machines. Machine Learning, 46:131- 159, 2002. [4] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273 297,1995. [5] D. DeCoste and B. Sch6lkopf. Training invariant support vector machines. Machine Learning, 2001. In press. [6] Todd K. Leen. From data distributions to regularization in invariant learning. In Nips, volume 7. The MIT Press, 1995. [7] P. Niyogi, T. Poggio, and F. Girosi. Incorporating prior information in machine learning by creating virtual examples. IEEE Proceedings on Intelligent Signal Processing, 86(11):2196-2209, November 1998. [8] John Platt. Probabilities for support vector machines. In A. Smola, P. Bartlett, B. Sch6lkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers. MIT Press, Cambridge, MA, 2000. [9] B. Sch6lkopf, C. Burges, and V. Vapnik. Extracting support data for a given task. In U. M. Fayyad and R. Uthurusamy, editors, First International Conference on Knowledge Discovery fj Data Mining. AAAI Press, 1995. [10] B. Sch6lkopf, C. Burges, and V. Vapnik. Incorporating invariances in support vector learning machines. In Artificial Neural Networks ICANN'96, volume 1112, pages 47- 52, Berlin, 1996. Springer Lecture Notes in Computer Science. [11] B. Sch6lkopf, P. Y. Simard, A. J. Smola, and V. N. Vapnik. Prior knowledge in support vector kernels. In MIT Press, editor, NIPS, volume 10, 1998. [12] B. Sch6lkopf, A. Smola, and K.-R. Muller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299- 1310, 1998. [13] P. Simard, Y. LeCun, J. Denker, and B. Victorri. Transformation invariance in pattern recognition, tangent distance and tangent propagation. In G. Orr and K. Muller, editors, Neural Networks: Tricks of the trade. Springer, 1998. [14] K. Tsuda. Support vector classifier with asymmetric kernel function. In M. Verleysen, editor, Proceedings of ESANN'99, pages 183- 188,1999. [15] V. Vapnik. Statistical Learning Theory. John Wiley & Sons, 1998.
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Partially labeled classification with Markov random walks Martin Szummer MIT AI Lab & CBCL Cambridge, MA 02139 szummer@ai.mit.edu Tommi Jaakkola MIT AI Lab Cambridge, MA 02139 tommi@ai.mit.edu Abstract To classify a large number of unlabeled examples we combine a limited number of labeled examples with a Markov random walk representation over the unlabeled examples. The random walk representation exploits any low dimensional structure in the data in a robust, probabilistic manner. We develop and compare several estimation criteria/algorithms suited to this representation. This includes in particular multi-way classification with an average margin criterion which permits a closed form solution. The time scale of the random walk regularizes the representation and can be set through a margin-based criterion favoring unambiguous classification. We also extend this basic regularization by adapting time scales for individual examples. We demonstrate the approach on synthetic examples and on text classification problems. 1 Introduction Classification with partially labeled examples involves a limited dataset of labeled examples as well as a large unlabeled dataset. The unlabeled examples to be classified provide information about the structure of the domain while the few labeled examples identify the classification task expressed in this structure. A common albeit tacit assumption in this context associates continuous high-density clusters in the data with pure classes. When this assumption is appropriate, we only require one labeled point for each cluster to properly classify the whole dataset. Data points are typically given relative to a global coordinate system with an associated metric. While the metric may provide a reasonable local similarity measure, it is frequently inadequate as a measure of global similarity. For example, the data may lie on a submanifold of the space, revealed by the density, and any global comparisons should preferably be made along the manifold structure. Moreover, we often wish to assign higher similarity values to examples contained in the same high-density regions or clusters implying that comparisons ought to incorporate the density in addition to the manifold structure. A representation of examples that satisfies these and other desiderata can be constructed through a Markov random walk similarly to [3]. The resulting global comparisons of examples integrate a “volume” of paths connecting the examples as opposed to shortest paths that are susceptible to noise. The time scale of the Markov process (the number of transitions) will permit us to incorporate the cluster structure in the data at different levels of granularity. We start by defining the representation and subsequently develop several classification methods naturally operating on such representations. 2 Representation based on Markov random walks We define a Markov random walk based on a locally appropriate metric [3]. The metric is the basis for the neighborhood graph, associated weights on the edges, and consequently the transition probabilities for the random walk. The new representation for the examples can be obtained naturally from the random walk. More formally, consider a set of points
with a metric . We first construct a symmetrized nearest neighbor graph over the points and assign a weight "! #$%
&'(&)+*
to each undirected edge in the graph. The weights are symmetric and ,- /. as we include self-loops; 01 32 for all non-neighbors. Note that the product of weights along a path in the graph relates to the total length of the path in the same way as the edge weights relate to the distances between the corresponding points. The one-step transition probabilities 4 ,' from 5 to 6 are obtained directly from these weights: 4 ,'7 8' 9 (1) ( 4 8': 2 for any non-neighbor 6 ). While the weights ,' are symmetric, the transition probabilities 4 ,' generally are not, because the normalization varies from node to node. We use ;=<?> @AB6
C 5 to denote the D step transition probabilities ( D here should be interpreted as a parameter, not as a random variable). If we organize the one step transition probabilities as a matrix E whose 5 6 -th entry is 4 8' , we can simply use a matrix power to calculate ;=<?> @FB6
C 5 G IH E <BJ 8' (2) The matrix E is row stochastic so that rows sum to 1. We assume that the starting point for the Markov random walk is chosen uniformly at random, i.e., ;KL5 M N.+)PO . We can now evaluate the probability that the Markov process started from 5 given that it ended up in 6 after D steps. These conditional probabilities ;=@> <PL5 C 6 define our new representation for the examples. In other words, each point 6 is associated with a vector of conditional probabilities ;@> <PL5 C 6 , 5 Q.A &O . The points in this representation are close whenever they have nearly the same distribution over the starting states. This representation is crucially affected by the time scale parameter D . When DSRUT , all the points become indistinguishable provided that the original neighborhood graph is connected. Small values of D , on the other hand, merge points in small clusters. In this representation D controls the resolution at which we look at the data points (cf [3]). The representation is also influenced by , * , and the local distance metric , which together define the one-step transition probabilities (see section 4). 3 Parameter estimation for classification Given a partially labeled data set V(XW Y P
Z[XW Y Z\
Z^]=+
S , we wish to classify the unlabeled points. The labels may come from two or more classes, and typically, the number of labeled points _ is a small fraction of the total points O . Our classification model assumes that each data point has a label or a distribution ;K Y C 5 over the class labels. These distributions are unknown and represent the parameters to be estimated. Now given a point 6 , which may be labeled or unlabeled, we interpret the point as a sample from the D step Markov random walk. Since labels are associated with the original (starting) points, the posterior probability of the label for point 6 is given by ; Y C 6 G ;K Y C 5 ;=@> <P 5PC 6 (3) To classify the 6 -th point, we choose the class that maximizes the posterior: ' arg
; Y C 6 . We will now discuss two techniques for estimating the unknown parameters ;K Y C 5 : maximum likelihood with EM, and maximum margin subject to constraints. 3.1 EM estimation The estimation criterion here is the conditional log-likelihood of the labeled points Z '= ;K W Y ' C 6 G Z 'V = ;K W Y ' C 5 ;@> <PL5 C 6 P (4) Since ;=@> < 5 C 6 are fixed for any specific D , this objective function is jointly concave in the free parameters and has a unique maximum value. The concavity also guarantees that this optimization is easily performed via the EM algorithm. Let ;K 5PC 6 FW Y ' be the soft assignment for component 5 given B6 AW Y ' , i.e., ;KL5 C 6 XW Y '+ ;K W Y ' C 5 ;=@> < 5 C 6 . The EM algorithm iterates between the E-step, where ;KL5 C 6 XW Y '+ are recomputed from the current estimates of ;K Y C 5 , and the M-step where we update ;K Y C 5 9 '! #" $% $ ;KL5 C 6 XW Y '+&)G9 ' ;K 5 C 6 AW Y '+ , (see [1]). 3.2 Margin based estimation An alternative discriminative formulation is also possible, one that is more sensitive to individual classification decisions rather than the product of their likelihoods. Define the margin of the classifier on labeled point 6 and class to be & '(' ;) * Y W Y ' C 6 # ; + Y C 6 . For correct classification, the margin should be nonnegative for all classes other than W Y ' , i.e. & '!'-, 2 , and be zero for the correct class & '." $% =0. During training, find the parameters ;K Y C 5 that maximize the average margin on the labeled points, thereby forcing most of them to be correctly classified. Unbalanced classes are handled by the per class margin, and we obtain the linear program
/.0 $ > 132 4 %65 . 7 7 # . Z '= 8 '9= . O 8 0 '1 & '(' subject to (5) ;) * Y W Y ' C 6 :, ; Y C 6 ; & '('=< 6?> . _ < @> .G 7 (6) 8 = ;K Y C 5 G . and 2
A ;K Y C 5 BA . < 5 (7) Here 7 denotes the number of classes and O 8 0 '1 gives the number of labeled points in the same class as 6 . The solution is achieved at extremal points of the parameter set and thus it is not surprising that the optimal parameters ;K Y C 5 reduce to hard values (0 or 1). The solution to this linear program can be found in closed form: ;K Y C 5 G DC . if arg E .F 9 'G " $ % ;=@> <P 5PC 6 2 otherwise. (8) unlabeled labeled +1 labeled −1 Figure 1: Top left: local connectivity for =5 neighbors. Below are classifications using Markov random walks for D =3, 10, and 30 (top to bottom, left to right), estimated with average margin. There are two labeled points (large cross, triangle) and 148 unlabeled points, classified (small crosses, triangles) or unclassified (small dots). The resulting posterior probabilities can also be written compactly as ;) + Y C 6 - 9 # /.0 $ > 1*= ;=@> <P 5PC 6 . The closed form solution for the label distributions facilitates an easy cross-validated setting of the various parameters involved in the example representations. The large margin restricts the 4 dimension of the classifier (section 3.4) and encourages generalization to correct classification of the unlabeled points as well. Note that the margins are bounded and have magnitude less than 1, reducing the risk that any single point would dominate the average margin. Moreover, this criterion maximizes a sum of probabilities, whereas likelihood maximizes a product of probabilities, which is easily dominated by low probability outliers. Other margin-based formulations are also possible. For separable problems, we can maximize the minimum margin instead of the average margin. In the case of only two classes, we then have only one global margin parameter & for all labeled points. The algorithm focuses all its attention at the site of the minimum margin, which unfortunately could be an outlier. If we tackled noisy or non-separable problems by adding a linear slack variable to each constraint, we would arrive at the average margin criterion given above (because of linearity). Average- and min-margin training yields hard parameters 0 or 1. The risk of overfitting is controlled by the smooth representation and can be regularized by increasing the time parameter D . If further regularization is desired, we have also applied the maximum entropy discrimination framework [2, 1] to bias the solution towards more uniform values. This additional regularization has resulted in similar classification performance but adds to the computational cost. 3.3 Examples Consider an example (figure 1) of classification with Markov random walks. We are given 2 labeled and 148 unlabeled points in an intertwining two moons pattern. This pattern has a manifold structure where distances are locally but not globally Euclidean, due to the curved arms. Therefore, the pattern is difficult to classify for traditional algorithms using global metrics, such as SVM. We use a Euclidean local metric, =5 and * =0.6 (the box has extent ), and show three different timescales. At D =3 the random walk has not connected all unlabeled points to some labeled point. The parameters for unconnected points do not affect likelihood or margin, so we assign them uniformly to both classes. The other points have a path to only one of the classes, and are therefore fully assigned to that class. At D =10 all points have paths to labeled points but the Markov process has not mixed well. Some paths do not follow the curved high-density structure, and instead cross between the two clusters. When the Markov process is well-mixed at D =30, the points are appropriately labeled. The parameter assignments are hard, but the class posteriors are weighted averages and remain soft. 3.4 Sample size requirements Here we quantify the sample size that is needed for accurate estimation of the labels for the unlabeled examples. Since we are considering a transduction problem, i.e., finding labels for already observed examples, the sample size requirements can be assessed directly in terms of the representation matrix. As before, the probabilities ;[@> < L5 C and ;=@> <PL5 C 6 denote the conditional probabilities of having started the random walk in 5 given that the process ends up in , 6 , respectively. For simplicity, we consider a binary problem with classes 1 and -1, and let M ;K Y /. C 5 # ;K Y # . C 5 . Classification decisions are then directly based on the sign of 6 G 9 = ;@> <PL5 C 6 . Lemma 1 Consider the absolute distance between the representations of two points &'S 9 = C ;=@> < 5 C # ;@> < L5 C 6 C . The 4 dimension [5] of the binary transductive classifier 6 is upper bounded by the number of connected components of a graph with O nodes and adjacency matrix E , where E &'S . if &' A & and zero otherwise. Proof: To evaluate 4 , a measure of the capacity of the classifier, we count the number of complete labelings Y ' consistent with the margin constraints Y ' B6 B, & for all 6 (labeled and unlabeled points). First, we establish that all examples and 6 for which &' A & must have the same label. This follows directly from C # 6 C A V C ;@> < L5 C # ;=@> < 5 C 6 CC ; Y . C 6 # ;) Y # . C 6 C (9) A V C ;@> < L5 C # ;=@> < 5 C 6 C &' (10) as this difference must be larger than & for the discriminant functions to have different signs. Since any pair of examples for which &'
A & share the same label, different labels can be assigned only to examples not connected by the &'@A & relation, i.e., examples in distinct connected components. This theorem applies more generally to any transductive classifier based on a weighted representation of examples so long as the weights are bounded in H # .X . J . To determine the sample size needed for a given dataset, and a desired classification margin & , let 4 dimension. With high probability we can correctly classify the unlabeled points given labeled examples [4]. This can also be helpful to determine timescale D since it is reflected in the 4 , for example 4 O for D =0 and 4 . for D = T for the full range of & > H 2^ J . 0 5 10 15 20 0.2 0.4 0.6 0.8 1 t average margin per class Class Mac Class Win 2 4 8 16 32 64 128 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 # labeled exampels error Markov avg margin Markov min margin Markov max ent SVM labeled only Figure 2: Windows vs. Mac text data. Left: Average per class margins for different D , 16 labeled documents. Right: Classification accuracy, between 2 and 128 labeled documents, for Markov random walks and best SVM. 4 Choices for , , , and The classifier is robust to rough heuristic choices of , , and * , as follows. The local similarity measure is typically given (Euclidean distance). The local neighborhood size should be on the order of the manifold dimensionality, sufficiently small to avoid introducing edges in the neighborhood graph that span outside the manifold. However, must be large enough to preserve local topology, and ideally large enough to create a singly connected graph, yielding an ergodic Markov process. The local scale parameter * trades off the emphasis on shortest paths (low * effectively ignores distant points), versus volume of paths (high * ). The smoothness of the random walk representation depends on D , the number of transitions. This is a regularization parameter akin to the kernel width of a density estimator. In the limiting case D =1, we employ only the local neighborhood graph. As a special case, we obtain the kernel expansion representation [1] by D =1, = O , and squared Euclidean distance. If all points are labeled, we obtain the -nearest neighbors classifier by D =1, * R T . In the limiting case D = T the representation for each node becomes a flat distribution over the points in the same connected component. We can choose D based on a few unsupervised heuristics, such as the mixing time to reach the stationary distribution, or dissipation of mutual information [3]. However, appropriate D depends on the classification task. For example, if classes change quickly over small distances, we want a sharper representation given by smaller D . Crossvalidation could provide a supervised choice of D but requires too many labeled points for good accuracy. Instead, we propose to choose D that maximizes the average margin per class, on both labeled and unlabeled data. Plot F 9 '! * 0 '1* 9 ' & '(' for each c, separately for labeled and unlabeled points to avoid issues of their relative weights. For labeled points,
6 S W Y ' , for unlabeled points, 6 is the class assigned by the classifier. Figure 2 shows the average margin as a function of D , for a large text dataset (section 5). We want large margins for both classes simultaneously, so D is a good choice, and also gave the best cross-validation accuracy. 4.1 Adaptive time scales So far, we have employed a single global value of D . However, the desired smoothness may be different at different locations (akin to adaptive kernel widths in kernel density estimation). At the simplest, if the graph has multiple connected components, we can set individual D for each component. Ideally, each point has its own time scale, and the choice of time scale is optimized jointly with the classifier parameters. Here we propose a restricted version of this criterion where we find individual time scales D ' for each unlabeled point but estimate a single timescale for labeled points as before. The principle by which we select the time scales for the unlabeled points encourages the node identities to become the only common correlates for the labels. More precisely, define ;K Y C 6 for any unlabeled point 6 as ;K Y C 6 G . ' # #" $ $ ;=@> < % L5 C 6 P (11) where ' 9 ;=@> < % L5 C 6 and both summations are only over the labeled points. Moreover, let ;K Y be the overall probability over the labels across the unlabeled points or ;K Y G ' ;KB6 ;K Y C 6 (12) where ;KB6 is uniform over the unlabeled points, corresponding to the start distribution. Note that ;K Y remains a function of all the individual time scales for the unlabeled points. With these definitions, the principle for setting the time scales reduces to maximizing the mutual information between the label and the node identity: D + D - arg E < 2 2 < Y
6 G arg E < 2 2 < Y # ;K 6 Y C 6 A (13) Y and Y C 6 are the marginal and conditional entropies over the labels and are computed on the basis of ;K Y and ;K Y C 6 , respectively. Note that the ideal setting of the time scales would be one that determines the labels for the unlabeled points uniquely on the basis of only the labeled examples while at the same time preserving the overall variability of the labels across the nodes. This would happen, for example, if the labeled examples fall on distinct connected components. We optimize the criterion by an axis parallel search, trying only discrete values of D ' large enough that at least one labeled point is reached from each unlabeled point. We initialize D ' to the smallest number of transitions needed to reach a labeled point. Empirically we have found that this initialization is close to the refined solution given by the objective. The objective is not concave, but separate random initializations generally yield the same answer, and convergence is rapid requiring about 5 iterations. 5 Experimental results We applied the Markov random walk approach to partially labeled text classification, with few labeled documents but many unlabeled ones. Text documents are represented by highdimensional vectors but only occupy low-dimensional manifolds, so we expect Markov random walk to be beneficial. We used the mac and windows subsets from the 20 newsgroups dataset1. There were 958 and 961 examples in the two classes, with 7511 dimensions. We estimated the manifold dimensionality to exceed 7, and a histogram of the distances to the 10 nearest neighbor is peaked at 1.3. We chose a Euclidean local metric, =10, which leads to a single connected component, and * =0.6 for a reasonable falloff. The average margin criterion indicated D , and we also cross-validated and plotted the decay of mutual information over D . We trained both the EM and the margin-based formulations, using between 2 and 128 labeled points, treating all remaining points as unlabeled. We trained on 20 random splits balanced for class labels, and tested on a fixed separate set of 987 points. Results in figure 2 show that Markov random walk based algorithms have 1Processed as 20news-18827, http://www.ai.mit.edu/˜jrennie/20Newsgroups/, removing rare words, duplicate documents, and performing tf-idf mapping. a clear advantage over the best SVM using only labeled data (which had a linear kernel and 7 =3), out of linear and Gaussian kernels, different kernel widths and values of 7 . The advantage is especially noticeable for few labeled points, but decreases thereafter. The average margin classifier performs best overall. It can handle outliers and mislabeled points, unlike the maximum min margin classifier that stops improving once 8 or more labeled points are supplied. The adaptive timescale criterion favors relatively small timescales for this dataset. For 90% of the unlabeled points, it picks the smallest timescale that reaches a labeled point, which is at most 8 for any point. As the number of labeled points increases, shorter times are chosen. For a few points, the criterion picks a maximally smooth representation (the highest timescale considered here, D =12), possibly to increase the Y criterion. However, our preliminary experiments suggest that the adaptive time scales do not have a special classification advantage for this dataset. 6 Discussion The Markov random walk representation of examples provides a robust variable resolution approach to classifying data sets with significant manifold structure and very few labels. The average margin estimation criterion proposed in this context leads to a closed form solution and strong empirical performance. When the manifold structure is absent or unrelated to the classification task, however, our method cannot be expected to derive any particular advantage. There are a number of possible extensions of this work. For example, instead of choosing a single overall resolution or time scale D , we may combine multiple choices. This can be done either by maintaining a few choices explicitly or including all time scales in a parametric form as in < ; D?E ; D E ) ; [7], but it is unclear whether the exponential decay is desirable. To facilitate continuum limit analysis (and establish better correspondence with the underlying density), we can construct the neighborhood graph on the basis of -balls rather than nearest neighbors. Acknowledgements The authors gratefully acknowledge support from Nippon Telegraph & Telephone (NTT) and NSF ITR grant IIS-0085836. References [1] Szummer, M; Jaakkola, T. (2000) Kernel expansions with unlabeled examples. NIPS 13. [2] Jaakkola, T; Meila, M; Jebara, T. (1999) Maximum entropy discrimination. NIPS 12. [3] Tishby, N; Slonim, N. (2000) Data clustering by Markovian relaxation and the Information Bottleneck Method. NIPS 13. [4] Blum, A; Chawla, S. (2001) Learning from Labeled and Unlabeled Data using Graph Mincuts. ICML. [5] Alon, N. et al (1997) Scale-sensitive Dimensions, Uniform Convergence, and Learnability. J. ACM, 44 (4) 615-631 [6] Tenenbaum, J; de Silva, V; Langford J. (2000) A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290 (5500): 2319-2323. [7] Kondor, I; Lafferty J; (2001) Diffusion kernels in continuous spaces. Tech report CMU, to appear.
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Dynamic Time-Alignment Kernel in Support Vector Machine Hiroshi Shimodaira School of Information Science, Japan Advanced Institute of Science and Technology sim@jaist.ac.jp Ken-ichi Noma School of Information Science, Japan Advanced Institute of Science and Technology knoma@jaist.ac.jp Mitsuru Nakai School of Information Science, Japan Advanced Institute of Science and Technology mit@jaist.ac.jp Shigeki Sagayama Graduate School of Information Science and Technology, The University of Tokyo sagayama@hil.t.u-tokyo.ac.jp Abstract A new class of Support Vector Machine (SVM) that is applicable to sequential-pattern recognition such as speech recognition is developed by incorporating an idea of non-linear time alignment into the kernel function. Since the time-alignment operation of sequential pattern is embedded in the new kernel function, standard SVM training and classification algorithms can be employed without further modifications. The proposed SVM (DTAK-SVM) is evaluated in speaker-dependent speech recognition experiments of hand-segmented phoneme recognition. Preliminary experimental results show comparable recognition performance with hidden Markov models (HMMs). 1 Introduction Support Vector Machine (SVM) [1] is one of the latest and most successful statistical pattern classifier that utilizes a kernel technique [2, 3]. The basic form of SVM classifier which classifies an input vector x ∈Rn is expressed as g(x) = N X i=1 αiyiφ(xi) · φ(x) + b = N X i=1 αiyiK(xi, x) + b, (1) where φ is a non-linear mapping function φ(x) : Rn 7→Rn′, (n ≪n′), “·” denotes the inner product operator, xi, yi and αi are the i-th training sample, its class label, and its Lagrange multiplier, respectively, K is a kernel function, and b is a bias. Despite the successful applications of SVM in the field of pattern recognition such as character recognition and text classification, SVM has not been applied to speech recognition that much. This is because SVM assumes that each sample is a vector of fixed dimension, and hence it can not deal with the variable length sequences directly. Because of this, most of the efforts that have been made so far to apply SVM to speech recognition employ linear time normalization, where input feature vector sequences with different lengths are aligned to same length [4]. A variant of this approach is a hybrid of SVM and HMM (hidden Markov model), in which HMM works as a pre-processor to feed time-aligned fixed-dimensional vectors to SVM [5]. Another approach is to utilize probabilistic generative models as a SVM kernel function. This includes the Fisher kernels [6, 7], and conditional symmetric independence (CSI) kernels [8], both of which employ HMMs as the generative models. Since HMMs can treat sequential patterns, SVM that employs the generative models based on HMMs can handle sequential patterns as well. In contrast to those approaches, our approach is a direct extension of the original SVM to the case of variable length sequence. The idea is to incorporate the operation of dynamic time alignment into the kernel function itself. Because of this, the proposed new SVM is called “Dynamic Time-Alignment Kernel SVM (DTAKSVM)”. Unlike the SVM with Fisher kernel that requires two training stages with different training criteria, one is for training the generative models and the second is for training the SVM, the DTAK-SVM uses one training criterion as well as the original SVM. 2 Dynamic Time-Alignment Kernel We consider a sequence of vectors X = (x1, x2, · · · , xL), where xi ∈Rn, L is the length of the sequence, and the notation |X| is sometimes used to represent the length of the sequence instead. For simplification, we at first assume the so-called linear SVM that does not employ non-linear mapping function φ. In such case, the kernel operation in (1) is identical to the inner product operation. 2.1 Formulation for linear kernel Assume that we have two vector sequences X and V . If these two patterns are equal in length, i.e. |X| = |V | = L, then the inner product between X and V can be obtained easily as a summation of each inner product between xk and vk for k = 1, · · · , L: X · V = L X k=1 xk · vk, (2) and therefore an SVM classifier can be defined as given in (1). On the other hand in case where the two sequences are different in length, the inner product can not be calculated directly. Even in such case, however, some sort of inner product like operation can be defined if we align the lengths of the patterns. To that end, let ψ(k), θ(k) be the time-warping functions of normalized time frame k for the pattern X and V , respectively, and let “◦” be the new inner product operator instead of the original inner product “·”. Then the new inner product between the two vector sequences X and V can be given by X ◦V = 1 L L X k=1 xψ(k) · vθ(k), (3) where L is a normalized length that can be either |X|, |V | or arbitrary positive integer. There would be two possible types of time-warping functions. One is a linear timewarping function and the other is a non-linear time-warping function. The linear time-warping function takes the form as ψ(k) = ⌈(|X|/L)k⌉, θ(k) = ⌈(|V |/L)k⌉, where ⌈x⌉is the ceiling function which gives the smallest integer that is greater than or equal to x. As it can be seen from the definition given above, the linear warping function is not suitable for continuous speech recognition, i.e. frame-synchronous processing, because the sequence lengths, |X| and |V |, should be known beforehand. On the other hand, non-linear time warping, or dynamic time warping (DTW) [9] in other word, enables frame-synchronous processing. Furthermore, the past research on speech recognition has shown that the recognition performance by the non-linear time normalization outperforms the one by the linear time normalization. Because of these reasons, we focus on the non-linear time warping based on DTW. Though the original DTW uses a distance/distortion measure and finds the optimal path that minimizes the accumulated distance/distortion, the DTW that is employed for SVM uses inner product or kernel function instead and finds the optimal path that maximizes the accumulated similarity: X ◦V = max ψ,θ 1 Mψθ L X k=1 m(k)xψ(k) · vθ(k), (4) subject to 1 ≤ψ(k) ≤ψ(k + 1) ≤|X|, ψ(k + 1) −ψ(k) ≤Q, (5) 1 ≤θ(k) ≤θ(k + 1) ≤|V |, θ(k + 1) −θ(k) ≤Q, where m(k) is a nonnegative (path) weighting coefficient, Mψθ is a (path) normalizing factor, and Q is a constant constraining the local continuity. In the standard DTW, the normalizing factor Mψθ is given as PL k=1 m(k), and the weighting coefficients m(k) are chosen so that Mψθ is independent of the warping functions. The above optimization problem can be solved efficiently by dynamic programming. The recursive formula in the dynamic programming employed in the present study is as follows G(i, j) = max ( G(i −1, j) + Inp(i, j), G(i −1, j −1) + 2 Inp(i, j), G(i, j −1) + Inp(i, j), ) (6) where Inp(i, j) is the standard inner product between the two vectors corresponding to point i and j. As a result, we have X ◦V = G(|X|, |V |)/(|X| + |V |). (7) 2.2 Formulation for non-linear kernel In the last subsection, a linear kernel, i.e. the inner product, for two vector sequences with different lengths has been formulated in the framework of dynamic time-warping. With a little constraint, similar formulation is possible for the case where SVM’s non-linear mapping function Φ is applied to the vector sequences. To that end, Φ is restricted to the one having the following form: Φ(X) = (φ(x1), φ(x2), · · · , φ(xL)), (8) where φ is a non-linear mapping function that is applied to each frame vector xi, as given in (1). It should be noted that under the above restriction Φ preserves the original length of sequence at the cost of losing long-term correlations such as the one between x1 and xL. As a result, a new class of kernel can be defined by using the extended inner product introduced in the previous section; Ks(X, V ) = Φ(X) ◦Φ(V ) (9) = max ψ,θ 1 Mψθ L X k=1 m(k)φ(xψ(k)) · φ(vθ(k)) (10) = max ψ,θ 1 Mψθ L X k=1 m(k)K(xψ(k), vθ(k)). (11) We call this new kernel “dynamic time-alignment kernel (DTAK)”. 2.3 Properties of the dynamic time-alignment kernel It has not been proven that the proposed function Ks(, ) is really an SVM’s admissible kernel which guarantees the existence of a feature space. This is because that the mapping function to a feature space is not independent but dependent on the given vector sequences. Although a class of data-dependent asymmetric kernel for SVM has been developed in [10], our proposed function is more complicated and difficult to analyze because the input data is a vector sequence with variable length and non-linear time normalization is embedded in the function. Instead, what have been known about the proposed function so far are (1) Ks is symmetric, (2) Ks satisfies the Cauchy-Schwartz like inequality described bellow: Proposition 1 Ks(X, V )2 ≤Ks(X, X)Ks(V, V ) (12) Proof For simplification, we assume that normalized length L is fixed, and omit m(k) and Mψθ in (11). Using the standard Cauchy-Schwartz inequality, the following inequality holds: Ks(X, V ) = max ψ,θ L X k=1 φ(xψ(k)) · φ(vθ(k)) = L X k=1 φ(xψ∗(k)) · φ(vθ∗(k)) (13) ≤ L X k=1 ∥φ(xψ∗(k)) ∥∥φ(vθ∗(k)) ∥, (14) where ψ∗(k), θ∗(k) represent the optimal warping functions that maximize the RHS of (13). On the other hand, Ks(X, X) = max ψ,θ L X k=1 φ(xψ(k)) · φ(xθ(k)) = L X k=1 φ(xψ+(k)) · φ(xθ+(k)). (15) Because here we assume that ψ+(k), θ+(k) are the optimal warping functions that maximize (15), for any warping functions including ψ ∗(k), the following inequality holds: Ks(X, X) ≥ L X k=1 φ(xψ∗(k)) · φ(xψ∗(k)) = L X k=1 ∥φ(xψ∗(k)) ∥2 . (16) In the same manner, the following holds: Ks(V, V ) ≥ L X k=1 φ(vθ∗(k)) · φ(vθ∗(k)) = L X k=1 ∥φ(vθ∗(k)) ∥2 . (17) Therefore, Ks(X, X)Ks(V, V ) −Ks(X, V )2 ≥ L X k=1 ∥φ(xψ∗(k)) ∥2 ! L X k=1 ∥φ(vθ∗(k)) ∥2 ! − L X k=1 ∥φ(xψ∗(k)) ∥∥φ(vθ∗(k)) ∥ !2 = L X i=1 L X j=i+1 ∥φ(xψ∗(i)) ∥∥φ(vθ∗(j)) ∥−∥φ(xψ∗(j)) ∥∥φ(vθ∗(i)) ∥ 2 ≥0 (18) ■ 3 DTAK-SVM Using the dynamic time-alignment kernel (DTAK) introduced in the previous section, the discriminant function of SVM for a sequential pattern is expressed as g(X) = N X i=1 αiyiΦ(X(i)) ◦Φ(X) + b (19) = N X i=1 αiyiKs(X(i), X) + b, (20) where X(i) represents the i-th training pattern. As it can be seen from these expressions, the SVM discriminant function for time sequence has the same form with the original SVM except for the difference in kernels. It is straightforward to deduce the learning problem which is given as min W,b,ξi 1 2W ◦W + C N X i=1 ξi, (21) subject to yi(W ◦Φ(X(i)) + b) ≥1 −ξi, (22) ξi ≥0, i = 1, · · · , N. Again, since the formulation of learning problem defined above is almost the same with that for the original SVM, same training algorithms for the original SVM can be used to solve the problem. 4 Experiments Speech recognition experiments were carried out to evaluated the classification performance of DTAK-SVM. As our objective is to evaluate the basic performance of the proposed method, very limited task, hand-segmented phoneme recognition task in which positions of target patterns in the utterance are known, was chosen. Continuous speech recognition task that does not require phoneme labeling would be our next step. 4.1 Experimental conditions The details of the experimental conditions are given in Table 1. The training and evaluation samples were collected from the ATR speech database: A-set (5240 Table 1: Experimental conditions Experiment-1 Experiment-2 Speaker dependency dependent dependent Phoneme classes 6 voiced consonants 5 vowels Speakers 5 males 5 males and 5 females Training samples 200 samples per phoneme 500 samples per phoneme Evaluation samples 2,035 samples in all per speaker 2500 samples in all per speaker Signal sampling 12kHz, 10ms frame-shift Feature values 13-MFCCs and 13-∆MFCCs Kernel type RBF (radial basis function): K(xi, xj) = exp(−∥xi−xj∥2 γ2 ) 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 Correct classification rate [%] RBF-sigma C=0.1 C=1.0 C=10 (a) Recognition performance 0 20 40 60 80 100 1 2 3 4 5 6 7 8 9 10 # SVs / # training samples [%] RBF-sigma C=0.1 C=1.0 C=10.0 (b) Number of SVs Figure 1: Experimental results for Experiment-1 (6 voiced-consonants recognition) showing (a) correct classification rate and (b) the number of SVs as a function of γ (the parameter of RBF kernel). Japanese words in vocabulary). In consonant-recognition task (Experiment-1), only six voiced-consonants /b,d,g,m,n,N/ were used to save time. The classification task of those 6 phonemes without using contextual information is considered as a relatively difficult task, whereas the classification of 5 vowels /a,i,u,e,o/ (Experiment-2) is considered as an easier task. To apply SVM that is basically formulated as a two-class classifier to the multiclass problem, “one against the others” type of strategy was chosen. The proposed DTAK-SVM has been implemented with the publicly available toolkit, SVMTorch [11]. 4.2 Experimental results Fig. 1 depicts the experimental results for Experiment-1, where average values over 5 speakers are shown. It can be seen in Fig. 1 that the best performance of 95.8% was achieved at γ = 2.0 and C = 10. Similar results were obtained for Experiment-2 as given in Fig. 2. 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 Correct classification rate [%] RBF-sigma (a) Recognition performance 0 20 40 60 80 100 1 2 3 4 5 6 7 8 9 10 # SVs / # training samples [%] RBF-sigma (b) Number of SVs Figure 2: Experimental results for Experiment-2 (5 vowels recognition) showing (a) correct classification rate and (b) the number of SVs as a function of γ (the parameter of RBF kernel). Table 2: Recognition performance comparison of DTAK-SVM with HMM. Results of Experiment-1 for 1 male and 1 female speakers are shown. (numbers represent correct classification rate [%]) # training samples/phoneme Model male female 50 100 200 50 100 200 HMM (1 mix.) 75.0 69.1 77.1 72.2 65.5 76.6 HMM (4 mix.) 83.3 84.7 90.9 77.3 76.4 86.4 HMM (8 mix.) 82.8 87.0 92.4 74.6 79.3 88.5 HMM (16 mix.) 79.9 85.0 93.2 72.9 78.7 89.8 DTAK-SVM 83.8 85.9 92.1 83.5 81.8 87.7 Next, the classification performance of DTAK-SVM was compared with that of the state-of-the-art HMM. In order to see the effect of generalization performance on the size of training data set and model complexity, experiments were carried out by varying the number of training samples (50, 100, 200), and mixtures (1,4,8,16) for each state of HMM. The HMM used in this experiment was a 3-states, continuous density, Gaussian-distribution mixtures with diagonal covariances, contextindependent model. HTK [12] was employed for this purpose. The parameters of DTAK-SVM were fixed to C = 10, γ = 2.0. The results for Experiment-1 with respect to 1 male and 1 female speakers are given in Table 2. It can be said from the experimental results that DTAK-SVM shows better classification performance when the number of training samples is 50, while comparable performance when the number of samples is 200. One might argue that the number of training samples used in this experiment is not enough at all for HMM to achieve best performance. But such shortage of training samples occurs often in HMMbased real-world speech recognition, especially when context-dependent models are employed, which prevents HMM from improving the generalization performance. 5 Conclusions A novel approach to extend the SVM framework for the sequential-pattern classification problem has been proposed by embedding a dynamic time-alignment operation into the kernel. Though long-term correlations between the feature vectors are omitted at the cost of achieving frame-synchronous processing for speech recognition, the proposed DTAK-SVMs demonstrated comparable performance in hand-segmented phoneme recognition with HMMs. The DTAK-SVM is potentially applicable to continuous speech recognition with some extension of One-pass search algorithm [9]. References [1] V. N. Vapnik, Statistical Learning Theory. Wiley, 1998. [2] B. Sch¨olkopf, C. J. Burges, and A. J. Smola, eds., Advances in Kernel Methods. The MIT Press, 1998. [3] “Kernel machine website,” 2000. http://www.kernel-machines.org/. [4] P. Clarkson, “On the Use of Support Vector Machines for Phonetic Classification,” in ICASSP99, pp. 585–588, 1999. [5] A. Ganapathiraju and J. Picone, “Hybrid SVM/HMM architectures for speech recognition,” in ICSLP2000, 2000. [6] Tommi S. Jaakkola and David Haussler, “Exploiting generative models in discriminative classifiers,” in Advances in Neural Information Processing Systems 11 (M. S. Kearns and S. A. Solla and D. A. Cohn, ed.), pp. 487–493, The MIT Press, 1999. [7] N. Smith and M. Niranjan, “Data-dependent Kernels in SVM classification of speech patterns,” in ICSLP-2000, vol. 1, pp. 297–300, 2000. [8] C. Watkins, “Dynamic Alignment Kernels,” in Advances in Large Margin Classifiers (A. J. Smola and P. L. Bartlett and B. Sch¨olkopf and D. Schuurmans, ed.), ch. 3, pp. 39–50, The MIT Press, 2000. [9] L. Rabiner and B. Juang, Fundamental of Speech Recognition. Prentice Hall, 1993. [10] K. Tsuda, “Support Vector Classifier with Asymmetric Kernel Functions,” in European Symposium on Artificial Neural Networks (ESANN), pp. 183–188, 1999. [11] R. Collobert, “SVMTorch: A Support Vector Machine for Large-Scale Regression and Classification Problems,” 2000. http://www.idiap.ch/learning/SVMTorch.html. [12] “The Hidden Markov Model Toolkit (HTK).” http://htk.eng.cam.ac.uk/.
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Associative memory in realistic neuronal networks P.E. Latham* Department of Neurobiology University of California at Los Angeles Los Angeles, CA 90095 pel@ucla.edu Abstract Almost two decades ago, Hopfield [1] showed that networks of highly reduced model neurons can exhibit multiple attracting fixed points, thus providing a substrate for associative memory. It is still not clear, however, whether realistic neuronal networks can support multiple attractors. The main difficulty is that neuronal networks in vivo exhibit a stable background state at low firing rate, typically a few Hz. Embedding attractor is easy; doing so without destabilizing the background is not. Previous work [2, 3] focused on the sparse coding limit, in which a vanishingly small number of neurons are involved in any memory. Here we investigate the case in which the number of neurons involved in a memory scales with the number of neurons in the network. In contrast to the sparse coding limit, we find that multiple attractors can co-exist robustly with a stable background state. Mean field theory is used to understand how the behavior of the network scales with its parameters, and simulations with analog neurons are presented. One of the most important features of the nervous system is its ability to perform associative memory. It is generally believed that associative memory is implemented using attractor networks - experimental studies point in that direction [4- 7], and there are virtually no competing theoretical models. Perhaps surprisingly, however, it is still an open theoretical question whether attractors can exist in realistic neuronal networks. The "realistic" feature that is probably hardest to capture is the steady firing at low rates - the background state - that is observed throughout the intact nervous system [8- 13]. The reason it is difficult to build an attractor network that is stable at low firing rates, at least in the sparse coding limit, is as follows [2,3]: Attractor networks are constructed by strengthening recurrent connections among sub-populations of neurons. The strengthening must be large enough that neurons within a sub-population can sustain a high firing rate state, but not so large that the sub-population can be spontaneously active. This implies that the neuronal gain functions - the firing rate of the post-synaptic neurons as a function of the average • http) / culture.neurobio.ucla.edu/ "'pel firing rate of the pre-synaptic neurons - must be sigmoidal: small at low firing rate to provide stability, high at intermediate firing rate to provide a threshold (at an unstable equilibrium), and low again at high firing rate to provide saturation and a stable attractor. In other words, a requirement for the co-existence of a stable background state and multiple attractors is that the gain function of the excitatory neurons be super linear at the observed background rates of a few Hz [2,3]. However - and this is where the problem lies - above a few Hz most realistic gain function are nearly linear or sublinear (see, for example, Fig. Bl of [14]). The superlinearity requirement rests on the implicit assumption that the activity of the sub-population involved in a memory does not affect the other neurons in the network. While this assumption is valid in the sparse coding limit, it breaks down in realistic networks containing both excitatory and inhibitory neurons. In such networks, activity among excitatory cells results in inhibitory feedback. This feedback, if powerful enough, can stabilize attractors even without a saturating nonlinearity, essentially by stabilizing the equilibrium (above considered unstable) on the steep part of the gain function. The price one pays, though, is that a reasonable fraction of the neurons must be involved in each of the memories, which takes us away from the sparse coding limit and thus reduces network capacity [15]. 1 The model A relatively good description of neuronal networks is provided by synaptically coupled, conductance-based neurons. However, because communication is via action potentials, such networks are difficult to analyze. An alternative is to model neurons by their firing rates. While this is unlikely to capture the full temporal network dynamics [16], it is useful for studying equilibria. In such simplified models, the equilibrium firing rate of a neuron is a function of the firing rates of all the other neurons in the network. Letting VEi and VIi denote the firing rates of the excitatory and inhibitory neurons, respectively, and assuming that synaptic input sums linearly, the equilibrium equations may be written ¢Ei (~Af;EVEj' ~Af;'V'j) ¢;; (~AifVEj, ~ Ai!V,j) . (la) (lb) Here ¢E and ¢I are the excitatory and inhibitory gain functions and Aij determines the connection strength from neuron j to neuron i. The gain functions can, in principle, be derived from conductance-based model equations [17]. Our goal here is to determine under what conditions Eq. (1) allows both attractors and a stable state at low firing rate. To accomplish this we will use mean field theory. While this theory could be applied to the full set of equations, to reduce complexity we make a number of simplifications. First, we let the inhibitory neurons be completely homogeneous (¢Ii independent of i and connectivity to and from inhibitory neurons all-to-all and uniform). In that case, Eq. (lb) becomes simply VI = ¢(VE' VI) where VE and VI are the average firing rates of the excitatory and inhibitory neurons. Solving for VI and inserting the resulting expression into Eq. (la) results in the expression VEi = ¢Ei(LjAijEVEj,AEIVI(VE)) where AEI == LjAijI. Second, we let cP Ei have the form cP Ei (u, v) = cP E( Xi + bu - ev) where Xi is a Gaussian random variable, and similarly for cPT (except with different constants band e and no dependence on i). Finally, we assume that cPT is threshold linear and the network operates in a regime in which the inhibitory firing rate is above zero. With these simplifications, and a trivial redefinition of constants, Eq. (la) becomes (2) We have dropped the sub and superscript E, since Eq. (2) refers exclusively to excitatory neurons, defined v to be the average firing rate, v == N-1 Li Vi, and rescaled parameters. We let the function cP be 0(1), so f3 can be interpreted as the gain. The parameter p is the number of memories. The reduction from Eq. (1) to Eq. (2) was done solely to simplify the analysis; the techniques we will use apply equally well to the general case, Eq. (1). Note that the gain function in Eq. (2) decreases with increasing average firing rate, since it's argument is -(1 + a)v and a is positive. This negative dependence on v arises because we are working in the large coupling regime in which excitation and inhibition are balanced [18,19]. The negative coupling to firing rate has important consequences for stability, as we will see below. We let the connectivity matrix have the form Here N is the number of excitatory neurons; Cij , which regulates the degree of connectivity, is lie with probability e and and 0 with probability (1 - e) (except Cii = 0, meaning no autapses); g(z) is an 0(1) clipping function that keeps weights from falling below zero or getting too large; (g) is the mean value of g(z), defined in Eq. (4) below; Wi j , which corresponds to background connectivity, is a random matrix whose elements are Gaussian distributed with mean 1 and variance 8w2 ; and Jij produces the attractors. We will follow the Hopfield prescription and write Jij as (3) where f is the coupling strength among neurons involved in the memories, and the patterns TJ",i determine which neurons participate in each memory. The TJ",i are a set of uncorrelated vectors with zero mean and unit variance. In simulations we use TJ",i = [(1 - 1)11]1/2 with probability 1 and -(f 1(1 - IW/2 with probability 1 - I, so a fraction 1 of the neurons are involved in each memory. Other choices are unlikely to significantly change our results. 2 Mean field equations The main difficulty in deriving the mean field equations from Eq. (2) is separating the signal from the noise. Our first step in this endeavor is to analyze the noise associated with the clipped weights. To do this we break Cijg(Wij + Jij ) into two pieces: Cijg(Wij + Jij) = (g) + (g')Jij + bCij where The angle brackets around 9 represent an average over the distributions of W ij and Jij, and a prime denotes a derivative. In the large p limit, bCij can be treated as a random matrix whose main role is to increase the effective noise [20]. The mean of bCij is zero and its variance normalized to (g)2 / c, which we denote (Y2, is given by For large p, the elements of Jij are Gaussian with zero mean and variance E2, so the averages involving 9 can be written (4) where k can be either an exponent or a prime and the "I" in g(1 + z) corresponds to the mean of Wij . In our simulations we use the clipping function g(z) = z if z is between 0 and 2, 0 if z ::::; 0 and 2 if z ;::: 2. Our main assumptions in the development of a mean field theory are that L;#i bCijvj is a Gaussian random variable, and that bCij and Vj are independent. Consequently, where (v2 ) == N- 1 L;i v; is the second moment of the firing rate. Letting 8i be a zero mean Gaussian random variable with variance 82 == (Y2 (v2) / cN, we can use the above assumptions along with the definition of Jij , Eq. (3), to write Eq. (20) as (5) We have defined the clipped memory strength, Ee, as Ee == E(g')/(g). While it is not totally obvious from the above equations, it can be shown that both (Y2 and Ee become independent of E for large E. This makes network behavior robust to changes in E, the strength of the memories, so long as E is large. Derivation ofthe mean field equations from Eq. (5) follow standard methods [21,22]. For definiteness we take ¢(x) to be threshold linear: ¢(x) = max(O, x). For the case of one active memory, the mean field equations may then be written in the form w 1 + r q {3Ec ( ) 1- r flF1 w,z (32E~ [1J2 1] a(l-r)2 CE~+(1-q)2 [F2(z)+jflF2(w,z)] {32B2a2/x2 (1 ~ r)2 a [Fl (z) + j flFl (w, zW a{3Ecq 1-q (3E~ [Fo(z) + jflFo(w,z)] 1 + a Ec (6a) (6b) (6c) (6d) where a == piN is the load parameter, Xo and B6/P are the mean and variance of of Xi (see Eq. (2)), and, recall, j is the fraction of neurons that participate in each memory. The functions Fk and flFk are defined by 1 00 d~ k 2 -z (27r )1/2 (z +~) exp( -~ /2) Fdw + z) - Fk(Z) . For large negative z, Fk(z) vanishes as exp(-z2/2), while for large positive z, Fk(Z) --+ zk /k!. The average firing rate, v, and strength of the memory, m == N- 1 2::i rJljVj (taken without loss of generality to be the overlap with pattern 1), are given in terms of z and was Xo v m 3 Results The mean field equations can be understood by examining Eqs. (6a) and (6b). The first of these, Eq. (6a), is a rescaled form of the equation for the overlap, m. (From the definition of flFt given above, it can be seen that m is proportional to w for small w). This equation always has a solution at w = 0 (and thus m = 0), which corresponds to a background state with no memories active. If {3Ec is large enough, there is a second solution with w (and thus m) greater than zero. This second solution corresponds to a memory. The other relevant equation, Eq. (6b), describes the behavior of the mean firing rate. This equation looks complicated only because the noise - the variation in firing rate from neuron to neuron - must be determined self-consistently. The solutions to Eqs. (6a) and (6b) are plotted in Fig. 1 in the z-w plane. The solid lines, including the horizontal line at w = 0, represents the solution to Eq. (6a), the w ~ , ',.: ... t t ... w=o z Figure 1: Graphical solution of Eqs. (6a) and (6b). Solid lines, including the one at w = 0: solution to Eq. (6a). Dashed line: solution to Eq. (6b). The arrows indicate approximate flow directions: vertical arrows indicate time evolution of w at fixed z; horizontal arrows indicate time evolution of z at fixed w. The black squares show potentially stable fixed points. Note the exchange of stability to the right of the solid curve, indicating that intersections too far to the right will be unstable. dashed line the solution to Eq. (6b), and their intersections solutions to both. While stability cannot be inferred from the equilibrium equations, a reasonable assumption is that the evolution equations for the firing rates, at least near an equilibrium, have the form Tdvi/dt = ¢i Vi. In that case, the arrows represent flow directions, and we see that there are potentially stable equilibria at the intersections marked by the solid squares. Note that in the sparse coding limit, f ---+ 0, z is independent of w, meaning that the mean firing rate, v , is independent of the overlap, m. In this limit there can be no feedback to inhibitory neurons, and thus no chance for stabilization. In terms of Fig. 1, the effect of letting f ---+ 0 is to make the dashed line vertical. This eliminates the possibility of the upper stable equilibrium (the solid square at w > 0), and returns us to the situation where a superlinear gain function is required for attractors to be embedded, as discussed in the introduction. Two important conclusions can be drawn from Fig. 1. First, the attractors can be stable even though the gain functions never saturate (recall that we used thresholdlinear gain functions). The stabilization mechanism is feedback to inhibitory neurons, via the -(1 + a)v term in Eq. (2). This feedback is what makes the dashed line in Fig. 1 bend, allowing a stable equilibrium at w > O. Second, if the dashed line shifts to the right relative to the solid line, the background becomes destabilized. This is because there is an exchange of stability, as indicated by the arrows. Thus, there is a tradeoff: w, and thus the mean firing rate of the memory neurons, can be increased by shifting the dashed line up or to the right, but eventually the background becomes destabilized. Shifting the dashed line to the left, on the other hand, will eventually eliminate the solution at w > 0, destroying all attractors but the background. For fixed load parameter Ct, fraction of neurons involved in a memory, f, and degree of connectivity, c, there are three parameters that have a large effect on the location of the equilibria in Fig. 1: the gain, {3, the clipped memory strength, fe, and the degree of heterogeneity in individual neurons, Bo. The effect of the first two can be seen in Fig. 2, which shows a stability plot in the f-{3 plane, determined by numerically solving the the equations Tdvi/dt = ¢i Vi (see Eq. (2)). The filled circles indicate regions where memories were embedded without destabilizing the background, open circles indicate regions where no memories could be embedded, and xs indicate regions where the background was unstable. As discussed above, fe becomes approximately independent of the strength of the memories, f, when f becomes large. This is seen in Fig. 2A, in which network behavior stabilizes when f becomes larger than about 4; increasing f beyond 8 would, presumably, produce no surprises. There is some sensitivity to gain, (3: when f > 4, memories co-existed with a stable background for (3 in a ±15% range. Although not shown, the same was true of eo: increasing it by about 20% eliminated the attractors; decreasing it by the same amount destabilized the background. However, more detailed analysis indicates that the stability region gets larger as the number of neurons in the network, N, increases. This is because fluctuations destabilize the background, and those fluctuations fall off as N - 1/ 2 . A B 70 N '.2[\momoo !:S 35 ~ I background 0 E: 11111",1 11 o 000000000000000 •••• o 2 4 0 4 8 ~ E Figure 2: A. Stability diagram, found by solving the set of equations Tdv;/dt = cPi Vi with the argument of cPi given in Eq. (2). Filled circles: memories co-exist with a stable background (also outlined with solid lines); open circles: memories could not be embedded; x s: background was unstable. The average background rate, when the background was stable, was around 3 Hz. The network parameters were eo = 6, Xo = 1.5, a = 0.5, c = 0.3, 0: = 2.5%, and 8w = 0.3. 2000 neurons were used in the simulations. These parameters led to an effective gain, pl/2 (3fc, of about 10, which is consistent with the gain in large networks in which each neuron receives "-'5-10,000 inputs. B . Plot of firing rate of memory neurons, m, when the memory was active (upper trace) and not active (lower trace) versus f at (3 = 2. 4 Discussion The main outcome of this analysis is that attractors can co-exist with a stable background when neurons have generic threshold-linear gain functions, so long as the sparse coding limit is avoided. The parameter regime for this co-existence is much larger than for attractor networks that operate in the sparse coding limit [2,23]. While these results are encouraging, they do not definitively establishing that attractors can exist in realistic networks. Future work must include inhibitory neurons, incorporate a much larger exploration of parameter space to ensure that the results are robust, and ultimately involve simulations with spiking neurons. 5 Acknowledgements This work was supported by NIMH grant #R01 MH62447. References [1] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci., 79:2554- 2558, 1982. [2] N. BruneI. Persistent activity and the single-cell frequency-current curve in a cortical network model. Network: Computation in Neural Systems, 11:261- 280, 2000. [3] P.E. Latham and S.N. Nirenberg. Intrinsic dynamics in cultured neuronal networks. Soc. Neuroscience Abstract, 25:2259, 1999. [4] J.M. Fuster and G.E. Alexander. Neuron activity related to short-term memory. Science, 173:652- 654, 1971. [5] Y. Miyashita. Inferior temporal cortex: where visual perception meets memory. Annu Rev Neurosci, 16:245- 263, 1993. [6] P.S. Goldman-Rakic. Cellular basis of working memory. Neuron, 14:477- 485, 1995. [7] R Romo, C.D. Brody, A. Hernandez, and L. Lemus. Neuronal correlates of parametric working memory in the prefrontal cortex. Nature, 399:470- 473, 1999. [8] C.D. Gilbert. Laminar differences in receptive field properties of cells in cat primary visual cortex. J. Physiol. , 268:391- 421, 1977. [9] Y. Lamour, P. Dutar, and A. Jobert. Cerebral neorcortical neurons in the aged rat: spontaneous activity, properties of pyramidal tract neurons and effect of acetylcholine and cholinergic drugs. Neuroscience, 16:835- 844, 1985. [10] M.B. Szente, A. Baranyi, and C.D. Woody. Intracellular injection of apamin reduces a slow potassium current mediating afterhyperpolarizations and IPSPs in neocortical neurons of cats. Brain Res., 461:64- 74, 1988. [11] I. Salimi, H.H. Webster, and RW. Dykes. Neuronal activity in normal and deafferented forelimb somatosensory cortex of the awake cat. Brain Res., 656:263- 273, 1994. [12] J.F. Herrero and P.M. Headley. Cutaneous responsiveness of lumbar spinal neurons in awake and halothane-anesthetized sheep. J. Neurophysiol. , 74:1549- 1562, 1997. [13] K. Ochi and J.J. Eggermont. Effects of quinine on neural activity in cat primary auditory cortex. Hear. Res., 105:105- 18, 1997. [14] P.E. Latham, B.J. Richmond, P.G. Nelson, and S.N. Nirenberg. Intrinsic dynamics in neuronal networks. I. Theory. J. Neurophysiol., 83:808- 827, 2000. [15] M.V. Tsodyks and M.V. Feigel'man. The enhanced storage capacity in neural networks with low activity level. Europhys. Lett., 6:101- 105, 1988. [16] A. Treves. Mean-field analysis of neuronal spike dynamics. Network, 4:259- 284, 1993. [17] O. Shriki, D. Hansel, and H. Sompolonski. Modeling neuronal networks in cortex by rate models using the current-frequency response properties of cortical cells. Soc. Neuroscience Abstract, 24:143, 1998. [18] C. van Vreeswijk and H. Sompolinsky. Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274: 1724- 1726, 1996. [19] C. van Vreeswijk and H. Sompolinsky. Chaotic balanced state in a model of cortical circuits. Neural Comput., 10:1321- 1371, 1998. [20] H. Sompolinsky. Neural networks with nonlinear synapses and a static noise. Phys. Rev. A, 34:2571- 2574, 1986. [21] J. Hertz, A. Krogh, and RG. Palmer. Introduction to the theory of neural computation. Addison Wesley, Redwood City, CA, 1991. [22] A.N. Burkitt. Retrieval properties of attractor neural that obey Dale's law using a self-consistent signal-to-noise analysis. Network: Computation in Neural Systems, 7:517- 531, 1996. [23] D.J. Amit and N. BruneI. Dynamics of a recurrent network of spiking neurons before and following learning. Network, 8:373- 404, 1997.
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A General Greedy Approximation Algorithm with Applications Tong Zhang IBM T.J. Watson Research Center Yorktown Heights, NY 10598 tzhang@watson.ibm.com Abstract Greedy approximation algorithms have been frequently used to obtain sparse solutions to learning problems. In this paper, we present a general greedy algorithm for solving a class of convex optimization problems. We derive a bound on the rate of approximation for this algorithm, and show that our algorithm includes a number of earlier studies as special cases. 1 Introduction The goal of machine learning is to obtain a certain input/output functional relationship from a set of training examples. In order to do so, we need to start with a model of the functional relationship. In practice, it is often desirable to find the simplest model that can explain the data. This is because simple models are often easier to understand and can have significant computational advantages over more complicated models. In addition, the philosophy of Occam’s Razor implies that the simplest solution is likely to be the best solution among all possible solutions, In this paper, we are interested in composite models that can be expressed as linear combinations of basic models. In this framework, it is natural to measure the simplicity of a composite model by the number of its basic model components. Since a composite model in our framework corresponds to a linear weight over the basic model space, therefore our measurement of model simplicity corresponds to the sparsity of the linear weight representation. In this paper, we are interested in achieving sparsity through a greedy optimization algorithm which we propose in the next section. This algorithm is closely related to a number of previous works. The basic idea was originated in [5], where Jones observed that if a target vector in a Hilbert space is a convex combination of a library of basic vectors, then using greedy approximation, one can achieve an error rate of with basic library vectors. The idea has been refined in [1] to analyze the approximation property of sigmoidal functions including neural networks. The above methods can be regarded as greedy sparse algorithms for functional approximation, which is the noise-free case of regression problems. A similar greedy algorithm can also be used to solve general regression problems under noisy conditions [6]. In addition to regression, greedy approximation can also be applied to classification problems. The resulting algorithm is closely related to boosting [2] under the additive model point of view [3]. This paper shows how to generalize the method in [5, 1] for analyzing greedy algorithms (in their case, for functional approximation problems) and apply it to boosting. Detailed analysis will be given in Section 4. Our method can also be used to obtain sparse kernel representations for regression problems. Such a sparse representation is what support vector regression machines try to achieve. In this regard, the method given in this paper complements some recently proposed greedy kernel methods for Gaussian processes such as [9, 10]. The proposed greedy approximation method can also be applied to other prediction problems with different loss functions. For example, in density estimation, the goal is to find a model that has the smallest negative log-likelihood. A greedy algorithm was analyzed in [7]. Similar approximation bounds can be directly obtained under the general framework proposed in this paper. We proceed as follows. Section 2 formalizes the general class of problems considered in this paper, and proposes a greedy algorithm to solve the formulation. The convergence rate of the algorithm is investigated in Section 3. Section 4 includes a few examples that can be obtained from our algorithm. Some final concluding remarks are given in Section 5. 2 General Algorithm In machine learning, our goal is often to predict an unobserved output value based on an observed input vector . This requires us to estimate a functional relationship from a set of example pairs of . Usually the quality of the predictor can be measured by a loss function that is problem dependent. In this paper, we are interested in the following scenario: given a family of basic predictors parameterized by , we want to obtain a good predictor that lies in the convex hull of with the fewest possible terms:
, where are nonnegative weights so that
. This family of models can be regarded as additive models in statistics [4]. Formally, each basic model can be regarded as a vector in a linear functional space. Our problem in its most general form can thus be described as to find a vector in the convex hull of to minimize a functional of that measures the quality of . This functional of plays the role of loss function for learning problems. More formally, we consider a linear vector space , and a subset ! . Denote by "$# the convex hull of : "$# %
'&)( * +-, +/.0+21 , +2354 ( * +6, +
.7+ 8 9:; =<?>A@B where we use < > to denote the set of positive integers. We consider the following optimization problem on "$# : CDFE GIHKJML$NOBP Q $R (1) In this paper, we assume that is a differentiable convex function on "/# . We propose the following algorithm to approximately solved (1). Algorithm 2.1 (Sparse greedy approximation) given Q 8"/# for
R R/R find Q = and 4 , that minimize , Q , Q
let Q
, MQ , Q end For simplicity, we assume that the minimization of in Algorithm 2.1 can be exactly achieved at each step. This assumption is not essential, and can be easily removed using a slightly more refined analysis. However due to the space limitation, we shall not consider this generalization. For convenience, we introduce the following quantity Q %
Q CD7E G HKJML/NOBP Q $R In the next section, we show that under appropriate regularity conditions, Q 4 as , where Q is computed from Algorithm 2.1. In addition, the convergence rate can be bounded as . 3 Approximation bound Given any convex function , we have the following proposition, which is a direct consequence of the definition of convexity. In convex analysis, The gradient can be replaced by the concept of subgradient, which we do not consider in this paper for simplicity. Proposition 3.1 Consider a convex function Q , and two vectors Q and Q , we have Q Q 3 Q Q Q where is the gradient of . The following lemma is the main theoretical result of the paper, which bounds the performance of each greedy solution step in Algorithm 2.1. We assume that is second order differentiable. Lemma 3.1 Let
"! G # G HKJML$NOBP Q $ Q Q where we assume that the Hessian $ of exists everywhere in "$# . For all vectors Q !"$# : if Q 3&% , we have CDFE ' H)( #
* # G H O ,+ MQ + Q if Q &% , we have CDFE ' H)( #
* # G H O .,+ MQ + Q Q Q $ / R Proof. Using Taylor expansion and the definition of , we have the following inequality for all Q 8"$# , Q ! , and + 0 4 21 , ,+ MQ +BQ Q + Q3 Q Q + $ % $R Now, consider two sequences , +2354 and Q + = (
R R/R : ), such that ( + , +
. Multiply the above inequality (with Q replaced by Q + ) by , + , and sum over , we obtain ( * + , + ,+ Q + Q + Q + ( * + , + Q + Q ) Q + $ R It is easy to see that this implies the inequality CDFE + + MQ + Q + Q + A( * + , + Q + Q Q + $ R Using Proposition 3.1, we obtain CDFE + + MQ + Q + Q + ( * + , + Q + Q + $ R Since in the above, , + and Q + are arbitrary, therefore ( + , + Q + can be used to express any vector Q 8"$# . This implies CDFE G H O ,+ MQ + Q Q + CDFE G Q Q + $ R Now by setting +
CD Q CDFE G Q % in the above inequality, we obtain the lemma. Using the above lemma and note that Q , it is easy to obtain the following theorem by induction. For space limitation, we skip the proof. Theorem 3.1 Under the assumptions of Lemma 3.1, Algorithm 2.1 approximately solves (1), and the rate of convergence for 3 is given by Q / R If Q &% , then we also have Q / NG
P R 4 Examples In this section, we discuss the application of Algorithm 2.1 in some learning problems. We show that the general formulation considered in this paper includes some previous formulations as special cases. We will also compare our results with similar results in the literature. 4.1 Regression In regression, we would like to approximate as so that the expected loss of
# $ is small, where we use the squared loss for simplicity (this choice is obviously not crucial in our framework). # is the expectation over and , which often corresponds to the empirical distribution of pairs. It may also represent the true distribution for some other engineering applications. Given a set of basis functions with , we may consider the following regression formulation that is slightly different from (1): CDFE # * $ (2) s.t. * where is a positive regularization parameter which is used to control the size of the weight vector . The above formulation can be readily converted into (1) by considering the following set of basic vectors:
& 1 @ R We may start with
4 ( Q
4 ) in Algorithm 2.1. Since the quantity in Lemma 3.1 can be bounded as
"! $ $ R This implies that the sparse solution Q in Algorithm 2.1, represented as weight and (
R/R R ), satisfies the following inequality: # * $ CDFE
# # ( * + + + $ $ 3! $ for all 3 . This leads to the original functional approximation results in [1, 5] and its generalization in [6]. The sparse regression algorithm studied in this section can also be applied to kernel methods. In this case, corresponds to the input training data space & R/R R 6@ , and the basis predictors are of the form
. Clearly, this corresponds to a special case of (2). A sparse kernel representation can be obtained easily from Algorithm 2.1 which leads to provably good approximation rate. Our sparse kernel regression formulation is related to Gaussian processes, where greedy style algorithms have also been proposed [9, 10]. The bound given here is comparable to the bound given in [10] where a sparse approximation rate of the form was obtained. 4.2 Binary classification and Boosting In binary classification, the output value & @ is a discrete variable. Given a continuous model , we consider the following prediction rule:
if 354 if 4 R The classification error (we shall ignore the point
4 , which is assumed to occur rarely) can be given by
if 54 4 if 4 R Unfortunately, this classification error function is not convex, which cannot be handled in our formulation. In fact, even in many other popular methods, such as logistic regression and support vector machines, some kind of convex formulations have to be employed. Although it is possible for us to analyze their formulations, in this section, we only consider the following form of loss that is closely related to Adaboost [2]:
D # ! (3) where is a scaling factor. Again, we consider a set of basis predictors 0 21 , which are often called weak learners in the boosting literature. We would like to find a strong learner as a convex combination of weak learners to approximately minimize the above loss: CD7E D # ! * (4) s.t. * 354 R (5) This can be written as formulation (1) with
& 1 4 @ R Using simple algebra, it is easy to verify that "! # HKJML$NOBP $ # ! $ # ! $ R We start with
4 in Algorithm 2.1. Theorem 3.1 implies that the sparse solution Q , represented as weight and (
R R/R ), satisfies the following inequality: # ! * CDFE
# # ! ( * + + + / $ (6) for all 3 . Weight in the above inequality is non-negative. Now we consider the special situation that there exists 4 such that CDFE
# # ! ( * + + + ! $R (7) This condition will be satisfied in the large margin linearly separable case where there exists + 354 + and 4 such that and for all data , ( * + + + 3 R Now, under (7), we obtain from (6) that * ! / $ $R Fix any 3 , we can choose
to obtain * ! $ R (8) This implies that the misclassification error rate decays exponentially. The exponential decay of misclassification error is the original motivation of Adaboost [2]. Boosting was later viewed as greedy approximation in the additive model framework [3]. From the learning theory perspective, the good generalization ability of boosting is related to its tendency to improve the misclassification error under a positive margin [8]. From this point of view, inequality (8) gives a much more explicit margin error bound (which decreases exponentially) than a related result in [8]. In the framework of additive models, Adaboost corresponds to the exponential loss (3) analyzed in this section. As pointed out in [3], other loss functions can also be used. Using our analysis, we may also obtain sparse approximation bounds for these different loss functions. However, it is also easy to observe that they will not lead to the exponential decay of classification error in the separable case. Although the exponential loss in (3) is attractive for separable problems due to the exponential decay of margin error, it is very sensitive to outliers in the non-separable case. We shall mention that an interesting aspect of boosting is the concept of adaptive resampling or sample reweighting. Although this idea has dominated the interpretation of boosting algorithms, it has been argued in [3] that adaptive resampling is only a computational by-product. The idea corresponds to a Newton step approximation in the sparse greedy solution of
in Algorithm 2.1 under the additive model framework which we consider here. Our analysis further confirmed that the greedy sparse solution of an additive model in (1), rather than reweighting itself is the key component in boosting. In our framework, it is also much easier to related the idea of boosting to the greedy function approximation method outlined in [1, 5]. 4.3 Mixture density estimation In mixture density estimation, the output is the probability density function of the input vector at . The following negative log-likelihood is commonly used as loss function:
D where 3 4 is a probability density function. Again, we consider a set of basis predictors , which are often called mixture components. We would like to find a mixture probability density model as a convex combination of mixture components to approximately minimize the negative log-likelihood: CDFE D * (9) s.t. *
%3 4 R (10) This problem was studied in [7]. The quantity defined in Lemma 3.1 can be computed as:
"! N P # N P HKJML/NOBP $ $ $
"! # $ $ $ R An approximation bound can now be directly obtained from Theorem 3.1. It has a form similar to the bound given in [7]. 5 Conclusion This paper studies a formalization of a general class of prediction problems in machine learning, where the goal is to approximate the best model as a convex combination of a family of basic models. The quality of the approximation can be measured by a loss function which we want to minimize. We proposed a greedy algorithm to solve the problem, and we have shown that for a variety of loss functions, a convergence rate of can be achieved using a convex combination of basic models. We have illustrated the consequence of this general algorithm in regression, classification and density estimation, and related the resulting algorithms to previous methods. References [1] A.R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory, 39(3):930–945, 1993. [2] Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. J. Comput. Syst. Sci., 55(1):119–139, 1997. [3] Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Additive logistic regression: A statistical view of boosting. The Annals of Statistics, 28(2):337–407, 2000. With discussion. [4] T. J. Hastie and R. J. Tibshirani. Generalized additive models. Chapman and Hall Ltd., London, 1990. [5] Lee K. Jones. A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. Ann. Statist., 20(1):608–613, 1992. [6] Wee Sun Lee, P.L. Bartlett, and R.C. Williamson. Efficient agnostic learning of neural networks with bounded fan-in. IEEE Transactions on Information Theory, 42(6):2118–2132, 1996. [7] Jonathan Q. Li and Andrew R. Barron. Mixture density estimation. In S.A. Solla, T.K. Leen, and K.-R. M¨uller, editors, Advances in Neural Information Processing Systems 12, pages 279–285. MIT Press, 2000. [8] Robert E. Schapire, Yoav Freund, Peter Bartlett, and Wee Sun Lee. Boosting the margin: a new explanation for the effectiveness of voting methods. Ann. Statist., 26(5):1651–1686, 1998. [9] Alex J. Smola and Peter Bartlett. Sparse greedy Gaussian process regression. In Advances in Neural Information Processing Systems 13, pages 619–625, 2001. [10] Tong Zhang. Some sparse approximation bounds for regression problems. In The Eighteenth International Conference on Machine Learning, pages 624–631, 2001.
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Reducing multiclass to binary by coupling probability estimates Bianca Zadrozny Department of Computer Science and Engineering University of California, San Diego La Jolla, CA 92093-0114 zadrozny@cs.ucsd.edu Abstract This paper presents a method for obtaining class membership probability estimates for multiclass classification problems by coupling the probability estimates produced by binary classifiers. This is an extension for arbitrary code matrices of a method due to Hastie and Tibshirani for pairwise coupling of probability estimates. Experimental results with Boosted Naive Bayes show that our method produces calibrated class membership probability estimates, while having similar classification accuracy as loss-based decoding, a method for obtaining the most likely class that does not generate probability estimates. 1 Introduction The two most well-known approaches for reducing a multiclass classification problem to a set of binary classification problems are known as one-against-all and all-pairs. In the one-against-all approach, we train a classifier for each of the classes using as positive examples the training examples that belong to that class, and as negatives all the other training examples. In the all-pairs approach, we train a classifier for each possible pair of classes ignoring the examples that do not belong to the classes in question. Although these two approaches are the most obvious, Allwein et al. [Allwein et al., 2000] have shown that there are many other ways in which a multiclass problem can be decomposed into a number of binary classification problems. We can represent each such decomposition by a code matrix M 1 0 1 k l, where k is the number of classes and l is the number of binary classification problems. If M c b
1 then the examples belonging to class c are considered to be positive examples for the binary classification problem b. Similarly, if M c b 1 the examples belonging to c are considered to be negative examples for b. Finally, if M c b 0 the examples belonging to c are not used in training a classifier for b. For example, in the 3-class case, the all-pairs code matrix is b1 b2 b3 c1 1 1 0 c2 1 0 1 c3 0 1 1 This approach for representing the decomposition of a multiclass problem into binary problems is a generalization of the Error-Correcting Output Codes (ECOC) scheme proposed by Dietterich and Bakiri [Dietterich and Bakiri, 1995]. The ECOC scheme does not allow zeros in the code matrix, meaning that all examples are used in each binary classification problem. Orthogonal to the problem of choosing a code matrix for reducing multiclass to binary is the problem of classifying an example given the labels assigned by each binary classifier. Given an example x, Allwein et al. [Allwein et al., 2000] first create a vector v of length l containing the -1,+1 labels assigned to x by each binary classifier. Then, they compute the Hamming distance between v and each row of M, and find the row c that is closest to v according to this metric. The label c is then assigned to x. This method is called Hamming decoding. For the case in which the binary classifiers output a score whose magnitude is a measure of confidence in the prediction, they use a loss-based decoding approach that takes into account the scores to calculate the distance between v and each row of M, instead of using the Hamming distance. This method is called loss-based decoding. Allwein et al. [Allwein et al., 2000] present theoretical and experimental results indicating that this method is better than Hamming decoding. However, both of these methods simply assign a class label to each example. They do not output class membership probability estimates ˆP C c X x for an example x. These probability estimates are important when the classification outputs are not used in isolation and must be combined with other sources of information, such as misclassification costs [Zadrozny and Elkan, 2001a] or the outputs of another classifier. Given a code matrix M and a binary classification learning algorithm that outputs probability estimates, we would like to couple the estimates given by each binary classifier in order to obtain class probability membership estimates for the multiclass problem. Hastie and Tibshirani [Hastie and Tibshirani, 1998] describe a solution for obtaining probability estimates ˆP C c X x in the all-pairs case by coupling the pairwise probability estimates, which we describe in Section 2. In Section 3, we extend the method to arbitrary code matrices. In Section 4 we discuss the loss-based decoding approach in more detail and compare it mathematically to the method by Hastie and Tibshirani. In Section 5 we present experimental results. 2 Coupling pairwise probability estimates We are given pairwise probability estimates rij x for every class i j, obtained by training a classifier using the examples belonging to class i as positives and the examples belonging to class j as negatives. We would like to couple these estimates to obtain a set of class membership probabilities pi x P C ci X x for each example x. The rij are related to the pi according to ri j x P C i C i C j X x pi x pi x pj x Since we additionally require that ∑i pi x 1, there are k 1 free parameters and k k 1 2 constraints. This implies that there may not exist pi satisfying these constraints. Let nij be the number of training examples used to train the binary classifier that predicts rij. In order to find the best approximation ˆrij x ˆpi x ˆpi x ˆp j x , Hastie and Tibshirani fit the Bradley-Terrey model for paired comparisons [Bradley and Terry, 1952] by minimizing the average weighted Kullback-Leibler distance l x between rij x and ˆrij x for each x, given by l x ∑ i j ni j ri j x logri j x ˆri j x 1 ri j x log1 ri j x 1 ˆri j x The algorithm is as follows: 1. Start with some guess for the ˆpi x and corresponding ˆri j x . 2. Repeat until convergence: (a) For each i 1 2 k ˆpi x ˆpi x ∑j i ni jri j x ∑j i ni j ˆri j x (b) Renormalize the ˆpi x . (c) Recompute the ˆri j x . Hastie and Tibshirani [Hastie and Tibshirani, 1998] prove that the Kullback-Leibler distance between rij x and ˆrij x decreases at each step. Since this distance is bounded below by zero, the algorithm converges. At convergence, the ˆrij are consistent with the ˆpi. The class predicted for each example x is ˆc x argmax ˆpi x . Hastie and Tibshirani also prove that the ˆpi x are in the same order as the non-iterative estimates ˜pi x ∑j i rij x for each x. Thus, the ˜pi x are sufficient for predicting the most likely class for each example. However, as shown by Hastie and Tibshirani, they are not accurate probability estimates because they tend to underestimate the differences between the ˆpi x values. 3 Extending the Hastie-Tibshirani method to arbitrary code matrices For an arbitrary code matrix M, instead of having pairwise probability estimates, we have an estimate rb x for each column b of M, such that rb x P
c I C c
c I J C c X x ∑c I pc x ∑c I J pc x where I and J are the set of classes for which M b 1 and M b 1, respectively. We would like to obtain a set of class membership probabilities pi x for each example x compatible with the rb x and subject to ∑i pi x 1. In this case, the number of free parameters is k 1 and the number of constraints is l 1, where l is the number of columns of the code matrix. Since for most code matrices l is greater than k 1, in general there is no exact solution to this problem. For this reason, we propose an algorithm analogous to the Hastie-Tibshirani method presented in the previous section to find the best approximate probability estimates ˆpi(x) such that ˆrb x ∑c I ˆpc x ∑c I J ˆpc x and the Kullback-Leibler distance between ˆrb x and rb x is minimized. Let nb be the number of training examples used to train the binary classifier that corresponds to column b of the code matrix. The algorithm is as follows: 1. Start with some guess for the ˆpi x and corresponding ˆrb x . 2. Repeat until convergence: (a) For each i 1 2 k ˆpi x ˆpi x ∑b s t M i b 1 nbrb x ∑b s t M i b 1 nb 1 rb x ∑b s t M i b 1 nbˆrb x ∑b s t M i b 1 nb 1 ˆrb x (b) Renormalize the ˆpi x . (c) Recompute the ˆrb x . If the code matrix is the all-pairs matrix, this algorithm reduces to the original method by Hastie and Tibshirani. Let B i be the set of matrix columns for which M i
1 and B i be the set of matrix columns for which M c 1. By analogy with the non-iterative estimates suggested by Hastie and Tibshirani, we can define non-iterative estimates ˜pi x ∑b B i rb x ∑b B i 1 rb x . For the all-pairs code matrix, these estimates are the same as the ones suggested by Hastie and Tibshirani. However, for arbitrary matrices, we cannot prove that the non-iterative estimates predict the same class as the iterative estimates. 4 Loss-based decoding In this section, we discuss how to apply the loss-based decoding method to classifiers that output class membership probability estimates. We also study the conditions under which this method predicts the same class as the Hastie-Tibshirani method, in the all-pairs case. The loss-based decoding method [Allwein et al., 2000] requires that each binary classifier output a margin score satisfying two requirements. First, the score should be positive if the example is classified as positive, and negative if the example is classified as negative. Second, the magnitude of the score should be a measure of confidence in the prediction. The method works as follows. Let f x b be the margin score predicted by the classifier corresponding to column b of the code matrix for example x. For each row c of the code matrix M and for each example x, we compute the distance between f and M c as dL x c l ∑ b 1 L M c b f x b (1) where L is a loss function that is dependent on the nature of the binary classifier and M c b = 0, 1 or 1. We then label each example x with the label c for which dL is minimized. If the binary classification learning algorithm outputs scores that are probability estimates, they do not satisfy the first requirement because the probability estimates are all between 0 and 1. However, we can transform the probability estimates rb x output by each classifier b into margin scores by subtracting 1 2 from the scores, so that we consider as positives the examples x for which rb x is above 1/2, and as negatives the examples x for which rb x is below 1/2. We now prove a theorem that relates the loss-based decoding method to the HastieTibshirani method, for a particular class of loss functions. Theorem 1 The loss-based decoding method for all-pairs code matrices predicts the same class label as the iterative estimates ˆpi x given by Hastie and Tibshirani, if the loss function is of the form L y ay, for any a
0. Proof: We first show that, if the loss function is of the form L y
ay, the loss-based decoding method predicts the same class label as the non-iterative estimates ˜pi x , for the all-pairs code matrix. Dataset #Training Examples #Test Examples #Attributes #Classes satimage 4435 2000 36 7 pendigits 7494 3498 16 10 soybean 307 376 35 9 Table 1: Characteristics of the datasets used in the experiments. The non-iterative estimates ˜pi x are given by ˜pc x ∑ b B c rb x ∑ b B c 1 rb x ∑ b B c rb x ∑ b B c rb x B c where B c and B c are the sets of matrix columns for which M c 1 and M c 1, respectively. Considering that L y ay and f x b rb x 1 2, and eliminating the terms for which M c b 0, we can rewrite Equation 1 as d x c ∑ b B c a rb x 1 2 ∑ b B c a rb x 1 2 a ∑ b B c rb x ∑ b B c rb x 1 2 B c B c For the all-pairs code matrix the following relationship holds: 1 2 B c B c B c k 1 2, where k is the number of classes. So, the distance d x c is d x c a ∑ b B c rb x ∑ b B c rb x B c k 1 2 It is now easy to see that the class c x which minimizes d x c for example x, also maximizes ˜pc x . Furthermore, if d x i d x j then p x i
p x j , which means that the ranking of the classes for each example is the same. Since the non-iterative estimates ˜pc x are in the same order as the iterative estimates ˆpc x , we can conclude that the Hastie-Tibshirani method is equivalent to the loss-based decoding method if L y ay, in terms of class prediction, for the all-pairs code matrix. Allwein et al. do not consider loss functions of the form L y ay, and uses non-linear loss functions such as L y e y. In this case, the class predicted by loss-based decoding may differ from the one predicted by the method by Hastie and Tibshirani. This theorem applies only to the all-pairs code matrix. For other matrices such that B c B c is a linear function of B c (such as the one-against-all matrix), we can prove that loss-based decoding (with L y ay) predicts the same class as the non-iterative estimates. However, in this case, the non-iterative estimates do not necessarily predict the same class as the iterative ones. 5 Experiments We performed experiments using the following multiclass datasets from the UCI Machine Learning Repository [Blake and Merz, 1998]: satimage, pendigits and soybean. Table 1 summarizes the characteristics of each dataset. The binary learning algorithm used in the experiments is boosted naive Bayes [Elkan, 1997], since this is a method that cannot be easily extended to handle multiclass problems directly. For all the experiments, we ran 10 rounds of boosting. Method Code Matrix Error Rate MSE Loss-based (L y y) All-pairs 0.1385 Loss-based (L y e y) All-pairs 0.1385 Hastie-Tibshirani (non-iterative) All-pairs 0.1385 0.0999 Hastie-Tibshirani (iterative) All-pairs 0.1385 0.0395 Loss-based (L y y) One-against-all 0.1445 Loss-based (L y e y) One-against-all 0.1425 Extended Hastie-Tibshirani (non-iterative) One-against-all 0.1445 0.1212 Extended Hastie-Tibshirani (iterative) One-against-all 0.1670 0.0396 Loss-based (L y y) Sparse 0.1435 Loss-based (L y e y) Sparse 0.1425 Extended Hastie-Tibshirani (non-iterative) Sparse 0.1480 0.1085 Extended Hastie-Tibshirani (iterative) Sparse 0.1330 0.0340 Multiclass Naive Bayes 0.2040 0.0651 Table 2: Test set results on the satimage dataset. We use three different code matrices for each dataset: all-pairs, one-against-all and a sparse random matrix. The sparse random matrices have 15 log2 k columns, and each element is 0 with probability 1/2 and -1 or +1 with probability 1/4 each. This is the same type of sparse random matrix used by Allwein et al.[Allwein et al., 2000]. In order to have good error correcting properties, the Hamming distance ρ between each pair of rows in the matrix must be large. We select the matrix by generating 10,000 random matrices and selecting the one for which ρ is maximized, checking that each column has at least one 1 and one 1, and that the matrix does not have two identical columns. We evaluate the performance of each method using two metrics. The first metric is the error rate obtained when we assign each example to the most likely class predicted by the method. This metric is sufficient if we are only interested in classifying the examples correctly and do not need accurate probability estimates of class membership. The second metric is squared error, defined for one example x as SE x ∑j tj x p j x 2, where p j x is the probability estimated by the method for example x and class j, and tj x is the true probability of class j for x. Since for most real-world datasets true labels are known, but not probabilities, t j x is defined to be 1 if the label of x is j and 0 otherwise. We calculate the squared error for each x to obtain the mean squared error (MSE). The mean squared error is an adequate metrics for assessing the accuracy of probability estimates [Zadrozny and Elkan, 2001b]. This metric cannot be applied to the loss-based decoding method, since it does not produce probability estimates. Table 2 shows the results of the experiments on the satimage dataset for each type of code matrix. As a baseline for comparison, we also show the results of applying multiclass Naive Bayes to this dataset. We can see that the iterative Hastie-Tibshirani procedure (and its extension to arbitrary code matrices) succeeds in lowering the MSE significantly compared to the non-iterative estimates, which indicates that it produces probability estimates that are more accurate. In terms of error rate, the differences between methods are small. For one-against-all matrices, the iterative method performs consistently worse, while for sparse random matrices, it performs consistently better. Figure 1 shows how the MSE is lowered at each iteration of the Hastie-Tibshirani algorithm, for the three types of code matrices. Table 3 shows the results of the same experiments on the datasets pendigits and soybean. Again, the MSE is significantly lowered by the iterative procedure, in all cases. For the soybean dataset, using the sparse random matrix, the iterative method again has a lower error rate than the other methods, which is even lower than the error rate using the all-pairs matrix. This is an interesting result, since in this case the all-pairs matrix has 171 columns (corresponding to 171 classifiers), while the sparse matrix has only 64 columns. 0 5 10 15 20 25 30 35 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 Iteration MSE Satimage all−pairs one−against−all sparse Figure 1: Convergence of the MSE for the satimage dataset. pendigits soybean Method Code Matrix Error Rate MSE Error Rate MSE Loss-based (L y y) All-pairs 0.0723 0.0665 Loss-based (L y e y) All-pairs 0.0715 0.0665 Hastie-Tibshirani (non-iterative) All-pairs 0.0723 0.0747 0.0665 0.0454 Hastie-Tibshirani (iterative) All-pairs 0.0718 0.0129 0.0665 0.0066 Loss-based (L y y) One-against-all 0.0963 0.0824 Loss-based (L y e y) One-against-all 0.0963 0.0931 Ext. Hastie-Tibshirani (non-it.) One-against-all 0.0963 0.0862 0.0824 0.0493 Ext. Hastie-Tibshirani (it.) One-against-all 0.1023 0.0160 0.0931 0.0073 Loss-based (L y y) Sparse 0.1284 0.0718 Loss-based (L y e y) Sparse 0.1266 0.0718 Ext. Hastie-Tibshirani (non-it.) Sparse 0.1484 0.0789 0.0798 0.0463 Ext. Hastie-Tibshirani (it.) Sparse 0.1261 0.0216 0.0636 0.0062 Multiclass Naive Bayes 0.2779 0.0509 0.0745 0.0996 Table 3: Test set results on the pendigits and soybean datasets. 6 Conclusions We have presented a method for producing class membership probability estimates for multiclass problems, given probability estimates for a series of binary problems determined by an arbitrary code matrix. Since research in designing optimal code matrices is still on-going [Utschick and Weichselberger, 2001] [Crammer and Singer, 2000], it is important to be able to obtain class membership probability estimates from arbitrary code matrices. In current research, the effectiveness of a code matrix is determined primarily by the classification accuracy. However, since many applications require accurate class membership probability estimates for each of the classes, it is important to also compare the different types of code matrices according to their ability of producing such estimates. Our extension of Hastie and Tibshirani’s method is useful for this purpose. Our method relies on the probability estimates given by the binary classifiers to produce the multiclass probability estimates. However, the probability estimates produced by Boosted Naive Bayes are not calibrated probability estimates. An interesting direction for future work is in determining whether the calibration of the probability estimates given by the binary classifiers improves the calibration of the multiclass probabilities. References [Allwein et al., 2000] Allwein, E. L., Schapire, R. E., and Singer, Y. (2000). Reducing multiclass to binary: A unifying approach for margin classifiers. Journal of Machine Learning Research, 1:113–141. [Blake and Merz, 1998] Blake, C. L. and Merz, C. J. (1998). UCI repository of machine learning databases. Department of Information and Computer Sciences, University of California, Irvine. http://www.ics.uci.edu/ mlearn/MLRepository.html. [Bradley and Terry, 1952] Bradley, R. and Terry, M. (1952). Rank analysis of incomplete block designs, I: The method of paired comparisons. Biometrics, pages 324–345. [Crammer and Singer, 2000] Crammer, K. and Singer, Y. (2000). On the learnability and design of output codes for multiclass problems. In Proceedings of the Thirteenth Annual Conference on Computational Learning Theory, pages 35–46. [Dietterich and Bakiri, 1995] Dietterich, T. G. and Bakiri, G. (1995). Solving multiclass learning problems via error-correcting output codes. Journal of Artificial Intelligence Research, 2:263– 286. [Elkan, 1997] Elkan, C. (1997). Boosting and naive bayesian learning. Technical Report CS97-557, University of California, San Diego. [Hastie and Tibshirani, 1998] Hastie, T. and Tibshirani, R. (1998). Classification by pairwise coupling. In Advances in Neural Information Processing Systems, volume 10. MIT Press. [Utschick and Weichselberger, 2001] Utschick, W. and Weichselberger, W. (2001). Stochastic organization of output codes in multiclass learning problems. Neural Computation, 13(5):1065–1102. [Zadrozny and Elkan, 2001a] Zadrozny, B. and Elkan, C. (2001a). Learning and making decisions when costs and probabilities are both unknown. In Proceedings of the Seventh International Conference on Knowledge Discovery and Data Mining, pages 204–213. ACM Press. [Zadrozny and Elkan, 2001b] Zadrozny, B. and Elkan, C. (2001b). Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers. In Proceedings of the Eighteenth International Conference on Machine Learning, pages 609–616. Morgan Kaufmann Publishers, Inc.
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Linear Time Inference in Hierarchical HMMs Kevin P. Murphy and Mark A. Paskin Computer Science Department University of California Berkeley, CA 94720-1776 murphyk,paskin @cs.berkeley.edu Abstract The hierarchical hidden Markov model (HHMM) is a generalization of the hidden Markov model (HMM) that models sequences with structure at many length/time scales [FST98]. Unfortunately, the original inference algorithm is rather complicated, and takes time, where is the length of the sequence, making it impractical for many domains. In this paper, we show how HHMMs are a special kind of dynamic Bayesian network (DBN), and thereby derive a much simpler inference algorithm, which only takes time. Furthermore, by drawing the connection between HHMMs and DBNs, we enable the application of many standard approximation techniques to further speed up inference. 1 Introduction The Hierarchical HMM [FST98] is an extension of the HMM that is designed to model domains with hierarchical structure, e.g., natural language, XML, DNA sequences [HIM 00], handwriting [FST98], plan recognition [BVW00], visual action recogntion [IB00, ME01, Hoe01], and spatial navigation [TRM01, BVW01]. HHMMs are less expressive than stochastic context free grammars (SCFGs), since they only allows hierarchies of bounded depth, but they are more efficient and easier to learn. Unfortunately, the original inference algorithm
described in [FST98] is somewhat complicated, and takes time, where is the length of the sequence, is the depth of the hierarchy, and is the (maximum) number of states at each level of the hierarchy. In this paper, we show how to represent an HHMM as a dynamic Bayesian network (DBN), and thereby derive a much simpler and faster inference algorithm, which takes at most time; empirically, we find it takes only
! time using the junction tree algorithm. Furthermore, by drawing the connection between HHMMs and DBNs, we enable the application of approximate inference techniques such as belief propagation, which can perform inference in time. " By inference, we mean offline smoothing, i.e., conditioning on a fixed-length observation sequence. This is needed as a subroutine for EM. Once the model has been learned, it will typically be used for online inference (filtering). end end end end 2 3 4 end 5 8 9 7 6 1 0 a b c d x y Figure 1: A 3-level hierarchical automaton representing the regular expression
. Solid lines represent horizontal transitions, dotted lines represent vertical transitions. Letters below a production state represent the symbol that is emitted. The unnumbered root node is considered level 0, and could be omitted if we fully interconnected states 0 and 1. We will describe HHMMs in Section 2, and the original inference algorithm in Section 3. The main contribution of the paper is in Section 4, where we show how to represent an HHMM as a DBN. In Section 5, we discuss how to do efficient inference in this DBN, and in Section 6, we discuss related work. In the full version of this paper, we discuss how to do parameter and structure learning using EM. 2 Hierarchical HMMs HHMMs are like HMMs except the states of the stochastic automaton can emit single observations or strings of observations. (For simplicity of exposition, we shall assume all observations are discrete symbols, but HHMMs can easily be generalized to handle continuous observations, as we discuss in Section 4.1.) Those that emit single symbols are called “production states”, and those that emit strings are termed “abstract states”. The strings emitted by abstract states are themselves governed by sub-HHMMs, which can be called recursively. When the sub-HHMM is finished, control is returned to wherever it was called from; the calling context is memorized using a depth-limited stack. We illustrate the generative process with Figure 1, which shows the state transition diagram of an example HHMM which models the regular expression . We start in the root state, and make a “vertical transition” to one of its children, say state 0. From here, we make another vertical transition to state 2. Since state 2 is a production state, it emits “a” and then makes a “horizontal transition” to state 3. State 3 calls its sub-HMM, which emits x’s and y’s until it enters its end state; then control is returned to the calling state, in this case state 3. We then make a horizontal transition to state 4, emit “b”, and enter the end state, thereby returning control to state 0. Finally, from state 0, we return control to the root, and optionally start again. Any HHMM can be converted to an HMM by creating a state for every possible legal stack configuration
! . If the HHMM transition diagram is a tree, there will be one HMM state for every HHMM production state. If the HHMM transition diagram has shared substructure (such as the sub-expression ), this structure must be duplicated in the HMM, generally resulting in a larger model. It is the ability to reuse sub-models in different contexts that makes HHMMs more powerful than standard HMMs. In particular, the parameters of such shared sub-models only need to be learned once. (Given segmented data, we can train the sub-HMMs separately, and then “glue them together”, but it is also possible to train the HHMM on unsegmented data; see the full version of this paper for details.) 3 Cubic-time inference The inference algorithm for HHMMs presented in [FST98] runs in time and is based on the Inside-Outside algorithm [LY90], an exact inference algorithm for stochastic context-free grammars (SCFGs) which we now describe. In an SCFG, sequences of observations are generated by a set of stochastic production rules. Each production rule stochastically rewrites a non-terminal symbol into either a symbol of the alphabet ( ) or a pair of nonterminal symbols ( ). Observation strings are generated by starting with the distinguished “start” nonterminal , and continually re-writing all non-terminals using stochastic production rules until, finally, only symbols of the alphabet remain. The Inside-Outside algorithm computes
! ! , where ! ! ! !
! is a subsequence. This can then be used to compute the expected sufficient statistics needed by the EM algorithm to learn the parameters of the model. If there are non-terminals in the grammar and the training sequence is of length , then the Inside-Outside algorithm requires time. To see why, note that we must compute
! ! for all end points and , and for all midpoints such that generates ! ! and generates !
! — the three degrees for freedom , and ! gives rise to the term. The term arises because we must consider all , and . The inference algorithm for HHMMs presented in [FST98] is based upon a straightforward adaptation of the Inside-Outside algorithm. The algorithm computes
in state "$# at time ! ! by assuming that sub-state " #
generates ! ! , that a transition to state % occurs, and that " #
generates !
! . Hence the algorithm takes time, where is the total number of states. We can always “flatten” an HHMM into a regular HMM and hence do inference in . Unfortunately, this flat model cannot represent the hierarchical structure, yet alone learn it. In the next section, we show how to represent the HHMM as a DBN, and thereby get the best of both worlds: low time complexity without losing hierarchical structure. 4 Representing the HHMM as a DBN We can represent the HHMM as a dynamic Bayesian network (DBN) as shown in Figure 2. (We assume for simplicity that all production states are at the bottom of the hierarchy; this restriction is lifted in the full version of this paper.) The state of the HMM at level and time is represented by &# ! . The state of the whole HHMM is encoded by the vector ' !
! ( ! ; intuitively, this encodes the contents of the stack, that specifies the complete “path” to take from the root to the leaf state. ) # ! is an indicator variable that is “on” if the HMM at level and time has just “finished” (i.e., is about to enter an end state), otherwise it is off. Note that if ) # ! +* , then ) # ! ,* )
)
)
) )
)
) ) )
*
Figure 2: An HHMM represented as a DBN. is the state at time , level ;
if the HMM at level has finished (entered its exit state), otherwise . Shaded nodes are observed; the remaining nodes are hidden. We may optionally clamp , where is the length of the observation sequence, to ensure all models have finished by the end of the sequence. (A similar trick was used in [Zwe97].) for all ; hence the number of ) nodes that are “off” represents the effective height of the “context stack”, i.e., which level of the hierarchy we are currently on. The downward going arcs between the variables represent the fact that a state “calls” a sub-state. The upward going arcs between the ) variables enforce the fact that a higherlevel HMM can only change state when the lower-level one is finished. This ensures proper nesting of the parse trees, and is the key difference between an HHMM and a hidden Markov decision tree [JGS96]. We will define the conditional probability distributions (CPDs) of each of the node types below, which will complete the definition of the model. We consider the bottom, middle and top layers of the hierarchy separately (since they have different local topology), as well as the first, middle and last time slices. 4.1 Definition of the CPDs Consider the bottom level of the hierarchy. follows a Markov chain with parameters determined by its position in the automaton, which is encoded by the vector of higher-up state variables
!
! (
! , which we will represent by the integer % . When If the topology is sparse, this distribution will be 0 for many values of . This will be discussed in Section 4.2. enters its end state, it will “turn on” ) , to mean it is finished; this will be a signal that higher-level HMMs can now change state. In addition, it will be a signal that the next value of should be drawn from its prior distribution (representing a vertical transition), instead of its transition matrix (representing a horizontal transition). Formally, we can write this as follows:
! !
) !
! % if if * where we have assumed
end. is the transition matrix for level given that the parent variables are in state % , and is just a rescaled version of . Similarly, is the initial distribution for level given that the parent variables are in state % . The equation for ) is simply
) ! *
! % ! end Now consider the intermediate levels. As before, # follows a Markov chain with parameters determined by
#
, and ) # specifies whether we should use the transition matrix or the prior. The difference is that we now also get a signal from below, ) #
, specifying whether the sub-model has finished or not; if it has, we are free to change state, otherwise we must remain in the same state. Formally, we can write this as follows:
# ! # !
) #
!
) # !
#
! % if # if * and # if * and * ) # should “turn on” only if # is “allowed” to enter a final state, the probability of which depends on the current context
#
. Formally, we can write this as follows:
) # ! * # !
#
! % ) #
! if # end if * The top level differs from the intermediate levels in that the node has no parent to specify which distribution to use. The equations are the same as above, except we eliminate the conditioning on
#
! % . (Equivalently, we can imagine a dummy top layer HMM, which is always in state 1: ! * . This is often how HHMMs are represented, so that this top-level state is the root of the overall parse tree, as in Figure 1.) The CPDs for the nodes in the first slice are as follows:
for the top level and
#
#
% # , for . If the observations are discrete symbols, we may represent
! ' ! as a multinomial (i.e., using a table), or by using any of the more parsimonious representations discussed in Section 4.2. If the observations are real-valued vectors, we can use a Gaussian for each value of ' ! , or a mixture of a smaller number of Gaussians, as in [GJ97]. Unlike the automaton representation, the DBN never actually enters an end state (i.e., can never taken on the value “end”), because if it did, it would not be able to emit the symbol . Instead, causes to turn on, and then enters a new (non-terminal) state at time . This means that the DBN and HHMM transition matrices are not identical, but satisfy the following relation: "!$# &% ('*) "+ "!$# ,! where represents the automaton transition matrix, represents the DBN transition matrix, and ) .-0/$1 "! end is the probability of terminating from state . The equations holds because the probability of each horizontal transition in the DBN gets multiplied by the probability that , which is 2'3) ; this product should match the original probability. It is easy to see that the new matrix is also stochastic, as required. 4.2 Parsimonious representations of the CPDs The number of parameters needed to represent
# ! &# !
#
! % as a multinomial is #
. If the state-transition diagram of the hierarchical automaton is sparse, many of the entries in this table will be 0. However, when we are learning a model, we do not know the structure of the state-transition diagram, and must therefore adopt a representation with fewer parameters. There are at least three possibilities: decision trees [BFGK96], softmax nodes, or representing
&# ! # !
#
! % as a mixture of smaller transition matrices at different depths c.f. [SJ99]. See the full version of this paper for details. 5 Linear-time inference We define inference to be computing
for all sets of nodes parents in the DBN. These “family” marginals are needed by EM. The simplest way to do this is to merge all the hidden nodes in each slice into a single “mega node”, ! , with possible values. (The term arises from the binary ) nodes.) We can then apply the forwards-backwards algorithm for HMMs, which takes time. Unfortunately, converting the DBN to an HMM in this way will not be tractable for reasonably large or . (Even storing the transition matrix is likely to consume too much space.) Fortunately, we can do better by exploiting the structure of the model. In [Mur01], we present a way of applying the junction tree (jtree) algorithm to variable-length DBNs; we give a brief sketch here. The algorithm works by performing a forwards-backwards sweep through a chain of jtrees. Each jtree is formed from a “ *
-slice DBN”; this is a DBN that contains all the nodes in slice 1 but only the interface nodes from slice 2. The interface nodes are those nodes in slice 2 that have an incoming temporal arc, plus parents of nodes that have incoming temporal arcs. In the case of an HHMM, the interface is all the nodes. The cost of doing inference in each jtree depends on the sizes of the cliques. Minimizing the maximal clique size is NP-hard, so we used a standard one-step look-ahead (greedy) algorithm [Kja90]. The resulting cliques are hard to interpret, but we can still analyze the complexity. Let be the number of nodes in clique , let be the number of ) nodes, and let
be the number of cliques. Then the cost of inference in a jtree is proportional to
!
"$#&%(' ) *
#&%(' ) +
Empirically we find that, for a wide range of ,
, ,-/.
10 2 3 4 and ,5-/.
! 6 "0 * 2 * 73 4 . Hence a crude upper bound on the cost of inference in each jtree is
, yielding an overall time and space complexity of
. We remind readers that the original algorithm has time complexity, since there can be up to states in the HHMM. The advantage of the new algorithm in practice is clearly illustrated in Figure 3. We can reduce the time (and space) complexity from
to by using approximate DBN inference techniques such as the “factored frontier (FF) algorithm” [MW01], which is equivalent to applying “loopy belief propagation” to the DBN using a left-right scheduling of the messages. (It is still exponential in because of the high fan-in of the nodes.) We can get a further speedup by using a mixture representation of the CPDs 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 running time (seconds) vs sequence length linear cubic Figure 3: Running time vs. sequence length. Both algorithms were implemented in Matlab. The HHMM has , . (see Section 4.2). In this case, we can exploit the form of the CPD to compute the required messages efficiently [Mur99], bringing the overall complexity down to . We remark that all of the above algorithms can also be used for online filtering. In addition, by replacing the sum operator with max, we can do Viterbi segmentation in the usual way. 6 Related work Hidden Markov decision trees (HMDT) [JGS96] are DBNs with a structure similar to Figure 2, but they lack the ) nodes and the upward going arcs; hence they are not able to represent the call-return semantics of the HHMM. Embedded HMMs [NI00] are a special case of HHMMs in which the ending “time” of the sub-HMMs is known in advance (e.g., the sub-HMM models exactly one row of pixels). ([Hoe01] calls these models “hierarchical mixture of Markov chains”.) A variable-duration HMM [Rab89] is a special case of a 2-level HHMM, where the bottom level counts how long we have been in a certain state; when the counter expires, the ) node turns on, and the parent can change state. [BVW00] describes the “Abstract HMM” (AHMM), which is very closely related to HHMMs. These authors are interested in inferring what abstract policy an agent is following by observing its effects in the world. An AHMM is equivalent to an HHMM if we consider # ! to represent the (abstract) policy being followed at level and time ; ! represents the concrete action, which causes the observation. We also need to add a hidden global state variable ! , which is a parent of the ! node, all the ) ! nodes and all the !
nodes. ( ! is hidden to us as observers, but not to the agent performing the actions.) [BVW00] consider abstract policies of the “options” kind [SPS99], which is equivalent to assuming that there are no horizontal transitions. (HAMs [PR97] generalize this by allowing horizontal transitions (i.e., internal state) within a controller.) In addition, they assume that # ! only depends on its immediate parent, #
! , but not its whole context,
#
! , so the nodes become connected by a chain. This enables them to use Rao-Blackwellized particle filtering for approximate online inference: conditioned on the ) nodes, the distribution over the nodes can be represented as a product of marginals, so they can be efficiently marginalized out. Acknowledgements I would like to thank Dr Christopher Schlick for giving me his Matlab implementation of the algorithm, which was used to create part of Figure 3. References [BFGK96] C. Boutilier, N. Friedman, M. Goldszmidt, and D. Koller. Context-Specific Independence in Bayesian Networks. In UAI, 1996. [BVW00] H. Bui, S. Venkatesh, and G. West. On the recognition of abstract Markov policies. In AAAI, 2000. [BVW01] H. Bui, S. Venkatesh, and G. West. Tracking and surveillance in wide-area spatial environments using the Abstract Hidden Markov Model. Intl. J. of Pattern Rec. and AI, 2001. [FST98] Shai Fine, Yoram Singer, and Naftali Tishby. The hierarchical Hidden Markov Model: Analysis and applications. Machine Learning, 32:41, 1998. [GJ97] Z. Ghahramani and M. Jordan. Factorial hidden Markov models. Machine Learning, 29:245–273, 1997. [HIM 00] M. Hu, C. Ingram, M.Sirski, C. Pal, S. Swamy, and C. Patten. A Hierarchical HMM Implementation for Vertebrate Gene Splice Site Prediction. Technical report, Dept. Computer Science, Univ. Waterloo, 2000. [Hoe01] J. Hoey. Hierarchical unsupervised learning of facial expression categories. In ICCV Workshop on Detection and Recognition of Events in Video, 2001. [IB00] Y. Ivanov and A. Bobick. Recognition of visual activities and interactions by stochastic parsing. IEEE Trans. on Pattern Analysis and Machine Intelligence, 22(8):852–872, 2000. [JGS96] M. I. Jordan, Z. Ghahramani, and L. K. Saul. Hidden Markov decision trees. In NIPS, 1996. [Kja90] U. Kjaerulff. Triangulation of graphs – algorithms giving small total state space. Technical Report R-90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990. [LY90] K. Lari and S. J. Young. The estimation of stochastic context-free grammars using the Inside-Outside algorithm. Computer Speech and Language, 4:35–56, 1990. [ME01] D. Moore and I. Essa. Recognizing multitasked activities using stochastic context-free grammar. In CVPR Workshop on Models vs Exemplars in Computer Vision, 2001. [Mur99] K. Murphy. Pearl’s algorithm and multiplexer nodes. Technical report, U.C. Berkeley, Dept. Comp. Sci., 1999. [Mur01] K. Murphy. Applying the junction tree algorithm to variable-length DBNs. Technical report, Comp. Sci. Div., UC Berkeley, 2001. [MW01] K. Murphy and Y. Weiss. The Factored Frontier Algorithm for Approximate Inference in DBNs. In UAI, 2001. [NI00] A. Nefian and M. Hayes III. Maximum likelihood training of the embedded HMM for face detection and recognition. In IEEE Intl. Conf. on Image Processing, 2000. [PR97] R. Parr and S. Russell. Reinforcement learning with hierarchies of machines. In NIPS, 1997. [Rab89] L. R. Rabiner. A tutorial on Hidden Markov Models and selected applications in speech recognition. Proc. of the IEEE, 77(2):257–286, 1989. [SJ99] L. Saul and M. Jordan. Mixed memory markov models: Decomposing complex stochastic processes as mixture of simpler ones. Machine Learning, 37(1):75–87, 1999. [SPS99] R.S. Sutton, D. Precup, and S. Singh. Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artificial Intelligence, 112:181–211, 1999. [TRM01] G. Theocharous, K. Rohanimanesh, and S. Mahadevan. Learning Hierarchical Partially Observed Markov Decision Process Models for Robot Navigation. In ICRA, 2001. [Zwe97] G. Zweig. Speech Recognition with Dynamic Bayesian Networks. PhD thesis, U.C. Berkeley, Dept. Comp. Sci., 1997.
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Sequential noise compensation by sequential Monte Carlo method Kaisheng Yao and Satoshi Nakamura ATR Spoken Language Translation Research Laboratories 2-2-2, Hikaridai Seika-cho, Souraku-gun, Kyoto, 619-0288, Japan E-mail: {kaisheng.yao, satoshi.nakamura}@slt.atr.co.jp Abstract We present a sequential Monte Carlo method applied to additive noise compensation for robust speech recognition in time-varying noise. The method generates a set of samples according to the prior distribution given by clean speech models and noise prior evolved from previous estimation. An explicit model representing noise effects on speech features is used, so that an extended Kalman filter is constructed for each sample, generating the updated continuous state estimate as the estimation of the noise parameter, and prediction likelihood for weighting each sample. Minimum mean square error (MMSE) inference of the time-varying noise parameter is carried out over these samples by fusion the estimation of samples according to their weights. A residual resampling selection step and a Metropolis-Hastings smoothing step are used to improve calculation efficiency. Experiments were conducted on speech recognition in simulated non-stationary noises, where noise power changed artificially, and highly non-stationary Machinegun noise. In all the experiments carried out, we observed that the method can have significant recognition performance improvement, over that achieved by noise compensation with stationary noise assumption. 1 Introduction Speech recognition in noise has been considered to be essential for its real applications. There have been active research efforts in this area. Among many approaches, model-based approach assumes explicit models representing noise effects on speech features. In this approach, most researches are focused on stationary or slow-varying noise conditions. In this situation, environment noise parameters are often estimated before speech recognition from a small set of environment adaptation data. The estimated environment noise parameters are then used to compensate noise effects in the feature or model space for recognition of noisy speech. However, it is well-known that noise statistics may vary during recognition. In this situation, the noise parameters estimated prior to speech recognition of the utterances is possibly not relevant to the subsequent frames of input speech if environment changes. A number of techniques have been proposed to compensate time-varying noise effects. They can be categorized into two approaches. In the first approach, timevarying environment sources are modeled by Hidden Markov Models (HMM) or Gaussian mixtures that were trained by prior measurement of environments, so that noise compensation is a task of identification of the underlying state sequences of the noise HMMs, e.g., in [1], by maximum a posterior (MAP) decision. This approach requires making a model representing different conditions of environments (signal-to-noise ratio, types of noise, etc.), so that statistics at some states or mixtures obtained before speech recognition are close to the real testing environments. In the second approach, environment model parameters are assumed to be timevarying, so it is not only an inference problem but also related to environment statistics estimation during speech recognition. The parameters can be estimated by Maximum Likelihood estimation, e.g., sequential EM algorithm [2][3][4]. They can also be estimated by Bayesian methods. In the Bayesian methods, all relevant information on the set of environment parameters and speech parameters, which are denoted as Θ(t) at frame t, is included in the posterior distribution given observation sequence Y (0 : t), i.e., p(Θ(t)|Y (0 : t)). Except for a few cases including linear Gaussian state space model (Kalman filter), it is formidable to evaluate the distribution updating analytically. Approximation techniques are required. For example, in [5], a Laplace transform is used to approximate the joint distribution of speech and noise parameters by vector Taylor series. The approximated joint distribution can give analytical formula for posterior distribution updating. We report an alternative approach for Bayesian estimation and compensation of noise effects on speech features. The method is based on sequential Monte Carlo method [6]. In the method, a set of samples is generated hierarchically from the prior distribution given by speech models. A state space model representing noise effects on speech features is used explicitly, and an extended Kalman filter (EKF) is constructed in each sample. The prediction likelihood of the EKF in each sample gives its weight for selection, smoothing, and inference of the time-varying noise parameter, so that noise compensation is carried out afterwards. Since noise parameter estimation, noise compensation and speech recognition are carried out frame-byframe, we denote this approach as sequential noise compensation. 2 Speech and noise model Our work is on speech features derived from Mel Frequency Cepstral Coefficients (MFCC). It is generated by transforming signal power into log-spectral domain, and finally, by discrete Cosine transform (DCT) to the cepstral domain. The following derivation of the algorithm is in log-spectral domain. Let t denote frame (time) index. In our work, speech and noise are respectively modeled by HMMs and a Gaussian mixture. For speech recognition in stationary additive noise, the following formula [4] has been shown to be effective in compensating noise effects. For Gaussian mixture kt at state st, the Log-Add method transforms the mean vector µl stkt of the Gaussian mixture by, ˆµl stkt = µl stkt + log(1 + exp(µl n −µl stkt)) (1) where µl n is the mean vector in the noise model. st ∈{1, · · · , S}, kt ∈{1, · · · , M}. S and M each denote the number of states in speech models and the number of mixtures at each state. Superscript l indicates that parameters are in the logspectral domain. After the transformation, the mean vector ˆµl stkt is further transformed by DCT, and then plugged into speech models for recognition of noisy speech. In case of time-varying noise, the µl n should be a function of time, i.e., µl n(t). Accordingly, the compensated mean is ˆµl stkt(t). ) 0 ( 0 0 l k s µ ) 0 ( l Y ) 0 ( l n µ ) 1 ( 1 1 − − − t l k s t t µ ) 1 ( − t Y l ) 1 ( − t l n µ ) (t l k s t t µ ) (t Y l ) (t l n µ ) (T l k s T T µ ) (T Y l ) (T l n µ 0 k 1 − tk tk T k 0s 1 − ts ts Ts Figure 1: The graphical model representation of the dependences of the speech and noise model parameters. st and kt each denote the state and Gaussian mixture at frame t in speech models. µl stkt(t) and µl n(t) each denote the speech and noise parameter. Y l(t) is the noisy speech observation. The following analysis can be viewed in Figure 1. In Gaussian mixture kt at state st of speech model, speech parameter µl stkt(t) is assumed to be distributed in Gaussian with mean µl stkt and variance Σl stkt. On the other hand, since the environment parameter is assumed to be time varying, the evolution of the environment mean vector can be modeled by a random walk function, i.e., µl n(t) = µl n(t −1) + v(t) (2) where v(t) is the environment driving noise in Gaussian distribution with zero mean and variance V . Then, we have, p(st, kt, µl stkt(t), µl n(t)|st−1, kt−1, µl st−1kt−1(t −1), µl n(t −1)) = ast−1stpstktN(µl stkt(t); µl stkt, Σl stkt)N(µl n(t); µl n(t −1), V ) (3) where ast−1st is the state transition probability from st−1 to st, and pstkt is the mixture weight. The above formula gives the prior distribution of the set of speech and noise model parameter Θ(t) = {st, kt, µl stkt(t), µl n(t)}. Furthermore, given observation Y l(t), assume that the transformation by (1) has modeling and measurement uncertainty in Gaussian distribution, i.e., Y l(t) = µl stkt(t) + log (1 + exp (µl n(t) −µl stkt(t))) + wstkt(t) (4) where wstkt(t) is Gaussian with zero mean and variance Σl stkt, i.e., N(·; 0, Σl stkt). Thus, the likelihood of observation Y l(t) at state st and mixture kt is p(Y l(t)|Θ(t)) = N(Y l(t); µl stkt(t) + log (1 + exp (µl n(t) −µl stkt(t))), Σl stkt) (5) Refereeing to (3) and (5), the posterior distribution of Θ(t) given Y l(t) is p(st, kt, µl stkt(t), µl n(t)|Y l(t)) ∝ p(Y l(t)|Θ(t))ast−1stpstktN(µl stkt(t); µl stkt, Σl stkt)N(µl n(t); µl n(t −1), V ) (6) The time-varying noise parameter is estimated by MMSE, given as, ˆµl n(t) = Z µl n(t) µl n(t) X st,kt Z µl stkt(t) p(Θ(t)|Y l(0 : t))dµl stkt(t)dµl n(t) (7) However, it is difficult to obtain the posterior distribution p(Θ(t)|Y l(0 : t)) analytically, since p(µl stkt(t), µl n(t)|Y l(t)) is non-Gaussian in µl stkt(t) and µl n(t) due to the non-linearity in (4). It is thus difficult, if possible, to assign conjugate prior of µl n(t) to the likelihood function p(Y l(t)|Θ(t)). Another difficulty is that the speech state and mixture sequence is hidden in (7). We thus rely on the solution by computational Bayesian approach [6]. 3 Time-varying noise parameter estimation by sequential Monte Carlo method We apply the sequential Monte Carlo method [6] for posterior distribution updating. At each frame t, a proposal importance distribution is sampled whose target is the posterior distribution in (7), and it is implemented by sampling from lower distributions in hierarchy. The method goes through the sampling, selection, and smoothing steps frame-by-frame. MMSE inference of the time-varying noise parameter is a by-product of the steps, carried out after the smoothing step. In the sampling step, the prior distribution given by speech models is set to the proposal importance distribution, i.e., q(Θ(t)|Θ(t −1)) = ast−1stpstktN(µl stkt(t); µl stkt, Σl stkt). The samples are then generated by sampling hierarchically of the prior distribution described as follows: set i = 1 and perform the following steps: 1. sample s(i) t ∼as(i) t−1st 2. sample k(i) t ∼ps(i) t kt 3. sample µl(i) s(i) t k(i) t (t) ∼N(; µl s(i) t k(i) t , Σl s(i) t k(i) t ), and set i = i + 1 4. repeat step 1 to 3 until i = N where superscript (i) denotes the index of samples and N denotes the number of samples. Each sample represents certain speech and noise parameter, which is denoted as Θ(i)(t) = (s(i) t , k(i) t , µl(i) s(i) t k(i) t (t), µl(i) n (t)). The weight of each sample is given by Qt τ=1 p(Θ(τ)(i)|Y l(τ)) q(Θ(τ)(i)|Θ(τ−1)(i)). Refereeing to (6), the weight is calculated by β(i)(t) = p(Y l(t)|Θ(i)(t))N(µl(i) n (t); µl(i) n (t −1), V )ˇβ(i)(t −1) (8) where ˇβ(i)(t −1) is the sample weight from previous frame. The remaining part in the right side of above equation, in fact, represents the prediction likelihood of the state space model given by (2) and (4) for each sample (i). This likelihood can be obtained analytically since after linearization of (4) with respect to µl n(t) at µl(i) n (t−1), an extended Kalman filter (EKF) can be obtained, where the prediction likelihood of the EKF gives the weight, and the updated continuous state of EKF gives µl(i) n (t). In practice, after the above sampling step, the weights of all but several samples may become insignificant. Given the fixed number of samples, this will results in degeneracy of the estimation, where not only some computational resources are wasted, but also estimation might be biased because of losing detailed information on some parts important to the parameter estimation. A selection step by residual resampling [6] is adopted after the sampling step. The method avoids the degeneracy by discarding those samples with insignificant weights, and in order to keep the number of the samples constant, samples with significant weights are duplicated. Accordingly, the weights after the selection step are also proportionally redistributed. Denote the set of samples after the selection step as ˜Θ(t) = {˜Θ(i)(t); i = 1 · · · N} with weights ˜β(t) = {˜β(i)(t); i = 1 · · · N}. After the selection step at frame t, these N samples are distributed approximately according to the posterior distribution in (7). However, the discrete nature of the approximation can lead to a skewed importance weights distribution, where the extreme case is all the samples have the same ˜Θ(t) estimated. A MetropolisHastings smoothing [7] step is introduced in each sample where the step involves sampling a candidate Θ⋆(i)(t) given the current ˜Θ(i)(t) according to the proposal importance distribution q(Θ⋆(t)|˜Θ(i)(t)). The Markov chain then moves towards Θ⋆(i)(t) with acceptance possibility as min{1, p(Θ⋆(i)|Y l(t))q( ˜Θ(i)|Θ⋆(i)) p( ˜Θ(i)|Y l(t))q(Θ⋆(i)| ˜Θ(i)) }, otherwise it remains at ˜Θ(i). To simplify calculation, we assume that the importance distribution q(Θ⋆(t)|˜Θ(i)(t)) is symmetric, and after some mathematical manipulation, it is shown that the acceptance possibility is given by min{1, β⋆(i)(t) ˜β(i)(t) }. Denote the obtained samples as ˇΘ(t) = {ˇΘ(i)(t); i = 1 · · · N} with weights ˇβ(t) = {ˇβ(i)(t); i = 1 · · · N}. Noise parameter µl n(t) is estimated via MMSE over the samples, i.e., ˆµl n(t) = N X i=1 ˇβ(i)(t) PN j=1 ˇβ(j)(t) ˇµl(i) n (t) where ˇµl(i) n (t) is the updated continuous state of the EKF in the sample after the smoothing step. Once the estimate ˆµl n(t) has been obtained, it is plugged into (1) to do non-linear transformation of clean speech models. 4 Experimental results 4.1 Experimental setup Experiments were performed on the TI-Digits database down-sampled to 16kHz. Five hundred clean speech utterances from 15 speakers and 111 utterances unseen in the training set were used for training and testing, respectively. Digits and silence were respectively modeled by 10-state and 3-state whole word HMMs with 4 diagonal Gaussian mixtures in each state. The window size was 25.0ms with a 10.0ms shift. Twenty-six filter banks were used in the binning stage. The features were MFCC + ∆MFCC. The baseline system had a 98.7% Word Accuracy under clean conditions. We compared three systems. The first was the baseline trained on clean speech without noise compensation, and the second was the system with noise compensation by (1) assuming stationary noise [4]. They were each denoted as Baseline and Stationary Compensation. The sequential method was un-supervised, i.e., without training transcript, and it was denoted according to the number of samples and variance of the environment driving noise V . Four seconds of contaminating noise was used in each experiment to obtain noise mean vector µl n in (1) for Stationary Compensation. It was also for initialization of µl n(0) in the sequential method. The initial µl(i) n (0) for each sample was sampled from N(µl n(0), 0.01) + N(µl n(0) + ζ(0), 10.0), where ζ(0) was flat distribution in [−1.0, 9.0]. 4.2 Speech recognition in simulated non-stationary noise White noise signal was multiplied by a Chirp signal and a rectangular signal, so that the noise power of the contaminating White noise changed continuously, denoted as experiment A, and dramatically, denoted as experiment B. As a result, signalto-noise ratio (SNR) of the contaminating noise ranged from 0dB to 20.4dB. We plotted the noise power in 12th filter bank versus frames in Figure 2, together with the estimated noise power by the sequential method with number of samples set to 120 and environment driving noise variance set to 0.0001. As a comparison, we also plotted the noise power and its estimate by the method with the same number of samples but larger driving noise variance to 0.001. By Figure 2 and Figure 3, we have the following observations. First, the method can track the evolution of the noise power. Second, the larger driving noise variance V will make faster convergence but larger estimation error of the method. In terms of recognition performance, Table 1 shows that the method can effectively improve system robustness to the time-varying noise. For example, with 60 samples, and the environment driving noise variance V set to 0.001, the method can improve word accuracy from 75.30% achieved by “Stationary Compensation”, to 94.28% in experiment A. The table also shows that, the word accuracies can be improved by increasing number of samples. For example, given environment driving noise variance V set to 0.0001, increasing number of samples from 60 to 120, can improve word accuracy from 77.11% to 85.84% in experiment B. Table 1: Word Accuracy (in %) in simulated non-stationary noises, achieved by the sequential Monte Carlo method in comparison with baseline without noise compensation, denoted as Baseline, and noise compensation assuming stationary noise, denoted as Stationary Compensation. Experiment Baseline Stationary # samples = 60 # samples = 120 Compensation V V 0.001 0.0001 0.001 0.0001 A 48.19 75.30 94.28 93.98 94.28 94.58 B 53.01 78.01 82.23 77.11 85.84 85.84 4.3 Speech recognition in real noise In this experiment, speech signals were contaminated by highly non-stationary Machinegun noise in different SNRs. The number of samples was set to 120, and the environment driving noise variance V was set to 0.0001. Recognition performances are shown in Table 2, together with “Baseline” and “Stationary Compensation”. Figure 2: Estimation of the time-varying parameter µl n(t) by the sequential Monte Carlo method at 12th filter bank in experiment A. Number of samples is 120. Environment driving noise variance is 0.0001. Solid curve is the true noise power. Dash-dotted curve is the estimated noise power. It is observed that, in all SNR conditions, the method can further improve system performance, compared to that obtained by “Stationary Compensation”, over “Baseline”. For example, in 8.86dB SNR, the method can improve word accuracy from 75.60% by “Stationary Compensation” to 83.13%. As a whole, the method can have a relative 39.9% word error rate reduction compared to “Stationary Compensation”. Table 2: Word Accuracy (in %) in Machinegun noise, achieved by the sequential Monte Carlo method in comparison with baseline without noise compensation, denoted as Baseline, and noise compensation assuming stationary noise, denoted as Stationary Compensation. SNR (dB) Baseline Stationary Compensation #samples = 120, V = 0.0001 28.86 90.36 92.77 97.59 14.88 64.46 76.81 88.25 8.86 56.02 75.60 83.13 1.63 50.0 68.98 72.89 5 Summary We have presented a sequential Monte Carlo method for Bayesian estimation of time-varying noise parameter, which is for sequential noise compensation applied to robust speech recognition. The method uses samples to approximate the posterior distribution of the additive noise and speech parameters given observation sequence. Figure 3: Estimation of the time-varying parameter µl n(t) by the sequential Monte Carlo method at 12th filter bank in experiment A. Number of samples is 120. Environment driving noise variance is 0.001. Solid curve is the true noise power. Dash-dotted curve is the estimated noise power. Once the noise parameter has been inferred, it is plugged into a non-linear transformation of clean speech models. Experiments conducted on digits recognition in simulated non-stationary noises and real noises have shown that the method is very effective to improve system robustness to time-varying additive noise. References [1] A. Varga and R.K. Moore, “Hidden markov model decomposition of speech and noise,” in ICASSP, 1990, pp. 845–848. [2] N.S. Kim, “Nonstationary environment compensation based on sequential estimation,” IEEE Signal Processing Letters, vol. 5, no. 3, March 1998. [3] K. Yao, K. K. Paliwal, and S. Nakamura, “Sequential noise compensation by a sequential kullback proximal algorithm,” in EUROSPEECH, 2001, pp. 1139–1142, extended paper submitted for publication. [4] K. Yao, B. E. Shi, S. Nakamura, and Z. Cao, “Residual noise compensation by a sequential em algorithm for robust speech recognition in nonstationary noise,” in ICSLP, 2000, vol. 1, pp. 770–773. [5] B. Frey, L. Deng, A. Acero, and T. Kristjansson, “Algonquin: Iterating laplace’s method to remove multiple types of acoustic distortion for robust speech recognition,” in EUROSPEECH, 2001, pp. 901–904. [6] J. S. Liu and R. Chen, “Sequential monte carlo methods for dynamic systems,” J. Am. Stat. Assoc, vol. 93, pp. 1032–1044, 1998. [7] W. K. Hastings, “Monte carlo sampling methods using markov chains and their applications,” Biometrika, vol. 57, pp. 97–109, 1970.
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Direct value-approxiIllation for factored MDPs Dale Schuurmans and ReIn Patrascll Department of Computer Science University of Waterloo {dale, rpatrasc} @cs.'Uwaterloo.ca Abstract We present a simple approach for computing reasonable policies for factored Markov decision processes (MDPs), when the optimal value function can be approximated by a compact linear form. Our method is based on solving a single linear program that approximates the best linear fit to the optimal value function. By applying an efficient constraint generation procedure we obtain an iterative solution method that tackles concise linear programs. This direct linear programming approach experimentally yields a significant reduction in computation time over approximate value- and policy-iteration methods (sometimes reducing several hours to a few seconds). However, the quality of the solutions produced by linear programming is weaker-usually about twice the approximation error for the same approximating class. Nevertheless, the speed advantage allows one to use larger approximation classes to achieve similar error in reasonable time. 1 Introduction Markov decision processes (MDPs) form a foundation for control in uncertain and stochastic environments and reinforcement learning. Standard methods such as value-iteration, policy-iteration and linear programming can be used to produce optimal control policies for MDPs that are expressed in explicit form; that is, the policy, value function and state transition model are all represented in a tabular manner that explicitly enumerates the state space. This renders the approaches impractical for all but toy problems. The real goal is to achieve solution methods that scale up reasonably in the size of the state description, not the size of the state space itself (which is usually either exponential or infinite). There are two basic premises on which solution methods can scale up: (1) exploiting structure in the MDP model itself (i.e. structure in the reward function and the state transition model); and (2) exploiting structure in an approximate representation of the optimal value function (or policy). Most credible attempts at scaling-up have generally had to exploit both types ofstructure. Even then, it is surprisingly difficult to formulate an optimization method that can handle large state descriptions and yet simultaneously produce value functions or policies with small approximation errors, or errors that can be bounded tightly. In this paper we investigate a simple approach to determining approximately .optimal policies based on a simple direct linear programming approach. Specifically, the idea is to approximate the optimal value function by formulating a single linear program and exploiting structure in the MDP and the value function approximation to solve this linear program efficiently. 2 Preliminaries We consider MDPs with finite state and action spaces and consider the goal of maximizing infinite horizon discounted reward. In this paper, states will be represented by vectors x of length n, where for simplicity we assume the state variables Xl, ..., X n are in {O, I}; hence the total nuniber of states is N == 2n . We also assume there is a small finite set of actions A == {aI, ...,al}. An MDP is defined by: (1) a state transition model P(x/lx, a) which specifies the probability of the next state Xl given the current state x and action a; (2) a reward function R(x, a) which specifies the immediate reward obtained by taking action a in state X; and (3) a discount factor " 0 :S , < 1. The problem is to determine an optimal control policy 1r* : X --7 A that achieves maximum expected future discounted reward in every state. To understand the standard solution methods it is useful to define some auxiliary concepts. For any policy 1r, the value function V 7r : X --7 JR denotes the expected future discounted reward achieved by policy 1r in each state x. It turns out that V 7r satisfies a fixed point relationship between the value of current states and the expected values of future states, given by a backup operator V 7r == B 7r V 7r , where B 7r operates on arbitrary functions over the state space according to (B 7r f) (x) == R(x, 1r(x)) + , E P(x'lx, 1r(x))f(x/) X' Another important backup operator is defined with respect to a fixed action a (Baf) (x) == R(x, a) +,E P(x/lx, a)f(x') X' The action-value function Q7r : X x A --7 JR denotes the expected future discounted reward achieved by taking action a in state x and following policy 1r thereafter; which must satisfy Q7r (x, a) == B a V 7r • Given an arbitrary function f over states, the greedy policy 1rgre(f) with respect to f is defined by 1rgre (I) (x) == arg max (B a f) (x) a Finally, if we let 1r* denote the optimal policy and V* denote its value function, we have the relationship V* == B*V*, where (B* f) (x) == maXa (Ba f) (x). If, in addition, we define Q*(x,a) == BaV* then we also have 1r*(x) == 1rgre (V*)(x) == arg maxa Q* (x, a). Given these definitions, the three fundamental methods for calculating 1r* can be formulated as: Policy iteration: Start with an arbitrary policy 1r(0). Iterate 1r(i+l) f- 1rgre(V 7r(i») until1r(i+l) == 1r(i). Return 1r* == 1r(i+I). Value iteration: Start with an arbitrary function f(O). Iterate f(i+l) f- B* f(i) untilllf(i+l) f(i) 1100 < tole Return 1r* == 1rgre(f(i+I)). Linear programming: Calculate V* == arg min] I:x I(x) subject to f(x) 2=: (B a f) (x) for all a and x. Return 1r* == 1rgre (V*). All three methods can be shown to produce optimal policies for the given MDP [1, 10] even though they do so in very different ways. However, all three approaches share the same fundamental limitation that they do not scale up feasibly in n, the size of the state descriptions. Instead, all of these approaches work with explicit representations of the policies and value functions that are exponential in n. 3 Exploiting structure To scale up to large state spaces it is necessary to exploit substantial structure in the MDP while also adopting some form of approximation for the optimal value function and policy. The two specific structural assumptions we consider in this paper are (1) factored MDPs and (2) linear value function approximations. Neither of these two assumptions alone is sufficient to permit efficient policy optimizationfor large MDPs. However, combined, the two assumptions allow approximate solutions to be obtained for problems involving trillions of states reasonably quickly. 3.1 Factored MDPs In the spirit of [7, 8, 6] we define a factored MDP to be one that can be represented compactly by an additive reward function and a factored state transition model. Specifically, we assume the reward function decomposes as R(x, a) == E~=l Ra,r (xa,r) where each local reward function Ra,r is defined on a small set of variables xa,r' We assume the state transition model P(x/lx, a) can be represented by a set of dynamic Bayesian networks (DBNs) on state variables-one for each action-where each DBN defines a compact transition model on a directed bipartite graph connecting state variables in consecutive time steps. Let Xa,i denote the parents of successor variable x~ in the DBN for action a. To allow efficient optimization we assume the patent set Xa,i contains a small number of state variables from the previous time step. Given this model, the probability of a successor state Xl given a predecessor state x and action a is given by the product P(x/lx, a) == Il7=1 P(X~IXa,i)' The main benefit of this factored representation is that it allows large MDPs to be encoded concisely: if the functions Ra,r(xa,r) and P(X~IXa,i) depend on a small number of variables, they can be represented by small tables and efficiently combined to determine R(x, a) and P(x'lx, a). Unfortunately, as pointed out in [7], a factored MDP does not by itself yield a feasible method to determining optimal policies. The main problem is that, even if P and R are factored, the optimal value function generally does not have a compact representation (nor does the optimal policy). Therefore, obtaining an exact solution appears to require a return to explicit representations. However, it turns out that the factored MDP representation interacts very well with linear value function approximations. 3.2 Linear approximation One of the central tenets to scaling up is to approximate the optimal value function rather than calculate it exactly. Numerous schemes have been investigated for approximating optimal value functions and policies in a compact representational framework, including: hierarchical decompositions [5], decision trees and diagrams [3, 12], generalized linear functions [1, 13, 4, 7, 8, 6], neural networks [2], and products of experts .[11]. However, the simplest of these is generalized linear functions, which is the form we investigate below. In this case, we consider functions of the form f(x) =2:;=1 wjbj(xj) where b1 , ••• , bk are a fixed set of basis functions, and Xj denotes the variables on which basis bj depends. Combining linear functions with, factored MDPs provides many opportunities for feasible approximation. The first main benefit of combining linear approximation with factored MDPs is that the result of applying the backup operator B a to a linear function results in a compact representation for the action-value function. Specifically if we define g(X, a) == (B a f) (x) then we can rewrite it as m k g(X, a) == L:Ra,r(xa,r) + L:WjCa,j(Xa,j) r=l j=l where Ca,j(Xa,j) ==1'L:P(xjla,xa,j)bj (xj) andxa,j == U Xa,i xj x~Exj That is, Xa,i are the parent variables of x~, and Xa,j is the union of the parent variables of x~ E xj. Thus, ca,j expresses the fact that in a factored MDP the expected future value of one component of the approximation depends only on the current state variables Xa,j that are direct parents of the variables xj in bj • If the MDP is sparsely connected then the variable sets in 9 will not be much larger than those in f. The ability to represent the state-action value function in a compact linear form immediately provides a feasible implementation of the greedy policy for f, since 1rgre (f) (x) == argmaXa g(~, a) by definition of 1rgre , and g(x, a) is efficiently determinable for each x and a. However, it turns out that this is not enough to permit feasible forms of approximate policy- and value-iteration to be easily implemented. The main problem is that even though Baf has a factored form for fixed a, B*f does not and (therefore) neither does 1rgre(f). In fact, even if a policy 1f were concisely represented, B 1r f would not necessarily have a compact form because 1f usually depends on all the state variables and thus P(x/lx, 1r(x)) == I17=1 P(x~IX1r(x),i) becomes a product of terms that depend on all the state variables. Here [8, 6] introduce an additional assumption that there is a special "default" action ad for the MDP such that all other actions a have a factored transition model P (·1·, a) that differs from P(·I·, ad) only on a small number of state variables. This allows the greedy policy 1rgre (f) to have a compact form and moreover allows B 1rgre(f) f to be concisely represented. With some effort, it then becomes possible to formulate feasible versions of approximate policy- and value-iteration [8, 6]. Approximate policy iteration: Start with default policy 1r(O)(x) == ad. Iterate f(i) +- arg minf maxx If(x) - (B 1r(i) f) (x)I , 1r(i+1) f- 1fgre (f(i)) until1r(i+1) == 1r(i). Approximate value iteration: Start with arbitrary f(O). Iterate 1r(i) +1rgre (f(i)) ,f(i+1) +- argminf maxx 1!(x)-(B1r(i) f)(x)1 until Ilf(i+1)_!(i) 1100 < tole The most expensive part of these iterative algorithms is determining arg minf maxx If(x) - (B7r(i) f) (x)I which involves solving a linear program minw,E E subject to -E :S !w(x) - (B 7r fw) (x) :S E for all x. This linear program is problematic because it involves an exponential number of constraints. A· central achievement of [6] is to show that this system of constraints can be encoded by an equivalent system of constraints that has a much more compact form. The idea behind this construction is to realize that searching for the max or a min of a linear function with a compact basis can be conducted in an organized fashion, and such an organized search can be encoded in an equally concise constraint system. This construction allows approximate solutions to MDPs with up to n == 40 state variables (1 trillion states) to be generated in under 7.5 hours using approximate policy iteration [6].1 1It turns out that approximate value iteration is less effective because it takes more iterations to converge, and in fact can diverge in theory [6, 13]. Our main observation is that if one has to solve linear programs to conduct the approximate iterations anyway, then it might be much simpler and more efficient to approximate the linear programming approach directly. 4 Approximate linear programming Our first idea is simply to observe that a factored MDP and linear value approximation immediately allow one to directly solve the linear programming approximation to the optimal value function, which is given by IIjinL f(x) subject to f(x) - (B af) (x) ;::: 0 for all x and a x where f is restricted to a linear form over a fixed basis. In fact, it is well known [1, 2] that this yields a linear program in the basis weights w. However, what had not been previously shown is that given a factored MDP, an equivalent linear program of feasible size could be formulated. Given the results of [6] outlined above this is now easy to do. First, one can show that the minimization objective can be encoded compactly Lf(x) x k LLWjbj(xj) x j=l k LWjYj where Yj == 2n-lxjl Lbj(xj) j=l ~ Here the Yj components can be easily precomputed by enumerating assignments to the small sets of variables in basis functions. Second, as we have seen, the exponentially many constraints have a structured form. Specifically f (x) - (B a f) (X) can be represented as f(x) - (B a f) (x) kL Wj (bj (Xj) - Ca,j (xa,j)) - L Ra,r (Xa,r) j=l r which has a simple basis representation that allows the technique of [6] to be used to encode a constraint system that enforces f(x) - (B a f) (x) 2:: 0 for all x and a without enumerating the state space for each action. We implemented this approach and tested it on some of the test problems from [6]. In these problems there is a directed network of computer systems Xl, ••• , X n where each system is either up (Xi == 1) or down (Xi == 0). Systems can spontaneously go down with some probability at each step, but this probability is increased if an immediately preceding machine in the network is down. There are n + 1 actions: do nothing (the default) and reboot machine i. The reward in a state is simply the sum of systems that are up, with a bonus reward of 1 if system 1 (the server) is up. I.e., R(x) == 2Xl +2:7=2 Xi. We considered the network architectures shown in Figure 1 and used the transition probabilities P(x~ == llxi, parent(Xi), a == i) == 0.95 and P(x~ == 11Xi, parent(Xi), a I- i) == 0.9 if Xi == parent(Xi) == 1; 0.67 if Xi == 1 and parent(xi) == 0; and 0.01 if Xi == o. The discount factor was 'I == 0.95. The first basis functions we considered were just the indicators on each variable Xi plus a constant basis function (as reported in [6]). The results for two network architectures are shown in Figure 1. Our approximate linear programming method is labeled ALP and is compared to the approxi.mate n= 12 16 20 24 28 32 36 40 N= 4e3 6e4 1e6 2e7 3e8 4e9 7e10 1e12 server API[6] 7m 30m 50m 1.3h 1.9h 3h 4.5h 7.5h 0 APIgen 39s 1.'5m 2.3m 4.0m 6.5m 13m 22m 28m time ALP 4.5s 23s 1.4m 4.1m 10m 23m 47m 2.4h ALPgen 0.7s 1.2s 1.8s 2.6s 3.5s 4.5s 5.9s 7.0s ALPgen2 14s 37s 102m 2.8m 4.7m 6.4m 12m 17m APIgen 420 777 921 1270 1591 2747 4325 4438 constraints ALP 1131 2023 3171 4575 6235 8151 10K 13K ALPgen 38 50 62 74 86 98 110 122 ALPgen2 166 321 514 914 1223 1433 1951 2310 API[6] 0.3Q' , 0.33 0.34 0.35 0.36 0.36 0.37 0.38 DB Bellman APIgen 0.36 0.34 0.33 0.33 0.32 0.32 0.32 0.31 / Rmax ALP (gen) 0.85 0.82 0.80 0.78 0.78 0.77 0.76 0.76 ALPgen2 0.12 0.14 0.08 0.08 0.10 0.08 0.07 0.07 n= 13 16 22 28 34 40 N= 8e4 6e4 4e6 3e8 2e10 1e12 API[6] 5m 15m 50m l.3h 2.Th 5h APIgen 28s 106m 3.9m 12m 23m 33m time ALP 0.7s 1.6s 6.0s 20s 56s 2.2m ALPgen 0.7s LOs 1.5s 2.4s 3.4s 4.7s ALPgen2 17s 338 1.9m 5.4m 9.6m 23m APIgen 363 952 1699 3792 6196 7636 constraints ALP 729 1089 2025 3249 4761 6561 ALPgen 50 69 90 114 135 162 ALPgen2 261 381 826 1505 1925 3034 API[6] 0.27 0.29 0.32 0.34 0.35 0.36 DB Bellman APIgen 0.50 0.46 0.42 0.39 0.38 0.37 / Rmax ALP(gen) 0.96 0.82 0.78 0.78 0.77 0.76 ALPgen2 0.21 0.22 0.15 0.06 0.07 0.03 Figure 1: Experimental results (timings on a 750MHz PIlI processor, except 2) policy iteration strategy API described in [6]. Since we did not have the specific probabilities used in [6] and could only estimate the numbers for API from graphs presented in the paper, this comparison is only meant to be loosely indicative of the general run times of the two methods on such problems. (Perturbing the probability values did not significantly affect our results, but we implemented APlgen for comparison.) As in [6] our implementation is based on Matlab, using CPLEX to solve linear programs. Our preliminary results appear to support the hypothesis that direct linear programming can be more efficient than approximate policy iteration on problems of this type. A further advantage of the linear programming approach is that it is simpler to program and involves solving only one LP. More importantly, the direct LP approach does not require the MDP to have a special default action since the action-value function can be directly extracted using 7rgre (f)(x) == argma:xay(x,a) and g is easily recoverable from f. Before discussing drawbacks, we note that it is possible to solve the linear program even more efficiently by iteratively generating constraints as needed. This is now possible because factored MDPs and linear value approximations allow an efficient search for the maximally violated constraints in the linear program, which provides an effective way of generating concise linear programs that can be solved much more efficiently than those formulated above. Specifically, the procedure ALPgen exploits the feasible search techniques for minimizing linear functions discussed previously to efficiently generate a small set of critical constraints, which is iteratively grown until the final solution is identified; see Figure 2. 2These numbers are estimated from graphs in [6]. The exact probabilities and computer used for the simulations were not reported in that paper, so we cannot assert an exact comparison. However, perturbed probabilities have little effect .on the performance of the methods we tried, and it seems that overall this is a loosely representative comparison of the general performance of the' various algorithms on these problems. ALPgen Start with f(O) = 0 and constraints = 0 Loop I For each a E A, compute x a t- arg minx f(i) (x) - (B a f(i)) (x) constraints t- constraints U{constraint(x a1 ), ••• , constraint(xak )} Solve f(i~l) t- minJ 2:~ f(x) subject to constraints Until minx f(~)(x) (Baf(~})(x) ~ 0 - tot for all a Return g(., a) = B af for each a, to represent the greedy policy Figure 2: ALPgen procedure The rationale for this procedure is that the main bottleneck in the previous methods is generating the constraints, not solving the linear programs [6]. Since only a small number of constraints are active at a solution and these are likely t.o be the most violated near the solution, adding only most violated constraints appears to be a useful way to proceed. Indeed, Figure 1 shows that ALPgen produces the same approximate solutions as ALP in a tiny fraction of the time. In the most extreme case ALPgen produces an approximate solution in 7 seconds while other methods take several hours on the same problem. The reason for this speedup is explained by the results which show the numbers of constraints generated by each method. Further investigation is also required to fully outline the robustness of the constraint generation method. In fact, one cannot guarantee that a greedy constraint generation scheme like the one proposed here will always produce a feasible number of constraints [9]. Nevertheless, the potential benefits of conservatively generating constraints as needed seem to be clear. Of course, the main drawback of the direct linear programming approach over approximate policy iteration is that ALP incurs larger approximation errors than API. 5 Bounding approximation error It turns out that neither API nor ALP are guaranteed to return the best linear approximation to the true value function. Nevertheless, it is possible to efficiently calculate bounds on the approximation errors of these methods, again by exploiting the structure of the problem: A well known result [14] asserts that maxx V* (x) - V 7rgre (J) (x) :S 12, maxx f(x) - (B* f) (x) (where in our case f ~ V*). This upper bound can in turn be bounded by a quantity that is feasible to calculate: maxx f(x)-(B* f) (x) = maxxmina f(x)-(Ba f) (x) :S mina maxx f(x)-(Baf)(x). Thus an upper bound on the error from the optimal value function can be calculated by performing an efficient search for maxx f(x) - (Baf) (x) for each a. Figure 1 shows that the measurable error quantity, maxx f(x) - (B a f) (x) (reported as UB Bellman) is about a factor of two larger for the linear programming approach than for approximate policy iteration on the same basis. In this respect, API appears to have an inherent advantage (although in the limit of an exhaustive basis both approaches converge to the same optimal value). To get an indication of the computational cost required for ALPgen to achieve a similar bound on approximation error, we repeated the same experiments with a larger basis set that included all four indicators between pairs of connected variables. The results for this model are reported as ALPgen2, and Figure 1 shows that, indeed, the bound on approximation error is reduced substantially-but at the predictable cost of a sizable increase in computation time. However, the run times are still appreciably smaller than the policy iteration methods. Paradoxically, linear programming seems to offer computational advantages over policy and value iteration in the context of approximation, even though it is widely held to be an inferior solution strategy for explicitly represented MDPs. References [1] D. Bertsekas. Dynamic Programming and Optimal Control, volume 2. Athena Scientific, 1995. [2] D. Bertsekas and J. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996. [3] C. Boutilier, R. Dearden, and M. Goldszmidt. Stochastic dynamic programming with factored representations. Artificial Intelligence, 2000. [4] J. Boyan. Least-squares temporal difference learning. In Proceedings ICML, 1999. [5J T. Dietterich.Hierarchical reinforcement learning vlith the 1\1AXQ value function decomposition. JAIR, 13:227-303,2000. [6] C. Guestrin, D. Koller, and R. Parr. Max-norm projection for factored MDPs. ·In Proceedings IJCAI, 2001. [7] D. Koller and R. Parr. Computing factored value functions for policies in structured MDPs. In Proceedings IJCAI, 1999. [8] D. Koller and R. Parr. Policy iteration for factored MDPs. In Proceedings UAI,2000. . [9] R. Martin. Large Scale Linear and Integer Optimization. Kluwer, 1999. [10] M. Puterman. Markov Decision Processes: Discrete Dynamic Programming. Wiley, 1994. [IIJ B. Sallans and G. Hinton. Using free energies to represent Q-values in a multiagent reinforcement learning task. In Proceedings NIPS, 2000. [12] R. St-Aubin, J. Hoey, and C. Boutilier. APRICODD: Approximating policy construction using decision diagrams. In Proceedings NIPS, 2000. [13J B. Van Roy. Learning and value function approximation in complex decision processes. PhD thesis, MIT, EECS, 1998. [14J R. Williams and L. Baird. Tight performance bounds on greedy policies based _on imperfect value functions. Technical report, Northeastern University, 1993.
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Perceptual Metamers in Stereoscopic Vision Benjamin T. Backus* Department of Psychology University of Pennsylvania Philadelphia, PA 19104-6196 backus@psych.upenn.edu Abstract Theories of cue combination suggest the possibility of constructing visual stimuli that evoke different patterns of neural activity in sensory areas of the brain, but that cannot be distinguished by any behavioral measure of perception. Such stimuli, if they exist, would be interesting for two reasons. First, one could know that none of the differences between the stimuli survive past the computations used to build the percepts. Second, it can be difficult to distinguish stimulus-driven components of measured neural activity from top-down components (such as those due to the interestingness of the stimuli). Changing the stimulus without changing the percept could be exploited to measure the stimulusdriven activity. Here we describe stimuli in which vertical and horizontal disparities trade during the construction of percepts of slanted surfaces, yielding stimulus equivalence classes. Equivalence class membership changed after a change of vergence eye posture alone, without changes to the retinal images. A formal correspondence can be drawn between these “perceptual metamers” and more familiar “sensory metamers” such as color metamers. 1 Introduction Two types of perceptual process might, in principle, map physically different visual stimuli onto the same percept. First, the visual system has a host of constancy mechanisms that extract information about the visual environment across uninteresting changes in the proximal stimulus. Some of these mechanisms could be “leak-proof,” leaving no trace of the original differences between the stimuli. Second, the visual system must combine information from redundant cues if it is to build percepts robustly. Recent cue conflict experiments have shown that the visual system’s estimate of a scene parameter, as evinced in a visual percept, is often simply a weighted average of the parameter as specified by each cue separately [1][2]. Thus, a properly balanced cue-conflict stimulus might come to evoke the same percept as a “natural” or cue-concordant stimulus. * http://psych.upenn.edu/~backus Here, random-dot stereograms will be used to argue that leak-proof versions of both types of process exist. When a vertical magnifier is placed before one eye, a truly frontoparallel surface appears slanted. Adding horizontal magnification in the same eye restores frontoparallel appearance. The original stimulus and the magnified stimulus therefore have different patterns of binocular disparity but give rise to similar judgments of surface slant [3]. We show here that such stimuli are perceptually indistinguishable to practiced observers in a psychophysical discrimination task, which implies the loss of some disparity information. This loss could occur, first, in a well-studied constancy mechanism that uses vertical disparity to correct the depth relief pattern associated with horizontal disparity [4]. However, the amount of horizontal magnification needed to null vertical magnification is less than would be predicted from use of this constancy mechanism alone; a second constancy mechanism exists that corrects horizontal disparities by using felt eye position, not vertical disparity [5]. Adding vertical magnification without changing eye position therefore creates a cue conflict stimulus. We show here that the amount of horizontal magnification needed to null the vertical magnification changes with the vergence posture of the eyes, which implies that both types of process (constancy and cue combination) are leak-proof across certain ranges of variation (magnifications) in these stereoscopic stimuli. 2 Stereoscopic slant perception: review of theory The stereo component of the perceived slant of a random-dot surface can be modeled as the visual system’s weighted average of two stereo slant estimates [5][ 6]. Horizontal disparity is ambiguous because it depends not only on surface slant, but also on surface patch location relative to the head. One stereo estimator resolves this ambiguity using vertical disparity (images are vertically larger in the closer eye), and the other resolves it using felt eye position. Vertical magnification in one eye thus creates a cue-conflict because it affects only the estimator that uses vertical disparity. The two stereo estimators have different relative reliability at different distances, so the weights assigned to them by the visual system changes as a function of distance [7]. Since vergence eye posture is a cue to distance [8], one might predict that “perceptually metameric” stereo stimuli, if they exist, will lose their metameric status after a pure change of vergence eye posture that preserves the metamers’ retinal images [9]. We shall now briefly describe the two stereoscopic slant estimators. This theory is covered elsewhere in greater detail [5]. Although surface slant has two components (slant and tilt [10]), we will consider only slant about a vertical axis. The arguments can be extended to slant about axes of arbitrary orientation [5]. The visual signals used in stereoscopic slant perception can be conveniently parameterized by four numbers [5]. Each can be considered a signal. A surface patch typically gives rise to all four signals. Two signals are the horizontal gradient of horizontal disparity, and the vertical gradient of vertical disparity, which we parameterize as horizontal size ratio (HSR) and vertical size ratio (VSR), respectively, in the manner of Rogers and Bradshaw [11]. They are defined as the horizontal (or vertical) size of the patch in the left eye, divided by the horizontal (or vertical) size in the right eye. These two signals must be measured from the retinal images. The two remaining signals are the headcentric azimuth and vergence of the surface patch. These signals can be known either by measuring the eyes’ version and vergence, respectively, or from the retinal images [12]. A very good approximation that relates surface slant to horizontal disparity and VSR is: S HSR,VSR -tan-1 [ 1 µ ln HSR VSR ] = Equation 1 where µ is the vergence of the surface patch in radians. We call this method of slant estimation slant from HSR and VSR. A very good approximation that relates surface slant to horizontal disparity and azimuth is: S HSR,EP -tan -1 [ 1 µ ln HSR - tanγ ] = Equation 2 where γ is the azimuth of the surface patch. We call this method of slant estimation slant from HSR and eye position on the supposition that azimuth per se is known to the visual system primarily through measurement of the eyes’ version. Each estimator uses three of the four signals available to estimate surface slant from horizontal disparity. Nonstereo slant estimates can be rendered irrelevant by the choice of task, in which case perceived slant is a weighted average of the slants predicted from these two stereoscopic slant estimates [5, 6]. In principle, the reliability of slant estimation by HSR and eye position is limited at short viewing distances (large µ) by error in the measurement of γ. Slant from HSR and VSR, on the other hand, continues to become more reliable as viewing distance decreases. If one assumes that the visual system knows how reliable each estimator is, one would predict that greater weight is given to the HSR and VSR estimate at near than at far distances, and this is in fact the case [7]. Whether each estimate is separately computed in its own neural process, and then given a weight, is not known. A maximum a posteriori Bayesian scheme that simply estimates the most likely slant given the observed signals behaves in a similar fashion as the weighted estimates model, though actual likelihood density (probability per deg of slant) is extraordinarily small in the case of stimuli that contain large cue conflicts [9]. The real visual system does not flinch, but instead produces a slant estimate that looks for all the world like a weighted average. It remains a possibility therefore that optimal slant estimation is implemented as a weighted combination of separate estimates. We have now developed the theory to explain why HSR and VSR trade with each other at the “constancy” level of a single estimator (Equation 1), and why natural stimuli might appear the same as cue conflict stimuli (weighted averaging of estimates derived from exploitation of Equations 1 and 2, respectively). We next describe experiments that tested whether magnified (cue conflict) stimuli are distinguishable from natural (concordant) stimuli. 3 Existence of stereoscopic metamers Stimuli were sparse random dot stereograms (RDS) on a black background, 28 deg in diameter, presented directly in front of the head using a haploscope. Observers performed a forced choice task with stimuli that contained different amounts of unilateral vertical and horizontal magnification. Vertical magnification was zero for the “A” stimuli, and 2% in the right eye for the “B” stimuli (1% minification in the left eye and 1% magnification in the right eye). Horizontal magnification was set at the value that nulled apparent slant in “A” stimuli (i.e. approximately 0%), and took on a range of values in “B” stimuli. Each trial consisted of two “A” stimuli and one “B” stimulus. The observer’s task was to determine whether the three stimuli were presented in AAB or BAA order [13], i.e., whether the stimulus with vertical magnification was first or last of the three stimuli. Each stimulus was presented for 0.5 sec. Each stimulus was generated using a fresh set of 200 randomly positioned dots. Each dot had a circular raised cosine luminance profile that was 30 arcmin in diameter. Three observers participated, including the author. Results are shown in Figure 1. Horiz mag in left eye in stimulus B (%) 40 60 80 100 Percent Correct -1.4 -1 -0.6 BTB -2 0 2 JRF -3 -2 -1 0 MJN vMags = 0% and 2% N = 40 trials per datapoint Figure 1. Observers are unable to distinguish 0% and 2% unilateral vertical magnification when unilateral horizontal magnification is added as well. Open squares show the horizontal magnification that evoked zero perceived slant under 2% vertical magnification. For each observer, there was a value of horizontal magnification that, when added to the “B” stimulus, rendered it indistinguishable from the “A” stimulus. This is shown in Figure 1 by the fact that performance drops to chance (50%) at some value of horizontal magnification. From this experiment it is evident that stimuli with very different disparity patterns can be made perceptually indistinguishable in a forcedchoice task with well-practiced observers. 3.1 Experimental conditions necessary for stereo metamers Several properties of the experiment were essential to the effect. First, the vertical magnification must not be to large. At large vertical magnifications it is still possible to null apparent slant, but the stimuli are distinguishable because the dots themselves look different (they look as though blurred in the vertical direction). Two out of three observers were able to distinguish the “A” and “B” stimuli 100% of the time when the vertical magnification was increased from 2% to 5%. Second, observers must be instructed to maintain fixation. If left and right saccades are allowed, the “B” stimulus appears slanted in the direction predicted by its horizontal magnification. This is a rather striking effect—the surface appears to change slant simply because one starts looking about. This effect was not found previously [14] but is predicted as a consequence of sequential stereopsis [15]. Finally, if the stimuli are shown for more than about 1 sec it is possible to distinguish “A” and “B” stimuli by making vertical saccades from the top to the bottom of the stimulus, by taking advantage of the fact that in forward gaze, vertical saccades have equal amplitude in the two eyes [16]. For “B” stimuli only, the dots are diplopic (seen in double vision) immediately after a saccade to the top (or bottom) of the stimulus. An automatic vertical vergence eye movement then brings the dots into register after about 0.5 sec. At that point a saccade to the bottom (or top) of the stimulus again causes diplopia. 4 Breaking metamerization though change of vergence eye posture In the haploscope it was possible to present unchanged retinal images across a range of vergence eye postures. Stimuli that were metameric to each other with the eyes verged at 100 cm were presented again with the eyes verged at 20 cm. For three out of four observers, the images were then distinguishable. Figure 2 illustrates this effect schematically, and Figure 3 quantifies it by plotting the amount of horizontal magnification that was needed to null apparent slant at each of the two vergence angles for one observer (left panel) and all four observers (right panel). HSR & eye pos HSR & VSR Percept Vmag Hmag Figure 2. Schematic illustration of the effect of distance in the slant-nulling task. First panel: both stereoscopic methods of estimating slant indicate that the surface is frontoparallel, and it appears so. Second panel: a vertical magnifier is placed before one eye, changing the estimate that uses vertical disparity, but not the estimate that uses eye position. The resulting percept is a weighted average of the two. Third panel: horizontal magnification is added until the surface appears frontoparallel again. At this point the two stereo estimates have opposite sign. Fourth panel: increasing the apparent distance to the stimulus (by decreasing the vergence) scales up both estimates by the same factor. The surface no longer appears frontoparallel because the weighting of the estimates has changed. Horiz magnification to null slant (%) MJ N BT B CS 100 cm 20 cm JRF Subject 0 5 10 15 20 25 30 35 40 1.2 1.6 2.0 2.4 2.8 Trial Vertical magnification: ±2% 20 cm 100 cm 100 cm Figure 3. When the eyes were verged at 100 or 20 cm distance, different amounts of horizontal magnification were needed to null the slant induced by vertical magnification. Left: 10 settings that nulled slant at 100 cm, followed by 20 settings at 20 cm, followed by 10 at 100 cm (observer BTB). Right: three out of four observers show an effect of vergence per se. Error bars are SEs of the mean. 5 Comparison of perceptual and sensory metamers The stimuli described here appear the same as a result of perceptual computations that occur well after transduction of light energy by the photoreceptors. Physically different stimuli that are transduced identically might be dubbed sensory metamers. One example of a sensory metamer is given by the trade between intensity and duration for briefly flashed lights (Bloch’s Law [17]): two flashes containing the same number of photons are indistinguishable if their durations are both less than 10 msec. Another example of sensory metamerization, that we will now consider in greater detail, is the traditional color metamer. The three cone photoreceptor types can support color vision because they are sensitive to different wavelengths of light. However, each cone type responds to a range of wavelengths, and two lights with different spectra may activate the three cone types identically. From that point on, the lights will be indistinguishable within the nervous system. (See [18] for a review of color metamers). Table 1 summarizes several properties of color metamers, and analogous properties of our new stereo metamers. We can approximate the visible spectrum of a light by sampling its power within N different wavelength intervals, where N is large. Thus light t can be represented by an Nx1 vector. Light t’ is metameric to t if Bt’ = Bt, where B is the 3xN matrix whose rows represent the spectral sensitivities of the three cone mechanisms [19]. The transformation that maps one stereo metamer to another is simply a scaling of one eyes’ image in the vertical and horizontal directions, with less scaling typically needed in the horizontal than vertical direction. Let u and v represent the x and y disparity, respectively, so that [u v] is a function of location (x,y) within the cyclopean image. Then two random-dot image pairs (representing flat surfaces slanted about a vertical axis) will be metameric if their disparity patterns, [u’ v’] and [u v], are related to each other by [u’ v’] = [u(1+m) v(1+n)], where m and n are small (on the order of 0.01), with m/n equal to the weight of SHSR,VSR in the final slant estimate. Table 1: properties of color and stereo metamers PROPERTY COLOR METAMERS STEREO METAMERS. Metamer type: Sensory Perceptual Site of loss: Peripheral Central (two places) Loss process: Transduction Computation Metameric class formation: Lights t’ and t are metameric iff Bt’ = Bt, where B is the 3xN matrix of cone spectral sensitivities Disparity map [u v] is metameric to [u’ v’] iff [u’ v’] = [u(1+m) v(1+n)] where m and n are small and in the proper ratio Dimensionality reduction: N → 3 loss of 1 degree of freedom Etiology: Capacity limit Recovery of scene parameter Computation of surface slant removes one dimension from the set of all physical stimuli. Depending how the problem is framed, this is a reduction from 2 dimensions (HSR and VSR) to one (slant), or from many dimensions (all physical stimuli that represent slanted surfaces) to one fewer dimensions. While color and stereo metamers can be described as sensory and perceptual, respectively, the boundary between these categories is fuzzy, as is the boundary between sensation and perception. Would motion metamers based on “early” motion detectors be sensory or perceptual? What of stimuli that look identical to retinal ganglion cells, after evoking different patterns of photoreceptor activity? While there is a real distinction to be made between sensory and perceptual metamers, but not all metamers need be easily categorized as one or the other. 5.1 The metamer hierarchy Loftus [20] makes a distinction reminiscent of the one made here, between “memory metamers” and “perceptual metamers,” with memory metamers being stimuli that evoke distinguishable percepts during live viewing, but that become indistinguishable after mnemonic encoding. Thus, Loftus classified as “perceptual” both our perceptual and sensory metamers. Figure 4 suggests how the three concepts are related. In this framework, color and stereo metamers are both perceptual metamers, but only color metamers are sensory metamers. Perceptual metamers (e.g. stereo) Sensory metamers (e.g. color) Memory metamers Figure 5. The metamer hierarchy. 6 Conclusions At each vergence eye posture it was possible to create stereoscopic stimuli with distinct disparity patterns that were nonetheless indistinguishable in a forced choice task. Stimuli that were metamers with the eyes in one position became distinguishable after a change of vergence eye posture alone, without changes to the retinal images. We can conclude that horizontal disparity per se is lost to the visual system after combination with the other signals that are used to interpret it as depth. Presumably, stereo metamers have distinguishable representations in primary visual cortex—one suspects this would be evident in evoked potentials or fMRI. The loss of information that renders these stimuli metameric probably occurs in two places. First, there appears to be a leak-proof “constancy” computation in which vertical disparity is used to correct horizontal disparity (Equation 1). The output of this computation is unaffected if equal amounts of horizontal and vertical magnification are added to one eyes’ image. However, the estimator that uses felt eye position can distinguish these stimuli, because their horizontal size ratios differ. Thus a second leak-proof step must occur, in which slant estimates are combined in a weighted average. It seems reasonable to call these stimuli “perceptual metamers,” by analogy with, and to distinguish them from, the traditional “sensory” metamerization of colored lights. Acknowledgments This work was supported by startup funds provided to the author by the University of Pennsylvania. The author thanks Mark Nolt for help conducting the experiments, Rufus Frazer for serving as an observer, and Jack Nachmias and David Brainard for comments on an earlier draft of this paper. References 1. Clark, J.J. and A.L. Yuille, Data fusion for sensory information processing systems. 1990, Boston: Kluwer. 2. Landy, M.S., et al., Measurement and modeling of depth cue combination: in defense of weak fusion. Vision Research, 1995. 35(3): p. 389-412. 3. Ogle, K.N., Induced size effect. I. A new phenomenon in binocular space perception associated with the relative sizes of the images of the two eyes. Archives of Ophthalmology, 1938. 20: p. 604-623. 4. Gårding, J., et al., Stereopsis, vertical disparity and relief transformations. Vision Res, 1995. 35(5): p. 703-22. 5. Backus, B.T., et al., Horizontal and vertical disparity, eye position, and stereoscopic slant perception. Vision Res, 1999. 39(6): p. 1143-70. 6. Banks, M.S. and B.T. Backus, Extra-retinal and perspective cues cause the small range of the induced effect. Vision Res, 1998. 38(2): p. 187-94. 7. Backus, B.T. and M.S. Banks, Estimator reliability and distance scaling in stereoscopic slant perception. Perception, 1999. 28(2): p. 217-42. 8. Foley, J.M., Binocular distance perception. Psychol Rev, 1980. 87(5): p. 411-34. 9. Backus, B.T. and M.J. Nolt, Analysis of stereoscopic metamers. Journal of Vision (Vision Sciences conference supplement), 2001. 1: p. in press. 10. Stevens, K.A., Slant-tilt: the visual encoding of surface orientation. Biol Cybern, 1983. 46(3): p. 183-95. 11. Rogers, B.J. and M.F. Bradshaw, Vertical disparities, differential perspective and binocular stereopsis. Nature, 1993. 361(6409): p. 253-5. 12. Mayhew, J.E. and H. Longuet-Higgins, C, A computational model of binocular depth perception. Nature, 1982. 297(5865): p. 376-378. 13. Calkins, D.J., J.E. Thornton, and E.N. Pugh, Jr., Monochromatism determined at a long-wavelength/middle-wavelength cone- antagonistic locus. Vision Res, 1992. 32(12): p. 2349-67. 14. van Ee, R. and C.J. Erkelens, Temporal aspects of binocular slant perception. Vision Res, 1996. 36(1): p. 43-51. 15. Enright, J.T., Sequential stereopsis: a simple demonstration. Vision Res, 1996. 36(2): p. 307-12. 16. Schor, C.M., J. Gleason, and D. Horner, Selective nonconjugate binocular adaptation of vertical saccades and pursuits. Vision Res, 1990. 30(11): p. 1827-44. 17. Barlow, H.B., Temporal and spatial summation in human vision at different backgound intensities. Journal of Physiology, 1958. 141: p. 337-350. 18. Wandell, B.A., Foundations of vision. 1995, Sunderland, MA: Sinauer Associates. 19. Baylor, D.A., B.J. Nunn, and J.L. Schnapf, Spectral sensitivity of cones of the monkey Macaca fascicularis. J Physiol, 1987. 390: p. 145-60. 20. Loftus, G.R. and E. Ruthruff, A theory of visual information acquisition and visual memory with special application to intensity-duration trade-offs. J Exp Psychol Hum Percept Perform, 1994. 20(1): p. 33-49.
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Scaling laws and local minima in Hebbian ICA Magnus Rattray and Gleb Basalyga Department of Computer Science, University of Manchester, Manchester M13 9PL, UK. magnus@cs.man.ac.uk, basalygg@cs.man.ac.uk Abstract We study the dynamics of a Hebbian ICA algorithm extracting a single non-Gaussian component from a high-dimensional Gaussian background. For both on-line and batch learning we find that a surprisingly large number of examples are required to avoid trapping in a sub-optimal state close to the initial conditions. To extract a skewed signal at least examples are required for -dimensional data and examples are required to extract a symmetrical signal with non-zero kurtosis. 1 Introduction Independent component analysis (ICA) is a statistical modelling technique which has attracted a significant amount of research interest in recent years (for a review, see Hyv¨arinen, 1999). The goal of ICA is to find a representation of data in terms of a combination of statistically independent variables. A number of neural learning algorithms have been applied to this problem, as detailed in the aforementioned review. Theoretical studies of ICA algorithms have mainly focussed on asymptotic stability and efficiency, using the established results of stochastic approximation theory. However, in practice the transient stages of learning will often be more significant in determining the success of an algorithm. In this paper a Hebbian ICA algorithm is analysed in both on-line and batch mode, highlighting the critical importance of the transient dynamics. We find that a surprisingly large number of training examples are required in order to avoid trapping in a sub-optimal state close to the initial conditions. To detect a skewed signal at least examples are required for -dimensional data, while
examples are required for a symmetric signal with non-zero kurtosis. In addition, for on-line learning we show that the maximal initial learning rate which allows successful learning is unusually low, being for a skewed signal and for a symmetric signal. In order to obtain a tractable model, we consider the limit of high-dimensional data and study an idealised data set in which a single non-Gaussian source is mixed into a large number of Gaussian sources. Recently, one of us considered a more general model in which an arbitrary, but relatively small, number of non-Gaussian sources were mixed into a high-dimensional Gaussian background (Rattray, 2002). In that work a solution to the dynamics of the on-line algorithm was obtained in closed form for learning iterations and a simple solution to the asymptotic dynamics under the optimal learning rate decay was obtained. However, it was noted there that modelling the dynamics on an timescale is not always appropriate, because the algorithm typically requires much longer in order to escape from a class of metastable states close to the initial conditions. In order to elucidate this effect in greater detail we focus here on the simplest case of a single non-Gaussian source and we will limit our analysis to the dynamics close to the initial conditions. In recent years a number of on-line learning algorithms, including back-propagation and Sanger’s PCA algorithm, have been studied using techniques from statistical mechanics (see, for example, Biehl (1994); Biehl and Schwarze (1995); Saad and Solla (1995) and contributions in Saad (1998)). These analyses exploited the “self-averaging” property of certain macroscopic variables in order to obtain ordinary differential equations describing the deterministic evolution of these quantities over time in the large limit. In the present case the appropriate macroscopic quantity does not self-average and fluctuations have to be considered even in the limit. In this case it is more natural to model the on-line learning dynamics as a diffusion process (see, for example Gardiner, 1985). 2 Data Model In order to apply the Hebbian ICA algorithm we must first sphere the data, ie. linearly transform the data so that it has zero mean and an identity covariance matrix. This can be achieved by standard transformations in a batch setting or for on-line learning an adaptive sphering algorithm, such as the one introduced by Cardoso and Laheld (1996), could be used. To simplify the analysis it is assumed here that the data has already been sphered. Without loss of generality it can also be assumed that the sources each have unit variance. Each data point is generated from a noiseless linear mixture of sources which are decomposed into a single non-Gaussian source and uncorrelated Gaussian components,
. We will also decompose the mixing matrix into a column vector and a rectangular matrix associated with the non-Gaussian and Gaussian components respectively, ! #"$% '& (1) We will consider both the on-line case, in which a new IID example )( is presented to the algorithm at each time * and then discarded, and also the batch case, in which a finite set of examples are available to the algorithm. To conform with the model assumptions the mixing matrix must be unitary, which leads to the following constraints, + ,- /. 10 %0 ,2 0 "3 0 4 (2)
0 50 + . 10 10 %0 ,6%0 %7 & (3) 3 On-line learning The goal of ICA is to find a vector 8 such that the projection 9;:<8=01 3>@?A . Defining the overlap B<:'8%0CC we obtain, 9 8 0 #"D BE#"GFIH JKJ 8LJMJ B where F NO P (4) where we have made use of the constraint in eqn. (2). This assumes zero correlation between 8 and which is true for on-line learning but is only strictly true for the first iteration of batch learning (see section 4). In the algorithm described below we impose a normalisation constraint on 8 such that JMJ 8LJKJQ . In this case we see that the goal is to find 8 such that BR>S? . A simple Hebbian (or anti-Hebbian) learning rule was studied by Hyv¨arinen and Oja (1998), who showed it to have a remarkably simple stability condition. We will consider the deflationary form in which a single source is learned at one time. The algorithm is closely related to Projection Pursuit algorithms, which seek interesting projections in highdimensional data. A typical criteria for an interesting projection is to find one which is maximally non-Gaussian in some sense. Maximising some such measure (simple examples would be skewness or kurtosis) leads to the following simple algorithm (see Hyv¨arinen and Oja, 1998, for details). The change in 8 at time * is given by, 8 9 ( ( followed by normalisation such that JKJ 8LJKJ & (5) Here, is the learning rate and 9 is some non-linear function which we will take to be at least three times differentiable. An even non-linearity, eg. 9 9 , is appropriate for detecting asymmetric signals while a more common choice is an odd function, eg. 9 9 or 9
9 , which can be used to detect symmetric non-Gaussian signals. In the latter case E P has to be chosen in order to ensure stability of the correct solution, as described by Hyv¨arinen and Oja (1998), either adaptively or using ´a priori knowledge. We set in the case of an even non-linearity. Remarkably, the same non-linearity can be used to separate both sub and super-Gaussian signals, in contrast to maximum likelihood methods for which this is typically not the case. We can write the above algorithm as, 8 ( 8%( " 9/( ,( H " 9 ( 9 ( " 9 ( JMJ ( JMJ & (6) For large and (two different scalings will be considered below) we can expand out to get a weight decay normalisation, 8 ( 8 ( " 9 ( ! ( 9 ( 8 (#" $ 9 ( 8 ( & (7) Taking the dot-product with gives the following update increment for the overlap B , % B< & 9 ( B ( 9 ( " 9 ( B ( (8) where we used the constraint in eqn. (3) to set ,0 . Below we calculate the mean and variance of % B for two different scalings of the learning rate. Because the conditional distribution for 9 given only depends on B (setting JKJ 8LJKJ in eqn. 4) these expressions will depend only on B and statistics of the non-Gaussian source distribution. 3.1 Dynamics close to the initial conditions If the entries in , and 8 are initially of similar order then one would expect B< ' . This is the typical case if we consider a random and uncorrelated choice for and the initial entries in 8 . Larger initial values of B could only be obtained with some prior knowledge of the mixing matrix which we will not assume. We will set (;: B*) in the following discussion, where ( is assumed to be an quantity. The discussion below is therefore restricted to describing the dynamics close to the initial conditions. For an account of the transient dynamics far from the initial conditions and the asymptotic dynamics close to an optimal solution, see Rattray (2002). 3.1.1 9 even, + -, N If the signal is asymmetrical then an even non-linearity can be used, for example 9 9 is a common choice. In this case the appropriate (ie. maximal) scaling for the learning rate is and we set /.10 where . is an scaled learning rate parameter. In even,
% odd,
% Figure 1: Close to the initial conditions (where (: 8=01 ) ) the learning dynamics is equivalent to diffusion in a polynomial potential. For asymmetrical source distributions we can use an even non-linearity in which case the potential is cubic, as shown on the left. For symmetrical source distributions with non-zero kurtosis we should use an odd non-linearity in which case the potential is quartic, as shown on the right. The dynamics is initially confined in a metastable state near (E N with a potential barrier % . this case we find that the mean and variance of the change in ( at each iteration are given by (to leading order in ), E + % ( . F . ( " + F .( " (9) Var + % ( . F . (10) where + is the third cumulant of the source distribution (third central moment), which measures skewness, and brackets denote averages over F NO . We also find that E + % ( . for integer ! . In this case the system can be described by a Fokker-Planck equation for large (see, for example, Gardiner, 1985) with a characteristic timescale of . The system is locally equivalent to a diffusion in the following cubic potential, ( " F . ( # + F .( (11) with a diffusion coefficient $ % F . which is independent of ( . The shape of this potential is shown on the left of fig. 1. A potential barrier of % must be overcome to escape a metastable state close to the initial conditions. 3.1.2 9 odd, + " , N If the signal is symmetrical, or only weakly asymmetrical, it will be necessary to use an odd non-linearity, for example 9 9 or 9
9 are popular choices. In this case a lower learning rate is required in order to achieve successful separation. The appropriate scaling for the learning rate is and we set 5 . 0 where again . is an scaled learning rate parameter. In this case we find that the mean and variance of the change in ( at each iteration are given by, E + % ( . F . ( " # + " F .( " (12) Var + % ( . F . (13) where + " is the fourth cumulant of the source distribution (measuring kurtosis) and brackets denote averages over F NO P . Again the system can be described by a Fokker-Planck equation for large but in this case the timescale for learning is , an order of slower than in the asymmetrical case. The system is locally equivalent to diffusion in the following quartic potential, ( " F . ( " J + " F J .( " (14) with a diffusion coefficient $ F . . We have assumed < Sign + " which is a necessary condition for successful learning. In the case of a cubic non-linearity this is also the condition for stability of the optimal fixed point, although in general these two conditions may not be equivalent (Rattray, 2002). The shape of this potential is shown on the right of fig. 1 and again a potential barrier of % must be overcome to escape a metastable state close to the initial conditions. 3.1.3 Escape times from a metastable state at B< N For large . the dynamics of ( corresponds to an Ornstein-Uhlenbeck process with a Gaussian stationary distribution of fixed unit variance. Thus, if one chooses too large . initially the dynamics will become localised close to B N (recall, B (0 ) ). As . is reduced the potential barrier confining the dynamics is reduced. The timescale for escape for large . (mean first passage time) is mainly determined by the effective size of the barrier (see, for example, Gardiner, 1985), escape * % $
(15) where % is the potential barrier, $ is the diffusion coefficient and * is a unit of time in the diffusion process. For the two cases considered above we obtain, even escape F . + F
for even 9 , + , N , + .5: . odd escape F . J + " F J for odd 9 , + " , N . + .%:& . (16) The constants of proportionality depend on the shape of the potential and not on . As the learning rate parameter is reduced so the timescale for escape is also reduced. However, the choice of optimal learning rate is non-trivial and cannot be determined by considering only the leading order terms in B as above, because although small . will result in a quicker escape from the unstable fixed point near B N , this in turn will lead to a very slow learning transient after escape. Notice that escape time is shortest when the cumulants + or + " are large, suggesting that deflationary ICA algorithms will tend to find these signals first. From the above discussion one can draw two important conclusions. Firstly, the initial learning rate must be less than initially in order to avoid trapping close to the initial conditions. Secondly, the number of iterations required to escape the initial transient will be greater than , resulting in an extremely slow initial stage of learning for large . The most extreme case is for symmetric source distributions with non-zero kurtosis, in which case learning iterations are required. In fig. 2 we show results of learning with an asymmetric source (top) and uniform source (bottom) for different scaled learning rates. As the learning rate is increased (left to right) we observe that the dynamics becomes increasingly stochastic, with the potential barrier becoming increasingly significant (potential maxima are shown as dashed lines). For the largest value of learning rate ( .$ ) the algorithm becomes trapped close to the initial conditions for the whole simulation time. From the time axis we observe that the learning timescale is for the asymmetrical signal and for the symmetric signal, as predicted by our theory. 0 5 10 −1 −0.5 0 0.5 1 t/N3 R φ(x)=x3, κ4≠ 0 0 5 10 −1 −0.5 0 0.5 1 t/N3 R 0 5 10 −1 −0.5 0 0.5 1 t/N3 R 0 5 10 15 0 0.5 1 t/N2 R φ(x)=x2, κ3≠ 0 ν=0.1 0 5 10 15 0 0.5 1 t/N2 R ν=1 0 5 10 15 0 0.5 1 t/N2 R ν=5 R Figure 2: 100-dimensional data ( NQN ) is produced from a mixture containing a single non-Gaussian source. In the top row we show results for a binary, asymmetrical source with skewness + & and . In the bottom row we show results for a uniformly distributed source and . Each row shows learning with the same initial conditions and data but with different scaled learning rates (left to right .R N & Q and ) where . : (top) or . : (bottom). Dashed lines are maxima of the potentials in fig. 1. 4 Batch learning The batch version of eqn. (5) for sufficiently small learning rates can be written, % 8 ( 9 ( ( 9 ( 8 ( " (17) where is the number of training examples. Here we argue that such an update requires at least the same order of examples as in the on-line case, in order to be successful. Less data will result in a low signal-to-noise ratio initially and the possibility of trapping in a sub-optimal fixed point close to the initial conditions. As in the on-line case we can write the update in terms of B , % B ( 9 ( ( 9 ( B ( " & (18) We make an assumption that successful learning is unlikely unless the initial increment in B is in the desired direction. For example, with an asymmetric signal and quadratic nonlinearity we require + % B N initially, while for a symmetric signal and odd non-linearity we require B % B% N . We have carried out simulations of batch learning which confirm that a relatively low percentage of runs in which the intial increment was incorrect result in successful learning compared to typical performance. As in the on-line case we observe that runs either succeed, in which case B> ? , or fail badly with B remaining ' . As before, B ' initially and we can therefore expand the right-hand side of eqn. (18) in orders of B for large . % B init ( % B at the first iteration) is a sum over randomly sampled terms, and the central limit theorem states that for large the distribution from which % B init is sampled will be Gaussian, with mean and variance given by (to leading order in B ), E % B init + F B " # + " F B (19) Var % B init F & (20) Notice that the + term disappears in the case of an asymmetrical non-linearity, which is why we have left both terms in eqn. (19). The algorithm will be likely to fail when the standard deviation of % B init is of the same order (or greater) than the mean. Since B ' initially, we see that this is true for R
in the case of an even nonlinearity and asymmetric signal, or for R
in the case of an odd non-linearity and a signal with non-zero kurtosis. We expect these results to be necessary but not necessarily sufficient for successful learning, since we have only shown that this order of examples is the minimum required to avoid a low signal-to-noise ratio in the first learning iteration. A complete treatment of the batch learning problem would require much more sophisticated formulations such as the mean-field theory of Wong et al. (2000). 5 Conclusions and future work In both the batch and on-line Hebbian ICA algorithm we find that a surprisingly large number of examples are required to avoid a sub-optimal fixed point close to the initial conditions. We expect simialr scaling laws to apply in the case of small numbers of non-Gaussian sources. Analysis of the square demixing problem appears to be much more challenging as in this case there may be no simple macroscopic description of the system for large . It is therefore unclear at present whether ICA algorithms based on Maximum-likelihood and Information-theoretic principles (see, for example, Bell and Sejnowski, 1995; Amari et al., 1996; Cardoso and Laheld, 1996), which estimate a square demixing matrix, exhibit similar classes of fixed point to those studied here. Acknowledgements: This work was supported by an EPSRC award (ref. GR/M48123). We would like to thank Jon Shapiro for useful comments on a preliminary version of this paper. References S-I Amari, A Cichocki, and H H Yang. In D S Touretzky, M C Mozer, and M E Hasselmo, editors, Neural Information Processing Systems 8, pages 757–763. MIT Press, Cambridge MA, 1996. A J Bell and T J Sejnowski. Neural Computation, 7:1129–1159, 1995. M Biehl. Europhys. Lett., 25:391–396, 1994. M Biehl and H Schwarze. J. Phys. A, 28:643–656, 1995. J-F Cardoso and B. Laheld. IEEE Trans. on Signal Processing, 44:3017–3030, 1996. C. W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, New York, 1985. A Hyv¨arinen. Neural Computing Surveys, 2:94–128, 1999. A Hyv¨arinen and E Oja. Signal Processing, 64:301–313, 1998. M Rattray. Neural Computation, 14, 2002 (in press). D Saad, editor. On-line Learning in Neural Networks. Cambridge University Press, 1998. D Saad and S A Solla. Phys. Rev. Lett., 74:4337–4340, 1995. K Y M Wong, S Li, and P Luo. In S A Solla, T K Leen, and K-R M¨uller, editors, Neural Information Processing Systems 12. MIT Press, Cambridge MA, 2000.
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Sampling Techniques for Kernel Methods Dimitris Achlioptas Microsoft Research optas@microsoft.com Frank McSherry University of Washington mcsherry@cs.washington.edu Bernhard Sch¨olkopf Biowulf Technologies NY bs@conclu.de Abstract We propose randomized techniques for speeding up Kernel Principal Component Analysis on three levels: sampling and quantization of the Gram matrix in training, randomized rounding in evaluating the kernel expansions, and random projections in evaluating the kernel itself. In all three cases, we give sharp bounds on the accuracy of the obtained approximations. Rather intriguingly, all three techniques can be viewed as instantiations of the following idea: replace the kernel function by a “randomized kernel” which behaves like in expectation. 1 Introduction Given a collection of training data
, techniques such as linear SVMs [13] and PCA extract features from by computing linear functions of this data. However, it is often the case that the structure present in the training data is not simply a linear function of the data representation. Worse, many data sets do not readily support linear operations such as addition and scalar multiplication (text, for example). In a “kernel method” is first mapped into some dot product space using . The dimension of can be very large, even infinite, and therefore it may not be practical (or possible) to work with the mapped data explicitly. Nonetheless, in many cases the dot products
! " can be evaluated efficiently using a positive definite kernel for , ı.e. a function so that #$ % '&()*+ ! " . Any algorithm whose operations can be expressed in terms of dot products can be generalized to an algorithm which operates on , , simply by presenting the Gram matrix . /& $ as the input covariance matrix. Note that at no point is the function explicitly computed; the kernel implicitly performs the dot product calculations between mapped points. While this “kernel trick” has been extremely successful, a problem common to all kernel methods is that, in general, is a dense matrix, making the input size scale as 021 . For example, in Kernel PCA such a matrix has to be diagonalized, while in SVMs a quadratic program of size 0 1 must be solved. As the size of training sets in practical applications increases, the growth of the input size rapidly poses severe computational limitations. Various methods have been proposed to deal with this issue, such as decomposition methods for SVM training (e.g., [10]), speedup methods for Kernel PCA [12], and other kernel methods [2, 14]. Our research is motivated by the need for such speedups that are also accompanied by strong, provable performance guarantees. In this paper we give three such speedups for Kernel PCA. We start by simplifying the Gram matrix via a novel matrix sampling/quantization scheme, motivated by spectral properties of random matrices. We then move on to speeding up classification, by using randomized rounding in evaluating kernel expansions. Finally, we consider the evaluation of kernel functions themselves and show how many popular kernels can be approximated efficiently. Our first technique relates matrix simplification to the stability of invariant subspaces. The other two are, in fact, completely general and apply to all kernel methods. What is more, our techniques suggest the notion of randomized kernels, whereby each evaluation of the kernel is replaced by an evaluation of a randomized function (on the same input pair). The idea is to use a function which for every input pair behaves like in expectation (over its internal coin-flips), yet confers significant computational benefits compared to using . In fact, each one of our three techniques can be readily cast as an appropriate randomized kernel, with no other intervention. 2 Kernel PCA Given 0 training points recall that is an 0 0 matrix with . & . For some choice of 0 , the Kernel PCA (KPCA) method [11] computes the largest eigenvalues, , and eigenvectors, of . Then, given an input point , the method computes the value of nonlinear feature extraction functions &
1
$# 2 There are several methods for computing the principal components of a symmetric matrix. The choice depends on the properties of the matrix and on how many components one is seeking. In particular, if relatively few principal components are required, as is the case in KPCA, Orthogonal Iteration is a commonly used method.1 Orthogonal Iteration 1. Let be a random 0 matrix with orthonormal columns. 2. While not converged, do (a) (b) Orthonormalize 3. Return It is worth looking closely at the complexity of performing Orthogonal Iteration on a matrix . Step 1 can be done in 0 steps, making step 2 the computational bottleneck. The orthonormalization step 2b takes time 0 1 and is overwhelmed by the cost of computing in step 2a which, generally, takes 0 1 . The number of iterations of the while loop is a somewhat complicated issue, but one can prove that the “error” in (with respect to the true principal components) decreases exponentially with the number of iterations. All in all, the running time of Orthogonal Iteration scales linearly with the cost of the matrix multiplication . If is sparse, ı.e., if roughly one out of every entries of is non-zero, then the matrix multiplication costs 0 1 . As mentioned earlier, the matrix used in Kernel PCA is almost never sparse. In the next section, we will show how to sample and quantize the entries of , obtaining a matrix which is sparser and whose entries have simpler data representation, yet has essentially the same spectral structure, i.e. eigenvalues/eigenvectors, as . 1Our discussion applies equally well to Lanczos Iteration which, while often preferable, is a more complicated method. Here we focus on Orthogonal Iteration to simplify exposition. 3 Sampling Gram Matrices In this section we describe two general “matrix simplification” techniques and discuss their implications for Kernel PCA. In particular, under natural assumptions on the spectral structure of , we will prove that applying KPCA to the simplified matrix yields subspaces which are very close to those that KPCA would find in . As a result, when we project vectors onto these spaces (as performed by the feature extractors) the results are provably close to the original ones. First, our sparsification process works by randomly omitting entries in . Precisely stated, we let the matrix be described entrywise as / & $ & with probability . with probability Second, our quantization process rounds each entry in to one of
, where & . , thus reducing the representation of each entry to a single bit. / & $ &
with probability . with probability . 2 Sparsification greatly accelerates the computation of eigenvectors by accelerating multiplication by . Moreover, both approaches greatly reduce the space required to store the matrix (and they can be readily combined), allowing for much bigger training sets to fit in main memory. Finally, we note that i) sampling also speeds up the construction of the Gram matrix since we need only compute those values of that remain in , while ii) quantization allows us to replace exact kernel evaluations by coarse unbiased estimators, which can be more efficient to compute. While the two processes above are quite different, they share one important commonality: in each case, . & / . Moreover, the entries of the error matrix, & , are independent random variables, having expectation zero and bounded variance. Large deviation extensions [5] of Wigner’s famous semi-circle law, imply that with very high probability such matrices have small L2 norm (denoted by ! throughout). Theorem 1 (Furedi and Komlos [5]) Let be an 0 0 symmetric matrix whose entries are independent random variables with mean 0, variance bounded above by " 1 , and magnitude bounded by "$# 0 &% '() 0 . With probability +* -, ./"1 0 0 , 1 2 " # 0 It is worth noting that this upper bound is within a constant factor of the lower bound on the L2 norm of any matrix where the mean squared entry equals " 1 . More precisely, it is easy to show that every matrix with Frobenius norm 3"#0 $1 has L2 norm at least "4# 0 . Therefore, we see that the L2 error introduced by is within a factor of 4 of the L2 error associated with any modification to that has the same entrywise mean squared error. We will analyze three different cases of spectral stability, corresponding to progressively stronger assumptions. At the heart of these results is the stability of invariant subspaces in the presence of additive noise. This stability is very strong, but can be rather technical to express. In stating each of these results, it is important to note that the eigenvectors correspond exactly to the feature extractors associated with Kernel PCA. For an input point , let 5 denote the vector whose 6 th coordinate is #$ and recall that # &
1 5%87 .5"2 Recall that in KPCA we associate features with the largest eigenvalues of , where is typically chosen by requiring , for some threshold . First, we consider what happens when is not large. Observe that in this case we cannot hope to equate all # and , as the th feature is very sensitive to small changes in . However, we can show that all features with far from are treated consistently in and . Theorem 2 Let be any matrix whose columns form an orthonormal basis for the space of features (eigenvectors) in whose eigenvalue is at least . Let be any matrix whose columns form an orthonormal basis for the orthogonal complement of . Let and be the analogous matrices for . For any ,
1 and 1
1 If we use the threshold for the eigenvalues of , the first equation asserts that the features KPCA recovers are not among the features of whose eigenvalues are less than . Similarly, the second equation asserts that KPCA will recover all the features of whose eigenvalues are larger than . Proof: We employ the techniques of Davis and Kahan [4]. Observe that & $
& & & where and are diagonal matrices whose entries (the eigenvalues of and ) are at least and at most , respectively. Therefore 1
1
1
1 which implies the first stated result. The second proof is essentially identical. In our second result we will still not be able to isolate individual features, as the error matrix can reorder their importance by, say, interchanging and . However, we can show that any such interchange will occur consistently in all test vectors. Let be the -dimensional vector whose th coordinate is 1 # , ı.e., here we do not normalize features to “equal importance”. Recall that 5 is the vector whose 6 th coordinate is # . Theorem 3 Assume that 1 1 for some . There is an orthonormal rotation matrix such that for all # & ! # 1 5 Proof: Instantiate Theorem 2 with & and & . Note that the rotation matrix becomes completely irrelevant if we are only concerned with differences, angles, or inner products of feature vectors. Finally, we prove that in the special case where a feature is well separated from its neighboring features in the spectrum of , we get a particularly strong bound. Theorem 4 If , 1 1 , and
1 , then # + # 5 Proof:(sketch) As before, we specialize Theorem 2, but first shift both and by . This does not change the eigenvectors, and allows us to consider in isolation. 4 Approximating Feature Extractors Quickly Having determined eigenvalues and eigenvectors, given an input point , the value of on each feature reduces to evaluating, for some unit vector , a function & $ 2 where we dropped the subscript , as well as the scaling by
1 . Assume that !. take values in an interval of width and let 1 1 be any unit vector. We will devise a fast, unbiased, small-variance estimator for , by sampling and rounding the expansion coefficients . Fix . For each 6 : if then let & ; if let & ( with probability ! otherwise. That is, after potentially keeping some large coefficients deterministically, we proceed to perform “randomized rounding” on the (remaining) coefficients of . Let # &
$* Clearly, we have & # . Moreover, using Hoeffding’s inequality [7], we can bound the behavior of # # arising from the terms subjected to probabilistic rounding. In particular, this gives
# & +* -, +1 0 1 (1) Note now that in Kernel PCA we typically expect # 0 , i.e., dense eigenvectors. This makes # 0 the natural scale for measuring # and suggests that using far fewer than 0 kernel evaluations we can get good approximations of . In particular, for a chosen (fixed) value of let us say that # is trivial if % '( 0 Having picked some threshold (for SVM expansions is related to the classification offset) we want to determine whether is non-trivial and, if so, we want to get a good relative error estimate for it. Theorem 5 For any and 0 +%1'( 0 set & # 0 . With probability at least 0 ) 1. There are fewer than 2
0 non-zero . 2. Either both % # and are trivial or % Proof: Let 0 denote the number of non-zero and let & 6 . Note that 0 equals plus the sum of 0 independent Bernoulli trials. It is not hard to show that the probability that the event in 1 fails is bounded by the corresponding probability for the case where all coordinates of are equal. In that case, 0 is a Binomial random variable with 0 trials and probability of success # 0 and, by our choice of , 0 & 0 . The Chernoff bound now implies that the event in 1 fails to occur with probability ! 0
) . For the enent in 2 it suffices to observe that failure occurs only if is at least " # % '( 0 . By (1), this also occurs with probability ! 0
) . 5 Quick batch approximations of Kernels In this section we devise fast approximations of the kernel function itself. We focus on kernels sharing the following two characteristics: i) they map -dimensional Euclidean space, and, ii) the mapping depends only on the distance and/or inner product of the considered points. We note that this covers some of the most popular kernels, e.g., RBFs and polynomial kernels. To simplify exposition we focus on the following task: given a sequence of (test) vectors 1 determine
for each of a fixed set of (training) vectors , where 0 . To get a fast batch approximdition, the idea is that rather than evaluating distances and inner products directly, we will use a fast, approximately correct oracle for these quantities offering the following guarantee: it will answer all queries with small relative error. A natural approach for creating such an oracle is to pick of the coordinates in input space and use the projection onto these coordinates to determine distances and inner products. The problem with this approach is that if & , any coordinate sampling scheme is bound to do poorly. On the other hand, if we knew that all coordinates contributed “approximately equally” to , then coordinate sampling would be much more appealing. We will do just this, using the technique of random projections [8], which can be viewed as coordinate sampling preceded by a random rotation. Imagine that we applied a spherically random rotation to ! (before training) and then applied the same random rotation to each input point as it became available. Clearly, all distances and inner products would remain the same and we would get exactly the same results as without the rotation. The interesting part is that any fixed vector that was a linear combination of training and/or input vectors, e.g. $ , after being rotated becomes a spherically random vector of length . As a result, the coordinates of are i.i.d. random variables, in fact # ! , enabling coordinate sampling. Our oracle amounts to multiplying each training and input point by the same projection matrix , where & %1'( , and using the resulting -dimensional points to estimate distances and inner products. (Think of as the result of taking a rotation matrix and keeping the first columns (sampling)). Before describing the choice of and the quality of the resulting approximations, let us go over the computational savings. 1. Rotating the 0 training vectors takes 0
%1'( . Note that This cost will be amortized over the sequence of input vectors. This rotation can be performed in the training phase. 2. The kernel evaluations for each now take 0 %1'( instead of 0 . 3. Rotating takes time % '( which is dominated by 0 %1'( . Having motivated our oracle as a spherically random rotation followed by coordinate sampling, we will actually employ a simpler method to perform the projection. Namely, we will rely on a recent result of [1], asserting that we can do at least as well by taking / & . # * where the */ are i.i.d. with
/ , each case having probability . Thus, postponing the scaling by # until the end, each of the new coordinates is formed as follows: split the coordinates randomly into two groups; sum the coordinates in each group; take the difference of the two sums. Regarding the quality of approximations we get Theorem 6 Consider any sets of points and #% in . Let & 0 and for given let & 2 1 ) % Let be a random matrix defined by . & */ # where the . are i.i.d. with . , each case having probability . For any let denote . With probability at least , for every pair of points # / 1 1 . 1 (2) and "& " 1 1 (3) Proof: We use Lemma 5 of [1], asserting that for any and any , 1 1 1 +* , 1 ) (4) By our choice of , the r.h.s. of (4) is 1 . Thus, by the union bound, with probability at least the lengths of all 0 0 1
vectors corresponding to and , 6 & , & 0 , are maintained within a factor of . This readily yields (2). For (3) we observe that " & 1 1 1 and thus if 1 1 and 1 are within of 1 1 and 1 , then (3) holds. 6 Conclusion We have described three techniques for speeding up kernel methods through the use of randomization. While the discussion has focused on Kernel PCA, we feel that our techniques have potential for further development and empirical evaluation in a more general setting. Indeed, the methods for sampling kernel expansions and for speeding up the kernel evaluation are universal; also, the Gram matrix sampling is readily applicable to any kernel technique based on the eigendecomposition of the Gram matrix [3]. Furthermore, it might enable us to speed up SVM training by sparsifying the Hessian and then applying a sparse QP solver, such as the ones described in [6, 9]. Our sampling and quantization techniques, both in training and classification, amount to repeatedly replacing single kernel evaluations with independent random variables that have appropriate expectations. Note, for example, that while we have represented the sampling of the kernel expansion as randomized rounding of coefficients, this rounding is also equivalent to the following process: consider each coefficients as is, but replace every kernel invocation ! with an invocation of a randomized kernel function, distributed as $ & $ with probability otherwise. Similarly, the process of sampling in training can be thought of as replacing with & with probability with probability while an analogous randomized kernel is the obvious choice for quantization. We feel that this approach suggests a notion of randomized kernels, wherein kernel evaluations are no longer considered as deterministic but inherently random, providing unbiased estimators for the corresponding inner products. Given bounds on the variance of these estimators, it seems that algorithms which reduce to computing weighted sums of kernel evaluations can exploit concentration of measure. Thus, randomized kernels appear promising as a general tool for speeding up kernel methods, warranting further investigation. Acknowledgments. BS would like to thank Santosh Venkatesh for detailed discussions on sampling kernel expansions. References [1] D. Achlioptas, Database-friendly random projections, Proc. of the 20th Symposium on Principle of Database Systems (Santa Barbara, California), 2001, pp. 274–281. [2] C. J. C. Burges, Simplified support vector decision rules, Proc. of the 13th International Conference on Machine Learning, Morgan Kaufmann, 1996, pp. 71–77. [3] N. Cristianini, J. Shawe-Taylor, and H. Lodhi, Latent semantic kernels, Proc. of the 18th International Conference on Machine Learning, Morgan Kaufman, 2001. [4] C. Davis and W. Kahan, The rotation of eigenvectors by a perturbation 3, SIAM Journal on Numerical Analysis 7 (1970), 1–46. [5] Z. F¨uredi and J. Koml´os, The eigenvalues of random symmetric matrices, Combinatorica 1 (1981), no. 3, 233–241. [6] N. I. M. Gould, An algorithm for large-scale quadratic programming, IMA Journal of Numerical Analysis 11 (1991), no. 3, 299–324. [7] W. Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 (1963), 13–30. [8] W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Conference in modern analysis and probability (New Haven, Conn., 1982), American Mathematical Society, 1984, pp. 189–206. [9] R. H. Nickel and J. W. Tolle, A sparse sequential quadratic programming algorithm, Journal of Optimization Theory and Applications 60 (1989), no. 3, 453–473. [10] E. Osuna, R. Freund, and F. Girosi, An improved training algorithm for support vector machines, Neural Networks for Signal Processing VII, 1997, pp. 276–285. [11] B. Sch¨olkopf, A. J. Smola, and K.-R. M¨uller, Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation 10 (1998), 1299–1319. [12] A. J. Smola and B. Sch¨olkopf, Sparse greedy matrix approximation for machine learning, Proc. of the 17th International Conference on Machine Learning, Morgan Kaufman, 2000, pp. 911–918. [13] V. Vapnik, The nature of statistical learning theory, Springer, NY, 1995. [14] C. K. I. Williams and M. Seeger, Using the Nystrom method to speed up kernel machines, Advances in Neural Information Processing Systems 2000, MIT Press, 2001.
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Online Learning with Kernels Jyrki Kivinen Alex J. Smola Robert C. Williamson Research School of Information Sciences and Engineering Australian National University Canberra, ACT 0200 Abstract We consider online learning in a Reproducing Kernel Hilbert Space. Our method is computationally efficient and leads to simple algorithms. In particular we derive update equations for classification, regression, and novelty detection. The inclusion of the -trick allows us to give a robust parameterization. Moreover, unlike in batch learning where the -trick only applies to the -insensitive loss function we are able to derive general trimmed-mean types of estimators such as for Huber’s robust loss. 1 Introduction While kernel methods have proven to be successful in many batch settings (Support Vector Machines, Gaussian Processes, Regularization Networks) the extension to online methods has proven to provide some unsolved challenges. Firstly, the standard online settings for linear methods are in danger of overfitting, when applied to an estimator using a feature space method. This calls for regularization (or prior probabilities in function space if the Gaussian Process view is taken). Secondly, the functional representation of the estimator becomes more complex as the number of observations increases. More specifically, the Representer Theorem [10] implies that the number of kernel functions can grow up to linearly with the number of observations. Depending on the loss function used [15], this will happen in practice in most cases. Thereby the complexity of the estimator used in prediction increases linearly over time (in some restricted situations this can be reduced to logarithmic cost [8]). Finally, training time of batch and/or incremental update algorithms typically increases superlinearly with the number of observations. Incremental update algorithms [2] attempt to overcome this problem but cannot guarantee a bound on the number of operations required per iteration. Projection methods [3] on the other hand, will ensure a limited number of updates per iteration. However they can be computationally expensive since they require one matrix multiplication at each step. The size of the matrix is given by the number of kernel functions required at each step. Recently several algorithms have been proposed [5, 8, 6, 12] performing perceptron-like updates for classification at each step. Some algorithms work only in the noise free case, others not for moving targets, and yet again others assume an upper bound on the complexity of the estimators. In the present paper we present a simple method which will allows the use of kernel estimators for classification, regression, and novelty detection and which copes with a large number of kernel functions efficiently. 2 Stochastic Gradient Descent in Feature Space Reproducing Kernel Hilbert Space The class of functions to be studied in this paper are elements of an RKHS . This means that there exists a kernel
and a dot product such that 1) "! # (reproducing property); 2) is the closure of the span of all with %$& . In other words, all &$ are linear combinations of kernel functions. Typically ' ' ()!* is used as a regularization functional. It is the “length of the weight vector in feature space” as commonly used in SV algorithms. To state our algorithm we need to compute derivatives of functionals defined on . For the regularizer +-, /.0 !21 ( ' '( we obtain 3546+-, /. ! . More general versions of +-, /. !87 ' ' lead to 3 4 +-, /. !97: ' ';' '=< 1 . For the evaluation functional >@?A, /.B ! we compute the derivative by using the reproducing property of and obtain 3 4 > ? , /. !%3 4 )!C # . Consequently for a function D EF G GHI which is differentiable in its third argument we obtain 3 4 D # KJL # !MD : # JL K # . Below D will be the loss function. Regularized Risk Functionals and Learning In the standard learning setting we are supplied with pairs of observations # LN J N $OP QG drawn according to some underlying distribution R # J . Our aim is to predict the likely outcome J at location . Several variants are possible: (i) R # J may change over time, (ii) the training sample # N KJ N may be the next observation on which to predict which leads to a true online setting, or (iii) we may want to find an algorithm which approximately minimizes a regularized risk functional on a given training set. We assume that we want to minimize a loss function D SG% TGU which penalizes the deviation between an observation J at location and the prediction , based on observations 1 KJ 1 V;VV # /W KJ W . Since R # J is unknown, a standard approach is to instead minimize the empirical risk XZY\[^] , /. ! _ ` W a Nb 1 D # cN KJ N # cN K (1) or, in order to avoid overly complex hypotheses, minimize the empirical risk plus an additional regularization term +-, /. . This sum is known as the regularized risk X"d#Y\e , /.f ! X"Y\[^] , /.5gih +-, /. ! _ ` W a Njb 1 D /N J N /N gkh +-, /. for hQl9m V (2) Common loss functions are the soft margin loss function [1] or the logistic loss for classification and novelty detection [14], the quadratic loss, absolute loss, Huber’s robust loss [9], or the -insensitive loss [16] for regression. We discuss these in Section 3. In some cases the loss function depends on an additional parameter such as the width of the margin n or the size of the -insensitive zone. One may make these variables themselves parameters of the optimization problem [15] in order to make the loss function adaptive to the amount or type of noise present in the data. This typically results in a term or o n added to D # KJL # . Stochastic Approximation In order to find a good estimator we would like to minimize X d#Y\e , /. . This can be costly if the number of observations is large. Recently several gradient descent algorithms for minimizing such functionals efficiently have been proposed [13, 7]. Below we extend these methods to stochastic gradient descent by approximating Xpd#Y\e , /. by X , .f ! D # KJ #
gkh +-, /. (3) and then performing gradient descent with respect to X , . . Here is either randomly chosen from _ ;VVV ` or it is the new training instance observed at time . Consequently the gradient of X , . with respect to is 3 4 X , . ! D : # J # K ; gSh 3 4 +-, /. ! D : J # K ; gSh/ V (4) The last equality holds if +-, /. ! 1 ( ' ' ( . Analogous results hold for general +-, /. ! 7 ' ' . The the update equations are hence straightforward: S o 354 X , . V (5) Here $
is the learning rate controlling the size of updates undertaken at each iteration. We will return to the issue of adjusting h at a later stage. Descent Algorithm For simplicity, assume that +-, /. !H1 ( ' ' ( . In this case (5) becomes o D : KJ #
K #
gihc E! _ o h o D : J #
K ; V (6) While (6) is convenient to use for a theoretical analysis, it is not directly amenable to computation. For this purpose we have to express as a kernel expansion E! a N N # cN (7) where the N are (previously seen) training patterns. Then (6) becomes _ o h o D : J # K (8) ! o D : # KJ for ! m (9) N _ o h N for ! V (10) Eq. (8) means that at each iteration the kernel expansion may grow by one term. Furthermore, the cost for training at each step is not larger than the prediction cost: once we have computed #
, is obtained by the value of the derivative of D at # J # K . Instead of updating all coefficients N we may simply cache the power series _ _ o h _ o h ( _ o h "!@V;VV and pick suitable terms as needed. This is particularly useful if the derivatives of the loss function D will only assume discrete values, say o _ m _ as is the case when using the soft-margin type loss functions (see Section 3). Truncation The problem with (8) and (10) is that without any further measures, the number of basis functions # will grow without bound. This is not desirable since # determines the amount of computation needed for prediction. The regularization term helps us here. At each iteration the coefficients N with ! are shrunk by _ o h . Thus after $ iterations the coefficient N will be reduced to _ o h "% N . Hence: Proposition 1 (Truncation Error) For a loss function D JL # K with its first derivative bounded by & and a kernel with bounded norm ' ; ;'('*) , the truncation error in incurred by dropping terms N from the kernel expansion of after $ update steps is bounded by _ o h % &) . Furthermore, the total truncation error by dropping all terms which are at least $ steps old is bounded by ' o d+-, ' ' <.% a Nb 1 _ o h < N &)0/ h < 1 _ o h % &) (11) Here d+ , ! Nb <.% 1 N /N . Obviously the approximation quality increases exponentially with the number of terms retained. The regularization parameter h can thus be used to control the storage requirements for the expansion. In addition, it naturally allows for distributions R KJ5 that change over time in which cases it is desirable to forget instances # N J N that are much older than the average time scale of the distribution change [11]. 3 Applications We now proceed to applications of (8) and (10) to specific learning situations. We utilize the standard addition of the constant offset to the function expansion, i.e. # ! g where O$ and $ . Hence we also update into o 3 X , . . Classification A typical loss function in SVMs is the soft margin, given by D JL ! m _ oJ
. In this situation the update equations become N K _ o h N J N Z g J N if J #
/ _ K _ o h N m otherwise. (12) In classification with the -trick we avoid having to fix the margin n by treating it as a variable [15]. The value of n is found automatically by using the loss function D # JL # ! m n oJ
Ko n (13) where m ' ' _ is another parameter. Since has a much clearer intuitive meaning than n , it is easier to tune. On the other hand, one can show [15] that the specific choice of h has no influence on the estimate in -SV classification. Therefore we may set h ! _ and obtain N = n5 _ o N KJ N " g J N n g _ o if J
/kn _ o N m = npo otherwise. (14) Finally, if we choose the hinge-loss, D # JL # ! m ;o J
N K _ o h N J N Z g J N if J # / m K _ o h N m otherwise. (15) Setting h ! m recovers the kernel-perceptron algorithm. For nonzero h we obtain the kernel-perceptron with regularization. Novelty Detection The results for novelty detection [14] are similar in spirit. The setting is most useful here particularly where the estimator acts as a warning device (e.g. network intrusion detection) and we would like to specify an upper limit on the frequency of alerts # (/Un . The relevant loss function is D KJc # Z! m n o # Ko n and usually [14] one uses 8$ rather than Tg where $i in order to avoid trivial solutions. The update equations are N n5 _ o N ZKn g _ o K if /8n _ o N m Kn o otherwise. (16) Considering the update of n we can see that on average only a fraction of observations will be considered for updates. Thus we only have to store a small fraction of the fN . Regression We consider the following four settings: squared loss, the -insensitive loss using the -trick, Huber’s robust loss function, and trimmed mean estimators. For convenience we will only use estimates *$ rather than 8! g where $% . The extension to the latter case is straightforward. We begin with squared loss where D is given by D # JL !P1 ( J o K( V Consequently the update equation is N _ o h N J o # K V (17) This means that we have to store every observation we make, or more precisely, the prediction error we made on the observation. The -insensitive loss D KJc # ! m JQo # /o @ avoids this problem but introduces a new parameter in turn — the width of the insensitivity zone . By making a variable of the optimization problem we have D JL # K! m J o # \o g 5V The update equations now have to be stated in terms of N , and which is allowed to change during the optimization process. This leads to N _ o h N J o g _ o if J o l _ o h N m "o otherwise. (18) This means that every time the prediction error exceeds , we increase the insensitivity zone by . Likewise, if it is smaller than , the insensitive zone is decreased by _ o . Next let us analyze the case of regression with Huber’s robust loss. The loss is given by D JL # K! J o # @o 1 ( if J o #
1 ( J o # K ( otherwise. (19) As before we obtain update equations by computing the derivative of D with respect to # . N _ o N J o # K if J o # l _ o N < 1 J o K otherwise. (20) Comparing (20) with (18) leads to the question whether might not also be adjusted adaptively. This is a desirable goal since we may not know the amount of noise present in the data. While the -setting allowed us to form such adaptive estimators for batch learning with the -insensitive loss, this goal has proven elusive for other estimators in the standard batch setting. In the online situation, however, such an extension is quite natural (see also [4]). All we need to do is make a variable of the optimization problem and set N K _ o N J o g _ o if J o l K _ o N < 1 J o # o otherwise. (21) 4 Theoretical Analysis Consider now the classification problem with the soft margin loss D KJc # ! m n o8J # K ; here n is a fixed margin parameter. Let denote the hypothesis of the online algorithm after seeing the first o _ observations. Thus, at time , the algorithm receives an input . , makes its prediction , receives the correct outcome J , and updates its hypothesis into 1 according to (5). We now wish to bound the cumulative risk W b 1 X , . . The motivation for such bounds is roughly as follows. Assume there is some fixed distribution R from which the examples # J are drawn, and define X , /.f ! ? ! , D # JL K .5gkh +-, /. V Then it would be desirable for the online hypothesis to converge towards " ! arg min 4 X , /. . If we can show that the cumulative risk is asymptotically ` X , " .Kg # ` , we see that at least in some sense does converge to " . Hence, as a first step in our convergence analysis, we obtain an upper bound for the cumulative risk. In all the bounds of this section we assume + E! 1 ( ' ' ( . Theorem 1 Let K# KJ W b 1 be an example sequence such that # ' ) ( for all . Fix lCm , and choose the learning rate *! ) ` 1 (; . Then for any such that ' '' we have W a b 1 X , . ' W a b 1 X , L . g ) ` 1 ( g _ V (22) Notice that the bound does not depend on any probabilistic assumptions. If the example sequence is such that some fixed predictor has a small cumulative risk, then the cumulative risk of the online algorithm will also be small. There is a slight catch here in that the learning rate must be chosen a priori, and the optimal setting depends on ` . The longer the sequence of examples, the smaller learning rate we want. We can avoid this by using a learning rate that starts from a fairly large value and decreases as learning progresses. This leads to a bound similar to Theorem 1 but with somewhat worse constant coefficients. Theorem 2 Let K# KJ K W b 1 be an example sequence such that # ' ) ( for all . Fix l*m , and use at update the learning rate ! _ h 1 ( . Then for any such that ' '' we have W a b 1 X , . ' W a b 1 X , L .5g hf g )
h ( ` 1 ( g _ V (23) Let us now consider the implications of Theorem 2 to a situation in which we assume that the examples J are i.i.d. according to some fixed distribution R . Theorem 3 Let R be a distribution over kG , such that ' ) ( holds with probability _ for # KJ R . Let W ! _ ` W < 1 b 1 where is the -th online hypothesis based on an example sequence J W b 1 that is drawn i.i.d. according to R . Fix l*m , and use at update the learning rate ! _ h 1 ( . Then for any such that ' '' we have 0, X , W .. ' X , . g hB g )
h ( ` < 1 ( g ` < 1 V (24) If we know in advance how many examples we are going to draw, we can use a fixed learning rate as in Theorem 1 and obtain somewhat better constants. 5 Experiments and Discussion In our experiments we studied the performance of online -SVM algorithms in various settings. They always yielded competitive performance. Due to space constraints we only report the findings in novelty detection as given in Figure 1 (where the training algorithm was fed the patterns sans class labels). Already after one pass through the USPS database (5000 training patterns, 2000 test patterns, each of them of size _ _ pixels), which took in MATLAB less than 15s on a 433MHz Celeron, the results can be used for weeding out badly written digits. The setting was used (with ! m V m _ ) to allow for a fixed fraction of detected “outliers.” Based on the theoretical analysis of Section 4 we used a decreasing learning rate with h
< . Conclusion We have presented a range of simple online kernel-based algorithms for a variety of standard machine learning tasks. The algorithms have constant memory requirements and are computationally cheap at each update step. They allow the ready application of powerful kernel based methods such as novelty detection to online and time-varying problems. Results after one pass through the USPS database. We used Gaussian RBF kernels with width (0! m V ! _ . The learning rate was adjusted to 1 W where ` is the number of iterations. Top: the first 50 patterns which incurred a margin error; bottom left: the 50 worst patterns according to o8n on the training set, bottom right: the 50 worst patterns on an unseen test set. Figure 1: Online novelty detection on the USPS dataset with ! m V m _ . Acknowledgments A.S. was supported by the DFG under grant Sm 62/1-1, J.K. & R.C.W. were supported by the ARC. The authors thank Paul Wankadia for help with the implementation. References [1] K. P. Bennett and O. L. Mangasarian. Robust linear programming discrimination of two linearly inseparable sets. Optimization Methods and Software, 1:23–34, 1992. [2] G. Cauwenberghs and T. Poggio. Incremental and decremental support vector machine learning. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 409–415. MIT Press, 2001. [3] L. Csat´o and M. Opper. Sparse representation for gaussian process models. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 444–450. MIT Press, 2001. [4] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of boosting. Technical report, Stanford University, Dept. of Statistics, 1998. [5] C. Gentile. A new approximate maximal margin classification algorithm. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 500–506. MIT Press, 2001. [6] T. Graepel, R. Herbrich, and R. C. Williamson. From margin to sparsity. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 210–216. MIT Press, 2001. [7] Y. Guo, P. Bartlett, A. Smola, and R. C. Williamson. Norm-based regularization of boosting. Submitted to Journal of Machine Learning Research, 2001. [8] M. Herbster. Learning additive models online with fast evaluating kernels. In Proc. 14th Annual Conference on Computational Learning Theory (COLT), pages 444–460. Springer, 2001. [9] P. J. Huber. Robust statistics: a review. Annals of Statistics, 43:1041, 1972. [10] G. S. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions. J. Math. Anal. Applic., 33:82–95, 1971. [11] J. Kivinen, A.J. Smola, and R.C. Williamson. Large margin classification for moving targets. Unpublished manuscript, 2001. [12] Y. Li and P.M. Long. The relaxed online maximum margin algorithm. In S. A. Solla, T. K. Leen, and K.-R. M¨uller, editors, Advances in Neural Information Processing Systems 12, pages 498–504. MIT Press, 1999. [13] L. Mason, J. Baxter, P. L. Bartlett, and M. Frean. Functional gradient techniques for combining hypotheses. In A. J. Smola, P. L. Bartlett, B. Sch¨olkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, Cambridge, MA, 2000. MIT Press. 221–246. [14] B. Sch¨olkopf, J. Platt, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson. Estimating the support of a high-dimensional distribution. Neural Computation, 13(7), 2001. [15] B. Sch¨olkopf, A. Smola, R. C. Williamson, and P. L. Bartlett. New support vector algorithms. Neural Computation, 12(5):1207–1245, 2000. [16] V. Vapnik, S. Golowich, and A. Smola. Support vector method for function approximation, regression estimation, and signal processing. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 281– 287, Cambridge, MA, 1997. MIT Press.
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3 state neurons for contextual processing Adam Kepecs* and Sridhar Raghavachari Volen Center for Complex Systems Brandeis University Waltham MA 02454 {kepecs,sraghava}@brandeis.edu Abstract Neurons receive excitatory inputs via both fast AMPA and slow NMDA type receptors. We find that neurons receiving input via NMDA receptors can have two stable membrane states which are input dependent. Action potentials can only be initiated from the higher voltage state. Similar observations have been made in several brain areas which might be explained by our model. The interactions between the two kinds of inputs lead us to suggest that some neurons may operate in 3 states: disabled, enabled and firing. Such enabled, but non-firing modes can be used to introduce context-dependent processing in neural networks. We provide a simple example and discuss possible implications for neuronal processing and response variability. 1 Introduction Excitatory interactions between neurons are mediated by two classes of synapses: AMPA and NMDA. AMPA synapses act on a fast time scale (TAMPA'" 5ms), and their role in shaping network dynamics has been extensively studied. The NMDA type receptors are slow ((TNMDA '" 150ms) and have been mostly investigated for their critical role in the induction of long term potentiation, which is thought to be the mechanism for storing long term memories. Crucial to this is the unique voltage dependence of NMDA receptors [6] that requires both the presynaptic neuron to be active and the post-synaptic neuron to be depolarized for the channel to open. However, pharamacological studies which block the NMDA receptors impair a variety of brain processes, suggesting that NMDA receptors also playa role in shaping the dynamic activity of neural networks [10, 3, 8, 11, 2]. Therefore, we wanted to examine the role of NMDA receptors in post-synaptic integration. Harsch and Robinson [4] have observed that injection of NMDA conductance that simulates synchronous synaptic input regularized firing while lowering response reliability. Our initial observations using a minimal model with 'The authors contributed equally to this work. large NMDA inputs in a leaky dendrite showed a large regenerative depolarization. Neurons however, also possess a variety of potassium currents that are able to limit these large excursions in voltage. In particular, recent observations show that A-type potassium currents are abundant in dendrites of a variety of neurons [7] . Combining these potassium currents with random NMDA inputs showed that the membrane voltage alternated between two distinct subthreshold states. Similar observations of two-state fluctuations have been made in vivo in several cortical areas and the striatum [17, 9, 1]. The origin and possible functional relevance of these fluctuations have remained a puzzle. We suggest that the NMDA type inputs combined with potassium currents are sufficient to produce such membrane dynamics. Our results lead us to suggest that the fluctuations could be used to represent contextual modulation of neuronal firing. 2 NMDA-type input causes 2 state membrane fluctuations 2.1 Model To examine the role of NMDA type inputs, we built a simple model of a cortical neuron receiving AMPA and NMDA type inputs. To capture the spatial extent of neuronal morphology we use a two-compartment model of pyramidal neurons [15]. We represent the soma, proximal dendrites and the axon lumped into one compartment containing the channels necessary for spike generation (INa and IK). The dendritic compartment includes two potassium currents, a fast activating IKA and the slower IKS along with a persistent sodium current INaP. The dendrite also receives synaptic input as INMDA and IAMPA . The membrane voltage of the neuron obeys the current balance equations: while the dendritic voltage, "\lid obeys: where em is the specific membrane capacitance which is taken to be 1 I1F / cm2 for both the dendrite and the soma for all cells and p =0.2, gc =0.05 determining the electrotonic structure of the neuron. The passive leak current in both the soma and dendrites were modeled as h eak = gleak(V Eleak ), where gleak was the leak conductance which was taken to be 0.3 mS/cm2 for the soma and dendrite. Eleak = -80mV was the leak reversal potential for both the compartments. The voltage-dependent currents were modeled according to the Hodgkin-Huxley formalism, with the gating variables obeying the equation: dx (xoo(V) - x) dt = ¢x(ax(V)(1 - x) - ,sx(V)x) = ¢x Tx(V) , (3) where x represents the activation/inactivation gates for the voltage-dependent currents. The sodium current, INa = gNam~ h(VS - ENa ), where gNa = 45 mS/cm2 and sodium reversal potential, ENa = 55 mV with moo(V) = a=(~)~~~(V). The activation variables, O::m(V) = -O.l(V + 32)/[exp( -(V + 32)/10) - 1], 'sm(V) = 4exp( -[V + 57]/18); O::h(V) = 0.07 exp( -[V + 48]/20) and 'sh (V) = l/[exp( -{V + 18}/10) + 1], with ¢m = ¢h = 2.5. The delayed rectifier potassium current, IKDr = gKn4(VS - EK), where gK = 9 mS/cm2 and potassium reversal potential, EK = -80 mV with O::n(V) = -O.Ol(V + 34)/[exp( -(V + 34)/10) - 1], 'sn(V) = 0.125 exp( -[V + 44]/80), with ¢n = 2.5. In the dendrite, the persistent sodium current, INaP = gNapr~(V)(V - VNa ), with roo(V) = 1/(1 + exp( -(V + 57)/5)) and gNaP =0.25 mS/cm2 • The two potassium currents were hs = gKsq(V - VK), with qoo (V) = 1/(1 + exp( -(V + 50)/2)) and Tq(V) = 200/(exp( -(V + 60)/10) + exp((V + 60)/10)) and gKS = 0.1 mS/cm2 ; and hA = gKAa~ (V)b(V - VK), with aoo(V) = 1/(1 + exp(-(V + 45)/6)), boo (V) = 1/(1 + exp(-(V + 56)/15)) and Tb(V) = 2.5(1 + exp((V + 60)/30)) and gKA = 10 mS/cm2 . The NMDA current, INMDA = fgNMDAS(V - ENMDA)/(l + 0.3[Mg] exp( -0.08V)), where S was the activation variable and f denoted the inactivation of NMDA channels due to calcium entry. AMPA and NMDA inputs were modeled as conductance kicks that decayed with TAMPA = 5 ms and TNMDA = 150 ms. Calcium dependent inactivation of the NMDA conductance was modeled as a negative feedback df /dt = (foo - f)/2 , where f oo was a shallow sigmoid function that was 1 below a conductance threshold of 2 ms/cm2 and was inversely proportional to the NMDA conductance above threshold. The coupling conductance is gc =0.1 mS/cm 2. The asymmetry between the areas of the two compartments is taken into account in the parameter p = somatic area/total area = 0.2. The temperature scaling factors are ¢h = ¢n = 3.33. Other parameter values are: gLeak =0.3, gNa =36, gK =6, gNaP =0.15, gKS =1, gKA =50 in mS/cm2 unless otherwise noted; ELeak = -75, ENa = +55, EK = -90, EKA = -80 in m V. Synchronous inputs were modeled as a compound Poisson process representing 100 inputs firing at a rate A each spiking with a probability of 0.1. Numerical integration was performed with a fourth-order Runge-Kutta method using a 0.01 ms time step. 2.2 NMDA induced two-state fluctuations Figure 1A shows the firing produced by inputs with high AMPA/NMDA ratio. Figure 1B shows that the same spike train input delivered via synapses with a high NMDA content results in robust two-state membrane behavior. We term the lower and higher voltage states as UP and DOWN states respectively. Spikes caused by AMPA-type inputs only occur during the up-state. In general, the same AMPA input can only elicit spikes in the postsynaptic neuron when the NMDA input switches that neuron into the up-state. Transitions from down to up-state occur when synchronous NMDA inputs depolarize the membrane enough to cause the opening of additional NMDA receptor channels (due to the voltage-dependence of their opening). This results in a regeneretive depolarization event, which is limited by the fast opening of IKA-type Time [s] Figure 1: Inputs with high AMPA-NMDA ratio cause the cell to spike (top trace, gAM PA =0.05, g N MDA =0.01). Strong NMDA inputs combined with potassium currents (for the same AMPA input) result in fluctuations of the membrane potential between two subthreshold states, with occasional firing due to the AMPA inputs (bottom trace, gAMPA =0.01, gNMDA =0.1) potassium channels. This up-state is stable because the regenerative nature and long lifetime of NMDA receptor opening keeps the membrane depolarized, while the slower I Ks potassium current prevents further depolarization. When input ceases, NMDA channels eventually (TNMDA ~ 150ms) close and the membrane jumps to the down-state. Note that while this bistable mechanism is intrinsic to the membrane, it is also conditional upon input. Since the voltage threshold for spike generation in the somal axon compartment is above the up-state, it acts as a barrier. Thus, synchronous AMPA input in the down-state has a low probability of eliciting a spike. A number of previous experimental studies have reported similar phenomena in various brain regions [16, 9, 1] where the two states persist even with all intrinsic inward currents blocked but the inputs left intact [17] . Pharmacological block of the potassium currents resulted in prolonged up-states [17]. These experimental results suggested a conceptual model in which two-state fluctuations are (i) input driven, (ii) the membrane states are stabilized by potassium currents. Nevertheless, there remained a puzzle that (iii) up-state transitions are abrupt and (iv) the the up-state is prolonged and restricted to a relatively narrow range of voltages. Our model suggests a plausible mechanism for this phenomenon consistent with all experimental constraints. Below, we examine the origins of the two-state fluctuations in light of these findings. 2.3 Analysis of two state fluctuations Figure 2A shows the histogram of membrane potential for a neuron driven by combined AMPA and NMDA input at 30 Hz. There are two clear modes corresponding to the up and down-states. The variability of the up-state and down-state voltages is very low (u = 1.4 mV and 2.4 mV respectively) as observed. Figure 2B shows the distribution of the up-state duration. The distribution of the up-state durations depend on the maximal NMDA conductance and the decay time constant of NMDA (not shown), as well as the mean rate of NMDA inputs (Figure 2C). A B C O. 40 400 >30 U) ~O. 300 :c E CQ '"" .c Q) £0. 200 E i= 100 500 1000 20 30 40 50 Time (ms) NMDA Rate (Hz) Figure 2: A. Histogram of the up and down states. B. Dwell times of the up states C. Mean duration of the up states increases with rate of NMDA inputs. Each histogram was calculated over a run of 120 seconds. Additionally, larger maximal potassium conductances shorten the duration of the up states. Thus, we predict that the NMDA receptors are intimately involved in shaping the firing characteristics of these neurons. Furthermore, our mechanistic explanation leads a strong prediction about the functional role for these fluctuations in neuronal processing. 3 Contextual processing with NMDA and AMPA pathways Since NMDA and AMPA pathways have distinct roles in respectively switching and firing our model neuron, we suggest the following conceptual model shown on Fig 3A. Without any input the neuron is at the rest or disabled state. Contextual input (via NMDA receptors) can bring the neuron into an enabled state. Informational (for instance, cue or positional) input (via AMPA receptors) can fire a neuron only from this enabled state. Where might such an architecture be used? In the CAl region of the hippocampus, pyramidal cells receive two distinct, spatially segregated input pathways: the perforant path from cortex and the Schaffer collaterals from the CA3 region. The perforant path has a very large NMDA receptor content [14] which is, interestingly, co-localized with high densities of I KA conductances [5]. Experimental [13] and theoretical [12] observations suggest that these two pathways carry distinct information. Lisman has suggested that the perforant path carries contextual information and the Schaffer collaterals bring sequence information [12]. Thus our model seems to apply biophysically as well as suggest a possible way for CAl neurons to carry out contextual computations. It is known that these cell can fire at specific places in specific contexts. How might these different signals interact? As shown on Fig3B, our model neuron can only fire spikes due to positional input when the right context enables it. We note that a requirement for contextual processing is that the two inputs be anatomically segregated, as they are in the CAl region. However, we stress that the phenomenon of 2-state fluctuations itself is independent of the location of the two kinds of inputs. Figure 4A shows a similar processing scheme adapted for higher-order language A B'5 Context off Context on Firing state .~ ~A/ a .... ;.~, ... , <;; ~ ~ 0 Q. "" 0 ~ ~ ~~'-) '5 .~ <;; Jll Contextual input ~ ~ 0 • Q. g> Down-state / Disabled ~ ~ Figure 3: A. Contextual input (high NMDA) switches the neuron from a rest state to an up state. Informational input (high AMPA) cause the neuron to spike only from the up state. B. Weak informational input can cause the cell to fire in conjunction with the contextual input, (left traces) while strong informational input will not fire the cell in the absence of contextual input (right traces). In this simulation, the soma/proximal dendrite compartment receives AMPA input, while the NMDA input targets the dendritic compartment. processing. We simulated 3 neurons each receiving the same AMPA, informational input. This might represent the word "green". Each of these neurons also receives distinct contextual input via NMDA type receptors. These might, for instance, represent specific noun groups: objects, people and fruit. The word "green" may have very different meanings in these different contexts such as the color green, a person who is a novice or an unripe fruit. We simulated this simple scenario shown in Figure 4C. Even though each neuron receives the same strong AMPA input, their firing seems uncorrelated. To evaluate the performance of the network in processing contextual conjunctions, we measured the correlations between the information and each contextual input. The most correlated at each moment was designated to be the correct meaning. We then measured the number of spikes emitted by each neuron during each "meaning" . Figure 4B shows that the neurons performed well, each tuned to fire preferentially in its appropriate context. This simple example illustrates the use of a plausible biophysical mechanism for computing conjuctions or multiplying with neurons. 4 Discussion Voltage fluctuations between two subthresold levels with similar properties are observed in vivo in a variety of brain regions. Our model is in accordance with these data and lead us to a new picture of how might these neuron operate in a functional manner. Figure 3A shows our model operating as a 3-state device. It has a stable low membrane state from which it cannot fire spikes, which we called disabled. It also has a stable depolarized state from which action potentials can be elicited, which this we call enabled state. Additionally, it has a firing state which is only reachable from the enabled state. What might be the role of the two non-firing states? We suggest that if high and low NMDA-content pathways carry separate information these neurons can compute A Contextual input: B C 90 0 frun "objects" "people" "fruit" objects people 111 0)(2)(3) Sensory input: "green" o 2 Time(s) 4 !6 o J ~5 • · o 4 o ~ • £ 3 0 0 0 0 0 0 0 I 0 1 2 3 1 2 3 1 2 3 Figure 4: A. Illustrative task for contextual processing in semantic inference. 3 neurons each receive independent contextual (NMDA) and common informational (AMPA) input. B. Voltage traces showing differences in firing patterns depending upon context. C. Each neuron is tuned to its defined context. Correlation was measured between the informational spike train and each contextual spike train smoothed with a gaussian filter (a = 60ms). The most correlated context was defined to be the right one and the spikes of all neurons were counted. conjuctions, a simple form of multiplication. If the high NMDA-content pathway carries contextual information then it would be in position to enable or disable a neuron. In the enabled state, AMPA-type informational input could then fire a neuron (Fig 3B). We have presented a biophysical model for two-state fluctuations that is strongly supported by data. One concern might be that most observations of 2-state fluctuations in vivo have been when the animal is anesthetized, implying that this kind of neuronal dynamics is an artifact of the anesthetized state. However, these fluctuations have been observed in several different kinds of anesthesia, including local anesthesia [16]. Furthermore, it has been shown that the duration of the up-states correlate with orientation selectivity in visual cortical neurons suggesting that these fluctuations might playa role in information processing. These observations suggest that this phenomenon may be more indicative of a natural state of the cortex rather than a by-product of anesthesia. When the inputs with different AMPA/NMDA content are anatomically segregated, the NMDA input alone generates voltage fluctuations between a resting and depolarized state, while the AMPA input causes the neuron to spike when in the up-state. This mechanism naturally leads to the suggestion that such two-state fluctuations could have a function in computing context/input conjuctions. In summary, we suggest the known biophysical mechanisms of some neurons can enable them two operate as 3-state devices. In this mode of operation, the neurons could be used for contextual processing. Acknowledgments We acknowledge John Lisman and John Fitzpatrick for useful discussion and suggestions. References [1] J. Anderson, 1. Lampl, 1. Reichova, M. Carandini, and D. Ferster. Stimulus dependence of two-state fluctuations of membrane potential in cat visual cortex. Nat Neurosci, 3:617- 21, 2000. [2] A. Compte, N. BruneI, P. Goldman-Rakic, and X.J. Wang. Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model. Cereb. Cortex, 10(9):910- 923, 2000. [3] S. Grillner, O. Ekeberg, A. Manira, A. Lansner, D. Parker, J. Tegner, and P. Wallen. Intrinsic function of a neuronal network - a vertebrate central pattern generator. Brain Res. Brain Res. Rev., 26:184- 197, 1998. [4] A. Harsch and H.P.C. Robinson. Postsynaptic variability of firing rates in rat cortical neurons: the role of input synchronization and synaptic nmda receptor conductance. J. Neurosci., 20:6181- 6192, 2000. [5] A Hoffman, JC Magee, CM Colbert, and D Johnston. K+ channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons. Nature, 387:869- 875, 1997. [6] C.E. Jahr and C.F. Stevens. Voltage dependence of nmda-activated macroscopic conductances predicted by single-channel kinetics. J Neurosci, 10:3178-82, 1990. [7] D. Johnston, D.A. Hoffman, J .C. Magee, N.P. Poolos, S. Watanabe, C.M. Colbert, and M. Migliore. Dendritic potassium channels in hippocampal pyramidal neurons. J Physiol, 15:75- 81, 2000. [8] O. Kiehn and T. Eken. Functional role of plateau potentials in vertebrate motor neurons. Curro Opin. Neurobiol., 8:746- 752, 1998. [9] B.L. Lewis and P. O'Donnell. Ventral tegmental area afferents to the prefrontal cortex maintain membrane potential 'up' states in pyramidal neurons via dl dopamine receptors. Cereb. Cortex, 10:1168- 1175, 2000. [10] Y.X. Li, R. Bertram, and J . Rinzel. Modeling N-methyl-D-aspartate induced bursting in dopamine neurons. Neuroscience, 71(2):397- 410, 1996. [11] J. Lisman, J.-M. Fellous, and X.J. Wang. A role for NMDA-receptor channels in working memory. Nat. Neurosci. , 1(4):273- 275, 1998. [12] J.E. Lisman. Relating hippocampal circuitry to function: recall of memory sequences by reciprocal dentate-CA3 interactions. Neuron, 22:233- 242, 1999. [13] B.L. McNaughton, C.A. Barnes, J. Meltzer, and R.J. Sutherland. Hippocampal granule cells are necessry for normal spatial learning but not for spatially-selective pyramidal cell discharge. Exp. Brain Res., 76:485- 496, 1989. [14] N.A. Otmakhova and Lisman J. Dopamine selectively inhibits the direct cortical pathway to the CAl hippocampal region. J Neurosci, 19:1437- 45, 1999. [15] P.F. Pinsky and J. Rinzel. Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons. J. Comput. Neurosci. , 1:39- 60, 1994. [16] C.J. Wilson and P.M. Groves. Spontaneous firing patterns of identified spiny neurons in the rat neostriatum. Brain Res, 220:67- 80, 1981. [17] C.J. Wilson and Y. Kawaguchi. The origins of two-state spontaneuous fluctuations of neostriatal spiny neurons. J. Neurosci., 16:2397- 2410, 1996.
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Convolution Kernels for Natural Language Michael Collins AT&T Labs–Research 180 Park Avenue, New Jersey, NJ 07932 mcollins@research.att.com Nigel Duffy Department of Computer Science University of California at Santa Cruz nigeduff@cse.ucsc.edu Abstract We describe the application of kernel methods to Natural Language Processing (NLP) problems. In many NLP tasks the objects being modeled are strings, trees, graphs or other discrete structures which require some mechanism to convert them into feature vectors. We describe kernels for various natural language structures, allowing rich, high dimensional representations of these structures. We show how a kernel over trees can be applied to parsing using the voted perceptron algorithm, and we give experimental results on the ATIS corpus of parse trees. 1 Introduction Kernel methods have been widely used to extend the applicability of many well-known algorithms, such as the Perceptron [1], Support Vector Machines [6], or Principal Component Analysis [15]. A key property of these algorithms is that the only operation they require is the evaluation of dot products between pairs of examples. One may therefore replace the dot product with a Mercer kernel, implicitly mapping feature vectors in into a new space , and applying the original algorithm in this new feature space. Kernels provide an efficient way to carry out these calculations when is large or even infinite. This paper describes the application of kernel methods to Natural Language Processing (NLP) problems. In many NLP tasks the input domain cannot be neatly formulated as a subset of . Instead, the objects being modeled are strings, trees or other discrete structures which require some mechanism to convert them into feature vectors. We describe kernels for various NLP structures, and show that they allow computationally feasible representations in very high dimensional feature spaces, for example a parse tree representation that tracks all subtrees. We show how a tree kernel can be applied to parsing using the perceptron algorithm, giving experimental results on the ATIS corpus of parses. The kernels we describe are instances of “Convolution Kernels”, which were introduced by Haussler [10] and Watkins [16], and which involve a recursive calculation over the “parts” of a discrete structure. Although we concentrate on NLP tasks in this paper, the kernels should also be useful in computational biology, which shares similar problems and structures. 1.1 Natural Language Tasks Figure 1 shows some typical structures from NLP tasks. Each structure involves an “observed” string (a sentence), and some hidden structure (an underlying state sequence or tree). We assume that there is some training set of structures, and that the task is to learn a) Lou Gerstner is chairman of IBM [S [NP Lou Gerstner ] [VP is [NP chairman [PP of [NP IBM ] ] ] ] ] b) Lou Gerstner is chairman of IBM Lou/SP Gerstner/CP is/N chairman/N of/N IBM/SC c) Lou/N Gerstner/N is/V chairman/N of/P IBM/N Figure 1: Three NLP tasks where a function is learned from a string to some hidden structure. In (a), the hidden structure is a parse tree. In (b), the hidden structure is an underlying sequence of states representing named entity boundaries (SP = Start person, CP = Continue person, SC = Start company, N= No entity). In (c), the hidden states represent part-of-speech tags (N = noun, V = verb, P = preposition,). the mapping from an input string to its hidden structure. We refer to tasks that involve trees as parsing problems, and tasks that involve hidden state sequences as tagging problems. In many of these problems ambiguity is the key issue: although only one analysis is plausible, there may be very many possible analyses. A common way to deal with ambiguity is to use a stochastic grammar, for example a Probabilistic Context Free Grammar (PCFG) for parsing, or a Hidden Markov Model (HMM) for tagging. Probabilities are attached to rules in the grammar – context-free rules in the case of PCFGs, state transition probabilities and state emission probabilities for HMMs. Rule probabilities are typically estimated using maximum likelihood estimation, which gives simple relative frequency estimates. Competing analyses for the same sentence are ranked using these probabilities. See [3] for an introduction to these methods. This paper proposes an alternative to generative models such as PCFGs and HMMs. Instead of identifying parameters with rules of the grammar, we show how kernels can be used to form representations that are sensitive to larger sub-structures of trees or state sequences. The parameter estimation methods we describe are discriminative, optimizing a criterion that is directly related to error rate. While we use the parsing problem as a running example in this paper, kernels over NLP structures could be used in many ways: for example, in PCA over discrete structures, or in classification and regression problems. Structured objects such as parse trees are so prevalent in NLP that convolution kernels should have many applications. 2 A Tree Kernel The previous section introduced PCFGs as a parsing method. This approach essentially counts the relative number of occurences of a given rule in the training data and uses these counts to represent its learned knowledge. PCFGs make some fairly strong independence assumptions, disregarding substantial amounts of structural information. In particular, it does not appear reasonable to assume that the rules applied at level in the parse tree are unrelated to those applied at level . As an alternative we attempt to capture considerably more structural information by considering all tree fragments that occur in a parse tree. This allows us to capture higher order dependencies between grammar rules. See figure 2 for an example. As in a PCFG the new representation tracks the counts of single rules, but it is also sensitive to larger sub-trees. Conceptually we begin by enumerating all tree fragments that occur in the training data . Note that this is done only implicitly. Each tree is represented by an dimensional vector where the ’th component counts the number of occurences of the ’th tree fragment. Let us define the function
to be the number of occurences of the ’th tree fragment in tree , so that is now represented as ! " #$ . a) S NP N Jeff VP V ate NP D the N apple b) NP D the N apple NP D N D the N apple NP D the N NP D N apple Figure 2: a) An example tree. b) The sub-trees of the NP covering the apple. The tree in (a) contains all of these sub-trees, and many others. We define a sub-tree to be any subgraph which includes more than one node, with the restriction that entire (not partial) rule productions must be included. For example, the fragment [NP [D the ]] is excluded because it contains only part of the production NP D N. Note that will be huge (a given tree will have a number of subtrees that is exponential in its size). Because of this we would like design algorithms whose computational complexity does not depend on . Representations of this kind have been studied extensively by Bod [2]. However, the work in [2] involves training and decoding algorithms that depend computationally on the number of subtrees involved. The parameter estimation techniques described in [2] do not correspond to maximum-likelihood estimation or a discriminative criterion: see [11] for discussion. The methods we propose show that the score for a parse can be calculated in polynomial time in spite of an exponentially large number of subtrees, and that efficient parameter estimation techniques exist which optimize discriminative criteria that have been well-studied theoretically. Goodman [9] gives an ingenious conversion of the model in [2] to an equivalent PCFG whose number of rules is linear in the size of the training data, thus solving many of the computational issues. An exact implementation of Bod’s parsing method is still infeasible, but Goodman gives an approximation that can be implemented efficiently. However, the method still suffers from the lack of justification of the parameter estimation techniques. The key to our efficient use of this high dimensional representation is the definition of an appropriate kernel. We begin by examining the inner product between two trees and under this representation, # # . To compute we first define the set of nodes in trees and as and respectively. We define the indicator function
to be if sub-tree is seen rooted at node and 0 otherwise. It follows that
#
and
. The first step to efficient computation of the inner product is the following property (which can be proved with some simple algebra): "!$#&% # '(% # ) *! + -,/.0 ! + 1,/. !2#43 # 6573 # 658 9 :! + 9,/.0 ! + ,;.2$< 65-=>5 - where we define ? @
A
. Next, we note that ? can be computed in polynomial time, due to the following recursive definition: B If the productions at and are different ? DC . B If the productions at and are the same, and and are pre-terminals, then ? . In training, a parameter is explicitly estimated for each sub-tree. In searching for the best parse, calculating the score for a parse in principle requires summing over an exponential number of derivations underlying a tree, and in practice is approximated using Monte-Carlo techniques. Pre-terminals are nodes directly above words in the surface string, for example the N, V, and D B Else if the productions at and are the same and and are not pre-terminals, ? $"? !
$ where is the number of children of in the tree; because the productions at / are the same, we have . The ’th child-node of is $ . To see that this recursive definition is correct, note that ? simply counts the number of common subtrees that are found rooted at both and . The first two cases are trivially correct. The last, recursive, definition follows because a common subtree for and can be formed by taking the production at / , together with a choice at each child of simply taking the non-terminal at that child, or any one of the common sub-trees at that child. Thus there are "? $$ possible choices at the ’th child. (Note that a similar recursion is described by Goodman [9], Goodman’s application being the conversion of Bod’s model [2] to an equivalent PCFG.) It is clear from the identity # ? , and the recursive definition of ? , that # can be calculated in
time: the matrix of ? values can be filled in, then summed. This can be a pessimistic estimate of the runtime. A more useful characterization is that it runs in time linear in the number of members such that the productions at and are the same. In our data we have found a typically linear number of nodes with identical productions, so that most values of ? are 0, and the running time is close to linear in the size of the trees. This recursive kernel structure, where a kernel between two objects is defined in terms of kernels between its parts is quite a general idea. Haussler [10] goes into some detail describing which construction operations are valid in this context, i.e. which operations maintain the essential Mercer conditions. This paper and previous work by Lodhi et al. [12] examining the application of convolution kernels to strings provide some evidence that convolution kernels may provide an extremely useful tool for applying modern machine learning techniques to highly structured objects. The key idea here is that one may take a structured object and split it up into parts. If one can construct kernels over the parts then one can combine these into a kernel over the whole object. Clearly, this idea can be extended recursively so that one only needs to construct kernels over the “atomic” parts of a structured object. The recursive combination of the kernels over parts of an object retains information regarding the structure of that object. Several issues remain with the kernel we describe over trees and convolution kernels in general. First, the value of $ will depend greatly on the size of the trees . One may normalize the kernel by using # $ which also satisfies the essential Mercer conditions. Second, the value of the kernel when applied to two copies of the same tree can be extremely large (in our experiments on the order of C! ) while the value of the kernel between two different trees is typically much smaller (in our experiments the typical pairwise comparison is of order 100). By analogy with a Gaussian kernel we say that the kernel is very peaked. If one constructs a model which is a linear combination of trees, as one would with an SVM [6] or the perceptron, the output will be dominated by the most similar tree and so the model will behave like a nearest neighbor rule. There are several possible solutions to this problem. Following Haussler [10] we may radialize the kernel, however, it is not always clear that the result is still a valid kernel. Radializing did not appear to help in our experiments. These problems motivate two simple modifications to the tree kernel. Since there will be many more tree fragments of larger size – say depth four versus depth three – and symbols in Figure 2. consequently less training data, it makes sense to downweight the contribution of larger tree fragments to the kernel. The first method for doing this is to simply restrict the depth of the tree fragments we consider. The second method is to scale the relative importance of tree fragments with their size. This can be achieved by introducing a parameter C , and modifying the base case and recursive case of the definitions of ? to be respectively ? and ? "? !
$$ This corresponds to a modified kernel #
# $
, where
is the number of rules in the ’th fragment. This kernel downweights the contribution of tree fragments exponentially with their size. It is straightforward to design similar kernels for tagging problems (see figure 1) and for another common structure found in NLP, dependency structures. See [5] for details. In the tagging kernel, the implicit feature representation tracks all features consisting of a subsequence of state labels, each with or without an underlying word. For example, the paired sequence Lou/SP Gerstner/CP is/N chairman/N of/N IBM/SC would include features such as SP CP , SP Gerstner/CP N , SP CP is/N N of/N and so on. 3 Linear Models for Parsing and Tagging This section formalizes the use of kernels for parsing and tagging problems. The method is derived by the transformation from ranking problems to a margin-based classification problem in [8]. It is also related to the Markov Random Field methods for parsing suggested in [13], and the boosting methods for parsing in [4]. We consider the following set-up: B Training data is a set of example input/output pairs. In parsing we would have training examples
where each
is a sentence and each
is the correct tree for that sentence. B We assume some way of enumerating a set of candidates for a particular sentence. We use
to denote the ’th candidate for the ’th sentence in training data, and
to denote the set of candidates for
. B Without loss of generality we take
to be the correct parse for
(i.e.,
). B Each candidate
is represented by a feature vector
in the space . The parameters of the model are also a vector ! . We then define the “ranking score” of each example as !
. This score is interpreted as an indication of the plausibility of the candidate. The output of the model on a training or test example is "$#&%(')"$* + $, ! . When considering approaches to training the parameter vector ! , note that a ranking function that correctly ranked the correct parse above all competing candidates would satisfy the conditions ! $ -
/.
104C3242 576 . It is simple to modify the Perceptron and Support Vector Machine algorithms to treat this problem. For example, the SVM optimization problem (hard margin version) is to find the !98 which minimizes ! subject to the constraints ! -
:.
;5 <2=2 5>6 . Rather than explicitly calculating ! , the perceptron algorithm and Support Vector Machines can be formulated as a search ? This can be achieved using a modified dynamic programming table where < 65 =>5 =@
stores the number of common subtrees at nodes 5)-=(58 of depth @ or less. The recursive definition of < can be modified appropriately. A A context-free grammar – perhaps taken straight from the training examples – is one way of enumerating candidates. Another choice is to use a hand-crafted grammar (such as the LFG grammar in [13]) or to take the 5 most probable parses from an existing probabilistic parser (as in [4]). Define:
-
- :.
- Initialization: Set dual parameters
DC For 6
If -
10 -
do nothing, Else
Figure 3: The perceptron algorithm for ranking problems. Depth 1 2 3 4 5 6 Score
Improvement !
Table 1: Score shows how the parse score varies with the maximum depth of sub-tree considered by the perceptron. Improvement is the relative reduction in error in comparison to the PCFG, which scored 74%. The numbers reported are the mean and standard deviation over the 10 development sets. for “dual parameters”
which determine the optimal weights ! 8 ! 8 !
-
:.
$ (1) (we use
as shorthand for
). It follows that the score of a parse can be calculated using the dual parameters, and inner products between feature vectors, without having to explicitly deal with feature or parameter vectors in the space : ! 8 !
.
For example, see figure 3 for the perceptron algorithm applied to this problem. 4 Experimental Results To demonstrate the utility of convolution kernels for natural language we applied our tree kernel to the problem of parsing the Penn treebank ATIS corpus [14]. We split the treebank randomly into a training set of size 800, a development set of size 200 and a test set of size 336. This was done 10 different ways to obtain statistically significant results. A PCFG was trained on the training set, and a beam search was used to give a set of parses, with PCFG probabilities, for each of the sentences. We applied a variant of the voted perceptron algorithm [7], which is a more robust version of the original perceptron algorithm with performance similar to that of SVMs. The voted perceptron can be kernelized in the same way that SVMs can but it can be considerably more computationally efficient. We generated a ranking problem by having the PCFG generate its top 100 candidate parse trees for each sentence. The voted perceptron was applied, using the tree kernel described previously, to this re-ranking problem. It was trained on 20 trees selected randomly from the top 100 for each sentence and had to choose the best candidate from the top 100 on the test set. We tested the sensitivity to two parameter settings: first, the maximum depth of sub-tree examined, and second, the scaling factor used to down-weight deeper trees. For each value of the parameters we trained on the training set and tested on the development set. We report the results averaged over the development sets in Tables 1 and 2. We report a parse score which combines precision and recall. Define
to be the number of correctly placed constituents in the ’th test tree, "
to be the number of constituents Scale 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Score
Imp. ! Table 2: Score shows how the parse score varies with the scaling factor for deeper sub-trees is varied. Imp. is the relative reduction in error in comparison to the PCFG, which scored 74%. The numbers reported are the mean and standard deviation over the 10 development sets. proposed, and
to be the number of constistuents in the true parse tree. A constituent is defined by a non-terminal label and its span. The score is then C C
!
6
"
The precision and recall on the ’th parse are
/ "
and
/
respectively. The score is then the average precision recall, weighted by the size of the trees
. We also give relative improvements over the PCFG scores. If the PCFG score is and the perceptron score is , the relative improvement is C C .
$ C C . , i.e., the relative reduction in error. We finally used the development set for cross-validation to choose the best parameter settings for each split. We used the best parameter settings (on the development sets) for each split to train on both the training and development sets, then tested on the test set. This gave a relative goodness score of
C with the best choice of maximum depth and a score of
C with the best choice of scaling factor. The PCFG scored on the test data. All of these results were obtained by running the perceptron through the training data only once. As has been noted previously by Freund and Schapire [7], the voted perceptron often obtains better results when run multiple times through the training data. Running through the data twice with a maximum depth of 3 yielded a relative goodness score of , while using a larger number of iterations did not improve the results significantly. In summary we observe that in these simple experiments the voted perceptron and an appropriate convolution kernel can obtain promising results. However there are other methods which perform considerably better than a PCFG for NLP parsing – see [3] for an overview – future work will investigate whether the kernels in this paper give performance gains over these methods. 5 A Compressed Representation When used with algorithms such as the perceptron, convolution kernels may be even more computationally attractive than the traditional radial basis or polynomial kernels. The linear combination of parse trees constructed by the perceptron algorithm can be viewed as a weighted forest. One may then search for subtrees in this weighted forest that occur more than once. Given a linear combination of two trees " which contain a common subtree, we may construct a smaller weighted acyclic graph, in which the common subtree occurs only once and has weight . This process may be repeated until an arbitrary linear combination of trees is collapsed into a weighted acyclic graph in which no subtree occurs more than once. The perceptron may now be evaluated on a new tree by a straightforward generalization of the tree kernel to weighted acyclic graphs of the form produced by this procedure. Given the nature of our data – the parse trees have a high branching factor, the words are chosen from a dictionary that is relatively small in comparison to the size of the training data, and are drawn from a very skewed distribution, and the ancestors of leaves are part of speech tags – there are a relatively small number of subtrees in the lower levels of the parse trees that occur frequently and make up the majority of the data. It appears that the approach we have described above should save a considerable amount of computation. This is something we intend to explore further in future work. 6 Conclusions In this paper we described how convolution kernels can be used to apply standard kernel based algorithms to problems in natural language. Tree structures are ubiquitous in natural language problems and we illustrated the approach by constructing a convolution kernel over tree structures. The problem of parsing English sentences provides an appealing example domain and our experiments demonstrate the effectiveness of kernel-based approaches to these problems. Convolution kernels combined with such techniques as kernel PCA and spectral clustering may provide a computationally attractive approach to many other problems in natural language processing. Unfortunately, we are unable to expand on the many potential applications in this short note, however, many of these issues are spelled out in a longer Technical Report [5]. References [1] Aizerman, M., Braverman, E., and Rozonoer, L. (1964). Theoretical Foundations of the Potential Function Method in Pattern Recognition Learning. Automation and Remote Control, 25:821–837. [2] Bod, R. (1998). Beyond Grammar: An Experience-Based Theory of Language. CSLI Publications/Cambridge University Press. [3] Charniak, E. (1997). Statistical techniques for natural language parsing. In AI Magazine, Vol. 18, No. 4. [4] Collins, M. (2000). Discriminative Reranking for Natural Language Parsing. Proceedings of the Seventeenth International Conference on Machine Learning. San Francisco: Morgan Kaufmann. [5] Collins, M. and Duffy, N. (2001). Parsing with a Single Neuron: Convolution Kernels for Natural Language Problems. Technical report UCSC-CRL-01-01, University of California at Santa Cruz. [6] Cortes, C. and Vapnik, V. (1995). Support–Vector Networks. Machine Learning, 20(3):273–297. [7] Freund, Y. and Schapire, R. (1999). Large Margin Classification using the Perceptron Algorithm. In Machine Learning, 37(3):277–296. [8] Freund, Y., Iyer, R.,Schapire, R.E., & Singer, Y. (1998). An efficient boosting algorithm for combining preferences. In Machine Learning: Proceedings of the Fifteenth International Conference. San Francisco: Morgan Kaufmann. [9] Goodman, J. (1996). Efficient algorithms for parsing the DOP model. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP 96), pages 143-152. [10] Haussler, D. (1999). Convolution Kernels on Discrete Structures. Technical report, University of Santa Cruz. [11] Johnson, M. The DOP estimation method is biased and inconsistent. To appear in Computational Linguistics. [12] Lodhi, H., Christianini, N., Shawe-Taylor, J., and Watkins, C. (2001). Text Classification using String Kernels. To appear in Advances in Neural Information Processing Systems 13, MIT Press. [13] Johnson, M., Geman, S., Canon, S., Chi, S., & Riezler, S. (1999). Estimators for stochastic ‘unification-based” grammars. In Proceedings of the 37th Annual Meeting of the Association for Computational Linguistics. San Francisco: Morgan Kaufmann. [14] Marcus, M., Santorini, B., & Marcinkiewicz, M. (1993). Building a large annotated corpus of english: The Penn treebank. Computational Linguistics, 19, 313-330. [15] Scholkopf, B., Smola, A.,and Muller, K.-R. (1999). Kernel principal component analysis. In B. Scholkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods – SV Learning, pages 327-352. MIT Press, Cambridge, MA. [16] Watkins, C. (2000). Dynamic alignment kernels. In A.J. Smola, P.L. Bartlett, B. Schlkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, pages 39-50, MIT Press.
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On the Convergence of Leveraging Gunnar R¨atsch, Sebastian Mika and Manfred K. Warmuth RSISE, Australian National University, Canberra, ACT 0200 Australia Fraunhofer FIRST, Kekul´estr. 7, 12489 Berlin, Germany University of California at Santa Cruz, CA 95060, USA raetsch@csl.anu.edu.au, mika@first.fhg.de, manfred@cse.ucsc.edu Abstract We give an unified convergence analysis of ensemble learning methods including e.g. AdaBoost, Logistic Regression and the Least-SquareBoost algorithm for regression. These methods have in common that they iteratively call a base learning algorithm which returns hypotheses that are then linearly combined. We show that these methods are related to the Gauss-Southwell method known from numerical optimization and state non-asymptotical convergence results for all these methods. Our analysis includes -norm regularized cost functions leading to a clean and general way to regularize ensemble learning. 1 Introduction We show convergence rates of ensemble learning methods such as AdaBoost [10], Logistic Regression (LR) [11, 5] and the Least-Square (LS) regression algorithm called LS-Boost [12]. These algorithms have in common that they iteratively call a base learning algorithm (also called weak learner) on a weighted training sample. The base learner is expected to return in each iteration a hypothesis from some hypothesis set of weak hypotheses that has small weighted training error. This is the weighted number of false predictions in classification and weighted estimation error in regression. These hypotheses are then linearly combined to form the final hypothesis ; in classification one uses the sign of for prediction. The hypothesis coefficient is determined at iteration , such that a certain objective is minimized or approximately minimized, and is fixed for later iterations. Here we will work out sufficient conditions on the base learning algorithm to achieve linear convergence to the minimum of an associated loss function . This means that for any starting condition the minimum can be reached with precision in only iterations. Relation to Previous Work In the original work on AdaBoost it has been shown that the optimization objective (which is an upper bound on the training error) converges exponentially fast to zero, if the base learner is consistently better than random guessing, i.e. its weighted training error is always smaller than some constant with . In this case the convergence is known to be linear (i.e. exponentially decreasing) [10]. One can easily show that this is the case when the data is separable:1 If the data is not separable, the Supported by DFG grants MU 987/1-1, JA 379/9-1 and NSF grant CCR 9821087; we gratefully acknowledge help from B. Borchers, P. Spellucci, R. Israel and S. Lemm. This work has been done, while G. R¨atsch was at Fraunhofer FIRST, Berlin. 1We call the data separable, if there exists such that separates the training examples. weighted training error cannot be upper bounded by a constant smaller , otherwise one could use AdaBoost to find a separation using the aforementioned convergence result. 2 For AdaBoost and Logistic Regression it has been shown [5] that they generate a combined hypothesis asymptotically minimizing a loss functional only depending on the output of the combined hypothesis . This holds for the non-separable case; however, the assumed conditions in [5] on the performance of the base learner are rather strict and can usually not be satisfied in practice. Although the analysis in [5] holds in principle for any strictly convex cost function of Legendre-type (e.g. [24], p. 258, and [1]), one needs to show the existence of a so-called auxiliary function [7, 5] for each cost function other than the exponential or the logistic loss. This can indeed be done [cf. 19, Section 4.2], but in any case only leads to asymptotical results. In the present work we can also show rates of convergence. In an earlier attempt to show the convergence of such methods for arbitrary loss functions [17], one needed to assume that the hypothesis coefficients are upper bounded by a rather small constant. For this case it has been shown that the algorithm asymptotically converges to a combined hypothesis minimizing . However, since the ’s need to be small, the algorithm requires many iterations to achieve this goal. In [9] it has been shown that for loss functions which are (essentially) exponentially decreasing (including the loss functions of AdaBoost and Logistic regression), the loss is in the first iterations and afterwards . This implies linear convergence. However, this only holds, if the loss reaches zero, i.e. if the data is separable. In our work we do not need to assume separability. An equivalent optimization problem for AdaBoost has also been considered in a paper that predates the formulation of AdaBoost [4]. This optimization problem concerns the likelihood maximization for some exponential family of distributions. In this work convergence is proven for the general non-separable case, however, only for the exponential loss, i.e. for the case of AdaBoost.3 The framework set up in this paper is more general and we are able to treat any strictly convex loss function. In this paper we propose a family of algorithms that are able to generate a combined hypothesis converging to the minimum of (if it exists), which is a functional depending on the outputs of the function evaluated on the training set. Special cases are AdaBoost, Logistic Regression and LS-Boost. While assuming mild conditions on the base learning algorithm and the loss function , we can show linear convergence rates [15] (beginning in the first iteration) of the type for some fixed . This means that the difference to the minimum loss converges exponentially fast to zero (in the number of iterations). A similar convergence has been proven for AdaBoost in the special case of separable data [10], although the constant shown in [10] can be considerable smaller [see also 9]. To prove the convergence of leveraging, we exploit results of Luo & Tseng [16] for a variant of the Gauss-Southwell method known from numerical optimization. Since in practice the hypothesis set can be quite large, ensemble learning algorithms without any regularization often suffer from overfitting [22, 12, 2, 19]. Here, the complexity can only be controlled by the size of the base hypothesis set or by early stopping after a few iterations. However, it has been shown that shrinkage regularization implied by penalizing some norm of the hypothesis coefficients is the favorable strategy [6, 12]. We therefore extend our analysis to the case of -norm regularized loss functions. With a slight modification this leads to a family of converging algorithms that e.g. includes the Leveraged Vector Machine [25] and a variant of LASSO [26]. In the following section we briefly review AdaBoost, Logistic Regression, and LS-Boost and cast them in a common framework. In Sec. 3 we present our main results. After re2This can also be seen when analyzing a certain linear program in the dual domain (cf. [23]) 3We will expand on this connection in the full paper (see also [14, 19]). lating these results to leveraging algorithms, we present an extension to regularized cost functions in Sec. 4 and finally conclude. 2 Leveraging algorithms revisited We first briefly review some of the most well known leveraging algorithms for classification and regression. For more details see e.g. [10, 11, 12, 8]. We work with Alg. 1 as a template for a generic leveraging algorithm, since these algorithms have the same algorithmical structure. Finally, we will generalize the problem and extend the notation. AdaBoost & Logistic Regression are designed for classification tasks. In each iteration they call a base learning algorithm on the training set (cf. step 3a in Alg. 1). Here a weighting on the sample is used that is recomputed in each iteration . The base learner is expected to return a hypothesis from some hypothesis space4 that has small weighted classification error5 [10, 11], where and . It is more convenient to work with the edge of , which is defined as . After selecting the hypothesis, its weight is computed such that it minimizes a certain functional (cf. step 3b). For AdaBoost this is (1) and for Logistic Regression it is (2) where is the combined hypothesis of the previous iteration given by . For AdaBoost it has been shown that minimizing (1) can be computed analytically [3]. This is true because we assumed that the hypotheses are binary valued. Similarly, for LR there exists an analytic solution of (2). The weighting on the sample is updated based on the new combined hypothesis : and for AdaBoost and Logistic Regression, respectively. Least-Square-Boost is an algorithm to solve regression tasks. In this case , and . It works in a similar way as AdaBoost and LR. It first selects a hypothesis solving (3) and then finds the hypothesis weight by minimizing the squared error of the new combined hypothesis: (4) The “weighting” of the sample is computed as , which is the residual of [12]. In a second version of LS-Boost, the base hypothesis and its weight are found simultaneously by solving [12]: (5) Since in (5) one reaches a lower loss function value than with (3) and (4), it might be the favorable strategy. 4Notice that always contains only a finite number of different hypotheses when evaluated on the training set and is effectively finite [2]. 5Different from common convention, we include the in to make the presentation simpler. Algorithm 1 – A Leveraging algorithm for the loss function . 1. Input: , No. of Iterations , Loss function 2. Initialize: , for all 3. Do for , (a) Train classifier on and obtain hypothesis (b) Set (c) Update and 4. Output: The General Case These algorithms can be summarized in Alg. 1 (where case (5) is slightly degenerated, cf. Sec. 3.2) for some appropriately defined functions and : plugin and choosing as for AdaBoost (cf. (1)), for Logistic Regression (cf. (2)) and for LS-Boost (cf. (4)). It can easily be verified that the function , used for computing the weights , is the derivative of with respect to the second argument [3, 12]. The Optimization Problem It has been argued in [3, 18, 11, 17] and finally shown in [5] that AdaBoost and Logistic Regression under certain condition asymptotically converge to a combined hypothesis minimizing the respective loss on the training sample, where is a linear combination of hypotheses from , i.e. . Thus, they solve the optimization problem: (6) where we defined a matrix with . To avoid confusions, note that hypotheses and coefficients generated during the iterative algorithm are marked by a hat. In the algorithms discussed so far, the optimization takes place by employing the leveraging scheme outlined in Alg. 1. The output of such an algorithm is a sequence of pairs and a combined hypothesis . With , , it is easy to verify that , which is in (note the missing hat). Other Preliminaries Throughout the paper we assume the loss function is of the form Although, this assumption is not necessary, the presentation becomes easier. In [7, 5, 19] a more general case of Legendre-type cost functions is considered. However, note that additive loss functions are commonly used, if one considers i.i.d.-drawn examples. We assume that each element and is finite ( , ) and does not contain a zero column. Furthermore, the function is assumed to be strictly convex for all . For simplicity we assume for the rest of the paper that is finite and complementation closed, i.e. for every there exists also . The assumption on the finiteness is not crucial for classification (cf. footnote 4). For regression problems the hypothesis space might be infinite. This case has explicitly been analyzed in [20, 19] and goes beyond the scope of this paper (see also [27]). 3 Main Result We now state a result known from the field of numerical optimization. Then we show how the reviewed leveraging algorithms fit into this optimization framework. 3.1 Coordinate Descent The idea of coordinate descent is to iteratively select a coordinate, say the -th, and find such that some functional is minimized with respect to . There exist several different strategies for selecting the coordinates [e.g. 15]; however, we are in particular interested in the Gauss-Southwell-type (GS) selection scheme: It selects the coordinate that has the largest absolute value in the gradient vector , i.e. . Instead of doing steps in the direction of the negative gradient as in standard gradient descent methods, one only changes the variable that has the largest gradient component. This can be efficient, if there are many variables and most of them are zero at the minimum. We start with the following general convergence result, which seemed to be fallen into oblivion even in the optimization community. It will be very useful in the analysis of leveraging algorithms. Due to a lack of space we omit proofs (see [21, 19]). Theorem 1 (Convergence of Coordinate Descent [16]). Suppose is twice continuously differentiable and strictly convex on . Assume that is open, the set of solutions to (7) is not empty, where is a fixed matrix having no zero column, fixed and is a (possibly unbounded) box-constrained set. Furthermore assume that the Hessian is a positive matrix for all . Let be the sequence generated by coordinate descent, where the coordinate selection satisfies (8) for some , where is the optimal value of if it would be selected, i.e. (9) Then converges to an element in . The coordinate selection in Thm. 1 is slightly different from the Gauss-Southwell selection rule described before. We therefore need the following: Proposition 2 (Convergence of GS on ). Assume the conditions on and as in Thm. 1. Let be a convex subset of such that . Assume and (10) holds for some fixed . Then a coordinate selection satisfies (8) of Thm. 1, if there exists a fixed such that (11) Thus the approximate Gauss-Southwell method on as described above converges. To show the convergence of the second variant of LS-Boost (cf. (5)) we need the following Proposition 3 (Convergence of the maximal improvement scheme on ). Let and as in Proposition 2 and assume (10) holds. Then a coordinate selection satisfies (8), if there exists a fixed with (12) Thus the maximal improvement scheme on as above converges in the sense of Thm.1. Finally we can also state a rate of convergence, which is surprisingly not worse than the rates for standard gradient descent methods: Theorem 4 (Rate of Convergence of Coordinate Descent, [16]). Assume the conditions of Thm. 1 hold. Let as in Prop. 2 and assume (10) holds for some . Then we have (13) where is the estimate after the -th coordinate descent step, denotes a optimal solution, and . Especially at iteration : . Following [16] one can show that the constant is , where is the Lipschitz constant of and is a constant that depends on and therefore on the geometry of the hypothesis set (cf. [16, 13] for details). While the upper bound on can be rather large, making the convergence slow, it is important to note (i) that this is only a rough estimate of the true constant and (ii) still guarantees an exponential decrease in the error functional with the number of iterations. 3.2 Leveraging and Coordinate Descent We now return from the abstract convergence results in Sec. 3.1 to our examples of leveraging algorithms, i.e. we show how to retrieve the Gauss-Southwell algorithm on as a part of Alg. 1. For now we set . The gradient of with respect to is given by (14) where is given as in step 3c of Alg. 1. Thus, the coordinate with maximal absolute gradient corresponds to the hypothesis with largest absolute edge (see definition). However, according to Proposition 2 and 3 we need to assume less on the base learner. It either has to return a hypothesis that (approximately) maximizes the edge, or alternatively (approximately) minimizes the loss function. Definition 5 ( -Optimality). A base learning algorithm is called -optimal, if it always returns hypotheses that either satisfy condition (11) or (12) for some fixed . Since we have assumed is closed under complementation, there always exist two hypotheses having the same absolute gradient ( and ). We therefore only need to consider the hypothesis with maximum edge as opposed to the maximum absolute edge. For classification it means: if the base learner returns the hypothesis with approximately smallest weighted training error, this condition is satisfied. It is left to show that we can apply the Thm. 1 for the loss functions reviewed in Sec. 2: Lemma 6. The loss functions of AdaBoost, Logistic regression and LS-Boost are bounded, strongly convex and fulfill the conditions in Thm. 1 on any bounded subset of . We can finally state the convergence result for leveraging algorithms: Theorem 7. Let be a loss function satisfying the conditions in Thm. 1. Suppose Alg.1 generates a sequence of hypotheses and weights using a -optimal base learner. Assume with is bounded. Then any limit point of is a solution of (6) and converges linearly in the sense of Thm. 4. Note that this result in particular applies to AdaBoost, Logistic regression and the second version of LS-Boost. For the selection scheme of LS-Boost given by (3) and (4), both conditions in Definition 5 cannot be satisfied in general, unless is constant for all hypotheses. Since , the base learner prefers hypotheses with small and could therefore stop improving the objective while being not optimal (see [20, Section 4.3] and [19, Section 5] for more details). 4 Regularized Leveraging approaches We have not yet exploited all features of Thm. 1. It additionally allows for box constraints and a linear function in terms of the hypothesis coefficients. Here, we are in particular interested in -norm penalized loss functions of the type , which are frequently used in machine learning. The LASSO algorithm for regression [26] and the PBVM algorithm for classification [25] are examples. Since we assumed complementation closeness of , we can assume without loss of generality that a solution satisfies . We can therefore implement the -norm regularization using the linear term , where and is the regularization constant. Clearly, the regularization defines a structure of nested subsets of , where the hypothesis set is restricted to a smaller set for larger values of . The constraint causes some minor complications with the assumptions on the base learning algorithm. However, these can easily be resolved (cf. [21]), while not assuming more on the base learning algorithm. The first step in solving the problem is to add the additional constraint to the minimization with respect to in step 3b of Alg. 1. Roughly speaking, this induces the problem that hypothesis coefficient chosen too large in a previous iteration, cannot be reduced again. To solve this problem one can check for each coefficient of a previously selected hypothesis whether not selecting it would violate the -optimality condition (11) or (12). If so, the Algorithm 2 – A Leveraging algorithm for -norm regularized loss . 1. Input: Sample , No. of Iterations , Loss function , Reg. const. 2. Initialize: , for all 3. Do for , (a) Train classifier on and obtain hypothesis (b) Let and for (c) , where and . (d) if then and else (e) Set (f) Update and 4. Output: algorithm selects such a coordinate for the next iteration instead of calling the base learning algorithm. This idea leads to Alg. 2 (see [21] for a detailed discussion). For this algorithm we can show the following: Theorem 8 (Convergence of -norm penalized Leveraging). Assume are as Thm. 1, is strictly convex, , and the base learner satisfies (15) for . Then Alg. 2 converges linearly to a minimum of the regularized loss function. This can also be shown for a maximum-improvement like condition on the base learner, which we have to omit due to space limitation. In [27] a similar algorithm has been suggested that solves a similar optimization problem (keeping fixed). For this algorithm one can show order one convergence (which is weaker than linear convergence), which also holds if the hypothesis set is infinite. 5 Conclusion We gave a unifying convergence analysis for a fairly general family of leveraging methods. These convergence results were obtained under rather mild assumptions on the base learner and, additionally, led to linear convergence rates. This was achieved by relating leveraging algorithms to the Gauss-Southwell method known from numerical optimization. While the main theorem used here was already proven in [16], its applications closes a central gap between existing algorithms and their theoretical understanding in terms of convergence. Future investigations include the generalization to infinite hypotheses spaces and an improvement of the convergence rate . Furthermore, we conjecture that our results can be extended to many other variants of boosting type algorithms proposed recently in the literature (cf. http://www.boosting.org). References [1] H.H. Bauschke and J.M. Borwein. Legendre functions and the method of random bregman projections. Journal of Convex Analysis, 4:27–67, 1997. [2] K.P. Bennett, A. Demiriz, and J. Shawe-Taylor. A column generation algorithm for boosting. In P. Langley, editor, Proceedings, 17th ICML, pages 65–72. Morgan Kaufmann, 2000. [3] L. Breiman. Prediction games and arcing algorithms. Neural Comp., 11(7):1493–1518, 1999. [4] N. Cesa-Bianchi, A. Krogh, and M. Warmuth. Bounds on approximate steepest descent for likelihood maximization in exponential families. IEEE Trans. Inf. Th., 40(4):1215–1220, 1994. [5] M. Collins, R.E. Schapire, and Y. Singer. Logistic Regression, Adaboost and Bregman distances. In Proc. COLT, pages 158–169, San Francisco, 2000. Morgan Kaufmann. [6] J. Copas. Regression, prediction and shrinkage. J.R. Statist. Soc. B, 45:311–354, 1983. [7] S. Della Pietra, V. Della Pietra, and J. Lafferty. Duality and auxiliary functions for bregman distances. TR CMU-CS-01-109, Carnegie Mellon University, 2001. [8] N. Duffy and D.P. Helmbold. A geometric approach to leveraging weak learners. In P. Fischer and H. U. Simon, editors, Proc. EuroCOLT ’99, pages 18–33, 1999. [9] N. Duffy and D.P. Helmbold. Potential boosters? In S.A. Solla, T.K. Leen, and K.-R. M¨uller, editors, NIPS, volume 12, pages 258–264. MIT Press, 2000. [10] Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 55(1):119–139, 1997. [11] J. Friedman, T. Hastie, and R.J. Tibshirani. Additive Logistic Regression: a statistical view of boosting. Annals of Statistics, 2:337–374, 2000. [12] J.H. Friedman. Greedy function approximation. Tech. rep., Stanford University, 1999. [13] A.J. Hoffmann. On approximate solutions of systems of linear inequalities. Journal of Research of the National Bureau of Standards, 49(4):263–265, October 1952. [14] J. Kivinen and M. Warmuth. Boosting as entropy projection. In Proc. 12th Annu. Conference on Comput. Learning Theory, pages 134–144. ACM Press, New York, NY, 1999. [15] D.G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley Publishing Co., Reading, second edition, May 1984. Reprinted with corrections in May, 1989. [16] Z.-Q. Luo and P. Tseng. On the convergence of coordinate descent method for convex differentiable minimization. Journal of Optimization Theory and Applications, 72(1):7–35, 1992. [17] L. Mason, J. Baxter, P.L. Bartlett, and M. Frean. Functional gradient techniques for combining hypotheses. In Adv. Large Margin Class., pages 221–247. MIT Press, 2000. [18] T. Onoda, G. R¨atsch, and K.-R. M¨uller. An asymptotic analysis of AdaBoost in the binary classification case. In L. Niklasson, M. Bod´en, and T. Ziemke, editors, Proc. of the Int. Conf. on Artificial Neural Networks (ICANN’98), pages 195–200, March 1998. [19] G. R¨atsch. Robust Boosting via Convex Optimization. PhD thesis, University of Potsdam, October 2001. http://mlg.anu.edu.au/˜raetsch/thesis.ps.gz. [20] G. R¨atsch, A. Demiriz, and K. Bennett. Sparse regression ensembles in infinite and finite hypothesis spaces. Machine Learning, 48(1-3):193–221, 2002. [21] G. R¨atsch, S. Mika, and M.K. Warmuth. On the convergence of leveraging. NeuroCOLT2 Technical Report 98, Royal Holloway College, London, 2001. [22] G. R¨atsch, T. Onoda, and K.-R. M¨uller. Soft margins for AdaBoost. Machine Learning, 42(3):287–320, March 2001. also NeuroCOLT Technical Report NC-TR-1998-021. [23] G. R¨atsch and M.K. Warmuth. Marginal boosting. NeuroCOLT2 Tech. Rep. 97, 2001. [24] R.T. Rockafellar. Convex Analysis. Princeton University Press, 1970. [25] Y. Singer. Leveraged vector machines. In S.A. Solla, T.K. Leen, and K.-R. M¨uller, editors, NIPS, volume 12, pages 610–616. MIT Press, 2000. [26] R.J. Tibshirani. Regression selection and shrinkage via the LASSO. Technical report, Department of Statistics, University of Toronto, June 1994. ftp://utstat.toronto.edu/pub/tibs/lasso.ps. [27] T. Zhang. A general greedy approximation algorithm with applications. In Advances in Neural Information Processing Systems, volume 14. MIT Press, 2002. in press.
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A Quantitative Model of Counterfactual Reasoning Daniel Yarlett Division of Informatics University of Edinburgh Edinburgh, Scotland dany@cogsci.ed.ac.uk Michael Ramscar Division of Informatics University of Edinburgh Edinburgh, Scotland michael@dai.ed.ac.uk Abstract In this paper we explore two quantitative approaches to the modelling of counterfactual reasoning – a linear and a noisy-OR model – based on information contained in conceptual dependency networks. Empirical data is acquired in a study and the fit of the models compared to it. We conclude by considering the appropriateness of non-parametric approaches to counterfactual reasoning, and examining the prospects for other parametric approaches in the future. 1 Introduction If robins didn’t have wings would they still be able to fly, eat worms or build nests? Previous work on counterfactual reasoning has tended to characterise the processes by which questions such as these are answered in purely qualitative terms, either focusing on the factors determining their onset and consequences (see Roese, 1997, for a review); the qualitative outline of their psychological characteristics (Kahneman and Miller, 1986; Byrne and Tasso, 1999); or else their logical or schematic properties (Lewis, 1973; Goodman, 1983). And although Pearl (2000) has described a formalism addressing quantitative aspects of counterfactual reasoning, this model has yet to be tested empirically. Furthermore, the non-parametric framework in which it is proposed means certain problems attach to it as a cognitive model, as discussed in 6. To date then, the quantitative processes underlying human counterfactual reasoning have proven surprisingly recalcitrant to philosophical, psychological and linguistic analysis. In this paper we propose two parametric models of counterfactual reasoning for a specific class of counterfactuals: those involving modifications to our conceptual knowledge. The models we present are intended to capture the constraints operative on this form of inference at the computational level. Having outlined the models, we present a study which compares their predictions with the judgements of participants about corresponding counterfactuals. Finally, we conclude by raising logistical and methodological doubts about a non-parametric approach to the problem, and considering future work to extend the current models. 2 Counterfactuals and Causal Dependencies One of the main difficulties in analysing counterfactuals is that they refer to alternative ways that things could be, but it’s difficult to specify exactly which alternatives they pick out. For example, to answer the counterfactual question we began this paper with we clearly need to examine the possible states of affairs in which robins don’t have wings in order to see whether they will still be able to fly, eat worms and build nests in them. But the problem is that we can imagine many possible ways in which robins can be without wings – for instance, at an extreme we can imagine a situation in which the robin genotype failed to evolve beyond the plankton stage – not all of which will be relevant when it comes to reasoning counterfactually. In the alternatives envisaged by a counterfactual some things are clearly going to differ from the way they are in the actual world, while others are going to remain unchanged. And specifying which things will be affected, and which things will be unaffected, by a counterfactual supposition is the crux of the issue. Counterfactual reasoning seems to revolve around causal dependencies: if something depends on a counterfactual supposition then it should differ from the way it is in the actual world, otherwise it should remain just as it is. The challenge is to specify exactly what depends on what in the world – and crucially to what degree, if we are interested in the quantitative aspects of counterfactual reasoning – in order that we can arrive at appropriate counterfactual inferences. Clearly some information about our representation of dependency relations is required. 3 Dependency Information Fortunately, data is available about people’s representations of dependencies, albeit in a limited domain. As part of an investigation into feature centrality, Sloman, Love and Ahn (1998) explored the idea that a feature is central to a concept to the degree that other features depend on it. To test this idea empirically they derived dependency networks for four concepts – robin, apple, chair and guitar – by asking people to rate on a scale of 0 to 3 how strongly they thought the features of the four concepts depended on one another. One of the dependency structures derived from this process is depicted in Figure 1. 4 Parametric Models The models we present here simulate counterfactual reasoning about a concept by operating on conceptual networks such as the one in Figure 1. A counterfactual supposition is entertained by setting the activation of the counterfactually manipulated feature to an appropriate level. Inference then proceeds via an iterative algorithm which propagates the effect of manipulating the selected feature throughout the network. In order to do this we make two main assumptions about cause-effect interactions. First we assume that a node representing an effect, , will be expected to change as a function of (i) the degree to which a node representing its cause, , has itself changed, and (ii) the degree to which depends on . Second, we also assume that multiple cause nodes, , will affect a target node, , independently of one another and in a cumulative fashion. This means that the proposed models do not attempt to deal with interactions between causes. The first assumption seems warranted by recent empirical work (Yarlett & Ramscar, in preparation). And while the second assumption is certainly not true in all instances – interaction effects are certainly possible – there do seem to be multiple schemas that can be adopted in causal reasoning (Kelley, 1967), and it may be that the parametric assumptions of the two models correspond to a form of reasoning that predominates. eats chirps wings two legs flies feathers beak small breast red worms eats lays eggs nests builds living moves Figure 1: Dependency network for the concept robin. An arrow drawn from feature A to feature B means that A depends on B. Note (i) that only the strongest dependency links are shown, but that all dependency information was used in the simulations; (ii) there is a numeric strength associated with every dependency connection, although this is not shown in the diagram; and (iii) the proposed models propagate information in the opposite direction to the dependency connections. 4.1 Causal Dependency Networks The dependency networks obtained by Sloman, Love and Ahn (1998) were collected by asking people to consider features in a pairwise fashion, independently of all other features. However, causal inference requires that the causal impact of multiple features on a target node be combined. Therefore some preprocessing needs to be done to the raw dependency networks to define a causal dependency network suitable for using in counterfactual inference. The original dependency networks can each be represented as a matrix , in which represents the strength with which feature depends on feature in concept , as judged by the original participants. The modified causal dependency networks, , are defined as follows:
(1) where where ; otherwise. (2) This transformation achieves two things. Firstly it normalises the weights to be in the range 0 to 1, instead of the range 0 to 3 that the original ratings occupied. Secondly it normalises the strength with which each input node is connected to a target node with respect to the sum of all other inputs to the target. This means that multiple inputs to a target node cannot activate the target any more than a single input. 4.2 Parametric Propagation Schemes We can now define how inference proceeds in the two parametric models: the linear and the noisy-OR models. Let denote the feature being counterfactually manipulated (‘has wings’ in our example), and let be a matrix in which each component ! #" $ represents the amount the model predicts feature to have changed as a result of the counterfactual modification to , after iterations. To initialise both models all predicted levels of change for features other than the manipulated feature, , are initialised to 0: ! " (3) 4.2.1 Linear Model The update rules for each iteration of the linear model are defined as follows. The manipulated feature is set to an initial activation level of 1, indicating it has been counterfactually modified1. All other features have their activations set as specified below: ! " $
! " $ (4) This condition states that a feature is expected to change in proportion to the degree to which the features that influence it have changed, given the initial alteration made to the manipulated feature , and the degree to which they affect it. The general robustness of linear models of human judgements (Dawes, 1979) provides grounds for expecting a good correlation between the linear model and human counterfactual judgements. 4.2.2 Noisy-OR Model The second model uses the noisy-OR gate (Pearl, 1988) to describe the propagation of information in causal inference. The noisy-OR gate assumes that each cause has an independent probability of failing to produce the effect, and that the effect will only be absent if all its associated causes fail to produce it. In the counterfactual model noisy-OR propagation is therefore formalised as follows: ! " $ ! " $ (5) The questions people were asked to validate the two models measured how strongly they would believe in different features of a concept, if a specific feature was subtracted. This can be interpreted as the degree to which their belief in the target feature would vary given the presence and the absence of the manipulated feature. Accordingly, the output of the noisy-OR model was the difference in activation of each node when the manipulated node was set to 1 and 0 respectively2. 4.2.3 Clamping Because of the existence of loops in the dependency networks, if the counterfactually manipulated node is not clamped to its initial value activation can feed back through the network and change this value. This is likely to be undesirable, because it will mean the network will converge to a state in which the required counterfactual manipulation has not been successfully maintained, and that therefore its consequences have not been properly assimilated. The empirical performance of the two models was therefore considered when 1Note that the performance of the linear model does not depend crucially on the activation of being set to 1, as solutions for at convergence are simply multiples of the initial value selected and hence will not affect the correlational results. 2This highlights an interesting difference in the output of the two models: the linear model outputs the degree to which a feature is expected to change as a result of a counterfactual manipulation directly, whereas the noisy-OR model outputs probabilities which need to be converted into an expected degree of change (expressed in Pearl’s causal calculus as !"$# %'&)(+*-, ./ 0 1 !"$# %'&23*-, ). the activation of the manipulated node was clamped to its initial value, and not clamped. The clamping constraint bears a close similarity to Pearl’s (2000) ‘ ’ operator, which prevents causes of a random variable affecting its value when an intervention has occurred in order to bring about. 4.2.4 Convergence Propagation continues in both models until the activations for the features converge: ! " $ ! " $
(6) The models thus offer a spreading activation account of the changes induced in a conceptual network as a result of a counterfactual manipulation, their iterative nature allowing the effect of non-local influences to be accommodated. 5 Testing the Models In order to test the validity of the two models we empirically studied people’s intuitions about how they would expect concepts to change if they no longer possessed characteristic features. For example, participants were asked to imagine that robins did not in fact have wings. They were then asked to rate how strongly they agreed or disagreed with statements such as ‘If robins didn’t have wings, they would still be able to fly’. The task clearly requires participants to engage in counterfactual reasoning: robins do in fact have wings – in normal contexts at least – so participants are required to modify their standard conceptual representation in order to find out how this affects their belief in the other aspects of robins. 5.1 Method Three features were chosen from each of the four concepts for which dependency information was available. These features were selected as having low, medium and high levels of centrality, as reported by Sloman, Love and Ahn (1998, Study 1). This was to ensure that counterfactuals revolving around more and less important features of a concept were considered in the study. Each selected feature formed the basis of a counterfactual manipulation. For example, if the concept was robin and the selected feature was ‘has wings’, then subjects were asked to imagine that robins didn’t have wings. Participants were then asked how strongly they believed that the concept in question would still possess each of its remaining features if it no longer possessed the selected feature. For example, they would read ‘If robins didn’t have wings, they would still be able to fly’ and be asked to rate how strongly they agreed with it. Ratings were elicited on a 1-7 point scale anchored by ‘strongly disagree’ at the lower end and ‘strongly agree’ at the upper end. The ratings provided by participants can be regarded as estimates of how much people expect the features of a concept to change if the concept were counterfactually modified in the specified way. If the models are good ones we would therefore expect there to be a correlation between their predictions and the judgements of the participants. 5.2 Design and Materials Participants were randomly presented with 4 of the 12 counterfactual manipulations, and were asked to rate their agreement with counterfactual statements about the remaining, Linear Model Noisy-OR Model Counterfactual Concept n Clamped Non-Clamped Clamped Non-Clamped robin-wings 13 -0.870** -0.044 -0.739** -0.062 robin-lays-eggs 13 -0.521* -0.105 -0.278 0.121 robin-eats-worms 13 -0.066 -0.069 -0.009 -0.017 chair-back 8 -0.451 0.191 -0.178 0.148 chair-arms 8 -0.530 0.042 -0.358 0.036 chair-holds-people 8 -0.815** -0.928** -0.917** -0.957** guitar-neck 8 -0.760* -0.242 -0.381 -0.181 guitar-makes-sound 8 -0.889** -0.920** -0.939** 0.895** guitar-used-by-music-groups 8 0.235 0.225 0.290 0.263 apple-grows-on-trees 8 -0.748* -0.838** -0.905** -0.921** apple-edible 8 -0.207 0.361 -0.288 0.000 apple-stem 8 -0.965** -0.948** -0.961** -0.893** Mean -0.549 -0.273 -0.472 -0.131 Table 1: The correlation between the linear and noisy-OR models, in the clamped and non-clamped conditions, with participants’ empirical judgements about corresponding inferences. All comparisons were one-tailed (* ; ** ). unmanipulated features of the concept. People read an introductory passage for each inference in which they were asked to ‘Imagine that robins didn’t have wings. If this was true, how much would you agree or disagree with the following statements...’ They were then asked to rate their agreement with the specific inferences. 5.3 Participants 38 members of the Division of Informatics, University of Edinburgh, took part in the study. All participants were volunteers, and no reward was offered for participation. 5.4 Results The correlation of the two models, in the clamped and non-clamped conditions, is shown in Table 1. A repeated-measures ANOVA revealed that there was a main effect of clamping (
, ), no main effect of propagation method ( , ), and no interaction effect ( ). The correlations of both the linear (Wilcoxon Test, Z , , one-tailed) and the noisy-OR model (Wilcoxon Test, Z , , one-tailed) differed significantly from 0 when clamping was used. 5.5 Discussion The simulation results show that clamping is necessary to the success of the counterfactual models; this thus constitutes an empirical validation of Pearl’s use of the ‘ ’ operator in modelling counterfactuals. In addition, both the models capture the empirical patterns with some degree of success, so further work is required to tease them apart. 6 Exploring Non-Parametric Approaches The models of counterfactual reasoning we have presented both make parametric assumptions. Although non-parametric models in general offer greater flexibility, there are two main reasons – one logistical and one methodological – why applying them in this context may be problematic. 6.1 A Logistical Reason: Conditional Probability Tables Bayesian Belief Networks (BBNs) define conditional dependence relations in terms of graph structures like the dependency structures used by the present model. This makes them an obvious choice of normative model for counterfactual inference. However, there are certain problems that make the application of a non-parametric BBN to counterfactual reasoning problematic. For non-parametric inference a joint conditional probability table needs to be defined for all the variables upon which a target node is conditioned. In other words, it’s not sufficient to know , , ..., alone; instead, is required. This leads to a combinatorial explosion in the number of parameters required. If is a vector of elements in which represents the number of discrete classes that the random variable can take, then the number of conditional probabilities required to compute the interaction between and in the general case is: (7) On the assumption that features can normally be represented by two classes (present or absent), the number of probability judgements required to successfully apply a nonparametric BBN to all four of Sloman, Love and Ahn’s (1998) concepts is 3888. Aside from the obvious logistical difficulties in obtaining estimates of this number of parameters from people, attribution theorists suggest that simplifying assumptions are often made in causal inference (Kelley, 1972). If this is the case then it should be possible to specify a parametric model which appropriately captures these patterns, as we have attempted to do with the models in this paper, thus obviating the need for a fully general non-parametric approach. 6.2 A Methodological Reason: Patterns of Interaction Parametric models are special cases of non-parametric models: this means that a nonparametric model will be able to capture patterns of interaction between causes that a parametric model may be unable to express. A risk concomitant with the generality of nonparametric models is that they can gloss over important limitations in human inference. Although a non-parametric approach, with exhaustively estimated conditional probability parameters, would likely fit people’s counterfactual judgements satisfactorily, it would not inform us about the limitations in our ability to process causal interactions. A parametric approach, however, allows one to adopt an incremental approach to modelling in which such limitations can be made explicit: parametric models can be generalised when there is empirical evidence that they fail to capture a particular kind of interaction. Parametric approaches go hand-in-hand, then, with an empirical investigation of our treatment of causal interactions. Obtaining a good fit with data is not of sole importance in cognitive modelling: it is also important for the model to make explicit the assumptions it is predicated on, and parametric approaches allow this to be done, hopefully making causal principles explicit which would otherwise lie latent in an exhaustive conditional probability table. 7 Closing Thoughts Given the lack of quantitative models of counterfactual reasoning, we believe the models we have presented in this paper constitute a significant contribution to our understanding of this process. Notably, the models achieved a significant correlation across a sizeable data-set (111 data-points), with no free parameters. However, there are limitations to the current models. As stated, the models both assume that causal factors contribute independently to a target factor, and this is clearly not always the case. Although a non-parametric Bayesian model with an exhaustive conditional probability table could accommodate all possible interaction effects between causal factors, as argued in the previous section, this would not necessarily be all that enlightening. It is up to further empirical work to unearth the principles underpinning our processing of causal interactions (e.g., Kelley, 1972); these principles can then be made explicit in future parametric models to yield a fuller understanding of human inference. In the future we intend to examine our treatment of causal interactions empirically, in order to reach a better understanding of the appropriate way to model counterfactual reasoning. Acknowledgements We would like to thank Tom Griffiths, Brad Love, Steven Sloman and Josh Tenenbaum for their discussion of the ideas presented in this paper. References [1] Byrne R.M.J. and Tasso A. (1999). Counterfactual Reasoning with Factual, Possible, and Counterfactual Conditionals, Memory & Cognition, 27(4), 726-740. [2] Dawes R.M. (1979). The Robust Beauty of Improper Linear Models in Decision Making, American Psychologist, 34, 571-582. [3] Goodman N. (1983; 4th edition). Fact, Fiction, and Forecast, Harvard University Press, Cambridge, Massachusetts. [4] Griffiths T. (August 2001). Assessing Interventions in Linear Belief Networks. [5] Kahneman D. and Miller D.T. (1986). Norm Theory: Comparing Reality to its Alternatives, Psychological Review, 93(2), 136-153. [6] Kahneman D., Slovic P. and Tversky A. (1982; eds.). Judgment Under Uncertainty: Heuristics and Biases, Cambridge University Press, Cambridge, UK. [7] Kelley H.H. (1972). Causal Schemata and the Attribution Process. In Jones, Kanouse, Kelley, Nisbett, Valins and Weiners (eds.), Attribution: Perceiving the Causes of Behavior, Chapter 9, 151174, General Learning Press, Morristown, New Jersey. [8] Lewis D.K. (1973). Counterfactuals, Harvard University Press, Cambridge, Massachusetts. [9] Murphy K.P., Weiss Y. and Jordan M.I. (1999). Loopy Belief Propagation for Approximate Inference: An Empirical Study, Proceedings of Uncertainty in AI, 467-475. [10] Pearl J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann, San Mateo, California. [11] Pearl J. (2000). Causality: Models, Reasoning, and Inference, Cambridge University Press, Cambridge. [12] Roese N.J. (1997). Counterfactual Thinking, Psychological Bulletin, 121, 133-148. [13] Sloman S., Love B.C. and Ahn W.K. (1998). Feature Centrality and Conceptual Coherence, Cognitive Science, 22(2), 189-228. [14] Yarlett D.G. and Ramscar M.J.A. (2001). Structural Determinants of Counterfactual Reasoning, Proceedings of the 23rd Annual Conference of the Cognitive Science Society, 1154-1159. [15] Yarlett D.G. and Ramscar M.J.A. (in preparation). Uncertainty in Causal and Counterfactual Inference.
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A Neural Oscillator Model of Auditory Selective Attention Stuart N. Wrigley and Guy J. Brown Department of Computer Science, University of Sheffield, Regent Court, 211 Portobello Street, Sheffield S1 4DP, UK. s.wrigley@dcs.shef.ac.uk, g.brown@dcs.shef.ac.uk Abstract A model of auditory grouping is described in which auditory attention plays a key role. The model is based upon an oscillatory correlation framework, in which neural oscillators representing a single perceptual stream are synchronised, and are desynchronised from oscillators representing other streams. The model suggests a mechanism by which attention can be directed to the high or low tones in a repeating sequence of tones with alternating frequencies. In addition, it simulates the perceptual segregation of a mistuned harmonic from a complex tone. 1 Introduction In virtually all listening situations, we are exposed to a mixture of sound energy from multiple sources. Hence, the auditory system must separate an acoustic mixture in order to create a perceptual description of each sound source. It has been proposed that this process of auditory scene analysis (ASA) [2] takes place in two conceptual stages: segmentation in which the acoustic mixture is separated into its constituent ‘atomic’ units, followed by grouping in which units that are likely to have arisen from the same source are recombined. The perceptual ‘object’ produced by auditory grouping is called a stream. Each stream describes a single sound source. Few studies have investigated the role of attention in ASA; typically, ASA is seen as a precursor to attentional mechanisms, which simply select one stream as the attentional focus. Recently, however, it has been suggested that attention plays a much more prominent role in ASA. Carlyon et al. [4] investigated how attention influences auditory grouping with the use of a rapidly repeating sequence of high and low tones. It is known that high frequency separations and/or high presentation rates encourage the high tones and low tones to form separate streams, a phenomenon known as auditory streaming [2]. Carlyon et al. demonstrated that auditory streaming did not occur when listeners attended to an alternative stimulus presented simultaneously. However, when they were instructed to attend to the tone sequence, auditory streaming occurred as normal. From this, it was concluded that attention is required for stream formation and not only for stream selection. It has been proposed that attention can be divided into two different levels [9]: low-level exogenous attention which groups acoustic elements to form streams, and a higher-level endogenous mechanism which performs stream selection. Exogenous attention may overrule conscious (endogenous) selection (e.g. in response to a sudden loud bang). The work presented here incorporates these two types of attention into a model of auditory grouping (Figure 1). The model is based upon the oscillatory correlation theory [10], which suggests that neural oscillations encode auditory grouping. Oscillators corresponding to grouped auditory elements are synchronised, and are desynchronised from oscillators encoding other groups. This theory is supported by neurobiological findings that report synchronised oscillations in the auditory system [6]. Within the oscillatory correlation framework, attentional selection can be implemented by synchronising attentional activity with the stream of interest. 2 The model 2.1 Auditory periphery Cochlear filtering is modelled by a bank of 128 gammatone filters with centre frequencies equally spaced on the equivalent rectangular bandwidth (ERB) scale between 50 Hz and 2.5 kHz [3]. Auditory nerve firing rate is approximated by half-wave rectifying and square root compressing the output of each filter. Input to the model is sampled at a rate of 8 kHz. 2.2 Pitch and harmonicity analysis It is known that a difference in fundamental frequency (F0) can assist the perceptual segregation of complex sounds [2]. Accordingly, the second stage of the model extracts pitch information from the simulated auditory nerve responses. This is achieved by computing the autocorrelation of the activity in each channel to form a correlogram [3]. At time t, the autocorrelation of channel i with lag τ is given by: (1) Here, r is the auditory nerve activity. The autocorrelation for channel i is computed using a 25 ms rectangular window w (P = 200) with lag steps equal to the sampling period, up to a maximum lag of 20 ms. When the correlogram is summed across frequency, the resulting ‘summary correlogram’ exhibits a large peak at the lag corresponding to the fundamental period of the stimulus. An accurate estimate of the F0 is found by fitting a parabolic curve to the three samples centred on the summary peak. The correlogram may also be used to identify formant and harmonic regions due to their similar patterns of periodicity [11]. This is achieved by computing the correlations between adjacent channels of the correlogram as follows: (2) Here, is the autocorrelation function of (1) which has been normalised to have zero mean and unity variance; L is the maximum autocorrelation lag in samples (L = 160). 2.3 Neural oscillator network The network consists of 128 oscillators and is based upon the two-dimensional locally excitatory globally inhibitory oscillator network (LEGION) of Wang [10], [11]. Within Network Correlation Neural Oscillator Correlogram Cross Hair cell Cochlear Filtering Signal Attentional Stream Channel ALI Figure 1: Schematic diagram of the model (the attentional leaky integrator is labelled ALI). A i t τ , , ( ) r i t k – , ( )r i t k – τ – , ( )w k ( ) k 0 = P 1 – ∑ = C i( ) 1 L--Aˆ i t τ , , ( )Aˆ i 1 t τ , , + ( ) τ 0 = L 1 – ∑ = Aˆ i t τ , , ( ) LEGION, oscillators are synchronised by placing local excitatory links between them. Additionally, a global inhibitor receives excitation from each oscillator, and inhibits every oscillator in the network. This ensures that only one block of synchronised oscillators can be active at any one time. Hence, separate blocks of synchronised oscillators - which correspond to the notion of a segment in ASA - arise through the action of local excitation and global inhibition. The model described here differs from Wang’s approach [10] in three respects. Firstly, the network is one-dimensional rather than two-dimensional; we argue that this is more plausible. Secondly, excitatory links can be global as well as local; this allows harmonically-related segments to be grouped. Finally, we introduce an attentional leaky integrator (ALI), which selects one block of oscillators to become the attentional stream (i.e., the stream which is in the attentional foreground). The building block of the network is a single oscillator, which consists of a reciprocally connected excitatory unit and inhibitory unit whose activities are represented by x and y respectively: (3a) (3b) Here, ε, γ and β are parameters. Oscillations are stimulus dependent; they are only observed when Io > 0, which corresponds to a periodic solution to (3) in which the oscillator cycles between an ‘active’ phase and a ‘silent’ phase. The system may be regarded as a model for the behaviour of a single neuron, or as a mean field approximation to a group of connected neurons. The input Io to oscillator i is a combination of three factors: external input Ir , network activity and global inhibition as follows: (4) Here, Wik is the connection strength between oscillators i and k; xk is the activity of oscillator k. The parameter θx is a threshold above which an oscillator can affect others in the network and Wz is the weight of inhibition from the global inhibitor z. Similar to θx, θz acts a threshold above which the global inhibitor can affect an oscillator. S is a squashing function which compresses oscillator activity to be within a certain range: (5) Here, K determines the sharpness of the sigmoidal function. The activity of the global inhibitor is defined as (6) where H is the Heaviside function (H(n) = 1 for n ≥ 0, zero otherwise). 2.3.1 Segmentation A block of channels are deemed to constitute a segment if the cross-channel correlation (2) is greater than 0.3 for every channel in the block. Cross-correlations are weighted by the energy of each channel in order to increase the contrast between spectral peaks and spectral dips. These segments are encoded by a binary mask, which is unity when a channel contributes to a segment and zero otherwise. To improve the resolution and separation of adjacent segments, x· 3x x3 – 2 y – Io + + = y· ε γ 1 x β--tanh + y – = Io Ir WzS z θz , ( ) – WikS xk θx , ( ) k i ≠∑ + = S n θ , ( ) 1 1 e K n θ – ( ) – + ------------------------------= z· H S xk θx , ( ) 0.1 – k∑ z – = the cross-frequency spread of a segment is restricted to 3 channels. Oscillators within a segment are synchronized by excitatory connections. The external input (Ir) of an oscillator whose channel is a member of a segment is set to Ihigh otherwise it is set to Ilow. 2.3.2 Harmonicity grouping Excitatory connections are made between segments if they are consistent with the current F0 estimate. A segment is classed as consistent with the F0 if a majority of its corresponding correlogram channels exhibit a significant peak at the fundamental period (ratio of peak height to channel energy greater than 0.46). A single connection is made between the centres of harmonically related segments subject to old-plus-new constraints. The old-plus-new heuristic [2] refers to the auditory system’s preference to ‘interpret any part of a current group of acoustic components as a continuation of a sound that just occurred’. This is incorporated into the model by attaching ‘age trackers’ to each channel of the network. Excitatory links are placed between harmonically related segments only if the two segments are of similar age. The age trackers are leaky integrators: (7) Here, [n]+ = n if n ≥ 0 and [n]+ = 0 otherwise. Mk is the (binary) value of the segment mask at channel k; small values of c and d result in a slow rise (d) and slow decay (c) for the integrator. g is a gain factor. Consider two segments that start at the same time; the age trackers for their constituent channels receive the same input, so the values of Bk will be the same. However, if two segments start at different times, the age trackers for the earlier segment will have already increased to a non-zero value when the second segment starts. This ‘age difference’ will dissipate over time, as the values of both sets of leaky integrators approach unity. 2.3.3 Attentional leaky integrator (ALI) Each oscillator is connected to the attentional leaky integrator (ALI) by excitatory links; the strength of these connections is modulated by endogenous attention. Input to the ALI is given by: (8) θALI is a threshold above which network activity can influence the ALI. Tk is an attentional weighting which is related to the endogenous interest at frequency k: (9) Here, Ak is the endogenous interest at frequency k and L is the leaky integrator defined as: (10) Small values of f and a result in a slow rise (a) and slow decay (f) for the integrator. b is a gain factor. where xmax is the largest output activity of the network. The build-up of attentional interest is therefore stimulus dependent. The attentional interest itself is modelled as a Gaussian according to the gradient model of attention [7]: (11) Here, Ak is the normalised attentional interest at frequency channel k and maxAk is the maximum value that Ak can attain. p is the channel at which the peak of attentional interest occurs, and σ determines the width of the peak. B· k d g Mk Bk – [ ]+ 1 H Mk Bk – ( ) – [ ]cBk – ( ) = ali· H S xk θx , ( )Tk θALI – k∑ ali – = Tk 1 1 Ak – ( )L – = L· a b R L – [ ]+ 1 H R L – ( ) – [ ]fL – ( ) = R H xmax ( ) = Ak maxAke k p – 2σ2 ----------– = A segment or group of segments are said to be attended to if their oscillatory activity coincides temporally with a peak in the ALI activity. Initially, the connection weights between the oscillator array and the ALI are strong: all segments feed excitation to the ALI, so all segments are attended to. During sustained activity, these weights relax toward the Ak interest vector such that strong weights exist for channels of high attentional interest and low weights exist for channels of low attentional interest. ALI activity will only coincide with activity of the channels within the attentional interest peak and any harmonically related (synchronised) activity outside the Ak peak. All other activity will occur within a trough of ALI activity. This behaviour allows both individual tones and harmonic complexes to be attended to using only a single Ak peak. The parameters for all simulations reported here were ε = 0.4, γ = 6.0, β = 0.1, Wz = 0.5, θz = 0.1, θx = -0.5 and K = 50, d = 0.001, c = 5, g = 3, a = 0.0005, f = 5, b = 3, maxAk = 1, σ = 3, θALI = 1.5, Ilow = -5.0, Ihigh = 0.2.The inter- and intra- segment connections have equal weights of 1.1. 3 Evaluation Throughout this section, output from the model is represented by a ‘pseudospectrogram’ with time on the abscissa and frequency channel on the ordinate. Three types of information are superimposed on each plot. A gray pixel indicates the presence of a segment at a particular frequency channel, which is also equivalent to the external input to the corresponding oscillator: gray signifies Ihigh (causing the oscillator to be stimulated) and white signifies Ilow (causing the oscillator to be unstimulated). Black pixels represent active oscillators (i.e. oscillators whose x value exceeds a threshold value). At the top of each figure, ALI activity is shown. Any oscillators which are temporally synchronised with the ALI are considered to be in the attentional foreground. 3.1 Segregation of a component from a harmonic complex Darwin et al. [5] investigated the effect of a mistuned harmonic upon the pitch of a 12 component complex tone. As the degree of mistuning of the fourth harmonic increased towards 4%, the shift in the perceived pitch of the complex also increased. This effect was less pronounced for mistunings of more than 4%; beyond 8% mistuning, little pitch shift was observed. Apparently, the pitch of a complex tone is calculated using only those channels which belong to the corresponding stream. When the harmonic is subject to mistunings below 8%, it is grouped with the rest of the complex and so can affect the pitch percept. Mistunings of greater than 8% cause the harmonic to be segregated into a second stream, and so it is excluded from the pitch percept. 0 2 4 6 8 0 0.5 1 1.5 Pitch shift (Hz) Mistuning of 4th harmonic (%) Time (ms) Channel 0 20 40 60 80 20 40 60 80 100 120 Time (ms) Channel 0 20 40 60 80 20 40 60 80 100 120 Time (ms) Channel 0 20 40 60 80 20 40 60 80 100 120 Figure 2: A,B,C: Network response to mistuning of the fourth harmonic of a 12 harmonic complex (0%, 6% and 8% respectively). ALI activity is shown at the top of each plot. Gray areas denote the presence of a segment and black areas denote oscillators in the active phase. Arrows show the focus of attentional interest. D: Pitch shift versus degree of mistuning. A Gaussian derivative is fitted to each data set. A B C D Darwin Model This behaviour is reproduced by our model (Figure 2). All the oscillators at frequency channels corresponding to harmonics are temporally synchronised for mistunings up to 8% (plots A and B) signifying that the harmonics belong to the same perceptual group. Mistunings beyond 8% cause the mistuned harmonic to become desychronised from the rest of the complex (plot C) - two distinct perceptual groups are now present: one containing the fourth harmonic and the other containing the remainder of the complex tone. A comparison of the pitch shifts found by Darwin et al. and the shifts predicted by the model is shown in plot D. The pitch of the complex was calculated by creating a summary correlogram (similar to that used in section 2.2) using frequency channels contained within the complex tone group. Only segment channels below 1.1 kHz were used for this summary since low frequency (resolved) harmonics are known to dominate the pitch percept [8]. Darwin et al. also showed that the effect of mistuning was diminished when the fourth harmonic was ‘captured’ from the complex by four preceding tones at the same frequency. In this situation, no matter how small the mistuning, the harmonic is segregated from the complex and does not influence the pitch percept. Figure 3 shows the capture of the harmonic with no mistuning. Attentional interest is focused on the fourth harmonic: oscillator activity for the captor tone segments is synchronised with the ALI activity. During the 550 ms before the complex tone onset, the age tracker activities for the captor tone channels build up. When the complex tone begins, there is a significant age difference between the frequency channels stimulated by the fourth harmonic and those stimulated by the remainder of the complex. Such a difference prevents excitatory harmonicity connections from being made between the fourth harmonic and the remaining harmonics. This behaviour is consistent with the old-plus-new heuristic; a current acoustic event is interpreted as a continuation of a previous stimulus. The old-plus-new heuristic can be further demonstrated by starting the fourth harmonic before the rest of the complex. Figure 4 shows the output of the model when the fourth harmonic is subject to a 50 ms onset asynchrony. During this time, the age trackers of channels excited by the fourth harmonic increase to a significantly higher value than those of the remaining harmonics. Once again, this prevents excitatory connections being made between the fourth harmonic and the other harmonically related segments. The early harmonic is desynchronised from the rest of the complex: two streams are formed. However, after a period of time, the importance of the onset asynchrony decreases as the channel ages approach their maximal values. Once this occurs, there is no longer any evidence to prevent excitatory links from being made between the fourth harmonic and the rest of the complex. Grouping by harmonicity then occurs for all segments: the complex and the early harmonic synchronise to form a single stream. 3.2 Auditory streaming Within the framework presented here, auditory streaming is an emergent property; all events which occur over time, and are subject to attentional interest, are implicitly grouped. Two temporally separated events at different frequencies must both fall under the Ak peak to be Time (ms) Channel 0 100 200 300 400 500 600 20 40 60 80 100 120 Figure 3: Captor tones preceding the complex capture the fourth harmonic into a separate stream. ALI activity (top) shows that this harmonic is the focus of attention and would be ‘heard out’. The attentional interest vector (Ak) is shown to the right of the figure. grouped. It is the width of the Ak peak that determines frequency separation-dependent streaming, rather than local connections between oscillators as in [10]. The build-up of streaming [1] is modelled by the leaky integrator in (9). Figure 5 shows the effect of two different frequency separations on the ability of the network to perform auditory streaming and shows a good match to experimental findings [1], [4]. At low frequency separations, both the high and low frequency segments fall under the attentional interest peak; this allows the oscillator activities of both frequency bands to influence the ALI and hence they are considered to be in the attentional foreground. At higher frequency separations, one of the frequency bands falls outside of the attentional peak (in this example, the high frequency tones fall outside) and hence it cannot influence the ALI. Such behaviour is not seen immediately, because the attentional interest vector is subject to a build up effect as described in (9). Initially the attentional interest is maximal across all frequencies; as the leaky integrator value increases, the interest peak begins to dominate and interest in other frequencies tends toward zero. 4 Discussion A model of auditory attention has been presented which is based on previous neural oscillator work by Wang and colleagues [10], [11] but differs in two important respects. Firstly, our network is unidimensional; in contrast, Wang’s approach employs a two-dimensional timefrequency grid for which there is weak physiological justification. Secondly, our model regards attention as a key factor in the stream segregation process. In our model, attentional interest may be consciously directed toward a particular stream, causing that stream to be selected as the attentional foreground. Few auditory models have incorporated attentional effects in a plausible manner. For example, Wang’s ‘shifting synchronisation’ theory [3] suggests that attention is directed towards a stream when its constituent oscillators reach the active phase. This contradicts experimental findings, which suggest that attention selects a single stream whose salience is increased for a sustained period of time [2]. Additionally, Wang’s model fails to account for exogenous reorientation of attention to a sudden loud stimulus; the shifting synchronisation approach would multiplex it as normal with no attentional emphasis. By ensuring that the minimum Ak value for the attentional interest is always non-zero, it is possible to weight activity outside of the attentional interest peak and force it to influence the ALI. Such weighting could be derived from a measure of the sound intensity present in each frequency channel. We have demonstrated the model’s ability to accurately simulate a number of perceptual phenomena. The time course of perception is well simulated, showing how factors such as mistuning and onset asynchrony can cause a harmonic to be segregated from a complex tone. It is interesting to note that a good match to Darwin’s pitch shift data (Figure 2D) was only found when harmonically related segments below 1.1 kHz were used. The dominance of lower (resolved) harmonics on pitch is well known [8], and our findings suggest that the correlogram does not accurately model this aspect of pitch perception. Time (ms) Channel 0 50 100 150 200 250 300 350 20 40 60 80 100 120 Figure 4: Asynchronous onset of the fourth harmonic causes it to segregate into a separate stream. The attentional interest vector (Ak) is shown to the right of the figure. The simulation of two tone streaming shows how the proposed attentional mechanism and its cross-frequency spread accounts for grouping of sequential events according to their proximity in frequency. A sequence of two tones will only stream if one set of tones fall outside of the peak of attentional interest. Frequency separations for streaming to occur in the model (greater than 3 to 4 semitones) are in agreement with experimental data, as is the timescale for the build-up of the streaming effect [1]. In summary, we have proposed a physiologically plausible model in which auditory streams are encoded by a unidimensional neural oscillator network. The network creates auditory streams according to grouping factors such as harmonicity, frequency proximity and common onset, and selects one stream as the attentional foreground. Current work is concentrating on expanding the system to include binaural effects, such as inter-ear attentional competition [4]. References [1] Anstis, S. & Saida, S. (1985) Adaptation to auditory streaming of frequency-modulated tones. J. Exp. Psychol. Human 11 257-271. [2] Bregman, A. S. (1990) Auditory Scene Analysis. Cambridge MA: MIT Press. [3] Brown, G. J. & Cooke, M. (1994) Computational auditory scene analysis. Comput. Speech Lang. 8, pp. 297-336. [4] Carlyon, R. P., Cusack, R., Foxton, J. M. & Robertson, I. H. (2001) Effects of attention and unilateral neglect on auditory stream segregation. J. Exp. Psychol. Human 27(1) 115-127. [5] Darwin, C. J., Hukin, R. W. & Al-Khatib, B. Y. (1995) Grouping in pitch perception: Evidence for sequential constraints. J. Acoust. Soc. Am. 98(2)Pt1 880-885. [6] Joliot, M., Ribary, U. & Llinás, R. (1994) Human oscillatory brain activity near 40 Hz coexists with cognitive temporal binding. Proc. Natl. Acad. Sci. USA 91 11748-51. [7] Mondor, T. A. & Bregman, A. S. (1994) Allocating attention to frequency regions. Percept. Psychophys. 56(3) 268-276. [8] Moore, B. C. J. (1997) An introduction to the psychology of hearing. Academic Press. [9] Spence, C. J., Driver, J. (1994) Covert spatial orienting in audition: exogenous and endogenous mechanisms. J. Exp. Psychol. Human 20(3) 555-574. [10] Wang, D. L. (1996) Primitive auditory segregation based on oscillatory correlation. Cognitive Sci. 20 409-456. [11] Wang, D. L. & Brown, G. J. (1999) Separation of speech from interfering sounds based on oscillatory correlation. IEEE Trans. Neural Networks 10 684-697. Time (ms) Channel 0 200 400 600 80 90 100 Time (ms) Channel 0 200 400 600 80 90 100 Figure 5: Auditory streaming at frequency separations of 5 semitones (left) and 3 semitones (right). Streaming occurs at the higher separation. The timescale of adaptation for the attentional interest has been reduced to aid the clarity of the figures.
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Self-regulation Mechanism of Temporally Asymmetric Hebbian Plasticity Narihisa Matsumoto Graduate School of Science and Engineering Saitama University: RIKEN Brain Science Institute Saitama 351-0198, Japan xmatumo@brain.riken.go.jp Masato Okada RIKEN Brain Science Institute Saitama 351-0198, Japan okada@brain.riken.go.jp Abstract Recent biological experimental findings have shown that the synaptic plasticity depends on the relative timing of the pre- and postsynaptic spikes which determines whether Long Term Potentiation (LTP) occurs or Long Term Depression (LTD) does. The synaptic plasticity has been called “Temporally Asymmetric Hebbian plasticity (TAH)”. Many authors have numerically shown that spatiotemporal patterns can be stored in neural networks. However, the mathematical mechanism for storage of the spatio-temporal patterns is still unknown, especially the effects of LTD. In this paper, we employ a simple neural network model and show that interference of LTP and LTD disappears in a sparse coding scheme. On the other hand, it is known that the covariance learning is indispensable for storing sparse patterns. We also show that TAH qualitatively has the same effect as the covariance learning when spatio-temporal patterns are embedded in the network. 1 Introduction Recent biological experimental findings have indicated that the synaptic plasticity depends on the relative timing of the pre- and post- synaptic spikes which determines whether Long Term Potentiation (LTP) occurs or Long Term Depression (LTD) does [1, 2, 3]. LTP occurs when a presynaptic firing precedes a postsynaptic one by no more than about 20ms. In contrast, LTD occurs when a presynaptic firing follows a postsynaptic one. A rapid transition occurs between LTP and LTD within a time difference of a few ms. Such a learning rule is called “Temporally Asymmetric Hebbian learning (TAH)” [4, 5] or “Spike Timing Dependent synaptic Plasticity (STDP)” [6]. Many authors have numerically shown that spatio-temporal patterns can be stored in neural networks [6, 7, 8, 9, 10, 11]. Song et al. discussed the variablity of spike generation about the network consisting of spiking neurons using TAH [6]. They found that the condition that the area of LTD was slightly larger than that of LTP was indispensable of the stability. Namely, the balance of LTP and LTD is crucial. Yoshioka also discussed the associative memory network consisting of spiking neurons using TAH [11]. He found that the area of LTP was needed to be equal to that of LTD for stable retrieval. Munro and Hernandez numerically showed that a network can retrieve spatio-temporal patterns even in a noisy environment owing to LTD [9]. However, they did not discuss the reason why TAH was effective in terms of the storage and retrieval of the spatio-temporal patterns. Since TAH has not only the effect of LTP but that of LTD, the interference of LTP and LTD may prevent retrieval of the patterns. To investigate this unknown mathematical mechanism for retrieval, we employ an associative memory network consisting of binary neurons. To simplify the dynamics of internal potential enables us to analyze the details of the retrieval process. We use a learning rule that is the similar formulation in the previous works. We show the mechanism that the spatio-temporal patterns can be retrieved in this network. There are many works concerned with associative memory networks that store spatio-temporal patterns by the covariance learning [12, 13]. Many biological findings imply that sparse coding schemes may be used in the brain [14]. It is wellknown that the covariance learning is indispensable when the sparse patterns are embedded in a network as attractors [15, 16]. The information on the firing rate for the stored patterns is not indispensable for TAH, although it is indispensable for the covariance learning. We theoretically show that TAH qualitatively has the same effect as the covariance learning when the spatio-temporal patterns are embedded in the network. This means that the difference in spike times induces LTP or LTD, and the effect of the firing rate information can be canceled out by this spike time difference. We conclude that this is the reason why TAH doesn’t require the information on the firing rate for the stored patterns. 2 Model We investigate a network consisting of N binary neurons that are connected mutually. In this paper, we consider the case of N →∞. We use a neuronal model with binary state, {0, 1}. We also use discrete time steps and the following synchronous updating rule, ui(t) = N X j=1 Jijxj(t), (1) xi(t + 1) = Θ(ui(t) −θ), (2) Θ(u) = 1, u ≥0 0, u < 0, (3) where xi(t) is the state of the i-th neuron at time t, ui(t) its internal potential, and θ a uniform threshold. If the i-th neuron fires at time t, its state is xi(t) = 1; otherwise, xi(t) = 0. The specific value of the threshold is discussed later. Jij is the synaptic weight from the j-th neuron to the i-th neuron. Each element ξµ i of the µ-th memory pattern ξµ = (ξµ 1 , ξµ 2 , · · ·, ξµ N) is generated independently by, Prob[ξµ i = 1] = 1 −Prob[ξµ i = 0] = f. (4) The expectation of ξµ is E[ξµ i ] = f, and thus, f can be considered as the mean firing rate of the memory pattern. The memory pattern is “sparse” when f →0, and this coding scheme is called “sparse coding”. The synaptic weight Jij follows the synaptic plasticity that depends on the difference in spike times between the i-th (post-) and j-th (pre-) neurons. The difference determines whether LTP occurs or LTD does. Such a learning rule is called “Temporally Asymmetric Hebbian learning (TAH)” or “Spike Timing Dependent synaptic Plasticity (STDP)”. This biological experimental finding indicates that LTP or LTD is induced when the difference in the pre- and post-synaptic spike times falls within about 20ms [3] (Figure 1(a)). We define that one time step in equations (1)–(3) corresponds to 20ms in Figure 1(a), and a time duration within 20ms is ignored (Figure 1(b)). Figure 1(b) shows that LTP occurs when the j-th neuron fires one time step before the i-th neuron does, ξµ+1 i = ξµ j = 1, and that LTD occurs when the j-th neuron fires one time step after the i-th neuron does, ξµ−1 i = ξµ j = 1. The previous work indicates the blance of LTP and LTD is significant [6]. Therefore, we define that the area of LTP is the (a) -100 -80 -60 -40 -20 0 20 40 60 80 100 -60 -40 -20 0 20 40 60 tpre - tpost (ms) Change in EPSP amplitude(%) LTP LTD (b) -2 -1 0 1 2 -1 0 1 tj - ti ∆Jij LTP LTD Figure 1: Temporally Asymmetric Hebbian plasticity. (a): The result of biological finding [3] and (b): the learning rule in our model. LTP occurs when the j-th neuron fires one time step before the i-th one. On the contrary, LTD occurs when the j-th neuron fires one time step after the i-th one. Synaptic weight Jij is followed by this rule. same as that of LTD, and that the amplitude of LTP is also the same as that of LTD. On the basis of these definitions, we employ the following learning rule, Jij = 1 Nf(1 −f) p X µ=1 (ξµ+1 i ξµ j −ξµ−1 i ξµ j ). (5) The number of memory patterns is p = αN where α is defined as the “loading rate”. There is a critical value αC of loading rate. If the loading rate is larger than αC, the pattern sequence becomes unstable. αC is called the “storage capacity”. The previous works have shown that the learning method of equation (5) can store spatio-temporal patterns, that is, pattern sequences [9, 10]. We show that p memory patterns are retrieved periodically like ξ1 →ξ2 →· · · →ξ p →ξ1 →· · ·. In other words, ξ1 is retrieved at t = 1, ξ2 at t = 2, and ξ1 at t = p + 1. Here, we discuss the value of threshold θ. It is well-known that the threshold value should be controlled according to the progress of the retrieval process timedependently [15, 16]. One candidate algorithm for controlling the threshold value is to maintain the mean firing rate of the network at that of memory pattern, f, as follows, f = 1 N N X i=1 xi(t) = 1 N N X i=1 Θ(ui(t) −θ(t)). (6) It is known that the obtained threshold value is nearly optimal, since it approximately gives a maximal storage capacity value [16]. 3 Theory Many neural network models that store and retrieve sequential patterns by TAH have been discussed by many authors [7, 8, 9, 10]. They have numerically shown that TAH is effective for storing pattern sequences. For example, Munro and Hernandez showed that their model could retrieve a stored pattern sequence even in a noisy environment [9]. However, the previous works have not mentioned the reason why TAH is effective. Exploring such a mechanism is the main purpose of our paper. Here, we discuss the mechanism that the network learned by TAH can store and retrieve sequential patterns. Before providing details of the retrieval process, we discuss a simple situation where the number of memory patterns is very small relative to the number of neurons, i.e., p ∼O(1). Let the state at time t be the same as the t-th memory pattern: x(t) = ξt. Then, the internal potential ui(t) of the equation (1) is given by, ui(t) = ξt+1 i −ξt−1 i . (7) ui(t) depends on two independent random variables, ξt+1 i and ξt−1 i , according to the equation (4). The first term ξt+1 i of the equation (7) is a signal term for the recall of the pattern ξt+1, which is designed to be retrieved at time t+1, and the second term ξt−1 i can interfere in retrieval of ξt+1. According to the equation (7), ui(t) takes a value of 0, −1 or +1. ξt−1 i = 1 means that the interference of LTD exists. If the threshold θ(t) is set between 0 and +1, ξt+1 i = 0 isn’t influenced by the interference of ξt−1 i = 1. When ξt+1 i = 1 and ξt−1 i = 1, the interference does influence the retrieval of ξt+1. We consider the probability distribution of the internal potential ui(t) to examine how the interference of LTD influences the retrieval of ξt+1. The probability of ξt+1 i = 1 and ξt−1 i = 1 is f2, that of ξt+1 i = 1 and ξt−1 i = 0 is f −f2, that of ξt+1 i = 0 and ξt−1 i = 1 is f −f2, and that of ξt+1 i = 0 and ξt−1 i = 0 is (1 −f)2. Then the probability distribution of ui(t) is given by this equation Prob(ui(t)) = (f −f2)δ(ui(t)−1)+(1−2f +2f2)δ(ui(t))+(f −f2)δ(ui(t)+1). (8) Since the threshold θ(t) is set between 0 and +1, the state xi(t + 1) is 1 with probability f −f 2 and 0 with 1 −f + f2. The overlap between the state x(t + 1) and the memory pattern ξt+1 is given by, mt+1(t + 1) = 1 Nf(1 −f) N X i=1 (ξt+1 i −f)xi(t + 1) = 1 −f. (9) In a sparse limit, f →0, the probability of ξ t+1 i = 1 and ξt−1 i = 1 approaches 0. This means that the interference of LTD disappears in a sparse limit, and the model can retrieve the next pattern ξt+1. Then the overlap mt+1(t + 1) approaches 1. Next, we discuss whether the information on the firing rate is indispensable for TAH or not. To investigate this, we consider the case that the number of memory patterns is extensively large, i.e., p ∼O(N). Using the equation (9), the internal potential ui(t) of the i-th neuron at time t is represented as, ui(t) = (ξt+1 i −ξt−1 i )mt(t) + zi(t), (10) zi(t) = p X µ̸=t (ξµ+1 i −ξµ−1 i )mµ(t). (11) zi(t) is called the “cross-talk noise”, which represents contributions from non-target patterns excluding ξt−1 and prevents the target pattern ξt+1 from being retrieved. This disappeared in the finite loading case, p ∼O(1). It is well-known that the covariance learning is indispensable when the sparse patterns are embedded in a network as attractors [15, 16]. Under sparse coding schemes, unless the covariance learning is employed, the cross-talk noise does diverge in the large N limit. Consequently, the patterns can not be stored. The information on the firing rate for the stored patterns is not indispensable for TAH, although it is indispensable for the covariance learning. We use the method of the “statistical neurodynamics” [17, 18] to examine whether the variance of cross-talk noise diverges or not. If a pattern sequence can be stored, the cross-talk noise is obeyed by a Gaussian distribution with mean 0 and time-dependent variance σ2(t). Otherwise, σ2(t) diverges. Since σ2(t) is changing over time, it is necessary to control a threshold at an appropriate value at each time step [15, 16]. According to the statistical neurodynamics, we obtain the recursive equations for the overlap mt(t) between the network state x(t) and the target pattern ξt and the variance σ2(t). The details of the derivation will be shown elsewhere. Here, we show the recursive equations for mt(t) and σ2(t), mt(t) = 1 −2f 2 erf(φ0) −1 −f 2 erf(φ1) + f 2 erf(φ2), (12) σ2(t) = t X a=0 2(a+1)C(a+1)αq(t −a) a Y b=1 U 2(t −b + 1), (13) U(t) = 1 √ 2πσ(t −1){(1 −2f + 2f2)e−φ2 0 + f(1 −f)(e−φ2 1 + e−φ2 2)}, (14) q(t) = 1 2 1 −(1 −2f + 2f2)erf(φ0) −f(1 −f)(erf(φ1) + erf(φ2)) , (15) erf(y) = 2 √π Z y 0 exp (−u2)du, bCa = b! a!(b −a)!, a! = a × (a −1) × · · · × 1, φ0 = θ(t −1) √ 2σ(t −1), φ1 = −mt−1(t −1) + θ(t −1) √ 2σ(t −1) , φ2 = mt−1(t −1) + θ(t −1) √ 2σ(t −1) . These equations reveal that the variance σ2(t) of cross-talk noise does not diverge as long as a pattern sequence can be retrieved. This result means that TAH qualitatively has the same effect as the covariance learning. Next, we discuss the mechanism that the variance of cross-talk noise does not diverge. Let us consider the equation (5). Synaptic weight Jij from j-th neuron to i-th neuron is also derived as follows, Jij = 1 Nf(1 −f) p X µ=1 (ξµ+1 i ξµ j −ξµ−1 i ξµ j ) = 1 Nf(1 −f) p X µ=1 (ξµ i ξµ−1 j −ξµ i ξµ+1 j ) = 1 Nf(1 −f) p X µ=1 ξµ i n (ξµ−1 j −f) −(ξµ+1 j −f) o (16) This equation implies that TAH has the information on the firing rate of the memory patterns when spatio-temporal patterns are embedded in a network. Therefore, the variance of cross-talk noise doesn’t diverge, and this is another factor for the network learned by TAH to store and retrieve a pattern sequence. We conclude that the difference in spike times induces LTP or LTD, and the effect of the firing rate information can be canceled out by this spike times difference. 4 Results We investigate the property of our model and examine the following two conditions: a fixed threshold and a time-dependent threshold, using the statistical neurodynamics and computer simulations. Figure 2 shows how the overlap mt(t) and the mean firing rate of the network, ¯x(t) = 1 N P i xi(t), depend on the loading rate α when the mean firing rate of the memory pattern is f = 0.1 and the threshold is θ = 0.52, where the storage capacity is maximum with respect to the threshold θ. The stored pattern sequence can be retrieved when the initial overlap m1(1) is greater than the critical value mC. The lower line indicates how the critical initial overlap mC depends on the loading rate α. In other words, the lower line represents the basin of attraction for the retrieved sequence. The upper line denotes a steady value of overlap mt(t) when the pattern sequence is retrieved. mt(t) is obtained by setting the initial state to the first memory pattern: x(1) = ξ1. In this case, the storage capacity is αC = 0.27. The dashed line shows a steady value of the normalized mean firing rate of network, ¯x(t)/f, for the pattern sequence. The data points and error bars indicate the results of the computer simulations with 5000 neurons: N = 5000. The former indicates mean values and the latter does variances in 10 trials. Since the results 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 1.2 loading rate overlap (solid), activity/f (dashed) Figure 2 !!The critical overlap (the lower line) and the overlap at the stationary state (the upper line). The dashed line shows the mean firing rate of the network divided firing rate which is 0.1. The threshold is 0.52 and the number of neurons is 5000. The data points and error bars show the means and variances, respectively, in 10 trials of computer simulations. The storage capacity is 0.27. of the computer simulations coincide with those of the statistical neurodynamics, hereafter, we show the results only of the statistical neurodynamics. Next, we examine the threshold control scheme in the equation (6), where the threshold is controlled to maintain the mean firing rate of the network at f. q(t) in equation (15) is equal to the mean firing rate because q(t) = 1 N PN i=1(xi(t))2 = 1 N PN i=1 xi(t) under the condition xi(t) = {0, 1}. Thus, the threshold is adjusted to satisfy the following equation, f = q(t) = 1 2 1 −(1 −2f + 2f2)erf(φ0) −f(1 −f)(erf(φ1) + erf(φ2)) . (17) Figure 3 shows the overlap mt(t) as a function of loading rate α with f = 0.1. The storage capacity is αC = 0.234. The basin of attraction becomes larger than that of the fixed threshold condition, θ = 0.52 (Figure 2). Thus, the network becomes robust against noise. This means that even if the initial state x(1) is different from the first memory pattern ξ1, that is, the state includes a lot of noise, the pattern sequence can be retrieved. 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 1.2 loading rate overlap (solid), activity/f (dashed) Figure 3 !!The critical overlap (the lower line) and the overlap at the stationary state (the upper line) when the threshold is changing over time to maintain mean firing rate of the network at f. The dashed line shows the mean firing rates of the network divided firing rate which is 0.1. The basin of attraction become larger than that of the fixed threshold condition: Figure 2. Finally, we discuss how the storage capacity depends on the firing rate f of the memory pattern. It is known that the storage capacity diverges as 1 f| log f| in a sparse limit, f →0 [19, 20]. Therefore, we investigate the asymptotic property of the storage capacity in a sparse limit. Figure 4 shows how the storage capacity depends on the firing rate where the threshold is controlled to maintain the network activity at f (symbol ◦). The storage capacity diverges as 1 f| log f| in a sparse limit. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.005 0.01 0.015 0.02 0.025 0.03 1/|log f| αC f 0.45 Figure 4 !!The storage capacity as a function of f in the case of maintaining activity at f (symbol ◦). Ths storage capacity diverges as 1 f| log f| in a sparse limit. 5 Discussion Using a simple neural network model, we have discussed the mechanism that TAH enables the network to store and retrieve a pattern sequence. First, we showed that the interference of LTP and LTD disappeared in a sparse coding scheme. This is a factor to enable the network to store and retrieve a pattern sequence. Next, we showed the mechanism that TAH qualitatively had the same effect as the covariance learning by analyzing the stability of the stored pattern sequence and the retrieval process by means of the statistical neurodynamics. Consequently, the variance of cross-talk noise didn’t diverge, and this is another factor for the network learned by TAH to store and retrieve a pattern sequence. We conclude that the difference in spike times induces LTP or LTD, and the effect of the firing rate information can be canceled out by this spike times difference. We investigated the property of our model. To improve the retrieval property of the basin of attraction, we introduced a threshold control algorithm where a threshold value was adjusted to maintain the mean firing rate of the network at that of a memory pattern. As a result, we found that this scheme enlarged the basin of attraction, and that the network became robust against noise. We also found that the loading rate diverged as 1 f| log f| in a sparse limit, f →0. Here, we compare the storage capacity of our model with that of the model using the covariance learning (Figure 5). The dynamical equations of the model using the covariance learning is derived by Kitano and Aoyagi [13]. We calculate the storage capacity αCOV C from their dynamical equations and compare these of our model, αTAH C , by the ratio of αTAH C /αCOV C . The threshold control method is the same as in this paper. As f decreases, the ratio of storage capacities approaches 0.5. The contribution of LTD reduces the storage capacity of our model to half. Therefore, in terms of the storage capacity, the covariance learning is better than TAH. But, as we discussed previously, the information of the firing rate is indispensable in TAH. In biological systems, to get the information of the firing rate is difficult. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 αC TAH/αC COV 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 log10f Figure 5 !!The comparison of the storage capacity of our model with that of the model using the covariance learning. As f decreases, the ratio of storage capacity approaches 0.5. References [1] G. Q. Bi and M. M. Poo. Synaptic modifications in cultured hippocampal neurons: Dependence on spike timing, synaptic strength, and postsynaptic cell type. The Journal of Neuroscience, 18:10464–10472, 1998. [2] H. Markram, J. L¨ubke, M. Frotscher, and B. Sakmann. Regulation of synaptic efficacy by coincidence of postsynaptic aps and epsps. Science, 275:213–215, 1997. [3] L. I. Zhang, H. W. Tao, C. E. Holt, W. A. Harris, and M. M. Poo. A critical window for cooperation and competition among developing retinotectal synapses. Nature, 395:37–44, 1998. [4] L. F. Abbott and S. Song. Temporally asymmetric hebbian learning, spike timing and neuronal response variability. In Advances in Neural Information Processing Systems 11, pages 69–75. MIT Press, 1999. [5] J. Rubin, D. D. Lee, and H. Sompolinsky. Equilibrium properties of temporally asymmetric hebbian plasticity. Physical Review Letters, 86:364–367, 2001. [6] S. Song, K. D. Miller, and L. F. Abbott. Competitive hebbian learning through spike-timing-dependent synaptic plasticity. Nature Neuroscience, 3:919–926, 2000. [7] W. Gerstner, R. Kempter, J. L. van Hemmen, and H. Wagner. A neuronal learning rule for sub-millisecond temporal coding. Nature, 383:76–78, 1996. [8] R. Kempter, W. Gerstner, and J. L. van Hemmen. Hebbian learning and spiking neurons. Physical Review E, 59:4498–4514, 1999. [9] P. Munro and G. Hernandez. LTD facilitates learning in a noisy environment. In Advances in Neural Information Processing Systems 12, pages 150–156. MIT Press, 2000. [10] R. P. N. Rao and T. J. Sejnowski. Predictive sequence learning in recurrent neocortical circuits. In Advances in Neural Information Processing Systems 12, pages 164–170. MIT Press, 2000. [11] M. Yoshioka. to be published in Physical Review E, 2001. [12] G. Chechik, I. Meilijson, and E. Ruppin. Effective learning requires neuronal remodeling of hebbian synapses. In Advances in Neural Information Processing Systems 11, pages 96–102. MIT Press, 1999. [13] K. Kitano and T. Aoyagi. Retrieval dynamics of neural networks for sparsely coded sequential patterns. Journal of Physics A: Mathematical and General, 31:L613–L620, 1998. [14] M. Miyashita. Neuronal correlate of visual associative long-term memory in the primate temporal cortex. Nature, 335:817–820, 1988. [15] S. Amari. Characteristics of sparsely encoded associative memory. Neural Networks, 2:1007–1018, 1989. [16] M. Okada. Notions of associative memory and sparse coding. Neural Networks, 9:1429–1458, 1996. [17] S. Amari and K. Maginu. Statistical neurodynamics of various versions of correlation associative memory. Neural Networks, 1:63–73, 1988. [18] M. Okada. A hierarchy of macrodynamical equations for associative memory. Neural Networks, 8:833–838, 1995. [19] M. V. Tsodyks and M. V. Feigle’man. The enhanced strage capacity in neural networks with low activity level. Europhysics Letters, 6:101–105, 1988. [20] C. J. Perez-Vicente and D. J. Amit. Optimized network for sparsely coded patterns. Journal of Physics A: Mathematical and General, 22:559–569, 1989.
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An Efficient, Exact Algorithm for Solving Tree-Structured Graphical Games Michael L. Littman AT&T Labs- Research Florham Park, NJ 07932-0971 mlittman©research.att.com Michael Kearns Department of Computer & Information Science University of Pennsylvania Philadelphia, PA 19104-6389 mkearns©cis.upenn.edu Satinder Singh Syntek Capital New York, NY 10019-4460 baveja©cs. colorado. edu Abstract We describe a new algorithm for computing a Nash equilibrium in graphical games, a compact representation for multi-agent systems that we introduced in previous work. The algorithm is the first to compute equilibria both efficiently and exactly for a non-trivial class of graphical games. 1 Introduction Seeking to replicate the representational and computational benefits that graphical models have provided to probabilistic inference, several recent works have introduced graph-theoretic frameworks for the study of multi-agent systems (La Mura 2000; Koller and Milch 2001; Kearns et al. 2001). In the simplest of these formalisms, each vertex represents a single agent, and the edges represent pairwise interaction between agents. As with many familiar network models, the macroscopic behavior of a large system is thus implicitly described by its local interactions, and the computational challenge is to extract the global states of interest. Classical game theory is typically used to model multi-agent interactions, and the global states of interest are thus the so-called Nash equilibria, in which no agent has a unilateral incentive to deviate. In a recent paper (Kearns et al. 2001), we introduced such a graphical formalism for multi-agent game theory, and provided two algorithms for computing Nash equilibria when the underlying graph is a tree (or is sufficiently sparse). The first algorithm computes approximations to all Nash equilibria, in time polynomial in the size of the representation and the quality of the desired approximation. A second and related algorithm computes all Nash equilibria exactly, but in time exponential in the number of agents. We thus left open the problem of efficiently computing exact equilibria in sparse graphs. In this paper, we describe a new algorithm that solves this problem. Given as input a graphical game that is a tree, the algorithm computes in polynomial time an exact Nash equilibrium for the global multi-agent system. The main advances involve the definition of a new data structure for representing "upstream" or partial Nash equilibria, and a proof that this data structure can always be extended to a global equilibrium. The new algorithm can also be extended to efficiently accommodate parametric representations of the local game matrices, which are analogous to parametric conditional probability tables (such as noisy-OR and sigmoids) in Bayesian networks. The analogy between graphical models for multi-agent systems and probabilistic inference is tempting and useful to an extent. The problem of computing Nash equilibria in a graphical game, however, appears to be considerably more difficult than computing conditional probabilities in Bayesian networks. Nevertheless, the analogy and the work presented here suggest a number of interesting avenues for further work in the intersection of game theory, network models, probabilistic inference, statistical physics, and other fields. The paper is organized as follows. Section 2 introduces graphical games and other necessary notation and definitions. Section 3 presents our algorithm and its analysis, and Section 4 gives a brief conclusion. 2 Preliminaries An n-player, two-action 1 game is defined by a set of n matrices Mi (1 ~ i ~ n), each with n indices. The entry Mi(Xl, ... ,xn ) = Mi(X) specifies the payoff to player i when the joint action of the n players is x E {O, I} n. Thus, each Mi has 2n entries. If a game is given by simply listing the 2n entries of each of the n matrices, we will say that it is represented in tabular form. The actions ° and 1 are the pure strategies of each player, while a mixed strategy for player i is given by the probability Pi E [0, 1] that the player will play 1. For any joint mixed strategy, given by a product distribution p, we define the expected payoff to player i as Mi(i/) = Ex~p[Mi(X)], where x'" pindicates that each Xj is 1 with probability Pj and ° with probability 1 - Pj. We use p[i : P:] to denote the vector that is the same as p except in the ith component, where the value has been changed to P:. A Nash equilibrium for the game is a mixed strategy p such that for any player i, and for any value P: E [0,1], Mi(i/) ::::: Mi(p[i : pm. (We say that Pi is a best response to jJ.) In other words, no player can improve its expected payoff by deviating unilaterally from a Nash equilibrium. The classic theorem of Nash (1951) states that for any game, there exists a Nash equilibrium in the space of joint mixed strategies (product distri butions). An n-player graphical game is a pair (G, M), where G is an undirected graph2 on n 1 At present, no polynomial-time algorithm is known for finding Nash equilibria even in 2-player games with more than two actions, so we leave the extension of our work to the multi-action setting for future work. 2The directed tree-structured case is trivial and is not addressed in this paper. vertices and M is a set of n matrices Mi (1 ::; i ::; n), called the local game matrices. Player i is represented by a vertex labeled i in G. We use N G (i) ~ {I, ... , n} to denote the set of neighbors of player i in G- those vertices j such that the undirected edge (i , j) appears in G. By convention, NG(i) always includes i itself. The interpretation is that each player is in a game with only his neighbors in G. Thus, if ING(i) I = k, the matrix Mi has k indices, one for each player in NG(i) , and if x E [0, Ilk, Mi(X) denotes the payoff to i when his k neighbors (which include himself) play x. The expected payoff under a mixed strategy jJ E [0, Ilk is defined analogously. Note that in the two-action case, Mi has 2k entries, which may be considerably smaller than 2n. Since we identify players with vertices in G, it will be easier to treat vertices symbolically (such as U, V and W) rather than by integer indices. We thus use Mv to denote the local game matrix for the player identified with vertex V. Note that our definitions are entirely representational, and alter nothing about the underlying game theory. Thus, every graphical game has a Nash equilibrium. Furthermore, every game can be trivially represented as a graphical game by choosing G to be the complete graph and letting the local game matrices be the original tabular form matrices. Indeed, in some cases, this may be the most compact graphical representation of the tabular game. However, exactly as for Bayesian networks and other graphical models for probabilistic inference, any game in which the local neighborhoods in G can be bounded by k « n, exponential space savings accrue. The algorithm presented here demonstrates that for trees, exponential computational benefits may also be realized. 3 The Algorithm If (G, M) is a graphical game in which G is a tree, then we can always designate some vertex Z as the root. For any vertex V, the single neighbor of Von the path from V to Z shall be called the child of V, and the (possibly many) neighbors of V on paths towards the leaves shall be called the parents of V. Our algorithm consists of two passes: a downstream pass in which local data structures are passed from the leaves towards the root, and an upstream pass progressing from the root towards the leaves. Throughout the ensuing discussion, we consider a fixed vertex V with parents UI , ... , Uk and child W. On the downstream pass of our algorithm, vertex V will compute and pass to its child W a breakpoint policy, which we now define. Definition 1 A breakpoint policy for V consists of an ordered set of W -breakpoints Wo = ° < WI < W2 < ... < Wt-I < Wt = 1 and an associated set of V-values VI , . .. ,Vt· The interpretation is that for any W E [0,1], if Wi-I < W < Wi for some index i and W plays w, then V shall play Vii and if W = Wi for some index i, then V shall play any value between Vi and Vi+I. We say such a breakpoint policy has t - 1 breakpoints. A breakpoint policy for V can thus be seen as assigning a value (or range of values) to the mixed strategy played by V in response to the play of its child W. In a slight abuse of notation, we will denote this breakpoint policy as a function Fv(w), with the understanding that the assignment V = Fv(w) means that V plays either the fixed value determined by the breakpoint policy (in the case that W falls between breakpoints), or plays any value in the interval determined by the breakpoint policy (in the case that W equals some breakpoint). Let G V denote the subtree of G with root V, and let M~=w denote the subset of the set of local game matrices M corresponding to the vertices in GV , except that the matrix M v is collapsed one index by setting W = w, thus marginalizing W out. On its downstream pass, our algorithm shall maintain the invariant that if we set the child W = w, then there is a Nash equilibrium for the graphical game (Gv , M~=w) (an upstream Nash) in which V = Fv(w). If this property is satisfied by Fv(w), we shall say that Fv(w) is a Nash breakpoint policy for V. Note that since (Gv, M~=w) is just another graphical game, it of course has (perhaps many) Nash equilibria, and V is assigned some value in each. The trick is to commit to one of these values (as specified by Fv (w)) that can be extended to a Nash equilibrium for the entire tree G, before we have even processed the tree below V . Accomplishing this efficiently and exactly is one of the main advances in this work over our previous algorithm (Kearns et al. 2001). The algorithm and analysis are inductive: V computes a Nash breakpoint policy Fv(w) from Nash breakpoint policies FUl (v), ... , FUk (v) passed down from its parents (and from the local game matrix Mv). The complexity analysis bounds the number of breakpoints for any vertex in the tree. We now describe the inductive step and its analysis. 3.1 Downstream Pass For any setting it E [0, l]k for -0 and w E [0,1] for W, let us define ~v(i1,w) == Mv(l,it,w) - Mv(O,it,w). The sign of ~v(it, w) tells us V's best response to the setting of the local neighborhood -0 = it, W = w; positive sign means V = 1 is the best response, negative that V = 0 is the best response, and 0 that V is indifferent and may play any mixed strategy. Note also that we can express ~v(it,w) as a linear function of w: ~v(it,w) = ~v(it, O) + w(~v(it, 1) ~v(it, 0)). For the base case, suppose V is a leaf with child W; we want to describe the Nash breakpoint policy for V. If for all w E [0,1], the function ~v(w) is non-negative (non-positive, respectively), V can choose 1 (0, respectively) as a best response (which in this base case is an upstream Nash) to all values W = w. Otherwise, ~ v (w) crosses the w-axis, separating the values of w for which V should choose 1, 0, or be indifferent (at the crossing point). Thus, this crossing point becomes the single breakpoint in Fv(w). Note that if V is indifferent for all values of w, we assume without loss of generality that V plays l. The following theorem is the centerpiece of the analysis. Theorem 2 Let vertex V have parents UI , ... ,Uk and child W, and assume V has received Nash breakpoint policies FUi (v) from each parent Ui . Then V can efficiently compute a Nash breakpoint policy Fv (w). The number of breakpoints is no more than two plus the total number of breakpoints in the FUi (v) policies. Proof: Recall that for any fixed value of v, the breakpoint policy FUi (v) specifies either a specific value for Ui (if v falls between two breakpoints of FUi (v)) , or a range of allowed values for Ui (if v is equal to a breakpoint). Let us assume without loss of generality that no two FUi (v) share a breakpoint, and let Vo = 0 < VI < ... < Vs = 1 be the ordered union of the breakpoints of the FUi (v). Thus for any breakpoint Vi, there is at most one distinguished parent Uj (that we shall call the free parent) for which Fu; (Vi) specifies an allowed interval of play for Uj . All other Ui are assigned fixed values by Fu; (ve). For each breakpoint Ve, we now define the set of values for the child W that, as we let the free parent range across its allowed interval, permit V to play any mixed strategy as a best response. Definition 3 Let Vo = 0 < VI < ... < Vs = 1 be the ordered union of the breakpoints of the parent policies Fu; (v). Fix any breakpoint Ve, and assume without loss of generality that UI is the free parent of V for V = Ve. Let [a, b] be the allowed interval ofUI specified by FUI (ve), and letui = Fu;(ve) for all 2 :::; i:::; k. We define We = {w E [0,1]: (:lUI E [a,b])6.v(UI,U2, ... ,Uk,W) = O}. In words, We is the set of values that W can play that allow V to play any mixed strategy, preserving the existence of an upstream Nash from V given W = w. The next lemma, which we state without proof and is a special case of Lemma 6 in Kearns et al. (2001), limits the complexity of the sets We. It also follows from the earlier work that We can be computed in time proportional to the size of V's local game matrix O(2k) for a vertex with k parents. We say that an interval [a, b] ~ [0, 1] is floating if both a -I- 0 and b -I- 1. Lemma 4 For any breakpoint Ve, the set We is either empty, a single interval, or the union of two intervals that are not floating. We wish to create the (inductive) Nash breakpoint policy Fv(w) from the sets We and the Fu; policies. The idea is that if w E We for some breakpoint index e, then by definition of We, if W plays wand the Uis play according to the setting determined by the Fu; policies (including a fixed setting for the free parent of V), any play by V is a best response-so in particular, V may play the breakpoint value Ve, and thus extend the Nash solution constructed, as the UiS can also all be best responses. For b E {O, I}, we define W b as the set of values w such that if W = w and the Uis are set according to their breakpoint policies for V = b, V = b is a best response. To create Fv (w) as a total function, we must first show that every w E [0, 1] is contained in some We or WO or WI. Lemma 5 Let Vo = 0 < VI < ... < Vs = 1 be the ordered union of the breakpoints of the Fu; (v) policies. Then for any value w E [0, 1], either w E wb for some bE {O, I} , or there exists an index e such that wE We. Proof: Consider any fixed value of w, and for each open interval (vi> vj+d determined by adjacent breakpoints, label this interval by V 's best response (0 or 1) to W = wand 0 set according to the Fu; policies for this interval. If either the leftmost interval [O,vd is labeled with 0 or the rightmost interval [vs-I ,I] is labeled with 1, then w is included in WO or WI , respectively (V playing 0 or 1 is a best response to what the Uis will play in response to a 0 or 1). Otherwise, since the labeling starts at 1 on the left and ends at 0 on the right, there must be a breakpoint Ve such that V's best response changes over this breakpoint. Let Ui be the free parent for this breakpoint. By continuity, there must be a value of Ui in its allowed interval for which V is indifferent to playing 0 or 1, so w E We. This completes the proof of Lemma 5. Armed with Lemmas 4 and 5, we can now describe the construction of Fv(w). Since every w is contained in some We (Lemma 5), and since every We is the union of at most two intervals (Lemma 4), we can uniquely identify the set WeI that covers the largest (leftmost) interval containing w = 0; let [0, a] be this interval. Continuing in the same manner to the right, we can identify the unique set We2 that contains v7r----- --- ----- --- ----- --- ----- --- ----- -------- - r-----v6 ------~ --------------------------- - -V v5 ------------------------------ - ,-------'--------------v4 ------------- ------------ --------------------v3f------.- ---------------------- ----------------------------v2 _______ --'-______ ----L _________________________________ _ vI ---------------------------------------------------w Figure 1: Example of the inductive construction of Fv(w). The dashed horizontal lines show the vrbreakpoints determined by the parent policies Fu; (v). The solid intervals along these breakpoints are the sets We. As shown in Lemma 4, each of these sets consists of either a single (possibly floating) interval, or two non-floating intervals. As shown in Lemma 5, each value of w is covered by some We. The construction of Fv(w) (represented by a thick line) begins on the left, and always next "jumps" to the interval allowing greatest progress to the right. w = a and extends farthest to the right of a. Any overlap between We 1 and We2 can be arbitrarily assigned coverage by We 1 , and We2 "trimmed" accordingly; see Figure 1. This process results in a Nash breakpoint policy Fv(w). Finally, we bound the number of breakpoints in the Fv (w) policy. By construction, each of its breakpoints must be the rightmost portion of some interval in WO, WI, or some We. After the first breakpoint, each of these sets contributes at most one new breakpoint (Lemma 4). The final breakpoint is at w = 1 and does not contribute to the count (Definition 1). There is at most one We for each breakpoint in each Fu; (v) policy, plus WO and WI, plus the initial leftmost interval and minus the final breakpoint, so the total breakpoints in Fv(w) can be no more than two plus the total number of breakpoints in the Fu; (v) policies. Therefore, the root of a size n tree will have a Nash breakpoint policy with no more than 2n breakpoints. This completes the proof of Theorem 2. 3.2 Upstream Pass The downstream pass completes when each vertex in the tree has had its Nash breakpoint policy computed. For simplicity of description, imagine that the root of the tree includes a dummy child with constant payoffs and no influence on the root, so the root's breakpoint policy has the same form as the others in the tree. To produce a Nash equilibrium, our algorithm performs an upstream pass over the tree, starting from the root. Each vertex is told by its child what value to play, as well as the value the child itself will play. The algorithm ensures that all downstream vertices are Nash (playing best response to their neighbors). Given this information, each vertex computes a value for each of its parents so that its own assigned action is a best response. This process can be initiated by the dummy vertex picking an arbitrary value for itself, and selecting the root's value according to its Nash breakpoint policy. Inductively, we have a vertex V connected to parents U1 , ... , Uk (or no parents if V is a leaf) and child W. The child of V has informed V to chose V = v and that it will play W = w. To decide on values for V's parents to enforce V playing a best response, we can look at the Nash breakpoint policies FUi (v), which provide a value (or range of values) for Ui as a function of v that guarantee an upstream Nash. The value v can be a breakpoint for at most one Ui . For each Ui , if v is not a breakpoint in FUi (v) , then Ui should be told to select Ui = FUi (v). If v is a breakpoint in FUi (v), then Ui's value can be computed by solving ~V(Ul "'" Ui,"" Uk, w) = 0; this is the value of Ui that makes V indifferent. The equation is linear in Ui and has a solution by the construction of the Nash breakpoint policies on the downstream pass. Parents are passed their assigned values as well as the fact that V = v. When the upstream pass completes, each vertex has a concrete choice of action such that jointly they have formed a Nash equilibrium. The total running time of the algorithm can be bounded as follows. Each vertex is involved in a computation in the downstream pass and in the upstream pass. Let t be the total number of breakpoints in the breakpoint policy for a vertex V with k parents. Sorting the breakpoints and computing the W£ sets and computing the new breakpoint policy can be completed in 0 (t log t + t2 k ). In the upstream pass, only one breakpoint is considered, so 0 (log t + 2k) is sufficient for passing breakpoints to the parents. By Theorem 2, t :S 2n, so the entire algorithm executes in time O(n2 10g n + n22k), where k is the largest number of neighbors of any vertex in the network. The algorithm can be implemented to take advantage of local game matrices provided in a parameterized form. For example, if each vertex's payoff is solely a function of the number of 1s played by the vertex's neighbors, the algorithm takes O(n2 10gn + n 2k), eliminating the exponential dependence on k. 4 Conclusion The algorithm presented in this paper finds a single Nash equilibrium for a game represented by a tree-structured network. By building representations of all equilibria, our earlier algorithm (Kearns et al. 2001) was able to select equilibria efficiently according to criteria like maximizing the total expected payoff for all players. The polynomial-time algorithm described in this paper throws out potential equilibria at many stages, most significantly during the construction of the Nash breakpoint policies. An interesting area for future work is to manipulate this process to produce equilibria with particular properties. References Michael Kearns, Michael L. Littman, and Satinder Singh. Graphical models for game theory. In Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence (UAI), pages 253- 260, 200l. Daphne Koller and Brian Milch. Multi-agent influence diagrams for representing and solving games. Submitted, 2001. Pierfrancesco La Mura. Game networks. In Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence (UAI), pages 335- 342, 2000. J. F. Nash. Non-cooperative games. Annals of Math ematics, 54:286- 295, 1951.
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Causal Categorization with Bayes Nets Bob Rehder Department of Psychology New York University New York, NY 10012 bob .rehder@nyu.edu Abstract A theory of categorization is presented in which knowledge of causal relationships between category features is represented as a Bayesian network. Referred to as causal-model theory, this theory predicts that objects are classified as category members to the extent they are likely to have been produced by a categorys causal model. On this view, people have models of the world that lead them to expect a certain distribution of features in category members (e.g., correlations between feature pairs that are directly connected by causal relationships), and consider exemplars good category members when they manifest those expectations. These expectations include sensitivity to higher-order feature interactions that emerge from the asymmetries inherent in causal relationships. Research on the topic of categorization has traditionally focused on the problem of learning new categories given observations of category members. In contrast, the theory-based view of categories emphasizes the influence of the prior theoretical knowledge that learners often contribute to their representations of categories [1]. However, in contrast to models accounting for the effects of empirical observations, there have been few models developed to account for the effects of prior knowledge. The purpose of this article is to present a model of categorization referred to as causal-model theory or CMT [2, 3]. According to CMT, people's know ledge of many categories includes not only features, but also an explicit representation of the causal mechanisms that people believe link the features of many categories. In this article I apply CMT to the problem of establishing objects category membership. In the psychological literature one standard view of categorization is that objects are placed in a category to the extent they have features that have often been observed in members of that category. For example, an object that has most of the features of birds (e.g., wings, fly, build nests in trees, etc.) and few features of other categories is thought to be a bird. This view of categorization is formalized by prototype models in which classification is a function of the similarity (i.e. , number of shared features) between a mental representation of a category prototype and a to-be-classified object. However, a well-known difficulty with prototype models is that a features contribution to category membership is independent of the presence or absence of other features. In contrast, consideration of a categorys theoretical knowledge is likely to influence which combinations of features make for acceptable category members. For example, people believe that birds have nests in trees because they can fly , and in light of this knowledge an animal that doesnt fly and yet still builds nests in trees might be considered a less plausible bird than an animal that builds nests on the ground and doesnt fly (e.g., an ostrich) even though the latter animal has fewer features typical of birds. To assess whether knowledge in fact influences which feature combinations make for good category members , in the following experiment undergraduates were taught novel categories whose four binary features exhibited either a common-cause or a common-effect schema (Figure 1). In the common-cause schema, one category feature (PI) is described as causing the three other features (F2, F3, and F4). In the common-effect schema one feature (F4) is described as being caused by the three others (F I, F2, and F3). CMT assumes that people represent causal knowledge such as that in Figure 1 as a kind of Bayesian network [4] in which nodes are variables representing binary category features and directed edges are causal relationships representing the presence of probabilistic causal mechanisms between features. Specifically , CMT assumes that when a cause feature is present it enables the operation of a causal mechanism that will, with some probability m , bring about the presence of the effect feature. CMT also allow for the possibility that effect features have potential background causes that are not explicitly represented in the network, as represented by parameter b which is the probability that an effect will be present even when its network causes are absent. Finally, each cause node has a parameter c that represents the probability that a cause feature will be present. Common-Cause Schema ~ ~ ® Common-Effect Schema Figure 1. . (~~) @ . .. : .... ~~:f·""""®1 ®"""::® " .... @/ ®'. F .• 3 Common-Cause Common-Effect Correlations Correlations Figure 2. The central prediction of CMT is that an object is considered to be a category member to the extent that its features were likely to have been generated by a category's causal mechanisms. For example, Table 1 presents the likelihoods that the causal models of Figure 1 will generate the sixteen possible combinations of F I, F2, F3, and F4. Each likelihood equation can be derived by the application of simple Boolean algebra operations. For example, the probability of exemplar 1101 (F I, F2, F4 present, F3 absent) being generated by a common-cause model is the probability that F I is present [c], times the probability that F2 was brought about by F I or its background cause [1- (lmj(l-b)], times the probability that F3 was brought about by neither F I nor its background cause [(l-m )(l-b)], times the probability that F 4 was brought about by F I or its background cause [1- (lmj(l-b)]. Likewise , the probability of exemplar 1011 (F I, F3, F 4 present, F2 absent) being generated by a common-effect model is the probability that FI and F3 are present [c 2], times the probability that F2 is absent [1-£], times the probability that F4 was brought about by F I, F3, or its background cause [1- (lmj(l-m )(l-b)] . Note that these likelihoods assume that the causal mechanisms in each model operate independently and with the same probability m, restrictions that can be relaxed in other applications. This formalization of categorization offered by CMT implies that peoples theoretical knowledge leads them to expect a certain distribution of features in category members , and that they use this information when assigning category membership. Thus , to gain insight into the categorization performance predicted by CMT, we can examine the statistical properties of category features that one can expect to be generated by a causal model. For example, dotted lines in Figure 2 represent the features correlations that are generated from the causal schemas of Figure 1. As one would expect, pairs of features directly linked by causal relationships are correlated in the common-cause schema F I is correlated with its effects and in the common-effect schema F4 is correlated with its causes. Thus, CMT predicts that combinations of features serve as evidence for category membership to the extent that they preserve these expected correlations (i.e. , both cause and effect present or both absent) , and against category membership to the extent that they break those correlations (one present and the other absent). Table 1: Likelihoods Equations and Observed and Predicted Values Common Cause Schema Common Effect Schema Control ExemElar Likelihood Observed Predicted Likelihood Observed Predicted Observed 0000 e'b ,3 60.0 61.7 e ,3 b , 70.0 69.3 70.7 0001 e 'b ,2 b 44.9 45 .7 e ,3 b 26.3 27 .8 67.0 0010 e'b ,2 b 46.1 45 .7 ee,2 m 'b ' 43.4 47 .7 65.6 0100 e 'b ,2 b 42.8 45 .7 ee ,2 m 'b ' 47.3 47 .7 66.0 1000 em ,3 b ,3 44.5 44.1 ee,2 m 'b ' 48.0 47 .7 67.0 0011 e 'b 'b 2 41.0 40.1 ee ,2 (1-m 'b ') 56.3 56.5 67.1 0101 e'b 'b 2 40.8 40.1 ee,2 (1-m 'b ') 56.5 56.5 66.5 0110 e 'b 'b 2 42.7 40.1 e2e 'm ,2 b , 38.3 39.2 65.6 1001 em ,2 b ,2 (1- m 'b ') 55.1 52.7 ee,2 (1-m 'b ') 57.7 56.5 68.0 1010 em ,2 b ,2 (1- m 'b ') 52.6 52.7 e2e 'm ,2 b , 43.0 39.2 67.6 1100 em ,2 b ,2 (1- m 'b ') 54.3 52.7 e2e 'm ,2 b , 41.9 39.2 69.9 0111 e 'b 3 39.4 38.1 e2e'(1-m ,2 b ,) 71.0 74.4 67.6 1011 em 'b '(1-m 'b ,)2 64.2 65 .6 e2e '(1-m ,2 b ,) 75 .7 74.4 67 .2 1101 em 'b '(1-m 'b ,)2 65 .3 65 .6 e2e'(1-m ,2 b ,) 74.7 74.4 70.2 1110 em 'b '(1-m 'b ,)2 62.0 65 .6 e3m ,3 b , 33.8 35 .8 72.2 1111 e(1-m 'b ,)3 90.8 89.6 e3(1-m ,3 b ,) 91.0 90.0 75.6 Note. e'=l- c . m '=l-m . b'=l-b. Causal networks not only predict pairwise correlations between directly connected features. Figure 2 indicates that as a result of the asymmetries inherent in causal relationships there is an important disanalogy between the common-cause and common-effect schemas: Although the common-cause schema implies that the three effects (F2 , F3 , F4) will be correlated (albeit more weakly than directly connected features) , the common-effect schema does not imply that the three causes (F I , F2 , F3) will be correlated. This asymmetry between common-cause and common-effect schemas has been the focus of considerable investigation in the philosophical and psychological literatures [3 , 5]. Use of these schemas in the following experiment enables a test of whether categorizers are sensitive the pattern of correlations between features directly-connected by causal laws, and also those that arise due to the asymmetries inherent in causal relationships shown in Figure 2. Moreover, I will show that CMT predicts, and humans exhibit, sensitivity to interactions among features of a higher-order than the pairwise interactions shown in Figure 2. Method Six novel categories were used in which the description of causal relationships between features consisted of one sentence indicating the cause and effect feature , and then one or two sentences describing the mechanism responsible for the causal relationship. For example, one of the novel categories, Lake Victoria Shrimp, was described as having four binary features (e.g. , A high quantity of ACh neurotransmitter. , Long-lasting flight response. , Accelerated sleep cycle. , etc.) and causal relationships among those features (e.g. , "A high quantity of ACh neurotransmitter causes a long-lasting flight response. The duration of the electrical signal to the muscles is longer because of the excess amount of neurotransmitter. "). Participants first studied several computer screens of information about their assigned category at their own pace. All participants were first presented with the categorys four features. Participants in the common-cause condition were additionally instructed on the common-cause causal relationships (F 1-;' F2 , F 1-;' F3 , F 1-;' F 4) , and participants in the common-effect condition were instructed on the common-effect relationships (F 1-;.F4 , F2-;.F4 , F3-;.F4). When ready , participants took a multiple-choice test that tested them on the knowledge they had just studied. Participants were required to retake the test until they committed 0 errors. Participants then performed a classification task in which they rated on a 0-100 scale the category membership of 16 exemplars, consisting of all possible objects that can be formed from four binary features. For example, those participants assigned to learn the Lake Victoria Shrimp category were asked to classify a shrimp that possessed "High amounts of the ACh neurotransmitter," "A normal flight response ," "Accelerated sleep cycle," and "Normal body weight." The order of the test exemplars was randomized for each participant. One hundred and eight University of Illinois undergraduates received course credit for participating in this experiment. They were randomly assigned in equal numbers to the three conditions, and to one of the six experimental categories. Results Categorization ratings for the 16 test exemplars averaged over partIclpants in the common-cause , common-effect, and control conditions are presented in Table 1. The presence of causal knowledge had a large effect on the ratings. For instance, exemplars 0111 and 0001 were given lower ratings in the common-cause and common-effect conditions, respectively (39.4 and 26.3) than in the control condition (67.6 and 67.0) presumably because in these exemplars correlations are broken (effect features are present even though their causes are absent). In contrast, exemplar 1111 received a significantly higher rating in the common-cause and common-effect conditions than in the control condition (90.8 and 9l.0 vs. 75.6) , presumably because in both conditions all correlations are preserved. To confirm that causal schemas induced a sensitivity to interactions between features, categorization ratings were analyzed by performing a multiple regression for each participant. Four predictor variables (f1 , f2, f3 , f4) were coded as -1 if the feature was absent, and + 1 if it was present. An additional six predictor variables were formed from the multiplicative interaction between pairs of features: f12 , f13 , f14 , f24 , f34 , and f23. For those feature pairs connected by a causal relationship the two-way interaction terms represent whether the causal relationship is confirmed (+ 1, cause and effect both present or both absent) or violated (-1 , one present and one absent). Finally , the four three-way interactions (f123 , f124 , f134, and f234) , and the single four-way interaction (f1234) were also included as predictors. Regression weights averaged over participants are presented in Figure 3 as a function of causal schema condition. Figure 3 indicates that the interaction terms corresponding to those feature pairs assigned causal relationships had significantly positive weights in both the common-cause condition (f12 , f13 , f14) , and the common-effect condition (f14, f24 , f34). That is , as predicted (Figure 2) an exemplar was rated a better category member when it preserved expected correlations (cause and effect feature either both present or both absent) , and a worse member when it broke those correlations (one absent and the other present). 12 (a) Common Cause vs. Control 10 • CC Observed .l: 8 Of) '0:; 6 ~ ~ 4 a Control Observed E9 CC Predicted '" 2 '" "'" 0 ~ (2) 12 (b) Common Effect vs. Control 10 .l: 8 Of) • CE Observed '0:; 6 ~ Control Observed ~ 4 a CE Predicted '" '" 2 "'" 0 (2) fl f2 f3 f4 fl2 fl3 fl4 f24 f34 f23 fl23 f124 f134 f234 f1234 Regression Term Figure 3 In addition, it was shown earlier (Figure 2) that because of their common-cause the three effect features in a common-cause schema will be correlated, albeit more weakly than directly-linked features. Consistent with this prediction, in this condition the three two-way interaction terms between the effect features (f24, f34, f23) are greater than those interactions in the control condition. In contrast, the common-effect schema does not imply that the three cause features will be correlated, and in fact in that condition the interactions between the cause attributes (f12, f13, f23) did not differ from those in the control condition (Figure 3). Figure 3 also reveals higher-order interactions among features in the common-effect condition: Weights on interaction terms f124, f134, f234, and f1234 (- 1.6,2.0 , -2.0, and 2.2) were significantly different from those in the control condition. These higher-order interactions arose because a common-effect schema requires only one cause feature to explain the presence of the common effect. Figures 7b presents the logarithm of the ratings in the common-effect condition for those test exemplars in which the common effect is present as a function of the number of cause features 4.5 bO 4.0 .= 'ill ~ 3.5 Of) 0 .....l 3.0 2.5 o • Observed (CC Present) 2 3 # of Effects o Figure 4 Observed (CE Present) Predicted 2 3 # of Causes present. Ratings increased more with the introduction of the first cause as compared to subsequent causes. That is, participants considered the presence of at least one cause explaining the presence of the common-effect to be sufficient grounds to grant an exemplar a relatively high category membership rating in a common-effect category. In contrast, Figure 7a shows a linear increase in (the logarithm of) categorization ratings for those exemplars in which the common cause is present as a function of the number of effect features. In the presence of the common cause each additional effect produced a constant increment to log categorization ratings. Finally , Figure 3 also indicates that the simple feature weights differed as a function of causal schema. In the common-cause condition, the common-cause (f1) carried greater weight than the three effects (f2, f3 , f4). In contrast, in the common-effect condition it was the common-effect (f4) that had greater weight than the three causes (f1 , f2, f3). That is , causal networks promote the importance of not only specific feature combinations, but the importance of individual features as well. Model Fitting To assess whether CMT accounts for the patterns of classification found in this experiment, the causal models of Figure 1 were fitted to the category membership ratings of each participant in the common-cause and common-effect conditions, respectively. That is , the ratings were predicted from the equation, Rating (X) = K ¥ Likelihood (X; c, m, b) where Likelihood (X; c, m, b) is the likelihood of exemplar X as a function of c, m , and b. The likelihood equations for the common-cause and common-effect models shown in Table 1 were used for common-cause and common-effect participants, respectively. K is a scaling constant that brings the likelihood into the range 0-100. For each participant, the values for parameters K , c, m, and b that minimized the squared deviation between the predicted and observed ratings was computed. The best fitting values for parameters K , c, m , and b averaged over participants were 846, .578 , .214, and .437 in the common-cause condition, and 876, .522, .325 , and .280 in the common-effect condition. The predicted ratings for each exemplar are presented in Table 1. The significantly positive estimate for m in both conditions indicates that participants categorization performance was consistent with them assuming the presence of a probabilistic causal mechanisms linking category features. Ratings predicted by CMT did not differ from observed ratings according to chi-square tests: )(\16)=3.0 for common cause, )(\16)=5.3 for common-effect. To demonstrate that CMT predicts participants sensitivity to particular combinations of features when categorizing, each participants predicted ratings were subjected to the same regressions that were performed on the observed ratings. The resulting regression weights averaged over participants are presented in Figure 3 superimposed on the weights from the observed data. First, Figure 3 indicates that CMT reproduces participants sensitivity to agreement between pairs of features directly connected by causal relationships (f12 , f13 , f14 in the common-cause condition, and f14 , f24 , f34 in the common-effect condition). That is , according to both CMT and human participants, category membership ratings increase when pairs of features confirm causal laws , and decrease when they violate those laws. Second, Figure 3 indicates that CMT accounts for the interactions between the effect features in the common-cause condition (f12, f13 , f23) and also for the higherorder feature interactions in the common-effect condition (f124 , f134, f234 , f1234) , indicating that that CMT is also sensitive to the asymmetries inherent in causal relationships. The predictions of CMT superimposed on the observed data in Figure 4 confirm that CMT, like the human participants, requires only one cause feature to explain the presence of a common effect (nonlinear increase in ratings in Figure 4b) whereas CMT predicts a linear increase in log ratings as one adds effect features to a common cause (Figure 4a). Finally, CMT also accounts for the larger weight given to the common cause and common-effect features (Figure 3). Discussion The current results support CMTs claims that people have a representation of the probabilistic causal mechanisms that link category features, and that they classify by evaluating whether an objects combination of features was likely to have been generated by those mechanisms. That is , people have models of the world that lead them to expect a certain distribution of features in category members , and consider exemplars good category members to the extent they manifest those expectations. One way this effect manifested itself is in terms of the importance of preserved correlations between features directly connected by causal relationships. An alternative model that accounts for this particular result assumes that the feature space is expanded to include configural cues encoding the confirmation or violation of each causal relationship [6]. However, such a model treats causal links as symmetric and does not consider interactions among links. As a result, it does not fit the common effect data as well as CMT (Figure 4b) , because it is unable to account for categorizers sensitivity to the higher-order feature interactions that emerge as a result of causal asymmetries in a complex network. CMT diverges from traditional models of categorization by emphasizing the knowledge people possess as opposed to the examples they observe. Indeed , the current experiment differed from many categorization studies in not providing examples of category members. As a result, CMT is applicable to the many realworld categories about which people know far more than they have observed first hand (e.g., scientific concepts). Of course, for many other categories people observe category members , and the nature of the interactions between knowledge and observations is an open question of considerable interest. Using the same materials as in the current study, the effects of knowledge and observations have been orthogonally manipulated with the finding that observations had little effect on classification performance as compared to the theories [7]. Thus , theories may often dominate categorization decisions even when observations are available. Acknowledgments Support for this research was provided by funds from the National Science Foundation (Grants Number SBR-98l6458 and SBR 97-20304) and from the National Institute of Mental Health (Grant Number ROl MH58362). References [1] Murphy, G . L., & Medin, D . L. (1985). The role of theories in conceptual coherence. Psychological Review , 92, 289-316. [2] Rehder, B. (1999). A causal model theory of categorization . In Proceedings of the 21st Annual Meeting of the Cognitive Science Society (pp. 595-600). Vancouver. [3] Waldmann , M .R., Holyoak, K.J ., & Fratianne, A. (1995). Causal models and the acquisition of category structure. Journal of Experimental Psychology: General, 124, 181-206. [4] Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Mateo, CA: Morgan Kaufman. [5] Salmon, W. C. (1984). Scientific explanation and the causal structure of the world. Princeton , NJ: Princeton University Press. [6] Gluck, M. A., & Bower, G. H . (1988). Evaluating an adaptive network model of human learning. Journal of Memory and Language, 27,166-195. [7] Rehder, B., & Hastie, R. (2001). Causal knowledge and categories: The effects of causal beliefs on categorization, induction, and similarity. Journal of Experimental Psychology: General, 130, 323-360.
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KLD-Sampling: Adaptive Particle Filters Dieter Fox Department of Computer Science & Engineering University of Washington Seattle, WA 98195 Email: fox@cs.washington.edu Abstract Over the last years, particle filters have been applied with great success to a variety of state estimation problems. We present a statistical approach to increasing the efficiency of particle filters by adapting the size of sample sets on-the-fly. The key idea of the KLD-sampling method is to bound the approximation error introduced by the sample-based representation of the particle filter. The name KLD-sampling is due to the fact that we measure the approximation error by the Kullback-Leibler distance. Our adaptation approach chooses a small number of samples if the density is focused on a small part of the state space, and it chooses a large number of samples if the state uncertainty is high. Both the implementation and computation overhead of this approach are small. Extensive experiments using mobile robot localization as a test application show that our approach yields drastic improvements over particle filters with fixed sample set sizes and over a previously introduced adaptation technique. 1 Introduction Estimating the state of a dynamic system based on noisy sensor measurements is extremely important in areas as different as speech recognition, target tracking, mobile robot navigation, and computer vision. Over the last years, particle filters have been applied with great success to a variety of state estimation problems (see [3] for a recent overview). Particle filters estimate the posterior probability density over the state space of a dynamic system [4, 11]. The key idea of this technique is to represent probability densities by sets of samples. It is due to this representation, that particle filters combine efficiency with the ability to represent a wide range of probability densities. The efficiency of particle filters lies in the way they place computational resources. By sampling in proportion to likelihood, particle filters focus the computational resources on regions with high likelihood, where things really matter. So far, however, an important source for increasing the efficiency of particle filters has only rarely been studied: Adapting the number of samples over time. While variable sample sizes have been discussed in the context of genetic algorithms [10] and interacting particle filters [2], most existing approaches to particle filters use a fixed number of samples during the whole state estimation process. This can be highly inefficient, since the complexity of the probability densities can vary drastically over time. An adaptive approach for particle filters has been applied by [8] and [5]. This approach adjusts the number of samples based on the likelihood of observations, which has some important shortcomings, as we will show. In this paper we introduce a novel approach to adapting the number of samples over time. Our technique determines the number of samples based on statistical bounds on the samplebased approximation quality. Extensive experiments using a mobile robot indicate that our approach yields significant improvements over particle filters with fixed sample set sizes and over a previously introduced adaptation technique. The remainder of this paper is organized as follows: In the next section we will outline the basics of particle filters and their application to mobile robot localization. In Section 3, we will introduce our novel technique to adaptive particle filters. Experimental results are presented in Section 4 before we conclude in Section 5. 2 Particle filters for Bayesian filtering and robot localization Particle filters address the problem of estimating the state of a dynamical system from sensor measurements. The goal of particle filters is to estimate a posterior probability density over the state space conditioned on the data collected so far. The data typically consists of an alternating sequence of time indexed observations and control measurements , which describe the dynamics of the system. Let the belief
denote the posterior at time . Under the Markov assumption, the posterior can be computed efficiently by recursively updating the belief whenever new information is received. Particle filters represent this belief by a set of weighted samples distributed according to
: "!$# &%(' ) +* !-,.,.,.! / Here each 0 is a sample (or state), and the # are non-negative numerical factors called importance weights, which sum up to one. The basic form of the particle filter updates the belief according to the following sampling procedure, often referred to as sequential importance sampling with re-sampling (SISR, see also [4, 3]): Re-sampling: Draw with replacement a random sample 1 324 from the sample set 32 4 according to the (discrete) distribution defined through the importance weights # 32 4 . This sample can be seen as an instance of the belief $ 324.
. Sampling: Use 32 4 and the control information 32 4 to sample 65 from the distribution 7 ' 324 ! 8324
, which describes the dynamics of the system. 195 now represents the density given by the product 7 ' 324 ! 324:
$ 32 4.
. This density is the proposal distribution used in the next step. Importance sampling: Weight the sample ;65 by the importance weight 7 ' 65
, the likelihood of the sample 065 given the measurement . Each iteration of these three steps generates a sample drawn from the posterior belief
. After iterations, the importance weights of the samples are normalized so that they sum up to 1. It can be shown that this procedure in fact approximates the posterior density, using a sample-based representation [4, 2, 3]. Particle filters for mobile robot localization We use the problem of mobile robot localization to illustrate and test our approach to adaptive particle filters. Robot localization is the problem of estimating a robot’s pose relative to a map of its environment. This problem has been recognized as one of the most fundamental problems in mobile robotics [1]. The mobile robot localization problem comes in different flavors. The simplest localization problem is position tracking. Here the initial robot pose is known, and localization seeks to correct small, incremental errors in a robot’s odometry. More challenging is the global localization problem, where a robot is not told its initial pose, but instead has to determine it from scratch. (a) Robot position Start (b) Robot position (c) Robot position (d) Fig. 1: a) Pioneer robot used throughout the experiments. b)-d) Map of an office environment along with a series of sample sets representing the robot’s belief during global localization using sonar sensors (samples are projected into 2D). The size of the environment is 54m 18m. b) After moving 5m, the robot is still highly uncertain about its position and the samples are spread trough major parts of the free-space. c) Even as the robot reaches the upper left corner of the map, its belief is still concentrated around four possible locations. d) Finally, after moving approximately 55m, the ambiguity is resolved and the robot knows where it is. All computation can be carried out in real-time on a low-end PC. In the context of robot localization, the state of the system is the robot’s position, which is typically represented in a two-dimensional Cartesian space and the robot’s heading direction. The state transition probability 7 ' 324 ! 8324
describes how the position of the robot changes using information collected by the robot’s wheel encoders. The perceptual model 7 '
describes the likelihood of making the observation given that the robot is at location . In most applications, measurements consist of range measurements or camera images (see [6] for details). Figure 1 illustrates particle filters for mobile robot localization. Shown there is a map of a hallway environment along with a sequence of sample sets during global localization. In this example, all sample sets contain 100,000 samples. While such a high number of samples might be needed to accurately represent the belief during early stages of localization (cf. 1(a)), it is obvious that only a small fraction of this number suffices to track the position of the robot once it knows where it is (cf. 1(c)). Unfortunately, it is not straightforward how the number of samples can be adapted on-the-fly, and this problem has only rarely been addressed so far. 3 Adaptive particle filters with variable sample set sizes The localization example in the previous section illustrates that the efficiency of particle filters can be greatly increased by changing the number of samples over time. Before we introduce our approach to adaptive particle filters, let us first discuss an existing technique. 3.1 Likelihood-based adaptation We call this approach likelihood-based adaptation since it determines the number of samples such that the sum of non-normalized likelihoods (importance weights) exceeds a prespecified threshold. This approach has been applied to dynamic Bayesian networks [8] and mobile robot localization [5]. The intuition behind this approach can be illustrated in the robot localization context: If the sample set is well in tune with the sensor reading, each individual importance weight is large and the sample set remains small. This is typically the case during position tracking (cf. 1(c)). If, however, the sensor reading carries a lot of surprise, as is the case when the robot is globally uncertain or when it lost track of its position, the individual sample weights are small and the sample set becomes large. The likelihood-based adaptation directly relates to the property that the variance of the importance sampler is a function of the mismatch between the proposal distribution and the distribution that is being approximated. Unfortunately, this mismatch is not always an accurate indicator for the necessary number of samples. Consider, for example, the ambiguous belief state consisting of four distinctive sample clusters shown in Fig. 1(b). Due to the symmetry of the environment, the average likelihood of a sensor measurement observed in this situation is approximately the same as if the robot knew its position unambiguously (cf. 1(c)). Likelihood-based adaptation would therefore use the same number of samples in both situations. Nevertheless, it is obvious that an accurate approximation of the belief shown in Fig. 1(b) requires a multiple of the samples needed to represent the belief in Fig. 1(c). 3.2 KLD-sampling The key idea of our approach is to bound the error introduced by the sample-based representation of the particle filter. To derive this bound, we assume that the true posterior is given by a discrete, piecewise constant distribution such as a discrete density tree or a multidimensional histogram [8, 9]. For such a representation we can determine the number of samples so that the distance between the maximum likelihood estimate (MLE) based on the samples and the true posterior does not exceed a pre-specified threshold . We denote the resulting approach the KLD-sampling algorithm since the distance between the MLE and the true distribution is measured by the Kullback-Leibler distance. In what follows, we will first derive the equation for determining the number of samples needed to approximate a discrete probability distribution (see also [12, 7]). Then we will show how to modify the basic particle filter algorithm so that it realizes our adaptation approach. To see, suppose that samples are drawn from a discrete distribution with different bins. Let the vector 4 !.,-,.,.!
denote the number of samples drawn from each bin. is distributed according to a multinomial distribution, i.e. Multinomial ! 7
, where 7 7 4 ,-,., 7 specifies the probability of each bin. The maximum likelihood estimate of 7 is given by 7 & 24 . Furthermore, the likelihood ratio statistic for testing 7 is
5 4 5 7 5 7 5 " 5 4 7 5
7 5 7 5 , (1) When 7 is the true distribution, the likelihood ratio converges to a chi-square distribution: ! 24 as "$# (2) Please note that the sum in the rightmost term of (1) specifies the K-L distance % 7 ! 7
between the MLE and the true distribution. Now we can determine the probability that this distance is smaller than , given that samples are drawn from the true distribution: &(' % 7 ! 7
*),+
&-' (% 7 ! 7
*) +:
, & . 24 ) +
(3) The second step in (3) follows by replacing (% 7 ! 7
with the likelihood ratio statistic, and by the convergence result in (2). The quantiles of the chi-square distribution are given by & 24 ) 240/ 4 221
+*4365 , (4) Now if we choose such that + is equal to 240/ 4:2 1 , we can combine (3) and (4) to get & ' % 7 ! 7
*),+:
, *4375 , (5) This derivation can be summarized as follows: If we choose the number of samples as * + . 240/ 4 221 ! (6) then we can guarantee that with probability * 3 5 , the K-L distance between the MLE and the true distribution is less than . In order to determine according to (6), we need to compute the quantiles of the chi-square distribution. A good approximation is given by the Wilson-Hilferty transformation [7], which yields * + 240/ 4 221 , 3 * + *43 3 *
3 *
4 221 ! (7) where 4 221 is the upper *4365 quantile of the standard normal
! *
distribution. This concludes the derivation of the sample size needed to approximate a discrete distribution with an upper bound on the K-L distance. From (7) we see that the required number of samples is proportional to the inverse of the bound, and to the first order linear in the number of bins with support. Here we assume that a bin of the multinomial distribution has support if its probability is above a certain threshold. This way the number will decrease with the certainty of the state estimation 1. It remains to be shown how to apply this result to particle filters. The problem is that we do not know the true posterior distribution (the estimation of this posterior is the main goal of the particle filter). Fortunately, (7) shows that we do not need the complete discrete distribution but that it suffices to determine the number of bins with support. However, we do not know this quantity before we actually generate the distribution. Our approach is to estimate by counting the number of bins with support during sampling. To be more specific, we estimate for the proposal distribution 7 ' 32 4 ! 32 4
$ 324
resulting from the first two steps of the particle filter update. The determination of can be done efficiently by checking for each generated sample whether it falls into an empty bin or not. Sampling is stopped as soon as the number of samples exceeds the threshold specified in (7). An update step of the resulting KLD-sampling particle filter is given in Table 1. The implementation of this modified particle filter is straightforward. The only difference to the original algorithm is that we have to keep track of the number of supported bins. The bins can be implemented either as a fixed, multi-dimensional grid, or more efficiently as tree structures [8, 9]. Please note that the sampling process is guaranteed to terminate, since for a given bin size , the maximum number of bins is limited. 4 Experimental results We evaluated our approach using data collected with one of our robots (see Figure 1). The data consists of a sequence of sonar scans and odometry measurements annotated with timestamps to allow systematic real-time evaluations. In all experiments we compared our KLDsampling approach to the likelihood-based approach discussed in Section 3.1, and to particle filters with fixed sample set sizes. Throughout the experiments we used different parameters for the three approaches. For the fixed approach we varied the number of samples, for the likelihood-based approach we varied the threshold used to determine the number of samples, and for our approach we varied , the bound on the K-L distance. In all experiments, we used a value of 0.99 for 5 and a fixed bin size of 50cm 50cm 10deg. We limited the maximum number of samples for all approaches to 100,000. 1This need for a threshold to determine (and to make vary over time) is not particularly elegant. However, it results in an efficient implementation that does not even depend on the value of the threshold itself (see next paragraph). We also implemented a version of the algorithm using the complexity of the state space to determine the number of samples. Complexity is measured by , where is the entropy of the distribution. This approach does not depend on thresholding at all, but it does not have a guarantee of approximation bounds and does not yield significantly different results. Inputs: 32 4 32 4 ! # 324 % ' ) +* !.,-,., ! / representing belief $ 324.
, control measurement 324 , observation , bounds and 5 , bin size 6 ! ! ! /* Initialize */ do /* Generate samples ,.,., */ Sample an index
from the discrete distribution given by the weights in 324 Sample 0 from 7 ' 324 ! 32 4
using 65 32 4 and 832 4 # 6 7 ' 0
; /* Compute importance weight */ 6 # /* Update normalization factor */ 6 !$# % / /* Insert sample into sample set */ if falls into empty bin ) then /* Update number of bins with support */ 9 * 9 non-empty 6" * /* Update number of generated samples */ while 4
240/ 4 221
/* ,-,., until K-L bound is reached */ for ) 6 * !.,-,., ! do /* Normalize importance weights */ # 9 # return Table 1: KLD-sampling algorithm. Approximation of the true posterior In the first set of experiments we evaluated how accurately the different methods approximate the true posterior density. Since the ground truth for these posteriors is not available, we compared the sample sets generated by the different approaches with reference sample sets. These reference sets were generated using a particle filter with a fixed number of 200,000 samples (far more than actually needed for position estimation). After each iteration, we computed the K-L distance between the sample sets and the corresponding reference sets, using histograms for both sets. Note that in these experiments the time-stamps were ignored and the algorithms was given as much time as needed to process the data. Fig. 2(a) plots the average K-L distance along with 95% confidence intervals against the average number of samples for the different algorithms (for clarity, we omitted the large error bars for KL distances above 1.0). Each data point represents the average of 16 global localization runs with different start positions of the robot (each run itself consists of approximately 150 sample set comparisons at the different points in time). As expected, the more samples are used, the better the approximation. The curves also illustrate the superior performance of our approach: While the fixed approach requires about 50,000 samples before it converges to a KL distance below 0.25, our approach converges to the same level using only 3,000 samples on average. This is also an improvement by a factor of 12 compared to the approximately 36,000 samples needed by the likelihood-based approach. In essence, these experiments indicate that our approach, even though based on several approximations, is able to accurately track the true posterior using significantly smaller sample sets on avarage than the other approaches. Real-time performance Due to the computational overhead for determining the number of samples, it is not clear that our approach yields better results under real-time conditions. To test the performance of our approach under realistic conditions, we performed multiple global localization experiments under real-time considerations using the timestamps in the data sets. Again, the −0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 20000 40000 60000 80000 100000 Fixed sampling Likelihood−based adaptation KLD−sampling KL distance Average number of samples (a) 0 50 100 150 200 0 20000 40000 60000 80000 Fixed sampling Likelihood−based adaptation KLD−sampling Average number of samples Localization error [cm] (b) Fig. 2: The -axis represents the average sample set size for different parameters of the three approaches. a) The -axis plots the K-L distance between the reference densities and the sample sets generated by the different approaches (real-time constraints were not considered in this experiment). b) The -axis represents the average localization error measured by the distance between estimated positions and reference positions. The U-shape in b) is due to the fact that under real-time conditions, an increasing number of samples results in higher update times and therefore loss of sensor data. different average numbers of samples for KLD-sampling were obtained by varying the bound. The minimum and maximum numbers of samples correspond to -bounds of 0.4 and 0.015, respectively. As a natural measure of the performance of the different algorithms, we determined the distance between the estimated robot position and the corresponding reference position after each iteration. 2 The results are shown in Fig. 2(b). The U-shape of all three graphs nicely illustrates the trade-off involved in choosing the number of samples under real-time constraints: Choosing not enough samples results in a poor approximation of the underlying posterior and the robot frequently fails to localize itself. On the other hand, if we choose too many samples, each update of the algorithm can take several seconds and valuable sensor data has to be discarded, which results in less accurate position estimates. Fig. 2(b) also shows that even under real-time conditions, our KLD-sampling approach yields drastic improvements over both fixed sampling and likelihood-based sampling. The smallest average localization error is 44cm in contrast to an average error of 79cm and 114cm for the likelihood-based and the fixed approach, respectively. This result is due to the fact that our approach is able to determine the best mix between more samples during early stages of localization and less samples during position tracking. Due to the smaller sample sets, our approach also needs significantly less processing power than any of the other approaches. 5 Conclusions and Future Research We presented a statistical approach to adapting the sample set size of particle filters onthe-fly. The key idea of the KLD-sampling approach is to bound the error introduced by the sample-based belief representation of the particle filter. At each iteration, our approach generates samples until their number is large enough to guarantee that the K-L distance between the maximum likelihood estimate and the underlying posterior does not exceed a prespecified bound. Thereby, our approach chooses a small number of samples if the density is focused on a small subspace of the state space, and chooses a large number of samples if the samples have to cover a major part of the state space. Both the implementational and computational overhead of this approach are small. Extensive experiments using mobile robot localization as a test application show that our approach yields drastic improvements over particle filters with fixed sample sets and over a previously introduced adaptation approach [8, 5]. In our experiments, KLD-sampling yields bet2Position estimates are extracted using histograming and local averaging, and the reference positions were determined by evaluating the robot’s highly accurate laser range-finder information. ter approximations using only 6% of the samples required by the fixed approach, and using less than 9% of the samples required by the likelihood adaptation approach. So far, KLDsampling has been tested using robot localization only. We conjecture, however, that many other applications of particle filters can benefit from this method. KLD-sampling opens several directions for future research. In our current implementation we use a discrete distribution with a fixed bin size to determine the number of samples. We assume that the performance of the filter can be further improved by changing the discretization over time, using coarse discretizations when the uncertainty is high, and fine discretizations when the uncertainty is low. Our approach can also be extended to the case where in certain parts of the state space, highly accurate estimates are needed, while in other parts a rather crude approximation is sufficient. This problem can be addressed by locally adapting the discretization to the desired approximation quality using multi-resolution tree structures [8, 9] in combination with stratified sampling. As a result, more samples are used in “important” parts of the state space, while less samples are used in other parts. Another area of future research is the thorough investigation of particle filters under real-time conditions. In many applications the rate of incoming sensor data is higher than the update rate of the particle filter. This introduces a trade-off between the number of samples and the amount of sensor data that can be processed (cf. 2(b)). In our future work, we intend to address this problem using techniques similar to the ones introduced in this work. Acknowledgments The author wishes to thank Jon A. Wellner and Vladimir Koltchinskii for their help in deriving the statistical background of this work. Additional thanks go to Wolfram Burgard and Sebastian Thrun for their valuable feedback on early versions of the technique. References [1] I. J. Cox and G. T. Wilfong, editors. Autonomous Robot Vehicles. Springer Verlag, 1990. [2] P. Del Moral and L. Miclo. Branching and interacting particle systems approximations of feynamkac formulae with applications to non linear filtering. In Seminaire de Probabilites XXXIV, number 1729 in Lecture Notes in Mathematics. Springer-Verlag, 2000. [3] A. Doucet, N. de Freitas, and N. Gordon, editors. Sequential Monte Carlo in Practice. SpringerVerlag, New York, 2001. [4] A. Doucet, S.J. Godsill, and C. Andrieu. On sequential monte carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3), 2000. [5] D. Fox, W. Burgard, F. Dellaert, and S. Thrun. Monte Carlo Localization: Efficient position estimation for mobile robots. In Proc. of the National Conference on Artificial Intelligence (AAAI), 1999. [6] D. Fox, S. Thrun, F. Dellaert, and W. Burgard. Particle filters for mobile robot localization. In Doucet et al. [3]. [7] N. Johnson, S. Kotz, and N. Balakrishnan. Continuous univariate distributions, volume 1. John Wiley & Sons, New York, 1994. [8] D. Koller and R. Fratkina. Using learning for approximation in stochastic processes. In Proc. of the International Conference on Machine Learning (ICML), 1998. [9] A. W. Moore, J. Schneider, and K. Deng. Efficient locally weighted polynomial regression predictions. In Proc. of the International Conference on Machine Learning (ICML), 1997. [10] M. Pelikan, D.E. Goldberg, and E. Cant-Paz. Bayesian optimization algorithm, population size, and time to convergence. In Proc. of the Genetic and Evolutionary Computation Conference (GECCO), 2000. [11] M. K. Pitt and N. Shephard. Filtering via simulation: auxiliary particle filters. Journal of the American Statistical Association, 94(446), 1999. [12] J.A. Rice. Mathematical Statistics and Data Analysis. Duxbury Press, second edition, 1995.
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Reinforcement Learning and Time Perception a Model of Animal Experiments J. L. Shapiro Department of Computer Science University of Manchester Manchester, M13 9PL U.K. jls@cs.man.ac.uk Abstract John Wearden Department of Psychology University of Manchester Manchester, M13 9PL U.K. Animal data on delayed-reward conditioning experiments shows a striking property the data for different time intervals collapses into a single curve when the data is scaled by the time interval. This is called the scalar property of interval timing. Here a simple model of a neural clock is presented and shown to give rise to the scalar property. The model is an accumulator consisting of noisy, linear spiking neurons. It is analytically tractable and contains only three parameters. When coupled with reinforcement learning it simulates peak procedure experiments, producing both the scalar property and the pattern of single trial covariances. 1 Introduction An aspect of delayed-reward reinforcement learning problem which has a long history of study in animal experiments, but has been overlooked by theorists, is the learning of the expected time to the reward. In a number of animal experiments, animals need to wait a given time interval after a stimulus before performing an action in order to receive the reward. In order to be able to do this, the animal requires an internal clock or mechanism for perceiving time intervals, as well as a learning system which can tackle more familiar aspects of delayed reward reinforcement learning problem. In this paper it is shown that a simple connectionist model of an accumulator used to measure time duration, coupled to a standard TD('\) reinforcement learning rule reproduces the most prominent features of the animal experiments. The reason it might be desirable for a learner to learn the expected time to receive a reward is that it allows it to perform the action for an appropriate length of time. An example described by Grossberg and Merrill [4] and modeled in animal experiments by Gibbon and Church [3] is foraging. An animal which had no sense of the typical time to find food might leave too often, thereby spending an inordinate amount of time flying between patches. Alternatively it could remain in a depleted patch and starve. The ability to learn times to rewards is an important aspect of intelligent behavior more generally. 1.1 Peak Procedure Experiments A typical type of experiment which investigates how animals learn the time between stimulus and reward is the peak procedure. In this, the animal is trained to respond after a given time interval tr has elapsed. Some stimulus (e.g. a light) is presented which stays on during the trial. The animal is able to respond at any time. The animal receives a reward for the first response after the length of time tr . The trial ends when the animal receives the reward. On some trials, however, no reward is given even when the animal responds appropriately. This is to see when the animal would stop responding. What happens in non-reward trials is that the animal typically will start responding at a certain time, will respond for a period, and then stop responding. Responses averaged over many trials, however, give a smooth curve. The highest response is at the time interval tr , and there is variation around this. The inaccuracy in the response (as measured by the standard deviation in the average response curves for non-reward trials) is also proportional to the time interval. In other words, the ratio of the standard deviation to the mean response time (the coefficient of variation) is a constant independent of the time interval. A more striking property of the timing curves is scalar property, of which the above are two consequences. When the average response rate for non-reward trials is multiplied by the time interval and plotted against the relative time (time divided by the time interval) the data from different time intervals collapse onto one curve. This strong form of the scalar property can be expressed mathematically as follows. Let T be the actual time since the start of the trial and T be subjective time. Subjective time is the time duration which the animal perceives to have occurred, (or at least appears to perceive judging from its behavior). The experiments show that T varies for a given T. This variation can be expressed as a conditional probability, the probability of acting as though the time is T given that the actual time is T, which is written P(TIT). The fact that the data collapses implies this probability depends on T and T in a special way, (1) Here Pinv is the function which describes the shape of the scaled curves. Thus, time acts as a scale factor. This is a strong and striking result. This has been seen in many species, including rats, pigeons, turtles; humans will show similar results if the time intervals are short or if they are prevented from counting through distracting tasks. For reviews of interval timing phenomena, see [5] and [3] . A key question which remains unanswered is: what is the origin of the scalar property. Since the scalar property is ubiquitous, it may be revealing something fundamental about the nature of an internal clock or time perception system. This is especially true if there are only a few known mechanisms which generate this phenomenon. It is well known that any model based on the accumulation of independent errors, such as a clock with a variable pulse-rate, does not produce the scalar property. In such a model it would be the ratio of the variance to the mean response time which would be independent of the time interval (a consequence of the law of large numbers). In section 2, a simple stochastic process will be presented which gives rise to scalar timing. In section 3 simulations of the model on the peak procedure are presented. The model reproduces experimental results on the mean responses and the covariation between responses on non-reward trials. 2 The model 2.1 An accumulator network of spiking neurons Here it is shown that a simple connectionist model of an accumulator can give rise to the strong scalar property. The network consists of noisy, linear, spiking neurons which are connected in a random, spatially homogeneous way. The network encodes time as the total activity in the network which grows during the measured time interval. Psychological aspects of the model will be presented elsewhere [8] The network consists of N identical neurons. The connectivity between neurons is random and defined by a connection matrix Cij which is random and sparse. The connection strength is the same between all connected neurons. An important parameter is the fan-out of the ith neuron Ci ; its average across the network is denoted C. Time is in discrete units of size T, the time required for a spike produced by a neuron to invoke a spike in a connected neuron. There is no refractory period. The neurons are linear the expected number of spikes produced by a neuron is "( times the number of pre-synaptic spikes. Let ai(t) denote the number of spikes produced by neuron i at time t. This obeys hi(t) ai(t + T) = L Va + Ii(t), (2) a=l where hi(t) is the number of spikes feeding into neuron i, hi(t) = E j CjiXj(t). Ii(t) is the external input at i, and V is a random variable which determines whether a pre-synaptic spike invokes one in a connected neuron. The mean of v is "( and the variance is denoted a~. So the spikes behave independently; saturation effects are ignored. The total activity of the network is N n(t) = L ai(t). (3) i = l At each time-step, the number of spikes will grow due to the fan-out of the neurons. At the same time, the number of spikes will shrink due to the fact that a spike invokes another spike with a probability less than 1. An essential assumption of this work is that these two processes balance each other, C"( = 1. Finally, in order for this network to act as an accumulator, it receives statistically stationary input during the time interval which is being measured, so I(t) is only present during the measured interval and statistically stationary then. 2.2 Derivation of the strong scalar property Here it is shown that the network activity obeys equation (1). Let y be the scaled network activity, y(t) = n(t)/t. (4) The goal here is the derive the probability distribution for y as a function of time, P(ylt). In order to do this, we use the cumulant generating function (or characteristic function). For any probability distribution, p(x), the generating function for cumulants is, G(8) (5) (6) where n is the domain of p(x), "'i is the ith cumulant of p(x), and 8 is just a dummy variable. Taking the nth derivative of G(8) with respect to 8 and setting 8 to 0 gives "'i. Cumulants are like moments, see [1] for some definitions and properties. We will derive a recursion relation for the cumulant generating function for y(t), denoted Gy (8; t). Let G y (8) denote the generating function for the distribution of v and G [(8) denote the generating functions for the distribution of inputs I(t). These latter two are assumed to be stationary, hence there is no time-dependence. From equation 2 it follows that, Gy(8;t+T) = G[C:T)+~LGy[tCiGYC:T);t]. (7) • In deriving the above, it was assumed that the activity at each node is statistically the same, and that the fan-out at i is uncorrelated with the activity at i (this requires a sufficiently sparsely connectivity, i.e. no tight loops). Differentiating the last equation n times with respect to 8 and setting 8 to zero produces a set recursion relations for the cumulants of y, denoted "'n. It is necessary to take terms only up to first order in lit to find the fixed point distribution. The recursion relations to this order are ( T) m[ 1-- "'l(t)+-t t+ T (8) ( T) 1n(n-1) 2 1 - n"'n(t) + C(J"y"'n-l(t) t t 2 + 0 C~) ;n > 1. (9) The above depends upon the mean total input activity m[ == G~(O) the average fan-out C, and the variance in the noise v, (J"~ == G~(O). In general it would depend upon the fan-out times the mean of the noise v, but that is 1 by assumption. Higher order statistics in C and v only contribute to terms which are higher order in lit. The above equations converge to a fixed point, which shows that n(t)lt has a timeindependent distribution for large t. The fixed point is found to be ~ 8n 2m[ ( (J"~) Gy (8,00) = ~ ,"'n(OO) = -2 log 1- 28 . n=O n. (J" T Equation 10 is the generating function for a gamma distribution, R (I b) = exp( -xlb)xa- 1 r x a, bar(a) with 2m[ a = C 2; (J"y Corrections to the fixed point are O(l/t). b = C(J"~. 2T (10) (11) (12) What this shows is that for large t, the distribution of neural activity, n is scalar, P(nlt) = ~ Pr (~ la, b) ; (13) with a and b defined above. 2.3 Reinforcement learning of time intervals The above model represents a way for a simple connectionist system to measure a time interval. In order to model behavior, the system must learn to association the external stimulus and the clock with the response and the reward. To do this, some additional components are needed. The ith stimulus is represented by a signal Si. The output of the accumulator triggers a set of clock nodes which convert the quantity or activity encoding of time used by the accumulator into a "spatial code" in which particular nodes represent different network activities. This was done because it is difficult to use the accumulator activity directly, as this takes a wide range of values. Each clock node responds to a particular accumulator activity. The output of the ith clock node at time t is denoted Xi(t) ; it is one if the activity is i, zero otherwise. It would be more reasonable to use a coarse coding, but this fine-grained encoding is particularly simple. The components of the learning model are shown schematically in figure 1. Vj(t) Stimulus si -----I Figure 1: The learning model. The accumulator feeds into a bank of clock nodes, Xi , which are tuned to accumulator activities. The response Vj is triggered by simultaneous presence of both the stimulus Si and the appropriate clock node. Solid lines denote weights which are fixed; dashed lines show weights which learn according to the TD(A) learning rule. The stimulus and the clock nodes feed into response nodes. The output of the jth response node, Vj(t) is given by (14) Here () is a threshold, Aij is the association between the stimulus and the response, and Wij is the association between a clock node and the response. Both the stimulus and the appropriate clock node must be present in order for there to be a reasonable probability of a response. The response probability is Vj (t) , unless that is negative, in which case there is no response, or is greater than 1, in which case there is definitely a response. Both Aij and the w's learn via a TD-A learning rule. TD-A is an important learning rule for modeling associative conditioning; it has been used to model aspects of classical conditioning including Pavlovian conditioning and blocking. For example, a model which is very effective at modeling Pavlovian eye-blink experiments and other classical conditioning results has been proposed by Moore et. al. [6] building on the model of Sutton, Barto, and Desmond (see description in [7]). This model represents time using a tapped delay line; at each time-step, a different node in the delay line is activated. Time acts as one of the conditioned stimuli. The conditioned stimsing temporal difference (TD) reinforcement learning is associated with the response through the unconditioned stimulus. These authors did not attempt to model the scalar property, and in their model time is represented accurately by the system. The model presented here is similar to these models. The clock nodes play the role of the tapped delay-line nodes in that model. However, here they are stimulated by the accumulator rather than each other, and they will follow a stochastic trajectory due to the fluctuating nature of the accumulator The learning rule for Wij couples to an "eligibility trace" for the clock nodes Xi(t) which takes time to build up and decays after the node is turned off. They obey the following equations, (15) The standard TD-A learning parameters, "( and A are used, see [9]. The learning equations are t:.Wij t:.Aij 8(t) a8(t + T)Xi(t), a8(t + T)Si' R(t) + "( Vj(t) - Vj(t - T). (16) (17) (18) Here a is a learning rate, 8 is the temporal difference component, R(t) is the reinforcement. The outputs Vj at both times use the current value of the weights. The threshold is set to a constant value (-1 in the simulations). It would make no difference if a eligibility trace were used for the stimulus Si, because that was held on during the learning. 3 Simulations The model has been used to simulate peak procedure. In the simulations, the model is forced to respond for the first set of trials (50 trials in the simulations); otherwise the model would never respond. This could represent shaping in real experiments. After that the model learns using reward trials for an additional number of trials (150 trials in these simulations). The system is then run for 1000 trials, every 10th trial is a non-reward trial; the system continues to learn during these trials. Figure 2 shows average over non-reward trials for different time intervals. The scalar property clearly holds. Gibbon and Church [3] have argued that the covariation between trials is a useful diagnostic to distinguish models of scalar timing. The methodology which they proposed is to fit the results of single non-reward trials from peak procedure experiments to a break-run-break pattern of response The animal is assumed to respond at a low rate until a start time is reached. The animal then responds at a high rate until a stop time is reached, whence it returns to the low response rate. The covariation between the start and stop times between trials is measured and compared to those predicted by theory. The question Gibbon and Church asked was, how does the start and stop time covary across trials. For example, if the animal starts responding early, does it stop Q) ~ O .5 cd l-< CJ) 0.4 rn .::: 0 0.3 P. rn Q) $-.( 0.2 time time/tr Figure 2: Left) Average response of the spatially encoded network for non-reward trials. The accumulator parameters are: mI = 10, Cu2 = 1 (Poisson limit); learning parameters are "( = 0.75, A = 1, learning rate 0: is 0.5. Right) Relative time plotted against response rate times time interval for reinforcement times of 40T, 80T, 160T, 240T, and 320T. All experiments are averages over 100 non-reward trials, which were every 10 trial in 1000 learning trials. responding early, as though it has a shifted estimate of the time interval? Or does it stop responding late, as though it has a more liberal view about what constitutes the particular interval. The covariance between start and stop parameters addresses this question. Comparable experiments can be carried out on the model proposed here. The procedure used is described in [2]. Figure 3 shows a comparison with data from reference [2] with simulations. The pattern of covariation found in the simulations is qualitatively similar to that of the animal data. The interesting quantity is the correlation between the start time and the spread (difference between stop and start times). This is negative in both. 0.5 0.5 I~ I~ I -0.5 '---~-____:-____:-___:_-___:_-___:_----' -0.5 '----~____:,----____:-___:_-___:_-___:_----' Figure 3: Left) Covariances across individual trials in experiments on rats. Data is taken from Table 2 of reference [2] averaged over the four conditions. The covariances are shown in the following order: 1. start-stop, 2. start-spread, 3. spreadmiddle, 4. start-middle, 5. stop-spread, 6. stop-middle. The black, gray, and white bars are for times of reinforcement tr of 15,30, and 60 seconds respectively. Right) Covariances across individual trials simulated by the model. The reinforcement times are 40T, 80T, and 160T. The covariances are given in the same order as in left figure. 4 Conclusion Previous models of interval timing fail to explain its most striking feature the collapse of the data when scaled by the time interval. We have presented a simple model of an accumulator clock based on spiking, noisy, linear neurons which produces this effect. It is a simple model, analytically tractable, based on a driven branching process. The parameters are: T the time for a spike on one neuron to excite spikes on connected neurons, mI the average number of spikes excited externally at each short time interval T, and the variance of the spike transmission process, which in this model is (}"~. A weakness of this model is that it requires fine-tuning of a pair of parameters, so that the expected number of spikes grows in with external excitation only. Once a scalar clock is produced, simple reinforcement learning can be used to associate the clock signal with appropriate responses. A set of intermediate clock nodes was used to encode time. TD-'\ reinforcement learning between the intermediate nodes at reinforcement and an eligibility trace simulates peak procedure and the individual trial covariances. References [1] M. Abramowitz and 1. A. Stegun, editors. Handbook of Mathematical Functions. New York: Dover Publications, 1967. [2] Russell M. Church, Walter H. Meck, and John Gibbon. Application of scalar timing theory to individual trials. Journal of Experimental Psychology - Animal Behavior Processes, 20(2):135- 155, 1994. [3] John Gibbon and Russell M. Church. Representation of time. Cognition, 37:23- 54, 1990. [4] Stephen Grossberg and John W. L. Merrill. A neural network model of adaptively timed reinforcement learning and hippocampal dynamics. Cognitive Brain Research, 1:3- 38, 1992. [5] S. C. Hinton and W . H. Meck. How time flies: Functional and neural mechansims of interval timing. In C. M. Bradshaw and E. Szadabi, editors, Tim e and Behaviour: Psychological and Neurobehavioural Analyses. Amsterdam: Elsevier Science, 1997. [6] J. W. Moore, J. E. Desmond, and N. E. Berthier. Adaptively timed conditioned responses and the cerebellum: A neural network approach. Biological Cybernetics, 62:17- 28, 1989. [7] John W. Moore, Neil D. Berthier, and Diana E. J. Blazis. Classical eye-blink conditioning: Brain systems and implementation of a computational model. In Michael Gabriel and John Moore, editors, Learning and Computational Neuroscience: Foundations of Adaptive Networks, A Bradford Book, pages 359- 387. The MIT Press, 1990. [8] J. L. Shapiro and John Wearden. Modelling scalar timing by an accumulator network of spiking neurons. In preparation, 200l. [9] Richard S. Sutton and Andrew G. Barto. Reinforcment Learning: An Introduction. A Bradford Book. The MIT Press, 1998.
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Unsupervised Learning of Human Motion Models Yang Song, Luis Goncalves, and Pietro Perona California Institute of Technology, 136-93, Pasadena, CA 9112 5, USA yangs,luis,perona @vision.caltech.edu Abstract This paper presents an unsupervised learning algorithm that can derive the probabilistic dependence structure of parts of an object (a moving human body in our examples) automatically from unlabeled data. The distinguished part of this work is that it is based on unlabeled data, i.e., the training features include both useful foreground parts and background clutter and the correspondence between the parts and detected features are unknown. We use decomposable triangulated graphs to depict the probabilistic independence of parts, but the unsupervised technique is not limited to this type of graph. In the new approach, labeling of the data (part assignments) is taken as hidden variables and the EM algorithm is applied. A greedy algorithm is developed to select parts and to search for the optimal structure based on the differential entropy of these variables. The success of our algorithm is demonstrated by applying it to generate models of human motion automatically from unlabeled real image sequences. 1 Introduction Human motion detection and labeling is a very important but difficult problem in computer vision. Given a video sequence, we need to assign appropriate labels to the different regions of the image (labeling) and decide whether a person is in the image (detection). In [8, 7], a probabilistic approach was proposed by us to solve this problem. To detect and label a moving human body, a feature detector/tracker (such as corner detector) is first run to obtain the candidate features from a pair of frames. The combination of features is then selected based on maximum likelihood by using the joint probability density function formed by the position and motion of the body. Detection is performed by thresholding the likelihood. The lower part of Figure 1 depicts the procedure. One key factor in the method is the probabilistic model of human motion. In order to avoid exponential combinatorial search, we use conditional independence property of body parts. In the previous work[8, 7], the independence structures were hand-crafted. In this paper, we focus on the the previously unresolved problem (upper part of Figure 1): how to learn the probabilistic independence structure of human motion automatically from unlabeled training data, meaning that the correspondence between the candidate features and the parts of the object is unknown. For example when we run a feature detector (such as LucasTomasi-Kanade detector [10]) on real image sequences, the detected features can be from Presence of Human? Localization of parts? Feature detector/ tracker Detection and Labeling Unlabeled Training Data Probabilistic Model of Human Motion Unsupervised Learning algorithm Feature detector/ tracker Testing: two frames Figure 1: Diagram of the system. target objects and background clutter with no identity attached to each feature. This case is interesting because the candidate features can be acquired automatically. Our algorithm leads to systems able to learn models of human motion completely automatically from real image sequences - unlabeled training features with clutter and occlusion. We restrict our attention to triangulated models, since they both account for much correlation between the random variables that represent the position and motion of each body part, and they yield efficient algorithms. Our goal is to learn the best triangulated model, i.e., the one that reaches maximum likelihood with respect to the training data. Structure learning has been studied under the graphical model (Bayesian network) framework ([2, 4, 5, 6]). The distinguished part of this paper is that it is an unsupervised learning method based on unlabeled data, i.e., the training features include both useful foreground parts and background clutter and the correspondence between the parts and detected features are unknown. Although we work on triangulated models here, the unsupervised technique is not limited to this type of graph. This paper is organized as follows. In section 2 we summarize the main facts about the triangulated probability model. In section 3 we address the learning problem when the training features are labeled, i.e., the parts of the model and the correspondence between the parts and observed features are known. In section 4 we address the learning problem when the training features are unlabeled. In section 5 we present some experimental results. 2 Decomposable triangulated graphs Discovering the probability structure (conditional independence) among variables is important since it makes efficient learning and testing possible, hence some computationally intractable problems become tractable. Trees are good examples of modeling conditional (in)dependence [2, 6]. The decomposable triangulated graph is another type of graph which has been demonstrated to be useful for biological motion detection and labeling [8, 1]. A decomposable triangulated graph [1] is a collection of cliques of size three, where there is an elimination order of vertices such that when a vertex is deleted, it is only contained in one triangle and the remaining subgraph is again a collection of triangles until only one triangle left. Decomposable triangulated graphs are more powerful than trees since each node can be thought of as having two parents. Similarly to trees, efficient algorithms allow fast calculation of the maximum likelihood interpretation of a given set of data. Conditional independence among random variables (parts) can be described by a decomposable triangulated graph. Let
be the set of parts, and "!$# , &%(')%* , is the measurement for +' . If the joint probability density function ,.-/ "!0 "!21 "!$3)4 can be decomposed as a decomposable triangulated graph, it can be written as,
"!$# %& ')(* +,(.-/(0
' (01
+ ( 2
( 43 ')5+)5-65
')5 2
+5 7
-65 (1) where 8 # 9 # : #<; , &% ' %>= @? ,
8 0 8 1 8BA C9 A : A * , and 8 0 9 0 : 0 4 8 1 C9 1 : 1 4 8 A C9 A : A 4 are the cliques. 8 0 8 1 8 A 4 gives the elimination order for the decomposable graph. 3 Optimization of the decomposable triangulated graph Suppose D
0 1 FE are i.i.d samples from a probability density function, where HG -/ G !0 G ! 3 4 , %JI %LK , are labeled data. We want to find the decomposable triangulated graph M , such that ,.MON D 4 is maximized. ,.MON D 4 is the probability of graph M being the ’correct’ one given the observed data D . Here we use M to denote both the decomposable graph and the conditional (in)dependence depicted by the graph. By Bayes’ rule, ,.MON D 4 ,.DHN M 4 ,.M 4P,.D 4 , therefore if we can assume the priors ,.M 4 are equal for different decompositions, then our goal is to find the structure M which can maximize ,.DHN M 4 . From the previous section, a decomposable triangulated graph M is represented by 8 0 C9 0 : 0 4 8 1 9 1 : 1 4 8 A C9 A : A 4 , then ,.DHN M 4 can be computed as follows, QSR T VU 1 W YX Z\[ 3 ! ] %& ,^
'(1
+,(
-( Z_[ 3 ^
+ 5 7
5 (2) where ` -7a 4 is differential entropy or conditional differential entropy [3] (we consider continuous random variables here). Equation (2) is an approximation which converges to equality for K b c due to the weak Law of Large numbers and definitions and properties of differential entropy [3, 2, 4, 5, 6]. We want to find the decomposition 8 0 9 0 : 0 4 8 1 C9 1 : 1 4 8BA C9 A : A 4 such that the above equation can be maximized. 3.1 Greedy search Though for tree cases, the optimal structure can be obtained efficiently by the maximum spanning tree algorithm [2, 6], for decomposable triangulated graphs, there is no existing algorithm which runs in polynomial time and guarantees to the optimal solution [9]. We develop a greedy algorithm to grow the graph by the property of decomposable graphs. For each possible choice of : A (the last vertex of the last triangle), find the best 9 A which can maximize ? ` ed 5 ef 5 4 , then get the best child of edge 9 A : A 4 as 8 A , i.e., the vertex (part) that can maximize ? ` -/ <g 5 N d 5 f 5 4 . The next vertex is added one by one to the existing graph by choosing the best child of all the edges (legal parents) of the existing graph until all the vertices are added to the graph. For each choice of : A , one such graph can be grown, so there are candidate graphs. The final result is the graph with the highest hji/k ,.DFN M 4 among the graphs. The above algorithm is efficient. The total search cost is ml n? porqts l =u? v 4 o 4 l v 4 4 , which is on the order of rw . The algorithm is a greedy algorithm, with no guarantee that the global optimal solution could be found. Its effectiveness will be explored through experiments. 3.2 Computation of differential entropy - translation invariance In the greedy search algorithm, we need to compute ` <g ( d ( f ( 4 and ` -/ d ( f ( 4 , .% v %r= . If we assume that they are jointly Gaussian, then the differential entropy can be computed by 0 1 hxiyk z|{ 4 G N }N , where I is the dimension and } is the covariance matrix. In our applications, position and velocity are used as measurements for each body part, but humans can be present at different locations of the scene. In order to make the Gaussian assumption reasonable, translations need to be removed. Therefore, we use local coordinate system for each triangle 8 s 9 s : s 4 , i.e., we can take one body part (for example 8 s ) as the origin, and use relative positions for other body parts. More formally, let denote a vector of positions g ( d ( f ( g ( d ( f ( 4 A . Then if we describe positions relative to 8 s , we obtain, d ( ? g ( f ( ? g ( d ( ? g ( f ( ? g ( 4 A . This can be written as , where [12]
, with Z Z . In the greedy search algorithm, the differential entropy of all the possible triplets are needed and different triplets are with different origins. To reduce computational cost, notice that [ ] & [ ] & 3 [ ] & (4) and ! (5) From the above equations, we can first estimate the mean and covariance } of G (including all the body parts and without removing translation), then take the dimensions corresponding to the triangle and use equations (4) and (5) to get the mean and covariance for eg ( d ( f ( 4 . Similar procedure can be applied to pairs (for example, 9 s can be taken as origin for ( 9 s : s )) to achieve translation invariant. 4 Unsupervised learning of the decomposable graph In this section, we consider the case when only unlabeled data are available. Assume we have a data set of K samples D
0 1 HE . Each sample G , %tI&% K , is a group of detected features which contains the target object, but G is unlabeled, which means the correspondence between the candidate features and the parts of the object is unknown. For example when we run a feature detector (such as Lucas-Tomasi-Kanade detector [10]) on real image sequences, the detected features can be from target objects and background clutter with no identity attached to each feature. We want to select the useful composite parts of the object and learn the probability structure from D . 4.1 All foreground parts observed Here we first assume that all the foreground parts are observed for each sample. If the labeling for each G is taken as a hidden variable, then the EM algorithm can be used to learn the probability structure and parameters. Our method was developed from [11], but here we learn the probabilistic independence structure and all the candidate features are with the same type. Let ` G denote the labeling for FG . If FG contains I features, then ` G is an I -dimensional vector with each element taken a value from !
9 M ( 9 M is the background clutter label). The observations for the EM algorithm are D
0 1 E , the hidden variables are "
` G E G$# 0 , and the parameters to optimize are the probability (in)dependence structure (i.e. the decomposable triangulated graph) and parameters for its associated probability density function. We use M to represent both the probability structure and the parameters. If we assume that G s are independent from each other and ` G only depends on G , then the likelihood function to maximize is, % QSR T VU W QSRT VU 1 W & QSR T W ] & QSRT ] ('*),+.'
^ ^ 0/ 1 W 1& Q RT W (6) where ` G # is the ' th possible labeling for FG , and G is the set of all such labelings. Optimization directly over equation (6) is hard, and the EM algorithm solves the optimization problem iteratively. In EM, for each iteration v , we will optimize the function, W % 1 W % # Q RT VU W % 1 U W % # ] & QSRT
^ W % 1
W % # ] & ] ' ) +.' ^ ^ 0/ 1
W % # 3 QSR T
^ ^ 0/ W % ] & ] (' ) +.' 0/ QSRT
^ ^ */ W % (7) where G # is the probability of ` G ` G # given the observation G and the decomposable probability structure M s 0 . For each iteration v , G # is a fixed number for a hypothesis ` G # . G # can be computed as, */ ^ */ 1
W % # 7 ^ */
W % #
] ('*) ^ */
W % # (8) We will discuss the computation of ,.` G # HG M s 04 below. Under the labeling hypothesis ` G ` G # , G is divided into the foreground features G , which are parts of the object, and background (clutter) G . If the foreground features G are independent of clutter G , then, ^ 0/
W
1 ^ */ W 2 ^ 0/ W
1 ^ 0/ W 2
1 ^ */ W 2 ^ 0/ 1 W 2 W (9) For simplicity, we will assume the priors ,.` G # N M 4 are the same for different ` G # , and ,.M 4 are the same for different graph structures. If we assume uniform background densities [11, 8], then ,. G N ` G # M 4 0 g 4 G 3 , where 8 is the volume of the space a background feature lies in, is the same for different ` G # . Under probability decomposition M , ,. HG N ` G # M 4 can be computed as in equation (1). Therefore the maximization of equation (7) is equivalent to maximizing, W % 1 W % # C X ] & ] ('*) 0/ QSRT
1 ^ */ W % ] & ] ('*) 0/ ! ] %& QSR T
0/ ' ( 1
*/ + ( 7
*/ ( 1& QSRT
*/ +)5
0/ -65 (10) For most problems, the number of possible labelings is very large (on the order of I 3 ), so it is computationally prohibitive to sum over all the possible ` G # as in equation (10). However, if there is one hypothesis labeling ` G # that is much better than other hypotheses,, i.e. G # corresponding to ` G # is much larger than other G # ’s, then G # can be taken as and other G # ’s as . Hence equation (10) can be approximated as, W % 1 W % # X ] & ! ] %& QSR T
0/! '( 1
*/" +(
*/" -( 1& QSR T
0/! +5
*/" -5 (11) where G # g ( G # d ( and G # f ( are measurements corresponding to the best labeling `# G # . Comparing with equation (2) and also by the weak law of large numbers, we know for iteration v , if the best hypothesis ` G # is used as the ’true’ labeling, then the decomposable triangulated graph structure M s can be obtained through the algorithm described in section 3. One approximation we make here is that the best hypothesis labeling ` G # for each G is really dominant among all the possible labelings so that hard assignment for labelings can be used. This is similar to the situation of K-means vs. mixture of Gaussian for clustering problems. We evaluate this approximation in experiments. The whole algorithm can be summarized as follows. Given some random initial guess of the decomposable graph structure M and its parameters, then for iteration v , ( v is from until the algorithm converges), E step: for each FG , use M s 0 to find the best labeling ` G # and then compute the differential entropies; M step: use the differential entropies to run the greedy graph growing algorithm described in section 3 and get M s . 4.2 Dealing with missing parts (occlusion) So far we assume that all the parts are observed. In the case of some parts missing, the measurements for the missing parts can be taken as additional hidden variables [11], and the EM algorithm can be modified to handle the missing parts. For each hypothesis ` G , let G denote the measurements of the observed parts, G be the measurements for the missing parts, and G G A G A A be the measurements of the whole object (to reduce clutter in the notation, we assume that the dimensions can be sorted in this way). For each EM iteration, we need to compute G and } G to obtain the differential entropies and then M s with its parameters. Taking ` G and G as hidden variables, we can get, 2 [ ]
(12) 2 [ ]
Z 2
Z 2 ! [ ]
! Z . ! 2 (13) Where
!
! ! , and
!
!
!
.
!
! . All the expectations -a 4 are conditional expectations with respect to G ` G ` G # and decomposable graph structure M s 0 . Therefore, G are the measurements of the observed foreground parts under ` G ` G # . Since M s 0 is Gaussian distributed, conditional expectation G 4 and G G A 4 can be computed from observed parts G and the mean and covariance matrix of M s 0 . 5 Experiments We tested the greedy algorithm on labeled motion capture data (Johansson displays) as in [8], and the EM-like algorithm on unlabeled detected features from real image sequences. 5.1 Motion capture data Our motion capture data consist of the 3-D positions of 14 markers fixed rigidly on a subject’s body. These positions were tracked with 1mm accuracy as the subject walked back and forth, and projected to 2-D. Under Gaussian assumption, we first estimated the joint probability density function (mean and covariance) of the data. From the estimated mean and covariance, we can compute differential entropies for all the possible triplets and pairs and further run the greedy search algorithm to find the approximated best triangulated model. Figure 2(a) shows the expected likelihood (differential entropy) of the estimated joint pdf, of the best triangulated model from the greedy algorithm, of the hand-constructed model from [8], and of randomly generated models. The greedy model is clearly superior to the hand-constructed model and the random models. The gap to the original joint pdf is partly due to the strong conditional independence assumptions of the triangulated model, which are an approximation of the true data’s pdf. Figure 2(b) shows the expected likelihood using 50 synthetic datasets. Since these datasets were generated from 50 triangulated models, the greedy algorithm (solid curve) can match the true model (dashed curve) extremely well. The solid line with error bars are the expected likelihoods of random triangulated models. 0 500 1000 1500 2000 2500 3000 −160 −150 −140 −130 −120 −110 expected log likelihood index of randomly generated triangulated models estimated joint pdf best trangulated model from greedy search triangulated model used in previous papers randomly generated triangulated models 0 10 20 30 40 50 60 −165 −160 −155 −150 −145 −140 −135 expected log likelihood index of randomly generated triangulated models (a) (b) Figure 2: Evaluation of greedy search. 5.2 Real image sequences There are three types of sequences used here: (I) a subject walks from left to right (Figure 3(a,b)); (II) a subject walks from right to left; (III) a subject rides a bike from left to right (Figure3(c,d)). Left-to-right walking motion models were learned from type I sequences and tested on all types of sequences to see if the learned model can detect left-to-right walking and label the body parts correctly. The candidate features were obtained from a Lucas-Tomasi-Kanade algorithm [10] on two frames. We used two frames to simulate the difficult situation, where due to extreme body motion or to loose and textured clothing and occlusion, tracking is extremely unreliable and each feature’s lifetime is short. Evaluation of the EM-like algorithm. As described in section 4.1, one approximation we made is taking the best hypothesis labeling instead of summing over all the possible hypotheses (equation (11)). This approximation was evaluated by checking how the loglikelihoods evolve with EM iterations and if they converge. Figure 4(a) shows the results of learning a 12-feature model. We used random initializations, and each curve of Figure 4(a) corresponds to one such random initialization. From Figure 4(a) we can see that generally the log-likelihoods grow and converge well with the iterations of EM. Models obtained. Figure 4 (b) and (c) show the best model obtained after we ran the EM algorithms for 11 times. Figure 4(b) gives the mean positions and mean velocities (shown in arrows) of the parts. Figure 4(c) shows the learned decomposable triangulated probabilistic structure. The letter labels show the body parts correspondence. Figure 3 shows samples of the results. The red dots (with letter labels) are the maximum likelihood configuration from the left-to-right walking model. The horizontal bar at the bottom left of each frame shows the likelihood of the best configuration. The short vertical bar gives the threshold where , *? , s s # G for all the test data. If A B C D E F G H I J L A B C D E F G H I J K L A B C D E F G H I L A B C D E F G H I L (a)walking detected (b)detected (c)not detected (d)not detected Figure 3: Sample frames from left-to-right walking (a-b) and biking sequences (c-d). The dots (either filled or empty) are the features selected by Tomasi-Kanade algorithm [10] on two frames. The filled dots (with letter labels) are the maximum likelihood configuration from the left-to-right walking model. The horizontal bar at the bottom left of each frame shows the likelihood of the best configuration. The short vertical bar gives the threshold for detection. 0 5 10 15 20 −220 −200 −180 −160 −140 −120 −100 iterations of EM log likelihood 100 150 200 250 40 60 80 100 120 140 160 180 200 A B C D E F G H I J K L A B C D E F G H I J K L (a) (b) (c) Figure 4: (a) Evaluation of the EM-like algorithm. Log-likelihood vs. iterations of EM for different random initializations. (b) and (c) show the best model obtained after we ran the EM-like algorithms for 11 times. the likelihood is greater than the threshold, a left-to-right walking person is detected.The detection rate is 100% for the left-to-right walking vs. right-to-left walking, and 87% for the left-to-right walking vs. left-to-right biking. 6 Conclusions In this paper we have described a method for learning the structure and parameters of a decomposable triangulated graph in an unsupervised fashion from unlabeled data. We have applied this method to learn models of biological motion that can be used to reliably detect and label biological motion. Acknowledgments Funded by the NSF Engineering Research Center for Neuromorphic Systems Engineering (CNSE) at Caltech (NSF9402726), and by an NSF National Young Investigator Award to PP (NSF9457618). We thank Charless Fowlkes for bringing the Chow and Liu’s paper to our attention. We thank Xiaolin Feng for providing the real image sequences. References [1] Y. Amit and A. Kong, “Graphical templates for model registration”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 18:225–236, 1996. [2] C.K. Chow and C.N. Liu, “Approximating discrete probability distributions with dependence trees”, IEEE Transactions on Information Theory, 14:462–467, 1968. [3] T.M. Cover and J.A. Thomas, Elements of Information Theory, John Wiley and Sons, 1991. [4] N. Friedman and M. Goldszmidt, “Learning bayesian networks from data”, Technical report, AAAI 1998 Tutorial, http://robotics.stanford.edu/people/nir/tutorial/, 1998. [5] M.I. Jordan, editor, Learning in Graphical Models, MIT Press, 1999. [6] M. Meila and M.I. Jordan, “Learning with mixtures of trees”, Journal of Machine Learning Rearch, 1:1–48, 2000. [7] Y. Song, X. Feng, and P. Perona, “Towards detection of human motion”, In Proc. IEEE CVPR 2000, volume 1, pages 810–817, June 2000. [8] Y. Song, L. Goncalves, E. Di Bernardo, and P. Perona, “Monocular perception of biological motion in johansson displays”, Computer Vision and Image Understanding, 81:303–327, 2001. [9] Nathan Srebro, “Maximum likelihood bounded tree-width markov networks”, In UAI, pages 504–511, San Francisco, CA, 2001. [10] C. Tomasi and T. Kanade, “Detection and tracking of point features”, Tech. Rep. CMU-CS-91132,Carnegie Mellon University, 1991. [11] M. Weber, M. Welling, and P. Perona, “Unsupervised learning of models for recognition”, In Proc. ECCV, volume 1, pages 18–32, June/July 2000. [12] Markus Weber, Unsupervised Learning of Models for Object Recognition, Ph.d. thesis, Caltech, May 2000.
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Orientational and geometric determinants place and headNeil Burgess & Tom Hartley Institute of Cognitive Neuroscience & Department of Anatomy, UCL 17 Queen Square, London WCIN 3AR, UK n. burgess@ucl.ac.uk. t.hartley@ucl.ac.uk Abstract We present a model of the firing of place and head-direction cells in rat hippocampus. The model can predict the response ofindividual cells and populations to parametric manipulations of both geometric (e.g. O'Keefe & Burgess, 1996) and orientational (Fenton et aI., 2000a) cues, extending a previous geometric model (Hartley et al., 2000). It provides a functional description of how these cells' spatial responses are derived from the rat's environment and makes easily testable quantitative predictions. Consideration of the phenomenon of remapping (Muller & Kubie, 1987; Bostock et aI., 1991) indicates that the model may also be consistent with nonparametric changes in firing, and provides constraints for its future development. 1 Introduction 'Place cells' recorded in the hippocampus of freely moving rats encode the rat's current location (O'Keefe & Dostrovsky, 1971; Wilson & McNaughton, 1993). In open environments a place cell will fire whenever the rat enters a specific portion of the environment (the 'place field'), independent of the rat's orientation (Muller et aI., 1994). This location-specific firing appears to be present on the rat's first visit to an environment (e.g. Hill, 1978), and does not depend on the presence of local cues such as odors on the floor or walls. The complementary pattern of firing has also been found in related brain areas: 'head-direction cells' that fire whenever the rat faces in a particular direction independent of its location (Taube et aI., 1990). Experiments involving consistent rotation of cues at or beyond the edge of the environment (referred to as 'distal' cues) produce rotation of the entire place (O'Keefe & Speakman, 1987; Muller et aI., 1987) or head-direction (Taube et aI., 1990) cell representation. Rotating cues within the environment does not produce this effect (Cressant et aI., 1997). Here we suggest a predicitive model of the mechanisms underlying these spatial responses. 2 Geometric influences given consistent orientation Given a stable directional reference (e.g. stable distal cues across trials), fields are determined by inputs tuned to detect extended obstacles or boundaries at particular (1) bearings. That is, they respond whenever a boundary or obstacle occurs at a given distance along a given allocentric direction, independent of the rat's orientation. These inputs are referred to below as putative 'boundary vector cells' (BVCs). The functional form of these inputs has been estimated by recording from the same place cell in several environments of differing geometry within the same set of distal orientation cu~s (O'Keefe & Burgess, 1996; Hartley et al., 2000). That is, for a BVC i tuned to a boundary at distance di and bearing <Pi relative to the rat, the response to a houndary segment at distance r and bearing 9, subtending an angle cfJ at the rat, is given by: Cli == gi(r, fJ)CfJ, ( fJ) exp[-(r - di )2/2a;ad(di )] exp[-(fJ - <Pi)2 /2a~ng] gi r, ex: X -----r===========--V21r0";ad(di ) - /21ru 2 V any where the angular width aang is a constant but the radial width Urad == uo(1+di //3) so that the width of tuning to distance increases with the distance of peak response diD Constants 0"0 and /3 determine width at zero distance and its rate of increase with distance. The firing rate of BVC i, when the rat is at a location z, is found by integrating eli over (1 (this is done numerically as the distance r to the nearest boundary in direction fJ is a function of z, fJ and the geometry of the environment). A place cell's firing rate F(Z) is then simply the thresholded linear sum of the firing rates of the n Bves connected to it, Le. where e(z) is the Heaviside function (S(z) == z if x > 0; Sex) == 0 otherwise). All simulations have /3 == 183cm, Uo == 12.2cm, Urad == 0.2rad, while the threshold T can vary between simulations (e.g. between Figs. 1 and 2) but not between cells, and A is an arbitrary constant as absolute firing rates are not shown. Thus, in this model, a place cell's response is simply determined by the parameters di and ifJi chosen for the set of BVes connected to it. Assuming a random selection of BVCs for each place cell, and a single value for T, the model provides a good fit to the characteristics of populations of place fields across different environments, such as the distribution of firing rates and field shapes and sizes. Inputs can also be chosen so as to fit a given place field so that its behavior in a new environment of different shape can be predicted. See Hartley et al. (2000) and Fig. 1. Like other models relying on the bearing to a landmark (Redish & Touretsky, 1996; McNaughton et al., 1996), the basic geometrical model assumes an accurate directionalreference, but does not state how this depends on the sensory input. Note that, as such, this model already captures effects of consistent rotation of orientation cues around an environment as a reorientation of the directional reference frame that in turn affects the directions along which BVCs are tuned to respond. Indeed, the effect of consistent rotation of orientation cues about a environment of fixed geometry is identical to the rotation of the environment within a fixed directional reference frame, and can be modelled in this way (see e.g. the square and diamond in Figs. 1b,c). 3 Model of geometric and orientation influences Models of head direction (Skaggs et al., 1995; Zhang, 1996) indicate how orientation might be derived. Internal inputs (e.g. vestibular or proprioceptive) maintain a consistent representation of heading within a ring of head-direction cells arranged to form a continuous attractor. Correlational learning of associations from visual inputs to head direction cells then allows the representation of head direction to be maintained in synch with the external world. These models account for the preferential influence of large cues at a stable bearing (i.e. at or beyond the edge of the environment), and effects of instability caused by continual movement of cues or disorientation of the rat. They also allow orientation to be maintained in the face of cue removal, unless all cues are removed in which case orientation is wholly reliant on internal inputs and will drift over time. In this paper we take a step towards providing a quantitative model for the combined influences of orientation cues and boundaries on the firing of place and head direction cells. Such a model should be able to predict the behaviour of these cells under arbitrary environmental manipulations, bearing in mind that some (extended) objects may be both orientation cues and boundaries. We focus on a series of experiments regarding inconsistent rotations of two extended cue cards (one white, one black) around the perimeter of a cylinder in the absence of any other orientation cues (Fenton et aI., 2000a). Each of these cards controls the orientation of the set of place fields when rotated together or alone (after removal of the other cue). When both are rotated inconsistently, place fields are displaced in a non-uniform manner, with the displacement of a field being a function of its location within the environment. These findings cannot be explained by a simple rotation of the reference frame. Fig. 2A shows how place fields are displaced following counter rotation of the two cue cards. Since the cue cards are orientation cues and also walls of the environment, explaining these data within the current framework requires two separate considerations: i) how the movement of the cards affects the BVC's directional reference frame, and ii) how the movement of the cards, acting as boundaries, directly affects the BVCs. We make the following assumptions: 1. The influence of a distal visual cue on the directional reference system is proportional to its proximity to the rat. 2. In the continued presence of color (or contrast) variation along a boundary to which a BVC responds, the BVC will become modulated by color: responding preferentially to, say, a white section of wall rather than the adjacent grey wall. In the absence of such variation it will revert to its unmodulated response. We note that assumption 1) is consistent with most implementations of the head direction model discussed above, in that the influence of an extended distal cue will increase with the angle subtended by it at the rat. We also note that assumption 2) implies the presence of synaptic learning (something not required by the rest of the model), albeit outside of the hippocampus. To avoid having to simulate enough random selections of BVCs to produce place fields at all locations within the environment and with all combinations of distance, bearing and color preferences, the model must be further simplified. To model the effect of cue manipulation on a place field in a location from which there are two cue cards at distances Di and bearings qli, we simulate a place cell for that location which receives inputs from two BVCs tuned to the distances D i and bearings qli, and to the most common color of boundary segments to which it respondes (across all positions of the rat). That is, di .= Di and 4>i == q>i in equation 1. For each location in the environment, we compute the shape of the place field formed by the thresholded sum of these BVCs, before and after the cue card manipulation. This simplification is broadly representative of the qualitative effect of the manipulation on the locations of place fields!. How does this model campare to the Fenton et aI. data? First we note that (due to assumption 1) each cue card can control the overall orientation of the place and head-direction representations. Similarly removing a cue card will have little effect, save for a slight rotation and/or transverse spreading of the Bve that responds to it (as it is no longer constrained by the color boundary, see assumption 2). When the cues are rotated inconsistently, the firing fields of the BVCs move relative to each other. The net effect of this on place fields and their centroids (Fig. 2B) compares well with the data (Fig. 2A) and is composed of two separate effects. First, the rotation of the cues produces a non-uniform distortion of the head direction system. The extent of rotation depends on the location of the animal relative to the cues as the closer a cue the more it affects the directional reference at that location (assumption 1) see Fig. 2C (ii). This distortion of the directional reference frame affects the orientations to which the boundary vector cells are tuned, and thus affects the location of place fields in an approximately rotational manner see Fig. 2C (iii).. Second, the movement of the cue cards directly affects the firing fields of the BVCs due to their color preferrence. This 'translational' effect is shown in Fig. 2C (iv). Note that neither translational nor rotational effects alone are sufficient to explain the observed data. Fenton et aI. (2000b) also make a distinction between translation and rotation in producing a phenomenological model of their data. However, as such, their model does not provide a mechanistic account at the level of cells, is specific to the cue-card manipulation they made and so does not make any prediction for head-direction cells or place cells in other experiments. 4 Non-parametric changes: 'remapping' Our model considers the pattern of firing of place cells when the rat is put into an environment of different shape, or when two very familiar landmarks are moved or removed. In these situations changes to patterns of firing tend to be parametric, and the model aims to capture the parametric relationships between firing pattern and environmental manipulation. However we note that, after several days or weeks of experience, the place cell representations of two environments of different shape gradually diverge (Lever et aI., 2002), such that the final representations can be said to have 'remapped' (MUller and Kubie, 1987). After 'remapping' a given cell might fire in only one of the environments, or might fire in both but in unrelated locations. Additionally, changing the color of the cue card in a grey cylinder from white to black can cause more rapid remapping such that the effect on the first day is probably best described as a slight rotation, with remapping occurring by the second day (Bostock et aI., 1991). Note that simply removing the cue card just causes the overall orientation of the place field representation to drift. Could the current model be extended to begin to understand these apparently nonparametric changes? The effects of replacing the cue card with a novel one are consistent with assumption 2 and the extra-hippocampal learning it implies: BVCs initially respond to the new color as they would upon removal of the cue card, with 1 Simulations of place fields with a larger number of BVCs indicate similar field movements, but of reduced magnitude in locations far from the cue cards. However the good match between the simple model and the data (Figs. 2A,B) suggests that the cue cards do provide the majority of BVC input. This might be due to learned salience over the extensive training period, and to the learning process implied by assumption 2. Against this, place fields formed by more the two BVC inputs (e.g. the four BVCs in Fig. Ic) generally give a better match to field shape, especially in locations far from the two cue cards. the slight rotation or spreading of the firing field noted above. Over time in the presence of the new color, the color modulation of BVCs sharpens such that those previously responding to white or grey no longer respond to black, while new BVCs that do respond to black begin to fire. Thus the original place fields (particularly those nearest the card and so most reliant on BVCs from that direction) will tend to fall below threshold, unless receiving a connection from a newly active blacksensitive BVC, in which case the field will change location. Equally, some previously silent place cells will become active due to input from a newly-active black-sensitive BVC. By contrast, the slow shape-dependent remapping would appear to require some additional mechanism. This may be related to the evidence of shorter-term learning of associations between place cells (M~hta et aI., 1997) or the NMDAdependent stability of place fields (Kentros et aI., 1998) or postulated processes of learned orthogonalisation of hippocampal representations (Marr, 1971; McClelland et aI., 1995; Treves & Rolls, 1992; Fuhs & Touretzky, 2000; Kali & Dayan, 2000). 5 Conclusion The model we have presented is consistent with a large body of detailed data on the effects of parametric environmental manipulations on place and head-direction cells. More importantly, it is a predictive model at the level of individual cells. Fig. 2C (ii) shows the prediction resulting from assumption 1) regarding the effect of the inconsistent cue card manipulation on head-direction cells. We note that there is an alternative to this location-dependent warping of head direction responses: a direction-dependent warping such that responses to north directions are tilted northwestwards while responses to south directions are tilted southwestwards. This would correspond to the alternative assumption that the influence of a distal visual cue on a head direction cell is proportional to the similarity of the average direction of the cue from the rat and the preferred direction of the cell. We chose to simulate the former (assumption 1) as this is consistent with current head-direction models in keeping the angular separation of preferred directions constant (but rotating all of them together as a function of the proximity ofthe rat to one or other cue card). The alternative assumption breaks this constancy, but would produce roughly equivalent results for place cell firing. Thus, on the basis of the Fenton et al. experiment on place cells we must predict one or other of the two effects on head-direction, or some combination of both. Beyond this, the model can predict the effect of essentially arbitrary parametric movements of cues and boundaries on place and head-direction 'cells over the short term. It also appears to be at least consistent with the nonparametric 'remapping' changes induced by color changes. Whether or not it can also predict the statistics of remapping over longer timescales in response to purely geometric changes is a question for future work. Acknowledgements: We thank John O'Keefe, Colin Lever and Bob Muller for many useful discussions. 6 References Bostock E, Muller RU, Kubie JL (1991) Experience-dependent modifications of hippocampal place cell firing Hippocampus 1, 193-206. Cressant A, Muller RU, Poucet B (1997) Failure of centrally placed objects to control the firing fields of hippocampal place cells. J. Neurosci. 17, 2531-2542. Fenton AA, Csizmadia G, & Muller RU (2000a). Conjoint control of hippocampal place cell firing by two visual stimuli. I. The effects of moving the stimuli on firing field positions. J. Gen. Physiol, 116, 191-209. Fenton AA, Csizmadia G, &. Muller RU (2000b). Conjoint control of hippocampal place cell firing by two visual stimuli. Ii. A vector-field theory that predicts modifications of the representation of the environment. J. Gen. Physiol, 116, 211-221. Fuhs MC, Touretzky DS (2000) Synaptic learning models of map separation in the hippocampus. Neurocomputing, 32:379-384. Hartley T, Burgess N, Lever C, Cacucci F, O'Keefe J (2000) Modeling place fields in terms of the cortical inputs to the hippocampus. Hippocampus, 10, 369-379. Hill AJ (1978) First occurrence of hippocampal spatial firing in a new environment. Exp. Neural 62, 282-297. Kali S, Dayan P (2000) The Involvement of Recurrent Connections in Area CA3 in Establishing the Properties of Place Fields: A Model. J. Neurosci. 20, 7463-7477. Kentros C, Hargreaves E, Hawkins RD, Kandel ER, Shapiro M, Muller RU (1998) Abolition of long-term stability of new hippocampal place cell maps by NMDA receptor blockade. Science, 280, 2121-2126. McNaughton BL, Knierim JJ, Wilson MA (1994) 'Vector encoding and the vestibular foundations of spatial cognition: a neurophysiological and computational hypothesis', In The Cognitive NeuroJJciences, (ed. Gazzaniga, M.) 585-596 (MIT Press, Boston, 1994). Lever CL, Wills T, Cacucci F, Burgess N, O'Keefe J (2002) Long-term plasticity in the hippocampal place cell representation of environmental geometry. Nature, in press. Marr D (1971) Simple memory: a theory for archicortex. Phil. Trans. Roy. Soc. Lond B 262, 23-81. McClelland JL, McNaughton BL, O'Reilly RC (1995) Why there are complementary learning-systems in the hippocampus and neocortex - insights from the successes and failures of connectionist models of learning and memory. Psychological Review 102, 419457. Mehta MR, Barnes CA, McNaughton BL (1997) Experience-dependent, asymmetric expansion of hippocampal place fields. Proc. Nat. Acad. Sci. 94, 8918-8921. Muller RU, Bostock E, Taube JS, Kubie JL (1994) On the directional firing properties of hippocampal place cells. J. Neurosci. 14 7235-7251. Muller RU, Kubie JL (1987) The effects of changes in the environment on the spatial firing of hippocampal complex-spike cells. J. Neurosci 7, 1951-1968. Muller RD, Kubie JL, Ranck JB (1987) Spatial firing patterns of hippocampal complexspike cells in a fixed environment. J. Neurosci., 7, 1935-1950. O'Keefe J, Burgess N (1996) Geometric Determinants of the Place Fields of Hippocampal Neurones. Nature 381, 425-428. O'Keefe J, Dostrovsky J (1971) The hippocampus as a spatial map: preliminary evidence from unit activity in the freely moving rat. Brain Res 34, 171-175. O'Keefe J, Speakman A (1987) Single unit activity in the rat hippocampus during a spatial memory task. Exp. Brain Res 68, 1-27. Redish AD, Touretzky DS (1996) Modeling interactions of the rat's place and head direction systems Advances in Neural Information Processing Systems, 8. D Touretzky, MC Mozer, ME Hasselmo (eds) pp. 61-67. MIT Press, Cambridge MA. Skaggs WE, Knierim JJ, Kudrimoti HS, McNaughton BL (1995) 'A model of the neural basis of the rat's sense of direction' Advances in Neural Information Processing Systems, 7. G Tesauro, D Touretzky &. TK Le'en (eds) pp. 51-58. MIT Press, Cambridge MA. Taube JS, Muller RD, Ranck JB (1990) Head-direction cells recorded from the postsubiculum in freely moving rats. 1. Description .and quantitative analysis. J. Neurosci 10, 420-435. Treves A, Rolls ET (1992) Computational constraints suggest the need for two distinct input systems to the hippocampal CA3 network. Hippocampus 2, 189-200. Wilson MA, McNaughton BL (1993) Dynamics of the hippocampal ensemble code for space. Science 261, 1055-1058. Zhang K (1996) Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. J.Neurosci., 16, 2112-2126. b) e) Figure 1: Model of the geometrical influence on place fields (adapted from Hartley et aI., 2000), assuming a stable directional reference frame. Place fields are composed from thresholded linear sums of the firing rates of 'boundary vector cells' (BVCs). a) Above: Each BVC has a Gaussian tuned response to the presence of a boundary at a given distance and bearing from the rat (independent of its orientation). Below: The sharpness of tuning of a BVC decreases as the distance to which it is tuned increases. The only free parameters of a BVC are the distance and direction of peak response. b) Place fields recorded from the same cell in four environments of different shape or orientation relative to distal cues. c) Simulation of the place fields in b) by the best fitting set of 4 BVCs constrained to be in orthogonal directions (BVCs shown on the left, simulated fields on the right). The simulated cell can now be used to predict firing in novel situations. Real and predicted data from three novel environments are shown in d) and e) respectIvely, showing good qualitative agreement. A B =:==-~" k"" , i) ii) '''t'-iii) :- --:q:::::'" c =:==-"~....., <It'" ,=:.~....., :- ....-?3:;:/·· .......: ~ . f t t . : ......~ \ /' I ~ i) ii) iii) iv) Figure 2: Changes to place fields in a cylinder following inconsistent rotation of two cue cards. A) Experimental data shown in a birds-eye view of the cyclinder including the black and white cue-cards (adapted from Fenton et aI., 2000a). i) A place field with the cue cards in the 'standard' condition (used throughout training). ii) The place field after inconsistent rotation of each cue card by 12.5° further apart ('apart' condition)~ iii) The movement of the centroid of place field from the standard condition (tail of arrow) to the apart condition (head of arrow). B) Simulation of 21 place fields in the cyclinder in standard and apart conditions. Cue card locations are indicated by a black line (initial card positions are indicated by a dotted line to illustrate changes from one condition to another). i) and ii) show the place field nearest in location to that shown in A) in standard and apart conditions. iii) shows the movement of the centroids of simulated place fields between standard and apart conditions. C) i) Simulation of the movement of place field centroids between the standard and 'together' conditions (cue cards rotated 12.5° closer together). ii) The distortion of the preferred direction of a head direction cell. Arrows show the preferred direction in the 'apart' condition, the preferred direction was 'up' in the standard condition. iii) the movement of place field centroids between the standard and apart condition due solely to the directional distortion shown in ii). iv) the movement of place field centroids due solely to the movement of the cue cards acting as distinct cues (without any directional distortion shown in ii). The net effect of fields iii) and iv) is that shown in B iii).
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Covariance Kernels from Bayesian Generative Models Matthias Seeger Institute for Adaptive and Neural Computation University of Edinburgh 5 Forrest Hill, Edinburgh EH1 2QL seeger@dai.ed.ac.uk Abstract We propose the framework of mutual information kernels for learning covariance kernels, as used in Support Vector machines and Gaussian process classifiers, from unlabeled task data using Bayesian techniques. We describe an implementation of this framework which uses variational Bayesian mixtures of factor analyzers in order to attack classification problems in high-dimensional spaces where labeled data is sparse, but unlabeled data is abundant. 1 Introduction Kernel machines, such as Support Vector machines or Gaussian processes, are powerful and frequently used tools for solving statistical learning problems. They are based on the use of a kernel function which encodes task prior knowledge in a Bayesian manner. In this paper, we propose the framework of mutual information (MI) kernels for learning covariance kernels from unlabeled task data using Bayesian techniques. This section introduces terms and concepts. We also discuss some general ideas for discriminative semi-supervised learning and kernel design in this context. In section 2, we define the general framework and give examples. We note that the Fisher kernel [4] is a special case of a MI kernel. MI kernels for mixture models are discussed in detail. In section 3, we describe an implementation for a MI kernel for variational Bayesian mixtures of factor analyzers models and show results of preliminary experiments. In the semi-supervised classification problem, a labeled dataset Dl {(Xl,tl), ... ,(Xm,tm)} as well as an unlabeled set Du = {xm+1 ,""Xm+n} are given for training, both i.i.d. drawn from the same unknown distribution, but the labels for Du cannot be observed. Here, Xi E I~.P and ti E {-1, +1}.1 Typically, m = IDll is rather small, and n = IDul »m. Our aim is to fit models to Du in a Bayesian way, thereby extracting (posterior) information, then use this information to build a covariance kernel K. Afterwards, K will be plugged into a supervised kernel machine, which is trained on the labeled data Dl to perform the classification task. 1 For simplicity, we only discuss binary labels here. It is important to distinguish very clearly between these two learning scenarios. For fitting Du, we use Bayesian density estimation. After having chosen a model family {p(xIOn and a prior distribution P(O) over parameters 0, the posterior distribution P(OIDu) ex P(DuIO)P(O), where P(DuIO) = rr::~'~l P(xiIO), encodes all information that Du contains about the latent (i.e. unobserved) parameters 0.2 The other learning scenario is supervised classification, using a kernel machine. Such architectures model a smooth latent function y (x) E ~ as a random process, together with a classification noise model P(tly).3 The covariance kernel K specifies the prior distribution for this process: namely, a-priori, y(x) is assumed to be a Gaussian process with zero mean and covariance function K , i.e. K(x(1) , X(2)) = E[y(x(1))Y(X(2))]; see e.g. [10] for details. In the following, we use the notation a = (ai)i = (al' ... ,aI)' for vectors, and A = (ai,j )i,j for matrices respectively. The prime denotes transposition. diag a is the matrix with diagonal a and 0 elsewhere. N(xlJ.t,~) denotes the Gaussian density with mean J.t and covariance matrix ~. Within the standard discriminative Bayesian classification scenario, unlabeled data cannot be used. However, it is rather straightforward to modify this scenario by introducing the concept of conditional priors (see [6]). If we have a discriminant model family {P(tlx; w n, a conditional prior P(w 10) allows to encode prior knowledge and assumptions about how information about P(x) (i.e. about 0) influences our assumptions about a-priori probabilities over discriminants w. For example, the P(wIO) could be Occam priors, expressing the intuitive fact that for many problems, the notion of "simplicity" of a discriminant function depends strongly on what is known about the input distribution P(x). For a given problem, it is in general not easy to come up with a useful conditional prior. However, once such a prior is specified, we can in principle use the same powerful techniques for approximate Bayesian inference that have been developed for supervised discriminative settings. Semi-supervised techniques that can be seen as employing conditional priors include co-training [1], feature selection based on clustering [7] and the Fisher kernel [4]. For a probabilistic kernel technique, P( w 10) is fully specified by a covariance function K(x(1) , X(2) 10) depending on O. The problem is therefore to find covariance kernels which (as GP priors) favour discriminants in some sense compatible with what we have learned about the input distribution P(x). Kernel techniques can be seen as nonparametric smoothers, based on the (prior) assumption that if two input points are "similar" (e.g. "close" under some distance), their labels (and latent outputs y) should be highly correlated. Thus, one generic way of learning kernels from unlabeled data is to learn a distance between input points from the information about P( x). A frequently used assumption about how classification labels may depend on P(x) is the cluster hypothesis: we assume discriminants whose decision boundaries lie between clusters in P(x) to be a-priori more likely than such that label clusters inconsistently. A general way of encoding this hypothesis is to learn a distance from P(x) which is consistent with clusters in P(x) , i.e. points within the same cluster are closer under this distance than points from different clusters. We can then try to embed the learned distance d(x(1), X(2)) approximately in an Euclidean space, i.e. learn a mapping ¢ : X r-+ ¢( x) E ~l such that d(x(1) , X(2)) :=;::j 11¢(x(1)) - ¢(X(2)) II for all pairs from Du. Then, a natural kernel function would be K(x(1) , X(2)) = exp( - ,BII¢(x(1)) - ¢(x(2))112). In this paper, however, we follow a simpler approach, by considering a similarity measure 2In practice, computation of P(OIDu) is hardly ever feasible, but powerful approximation techniques can be used. 3 A natural choice for binary classification is to represent the log odds log(P(t = +1Ix)/P(t = -1Ix)) by y(x) . which immediately gives rise to a covariance kernel, without having to compute an approximate Euclidean embedding. Remark: Our main aim in this paper is to construct kernels that can be learned from unlabeled data only. In contrast to this, the task of learning a kernel from labeled data is somewhat simpler and can be approached in the following generic way: start with a parametric model family {y(x; w)} , with the interpretation that y(x;w) models the log odds log(P(t = +llx)/P(t = -llx)). Fitting these models to labeled data D[ , we obtain a posterior P(wIDI) . Now, a natural covariance kernel for our problem is simply K(x(1),X(2)) = Jy(x(1);w)y(x(2 );w)Q(w)dw, where (say) Q(w) <X P(wID[)AP(W)l - A (or an approximation thereof). For A = 0, we obtain the prior covariance kernel for our model, while for larger A the kernel incorporates more and more posterior information. The kernel proposed in [8] can be seen as approximation to this approach. 2 Mutual Information Kernels In this section, we begin by introducing the framework of mutual information kernels. Given a mediator distribution Pm ed (()) over parameters (), we define the joint distribution Q(x(1) , X(2)) mediated by Pm ed (()) as Q(x(1) , X(2)) = J Pmed(())P(x(1)I())P(x(2)1())d(). (1) The sample mutual information between x(1) and X(2) under this distribution is (1) (2) _ Q(X(l) , X(2)) I(x ,x ) - log Q(x(1))Q(X(2)) ' (2) where Q(x) = JQ(x ,x)dx. I(x(1) , x(2)) is called the mutual information (MI) score. In a very concrete sense, it measures the similarity between x(1) and X(2) with respect to the generative process represented by the mediator distribution Pm ed (()): it is the amount of information they share via the mediator variable () ~ Pm ed (()) . Note that Q(x(1), X(2)) can be seen as inner product in a space of functions () f-t R, the features of X(k) being (P(x(k)I()))o, weighted by the distribution Pm ed .4 X(k) is represented by its likelihood under all possible models. Covariance kernels have to satisfy the property of positive definiteness5 , and the MI score I does not. However, applying a standard transformation (called exponential embedding (EE) here), we arrive at K(x(1) X(2)) = e - (I(x(l) ,x(1))+I(x(2) ,x(2)))/2+I(x(1) ,x(2)) = Q(x(1), X(2)) , vQ(x(1) , x(1))Q(X(2), X(2)) (3) EE becomes familiar if we note that it transforms the standard inner product x(1)' X(2) into the well-known Radial Basis Function (RBF) kernel6 (4) 4When comparing X (l) , X (2) via the inner product, we are not interested in correlating their features uniformly, but rather focus on regions of high volume under Pm e d . 5 K is positive definite if the matrix (K(X(kll , X(k2»)hl ,k2 is positive definite for every set {x(1 ), ... , X (K ) } of distinct points. 60ne can show that if j is itself a kernel, and j -+ I< under EE, then 1<(3 is also a kernel for all (3 > 0 (see e.g. [3]) . or the weighted inner product x(1)'VX(2) into the squared-exponential kernel (e.g. [10]). It is easy to show that K in (3) is a valid covariance kernel7 , and we refer to it as mutual information (MI) kernel. Example: Let P(xIO) = N(xIO, (p/2)I) (spherical Gaussian with mean 0), Prned(O) = N(OIO, aI). Then, the MI kernel K is the RBF kernel (4) with (3 = 4/(p(4 + pia)). Thus, the RBF kernel is a special case of a MI kernel. 2.1 Mediator distribution. Model-trust scaling. The mediator distribution Prned(O), motivated earlier in this section, should ideally encode information about the x generation process, just as the Bayesian posterior P(OIDu). On the other hand, we need to be able to control the influence that information from sources such as unlabeled data Du can have on the kernel (relying too much on such sources results in lack of robustness, see e.g. [6] for details). Here, we propose model-trust scaling (MTS) , by setting (5) Prned varies with A from the (usually vague) prior P(O) (A = 0) towards the sharp posterior P(OIDu) (A = n), rendering the Du information (via the model) more and more influence upon the kernel K. The concrete effect of MTS on the kernel depends on the model family. Example (continued): Again, P(xIO) = N(xIO , (p/2)I) , with a flat prior P(O) = 1 on the mean. Then, P(OIDu) = N(Olx , (p/2n)I), where x = n- 1 L:;~;>~l Xi, and Prned(O) = N(Olx, (p/2A)I) (after (5)). Thus, the MI kernel is again the RBF kernel (4) with (3 = 2/(p(2 + A)) . For the more flexible model P(xIO) = N(xIJL, ~), ° = (JL,~) and the conjugate Jeffreys prior, the MI kernel is computed in [5]. If the Bayesian analysis is done with conjugate prior-model pairs, the corresponding MI kernel can be computed easily, and for many of these cases, MTS has a very simple, analytic form (see [5]). In general, approximation techniques developed for Bayesian analysis have to be applied. For example, applying the Laplace approximation to the computations on a model with flat prior P(O) = 1 results in the Fisher kernel [4]8, see e.g. [5]. However, in this paper we favour another approximation technique (see section 3). 2.2 Mutual Information Kernels for Mixture Models If we apply the MI kernel framework to mixture models P(x 10, 7T") = Ls 7fsP(x lOs), we run into a problem. As mentioned in section 1, we would like our kernel at least partly to encode the cluster hypothesis, i.e. K(x(1), X(2)) should be small if x(1), X(2) come from different clusters in P(x ),9 but the opposite is true (for not too small 7 Q(x(1 ), X (2)) is an inner product (therefore a kernel), for the rest of the argument see [3], section 5. 8This was essentially observed by the authors of [4] on workshop talks, but has not been published to our knowledge. The fascinating idea of the Fisher kernel has indeed been the main motivation and inspiration for this paper. 9This does not mean that we (a-priori) believe they should have different labels, but only that the label (or better: the latent yO) at one of them should not depend strongly on yO at the other. A). To overcome this problem, we generalize Q(x(1), X(2)): S Q(X(1),X(2)) = L WS1 ,S2 J P(x(1) IOsJP(X(2) IOs2)Prned(O) dO, (6) 8} , 82=1 where W = (WS1 ,S2)Sl ,S2 is symmetric with nonnegative entries and positive elements on the diagonal. The MI kernel K is defined as before by (3), based on the new Q. If Prned(O,rr) = ITsPrned(Os)Prned(rr) (which is true for the cases we will be interested in), we see that the original MI kernel arises as special case WS1,S2 = EPmed[7fS17fs2]' Now, by choosing W = diag(Epmed[7fs])s, we arrive at a MI kernel K which (typically) behaves as expected w.r.t. cluster separation (see figure 1), but does not exhibit long-range correlations between joined components. In the present work, we restrict ourselves to this diagonal mixture kernel. Note that this kernel can be seen as (normalized) mixture of MI kernels over the component models. Figure 1: Kernel contours on 2-cluster dataset (A = 5,100,30) Figure 1 shows contour plots10 of the diagonal mixture kernel for VB-MoFA (see section 3), learned on a 500 cases dataset sampled from two Gaussians with equal covariance (see subsection 3.1). We plot K(a,x) for fixed a (marked by a cross) against all x , the height between contour lines is 0.1. The left and middle plot have the lower cluster's centre as a, with A = 5, A = 100 respectively, the right plot's a lies between the cluster centres, A = 30. The effect of MTS can be seen by comparing left and middle, note the different sharpness of the slopes towards the other cluster and the different sizes and shapes of the "high correlation" regions. As seen on the right, points between clusters have highest correlation with other such inter-cluster points, a feature that may be very useful for successful discrimination. 3 Experiments with Mixtures of Factor Analyzers In this section, we describe an implementation of a MI kernel, using variational Bayesian mixtures of factor analyzers (VB-MoFA) [2] as density models. These are able to combine local dimensionality reduction (using noisy linear transformations u -+ x from low-dimensional latent spaces) with good global data fit using mixtures. VB-MoFA is a variational approximation to Bayesian analysis on these models, able to deliver the posterior approximations we require for an MI kernel. We employ the diagonal mixture kernel (see subsection 2.2). Instead of implementing MTS analytically, we compute the VB approximation to the true posterior (i.e. A = n), then simply apply the scaling to this distribution. P rned (0, rr) factorizes as required in subsection 2.2. The integrals J P(x(1) IOs)p(X(2) IOs)Prned(Os) dOs in (6) lOProduced using the first-order approximation (see 3) to the MI kernel. Plots using the one-step variational approximation (see 3) have a somewhat richer structure. are not analytically tractable. Our first idea was to approximate them by applying the VB technique once more, ending up with what we called one-step variational approximations. Unfortunately, the MI kernel approximation based on these terms cannot be shown to be positive definite anymorell! Thus, in the moment we use a less elegant and, we feel, less accurate approximation (details can be found in [5]) based on first-order Taylor expansions. In the remainder of this section we compare the VB-MoFA kernel with the RBF kernel (4) on two datasets, using a Laplace GP classifier (see [10]). In each case we sample a training pool, a kernel dataset Du and a test set (mutually exclusive). The VB-MoFA diagonal mixture kernel is learned on Du. For a given training set size m, a run consists of sampling a training set Dl and a holdout set Dh (both of size m) from the training pool, tuning kernel parameters by validation on D h , then testing on the test set. We use the same Dl, Du for both kernels. For each training set size, we do L = 30 runs. Results are presented by plotting means and 95% t-test confidence intervals of test errors over runs. 3.1 Two Gaussian clusters The dataset is sampled from two 2-d Gaussians with same non-spherical covariance (see figure 1), one for each class (the Bayes error is 2.64%) . We use n = 500 points for D u , a training pool of 100 and a test set of 500 points. The learning curves in figure 2 show that on this simple toy problem, on which the fitted VB-MoFA model represents the cluster structure in P(x) almost perfectly, the VB-MoFA MI kernel outperforms the RBF kernel for samples sizes n :::; 40. ~ 02 ~ 0.175 ~ 0.15 IL _ 0.225 , ~ 02 ~O . 175 ~ 0.15 ~O.125 rI I I ~~~~~---7---~~~~~~~--~ ',~~~~---7---.~ , --~--~--~~ Training set size n Trair>ngsets;zen Figure 2: Learning curves on 2-cluster dataset. Left: RBF kernel; right: MI kernel 3.2 Handwritten Digits (MNIST): Twos against threes We report results of preliminary experiments using the subset of twos and threes of the MNIST Handwritten Digits database12 . Here, n = IDul = 2000, the training pool contains 8089, the test set 2000 cases. We employ a VB-MoFA model with 20 components, fitted to Du. We use a very simple baseline (BL) algorithm (see [6], section 2.3) based on the component densities from the VB-MoFA model13 , which llThanks to an anonymous reviewer for pointing out this flaw. 12The 28 x 28 images were downsampled to size 8 x 8. 13The estimates P(xls) are obtained by integrating out the parameters (}s using the variational posterior approximation. The integral is not analytic, and we use a one-step variational approximation to it. allows us to assess the "purity" of the component clusters w.r.t. the labels1\ this algorithm is the only one not based on a kernel. Furthermore, we show results for the one-step variational approximation to the MI kernel15 (MIOLD). The learning curves are shown in figure 3. I r II , II to.> ~ 0.' j" 1 • i ~ ... t· .. I .. I I ! •• I ! I ..... = • T.-.",",_ , ,,_ ... _, T,_ ... _, , I j" i I I .. ! [ Figure 3: Learning curves on MNIST twos/threes. Upper left: RBF kernel; upper middle: Baseline method; upper right: VB-MoFA MI kernel (first-order approx.); lower left: VB-MoFA MI "kernel" (one-step var. approx.) The results are disappointing. The fact that the first-order approximation to the MI kernel performs worse than the one-step variational approximation (although the latter may fail to be positive definite) , indicates that the former is a poorer approximation. The latter renders results close to the baseline method, while the smoothing RBF kernel makes much better use of a growing number of labeled examples16 This indicates that the conditional prior, as represented by the VB-MoFA MI kernel, behaves nonsmooth and overrides label information in regions where it should not. We suspect this problem to be related to the high dimensionality of the input space, in which case probability densities tend to have a large dynamic range, and mixture component responsibility estimates tend to behave very nonsmooth. Thus, it seems to be necessary to extend the basic MI kernel framework by new scaling mechanisms in order to produce a smoother encoding of the prior assumptions. 14The baseline algorithm is based on the assumption that, given the component index s, the input point x and the label t are independent. Only the conditional probabilities P(tls) are learned, while P(xls) and pes) is obtained from the VB-MoFA model fitted to unlabeled data only. Thus, success/failure of this method should be closely related to the degree of purity of the component clusters w.r.t . the labels. 15This is somewhat inconsistent, since we use a kernel function which might not be positive definite in a context (GP classification) which requires a covariance function. 16Note also that RBF kernel matrices can be evaluated significantly faster than such using the VB-MoFA MI kernel. 4 Related work. Discussion The present work is probably most closely related to the Fisher kernel (see subsection 2.1). The arguments concerning mixture models (see subsection 2.2) apply there as well. Haussler [3] contains a wealth of material about kernel design for discrete objects x. Watkins [9] mentions that expressions like Q in (1) are valid kernels for discrete x and countable parameter spaces. Very recently we came across [11], which essentially describes a special case of the diagonal mixture kernel (see subsection 2.2) for Gaussian components with diagonal covariances17 . The author calls Q a stochastic equivalence predicate. He is interested in distance learning, does not apply his method to kernel machines and does not give a Bayesian interpretation. We have presented a general framework for kernel learning from unlabeled data and described an approximate implementation using VB-MoFA models. A straightforward application of this technique to high-dimensional real-world data did not prove successful, and in future work we will explore new ideas for extending the basic MI kernel framework in order to be able to deal with high-dimensional input spaces. Acknowledgments We thank Chris Williams for many inspiring discussions, furthermore Ralf Herbrich, Amos Storkey, Hugo Zaragoza and Neil Lawrence. Matt Beal helped us a lot with VB-MoFA. The author gratefully acknowledges support through a research studentship from Microsoft Research Ltd. References [1] Avrim Blum and Tom Mitchell. Combining labeled and unlabeled data with CoTraining. In Proceedings of COLT, 1998. [2] Z. Ghahramani and M. Beal. Variational inference for Bayesian mixtures of factor analysers. In Advances in NIPS 12. MIT Press, 1999. [3] David Haussler. Convolution kernels on discrete structures. Technical Report UCSCCRL-99-10, University of California, Santa Cruz, July 1999. [4] Tommi S. Jaakkola and David Haussler. Exploiting generative models in discriminative classifiers. In Advances in Neural Information Processing Systems 11, 1998. [5] Matthias Seeger. Covariance kernels from Bayesian generative models. Technical report, 2000. Available at http : //yyy . dai . ed. ac . ukr seeger /papers . html. [6] Matthias Seeger. Learning with labeled and unlabeled data. Technical report, 2000. Available at http://yyy .dai. ed. ac. ukrseeger/papers .html. [7] Martin Szummer and Tommi Jaakkola. Partially labeled classification with Markov random walks. In Advances in NIPS 14. MIT Press, 200l. [8] Koji Tsuda, Motoaki Kawanabe, Gunnar Ratsch, Soeren Sonnenburg, and KlausRobert Muller. A new discriminative kernel from probabilistic models. In Advances in NIPS 14. MIT Press, 200l. [9] Chris Watkins. Dynamic alignment kernels. Technical Report CSD-TR-98-11 , Royal Holloway, University of London, 1999. [10] Christopher K.1. Williams and David Barber. Bayesian classification with Gaussian processes. IEEE Trans. PAMI, 20(12):1342- 1351, 1998. [11] Peter Yianilos. Metric learning via normal mixtures. Technical report, NEC Research, Princeton, 1995. 17The a parameter in this work is related to MTS in this case.
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Modeling Temporal Structure in Classical Conditioning Aaron C. Courville1,3 and David S. Touretzky 2,3 1 Robotics Institute, 2Computer Science Department 3Center for the Neural Basis of Cognition Carnegie Mellon University, Pittsburgh, PA 15213-3891 { aarone, dst} @es.emu.edu Abstract The Temporal Coding Hypothesis of Miller and colleagues [7] suggests that animals integrate related temporal patterns of stimuli into single memory representations. We formalize this concept using quasi-Bayes estimation to update the parameters of a constrained hidden Markov model. This approach allows us to account for some surprising temporal effects in the second order conditioning experiments of Miller et al. [1, 2, 3], which other models are unable to explain. 1 Introduction Animal learning involves more than just predicting reinforcement. The well-known phenomena of latent learning and sensory preconditioning indicate that animals learn about stimuli in their environment before any reinforcement is supplied. More recently, a series of experiments by R. R. Miller and colleagues has demonstrated that in classical conditioning paradigms, animals appear to learn the temporal structure of the stimuli [8]. We will review three of these experiments. We then present a model of conditioning based on a constrained hidden Markov model, using quasiBayes estimation to adjust the model parameters online. Simulation results confirm that the model reproduces the experimental observations, suggesting that this approach is a viable alternative to earlier models of classical conditioning which cannot account for the Miller et al. experiments. Table 1 summarizes the experimental paradigms and the results. Expt. 1: Simultaneous Conditioning. Responding to a conditioned stimulus (CS) is impaired when it is presented simultaneously with the unconditioned stimulus (US) rather than preceding the US. The failure of the simultaneous conditioning procedure to demonstrate a conditioned response (CR) is a well established result in the classical conditioning literature [9]. Barnet et al. [1] reported an interesting Phase 1 Phase 2 Test => Result Test => Result Expt. 1 (4)T+ US (4)C -+ T T=> C =>CR Expt.2A (12)T -+ C (8)T -+ US C=> Expt. 2B (12)T -+ C (8)T ---+ US C =>CR Expt. 3 (96)L -+ US -+ X (8)B -+ X X=> B =>CR Table 1: Experimental Paradigms. Phases 1 and 2 represent two stages of training trials, each presented (n) times. The plus sign (+ ) indicates simultaneous presentation of stimuli; the short arrow (-+) indicates one stimulus immediately following another; and the long arrow (-----+) indicates a 5 sec gap between stimulus offset and the following stimulus onset. For Expt. 1, the tone T, click train C, and footshock US were all of 5 sec duration. For Expt. 2, the tone and click train durations were 5 sec and the footshock US lasted 0.5 sec. For Expt. 3, the light L, buzzer E , and auditory stimulus X (either a tone or white noise) were all of 30 sec duration, while the footshock US lasted 1 sec. CR indicates a conditioned response to the test stimulus. second-order extension of the classic paradigm. While a tone CS presented simultaneously with a footshock results in a minimal CR to the tone, a click train preceding the tone (in phase 2) does acquire associative strength, as indicated by a CR. Expt. 2: Sensory Preconditioning. Cole et al. [2] exposed rats to a tone T immediately followed by a click train C. In a second phase, the tone was paired with a footshock US that either immediately followed tone offset (variant A), or occurred 5 sec after tone offset (variant B). They found that when C and US both immediately follow T , little conditioned response is elicited by the presentation of C. However, when the US occurs 5 sec after tone offset, so that it occurs later than C (measured relative to T), then C does come to elicit a CR. Expt. 3: Backward Conditioning. In another experiment by Cole et al. [3], rats were presented with a flashing light L followed by a footshock US, followed by an auditory stimulus X (either a tone or white noise). In phase 2, a buzzer B was followed by X. Testing revealed that while X did not elicit a CR (in fact, it became a conditioned inhibitor), X did impart an excitatory association to B. 2 Existing Models of Classical Conditioning The Rescorla-Wagner model [11] is still the best-known model of classical conditioning, but as a trial-level model, it cannot account for within-trial effects such as second order conditioning or sensitivity to stimulus timing. Sutton and Barto developed V-dot theory [14] as a real-time extension of Rescorla-Wagner. Further refinements led to the Temporal Difference (TD) learning algorithm [14]. These extensions can produce second order conditioning. And using a memory buffer representation (what Sutton and Barto call a complete serial compound), TD can represent the temporal structure of a trial. However, TD cannot account for the empirical data in Experiments 1- 3 because it does not make inferences about temporal relationships among stimuli; it focuses solely on predicting the US. In Experiment 1, some versions of TD can account for the reduced associative strength of a CS when its onset occurs simultaneously with the US, but no version of TD can explain why the second-order stimulus C should acquire greater associative strength than T. In Experiment 2, no learning occurs in Phase 1 with TD because no prediction error is generated by pairing T with C. As a result, no CR is elicited by C after T has been paired with the US in Phase 2. In Experiment 3, TD fails to predict the results because X is not predictive of the US; thus X acquires no associative strength to pass on to B in the second phase. Even models that predict future stimuli have trouble accounting for Miller et al. 's results. Dayan's "successor representation" [4], the world model of Sutton and Pinette [15], and the basal ganglia model of Suri and Schultz [13] all attempt to predict future stimulus vectors. Suri and Schultz's model can even produce one form of sensory preconditioning. However, none of these models can account for the responses in any of the three experiments in Table 1, because they do not make the necessary inferences about relations among stimuli. Temporal Coding Hypothesis The temporal coding hypothesis (TCH) [7] posits that temporal contiguity is sufficient to produce an association between stimuli. A CS does not need to predict reward in order to acquire an association with the US. Furthermore, the association is not a simple scalar quantity. Instead, information about the temporal relationships among stimuli is encoded implicitly and automatically in the memory representation of the trial. Most importantly, TCH claims that memory representations of trials with similar stimuli become integrated in such a way as to preserve the relative temporal information [3]. If we apply the concept of memory integration to Experiment 1, we get the memory representation, C ---+ T + US. If we interpret a CR as a prediction of imminent reinforcement, then we arrive at the correct prediction of a strong response to C and a weak response to T. Integrating the hypothesized memory representations of the two phases of Experiment 2 results in: A) T ---+ C+US and B) T ---+ C ---+ US. The stimulus C is only predictive ofthe US in variant B, consistent with the experimental findings. For Experiment 3, an integrated memory representation of the two phases produces L+ B ---+ US ---+ X. Stimulus B is predictive of the US while X is not. Thus, the temporal coding hypothesis is able to account for the results of each of the three experiments by associating stimuli with a timeline. 3 A Computational Model of Temporal Coding A straightforward formalization of a timeline is a Markov chain of states. For this initial version of our model, state transitions within the chain are fixed and deterministic. Each state represents one instant of time, and at each timestep a transition is made to the next state in the chain. This restricted representation is key to capturing the phenomena underlying the empirical results. Multiple timelines (or Markov chains) emanate from a single holding state. The transitions out of this holding state are the only probabilistic and adaptive transitions in the simplified model. These transition probabilities represent the frequency with which the timelines are experienced. Figure 1 illustrates the model structure used in all simulations. Our goal is to show that our model successfully integrates the timelines of the two training phases of each experiment. In the context of a collection of Markov chains, integrating timelines amounts to both phases of training becoming associated with a single Markov chain. Figure 1 shows the integration of the two phases of Expt. 2B. Figure 1: A depiction of the state and observation structure of the model. Shown are two timelines, one headed by state j and the other headed by state k. State i, the holding state, transitions to states j and k with probabilities aij and aik respectively. Below the timeline representations are a sequence of observations represented here as the symbols T, C and US. The T and C stimuli appear for two time steps each to simulate their presentation for an extended duration in the experiment. During the second phase of the experiments, the second Markov chain (shown in Figure 1 starting with state k) offers an alternative to the chain associated with the first phase of learning. If we successfully integrate the timelines, this second chain is not used. As suggested in Figure 1, associated with each state is a stimulus observation. "Stimulus space" is an n-dimensional continuous space, where n is the number of distinct stimuli that can be observed (tone, light, shock, etc.) Each state has an expectation concerning the stimuli that should be observed when that state is occupied. This expectation is modeled by a probability density function, over this space, defined by a mixture of two multivariate Gaussians. The probability density at stimulus observation xt in state i at time tis, where Wi is a mixture coefficient for the two Gaussians associated with state i. The Gaussian means /tiD and /til and variances ufo and ufl are vectors of the same dimension as the stimulus vector xt. Given knowledge of the state, the stimulus components are assumed to be mutually independent (covariance terms are zero). We chose a continuous model of observations over a discrete observation model to capture stimulus generalization effects. These are not pursued in this paper. For each state, the first Gaussian pdf is non-adaptive, meaning /tiO is fixed about a point in stimulus space representing the absence of stimuli. ufo is fixed as well. For the second Gaussian, /til and Ufl are adaptive. This mixture of one fixed and one adaptive Gaussian is an approximation to the animal's belief distribution about stimuli, reflecting the observed tolerance animals have to absent expected stimuli. Put another way, animals seem to be less surprised by the absence of an expected stimulus than by the presence of an unexpected stimulus. We assume that knowledge of the current state st is inaccessible to the learner. This information must be inferred from the observed stimuli. In the case of a Markov chain, learning with hidden state is exactly the problem of parameter estimation in hidden Markov models. That is, we must update the estimates of w, /tl and ur for each state, and aij for each state transition (out of the holding state), in order to maximize the likelihood of the sequence of observations The standard algorithm for hidden Markov model parameter estimation is the Baum-Welch method [10]. Baum-Welch is an off-line learning algorithm that requires all observations used in training to be held in memory. In a model of classical conditioning, this is an unrealistic assumption about animals' memory capabilities. We therefore require an online learning scheme for the hidden Markov model, with only limited memory requirements. Recursive Bayesian inference is one possible online learning scheme. It offers the appealing property of combining prior beliefs about the world with current observations through the recursive application of Bayes' theorem, p(Alxt) IX p(xtlxt- 1 , A)p(AIXt- 1 ). The prior distribution, p(AIXt- 1 ) reflects the belief over the parameter A before the observation at time t, xt. X t- 1 is the observation history up to time t-l, i.e. X t- 1 = {xt- 1,xt- 2 , ... }. The likelihood, p(xtIXt-l,A) is the probability density over xt as a function of the parameter A. Unfortunately, the implementation of exact recursive Bayesian inference for a continuous density hidden Markov model (CDHMM) is computationally intractable. This is a consequence of there being missing data in the form of hidden state. With hidden state, the posterior distribution over the model parameters, after the observation, is given by N p(Alxt) IX LP(xtlst = i, X t- 1 , A)p(st = iIXt- 1 , A)p(AIXt- 1 ), (2) i=1 where we have summed over the N hidden states. Computing the recursion for multiple time steps results in an exponentially growing number of terms contributing to the exact posterior. We instead use a recursive quasi-Bayes approximate inference scheme developed by Huo and Lee [5], who employ a quasi-Bayes approach [12]. The quasi-Bayes approach exploits the existence of a repeating distribution (natural conjugate) over the parameters for the complete-data CDHMM. (i.e. where missing data such as the state sequence is taken to be known). Briefly, we estimate the value of the missing data. We then use these estimates, together with the observations, to update the hyperparameters governing the prior distribution over the parameters (using Bayes' theorem). This results in an approximation to the exact posterior distribution over CDHMM parameters within the conjugate family of the complete-data CDHMM. See [5] for a more detailed description of the algorithm. Estimating the missing data (hidden state) involves estimating transition probabilities between states, ~0 = Pr(sT = i, ST+1 = jlXt , A), and joint state and mixture component label probabilities ([k = Pr(sT = i, IT = klXt , A). Here zr = k is the mixture component label indicating which Gaussian, k E {a, I}, is the source of the stimulus observation at time T. A is the current estimate of all model parameters. We use an online version of the forward-backward algorithm [6] to estimate ~0 and ([1. The forward pass computes the joint probability over state occupancy (taken to be both the state value and the mixture component label) at time T and the sequence of observations up to time T. The backward pass computes the probability of the observations in a memory buffer from time T to the present time t given the state occupancy at time T. The forward and backward passes over state/observation sequences are combined to give an estimate of the state occupancy at time T given the observations up to the present time t. In the simulations reported here the memory buffer was 7 time steps long (t T = 6). We use the estimates from the forward-backward algorithm together with the observations to update the hyperparameters. For the CDHMM, this prior is taken to be a product of Dirichlet probability density functions (pdfs) for the transition probabilities (aij) , beta pdfs for the observation model mixture coefficients (Wi) and normal-gamma pdfs for the Gaussian parameters (Mil and afl)' The basic hyperparameters are exponentially weighted counts of events, with recency weighting determined by a forgetting parameter p. For example, "'ij is the number of expected transitions observed from state i to state j, and is used to update the estimate of parameter aij. The hyperparameter Vik estimates the number of stimulus observations in state i credited to Gaussian k , and is used to update the mixture parameter Wi. The remaining hyperparameters 'Ij;, ¢, and () serve to define the pdfs over Mil and afl' The variable d in the equations below indexes over stimulus dimensions. Si1d is an estimate of the sample variance, and is a constant in the present model. T _ ((T-1) 1) 1 (:T "'ij P "'ij + + '>ij T _ ((T-1) 1) 1 r T v ik P v ik + + '>ik .I,T .1,(T-1) r T 'l'i1d = P 'I'i1d + '>i1 ,/,T _ p(,/,(T-1) _ 1) + 1H[1 'l'i 1d 'l'i1d 2 2 ( ) 7" 0,,(7"-1 ) ,7" () ()T _ p() T- 1 + (i1 Sild + Po/ i 1d 'il (xT _ II. T- 1 )2 i1d i1d 2 2(p1/Ji;d 1) H[1) d f"'i 1d In the last step of our inference procedure, we update our estimate of the model parameters as the mode of their approximate posterior distribution. While this is an approximation to proper Bayesian inference on the parameter values, the mode of the approximate posterior is guaranteed to converge to a mode of the exact posterior. In the equations below, N is the number of states in the model. T_ v[1- 1 Wi vio + viI -2 4 Results and Discussion The model contained two timelines (Markov chains). Let i denote the holding state and j, k the initial states of the two chains. The transition probabilities were initialized as aij = aik = 0.025 and aii = 0.95. Adaptive Gaussian means Mild were initialized to small random values around a baseline of 10-4 for all states. The exponential forgetting factor was P = 0.9975, and both the sample variances Si1d and the fixed variances aIOd were set to 0.2. We trained the model on each of the experimental protocols of Table 1, using the same numbers of trials reported in the original papers. The model was run continuously through both phases of the experiments with a random intertrial interval. 4 5 '+ noCR CR '+ noCR '+ -:::: -:::: -:g4 C ~3 OJ OJ noCR E E E OJ OJ ~3 ~ () .E 02 .E c C ~2 &!1 "Qi ~1 a: Oi g;1 0 0 trr trr trr T C 0 (A)C (B)C 0 X B Experiment 1 Experiment 2 Experiment 3 Figure 2: Results from 20 runs of the model simulation with each experimental paradigm. On the ordinate is the total reinforcement (US), on a log scale, above the baseline (an arbitrary perception threshold) expected to occur on the next time step. The error bars represent two standard deviations away from the mean. Figure 2 shows the simulation results from each of the three experiments. If we assume that the CR varies monotonically with the US prediction, then in each case, the model's predicted CR agreed with the observations of Miller et al. The CR predictions are the result of the model integrating the two phases of learning into one timeline. At the time of the presentation of the Phase 2 stimuli, the states forming the timeline describing the Phase 1 pattern of stimuli were judged more likely to have produced the Phase 2 stimuli than states in the other timeline, which served as a null hypothesis. In another experiment, not shown here, we trained the model on disjoint stimuli in the two phases. In that situation it correctly chose a separate timeline for each phase, rather than merging the two. We have shown that under the assumption that observation probabilities are modeled by a mixture of Gaussians, and a very restrictive state transition structure, a hidden Markov model can integrate the memory representations of similar temporal stimulus patterns. "Similarity" is formalized in this framework as likelihood under the timeline model. We propose this model as a mechanism for the integration of memory representations postulated in the Temporal Coding Hypothesis. The model can be extended in many ways. The current version assumes that event chains are long enough to represent an entire trial, but short enough that the model will return to the holding state before the start of the next trial. An obvious refinement would be a mechanism to dynamically adjust chain lengths based on experience. We are also exploring a generalization of the model to the semi-Markov domain, where state occupancy duration is modeled explicitly as a pdf. State transitions would then be tied to changes in observations, rather than following a rigid progression as is currently the case. Finally, we are experimenting with mechanisms that allow new chains to be split off from old ones when the model determines that current stimuli differ consistently from the closest matching timeline. Fitting stimuli into existing timelines serves to maximize the likelihood of current observations in light of past experience. But why should animals learn the temporal structure of stimuli as timelines? A collection of timelines may be a reasonable model of the natural world. If this is true, then learning with such a strong inductive bias may help the animal to bring experience of related phenomena to bear in novel situations- a desirable characteristic for an adaptive system in a changing world. Acknowledgments Thanks to Nathaniel Daw and Ralph Miller for helpful discussions. This research was funded by National Science Foundation grants IRI-9720350 and IIS-997S403. Aaron Courville was funded in part by a Canadian NSERC PGS B fellowship. References [1] R. C. Barnet, H. M. Arnold, and R. R. Miller. Simultaneous conditioning demonstrated in second-order conditioning: Evidence for similar associative structure in forward and simultaneous conditioning. Learning and Motivation, 22:253- 268, 1991. [2] R. P. Cole, R. C. Barnet, and R. R. Miller. Temporal encoding in trace conditioning. Animal Learning and Behavior, 23(2):144- 153, 1995. [3] R. P. Cole and R. R. Miller. Conditioned excitation and conditioned inhibition acquired through backward conditioning. Learning and Motivation, 30:129- 156, 1999. [4] P. Dayan. Improving generalization for temporal difference learning: the successor representation. Neural Computation, 5:613- 624, 1993. [5] Q. Huo and C.-H. Lee. On-line adaptive learning of the continuous density hidden Markov model based on approximate recursive Bayes estimate. IEEE Transactions on Speech and Audio Processing, 5(2):161- 172, 1997. [6] V. Krishnamurthy and J. B. Moore. On-line estimation of hidden Markov model parameters based on the Kullback-Leibler information measure. IEEE Transactions on Signal Processing, 41(8):2557- 2573, 1993. [7] L. D. Matzel, F. P. Held, and R. R. Miller. Information and the expression of simultaneous and backward associations: Implications for contiguity theory. Learning and Motivation, 19:317- 344, 1988. [8] R. R. Miller and R. C. Barnet. The role of time in elementary associations. Current Directions in Psychological Science, 2(4):106- 111, 1993. [9] 1. P. Pavlov. Conditioned Reflexes. Oxford University Press, 1927. [10] L. R. Rabiner. A tutorial on hidden Markov models and selected applications III speech recognition. Proceedings of the IEEE, 77(2):257- 285, 1989. [11] R. A. Rescorla and A. R. Wagner. A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement. In A. H. Black and W. F. Prokasy, editors, Classical Conditioning II. Appleton-Century-Crofts, 1972. [12] A. F . M. Smith and U. E. Makov. A quasi-Bayes sequential procedure for mixtures. Journal of the Royal Statistical Society, 40(1):106- 112, 1978. [13] R. E. Suri and W. Schultz. Temporal difference model reproduces anticipatory neural activity. Neural Computation, 13(4):841- 862, 200l. [14] R. S. Sutton and A. G. Barto. Time-derivative models of Pavlovian reinforcement. In M. Gabriel and J. Moore, editors, Learning and Computational Neuroscience: Foundations of Adaptive Networks, chapter 12, pages 497- 537. MIT Press, 1990. [15] R. S. Sutton and B. Pinette. The learning of world models by connectionist networks. In L. Erlbaum, editor, Proceedings of the seventh annual conference of the cognitive science society, pages 54- 64, Irvine, California, August 1985.
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Modeling the Modulatory Effect of Attention on Human Spatial Vision Laurent Itti Computer Science Department, Hedco Neuroscience Building HNB-30A, University of Southern California, Los Angeles, CA 90089-2520, U.S.A. J oehen Braun nstitute of Neuroscience and School of Computing, University of Plymouth, Plymouth Devon PL4 8AA, U.K. Christof Koch Computation and Neural Systems Program, MC 139-74, California Institute of Technology, Pasadena, CA 91125, U.S.A. Abstract We present new simulation results, in which a computational model of interacting visual neurons simultaneously predicts the modulation of spatial vision thresholds by focal visual attention, for five dual-task human psychophysics experiments. This new study complements our previous findings that attention activates a winnertake-all competition among early visual neurons within one cortical hypercolumn. This "intensified competition" hypothesis assumed that attention equally affects all neurons, and yielded two singleunit predictions: an increase in gain and a sharpening of tuning with attention. While both effects have been separately observed in electrophysiology, no single-unit study has yet shown them simultaneously. Hence, we here explore whether our model could still predict our data if attention might only modulate neuronal gain, but do so non-uniformly across neurons and tasks. Specifically, we investigate whether modulating the gain of only the neurons that are loudest, best-tuned, or most informative about the stimulus, or of all neurons equally but in a task-dependent manner, may account for the data. We find that none of these hypotheses yields predictions as plausible as the intensified competition hypothesis, hence providing additional support for our original findings. 1 INTRODUCTION Psychophysical studies as well as introspection indicate that we are not blind outside the focus of attention, and that we can perform simple judgments on objects not being attended to [1], though those judgments are less accurate than in the presence of attention [2, 3]. While attention thus appears not to be mandatory for early vision, there is mounting experimental evidence from single-neuron electrophysiology [4, 5, 6, 7, 8, 9, 10], human psychophysics [11, 12, 13, 14,3, 2, 15, 16] and human functional imaging experiments [17, 18, 19, 20, 21, 22, 23] that focal visual attention modulates, top-down, activity in early sensory processing areas. In the visual domain, this modulation can be either spatially-defined (i.e., neuronal activity only at the retinotopic location attended to is modulated) or feature-based (i.e., neurons with stimulus preference matching the stimulus attended to are enhanced throughout the visual field), or a combination of both [7, 10, 24]. Computationally, the modulatory effect of attention has been described as enhanced gain [8, 10], biased [4] or intensified [14, 2] competition, enhanced spatial resolution [3], sharpened neuronal tuning [5, 25] or as modulated background activity [19], effective stimulus strength [26] or noise [15]. One theoretical difficulty in trying to understand the modulatory effect of attention in computational terms is that, although attention profoundly alters visual perception, it is not equally important to all aspects of vision. While electrophysiology demonstrates "increased firing rates" with attention for a given task, psychophysics show "improved discrimination thresholds" on some other tasks, and functional magnetic resonance imaging (fMRI) reports "increased activation" for yet other tasks, the computational mechanism at the origin of these observations remains largely unknown and controversial. While most existing theories are associated to a specific body of data, and a specific experimental task used to engage attention in a given experiment, we have recently proposed a unified computational account [2] that spans five such tasks (32 thresholds under two attentional conditions, i.e., 64 datapoints in total). This theory predicts that attention activates a winner-take-all competition among neurons tuned to different orientations within a single hyper column in primary visual cortex (area VI). It is rooted in new information-theoretic advances [27], which allowed us to quantitatively relate single-unit activity in a computational model to human psychophysical thresholds. A consequence of our "intensified competition hypothesis" is that attention both increases the gain of early visual neurons (by a factor 3.3), and sharpens their tuning for the orientation (by 40%) and spatial frequency (by 30%). While gain modulation has been observed in some of the single-unit studies mentioned above [8, 10] (although much smaller effects are typically reported, on the order of 10-15%, probably because these studies do not use dual-task paradigms and thus poorly engage the attention of the animal towards or away from the stimulus of interest), and tuning modulation has been observed in other single-unit studies [5, 25], both gain and tuning modulation have not been simultaneously observed in a single electrophysiological set of experiments [10]. In the present study, we thus investigate alternatives to our intensified competition hypothesis which only involve gain modulation. Our previous results [2] have shown that both increased gain and sharper tuning were necessary to simultaneously account for our five pattern discrimination tasks, if those modulatory effects were to equally affect all visual neurons at the location of the stimulus and to be equal for all tasks. Thus, we here extend our computational search space under two new hypotheses: First, we investigate whether attention might only modulate the gain of selected sub-populations of neurons (responding the loudest, best tuned, or most informative about the stimulus) in a task-independent manner. Second, we investigate whether attention might equally modulate the gain of all visual neurons responding to the stimulus, but in a task-dependent manner. Thus, the goal of the present study is to determine, using new computational simulations, whether the modulatory effect of attention on early visual processing might be explained by gain-only modulations, if such modulations are allowed to be sufficiently complex (affecting only select visual neurons, or task-dependent). Although attention certainly affects most stages of visual processing, we here continue to focus on early vision, as it is widely justified by electrophysiological and fMRI evidence that some modulation does happen very early in the processing hierarchy [5, 8, 9, 23]. 2 PSYCHOPHYSICAL DATA Our recent study [2] measured psychophysical thresholds for three pattern discrimination tasks (contrast, orientation and spatial frequency discriminations), and two spatial masking tasks (32 thresholds) . We used a dual-task paradigm to measure thresholds either when attention was fully available to the task of interest (presented in the near periphery), or when it was poorly available because engaged elsewhere by a concurrent attention-demanding task (a letter discrimination task at the center of the display). The results are summarized in Fig. 1 and [2]. 'I' A c:: B C 0 4:: c:: c:: ~ .g .g .S E ~ ~ .;: .S .S ... .§ .~ [' . ~ Q .. 8 I • Fully Attended I ... ~ 0 Poorly Attended ~ ",0.3 I I .~ .~ c:: Q) Q 6 Q I > c:: 810.1 t ; CU oJ ... ::s 1:3 0.2 .g ~ 4j '" E W I I 0£- I ~ : ~ : t~ ; ~ ~ 0.1 I I .s ! c:: c:: 2 ~ f!! a .~ 10-2 /1 4:: 00 0 00 0 10'2 10" ~ 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Mask contrast ~ Contrast Contrast 'I' D E I't I:J\ I:J\ .S .S '" .lC .lC '" 0.4 h '" 0.4 I ~ • Fully Attended ~ :E 1;; :E 1;; ... l!! 0.3 o Poorly Attended ... l!! 0.3 '" C '" c E 8 0.2 1+ E 0 0.2 " c:: W c:: W ~ ~ 0.1 I ~ ~ 0.1 fI I f0 0 20 40 60 80 0 0.5 2 Mask e - Target e C) Mask w / Target w Figure 1: Psychophysical data from Lee et ai. Central targets appeared at 0 - 0.8° eccentricity and measured 0.4° across. Peripheral targets appeared for 250 ms at 4° eccentricity, in a circular aperture of 1.50 • They were either sinusoidal gratings (B, C) or vertical stripes whose luminance profile was given by the 6th derivative of a Gaussian (A, D, E) . Mask patterns were generated by superimposing 100 Gabor filters, positioned randomly within the circular aperture (A, D, E). Thresholds were established with an adaptive staircase method (80 trials per block). A complex pattern of effects is observed, with a strong modulation of orientation and spatial frequency discriminations (B, C) , smaller modulation of contrast discriminations (A) , and modulation of contrast masking that depends on stimulus configurations (D, E). These complex observations can be simultaneously accounted for by our computational model of one hypercolumn in primary visual cortex. 3 COMPUTATIONAL MODEL Linear filters Divisive inhibition Decision The model developed to quantitatively account for this data comprises three successive stages [14, 27]. In the first stage, a bank of Gabor-like linear filters (12 orientations and 5 spatial scales) analyzes a given visual location, similarly to a cortical hyper column. In the second stage, filters nonlinearly interact through both a selfexcitation component, and a divisive inhibition component that is derived from a pool of similarly-tuned units. With E)."o being the linear response from a unit tuned to spatial period A and orientation (), the response R)."o after interactions is given by (see [27] for additional details): R (A.E)."o)' + B ).,,0 (S)O + L W)."o(A',()') (A.E)..!,O')O ' (1) ()..! ,O') EA x 8 where: W (A' ()') = (_ (log(A') -log(A))2 _ (()' - ())2) ).,,0, exp 2A2 2A2 )., 0 (2) is a 2D Gaussian weighting function centered around (A, ()) whose widths are determined by the scalars Ao and A).,. The neurons are assumed to be noisy, with noise variance V{o given by a generalized Poisson model: V{o = (3(R)."o + <:). The third stage relates activity in the population of interacting noisy units to behavioral discrimination performance. To allow us to quantitatively predict thresholds from neural activity for any task, our decision stage assumes that observers perform close to an unbiased efficient statistic, that is, the best possible estimator (in the statistical estimation sense) of the characteristics of the stimulus given the noisy neuronal responses. This methodology (described further in [27]) allows us to quantitatively compute thresholds in any behavioral situation, and eliminates the need for task-dependent assumptions about the decision strategy used by the observers. 4 RESULTS and DISCUSSION The 10 free model parameters (Fig. 2) were automatically adjusted to best fit the psychophysical data from all experiments, using a multidimensional downhill simplex with simulated annealing overhead (see [27]), running on our 16CPU Linux Beowulf system (16 x 733 MHz, 4 GB RAM, 0.5 TB disk; see http://iLab . usc. edu/beo/). Parameters were simultaneously adjusted for both attentional conditions; that is, the total fit error was the sum of the error obtained with the baseline set of parameters on the poorly attended data, and of the error obtained with the same parameters plus some attentional perturbation on the fully attended data. Thus, no bias was given to any of the two attentional conditions. For the "separate fits" (Fig. 2), all parameters were allowed to differ with attention [2], while only the interaction parameters b, 8) could differ in the "intensified competition" case. The "loudest filter" was the one responding loudest to the entire visual pattern presented (stimulus + mask), the "best-tuned filter" was that responding best to the stimulus component alone, and the "most informative filter" was that for which the Fisher information about the stimulus was highest (see (I) ~ Ul CU:!:: Q.LL (I) en "5 (1)..---..-Ul (I) co. ,SE c 0 -0 ~ ... ",S :::l= OLL -I " (I) c ... :::l (I) != UlLL (I) m c ~(I) Ul" CU c 1-8. (I) c Attentional manipulations y Top-down Attention Top-down Attention Top-down Attention [stimulus-dependent; only affects filter responding most to whole (targr+maSk) stimulus] ~. Q ~~ Top-down Attention [stimulus-dependent; only affects filter beslluned to/arget stimulus] ~~Q ~~ Top-down Attention [stimulus-dependent; only affects filter most infonnative a~out target stimulus] ~~ Q ~Q Top-down Attention [affects all filters, but differently for each task] .~ I ~. ... QQQ ... ,,,,,,. . 'y" .... -......... / -........ ..Model Parameters y,o: Interaction strength ~ : ~~; ~::~:~ ~~~~g S, £: Dark noise 11 Light noise S: Semisaturation Fully Poorly Parameters 3.78 1.85 3.421 .80 0 0(0) 9.6913.19 " ,,(oct) 0.440.36 r Q ( " ) 23.01 23.90 :!: ,,(oct) 0.810.18 B / A 0.30 1.10 S / A 10.12 8.05 11 0.17 0.01 £ / Rmu, 0.03 0.11 ***** Fully Poorly Parameters 3.941.40 3.551.00 0 0 (0) 8.92 O,,(OCI) 0.41 I: 9(") 20.80 I: ,,(oct) 0.31 B I A 0.96 S / A 5.39 P 0.56 EI R""" 0.02 ***** Parameters Parameters Fully Poorly 2.22~56 1.03 ~ 6.62 \'""i. 1 ~:~ \.'\ 1.09 0.21 12.16 0.65 0.01 * Fully Poorly 1.40~~ 16:: '\\ 14.71 V-"\. 7.03 0.24 3.00 1.5e-9 8635.52 **** Parameters °9 (" ) o .. (OCI) I: 9(") I: .. (OCI) B I A S I A ~ EI R""" Fully Poorly 1.45 ~05 ~:~~~ * 0.31 \~"\ 3~:!~ .. ~ ~:: '\, 3.4<-4 0.01 Parameters Fully Poorly (1"9(") (I".,(oct) I:9(") r .,(oct) B I A S / A , £/R""", Discussion - very good fit overall - all parameters biologically plausible - attention significantly modulates interactions and noise ***** - very good fit overall - all parameters biologically plausible - modulation of orientation thresholds slightly underestimated - contrast masking with variable mask orientation not perfectly predicted - no modulation of contrast detection threshold - no modulation of orientation thresholds - no modulation of period thresholds - poor prediction of masking - filter tuning too narrow - gain modulation too large - no modulation of orientation thresholds - no modulation of period thresholds - contrast discrimination and masking well fit - only fit predicting broad pooling in spatial period - noise parameters unrealistic - no contrast discrimination "dipper" - power-law rather than sigmoidal contrast response (S=O) - modulation of orientation thresholds slightly underestimated - noise parameter unrealistic *** - very good fit overall - gain modulation unrealistically high, especially for orientation discrimination (filter gain when attending to orientation is > 20x poorly attended) Figure 2: Attentional modulation hypotheses and corresponding model parameters. See next page for the corresponding model predictions on our five tasks, for the hypotheses shown. The middle column shows which parameters were allowed to differ with attention, and the best-fit values for both attentional conditions. ~ 810" " ~ Contrast Increment **** 10 -, "----'--,--"'7----l o 10.2 10" ~ 8'0" ~ !!'10~ 0; • §10" " l 10 ~ ~ ~'0" " l 10'> Mask contrast **** - Fully attended Poorly attended 10"2 10" Mask contrast *** - Fully attended Poorly attended 10'> 10-' Mask contrast **** - Fully attended Poorly attended 10" 10" Mask contrast ** 10.2 10" Mask contrast *** Spatial Frequency Discrimination lM~ R 0.2 ! .0.1 °0 0.2 0.4 0.6 0.8 1 Contrast ***** lM~ R 02 I . ~ 0.1 ••• • °0 0.2 0.4 0.6 0.8 1 Contrast **** !:~~ °0 0.2 0.4 0.6 0.8 1 Contrast Orientation Discrimination .~ C 6 I ! ~ , 2 °0 0.2 0.4 0.6 0.8 1 Contrast ***** .~ C 6 I I ~ , 2 ••• • °0 0.2 0.4 0.6 0.8 1 Contrast *** .~ \ ~ I I I o 0.2 0.4 0.6 0.8 1 Contrast f3~ 1!! c:BI R 0.2 f % 4 • ~ 0.1 • 2 '.. • 00 0.2 0.4 0.6 0.8 1 00 0.2 0.4 0.6 0.8 1 Contrast Contrast f.3~ r R 0.2 j i 4 . ~ 01 '. • 2 • 0 0 0.2 0.4 0.6 0.8 1 00 0.2 0.4 0.6 0.8 1 Contrast Contrast ***** *** ~ 8~ ~ 0.3 ~ 6 !o,\ ~~ i, t I I ~0.1 .~ 2 ' • • 0 0 0.2 0.4 0.6 0.8 1 00 0.2 0.4 0.6 0.8 1 Contrast Contrast ***** ***** Contrast Masking, Variable Mask e • Fully attended Poorly attended °020406080 Mask6 - Target6(O) **** • Fullyattended Poorly attended °020406080 Mask6 - Target6(O) *** • Fullyattended POOrly attended 60 80 Mask6 - Target6(O) • Fully attended Poorly attended °020406080 Mask O- TargetOr} ***** • Fully attended Poorly attended 0020406080 Mask O- TargetOr} ***** • Fully attended Poorly attended 0020406080 MaskO - TargetO(O} ***** Contrast Masking, Variable Mask 00 0.4[8 ~ 0.3 I C ' 8 0.2 L ~ • + ~ 0.1 o 0.5 1 2 MaskwlTargetw ***** 0.4~ ~ 0.3 ~ 0.2 j . . . ~ 0.1 t o 0.5 1 2 MaskwlTargetw **** 0.4~ t: I . I " ! $ 0.1 • o 0.5 1 2 MaskwlTargetw * 0.4~ ~ 0 .. 3 I 8 02 I . Qj • + $ 0.1 t o 0.5 1 2 MaskwfTarget (J) **** 0.4[8 ~ 03 , I 8 0.2 '. Qj • + $ 0.1 o 0.5 1 2 MaskwfTarget (J) ***** 0.4[8 I 0.3 t o 0.2 • ~ • + ~ 0.1 o 0.5 1 2 MaskwlTargetw ***** Figure 3: Model predictions for the different attentional modulation hypotheses studied. The different rows correspond to the different attentional manipulations studied, as labeled in the previous figure. Ratings (stars below the plots) were derived from the residual error of the fits. [14, 27]). Finally, in the "task-dependent" case, the gain of all filters was affected equally (parameter ')'), but with three different values for the contrast (discrimination and masking), orientation and spatial frequency tasks. Overall, very good fits were obtained in the "separate fits" and "intensified competition" conditions (as previously reported), as well as in the "most informative filter" and "taskdependent" conditions (Fig. 3) , while the two remaining hypotheses yielded very poor predictions of orientation and spatial frequency discriminations. In the "most informative filter" case, the dipper in the contrast increment thresholds was missing because the nonlinear response function of the neurons converged to a power law rather than the usually observed sigmoid [27]; thus, this hypothesis lost some of its appeal because of its lower biological plausibility. More importantly, a careful analysis of the very promising results for the "task-dependent" case also revealed their low biological plausibility, with a gain modulation in excess of 20-fold being necessary to explain the orientation discrimination data (Fig. 2). In summary, we found that none of the simpler (gain only) attentional manipulations studied here could explain as well the psychophysical data as our previous manipulation, "intensified competition," which implied that attention both increases the gain and sharpens the tuning of early visual neurons. Two of the four new manipulations studied yielded good quantitative model predictions: affecting the gain of the filter most informative about the target stimulus, and affecting the gain of all filters in a task-dependent manner. In both cases, however, some of the internal model parameters associated with the fits were biologically unrealistic, thus reducing the plausibility of these two hypotheses. In all manipulations studied, the greatest difficulty was in trying to account for the orientation and spatial frequency discrimination data without unrealistically high gain changes (greater than 20-fold). Our results hence provide additional evidence for the hypothesis that sharpening of tuning may be necessary to account for these thresholds, as was originally suggested by our separate fits and our intensified competition hypothesis and has been recently supported by new investigations [16]. Acknowledgements This research was supported by the National Eye Institute, the National Science Foundation, the NSF-supported ERC center at Caltech, the National Institutes for Mental Health, and startup funds from the Charles Lee Powell Foundation and the USC School of Engineering. References [1] Braun J & Sagi D. P ercept Psychophys , 1990;48(1):45- 58. [2] Lee DK, Itti L, Koch C et al. Nat Neurosci, 1999;2(4):375-81. [3] Yeshurun Y & Carrasco M. Nature, 1998;396(6706) :72- 75 . [4] Moran J & Desimone R. Science , 1985;229(4715):782- 4. [5] Spitzer H, Desimone R & Moran J. Science, 1988;240(4850):338- 40. [6] Chelazzi L, Miller EK, Duncan J et al. Nature , 1993;363(6427):345- 7. [7] Motter BC. J Neurosci, 1994;14(4):2178-89. [8] Treue S & Maunsell JH. Nature, 1996;382(6591):539- 41. [9] Luck SJ, Chelazzi L, Hillyard SA et al. J Neurophysiol, 1997;77(1):24- 42. [10] Treue S & Trujillo JCM. Nature, 1999;399(6736) :575- 579. [11] Nakayama K & Mackeben M. Vision Res, 1989;29(11):1631- 47. [12] Bonnel AM, Stein JF & Bertucci P. Q J Exp Psychol A, 1992;44(4):601- 26. [13] Lee DK, Koch C & Braun J. Vision Res, 1997;37(17):2409- 18. [14] Itti L, Braun J, Lee DK et al. In NIPS*ll. MIT Press, 1999; pp. 789- 795. [15] Dosher BA & Lu ZL. Vision Res, 2000;40(10-12):1269- 1292. [16] Carrasco M, Penpeci-Talgar C & Eckstein M. Vision Res, 2000;40(10-12):1203- 1215. [17] Corbett a M, Miezin FM, Dobmeyer S et al. Science, 1990;248(4962):1556- 9. [18] Rees G, Frackowiak R & Frith C. Science, 1997;215(5301):835- 8. [19] Chawla D, Rees G & Friston KJ. Nat Neurosci, 1999;2(7):671- 676. [20] Brefczynski JA & DeYoe EA. Nat Neurosci, 1999;2(4):370- 374. [21] Corbetta M, Kincade JM, Ollinger JM et al. Nat Neurosci, 2000;3(3):292- 297. [22] Kanwisher N & Wojciulik E. Nat Rev Neurosci, 2000;1:91- 100. [23] Ress D, Backus BT & Heeger DJ. Nat Neurosci, 2000;3(9):940- 945. [24] Barcelo F, Suwazono S & Knight RT. Nat Neurosci, 2000;3(4):399- 403. [25] Desimone R & Duncan J. Annu Rev Neurosci, 1995;18:193- 222. [26] Reynolds JH, Pasternak T & Desimone R. Neuron, 2000;26(3):703- 714. [27] Itti L, Koch C & Braun J. J Opt Soc Am A, 2000;11(11):1899- 1917.
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Approximate Dynamic Programming via Linear Programming Daniela P. de Farias Department of Management Science and Engineering Stanford University Stanford, CA 94305 pucci@stanford.edu Benjamin Van Roy Department of Management Science and Engineering Stanford University Stanford, CA 94305 bvr@stanford. edu Abstract The curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of large- scale stochastic control problems. We study an efficient method based on linear programming for approximating solutions to such problems. The approach "fits" a linear combination of pre- selected basis functions to the dynamic programming cost- to- go function. We develop bounds on the approximation error and present experimental results in the domain of queueing network control, providing empirical support for the methodology. 1 Introduction Dynamic programming offers a unified approach to solving problems of stochastic control. Central to the methodology is the cost- to- go function, which can obtained via solving Bellman's equation. The domain of the cost- to- go function is the state space of the system to be controlled, and dynamic programming algorithms compute and store a table consisting of one cost- to- go value per state. Unfortunately, the size of a state space typically grows exponentially in the number of state variables. Known as the curse of dimensionality, this phenomenon renders dynamic programming intractable in the face of problems of practical scale. One approach to dealing with this difficulty is to generate an approximation within a parameterized class of functions, in a spirit similar to that of statistical regression. The focus of this paper is on linearly parameterized functions: one tries to approximate the cost- to- go function J* by a linear combination of prespecified basis functions. Note that this scheme depends on two important preconditions for the development of an effective approximation. First, we need to choose basis functions that can closely approximate the desired cost-to-go function. In this respect, a suitable choice requires some practical experience or theoretical analysis that provides rough information on the shape of the function to be approximated. "Regularities" associated with the function, for example, can guide the choice of representation. Second, we need an efficient algorithm that computes an appropriate linear combination. The algorithm we study is based on a linear programming formulation, originally proposed by Schweitzer and Seidman [5], that generalizes the linear programming approach to exact dynamic programming, originally introduced by Manne [4]. We present an error bound that characterizes the quality of approximations produced by the linear programming approach. The error is characterized in relative terms, compared against the "best possible" approximation of the optimal cost-to-go function given the selection of basis functions. This is the first such error bound for any algorithm that approximates cost- to- go functions of general stochastic control problems by computing weights for arbitrary collections of basis functions. 2 Stochastic control and linear programming We consider discrete- time stochastic control problems involving a finite state space S of cardinality lSI = N. For each state XES, there is a finite set of available actions Ax. Taking action a E Ax when the current state is x incurs cost 9a(X). State transition probabilities Pa(x,y) represent, for each pair (x,y) of states and each action a E Ax, the probability that the next state will be y given that the current state is x and the current action is a E Ax. A policy u is a mapping from states to actions. Given a policy u, the dynamics of the system follow a Markov chain with transition probabilities Pu(x)(x, y). For each policy u, we define a transition matrix Pu whose (x,y)th entry is Pu(x)(x,y). The problem of stochastic control amounts to selection of a policy that optimizes a given criterion. In this paper, we will employ as an optimality criterion infinitehorizon discounted cost of the form Ju(x) =E [~(i9U(Xd lxo =x] , where 9u(X) is used as shorthand for 9u(x)(X) and the discount factor a E (0,1) reflects inter- temporal preferences. Optimality is attained by any policy that is greedy with respect to the optimal cost-to-go function J*(x) = minu Ju(x) (a policy u is called greedy with respect to J if TuJ = T J). Let us define operators Tu and T by TuJ = 9u +aPuJ and T J = minu (9u + aPuJ). The optimal cost-to-go function solves uniquely Bellman's equation J = T J. Dynamic programming offers a number of approaches to solving this equation; one of particular relevance to our paper makes use of linear programming, as we will now discuss. Consider the problem max clJ (1) S.t. T J;::: J, where c is a vector with positive components, which we will refer to as staterelevance wei9hts. It can be shown that any feasible J satisfies J :::; J*. It follows that, for any set of positive weights c, J* is the unique solution to (1). Note that each constraint (T J)(x) ;::: J(x) is equivalent to a set of constraints 9a(X) + a L.YEs Pa(X,y) J(y) ;::: J(x), Va E Ax, so that the optimization problem (1) can be represented as an LP, which we refer to as the exact LP. As mentioned in the introduction, state spaces for practical problems are enormous due to the curse of dimensionality. Consequently, the linear program of interest involves prohibitively large numbers of variables and constraints. The approximation algorithm we study reduces dramatically the number of variables. Let us now introduce the linear programming approach to approximate dynamic programming. Given pre-selected basis functions (Pl, .. . , cPK, define a matrix If> = [ cPl cPK ]. With an aim of computing a weight vector f E ~K such that If>f is a close approximation to J*, one might pose the following optimization problem: max c'lf>r (2) s.t. Tlf>r 2:: If>r. Given a solution f, one might then hope to generate near- optimal decisions by using a policy that is greedy with respect to If>f. As with the case of exact dynamic programming, the optimization problem (2) can be recast as a linear program. We will refer to this problem as the approximate LP. Note that, though the number of variables is reduced to K, the number of constraints remains as large as in the exact LP. Fortunately, we expect that most of the constraints will become irrelevant, and solutions to the linear program can be approximated efficiently, as demonstrated in [3] . 3 Error Bounds for the Approximate LP When the optimal cost- to- go function lies within the span of the basis functions, solution of the approximate LP yields the exact optimal cost-to-go function. Unfortunately, it is difficult in practice to select a set of basis functions that contains the optimal cost- to- go function within its span. Instead, basis functions must be based on heuristics and simplified analyses. One can only hope that the span comes close to the desired cost- to- go function. For the approximate LP to be useful, it should deliver good approximations when the cost- to- go function is near the span of selected basis functions. In this section, we present a bound that ensure desirable results of this kind. To set the stage for development of an error bound, let us establish some notation. First, we introduce the weighted norms, defined by 1IJ111 ~ = '"' ')'(x) IJ(x)l , IIJlloo ~ = max')'(x) IJ(x)l, " ~ " xES xES for any ')' : S f-t ~+. Note that both norms allow for uneven weighting of errors across the state space. We also introduce an operator H, defined by (HV)(x) = max L Pa(x, y)V(y), aEAz y for all V : S f-t R For any V , (HV)(x) represents the maximum expected value of V (y) if the current state is x and y is a random variable representing the next state. Based on this operator, we define a scalar V(x) kv = m,:x V(x) - a(HV)(x) , (3) for each V : S f-t ~. We interpret the argument V of H as a "Lyapunov function," while we view kv as a "Lyapunov stability factor," in a sense that we will now explain. In the upcoming theorem, we will only be concerned with functions V that are positive and that make kv nonnegative. Also, our error bound for the approximate LP will grow proportionately with kv, and we therefore want kv to be small. At a minimum, kv should be finite, which translates to a condition a(HV)(x) < V(x) , "Ix ES. (4) If a were equal to 1, this would look like a Lyapunov stability condition: the maximum expected value (HV)(x) at the next time step must be less than the current value V(x). In general, a is less than 1, and this introduces some slack in the condition. Note also that kv becomes smaller as the (HV)(x)'s become small relative to the V(x)'s. Hence, kv conveys a degree of "stability," with smaller values representing stronger stability. We are now ready to state our main result. For any given function V mapping S to positive reals, we use l/V as shorthand for a function x I-t l/V(x). Theorem 3.1 {2} Let f be a solution of the approximate LP. Then, for any v E 3rK such that (<T>v) (x) > 0 for all xES and aH <T>v < <T>v, IIJ* - <T>flkc :::; 2k<I>v(c'<T>v) min IIJ* - <T>rll oo,l/<I>v· r (5) A proof of Theorem 3.1 can be found in the long version of this paper [2]. We highlight some implications of Theorem 3.1. First, the error bound (5) tells that the the approximation error yielded by the approximate LP is proportional to the error associated with the best possible approximation relative to a certain norm 11·lll,l/<I>v. Hence we expect that the approximate LP will have reasonable behavior - if the choice of basis functions is appropriate, the approximate LP should yield a relatively good approximation to the cost-to-go function, as long as the constants k<I>v and c' <T>v remain small. Note that on the left-hand side of (5), we measure the approximation error with the weighted norm 11·lkc. Recall that the weight vector c appears in objective function of the approximate LP (2) and must be chosen. In approximating the solution to a given stochastic control problem, it seems sensible to weight more heavily portions of the state space that are visited frequently, so that accuracy will be emphasized in such regions. As discussed in [2], it seems reasonable that the weight vector c should be chosen to reflect the relative importance of each state. Finally, note that the Lyapunov function <T>v plays a central role in the bound of Theorem 3.1. Its choice influences three terms on the right-hand-side of the bound: 1. the error minr IIJ* - <T>rlloo,l/<I>v; 2. the Lyapunov stability factor k<I>v; 3. the inner product c' <T>v with the state- relevance weights. An appropriately chosen Lyapunov function should make all three of these terms relatively small. Furthermore, for the bound to be useful in practical contexts, these terms should not grow much with problem size. We now illustrate with an application in queueing problems how a suitable Lyapunov function could be found and show how these terms scale with problem size. 3.1 Example: A Queueing Network Consider a single reentrant line with d queues and finite buffers of size B. We assume that exogenous arrivals occur at queue 1 with probability p < 1/2. The state x E ~d indicates the number of jobs in each queue. The cost per stage incurred at state x is given by the average number of jobs per queue. As discussed in [2], under certain stability assumptions we expect that the optimal cost-to-go function should satisfy O J* () P2 I Pl I ::::; x::::; dX x + de x + Po, for some positive scalars Po, Pl and P2 independent of d. We consider a Lyapunov function V(x) = ~XIX + C for some constant C > 0, which implies m}n IIJ* -lJ>rll oo,l/V < IIJ*lloo,l/V < P2XlX + Plelx + dpo max '-----'---..,-----'-x2:O XiX + dC Po < P2 + Pl + C' and the above bound is independent of the number of queues in the system. Now let us study kv. We have a(HV)(x) < a [p (~XIX + 2X1/ 1 + C) + (1- p) (~XIX + C)] < V(x) (a+ap:;~:~), and it is clear that, for C sufficiently large and independent of d, there is a j3 < 1 independent of d such that aHV ::::; j3V, and therefore kv ::::; 1 ~ ,6 . Finally, let us consider ciV. Discussion presented in [2] suggests that one might want to choose c so as to reflect the stationary state distribution. We expect that under some stability assumptions, the tail of the stationary state distribution will have an upper bound with geometric decay [1]. Therefore we let c(x) = (l!;l+l)d plxl, for some 0 < P < 1. In this case, c is equivalent to the conditional joint distribution of d independent and identically distributed geometric random variables conditioned on the event that they are less than B + 1, and we have clV = E [~t, xl + C I Xi < B + 1, i = 1, ... , d] < 2 (1 ~2p)2 + 1 ~ P + C, where Xi, i = 1, ... , d are identically distributed geometric random variables with parameter 1 - p. It follows that clV is uniformly bounded over the number of queues. This example shows that the terms involved in the error bound (5) are uniformly bounded both in the number of states in the system and in the number of state variables, hence the behavior of the approximate LP does not deteriorate as the problem size increases. We finally present a numerical experiment to further illustrate the performance of the approximate LP. L ", - 3/11.5 J =-r IJ.z - 2/ 11.5 "" - 3/11.5 Al - 1/11.5 ) ~ - 4 /11.5 ) ~5 - 3/ 11.5 ~ -----':7 IC>I"'-" 'U A 2 - 1/11.5 1A8 - 2.5/11.5 ) 1"4 - 3 .1/11.5 l Figure 1: System for Example 3.2. Policy A verage Cost Table 1: Average number of jobs after 50,000,000 simulation steps 3.2 An Eight-Dimensional Queueing Network We consider a queueing network with eight queues. The system is depicted in Figure 1, with arrival P'i, i = 1,2) and departure (J.Li, i = 1, ... ,8) probabilities indicated. The state x E ~8 represents the number of jobs in each queue. The cost-per-state is g(x) = lxi, and the discount factor 0:: is 0.995. Actions a E {O, 1}8 indicate which queues are being served; ai = 1 iff a job from queue i is being processed. We consider only non-iddling policies and, at each time step, a server processes jobs from one of its queues exclusively. We choose c of the form c(x) = (1 - p)8 plxl. The basis functions are chosen to span all polynomials in x of degree 2; therefore, the approximate LP has 47 variables. Constraints (T<I>r)(x) 2: (<I>r)(x) for the approximate LP are generated by sampling 5000 states according to the distribution associated with c. Experiments were performed for p = 0.85,0.9 and 0.95, and p = 0.9 yielded the policy with smallest average cost. We compared the performance of the policy yielded by the approximate LP (ALP) with that of first-in-first-out (FIFO), last-buffer-first-serve (LBFS)l and a policy that serves the longest queue in each server (LONG). The average number of jobs in the system for each policy was estimated by simulation. Results are shown in Table 1. The policy generated by the approximate LP performs significantly better than each of the heuristics, yielding more than 10% improvement over LBFS, the second best policy. We expect that even better results could be obtained by refining the choice of basis functions and state-relevance weights. 4 Closing Remarks and Open Issues In this paper we studied the linear programming approach to approximate dynamic programming for stochastic control problems as a means of alleviating the curse of 1 LBFS serves the job that is closest to leaving the system; for example, if there are jobs in queue 2 and in queue 6, a job from queue 2 is processed since it will leave the system after going through only one more queue, whereas the job from queue 6 will still have to go through two more queues. We also choose to assign higher priority to queue 8 than to queue 3 since queue 8 has higher departure probability. dimensionality. We provided an error bound based on certain assumptions on the basis functions. The bounds were shown to be uniformly bounded in the number of states and state variables in certain queueing problems. Several questions remain open and are the object of future investigation: Can the state-relevance weights in the objective function be chosen in some adaptive way? Can we add robustness to the approximate LP algorithm to account for errors in the estimation of costs and transition probabilities, i.e., design an alternative LP with meaningful performance bounds when problem parameters are just known to be in a certain range? How do our results extend to the average cost case? How do our results extend to the infinite-state case? How does the quality of the approximate value function, measure by the weighted L1 norm, translate into actual performance of the associated greedy policy? Acknowledgements This research was supported by NSF CAREER Grant ECS-9985229, by the ONR under Grant MURI N00014-00-1-0637, and by an IBM Research Fellowship. References [1] Bertsimas, D. , Gamarnik, D. & Tsitsiklis, J., "Performance of Multiclass Markovian Queueing Networks via Piecewise Linear Lyapunov Functions," submitted to Annals of Applied Probability, 2000. [2] de Farias, D.P. & Van Roy, B., "The Linear Programming Approach to Approximate Dynamic Programming," submitted to publication, 200l. [3] de Farias, D.P. & Van Roy, B., "On Constraint Sampling for Approximate Linear Programming," , submitted to publication, 200l. [4] Manne, A.S., "Linear Programming and Sequential Decisions," Management Science 6, No.3, pp. 259-267, 1960. [5] Schweitzer, P. & Seidmann, A., "Generalized Polynomial Approximations in Markovian Decision Processes," Journal of Mathematical Analysis and Applications 110, pp. 568582, 1985.
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A New Discriminative Kernel From Probabilistic Models K. Tsuda,*tM. Kawanabe,* G. Ratsch,§*S. Sonnenburg,* and K.-R. Muller*+ t AIST CBRC, 2-41-6, Aomi, Koto-ku, Tokyo, 135-0064, Japan * Fraunhofer FIRST, Kekulestr. 7, 12489 Berlin, Germany § Australian National U ni versi ty, Research School for Information Sciences and Engineering, Canberra, ACT 0200, Australia + University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany ko ji. tsuda@aist.go.jp, nabe@first.fraunhofer.de , Gunnar.Raetsch@anu.edu.au, {sonne, klaus }@first.fraunhofer.de Abstract Recently, Jaakkola and Haussler proposed a method for constructing kernel functions from probabilistic models. Their so called "Fisher kernel" has been combined with discriminative classifiers such as SVM and applied successfully in e.g. DNA and protein analysis. Whereas the Fisher kernel (FK) is calculated from the marginal log-likelihood, we propose the TOP kernel derived from Tangent vectors Of Posterior log-odds. Furthermore we develop a theoretical framework on feature extractors from probabilistic models and use it for analyzing FK and TOP. In experiments our new discriminative TOP kernel compares favorably to the Fisher kernel. 1 Introduction In classification tasks, learning enables us to predict the output y E {-1, + 1} of some unknown system given the input a! E X based on the training examples {a!i ' y;}i=l' The purpose of a feature extractor f : X --+ ]RD is to convert the representation of data without losing the information needed for classification [3]. When X is a vector space like ]Rd , a variety of feature extractors have been proposed (e.g. Chapter 10 in [3]). However, they are typically not applicable when X is a set of sequences of symbols and does not have the structure of a vector space as in DNA or protein analysis [2]. Recently, the Fisher kernel (FK) [6] was proposed to compute features from a probabilistic model p( a!, Y 18). At first, the parameter estimate 9 is obtained from training examples. Then, the tangent vector of the log marginal likelihood log p( ~ 19) is used as a feature vector. The Fisher kernel refers to the inner product in this feature space, but the method is effectively a feature extractor (also since the features are computed explicitly). The Fisher kernel was combined with a discriminative classifier such as SVM and achieved excellent classification results in several fields, for example in DNA and protein analysis [6, 5]. Empirically, it is reported that the FK-SVM system often outperforms the classification performance of the plug-in estimate. 1 Note that the Fisher kernel is only one possible member in the family of feature extractors f iJ (re) : X --+ ]RD that can be derived from probabilistic models. We call this family "model-dependent feature extractors" . Exploring this family is a very important and interesting subject. Since model-dependent feature extractors have been newly developed, the performance measures for them are not yet established. We therefore first propose two performance measures. Then, we define a new kernel (or equivalently a feature extractor) derived from the Tangent vector Of Posterior log-odds - that we denote as TOP kernel. vVe will analyze the performance of the TOP kernel and the Fisher kernel in terms of our performance measures. Then the TOP kernel is compared favorably to the Fisher kernel in a protein classification experiment. 2 Performance Measures To begin with, let us describe the notations. Let re E X be the input 'point' and y E {-1, +1} be the class label. X may be a finite set or an infinite set like ]Rd. Let us assume that we know the parametric model of the joint probability p( re, y19) where 9 E]RP is the parameter vector. Assume that the model p(re,yI9) is regular [7] and contains the true distribution. Then the true parameter 9 * is uniquely determined. Let iJ be a consistent estimator [1] of 9*, which is obtained by n training examples drawn i.i.d. from p(re , YI9*). Let oed = of 108i , Vof = (OeJ, ... ,Oep !) T , and V~f denote a p X P matrix whose (i,j) element is 02 f 1(08i08j ) . As the Fisher kernel is commonly used in combination with linear classifiers such as SVMs, one reasonable performance measure is the classification error of a linear classifier wTfiJ (re) + b (w E]RD and b E]R) in the feature space. Usually wand b are learned, so the optimal feature extractor is different with regard to each learning algorithm. To cancel out this ambiguity and to make a theoretical analysis possible, we assume the optimal learning algorithm is used. When wand b are optimally chosen, the classification error is R(fiJ) = min E""y<I>[-y(w T fiJ(re ) + b)], wES ,bE~ (2.1 ) where S = {w l llwil = 1,w E ]RD}, <I> [a] is the step function which is 1 when a > 0 and otherwise 0, and E""y denotes the expectation with respect to the true distribution p( re, y 19*). R(f iJ) is at least as large as the Bayes error L * [3] and R(f iJ) = L * only if the linear classifier implements the same decision rule as the Bayes optimal rule. As a related measure, we consider the estimation error of the posterior probability by a logistic regressor F(w T fiJ(re ) + b), with e.g. F(t) = 1/(1 + exp( -t)): D(fiJ) = min E",IF(w T fo(re ) + b) - P(y = +1Ire,9*)I. (2.2) wE~D ,bE~ The relationship between D(fiJ ) and R(fiJ) is illustrated as follows: Let L be the classification error rate of a posterior probability estimator P(y + lire). With regard to L, the following inequality is known[l]: L - L* :s; 2E.,IP(y = +l lre) - P(y = +1 Ire ,9*)I. (2.3) When P(y +llre):= F(w T fiJ(re) + b) , this inequality leads to the following relationship between the two measures R(fiJ) - L* :s; 2D(fiJ)· (2.4) --------------------------1 In classification by plug-in estimate, re is classified by t hresholding the posterior probability fj = sign(P(y = +llre, 0) - 0.5) [1]. Since D(fo ) is an upper bound on R(fo), it is useful to derive a new kernel to minimize D(f 0)' as will be done in Sec. 4. 3 The Fisher kernel The Fisher kernel (FK) is defined 2 as K (;e, ;e') = s(;e ,iJ)TZ-1(iJ)s(;e',iJ) , where s is the Fisher score s(;e,iJ) = (otl1logp(;eliJ) , ... ,Otlp 10gp(;eliJ)) T = \7e logp(;e ,iJ), and Z is the Fisher information matrix: Z(9) = E", [s(;e,9)s(;e,9)TI9]. The theoretical foundation of FK is described in the following theorem [6]: "a kernel classifier employed the Fisher kernel derived from a model that contains the label as a latent variable is, asymptotically, at least as good a classifier as the MAP labeling based on the model" . The theorem says that the Fisher kernel can perform at least as well as the plug-in estimate, if the parameters of linear classifier are properly determined (cf. Appendix A of [6]). With our performance measure, this theorem can be represented more concisely: R(f 0) is bounded by the classification error of the plug-in estimate R(fo) :S; E""y<I>[- y(P(y = + ll;e,iJ) - 0.5)]. (3.1 ) Note that the classification rule constructed by the plug-in estimate P(y = + 11;e, iJ) can also be realized by a linear classifier in feature space. Property (3.1) is important since it guarantees that the Fisher kernel performs better when the optimal w and b are available. However, the Fisher kernel is not the only one to satisfy this inequality. In the next section, we present a new kernel which satisfies (3.1) and has a more appealing theoretical property as well. 4 The TOP Kernel Definition Now we proceed to propose a new kernel: Our aim is to obtain a feature extractor that achieves small D(f 0). When a feature extractor! 0 (;e) satisfies3 W T !o(;e ) + b = p -1(p(y = + 11;e, 9 *)) for all;e E X (4.1 ) with certain values of w and b, we have D(f 0) = O. However, since the true parameter 9 * is unknown, all we can do is to construct ! 0 which approximately satisfies (4.1). Let us define v(;e,9) = p-1 (p (y = +11;e , 9 )) = 10g(P(y = +11;e,9)) -log(P(y = -11;e,9) ), which is called the posterior log-odds of a probabilistic model [1]. By Taylor expansion around the estimate iJ up to the first order4 , we can approximate v(;e,9*) as l' v(;e,9*) ~ v(;e,iJ) + L0tliv( ;e,iJ)(e: - ad. ( 4.2) i=l 2In practice, some variants of the Fisher kernel are used. For example, if the derivative of each class distribution , not marginal, is taken, the feature vector of FK is quite similar to that of our kernel. However, these variants should be deliberately discriminated from the Fisher kernel in theoretical discussions. Throughout this paper including experiments, we adopt the original definition of the Fisher kernel from [6] . 3Notice that p-l (t) = log t - log(l - t) 40 ne can easily derive TOP kernels from higher order Taylor expansions. However, we will only deal with t he first order expansion here, because higher order expansions would induce extremely high dimensional feature vectors in practical cases. Thus, by setting ( 4.3) and w:= w* = (1, 8; - el , ··· , 8; - ep)T, b = 0, (4.4) equation (4.1) is approximately satisfied. Since a Tangent vector Of the Posterior log-odds constitutes the main part of the feature vector, we call the inner product of the two feature vectors "TOP kernel" : ( 4.5) It is easy to verify that the TOP kernel satisfies (3.1) , because we can construct the same decision rule as the plug-in estimate by using the first element only (i.e. w = (1,0, . .. ,0), b = 0). A Theoretical Analysis In this section, we compare the TOP kernel with the plugin estimate in terms of performance measures. Later on, we assume that 0 < P (y = +1Ial,8) < 1 to prevent IV(al,8)1 from going to infinity. Also, it is assumed that VeP (y = +1Ial , 8) and V~P ( y = +1Ial,8) are bounded. Substituting the plug-in estimate denoted by the subscript IT into D(fo ), we have Define 68 = 8 - 8*. By Taylor expansion around 8* , we have where 8 0 = 8* + "(68 (O :S "( :S 1). When the TOP kernel is used, D(fo) :S E",IF((w* )T fo(al)) - P (y = +1 Ial,8*)I , ( 4.7) where w* is defined as in (4.4). Since P is Lipschitz-continuous, there is a finite positive constant M such that IP(a) - P(b)1 :S Mia - bl. Thus, D(fo) :S ME",I(w *)T fo (rn ) - P-l (P(y = + 1Irn, 8* )) I· (4.8) Since (w* ) T f 0 (al) is the Taylor expansion of p - 1 (P(y = + 11al, 8 *)) up to the first order (4.2) , the first order terms of 68 are excluded from the right side of (4.8), thus D(fo ) = 0 (1168 112). Since both, the plug-in and the TOP kernel, depend on the parameter estimate 8, the errors D,,(8) and D(fo) become smaller as 116811 decreases. This shows that if w and b are optimally chosen, the rate of convergence of the TOP kernel is much faster than that of the plug-in estimate. This result is closely related to large sample performances: Assuming that 8 is a n 1/2 consistent estimator with asymptotic normality (e.g. the maximum likelihood estimator) , we have 116811 = Op(n- l / 2)[7J, where 01' denotes stochastic order cf. [1]. So we can directly derive the convergence order as D,,(8) = Op (n- l / 2 ) and D(f 0) = Op(n- l ). By using the relation (2.4) , it follows that R,,(8) - L* = Op(n- l / 2 ) and R(f 0) - L * = Op (n- l ).5 Therefore, the TOP kernel has a much better convergence rate in R(f 0)' which is a strong motivation to use the TOP kernel instead of plug-in estimate. 5For detailed discussion about t he convergence orders of classification error, see C hapter 6 of [1] However, we must notice that this fast rate is possible only when the optimal linear classifier is combined with the TOP kernel. Since non-optimal linear classifiers typically have the rate Op(n- 1/ 2 ) [1], the overall rate is dominated by the slower rate and turns out to be Op(n- 1/ 2 ) . But this theoretical analysis is still meaningful, because it shows the existence of a very efficient linear boundary in the TOP feature space. This result encourages practical efforts to improve linear boundaries by engineering loss functions and regularization terms with e.g. cross validation, bootstrapping or other model selection criteria [1]. Exponential Family: A Special Case ·When the distribution of two classes belong to the exponential family, the TOP kernel can achieve an even better result than shown above. Distributions of the exponential family can be written as q( re , 11) = exp( 11 T t (;I!) +~( 11)) , where t (;I!) is a vector-valued function called sufficient statistics and ~ ( 11) is a normalization factor [4]. Let 0' denote the parameter for class prior probability of the positive model P(y = +1). Then, the probabilistic model IS described as where 8 = {O', 11+1 ' 11- 1}· The posterior log-odds reads The TOP feature vector is described as A A AT ATT iiJ(;I! ) = (v( ;I!,8) ,Oav(re ,8), V'7+1 v(re , 8 ) , V'7 _1 v(;I!,8) ) . where V'7,v(;I!,iJ) = s(ts(re) + V'7,~ s(f,s)) for s = {+1,-1}. So, when w = (1,0, 11+1 - "'+1,11"-1 - "'- 1) T and b is properly set, the true log-odds p - l (P(y = +11;I!,8*)) can be constructed as a linear function in the feature space (4.1). Thus DUiJ) = 0 and RUiJ) = L*. Furthermore, since each feature is represented as a linear function of sufficient statistics t+1 (re) and t - l (re), one can construct an equivalent feature space as (t+ l (re) T, Ll (re) T) T without knowing iJ. This result is important because all graphical models without hidden states can be represented as members of the exponential family, for example markov models [4]. 5 Experiments on Protein Data In order to illustrate that the TOP kernel works well for real-world problems, we will show the results on protein classification. The protein sequence data is obtained from the Superfamily website. 6 This site provides sequence files with different degrees of redundancy filtering; we used the one with 10% redundancy filtering. Here, 4541 sequences are hierarchically labeled into 7 classes, 558 folds, 845 superfamilies and 1343 families according to the SCOP(1.53) scheme. In our experiment, we determine the top category "classes" as the learning target. The numbers of sequences in the classes are listed as 791, 1277, 1015, 915,84,76,383. We only use the first 4 classes, and 6 two-class problems are generated from all pairs among the 4 classes. The 5th and 6th classes are not used because the number of examples is too small. Also, the 7th class is not used because this class is quite different from the others and too easy to classify. In each two-class problem , the examples are randomly divided into 25% training set, 25% validation set and 50% test set. The validation set is used for model selection. 6http://stash.mrc-lmb.cam.ac.uk/SUPERFAMILY / As a probabilistic model for protein sequences, we make use of hidden markov models [2] with fully connected states.7 The Baum-Welch algorithm (e.g. [2]) is used for maximum likelihood training. To construct FK and TOP kernels, the derivatives with respect to all parameters of the HMMs from both classes are included. The derivative with respect to the class prior probability is included as well: Let q( OIl , e) be the probability density function of a HMM. Then, the marginal distribution is written as p(ocI8) = aq( oc, e+1 ) + (1- a)q( oc, L1), where a is a parameter corresponding to the class prior. The feature vector of FK consists of the following: V'e, logp( oc I8) 00: logp(oc I8) P(y=s loc ,8)V'e, logq(oc ,es) SE {-l,+l} 1 , 1 ' --;;-P(y = +1 1°c, 9) - --, P(y = -11°c, 9), a I -a (5.1 ) (5.2) while the feature vector of TOP includes V'e,v( oc ,8) sV'e, logq( oc ,es) s = {+ 1, _ 1}.8 We obtained e+1 and e-1 from the training examples of respective classes and set a = 0.5. In the definition of the TOP kernel (4.5), we did not include any normalization of feature vectors. However, in practical situations, it is effective to normalize features for improving classification performance. Here, each feature of the TOP kernel is normalized to have mean 0 and variance 1. Also in FK, we normalized the features in the same way instead of using the Fisher information matrix, because it is difficult to estimate it reliably in a high dimensional parameter space. Both, the TOP kernel and FK are combined with SVMs with bias terms. When classifying with HMMs, one observes the difference of the log-likelihoods for the two classes and discriminates by thresholding at an appropriate value. Theoretically, this threshold should be determined by the (true) class prior probability. But, this is typically not available. Furthermore the estimation of the prior probability from training data often leads to poor results [2]. To avoid this problem, the threshold is determined such that the false positive rate and the false negative rate are equal in the test set. This threshold is determined in the same way for FK-SVMs and TOP-SVMs. The hybrid HMM-TOP-SVM system has several model parameters: the number of HMM states, the pseudo count value [2] and the regularization parameter C of the SVM. vVe determine these parameters as follows: First, the number of states and the pseudo count value are determined such that the error of the HMM on the validation set (i.e. validation error) is minimized. Based on the chosen HMM model, the parameter C is determined such that the validation error of TOP-SVM is minimized. Here, the number of states and the pseudo count value are chosen from {3, 5,7,10,15,20,30,40, 60} and {l0-10, 10- 7 , 10- 5 , 10- 4 ,10- 3 , 1O- 2 }, respectively. For C, 15 equally spaced points on the log scale are taken from [10-4 ,101]. Note that the model selection is performed in the same manner for the Fisher kernel as well. The error rates over 15 different training/validation/test divisions are shown in Figure 1 and 2. The results of statistical tests are shown in Table 1 as well. Compared with the plug-in estimate, the Fisher kernel performed significantly better in several settings (i.e. 1-3, 2-3, 3-4). This result partially agrees with observations in [6]. However, our TOP approach significantly outperforms the Fisher kernel: According to the Wilcoxson signed ranks test, the TOP kernel was significantly better 7Several HMM models have been engineered for protein classification [2]. However, we do not use such HMMs because the main purpose of experiment is to compare FK and TOP. 8 0aV (OC, 0) is a constant which does not depend on OIl. So it is not included in the feature vector. 1-2 1-3 1-4 0.1 6 0.3 0.36 1 0.1 4 ~ ~ ~ ""I ~ ~ 0.25 0.32 ~ 0.1 2 0.3 0.1 0.2 $ 0.281 0.26 0.08 0.24 0.15 P FK TOP P FK TOP P FK TOP 2-3 2-4 3-4 0.18 0.32 0 4 1 ~ t8 0.3 ! :::1 , ~ 0.16 ~ ffi 0.28 ~ 0.14 0.26 034 ~ 0.24 032, 0.12 0 3 1 0.22 0.1 0 28 0.2 I P FK TOP P FK TOP P FK TOP Figure 1: The error rates of SVMs with two feature extractors in the protein classification experiments. T he labels 'P','FK','TOP' denote the plug-in estimate, the Fisher kernel and the TOP kernel, respectively. The title on each subfigure shows the two protein classes used for the experiment. 1-2 0.14 0.12 0.1 0.08 0·ct~06 0.08 0.1 0.12 0.14 TOP Figure 2: Comparison of the error rates of the Fisher kernel and the TOP kernel in discrimination between class 1 and 2. Every point corresponds to one of 15 different training/validation/test set splits. Except two cases, t he T OP kernel achieves smaller error rates. in all settings. Also, the t-test judged that the difference is significant except for 1-4 and 2-4. This indicates that the T OP kernel was able to capture discriminative information better than the Fisher kernel. 6 Conclusion In this study, we presented the new discriminative TOP kernel derived from probabilistic models. Since the theoretical framework for such kernels has so far not been established, we proposed two performance measures to analyze them and gave bounds and rates to gain a better insight into model dependent feature extractors from probabilistic models. Experimentally, we showed that the T OP kernel compares favorably to FK in a realistic protein classification experiment . Note that Smith and Gales[8] have shown that a similar approach works excellently in speech recognition tasks as well. Future research will focus on constructing small sample bounds for the T OP kernel to extend the validity of this work. Since other nonlinear transformations F are possible for obtaining different and possibly better features, we will furthermore consider to learn t he nonlinear transformation F from training samples. An interesting point is that so far T OP kernels perform local linear approximations, it would be interesting to move in the direction of local or even Table 1: P-values of statistical tests in the protein classification experiments. Two kinds of tests, t-test (denoted as T in the table) and Wilcoxson signed ranks test (i.e. WX) , are used. When the difference is significant (p-value < 0.05), a single star * is put beside the value. Double stars ** indicate that the difference is very significant (p-value < 0.01). I Methods I Test II 1-2 1-3 1-4 P, FK T 0.95 0.14 0.78 WX 0.85 0.041 * 0.24 P, TOP T 0.015* 1.7 X 10 - ~** 0.11 WX 4.3 X 10-4** 6.1 X 10-5** 0.030* FK,TOP T 0.0093** 2.2 X 10 -4** 0.21 WX 8.5 X 10- 4** 6.1 X 10- 5** 0.048* I Methods I Test II 2-3 2-4 3-4 P, FK T 0.0032** 0.79 0.12 WX 0.0040** 0.80 0.026* P, TOP T 3.0 X 10 -1~** 0.059 5.3 X 10 -0** WX 6.1 X 10- 5** 0.035* 3.1 X 10- 4** FK,TOP T 2.6 X 10 -M* 0.079 0.0031 ** WX 6.1 X 10- 5** 0.0034** 1.8 X 10- 4** global nonlinear expansions. Acknowledgments vVe thank T. Tanaka, M. Sugiyama, S.-I. Amari, K. Karplus, R. Karchin, F. Sohler and A. Zien for valuable discussions. Moreover, we gratefully acknowledge partial support from DFG (JA 379/9-1, MU 987/1-1) and travel grants from EU (Neurocolt II). References [1] L. Devroye, L. Gyorfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer, 1996. [2] R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, 1998. [3] K. Fukunaga. Introduction to Statistical Pattern Recognition. Academic Press, San Diego, 2nd edition, 1990. [4] D. Geiger and C. Meek. Graphical models and exponential families. Technical Report MSR-TR-98-10, Microsoft Research, 1998. [5] T.S. Jaakkola, M. Diekhans, and D. Haussler. A discriminative framework for detecting remote protein homologies. J. Compo Biol., 7:95-114, 2000. [6] T.S. Jaakkola and D. Haussler. Exploiting generative models in discriminative classifiers. In M.S. Kearns, S.A. SoHa, and D.A. Cohn, editors, Advances in Neural Information Processing Systems 11, pages 487- 493. MIT Press, 1999. [7] N. Murata, S. Yoshizawa, and S. Amari. Network information criterion determining the number of hidden units for an artificial neural network model. IEEE Trans. Neural Networks, 5:865- 872, 1994. [8] N. Smith and M. Gales. Speech recognition using SVMs. In T.G. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems 14. MIT Press, 2002. to appear.
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Kernel Feature Spaces and Nonlinear Blind Source Separation Stefan Harmeling1∗, Andreas Ziehe1, Motoaki Kawanabe1, Klaus-Robert Müller1,2 1Fraunhofer FIRST.IDA, Kekuléstr. 7, 12489 Berlin, Germany 2University of Potsdam, Department of Computer Science, August-Bebel-Strasse 89, 14482 Potsdam, Germany {harmeli,ziehe,kawanabe,klaus}@first.fhg.de Abstract In kernel based learning the data is mapped to a kernel feature space of a dimension that corresponds to the number of training data points. In practice, however, the data forms a smaller submanifold in feature space, a fact that has been used e.g. by reduced set techniques for SVMs. We propose a new mathematical construction that permits to adapt to the intrinsic dimension and to find an orthonormal basis of this submanifold. In doing so, computations get much simpler and more important our theoretical framework allows to derive elegant kernelized blind source separation (BSS) algorithms for arbitrary invertible nonlinear mixings. Experiments demonstrate the good performance and high computational efficiency of our kTDSEP algorithm for the problem of nonlinear BSS. 1 Introduction In a widespread area of applications kernel based learning machines, e.g. Support Vector Machines (e.g. [19, 6]) give excellent solutions. This holds both for problems of supervised and unsupervised learning (e.g. [3, 16, 12]). The general idea is to map the data xi (i = 1, . . . , T) into some kernel feature space F by some mapping Φ : ℜn →F. Performing a simple linear algorithm in F, then corresponds to a nonlinear algorithm in input space. Essential ingredients to kernel based learning are (a) VC theory that can provide a relation between the complexity of the function class in use and the generalization error and (b) the famous kernel trick k(x, y) = Φ(x) · Φ(y), (1) which allows to efficiently compute scalar products. This trick is essential if e.g. F is an infinite dimensional space. Note that even though F might be infinite dimensional the subspace where the data lies is maximally T -dimensional. However, the data typically forms an even smaller subspace in F (cf. also reduced set methods [15]). In this work we therefore propose a new mathematical construction that allows us to adapt to the intrinsic dimension and to provide an orthonormal basis of this submanifold. Furthermore, this makes computations much simpler and provides the basis for a new set of kernelized learning algorithms. ∗To whom correspondence should be addressed. To demonstrate the power of our new framework we will focus on the problem of nonlinear BSS [2, 18, 9, 10, 20, 11, 13, 14, 7, 17, 8] and provide an elegant kernel based algorithm for arbitrary invertible nonlinearities. In nonlinear BSS we observe a mixed signal of the following structure xt = f(st), (2) where xt and st are n×1 column vectors and f is a possibly nonlinear function from ℜn to ℜn. In the special case where f is an n×n matrix we retrieve standard linear BSS (e.g. [8, 4] and references therein). Nonlinear BSS has so far been only applied to industrial pulp data [8], but a large class of applications where nonlinearities can occur in the mixing process are conceivable, e.g. in the fields of telecommunications, array processing, biomedical data analysis (EEG, MEG, EMG, . . .) and acoustic source separation. Most research has so far centered on post-nonlinear models, i.e. xt = f(Ast), (3) where A is a linear mixing matrix and f is a post-nonlinearity that operates componentwise. Algorithmic solutions of eq.(3) have used e.g. self-organizing maps [13, 10], extensions of GTM [14], neural networks [2, 11] or ensemble learning [18] to unfold the nonlinearity f. Also a kernel based method was tried on very simple toy signals; however with some stability problems [7]. Note, that all existing methods are of high computational cost and depending on the algorithm are prone to run into local minima. In our contribution to the general invertable nonlinear BSS case we apply a standard BSS technique [21, 1] (that relies on temporal correlations) to mapped signals in feature space (cf. section 3). This is not only mathematically elegant (cf. section 2), but proves to be a remarkably stable and efficient algorithm with high performance, as we will see in the experiments on nonlinear mixtures of toy and speech data (cf. section 4). Finally, a conclusion is given in section 5. 2 Theory An orthonormal basis for a subspace in F In order to establish a linear problem in feature space that corresponds to some nonlinear problem in input space we need to specify how to map inputs x1, . . . , xT ∈ℜn into the feature space F and how to handle its possibly high dimensionality. In addition to the inputs, consider some further points v1, . . . , vd ∈ℜn from the same space, that will later generate a basis in F. Alternatively, we could use kernel PCA [16]. However, in this paper we concentrate on a different method. Let us denote the mapped points by Φx := [Φ(x1) · · · Φ(xT )] and Φv := [Φ(v1) · · · Φ(vd)]. We assume that the columns of Φv constitute a basis of the column space1 of Φx, which we note by span(Φv) = span(Φx) and rank(Φv) = d. (4) Moreover, Φv being a basis implies that the matrix Φ⊤ v Φv has full rank and its inverse exists. So, now we can define an orthonormal basis Ξ := Φv(Φ⊤ v Φv)−1 2 (5) the column space of which is identical to the column space of Φv. Consequently this basis Ξ enables us to parameterize all vectors that lie in the column space of Φx by some vectors in ℜd. For instance for vectors PT i=1 αΦiΦ(xi), which we write more compactly as ΦxαΦ, and ΦxβΦ in the column space of Φx with αΦ and βΦ in ℜT there exist αΞ and βΞ in ℜd such that ΦxαΦ = ΞαΞ and ΦxβΦ = ΞβΞ. The orthonormality implies α⊤ ΦΦ⊤ x ΦxβΦ = α⊤ Ξ Ξ⊤ΞβΞ = α⊤ ΞβΞ (6) F feature space ℜn ℜd parameter space input space span(Ξ) Figure 1: Input data are mapped to some submanifold of F which is in the span of some ddimensional orthonormal basis Ξ. Therefore these mapped points can be parametrized in ℜd. The linear directions in parameter space correspond to nonlinear directions in input space. which states the remarkable property that the dot product of two linear combinations of the columns of Φx in F coincides with the dot product in ℜd. By construction of Ξ (cf. eq.(5)) the column space of Φx is naturally isomorphic (as vector spaces) to ℜd. Moreover, this isomorphism is compatible with the two involved dot products as was shown in eq.(6). This implies that all properties regarding angles and lengths can be taken back and forth between the column space of Φx and ℜd. The space that is spanned by Ξ is called parameter space. Figure 1 pictures our intuition: Usually kernel methods parameterize the column space of Φx in terms of the mapped patterns {Φ(xi)} which effectively corresponds to vectors in ℜT . The orthonormal basis from eq.(5), however enables us to work in ℜd i.e. in the span of Ξ, which is extremely valuable since d depends solely on the kernel function and the dimensionality of the input space. So d is independent of T . Mapping inputs Having established the machinery above, we will now show how to map the input data to the right space. The expressions (Φ⊤ v Φv)ij = Φ(vi)⊤Φ(vj) = k(vi, vj) with i, j = 1 . . . d are the entries of a real valued d × d matrix Φ⊤ v Φv that can be effectively calculated using the kernel trick and by construction of v1, . . . , vd, it has full rank and is thus invertible. Similarly we get (Φ⊤ v Φx)ij = Φ(vi)⊤Φ(xj) = k(vi, xj) with i = 1 . . . d, j = 1 . . . T, which are the entries of the real valued d × T matrix Φ⊤ v Φx. Using both matrices we compute finally the parameter matrix Ψx := Ξ⊤Φx = (Φ⊤ v Φv)−1 2 Φ⊤ v Φx (7) 1The column space of Φx is the space that is spanned by the column vectors of Φx, written span(Φx). which is also a real valued d × T matrix; note that (Φ⊤ v Φv)−1 2 is symmetric. Regarding computational costs, we have to evaluate the kernel function O(d2) + O(dT ) times and eq.(7) requires O(d3) multiplications; again note that d is much smaller than T . Furthermore storage requirements are cheaper as we do not have to hold the full T × T kernel matrix but only a d × T matrix. Also, kernel based algorithms often require centering in F, which in our setting is equivalent to centering in ℜd. Fortunately the latter can be done quite cheaply. Choosing vectors for the basis in F So far we have assumed to have points v1, . . . , vd that fulfill eq.(4) and we presented the beneficial properties of our construction. In fact, v1, . . . , vd are roughly analogous to a reduced set in the support vector world [15]. Note however that often we can only approximately fulfill eq.(4), i.e. span(Φv) ≈span(Φx). (8) In this case we strive for points that provide the best approximation. Obviously d is finite since it is bounded by T , the number of inputs, and by the dimensionality of the feature space. Before formulating the algorithm we define the function rk(n) for numbers n by the following process: randomly pick n points v1, . . . , vn from the inputs and compute the rank of the corresponding n × n matrix Φ⊤ v Φv. Repeating this random sampling process several times (e.g. 100 times) stabilizes this process in practice. Then we denote by rk(n) the largest achieved rank; note that rk(n) ≤n. Using this definition we can formulate a recipe to find d (the dimension of the subspace of F): (1) start with a large d with rk(d) < d. (2) Decrement d by one as long as rk(d) < d holds. As soon as we have rk(d) = d we found the d. Choose v1, . . . , vd as the vectors that achieve rank d. As an alternative to random sampling we have also employed k-means clustering with similar results. 3 Nonlinear blind source separation To demonstrate the use of the orthonormal basis in F, we formulate a new nonlinear BSS algorithm based on TDSEP [21]. We start from a set of points v1, . . . , vd, that are provided by the algorithm from the last section such that eq.(4) holds. Next, we use eq.(7) to compute Ψx[t] := Ξ⊤Φ(x[t]) = (Φ⊤ v Φv)−1 2 Φ⊤ v Φ(x[t]) ∈ℜd. Hereby we have transformed the time signals x[t] from input space to parameter space signals Ψx[t] (cf. Fig.1). Now we apply the usual TDSEP ([21]) that relies on simultaneous diagonalisation techniques [5] to perform linear blind source separation on Ψx[t] to obtain d linear directions of separated nonlinear components in input space. This new algorithm is denoted as kTDSEP (kernel TDSEP); in short, kTDSEP is TDSEP on the parameter space defined in Fig.1. A key to the success of our algorithm are the time correlations exploited by TDSEP; intuitively they provide the ‘glue’ that yields the coherence for the separated signals. Note that for a linear kernel functions the new algorithm performs linear BSS. Therefore linear BSS can be seen as a special case of our algorithm. Note that common kernel based algorithms which do not use the d-dimensional orthonormal basis will run into computational problems. They need to hold and compute with a kernel matrix that is T × T instead of d × T with T ≫d in BSS problems. A further problem is that manipulating such a T × T matrix can easily become unstable. Moreover BSS methods typically become unfeasible for separation problems of dimension T .
! Figure 2: Scatterplot of x1 vs x2 for nonlinear mixing and demixing (upper left and right) and linear demixing and true source signals (lower left and right). Note, that the nonlinear unmixing agrees very nicely with the scatterplot of the true source signal. 4 Experiments In the first experiment the source signals s[t] = [s1[t] s2[t]]⊤are a sinusoidal and a sawtooth signal with 2000 samples each. The nonlinearly mixed signals are defined as (cf. Fig.2 upper left panel) x1[t] = exp(s1[t]) −exp(s2[t]) x2[t] = exp(−s1[t]) + exp(−s2[t]). A dimension d = 22 of the manifold in feature space was obtained by kTDSEP using a polynomial kernel k(x, y) = (x⊤y + 1)6 by sampling from the inputs. The basisgenerating vectors v1, . . . , v22 are shown as big dots in the upper left panel of Figure 2. Applying TDSEP to the 22 dimensional mapped signals Ψx[t] we get 22 components in parameter space. A scatter plot with the two components that best match the source signals are shown in the right upper panel of Figure 2. The left lower panel also shows for comparison the two components that we obtained by applying linear TDSEP directly to the mixed signals x[t]. The plots clearly indicate that kTDSEP has unfolded the nonlinearity successfully while the linear demixing algorithm failed. In a second experiment two speech signals (with 20000 samples, sampling rate 8 kHz) that are nonlinearly mixed by x1[t] = s1[t] + s3 2[t] x2[t] = s3 1[t] + tanh(s2[t]). This time we used a Gaussian RBF kernel k(x, y) = exp(−|x −y|2). kTDSEP identified d = 41 and used k-means clustering to obtain v1, . . . , v41. These points are marked as ’+’ in the left panel of figure 4. An application of TDSEP to the 41 dimensional parameter mixture kTDSEP TDSEP x1 x2 u1 u2 u1 u2 s1 0.56 0.72 0.89 0.07 0.09 0.72 s2 0.63 0.46 0.04 0.86 0.31 0.55 Table 3: Correlation coefficients for the signals shown in Fig.4. space yields nonlinear components whose projections to the input space are depicted in the right lower panel. We can see that linear TDSEP (right middle panel) failed and that the directions of best matching kTDSEP components closely resemble the sources. To confirm this visual impression we calculated the correlation coefficients of the kTDSEP and TDSEP solution to the source signals (cf. table 3). Clearly, kTDSEP outperforms the linear TDSEP algorithm, which is of course what one expects. 5 Conclusion Our work has two main contributions. First, we propose a new formulation in the field of kernel based learning methods that allows to construct an orthonormal basis of the subspace of kernel feature space F where the data lies. This technique establishes a highly useful (scalar product preserving) isomorphism between the image of the data points in F and a d-dimensional space ℜd. Several interesting things follow: we can construct a new set of efficient kernel-based algorithms e.g. a new and eventually more stable variant of kernel PCA [16]. Moreover, we can acquire knowledge about the intrinsic dimension of the data manifold in F from the learning process. Second, using our new formulation we tackle the problem of nonlinear BSS from the viewpoint of kernel based learning. The proposed kTDSEP algorithm allows to unmix arbitrary invertible nonlinear mixtures with low computational costs. Note, that the important ingredients are the temporal correlations of the source signals used by TDSEP. Experiments on toy and speech signals underline that an elegant solution has been found to a challenging problem. Applications where nonlinearly mixed signals can occur, are found e.g. in the fields of telecommunications, array processing, biomedical data analysis (EEG, MEG, EMG, . . .) and acoustic source separation. In fact, our algorithm would allow to provide a softwarebased correction of sensors that have a nonlinear characteristics e.g. due to manufacturing errors. Clearly kTDSEP is only one algorithm that can perform nonlinear BSS; kernelizing other ICA algorithms can be done following our reasoning. Acknowledgements The authors thank Benjamin Blankertz, Gunnar Rätsch, Sebastian Mika for valuable discussions. This work was partly supported by the EU project (IST1999-14190 – BLISS) and DFG (JA 379/9-1, MU 987/1-1). 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 2 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 2 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 2 1 −6 −4 −2 0 2 4 −6 −4 −2 0 2 4 Figure 4: A highly nonlinear mixture of two speech signals: Scatterplot of x1 vs x2 and the waveforms of the true source signals (upper panel) in comparison to the best matching linear and nonlinear separation results are shown in the middle and lower panel, respectively. References [1] A. Belouchrani, K. Abed Meraim, J.-F. Cardoso, and E. Moulines. A blind source separation technique based on second order statistics. IEEE Trans. on Signal Processing, 45(2):434–444, 1997. [2] G. Burel. Blind separation of sources: a nonlinear neural algorithm. Neural Networks, 5(6):937–947, 1992. [3] C.J.C. Burges. A tutorial on support vector machines for pattern recognition. Knowledge Discovery and Data Mining, 2(2):121–167, 1998. [4] J.-F. Cardoso. Blind signal separation: statistical principles. Proceedings of the IEEE, 9(10):2009–2025, 1998. [5] J.-F. Cardoso and A. Souloumiac. Jacobi angles for simultaneous diagonalization. SIAM J.Mat.Anal.Appl., 17(1):161 ff., 1996. [6] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge University Press, Cambridge, UK, 2000. [7] C. Fyfe and P. L. Lai. ICA using kernel canonical correlation analysis. In Proc. Int. Workshop on Independent Component Analysis and Blind Signal Separation (ICA2000), pages 279–284, Helsinki, Finland, 2000. [8] A. Hyvarinen, J. Karhunen, and E. Oja. Independent Component Analysis. Wiley, 2001. [9] T.-W. Lee, B.U. Koehler, and R. Orglmeister. Blind source separation of nonlinear mixing models. In Neural Networks for Signal Processing VII, pages 406–415. IEEE Press, 1997. [10] J. K. Lin, D. G. Grier, and J. D. Cowan. Faithful representation of separable distributions. Neural Computation, 9(6):1305–1320, 1997. [11] G. Marques and L. Almeida. Separation of nonlinear mixtures using pattern repulsion. In Proc. Int. Workshop on Independent Component Analysis and Signal Separation (ICA’99), pages 277– 282, Aussois, France, 1999. [12] K.-R. Müller, S. Mika, G. Rätsch, K. Tsuda, and B. Schölkopf. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 12(2):181–201, 2001. [13] P. Pajunen, A. Hyvärinen, and J. Karhunen. Nonlinear blind source separation by selforganizing maps. In Proc. Int. Conf. on Neural Information Processing, pages 1207–1210, Hong Kong, 1996. [14] P. Pajunen and J. Karhunen. A maximum likelihood approach to nonlinear blind source separation. In Proceedings of the 1997 Int. Conf. on Artificial Neural Networks (ICANN’97), pages 541–546, Lausanne, Switzerland, 1997. [15] B. Schölkopf, S. Mika, C.J.C. Burges, P. Knirsch, K.-R. Müller, G. Rätsch, and A.J. Smola. Input space vs. feature space in kernel-based methods. IEEE Transactions on Neural Networks, 10(5):1000–1017, September 1999. [16] B. Schölkopf, A.J. Smola, and K.-R. Müller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299–1319, 1998. [17] A. Taleb and C. Jutten. Source separation in post-nonlinear mixtures. IEEE Trans. on Signal Processing, 47(10):2807–2820, 1999. [18] H. Valpola, X. Giannakopoulos, A. Honkela, and J. Karhunen. Nonlinear independent component analysis using ensemble learning: Experiments and discussion. In Proc. Int. Workshop on Independent Component Analysis and Blind Signal Separation (ICA2000), pages 351–356, Helsinki, Finland, 2000. [19] V.N. Vapnik. The nature of statistical learning theory. Springer Verlag, New York, 1995. [20] H. H. Yang, S.-I. Amari, and A. Cichocki. Information-theoretic approach to blind separation of sources in non-linear mixture. Signal Processing, 64(3):291–300, 1998. [21] A. Ziehe and K.-R. Müller. TDSEP—an efficient algorithm for blind separation using time structure. In Proc. Int. Conf. on Artificial Neural Networks (ICANN’98), pages 675–680, Skövde, Sweden, 1998.
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The Unified Propagation and Scaling Algorithm Yee Whye Teh Department of Computer Science University of Toronto 10 King’s College Road Toronto M5S 3G4 Canada ywteh@cs.toronto.edu Max Welling Gatsby Computational Neuroscience Unit University College London 17 Queen Square London WC1N 3AR U.K. welling@gatsby.ucl.ac.uk Abstract In this paper we will show that a restricted class of constrained minimum divergence problems, named generalized inference problems, can be solved by approximating the KL divergence with a Bethe free energy. The algorithm we derive is closely related to both loopy belief propagation and iterative scaling. This unified propagation and scaling algorithm reduces to a convergent alternative to loopy belief propagation when no constraints are present. Experiments show the viability of our algorithm. 1 Introduction For many interesting models, exact inference is intractible. Trees are a notable exception where Belief Propagation (BP) can be employed to compute the posterior distribution [1]. BP on loopy graphs can still be understood as a form of approximate inference since its fixed points are stationary points of the Bethe free energy [2]. A seemingly unrelated problem is that of finding the distribution with minimim KL divergence to a prior distribution subject to some constraints. This problem can be solved through the iterative scaling (IS) procedure [3]. Although a lot of work has been done on approximate inference, there seems to be no counterpart in the literature on approximate minimum divergence problems. This paper shows that the Bethe free energy can be used as an approximation to the KL divergence and derives a novel approximate minimum divergence algorithm which we call unified propagation and scaling (UPS). In section 2 we introduce generalized inference and the iterative scaling (IS) algorithm. In section 3, we approximate the KL divergence with the Bethe free energy and derive fixed point equations to perform approximate generalized inference. We also show in what sense our fixed point equations are related to loopy BP and IS. Section 4 describes unified propagation and scaling (UPS), a novel algorithm to minimize the Bethe free energy, while section 5 shows experiments on the efficiency and accuracy of UPS. 2 Generalized Inference In this section we will introduce generalized inference and review some of the literature on iterative scaling (IS). Let where is the variable associated with node . Consider an undirected graphical model with single and pairwise potentials
,
. Let be the distribution represented by , i.e.
(1) where
,
, ranges over the edges of , ranges over the nodes of and is the number of neighbours of . Let be a subset of nodes. For let !" be a fixed distribution over . Given these “observed distributions” on , define the generalized posterior as the distribution # which minimizes the KL divergence $&% #(') *,+ # .-0/1 # 324-0/1 (2) subject to the constraints that # !" for each 5 . We call these constraints observational (Obs) constraints. Generalized inference is the process by which we determine the generalized posterior1. Let 6 be the set of unobserved nodes, i.e. all nodes not in . Theorem 1 If ! " 87 2 ! for each 9 then the generalized posterior is # :<; ! >= ? A@ = 7 2 ! . where B A@B for a subset of nodes C . Similarly if C is a subgraph of . The above theorem shows that if the constrained marginals are delta functions, i.e. the observations are hard, then the generalized posterior reduces to a trivial extension of the ordinary posterior, hence explaining our use of the term generalized inference. Since generalized inference is a constrained minimum divergence problem, a standard way of solving it is using Lagrange multipliers. For each D and , let E be the Lagrange multiplier enforcing # !" . Then the generalized posterior is # A@ =GF,H +
5 @ :
I A@ =
FJH + (3) where we chose E to satisfy the Obs constraints. Iterative scaling (IS) can now be used to solve for E [3]. At each iteration of IS, the Lagrange multiplier E is updated using the IS scaling update F H + 3K F H + ! " # for each (4) Intuitively, (4) updates the current posterior so that the marginal # for node match the given constraint ! " . IS is a specific case of the generalized iterative scaling (GIS) algorithm [4], which updates the Lagrange multipliers for a subset LNM of nodes using F H + K F H + PO Q + R + SUT V3T . Parallel GIS steps can be understood as performing IS updates in parallel, but damping the steps such that the algorithm is still guaranteed to converge. Ordinary inference is needed to compute the current marginals # required by (4). If is singly connected, then belief propagation (BP) can be used to compute the required marginals. Otherwise, exact inference or sampling algorithms like Markov chain Monte Carlo can be used, but usually are computationally taxing. Alternative approximate inference algorithms like variational methods and loopy BP can be used instead to estimate the 1To avoid confusion, we will explicitly use “ordinary inference” for normal inference, but when there is no confusion “inference” by itself will mean generalized inference. Ditto for posteriors. required marginals. Although being much more efficient, they can also produce biased estimates, potentially leading to the overall IS not converging2. Even if IS did converge, we do not have much theoretical understanding of the accuracy of the overall algorithm. A more principled approach is to first approximate the KL divergence, then derive algorithms to minimize the approximation. In the next section, we describe a Bethe free energy approximation to the KL divergence. Fixed point equations for minimizing the Bethe approximation can then be derived. The fixed point equations reduce to BP propagation updates at hidden nodes, and to IS scaling updates at observed nodes. As a consequence, using loopy BP to approximate the required marginals turns out to be a particular scheduling of the fixed point equations. Because the Bethe free energy is fairly well understood, and is quite accurate in many regimes [5, 2, 6], we conclude that IS with loopy BP is a viable approximate generalized inference technique. However, in section 4 we describe more efficient algorithms for approximate generalized inference based upon the Bethe free energy. 3 Approximate Generalized Inference Let
and be estimates of the pair-wise and single site marginals of the generalized posterior.
and are called beliefs. The beliefs need to satisfy the following marginalization and normalization (MN) constraints: * + *A+ (5) Let #
. The Bethe free energy is defined as # * / 1
* 2 / 1 (6) is an approximation to the KL divergence which only accounts for pair-wise correlations between neighbouring variables and is exact if is singly connected. We wish to minimize # subject to the MN and Obs constraints. We use Lagrange multipliers E to impose the marginalization constraints. We can also use Lagrange multipliers to impose the normalization and observational constraints as well, but this reduces to simply keeping
and normalized, and keeping !" fixed for . We shall ignore these for clarity. The resulting Lagrangian is % # 2 * * )@ E * +
I2 (7) where denotes the set of neighbours of node . Setting derivatives of % with respect to ) and E to 0, we get Theorem 2 Subject to the MN and Obs constraints, every stationary point of is given by 5
F H + H + F ! #"%$ '& H + (8) 2For a quick example, consider a two node Boltzmann machine, with weight ( and biases )+*-, ( , and the desired means on both nodes are ., / . Then using either naive mean field or naive TAP equations to estimate the marginals required by IS will not converge. where the Lagrange multipliers are fixed points of the following updates: F H + K @ 0 * + F H + for , (9) FJH + K !" +
F H + for 5 , (10) Equation (9) is equivalent to the BP propagation updates by identifying the messages as + + + + F H + 3. Rewriting (10) in terms of messages as well we find,
K * +
! " for 5 (11) We can extend the analogy and understand (11) as a message “bouncing” step, in which messages going into an observed node get bounced back and are altered in the process. If !" 7 ! is a delta function, then (11) reduces to K
! so that instead of bouncing back, messages going into node get absorbed. An alternative description of (10) is given by the following theorem. Theorem 3 Let G . Updating each E for using (10) is equivalent to updating E using (4), where we identify # )@ 0 FJH +
! " )@ 0 F,H + (12) Theorem 3 states the unexpected result that scaling updates (4) are just fixed point equations to minimize ! . Further, the required marginals # are computed using (9), which is exactly loopy BP. Hence using loopy BP to approximate the marginals required by IS is just a particular scheduling of the fixed point equations (9,10). 4 Algorithms to Minimize the Bethe Free Energy Inspired by [2], we can run run the fixed point equations (9,10) and hope that they converge to a minimum of . We call this algorithm loopy IS. Theorem 2 states that if loopy IS converges it will converge to stationary points of . In simulations we find that it always gets to a good local minimum, if not the global minimum. However loopy IS does not necessarily converge, especially when the variables are strongly correlated. There are two reasons why it can fail to converge. Firstly, the loopy BP component (9) may fail to converge. However this is not serious as past results indicate that loopy BP often fails only when the Bethe approximation is not accurate [6]. Secondly, the IS component (10) may fail to converge, since it is not run sequentially and the estimated marginals are inaccurate. We will show in section 5 that this is a serious problem for loopy IS. One way to mitigate the second problem is to use the scaling updates (4), and approximate the required marginals using an inner phase of loopy BP (call this algorithm IS+BP). Theorem 3 shows that IS+BP is just a particular scheduling of loopy IS, hence it inherits the accuracy of loopy IS while converging more often. However because we have to run loopy BP until convergence for each scaling update, IS+BP is not particularly efficient. Another way to promote convergence is to damp the loopy IS updates. This works well in practice. In this section, we describe yet another possibility – an efficient algorithm based on the 3This was first shown in [2], with a different but equivalent identification of and . same fixed point equations (9,10) which is guaranteed to converge without damping. In subsection 4.1 we describe UPS-T, an algorithm which applies when is a tree and the Obs constraints are on the leaves of . In subsection 4.2 we describe UPS for the general case, which will make use of UPS-T as a subroutine. 4.1 Constraining the leaves of trees Suppose that is a tree, and all observed nodes are leaves of . Since is a tree, the Bethe free energy is exact, i.e. if the MN constraints are satisfied then $&% #(') where # ?
? . As a consequence, ! is convex in the subspace defined by the MN constraints. Therefore if the fixed point equations (9,10) converge, they will converge to the unique global minimum. Further, since (9) is exactly a propagation update, and (10) is exactly a scaling update, the following scheduling of (9,10) will always converge: alternately run (9) until convergence and perform a single (10) update. The schedule essentially implements the IS+BP procedure, except that loopy BP is exact for a tree. Our algorithm essentially implements the scheduling, except that unnecessary propagation updates are not performed. Algorithm UPS-T Unified Propagation and Scaling on Trees 1. Run propagation updates (9) until convergence. 2. Let , , ,
be such that every node occurs infinitely often. 3. For */ , , , until convergence criterion is met: 4. Perform scaling update (10) for , where is the unique neighbour of . 5. For each edge ! on path from to #"$ , apply propagation update (9) for &%(')' . 6. Run propagation updates (9) until convergence. 4.2 Graphs with cycles For graphs with cycles, is not exact nor convex. However we can make use of the fact that it is exact on trees to find a local minimum (or saddle point). The idea is that we clamp a number of hidden nodes to their current marginals such that the rest of the hidden nodes become singly connected, and apply UPS-T. Once UPS-T has converged, we clamp a different set of hidden nodes and apply UPS-T again. The algorithm can be understood as coordinate descent where we minimize with respect to the unclamped nodes at each iteration. Let * M 6 be a set of clamped nodes such that every loop in the graph contains a node from L ,+-* . Define /. to be the graph obtained from as follows. For each node G L replicate it times, and connect each replica to one neighbour of and no other nodes. This is shown in figures 1(c) and 1(d) for the graph in 1(a). Clearly 0. will be singly connected. Let 1 M /. denote the trees in 2. . Define4 . 346587 . 3 3 346587 7 7 @ 3
7 7 7 7 7 @ 3
7 7 7 (13) # . 3465 7 # . 3 3 394$5 7 7 7 @ 3 7 7 7
7 7 @ 3 7 7 7 (14) where 7 is the number of neighbours of node . in 2. . By regrouping terms in we can show the following: 4For ;:; :< >=?: define @ 7 7 7 7 AB@ where and are the original nodes in = . Similarly for @ 7 7 CED 7 7 7 7 and D 7 7 . Theorem 4 Let ! be a distribution over for * . Then in the subspace defined by ! for 5 * and by the MN and Obs constraints, we have * 3465 7 $D% # . 3 'P . 3
* @ V 2 * + ! / 1 ! (15) To minimize ! , now all we have to do is to minimize each $D% # . 3 'P2. 3 individually. We can already solve this using UPS-T. By clamping the marginals of nodes in * , we have reduced the problem to one solved by UPS-T, where the observed nodes are taken to include those in * . The overall algorithm is Algorithm UPS Unified Propagation and Scaling 1. Initialize beliefs D E CD . 2. For */ , , , until convergence criteria is met: 3. Find a set of nodes such that every loopy is broken by
. 4. Using UPS-T, set "! D D #6 for $ , and MN and Obs constraints are satisfied . It is clear that # &% ! # for all ' . Now by using the fact that both scaling and propagation updates are fixed point equations for finding stationary points of we have, Theorem 5 If for all ' and 6 there is a ' )( ' with * , then # will converge to a local minimum (or saddle point) of ! with MN and Obs constraints satisfied. 5 Experiments In this section we report on two experiments on the feasibility of UPS. In the first experiment we compared the speed of convergence against other methods which minimize . In the second experiment we compared the accuracy of UPS against loopy IS. In both experiments we used *,+-* Boltzmann machines with states .# and structure as shown in figure 1a. The weights are sampled randomly from a Gaussian with mean 0 and standard deviation /10 and the biases are sampled from a Gaussian with standard deviation / and mean 2 incoming weights 3254 . The means of the biases are shifted so that if / is small, the mean values of will be approximately .76 * . The desired marginals are ! " 32
F 8 where 9 are sampled from a Gaussian with mean 0 and standard deviation / 8 . Experiment 1 Speed of Convergence We compared the speed of convergence for the following algorithms: loopy IS, IS+BP, GIS+BP (parallel GIS with marginals estimated by loopy BP), UPS-H (clamping rows of nodes every iteration as in figure 1(b) and UPS-HV (alternatingly clamping rows and columns as in figures 1(b) and 1(c)). We tested the algorithms on 100 networks, with / 0 * , / and / 8 ;: . We find that the result is not sensitive to the settings of / 0 / and / 8 so long as the algorithms are able to converge without damping. The result is shown in figure 1e. IS+BP and GIS+BP are slow because the loopy BP phase is expensive. UPS-H and UPS-HV both do better than IS+BP and GIS+BP because the inner loops are cheaper, and the Lagrange multipliers E are updated more frequently. Further we see that UPS-HV is faster than UPS-H since information is propagated faster throughout the network. loopy IS is the fastest. However the next experiment shows that it also converges less frequently and there is a trade off between the speed of loopy IS and the stability of UPS. GIS+BP IS+BP UPS−H UPS−HV loopy IS 10 3 10 4 10 5 Number of Updates (a) (d) (e) (b) (c) Figure 1: (a) Network structure. Circles are hidden nodes and black squares are observationally constrained nodes. (b) Clamping rows of nodes. Black circles are the clamped nodes. (c) Clamping columns of nodes. (d) Replicating each clamped and observed node in (c). (e) Speed of convergence. The box lines are at the median and upper and lower quartiles, and the whiskers describe the extent of data. An algorithm or subroutine is considered converged if the beliefs change by less than . # . Experiment 2 Accuracy of Estimated Marginals We compared the accuracy of the posterior marginals obtained using UPS-HV and loopy IS for four possible types of constraints, as shown in figure 2. In case (a), the constraint marginals are delta functions, so that generalized inference reduces down to ordinary inference, loopy IS reduces to loopy BP and UPS becomes a convergent alternative to loopy BP. In case (b), we did not enforce any Obs constraints so that the problem is one of estimating the marginals of the prior . The general trend is that loopy BP and UPS are comparable, and they perform worse as weights get larger, biases get smaller or there is less evidence. This confirms the results in [6]. Further, we see that when loopy BP did not converge, UPS’s estimates are not better than loopy BP’s estimates. The reason this is happening is described in [6]. In cases (c) and (d) we set / 8 .76 4
46 . , corresponding to .76 * and spread out over . respectively. In these cases UPS and loopy IS did equally well when the latter converged, but UPS continued to perform well even when loopy IS did not converge. Since loopy BP always converged when UPS performed well (for cases (a) and (b)), and we used very high damping, we conclude that loopy IS’s failure to converge must be due to performing scaling updates before accurate marginals were available. Concluding, we see that UPS is comparable to loopy IS when generalized inference reduces to ordinary inference, but in the presence of Obs constraints it is better. 6 Discussion In this paper we have shown that approximating the KL divergence with the Bethe free energy leads to viable algorithms for approximate generalized inference. We also find that there is an interesting and fruitful relationship between IS and loopy BP. Our novel algorithm UPS can also be used as a convergent alternative to loopy BP for ordinary inference. Interesting extensions are to cluster nodes together to get more accurate approximations to the KL divergence analogous to the Kikuchi free energy, and to handle marginal constraints over subsets of nodes. This will again lead to a close relationship between IS and junction tree propagation, but the details are to be worked out. We can also explore other algorithms to minimize ! , including the CCCP algorithm [7]. Another interesting direction for future work is algorithms for learning in log linear models by approximating the free energy. References [1] J. Pearl. Probabilistic reasoning in intelligent systems : networks of plausible inference. Morgan Kaufmann Publishers, San Mateo CA, 1988. 0 0.2 0.4 1 3 5 7 9 1 3 5 7 9 UPS 1 3 5 7 9 loopy IS (a) Ordinary Inference 1 3 5 7 9 (b) No Obs Constraints 1 3 5 7 9 (c) σ(α) = 0.2 1 3 5 7 9 (d) σ(α) = 2.0 Figure 2: Each plot shows the mean absolute errors for various settings of (x-axis) and (yaxis). The top plots show errors for loopy IS and bottom plots show errors for UPS. The inset shows the cases (black) when loopy IS did not converge within 2000 iterations, with linear damping slowly increasing to ., . [2] J.S. Yedidia, W. Freeman, and Y. Weiss. Generalized belief propagation. In Advances in Neural Information Processing Systems, volume 13, 2000. [3] W. E. Deming and F. F. Stephan. On a least square adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics, 11:427–444, 1940. [4] J. Darroch and D. Ratcliff. Generalized iterative scaling for log-linear models. Annals of Mathematical Statistics, 43:1470–1480, 1972. [5] K. Murphy, Y. Weiss, and M. Jordan. Loopy belief propagation for approximate inference : An empirical study. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, volume 15. Morgan Kaufmann Publishers, 1999. [6] M. Welling and Y. W. Teh. Belief optimization for binary networks : A stable alternative to loopy belief propagation. In Uncertainty in Artificial Intelligence, 2001. [7] A. L. Yuille. CCCP algorithms to minimize the Bethe and Kikuchi free energies: Convergent alternatives to belief propagation. 2002.
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Receptive field structure of flow detectors for heading perception Jaap A. B eintema Dept. Zoology & Neurobiology Ruhr University Bochum, Germany, 44780 beintema@neurobiologie.ruhr-uni-bochum.de Albert V. van den Berg Dept. of Neuro-ethology, Helmholtz Institute, Utrecht University, The Netherlands a. v. vandenberg@bio.uu.nl Markus Lappe Dept. Zoology & Neurobiology Ruhr University Bochum, Germany, 44780 lappe@neurobiologie.ruhr-uni-bochum.de Abstract Observer translation relative to the world creates image flow that expands from the observer's direction of translation (heading) from which the observer can recover heading direction. Yet, the image flow is often more complex, depending on rotation of the eye, scene layout and translation velocity. A number of models [1-4] have been proposed on how the human visual system extracts heading from flow in a neurophysiologic ally plausible way. These models represent heading by a set of neurons that respond to large image flow patterns and receive input from motion sensed at different image locations. We analysed these models to determine the exact receptive field of these heading detectors. We find most models predict that, contrary to widespread believe, the contributing motion sensors have a preferred motion directed circularly rather than radially around the detector's preferred heading. Moreover, the results suggest to look for more refined structure within the circular flow, such as bi-circularity or local motion-opponency. Introduction The image flow can be considerably more complicated than merely an expanding pattern of motion vectors centered on the heading direction (Fig. 1). Flow caused by eye rotation (Fig. 1 b) causes the center of flow to be displaced (compare Fig. 1a and c). The effect of rotation depends on the ratio ofrotation and translation speed. A Translational flow B Rotational flow C Combined flow f~~~~; <If:.. 1J: ~ \\ ~1 ... " '\ ,,~ t // ~ ........ ... ',,,, ....... .. -... ... 4i...... .... ...... ... .....-1 ........ ....... 1 • 0 .. -+ .. +... 0 ~ .. " • ... -+~ ... +-..,... ...... • • "t +- ., .... +'# " ~ --... • ... ~ .,. .. ... ,. " II .J + • ... ....-.... +-....... I' 4 ~ .. .,.. ~ ~~~ .. ~\ + J ~ .. S:: .. * :: ~ .1, .... Figure 1: Flow during a) observer translation through a 3D-cloud of dots, headed 10° towards the left, during b) observer rotation about the vertical towards the right, and during c) the combination of both. Also, since the image motions caused by translation depend on point distance and the image motions caused by rotation do not, the combined movement results in flow that is no longer purely expanding for scenes containing depth differences (Fig. lc). Heading detection can therefore not rely on a simple extrapolation mechanism that determines the point of intersection of motion vectors. A number of physiologically-based models [1-4] have been proposed on how the visual system might arrive at a representation of heading from flow that is insensitive to parameters other than heading direction. These models assume heading is encoded by a set of units that each respond best to a specific pattern of flow that matches their preferred heading. Such units resemble neurons found in monkey brain area MST. MST cells have large receptive fields (RF), typically covering one quart or more of the visual field, and receive input from several local motion sensors in brain area MT. The receptive field of MST neurons may thus be defined as the preferred location, speed and direction of all input local motion sensors. Little is known yet about the RF structure of MST neurons. We looked for similarities between current models at the level of the RF structure. First we explain the RF structure of units in the velocity gain model, because this model makes clear assumptions on the RF structure. Next, we we show the results of reconstructing RF structure of units in the population model[2] . Finally, we analyse the RF structure of the template model[3] and motion-opponency model[4]. Velocity gain field model The velocity gain field model[l] is based on flow templates. A flow template, as introduced by Perrone and Stone[3] , is a unit that evaluates the evidence that the flow fits the unit's preferred flow field by summing the responses of local motion sensors outputs. Heading is then represented by the preferred heading direction of the most active template(s) . The velocity gain field model[l] is different from Perrone and Stone's template model[2] in the way it acquires invariance for translation speed, point distances and eye rotation. Whereas the template model requires a different template for each possible combination of heading direction and rotation, the velocity gain field model obtains rotation invariance using far less templates by exploiting eye rotation velocity signals. The general scheme applied in the velocity gain field model is as follows. In a set of flow templates, each tuned to pure expansion with specific preferred heading, A Circular component B Radial component <i!tl t r 1'f ;J? -. ... .... ~ ~ ,\ t "" .J' 4t"4~ ~~ ,-' Jr .......... 4.... ~ .t. y II ~ ~ . ~ • y 9' ... 0 9'9' • + 0 -+ • 9'Y " ... ... -.II ,9' 9' • ~. .... ............ "" • ,//, .. •• +-4••• .~ II \ .. -...; #= " J ¥s.. ~-. .. .. ¥ II Figure 2: The heading-centered circular (a) and radial (b) component of the flow during combined translation and rotation as in Fig. 2c. the templates would change their activity during eye rotation. Simply subtracting the rotation velocity signal for each flow template would not suffice to compensate because each template is differently affected by rotational flow. However, each flow template can become approximately rotation-invariant by subtracting a gain field activity that is a multiplication of the eye velocity t with a derivative template activity 80/ 8R that is specific for each flow template. The latter reflects the change in flow template activity 0 given a change in rotational flow 8R. Such derivative template 80/ 8R can be constructed from the activity difference of two templates tuned to the same heading, but opposite rotation. Thus, in the velocity gain field model, templates tuned to heading direction and a component of rotation play an important role. To further appreciate the idea behind the RF structure in the velocity gain field model, note that the retinal flow can be split into a circular and radial component, centered on the heading point (Fig. 2). Translation at different speeds or through a different 3D environment will alter the radial component only. The circular component contains a rotational component of flow but does not change with point distances or translational speed. This observation lead to the assumption implemented in the velocity gain field model that templates should only measure the flow along circles centered on the point of preferred heading. An example of the RF structure of a typical unit in the velocity gain field model, tuned to heading and rightward rotation is shown in Fig. 3. This circular RF structure strongly reduces sensitivity to variations in depth structure or the translational speed, while the template's tuning to heading direction is preserved, because its preferred structure is centered on its preferred heading direction [1] . Interestingly, the RF structure of the typical rotation-tuned heading units is bi-circular, because the direction of circular flow is opponent in the hemifields to either side of an axis (in this case the horizontal axis) through the heading point. Moreover, the structure contains a gradient in magnitude along the circle, decreasing towards the horizontal axis. F: I\. 1\ ..... ~ ... - ..... I\. It • . .. .rI\. • • • I 0 It "'.r .. • • ..... .... Of' " .. .. .... -- ... ~ " rJ Figure 3: Bi-circular RF structure of a typical unit in the velocity gain field model, tuned to leftward heading and simultaneous rightward rotation about the vertical. Individual vectors show the preferred direction and velocity of the input motion sensors. Population model The population model [2] derives a representation of heading direction that is invariant to the other flow parameters using a totally different approach. This model does not presume an explicit RF structure. Instead, the connections strengths and preferred directions of local motion inputs to heading-specific flow units are computed according to an optimizing algorithm[5]. We here present the results obtained for a restricted version of the model in which eye rotation is assumed to be limited to pursuit that keeps the eye fixated on a stationary point in the scene during the observer translation. Specifically, we investigated whether a circular or bi-circular RF structure as predicted by the velocity gain model emerges in the population model. The population model [2,6] is an implementation of the subspace algorithm by Heeger and Jepson [5] into a neural network. The subspace algorithm computes a residual function R(T j) for a range of possible preferred heading directions. The residual function is minimized when flow vectors measured at m image locations, described as one array, are perpendicular to the vectors that form columns of a matrix C ~ (T j). This matrix is computed from the preferred 3-D translation vector T j and the m image locations. Thus, by finding the matrix that minimizes the residue, the algorithm has solved the heading, irrespective of the 3D-rotation vector, unknown depths of points and translation speed. To implement the subspace algorithm in a neurophysiologically plausible way, the population model assumes two layers of units. The first MT-like layer contains local motion sensors that fire linearly with speed and have cosine-like direction tuning. These sensors connect to units in the second MST-like layer. The activity in a 2nd layer unit, with specific preferred heading T j, represents the likelihood that the residual function is zero. The connection strengths are determined by the C ~ (T j) matrix. As not to have too many motion inputs per 2nd layer unit, the residual function R(T j) is partitioned into smaller sub residues that take only a few motion inputs. The likelihood for a specific heading is then given by the sum of responses in a population with same preferred heading. Given the image locations and the preferred heading, one can reconstruct the RF structure for 2nd layer units with the same preferred heading. The preferred motion inputs to a second layer unit are given by vectors that make up each column of C ~ (T j). Hereby, the vector direction represents the preferred motion direction, A B "? \ t I Figure 4: Examples of receptive field structure of a population that encodes heading 100 towards the left (circle) . a-b) Five pairs of MT-like sensors, where the motion sensors of each pair are at a) the same image location, or b) at image locations one quarter of a cycle apart. c) Distribution of multiple pairs leading to bi-circular pattern. and the vector magnitude represents the strength of the synaptic connection. The matrix C l..(Tj) is computed from the orthogonal complement of a (2m x m + 3) matrix C(Tj) [5]. On the assumption that only fixational eye movements occur, the matrix reduces to (2m x m + 1)[6]. Given only two flow vector inputs (m = 2), the matrix C l.. (T j) reduces to one column of length m = 4. The orthogonal complement of this 4 x 3 matrix was solved in Mathematica by first computing the nullspace of the inverse matrix of C (T j), and then constructing an orthonormal basis for it using Gram-Schmidt orthogonalisation. We computed the orientation and magnitude of the two MT-inputs analytically. Instead of giving the mathematics, we here describe the main results. Circularity Independent of the spatial arrangement of the two MT-inputs to a 2nd-layer unit, their preferred motions turned out to be always directed along a circle centered on the preferred heading point. Fig. 4 shows examples of the circular RF structures, for different distributions of motion pairs that code for the same heading direction. Motion-opponency For pairs of motion sensors at overlapping locations, the vectors of each pair always turned out to be opponent and of equal magnitude (Fig. 4a). For pairs of spatially separated motion sensors, the preferred magnitude and direction of the two motion inputs depend on their location with respect to the hemispheres divided by the line through heading and fixation point. We find that preferred motion directions are opponent if the pair is located within the same hemifield, but uni-directional if the pair is split across the two hemifields as in Fig. 4b. Bi-circularity Interestingly, if pairs of motion sensors are split across hemi fields, with partners at image locations 900 rotated about the heading point, a magnitude gradient appears in the RF structure (Fig. 4b). Thus, with these pairs a bi-circular RF structure can be constructed similar to units tuned to rotation about the vertical in the velocity gain field model (compare with Fig. 3). Note, that the bi-circular RF structures do differ since the axis along which the largest magnitude occurs is horizontal for the population model and vertical for the velocity gain field model. The RF structure of the population model unit resembles a velocity gain field unit tuned to rotation about the horizontal axis, implying a A Adapted from Perrone and Stone (1994) Direction and speed tuned motion sensors ""","" .,"',,", ~ (j Effective RF structure B Figure 5: Adapted from Perrone and Stone 1994). a) Each detector sums the responses of the most active sensor at each location. This most active motion sensor is selected from a pool of sensors tuned to different depth planes (Ca, Cb, etc). These vectors are the vector sums of preferred rotation component Rand translational components Ta, Tb, etc. b) Effective RF structure. large sensitivity to such rotation. This, however, does not conflict with the expected performance of the population model. Because in this restricted version rotation invariance is expected only for rotation that keeps the point of interest in the center of the image plane (in this case rotation about the vertical because heading is leftward) units are likely to be sensitive to rotation about the horizontal and torsional axis. Template model The template model and the velocity gain field model differ in how invariance for translation velocities, depth structure and eye rotation is obtained. Here, we investigate whether this difference affects the predicted RF structure. In the template model of Perrone and Stone [3], a template invariant to translation velocity or depth structure is obtained by summing the responses of the most active sensor at each image location. This most active sensor is selected from a collection of motion sensors, each tuned to a different ego-translation speed (or depth plane), but with the same preferred ego-rotation and heading direction (Fig. 5a). Given a large range of depth planes, it follows that a different radial component of motion will stimulate another sensor maximally, but that activity nevertheless remains the same. The contributing response will change only due to a component of motion along a circle centered on the heading, such as is the case when heading direction or rotation is varied. Thus, the contributing response will always be from the motion sensor oriented along the circle around the template's preferred heading. Effectively, this leads to a bi-circular RF structure for units tuned to heading and rotation (Fig. 5b). Motion-opponency model Royden[4] proposed that the effect of rotation is removed at local motion detection level before the motion signals are received by flow detectors. This is achieved by MT-like sensors that compute the difference vector between spatially neighbouring motion vectors. Such difference vector will always be oriented along lines intersecting at the heading point (Fig. 6). Thus, the resulting input to flow detectors will be oriented radially. Indeed, Royden's results[4] show that the preferred directions of the operators with the largest response will be radially, not circularly, oriented. A Translational flow B Rotational flow L-________ ~.~ ______ ~ Figure 6: Motion parallax, the difference vector between locally neighbouring motion vectors. For translation flow (a) the difference vector will be oriented along line through the heading point, whereas for rotational flow (b) the difference vector vanishes (compare vectors within square). Summary and Discussion We showed that a circular RF structure, such as proposed by the velocity gain field model[l] , is also found in the population model[2] and is effectively present in the template model[3] as well. Only the motion-opponent model [4] prefers radial RF structures. Furthermore, we find that under certain restrictions, the population model reveals local motion-opponency and bi-circularity, properties that can be found in the other models as well. A circular RF structure turns out to be a prominent property in three models. This supports the counterintuitive, but computationally sensible idea, that it is not the radial flow structure, but the structure perpendicular to it, that contributes to the response of heading-sensitive units in the human brain. Studies on area MST cells not only report selectivity for expanding motion patterns, but also a significant proportion of cells that are selective to rotation patterns [7-10]. These models could explain why cells respond so well to circular motion, in particular to the high rotation speeds (up to about 80 deg/s) not experienced in daily life. This model study suggests that selectivity for circular flow has a direct link to heading detection mechanisms. It also suggests that testing selectivity for expanding motion might be a bad indicator for determining a cell's preferred heading. This point has been noted before, as MST seems to be systematically tuned to the focus of rotation, exactly like model neurons [9]. Little is still known about the receptive field structure of MST cells. So far the receptive field structure of MST cells has only been roughly probed [10], and the results neither support a radial nor a circular structure. Also, so far only uni-circular motion has been tested. Our analyses points out that it would be worthwhile to look for more refined circular structure such as local motion-opponency. Local motion opponency has already been found in area MT, where some cells respond only if different parts of their receptive field are stimulated with different motion [11]. Another promising structure to look for would be bi-circularity, with gradients in magnitude of preferred motion along the circles. Acknowledgments Supported by the German Science Foundation and the German Federal Ministry of Education and Research. References [1] Beintema, J . A. & van den Berg A. V. (1998) Heading detection using motion templates and eye velocity gain fields. Vision Research, 38(14):2155-2179. [2] Lappe M., & Rauschecker J . P. (1993) A neural network for the processing of optic flow from ego-motion in man and higher mammals. Neural Computation, 5:374-39l. [3] Perrone J. A. & Stone L. S. (1994) A model for the self-motion estimation within primate extrastriate visual cortex. Vision Research, 34:2917-2938. [4] Royden C. S. (1997) Mathematical analysis of motion-opponent mechanisms used in the determination of heading and depth. Journal of the Optical Society of America A, 14(9):2128-2143. [5] Heeger D. J . & Jepson A. D. (1992) Subspace methods for recovering rigid motion I: Algorithm and implementation. International Journal of Computational Vision, 7:95-117. [6] Lappe M. & Rauschecker J.P. (1993) Computation of heading direction from optic flow in visual cortex. In C.L. Giles, S.J. Hanson and J.D. Cowan (eds.), Advances in Neural Information Processing Systems 5, pp. 433-440. Morgan Kaufmann. [7] Tanaka K. & Saito H. (1989) Analysis of the visual field by direction, expansion/contraction, and rotation cells clustered in the dorsal part of the medial superior temporal area of the macaque monkey Journal of Neurophysiology, 62(3):626-64l. [8] Duffy C. J. & Wurtz R. H. (1991) Sensitivity of MST neurons to optic flow stimuli. I. A continuum of response selectivity to large-field stimuli. Journal of Neurophysiology, 65(6):1329-1345. [9] Lappe M., Bremmer F., Pekel M., Thiele A., Hoffmann K.-P. (1996) Optic flow processing in monkey STS: a theoretical and experimental approach. the Journal of Neuroscience, 16(19):6265-6285. [10] Duffy C. J. & Wurtz R. H. (1991) Sensitivity of MST neurons to optic flow stimuli. II. Mechanisms of response selectivity revealed by small-field stimuli. Journal of Neurophysiology, 65(6):1346-1359. [11] Allman J., Miezin F. & McGuinness E. (1985) Stimulus specific responses from beyond the classical receptive field: Neurophysiological mechanisms for local-global comparisons in visual neurons. Ann. Rev. Neurosci., 8:407-430.
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Hyperbolic Self-Organizing Maps for Semantic Navigation J¨org Ontrup Neuroinformatics Group Faculty of Technology Bielefeld University D-33501 Bielefeld, Germany jontrup@techfak.uni-bielefeld.de Helge Ritter Neuroinformatics Group Faculty of Technology Bielefeld University D-33501 Bielefeld, Germany helge@techfak.uni-bielefeld.de Abstract We introduce a new type of Self-Organizing Map (SOM) to navigate in the Semantic Space of large text collections. We propose a “hyperbolic SOM” (HSOM) based on a regular tesselation of the hyperbolic plane, which is a non-euclidean space characterized by constant negative gaussian curvature. The exponentially increasing size of a neighborhood around a point in hyperbolic space provides more freedom to map the complex information space arising from language into spatial relations. We describe experiments, showing that the HSOM can successfully be applied to text categorization tasks and yields results comparable to other state-of-the-art methods. 1 Introduction For many tasks of exploraty data analysis the Self-Organizing Maps (SOM), as introduced by Kohonen more than a decade ago, have become a widely used tool [1, 2]. So far, the overwhelming majority of SOM approaches have taken it for granted to use a flat space as their data model and, motivated by its convenience for visualization, have favored the (suitably discretized) euclidean plane as their chief “canvas” for the generated mappings. However, even if our thinking is deeply entrenched with euclidean space, an obvious limiting factor is the rather restricted neighborhood that “fits” around a point on a euclidean 2D surface. Hyperbolic spaces in contrast offer an interesting loophole. They are characterized by uniform negative curvature, resulting in a geometry such that the size of a neighborhood around a point increases exponentially with its radius . This exponential scaling behavior allows to create novel displays of large hierarchical structures that are particular accessible to visual inspection [3, 4]. Consequently, we suggest to use hyperbolic spaces also in conjunction with the SOM. The lattice structure of the resulting hyperbolic SOMs (HSOMs) is based on a tesselation of the hyperbolic space (in 2D or 3D) and the lattice neighborhood reflects the hyperbolic distance metric that is responsible for the non-intuitive properties of hyperbolic spaces. After a brief introduction to the construction of hyperbolic spaces we describe several computer experiments that indicate that the HSOM offers new interesting perspectives in the field of text-mining. 2 Hyperbolic Spaces Hyperbolic and spherical spaces are the only non-euclidean geometries that are homogeneous and have isotropic distance metrics [5, 6]. The geometry of H2 is a standard topic in Riemannian geometry (see, e.g. [7]), and the relationships for the area and the circumference of a circle of radius are given by
(1) These formulae exhibit the highly remarkable property that both quantities grow exponentially with the radius . It is this property that was observed in [3, 4] to make hyperbolic spaces extremely useful for accommodating hierarchical structures. To use this potential for the SOM, we must solve two problems: we must find suitable discretization lattices on H2 to which we can “attach” the SOM prototype vectors. ! after having constructed the SOM, we must somehow project the (hyperbolic!) lattice into “flat space” in order to be able to inspect the generated maps. 2.1 Projections of Hyperbolic Spaces To construct an isometric (i.e., distance preserving) embedding of the hyperbolic plane into a “flat” space, we may use a Minkowski space [8]. In such a space, the squared distance " between two points $# %& '& and #&( % ( ' ( is given by " )$#+*,# ( . % * % ( */ ' * ' ( (2) i.e., it ceases to be positive definite. Still, this is a space with zero curvature and its somewhat peculiar distance measure allows to construct an isometric embedding of the hyperbolic plane H2, given by #0
1 2 435 687 9% :$2 :87 9' 365 ; 2 (3) where $2 7 are polar coordinates on the H2. Under this embedding, the hyperbolic plane appears as the surface < swept out by rotating the curve ' )= # % about the ' -axis. A D C B u S O 1 N 1 M Figure 1: Construction steps underlying Klein and Poincar´e-models of the space H2 From this embedding, we can construct two further ones, the so-called Klein model and the Poincar´e model [5, 9] (we will use the latter to visualize HSOMs below). Both achieve a projection of the infinite H2 into the unit disk, however, at the price of distorting distances. The Klein model is obtained by projecting the points of < onto the plane ' >= along rays passing through the origin ? (see Fig. 1). Obviously, this projects all points of < into the “flat” unit disk # % A@ = of B C . (e.g., EDF G ). The Poincar´e Model results if we add two further steps: first a perpendicular projection of the Klein Model onto the (“northern”) surface of the unit sphere centered at the origin (e.g., GHDF ), and then a stereographic projection of the “northern” hemisphere onto the unit circle about the origin in the ground plane ' JI (point K ). It turns out that the resulting projection of H2 has a number of pleasant properties, among them the preservation of angles and the mapping of shortest paths onto circular arcs belonging to circles that intersect the unit disk at right angles. Distances in the original H2 are strongly distorted in its Poincar´e (and also in the Klein) image (cf. Eq. (5)), however, in a rather useful way: the mapping exhibits a strong “fish-eye”-effect. The neighborhood of the H2 origin is mapped almost faithfully (up to a linear shrinkage factor of 2), while more distant regions become increasingly “squeezed”. Since asymptotically the radial distances and the circumference grow both according to the same exponential law, the squeezing is “conformal”, i.e., (sufficiently small) shapes painted onto H2 are not deformed, only their size shrinks with increasing distance from the origin. By translating the original H2, the fish-eye-fovea can be moved to any other part of H2, allowing to selectively zoom-in on interesting portions of a map painted on H2 while still keeping a coarser view of its surrounding context. 2.2 Tesselations of the Hyperbolic Plane To complete the set-up for a hyperbolic SOM we still need an equivalent of a regular grid in the hyperbolic plane. For the hyperbolic plane there exist an infinite number of tesselations with congruent polygons such that each grid point is surrounded by the same number of neighbors [9, 10]. Fig. 2 shows two example tesselations (for the minimal value of and for )= I ), using the Poincar´e model for their visualization. While these tesselations appear non-uniform, this is only due to the fish-eye effect of the Poincar´e projection. In the original H2, each tesselation triangle has the same size. One way to generate these tesselations algorithmically is by repeated application of a suitable set of generators of their symmetry group to a (suitably sized, cf. below) “starting triangle”, for more details cf. [11]. Figure 2: Regular triangle tesselations of the hyperbolic plane, projected into the unit disk using the Poincar´e mapping. The left tesselation shows the case where the minimal number ( ) of equilateral triangles meet at each vertex, the right figure was constructed with . In the Poincar´e projection, only sides passing through the origin appear straight, all other sides appear as circular arcs, although in the original space all triangles are congruent. 3 Hyperbolic SOM Algorithm We have now all ingredients required for a “hyperbolic SOM”. We organize the nodes of a lattice as described above in “rings” around an origin node. The numbers of nodes of such a lattice grows very rapidly (asymptotically exponentially) with the chosen lattice radius C (its number of rings). For instance, a lattice with
C contains 1625 nodes. Each lattice node carries a prototype vector B C from some K -dimensional feature space (if we wish to make any non-standard assumptions about the metric structure of this space, we would build this into the distance metric that is used for determining the best-match node). The SOM is then formed in the usual way, e.g., in on-line mode by repeatedly determining the winner node and adjusting all nodes in a radial lattice neighborhood around according to the familiar rule #+* (4) with
:* " . However, since we now work on a hyperbolic lattice, we have to determine both the neighborhood and the (squared) node distance " according to the natural metric that is inherited by the hyperbolic lattice. The simplest way to do this is to keep with each node a complex number to identify its position in the Poincar´e model. The node distance is then given (using the Poincar´e model, see e.g. [7]) as " arctanh * = * (5) The neighborhood can be defined as the subset of nodes within a certain graph distance (which is chosen as a small multiple of the neighborhood radius ) around . 4 Experiments Some introductory experiments where several examples illustrate the favorable properties of the HSOM as compared to the “standard” euclidean SOM can be found in [11, 12]. A major example of the use of the SOM for text mining is the WEBSOM project [2]. 4.1 Text Categorization In order to apply the HSOM to natural text categorization, i.e. the assignment of natural language documents to a number of predefined categories, we follow the widely used vector-space-model of Information Retrieval (IR). For each document " we construct a feature vector " , where the components are determined by the frequency of which term occurs in that document. Following standard practice [13] we choose a term frequency inverse document frequency weighting scheme: ! #"%$'& " (6) where the term frequency ! denotes the number of times term occurs in ")( , the number of documents in the training set and " the document frequency of , i.e. the number of documents occurs in. The HSOM can be utilized for text categorization in the following manner. In a first step, the training set is used to adapt the weight vectors according to (4). During the second step, the training set is mapped onto the HSOM lattice. To this end, for each training example " ( its best match node is determined such that " ( * +* " ( * -, (7) where ")( denotes the feature vector of document "( , as described above. After all examples have been presented to the net, each node is labelled with the union . of all categories that belonged to the documents that were mapped to this node. A new, unknown text is then classified into the union . of categories which are associated with its winner node selected in the HSOM. Text Collection. We used the Reuters-215781 data set since it provides a well known baseline which is also used by other authors to evaluate their approaches, c.f. [14, 15]. We 1As compiled by David Lewis from the AT&T Research Lab in 1987. The data can be found at http://www.research.att.com/ / lewis/ have used the “ModApte” split, leading to 9603 training and 3299 test documents. After preprocessing, our training set contained 5561 distinct terms. Performance Evaluation. The classification effectiveness is commonly measured in terms of precision and recall C [16], which can be estimated as C
where and are the numbers of documents correctly classified, and correctly not classified to category , respectively. Analogous, and are the corresponding numbers of falsely classified documents. For each node and each category a confidence value is determined. It describes the number of training documents belonging to class which were mapped to node . When retrieving documents from a given category , we compare for each node its associated against a threshold . Documents from nodes with become then included into the retrieval set. For nodes which contain a set of documents K , the order of the retrieval set is ranked by $ " ( , where " ( K . In this way the number of retrieved documents can be controlled and we obtain the precision-recall-diagrams as shown in Fig. 3. In order to compare the HSOM’s performance for text categorization, we also evaluated a -nearest neighbor ( -NN) classifier with our training set. Apart from boosting methods [16] only support vector machines [14] have shown better performances. The confidence level of a -NN classifier to assign document "( to class is -NN " (
! #"%$'& )( $ " ( " ( (8) where " ( is the set of documents " ( for which $ " ( " ( is maximum. The assignment factor & )( is 1, if " ( belongs to category and 0 otherwise. According to [14, 17] we have chosen the +*I nearest neighbors. Text Categorization Results. The results of three experiments are shown in Table 1. We have compared a HSOM with C ) rings and a tesselation with -, neighbors (summing up to 1306 nodes) to a spherical standard euclidean SOM as described in [11] with approx. 1300 nodes, and the -NN classifier. Our results indicate that the HSOM does not perform better than a -NN classifier, but to a certain extent also does not play significantly worse either. It is noticable that for less dominant categories the HSOM yields superior results to those of the standard SOM. This is due to the fact, that the nodes in H2 cover a much broader space and therefore offer more freedom to map smaller portions of the original dataspace with less distortions as compared to euclidean space. As the -NN results suggest, other state-of-the-art techniques like support vector machines will probably lead to better numerical categorization results than the HSOM. However, since the main purpose of the HSOM is the visualization of relationships between texts and text categories, we believe that the observed categorization performance of the HSOM compares sufficiently well with the more specialized (non-visualization) techniques to warrant its efficient use for creating insightful maps of large bodies of document data. Table 1: Precision-recall breakeven points for the ten most prominent categories. earn acq mny-fx crude grain trade interest wheat ship corn SOM 90.0 81.2 61.7 70.3 69.4 48.8 57.1 61.9 54.8 50.3 HSOM 90.2 81.6 68.7 78.8 76.2 56.8 66.4 69.3 61.8 53.6 . -NN 93.8 83.7 69.3 84.7 81.9 61.9 71.0 69.0 77.5 67.9 0.4 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 acq earn money−fx (a) . -NN 0.4 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 2: 0.69 (b) HSOM Figure 3: Precision-recall curves for the three most frequent categories earn, acq and money-fx. 4.2 Text Mining & Semantic Navigation A major advantage of the HSOM is its remarkable capability to map high-dimensional similarity relationships to a low-dimensional space which can be more easily handled and interpreted by the human observer. This feature and the particular “fish-eye” capability motivates our approach to visualize whole text collections with the HSOM. It can be regarded as an interface capturing the semantic structure of a text database and provides a way to guide the users attention. In preliminary experiments we have labelled the nodes with glyphs corresponding to the categories of the documents mapped to that node. In Fig. 4 two HSOM views of the Reuters data set are shown. Note, that the major amount of data gets mapped to the outermost region, where the nodes of the HSOM make use of the large space offered by the hyperbolic geometry. During the unsupervised training process, the document’s categories were not presented to the HSOM. Nevertheless, several document clusters can be clearly identified. The two most prominent are the earn and acquisition region of the map, reflecting the large proportion of these categories in the Reuters-21578 collection. Note, that categories which are semantically similar are located beside each other, as can be seen in the corn, wheat, grain the interest, money-fx or the crude, ship area of the map. Additional to the category (glyph type) and the number of training documents per node (glyph size), the number of test documents mapped to each node is shown as the height of the symbol above the ground plane. In this way the HSOM can be used as a novelty detector in chronological document streams. For the Reuters-21578 dataset, a particular node strikes out. It corresponds to the small glyph tagged with the “ship” label in Fig. 4. Only a few documents from the training collection are mapped to that node as shown by it’s relatively small glyph size. The large -value on the other hand indicates that it contains a large number of test documents, and is therefore probably semantically connected to a significant, novel event only contained in the test collection. The right image of Fig. 4 shows the same map, but the focal view now moved into the direction of the conspicious “ship” node, resulting in a magnification of the corresponding area. A closer inspection reveals, that the vast majority (35 of 40) of the test documents describe an incident where an Iranian oil rig was attacked in the gulf. Although no document of the training set describes this incident (because the text collection is ordered by time and the attack took place “after” the split into train and test set), the HSOM generalizes well and maps the semantic content of these documents to the proper area of the map, located between the regions for crude and ship. The next example illustrates that the HSOM can provide more information about an unknown text than just it’s category. For this experiment we have taken movie reviews from the rec.art.movies.reviews newsgroup. Since all the reviews describe a certain movie, we retrieved their associated genres from the Internet Movie Database (http://www.imdb.com) to build a set of category labels for each document. The training set contained 8923 rangrain ship crude interest money−fx trade corn wheat acq earn Figure 4: The left figure shows a central view of the Reuters data. We used a HSOM with rings and a tesselation with neighbors. Ten different glyphs were used to visualize the ten most frequent categories. They were manually tagged to indicate the correspondence between category and symbol type. The glyph sizes and the -values (height above ground plane) reflect the number of training and test documents mapped to the corresponding node, respectively. domly selected reviews (without their genre information) from films released before 2000. We then presented the system with five reviews from the film “Atlantis”, a Disney cartoon released in 2001. The HSOM correctly classified all of the five texts as reviews for an animation movie. In Fig. 5 the projection of the five new documents onto the map with the previously acquired text collection is shown. It can be seen that there exist several clusters related to the animation genre. By moving the fovea of the HSOM we can now “zoom” into that region which contains the five new texts. In the right of Fig. 5 it can be seen that all of the “Atlantis” reviews where mapped to a node in immediate vicinity of documents describing other Disney animation movies. This example motivates the approach of “semantic navigation” to rapidly visualize the linkage between unknown documents and previously acquired semantic concepts. Mulan A Bug´s Life Antz The Iron Giant Tarzan South Park Chicken Run The Prince of Egypt Dinosaur Pocahontas the beast Beauty andAnastasia Mulan Tarzan Hercules Aladin Atlantis Figure 5: A HSOM with and a tesselation with neighbors was used to map movie rewies from newsgroup channels. In both figures, glyph size and -value indicate the number of texts related to the animation genre mapped to the corresponding node. Nodes exceeding a certain threshold were labelled with the title corresponding to the most frequently occuring movie mapped to that node. The underlined label in the right figure indicates the position of the node to which five new documents were mapped to. 5 Conclusion Efficient navigation in “Sematic Space” requires to address two challenges: (i) how to create a low dimensional display of semantic relationship of documents, and (ii) how to obtain these relationships by automated text categorization. Our results show that the HSOM can provide a good solution to both demands simultaneously and within a single framework. The HSOM is able to exploit the peculiar geometric properties of hyperbolic space to successfully compress complex semantic relationships between text documents. Additionally, the use of hyperbolic lattice topology for the arrangement of the HSOM nodes offers new and attractive features for interactive “semantic navigation”. Large document databases can be inspected at a glance while the HSOM provides additional information which was captured during a previous training step, allowing e.g. to rapidly visualize relationships between new documents and previously acquired collections. Future work will address more sophisticated visualization strategies based on the new approach, as well as the exploration of other text representations which might take advantage of hyperbolic space properties. References [1] T. Kohonen. Self-Organizing Maps. Springer Series in Information Sciences. 3rd edition, 2001. [2] Teuvo Kohonen, Samuel Kaski, Krista Lagus, Jarkko Saloj¨arvi, Vesa Paatero, and Antti Saarela. Organization of a massive document collection. IEEE Transactions on Neural Networks, Special Issue on Neural Networks for Data Mining and Knowledge Discovery, 11(3):574–585, May 2000. [3] John Lamping and Ramana Rao. Laying out and visualizing large trees using a hyperbolic space. In Proceedings of UIST’94, pages 13–14, 1994. [4] T. Munzer. Exploring large graphs in 3D hyperbolic space. IEEE Computer Graphics and Applications, 18(4):18–23, July/August 1998. [5] H. S. M. Coxeter. Non Euclidean Geometry. Univ. of Toronto Press, Toronto, 1957. [6] J.A. Thorpe. Elementary Topics in Differential Geometry. Springer-Verlag, New York, 1979. [7] Frank Morgan. Riemannian Geometry: A Beginner’s Guide. Jones and Bartlett Publishers, Boston, London, 1993. [8] Charles W. Misner, J. A. Wheeler, and Kip S. Thorne. Gravitation. Freeman, 1973. [9] R. Fricke and F. Klein. Vorlesungen ¨uber die Theorie der automorphen Funktionen, volume 1. Teubner, Leipzig, 1897. Reprinted by Johnson Reprint, New York, 1965. [10] W. Magnus. Noneuclidean Tesselations and Their Groups. Academic Press, 1974. [11] Helge Ritter. Self-organizing maps in non-euclidian spaces. In E. Oja and S. Kaski, editors, Kohonen Maps, pages 97–108. Amer Elsevier, 1999. [12] J. Ontrup and H. Ritter. Text categorization and semantic browsing with self-organizing maps on non-euclidean spaces. In Proc. of the PKDD-01, 2001. [13] G. Salton and C. Buckley. Term-weighting approaches in automatic text retrieval. Information Processing and Management, 24(5):513–523, 1988. [14] T. Joachims. Text categorization with support vector machines: learning with many relevant features. In Proc. of ECML-98, number 1398, pages 137–142, Chemnitz, DE, 1998. [15] Huma Lodhi, John Shawe-Taylor, Nello Cristianini, and Chris Watkins. Text classification using string kernels. In Todd K. Leen, Thomas G. Dietterich, and Volker Tresp, editors, Advances in Neural Information Processing Systems 13, pages 563–569. MIT Press, 2001. [16] F. Sebastiani, A. Sperduti, and N. Valdambrini. An improved boosting algorithm and its application to automated text categorization. In Proc. of CIKM-00, pages 78–85, 2000. [17] Y. Yang. An evaluation of statistical approaches to text categorization. Information Retrieval, 1-2(1):69–90, 1999.
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1 Bayesian morphometry of hippocampal cells suggests same-cell somatodendritic repulsion Giorgio A. Ascoli * Alexei Samsonovich Krasnow Institute for Advanced Study at George Mason University Fairfax, VA 22030-4444 ascoli@gmu.edu asamsono@gmu.edu Abstract Visual inspection of neurons suggests that dendritic orientation may be determined both by internal constraints (e.g. membrane tension) and by external vector fields (e.g. neurotrophic gradients). For example, basal dendrites of pyramidal cells appear nicely fan-out. This regular orientation is hard to justify completely with a general tendency to grow straight, given the zigzags observed experimentally. Instead, dendrites could (A) favor a fixed (“external”) direction, or (B) repel from their own soma. To investigate these possibilities quantitatively, reconstructed hippocampal cells were subjected to Bayesian analysis. The statistical model combined linearly factors A and B, as well as the tendency to grow straight. For all morphological classes, B was found to be significantly positive and consistently greater than A. In addition, when dendrites were artificially re-oriented according to this model, the resulting structures closely resembled real morphologies. These results suggest that somatodendritic repulsion may play a role in determining dendritic orientation. Since hippocampal cells are very densely packed and their dendritic trees highly overlap, the repulsion must be cellspecific. We discuss possible mechanisms underlying such specificity. 1 Introduction The study of brain dynamics and development at the cellular level would greatly benefit from a standardized, accurate and yet succinct statistical model characterizing the morphology of major neuronal classes. Such model could also provide a basis for simulation of anatomically realistic virtual neurons [1]. The model should accurately distinguish among different neuronal classes: a morphological difference between classes would be captured by a difference in model parameters and reproduced in generated virtual neurons. In addition, the model should be self-consistent: there should be no statistical difference in model parameters measured from real neurons of a given class and from virtual neurons of the same class. The assumption that a simple statistical model of this sort exists relies on the similarity of average environmental and homeostatic conditions encountered by individual neurons during development and on the limited amount of genetic information that underlies differentiation of neuronal classes. Previous research in computational neuroanatomy has mainly focused on the topology and internal geometry of dendrites (i.e., the properties described in “dendrograms”) [2,3]. Recently, we attempted to include spatial orientation in the models, thus generating 2 virtual neurons in 3D [4]. Dendritic growth was assumed to deviate from the straight direction both randomly and based on a constant bias in a given direction, or “tropism”. Different models of tropism (e.g. along a fixed axis, towards a plane, or away from the soma) had dramatic effects on the shape of virtual neurons [5]. Our current strategy is to split the problem of finding a statistical model describing neuronal morphology in two parts. First, we maintain that the topology and the internal geometry of a particular dendritic tree can be described independently of its 3D embedding (i.e., the set of local dendritic orientations). At the same time, one and the same internal geometry (e.g., the experimental dendrograms obtained from real neurons) may have many equally plausible 3D embeddings that are statistically consistent with the anatomical characteristics of that neuronal class. The present work aims at finding a minimal statistical model describing local dendritic orientation in experimentally reconstructed hippocampal principal cells. Hippocampal neurons have a polarized shape: their dendrites tend to grow from the soma as if enclosed in cones. In pyramidal cells, basal and apical dendrites invade opposite hemispaces (fig. 1A), while granule cell dendrites all invade the same hemispace. This behavior could be caused by a tendency to grow towards the layers of incoming fibers to establish synapses. Such tendency would correspond to a tropism in a direction roughly parallel to the cell main axis. Alternatively, dendrites could initially stem in the appropriate (possibly genetically determined) directions, and then continue to grow approximately in a radial direction from the soma. A close inspection of pyramidal (basal) trees suggests that dendrites may indeed be repelled from their soma (Fig. 1B). A typical dendrite may reorient itself (arrow) to grow nearly straight along a radius from the soma. Remarkably, this happens even after many turns, when the initial direction is lost. Such behavior may be hard to explain without tropism. If the deviations from straight growth were random, one should be able to “remodel” th e trees by measuring and reproducing the statistics of local turn angles, assuming its independence of dendritic orientation and location. Figure 1C shows the cell from 1A after such remodeling. In this case basal and apical dendrites retain only their initial (stemming) orientations from the original data. The resulting “cotton ball” s uggests that dendritic turns are not in dependent of dendritic orientation. In this paper, we use Bayesian analysis to quantify the dendritic tropism. 2 Methods Digital files of fully reconstructed rat hippocampal pyramidal cells (24 CA3 and 23 CA1 neurons) were kindly provided by Dr. D. Amaral. The overall morphology of these cells, as well as the experimental acquisition methods, were extensively described [6]. In these files, dendrites are represented as (branching) chains of cylindrical sections. Each section is connected to one other section in the path to the soma, and may be connected on the other extremity to two other sections (bifurcation), one other section (continuation point), or no other section (terminal tip). Each section is described in the file by its ending point coordinates, its diameter and its "parent", i.e., the attached section in the path to the soma [5,7]. In CA3 cells, basal dendrites had an average of 687(±216) continuation points and 72(±17) bifurcations per cell, while apical dendrites had 717(±156) continuation points and 80(±21) bifurcations per cell. CA1 cells had 462(±138) continuation points and 52(±12) bifurcations (basal), 860(±188) continuation points and 120(±22) bifurcations (apical). In the present work, basal and apical trees of CA3 and CA1 pyramidal cells were treated as 4 different classes. Digital data of rat dentate granule cells [8] are kindly made available by Dr. B. Claiborne through the internet (http://cascade.utsa.edu/bjclab). Only 36 of the 42 cells in this archive were used: in 6 cases numerical processing was not accomplished due to minor inconsistencies in the data files. The “rejected” cells were 1208875, 3319201, 411883, 411884A, 411884B, 803887B. Granule dendrites had 3 549(±186) continuation points and 30(±6) bifurcations per cell. Cells in these or similar formats can be rendered, rotated, and zoomed with a java applet available through the internet (www.cns.soton.ac.uk) [7]. Figure 1: A: A pyramidal cell (c53063) from Amaral’s archive. B: A zoom-in from panel A (arrows point to the same basal tree location). Dotted dendrites are behind the plane. C: Same cell (c53063) with its dendritic orientation remodeled assuming zero tropism and same statistics of all turn angles (see Results). In agreement with the available format of morphological data (described above), the process of dendritic growth1 can be represented as a discrete stochastic process consisting of sequential attachment of new sections to each growing dendrite. Here we keep the given internal geometry of the experimental data while remodeling the 3D embedding geometry (dendritic orientation). The task is to make a remodeled geometry statistically consistent with the original structure. The basic assumption is that neuronal development1 is a Markov process governed by local rules [4]. Specifically, we assume that (i) each step in dendritic outgrowth only depends on the preceding step and on current local conditions; and (ii) dendrites do not undergo geometrical or topological modification after their formation (see, however, Discussion). In this Markov approximation, a plausible 3D embedding can be found by sequentially orienting individual sections, starting from the soma and moving toward the terminals. We are implementing this procedure in two-step iterations (1). First, at a given node i with coordinates ri we select a section i+1, disregard its given orientation, and calculate its most likely expected direction n'i+1 based on the model (here section i+1 connects nodes i and i+1, and n stands for a unit vector). For a continuation point, the most likely direction n'i+1 is computed as the direction of the vector sum ni + vi. The first term is the direction of the parent section ni, and reflects the tendency dendrites exhibit to grow relatively straight due to membrane tension, mechanical properties of the cytoskeleton, etc. The second term is a local value of a vector field: vi = v(ri), which comprises the influence of external local conditions on the direction of growth (as specified below). Finally, we generate a perturbation of the most likely direction n'i+1 to produce a particular plausible instance of a new direction. In summary, the new direction ni+1 is generated as: 1 Although we resort to a developmental metaphor, our goal is to describe accurately the result of development rather than the process of development. 4 . || ' , ' 1 1 1 i i i i i i v n n n T n + = + + + (1) Here Ti is an operator that deflects n'i+1 into a random direction. If we view each deflection as a yaw of angle αi, then the corresponding rolling angle (describing rotation around the axis of the parent dendrite) is distributed uniformly between 0 and 2π. The probability distribution function for deflections as a function of αi is taken in a form that, as we found, well fits experimental data: ( ) , σ αi e P i − ∝ T (2) where σ << 1 is a parameter of the model. At bifurcation points, the same rule (1), (2) is applied for each daughter independently. A more accurate and plausible description of dendritic orientation at bifurcations might require a more complex model. However, our simple choice yields surprisingly good results (see below). The model (1), (2) can be used in the simulation of virtual neuronal morphology. In this case one would first need to generate the internal geometry of the dendrites [1-5]. Most importantly, model (1), (2) can be used to quantitatively assess the significance of the somatocentric (radial) tropism of real dendrites. Assuming that there is a significant preferential directionality of growth in hippocampal dendrites, the two main alternatives are (see Introduction): HA: The dominating tropic factor is independent of the location of the soma. HB: The dominating tropic factor is radial with respect to the soma. The simplest model for the vector field v that discriminates between these alternative hypotheses includes both factors, A and B, linearly: . r i i bn a v + = (3) Here a = (ax, ay, az) is a constant vector representing global directionality of cellindependent environmental factors (chemical gradients, density of neurites, etc.) influencing dendritic orientation. nr i is the unit vector in the direction connecting the soma to node i, thus representing a somatocentric tropic factor. In summary, ax, ay, az, b and σ are the parameters of the model. Finding that the absolute value a = |a| is significantly greater than b would support HA. On the contrary, finding that b is greater than a would support HB. Based on a Bayesian approach, we compute the most likely values of a, b and σ by maximization of the likelihood of all experimentally measured orientations (taken at continuation points only) of a given dendritic tree: ( ) { } ( ) { } , min arg max arg * *, , , , i b i i b P b α σ a a T a = = ∏ (4) where αi is given by (1)-(3) with experimental section orientations substituted for ni, ni+1, asterisk denotes most likely values, and the average is over all continuation points. Given a* and b*, the value of σ* can be found from the average value of αi computed with a = a* and b = b*. The relation results from differentiation of (4) by σ. The same relation holds for the average value of α computed based on the probability distribution function (2) with σ = σ*. Therefore, <αi> computed from the neurometric data with a = a* and b = b* is equal to <α> based on (2) with σ = σ*. The model is thus self-consistent: the measured value of σ* in a remodeled neuron is guaranteed to coincide on average with the input parameter σ used for simulation. In addition, our numerical analysis indicates self-consistency of the model with respect to a and b, when their values are within a practically meaningful range. 5 3 Results Results of the Bayesian analysis are presented in Table 1. Parameters a and b were optimized for each cell individually, then the absolute value a = |a| was taken for each cell. The mean value and the standard deviation of a in Table 1 were computed based on the set of the individual absolute values, while each individual value of b was taken with its sign (which was positive in all cases but one). The most likely direction of a varied significantly among cells, i.e., no particular fixed direction was generally preferred. Table 1: Results from Bayesian analysis (mean ± standard deviation). α is the minimized deflection angle, a and b are parameters of the model (1)-(3) computed according to (4). Original data Z coordinate set to zero Dataset α B a α b A CA3-bas 16.4 ± 2.3 0.49 ± 0.17 0.08 ± 0.05 12.0 ± 2.4 0.42 ± 0.15 0.06 ± 0.05 CA3-apic 15.2 ± 1.9 0.36 ± 0.16 0.12 ± 0.07 12.0 ± 2.9 0.29 ± 0.23 0.10 ± 0.14 CA1-bas 16.6 ± 1.6 0.49 ± 0.26 0.14 ± 0.10 14.2 ± 1.9 0.48 ± 0.31 0.16 ± 0.12 CA1-apic 19.1 ± 2.0 0.30 ± 0.20 0.16 ± 0.15 17.3 ± 2.4 0.22 ± 0.17 0.11 ± 0.10 Granule 19.1 ± 2.7 1.01 ± 0.64 0.17 ± 0.11 11.0 ± 1.9 0.36 ± 0.16 0.07 ± 0.05 The key finding is that a is not significantly different from zero, while b is significantly positive. The slightly higher coefficient of variation obtained for granule cells could be due to a larger experimental error in the z coordinate (orthogonal to the slice). In several granule cells (but in none of the pyramidal cells) the greater noise in z was apparent upon visual inspection of the rendered structures. Therefore, we re-ran the analysis discarding the z coordinate (right columns). Results changed only minimally for pyramidal cells, and the granule cell parameters became more consistent with the pyramidal cells. The measured average values of the model parameters were used for remodeling of experimental neuronal shapes, as described above. In particular, b was set to 0.5, while a was set to zero. We kept the internal geometry and the initial stemming direction of each tree from the experimental data, and simulated dendritic orientation at all nodes separated by more than 2 steps from the soma. A typical result is shown in Figure 2. Generally, the artificially re-oriented dendrites looked better than one could expect for a model as simple as (1) – (3). This result may be compared with figure 1C, which shows an example of remodeling based on the same model in the absence of tropism (a = b = 0). Although in this case the shape can be improved by reducing σ, the result never gets as close to a real shape as in Fig. 2 C, D, even when random, uncorrelated local distortions ("shuffling") are applied to the generated geometry. Thus, although the tendency to grow straight represents the dominant component of the model (i.e., b<1), somatocentric tropism may exert a dramatic effect on dendritic shape. Surprisingly, even the asymmetry of the dendritic spread (compare front and side views) is preserved after remodeling. However, two details are difficult to reproduce with this model: the uniform distribution of dendrites in space and other subtle medium-distance correlations among dendritic deflections. In order to account for these properties, we may need to consider spatially correlated inhomogeneities of the medium and possible short range dendrodendritic interactions. 6 4 Discussion The key results of this work is that, according to Bayesian analysis, dendrites of hippocampal principal cells display a significant radial tropism. This means that the spatial orientation of these neuronal trees can be statistically described as if dendrites were repelled from their own soma. This preferential direction is stronger than any tendency to grow along a fixed direction independent of the location of the soma. These results apply to all dendritic classes, but in general pyramidal cell basal trees (and granule cell dendrites) display a bigger somatocentric tropism than apical trees. Figure 2: Dendritic remodeling with somatocentric tropism. A, B: front and side views of cell 10861 from Amaral' s archive. C, D: Same views after remodeling with parameters a = 0, b = 0.5, σ = 0.15 (corresponding to <α> = 17
! #" $ # stem were taken in their original orientations; all subsequent experimental orientations were disregarded and regenerated from scratch according to the model. Assuming that dendrites are indeed repelled from their soma during development, what could be a plausible mechanism? Principal cells are very densely packed in the hippocampus, and their dendrites highly overlap. If repulsion were mediated by a diffusible chemical factor, in order for dendrites to be repelled radially from their own soma, each neuron should have its own specific chemical marker (a fairly unlikely possibility). If the same repulsive factor were released by all neurons, each dendrite would be repelled by hundreds of somata, and not just by their own. The resulting tropism would be perpendicular to the principal cell layer, i.e. each dendrite would be pushed approximately in the same direction, independent of the location of its soma. This scenario is in clear contrast with the result of our statistical analysis. Thus, how can a growing dendrite sense the location of its own soma? One possibility involves the spontaneous spiking activity of neurons during development. A cell that spikes becomes unique in its neighborhood for a short period of time. The philopodia of dendritic growth 7 cones could possess voltage-gated receptors to sense transient chemical gradients (e.g., pH) created by the spiking cell. Only dendrites that are depolarized during the transient chemical gradient (i.e., those belonging to the same spiking cell) would be repelled by it. Alternatively, depolarized philopodia could be sensitive to the small voltage difference created by the spike in the extracellular space (a voltage that can be recorded by tetrodes). The main results obtained with the simple model presented in this work are independent of the z coordinate in the morphometric files, i.e. the most error-prone measurement in the experimental reconstruction. However, it is important to note that any observed deviation of dendritic path from a straight line, including that due to measurement errors, causes an increase in the most likely values of parameters a and b. Another possibility is that dendrites do grow almost precisely in straight lines, and the measured values of a and b reflect distortions of dendritic shapes after development. In order to assess the effect of these factors on a and b, we pre-processed the experimental data by adding a gradually increasing noise to all coordinates of dendritic sections. Then we were able to extrapolate the dependence of a*, b* and <α>* on the amplitude of noise in order to estimate the parameter values in the absence of the experimental error (which was conservatively taken to be of 0.5 µm). For basal trees of CA3 pyramidal cells, this analysis yielded an estimated “corrected” value of b between 0.14 and 0.25, with a remaining much smaller than b. Interestingly, our analysis based on extrapolation shows that, regardless of the assumed amount of distortion present in the experimental data, given the numbers measured for CA3 basal trees, positive initial <α> implies positive initial b. In other words, not only measurement errors, but also possible biological distortions of the dendritic tree may not be capable of accounting for the observed positivity of the parameter b. Although these factors affect our results quantitatively, they do not change the statistical significance nor the qualitative trends. However, a more rigorous analysis needs to be carried out. Nevertheless, artificially reoriented dendrites according to our simple model appear almost as realistic as the original structures, and we could not achieve the same result with any choice of parameters in models of distortion without a somatocentric tropism. In conclusion, whether the present Bayesian analysis reveals a phenomenon of somatodendritic repulsion remains an (experimentally testable) open question. What is unquestionable is that the presented model is a significant step forward in the formulation of an accurate statistical description of dendritic morphology. Acknowledgments This work was supported in part by Human Brain Project Grant R01 NS39600, funded jointly by NINDS and NIMH. References [1] Ascoli G.A. (1999) Progress and perspectives in computational neuroanatomy. Anat. Rec. 257(6):195-207. [2] van Pelt J. (1997) Effect of pruning on dendritic tree topology. J. Theor. Biol. 186(1):17-32. [3] Burke R.E., W. Marks, B. Ulfhake (1992) A parsimonious description of motoneurons dendritic morphology using computer simulation. J. Neurosci. 12(6):2403-2416. [4] Ascoli G.A., J. Krichmar (2000) L-Neuron: a modeling tool for the efficient generation and parsimonious description of dendritic morphology. Neurocomputing 32-33:1003-1011. [5] Ascoli G.A., J. Krichmar, S. Nasuto, S. Senft (2001) Generation, description and storage of dendritic morphology data. Phil. Trans. R. Sci. B, In Press. [6] Ishizuka N., W. Cowan, D. Amaral (1995) A quantitative analysis of the dendritic organization of pyramidal cells in the rat hippocampus. J. Comp. Neurol. 362(1):17-45. [7] Cannon R.C., D. Turner, G. Pyapali, H. Wheal (1998) An on-line archive of reconstructed hippocampal neurons. J Neurosci. Meth. 84(1-2):49-54. [8] Rihn L.L., B. Claiborne (1990) Dendritic growth and regression in rat dentate granule cells during late postnatal development. Dev. Brain Res. 54(1):115-124
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Entropy and Inference, Revisited Ilya Nemenman,1,2 Fariel Shafee,3 and William Bialek1,3 1NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540 2Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 3Department of Physics, Princeton University, Princeton, New Jersey 08544 nemenman@itp.ucsb.edu, {fshafee/wbialek}@princeton.edu Abstract We study properties of popular near–uniform (Dirichlet) priors for learning undersampled probability distributions on discrete nonmetric spaces and show that they lead to disastrous results. However, an Occam–style phase space argument expands the priors into their infinite mixture and resolves most of the observed problems. This leads to a surprisingly good estimator of entropies of discrete distributions. Learning a probability distribution from examples is one of the basic problems in data analysis. Common practical approaches introduce a family of parametric models, leading to questions about model selection. In Bayesian inference, computing the total probability of the data arising from a model involves an integration over parameter space, and the resulting “phase space volume” automatically discriminates against models with larger numbers of parameters—hence the description of these volume terms as Occam factors [1, 2]. As we move from finite parameterizations to models that are described by smooth functions, the integrals over parameter space become functional integrals and methods from quantum field theory allow us to do these integrals asymptotically; again the volume in model space consistent with the data is larger for models that are smoother and hence less complex [3]. Further, at least under some conditions the relevant degree of smoothness can be determined self–consistently from the data, so that we approach something like a model independent method for learning a distribution [4]. The results emphasizing the importance of phase space factors in learning prompt us to look back at a seemingly much simpler problem, namely learning a distribution on a discrete, nonmetric space. Here the probability distribution is just a list of numbers {qi}, i = 1, 2, · · · , K, where K is the number of bins or possibilities. We do not assume any metric on the space, so that a priori there is no reason to believe that any qi and qj should be similar. The task is to learn this distribution from a set of examples, which we can describe as the number of times ni each possibility is observed in a set of N = PK i=1 ni samples. This problem arises in the context of language, where the index i might label words or phrases, so that there is no natural way to place a metric on the space, nor is it even clear that our intuitions about similarity are consistent with the constraints of a metric space. Similarly, in bioinformatics the index i might label n–mers of the the DNA or amino acid sequence, and although most work in the field is based on metrics for sequence comparison one might like an alternative approach that does not rest on such assumptions. In the analysis of neural responses, once we fix our time resolution the response becomes a set of discrete “words,” and estimates of the information content in the response are determined by the probability distribution on this discrete space. What all of these examples have in common is that we often need to draw some conclusions with data sets that are not in the asymptotic limit N ≫K. Thus, while we might use a large corpus to sample the distribution of words in English by brute force (reaching N ≫K with K the size of the vocabulary), we can hardly do the same for three or four word phrases. In models described by continuous functions, the infinite number of “possibilities” can never be overwhelmed by examples; one is saved by the notion of smoothness. Is there some nonmetric analog of this notion that we can apply in the discrete case? Our intuition is that information theoretic quantities may play this role. If we have a joint distribution of two variables, the analog of a smooth distribution would be one which does not have too much mutual information between these variables. Even more simply, we might say that smooth distributions have large entropy. While the idea of “maximum entropy inference” is common [5], the interplay between constraints on the entropy and the volume in the space of models seems not to have been considered. As we shall explain, phase space factors alone imply that seemingly sensible, more or less uniform priors on the space of discrete probability distributions correspond to disastrously singular prior hypotheses about the entropy of the underlying distribution. We argue that reliable inference outside the asymptotic regime N ≫K requires a more uniform prior on the entropy, and we offer one way of doing this. While many distributions are consistent with the data when N ≤K, we provide empirical evidence that this flattening of the entropic prior allows us to make surprisingly reliable statements about the entropy itself in this regime. At the risk of being pedantic, we state very explicitly what we mean by uniform or nearly uniform priors on the space of distributions. The natural “uniform” prior is given by Pu({qi}) = 1 Zu δ 1 − K X i=1 qi ! , Zu = Z A dq1dq2 · · · dqK δ 1 − K X i=1 qi ! (1) where the delta function imposes the normalization, Zu is the total volume in the space of models, and the integration domain A is such that each qi varies in the range [0, 1]. Note that, because of the normalization constraint, an individual qi chosen from this distribution in fact is not uniformly distributed—this is also an example of phase space effects, since in choosing one qi we constrain all the other {qj̸=i}. What we mean by uniformity is that all distributions that obey the normalization constraint are equally likely a priori. Inference with this uniform prior is straightforward. If our examples come independently from {qi}, then we calculate the probability of the model {qi} with the usual Bayes rule: 1 P({qi}|{ni}) = P({ni}|{qi})Pu({qi}) Pu({ni}) , P({ni}|{qi}) = K Y i=1 (qi)ni. (2) If we want the best estimate of the probability qi in the least squares sense, then we should compute the conditional mean, and this can be done exactly, so that [6, 7] ⟨qi⟩= ni + 1 N + K . (3) Thus we can think of inference with this uniform prior as setting probabilities equal to the observed frequencies, but with an “extra count” in every bin. This sensible procedure was first introduced by Laplace [8]. It has the desirable property that events which have not been observed are not automatically assigned probability zero. 1If the data are unordered, extra combinatorial factors have to be included in P({ni}|{qi}). However, these cancel immediately in later expressions. A natural generalization of these ideas is to consider priors that have a power–law dependence on the probabilities, the so called Dirichlet family of priors: Pβ({qi}) = 1 Z(β)δ 1 − K X i=1 qi ! K Y i=1 qβ−1 i , (4) It is interesting to see what typical distributions from these priors look like. Even though different qi’s are not independent random variables due to the normalizing δ–function, generation of random distributions is still easy: one can show that if qi’s are generated successively (starting from i = 1 and proceeding up to i = K) from the Beta–distribution P(qi) = B qi 1 −P j<i qj ; β, (K −i)β ! , B (x; a, b) = xa−1(1 −x)b−1 B(a, b) , (5) 0 0.8 q β = 0.0007, S = 1.05 bits 0 0.2 q β = 0.02, S = 5.16 bits 0 200 400 600 800 1000 0 0.01 q β = 1, S = 9.35 bits bin number Figure 1: Typical distributions, K = 1000. then the probability of the whole sequence {qi} is Pβ({qi}). Fig. 1 shows some typical distributions generated this way. They represent different regions of the range of possible entropies: low entropy (∼1 bit, where only a few bins have observable probabilities), entropy in the middle of the possible range, and entropy in the vicinity of the maximum, log2 K. When learning an unknown distribution, we usually have no a priori reason to expect it to look like only one of these possibilities, but choosing β pretty much fixes allowed “shapes.” This will be a focal point of our discussion. Even though distributions look different, inference with all priors Eq. (4) is similar [6, 7]: ⟨qi⟩β = ni + β N + κ , κ = Kβ. (6) This simple modification of the Laplace’s rule, Eq. (3), which allows us to vary probability assigned to the outcomes not yet seen, was first examined by Hardy and Lidstone [9, 10]. Together with the Laplace’s formula, β = 1, this family includes the usual maximum likelihood estimator (MLE), β →0, that identifies probabilities with frequencies, as well as the Jeffreys’ or Krichevsky–Trofimov (KT) estimator, β = 1/2 [11, 12, 13], the Schurmann–Grassberger (SG) estimator, β = 1/K [14], and other popular choices. To understand why inference in the family of priors defined by Eq. (4) is unreliable, consider the entropy of a distribution drawn at random from this ensemble. Ideally we would like to compute this whole a priori distribution of entropies, Pβ(S) = Z dq1dq2 · · · dqK Pβ({qi}) δ " S + K X i=1 qi log2 qi # , (7) but this is quite difficult. However, as noted by Wolpert and Wolf [6], one can compute the moments of Pβ(S) rather easily. Transcribing their results to the present notation (and correcting some small errors), we find: ξ(β) ≡⟨S[ni = 0] ⟩β = ψ0(κ + 1) −ψ0(β + 1) , (8) σ2(β) ≡⟨(δS)2[ni = 0]⟩β = β + 1 κ + 1 ψ1(β + 1) −ψ1(κ + 1) , (9) where ψm(x) = (d/dx)m+1 log2 Γ(x) are the polygamma functions. 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β ξ(β) / log2K K=10 K=100 K=1000 1e−7 1e−5 1e−3 .25 1.0 1.5 2.0 0 0.2 0.4 0.6 0.8 β σ(β) Figure 2: ξ(β)/ log2 K and σ(β) as functions of β and K; gray bands are the region of ±σ(β) around the mean. Note the transition from the logarithmic to the linear scale at β = 0.25 in the insert. This behavior of the moments is shown on Fig. 2. We are faced with a striking observation: a priori distributions of entropies in the power–law priors are extremely peaked for even moderately large K. Indeed, as a simple analysis shows, their maximum standard deviation of approximately 0.61 bits is attained at β ≈1/K, where ξ(β) ≈ 1/ ln 2 bits. This has to be compared with the possible range of entropies, [0, log2 K], which is asymptotically large with K. Even worse, for any fixed β and sufficiently large K, ξ(β) = log2 K −O(K0), and σ(β) ∝ 1/√κ. Similarly, if K is large, but κ is small, then ξ(β) ∝κ, and σ(β) ∝√κ. This paints a lively picture: varying β between 0 and ∞results in a smooth variation of ξ, the a priori expectation of the entropy, from 0 to Smax = log2 K. Moreover, for large K, the standard deviation of Pβ(S) is always negligible relative to the possible range of entropies, and it is negligible even absolutely for ξ ≫1 (β ≫1/K). Thus a seemingly innocent choice of the prior, Eq. (4), leads to a disaster: fixing β specifies the entropy almost uniquely. Furthermore, the situation persists even after we observe some data: until the distribution is well sampled, our estimate of the entropy is dominated by the prior! Thus it is clear that all commonly used estimators mentioned above have a problem. While they may or may not provide a reliable estimate of the distribution {qi}2, they are definitely a poor tool to learn entropies. Unfortunately, often we are interested precisely in these entropies or similar information–theoretic quantities, as in the examples (neural code, language, and bioinformatics) we briefly mentioned earlier. Are the usual estimators really this bad? Consider this: for the MLE (β = 0), Eqs. (8, 9) are formally wrong since it is impossible to normalize P0({qi}). However, the prediction that P0(S) = δ(S) still holds. Indeed, SML, the entropy of the ML distribution, is zero even for N = 1, let alone for N = 0. In general, it is well known that SML always underestimates the actual value of the entropy, and the correction S = SML + K∗ 2N + O 1 N 2 (10) is usually used (cf. [14]). Here we must set K∗= K −1 to have an asymptotically correct result. Unfortunately in an undersampled regime, N ≪K, this is a disaster. To alleviate the problem, different authors suggested to determine the dependence K ∗= K∗(K) by various (rather ad hoc) empirical [15] or pseudo–Bayesian techniques [16]. However, then there is no principled way to estimate both the residual bias and the error of the estimator. The situation is even worse for the Laplace’s rule, β = 1. We were unable to find any results in the literature that would show a clear understanding of the effects of the prior on the entropy estimate, SL. And these effects are enormous: the a priori distribution of the entropy has σ(1) ∼1/ √ K and is almost δ-like. This translates into a very certain, but nonetheless possibly wrong, estimate of the entropy. We believe that this type of error 2In any case, the answer to this question depends mostly on the “metric” chosen to measure reliability. Minimization of bias, variance, or information cost (Kullback–Leibler divergence between the target distribution and the estimate) leads to very different “best”estimators. (cf. Fig. 3) has been overlooked in some previous literature. The Schurmann–Grassberger estimator, β = 1/K, deserves a special attention. The variance of Pβ(S) is maximized near this value of β (cf. Fig. 2). Thus the SG estimator results in the most uniform a priori expectation of S possible for the power–law priors, and consequently in the least bias. We suspect that this feature is responsible for a remark in Ref. [14] that this β was empirically the best for studying printed texts. But even the SG estimator is flawed: it is biased towards (roughly) 1/ ln 2, and it is still a priori rather narrow. 10 30 100 300 1000 3000 10000 −3 −2 −1 0 1 2 3 4 5 N <S>β − S β = 0.001 β = 0.02 β = 1 Figure 3: Learning the β = 0.02 distribution from Fig. 1 with β = 0.001, 0.02, 1. The actual error of the estimators is plotted; the error bars are the standard deviations of the posteriors. The “wrong” estimators are very certain but nonetheless incorrect. Summarizing, we conclude that simple power–law priors, Eq. (4), must not be used to learn entropies when there is no strong a priori knowledge to back them up. On the other hand, they are the only priors we know of that allow to calculate ⟨qi⟩, ⟨S⟩, ⟨χ2⟩, ...exactly [6]. Is there a way to resolve the problem of peakedness of Pβ(S) without throwing away their analytical ease? One approach would be to use Pflat β ({qi}) = Pβ({qi}) Pβ(S[qi]) Pactual(S[qi]) as a prior on {qi}. This has a feature that the a priori distribution of S deviates from uniformity only due to our actual knowledge Pactual(S[qi]), but not in the way Pβ(S) does. However, as we already mentioned, Pβ(S[qi]) is yet to be calculated. Another way to a flat prior is to write P(S) = 1 = R δ(S −ξ)dξ. If we find a family of priors P({qi}, parameters) that result in a δ-function over S, and if changing the parameters moves the peak across the whole range of entropies uniformly, we may be able to use this. Luckily, Pβ(S) is almost a δ-function! 3 In addition, changing β results in changing ξ(β) = ⟨S[ni = 0] ⟩β across the whole range [0, log2 K]. So we may hope that the prior 4 P({qi}; β) = 1 Z δ 1 − K X i=1 qi ! K Y i=1 qβ−1 i dξ(β) dβ P(β) (11) may do the trick and estimate entropy reliably even for small N, and even for distributions that are atypical for any one β. We have less reason, however, to expect that this will give an equally reliable estimator of the atypical distributions themselves.2 Note the term dξ/dβ in Eq. (11). It is there because ξ, not β, measures the position of the entropy density peak. Inference with the prior, Eq. (11), involves additional averaging over β (or, equivalently, 3The approximation becomes not so good as β →0 since σ(β) becomes O(1) before dropping to zero. Even worse, Pβ(S) is skewed at small β. This accumulates an extra weight at S = 0. Our approach to dealing with these problems is to ignore them while the posterior integrals are dominated by β’s that are far away from zero. This was always the case in our simulations, but is an open question for the analysis of real data. 4Priors that are formed as weighted sums of the different members of the Dirichlet family are usually called Dirichlet mixture priors. They have been used to estimate probability distributions of, for example, protein sequences [17]. Equation (11), an infinite mixture, is a further generalization, and, to our knowledge, it has not been studied before. ξ), but is nevertheless straightforward. The a posteriori moments of the entropy are c Sm = R dξ ρ(ξ, {ni})⟨Sm[ni] ⟩β(ξ) R dξ ρ(ξ, [ni]) , where (12) ρ(ξ, [ni]) = P (β (ξ)) Γ(κ(ξ)) Γ(N + κ(ξ)) K Y i=1 Γ(ni + β(ξ)) Γ(β(ξ)) . (13) Here the moments ⟨Sm[ni] ⟩β(ξ) are calculated at fixed β according to the (corrected) formulas of Wolpert and Wolf [6]. We can view this inference scheme as follows: first, one sets the value of β and calculates the expectation value (or other moments) of the entropy at this β. For small N, the expectations will be very close to their a priori values due to the peakedness of Pβ(S). Afterwards, one integrates over β(ξ) with the density ρ(ξ), which includes our a priori expectations about the entropy of the distribution we are studying [P (β (ξ))], as well as the evidence for a particular value of β [Γ-terms in Eq. (13)]. The crucial point is the behavior of the evidence. If it has a pronounced peak at some βcl, then the integrals over β are dominated by the vicinity of the peak, bS is close to ξ(βcl), and the variance of the estimator is small. In other words, data “selects” some value of β, much in the spirit of Refs. [1] – [4]. However, this scenario may fail in two ways. First, there may be no peak in the evidence; this will result in a very wide posterior and poor inference. Second, the posterior density may be dominated by β →0, which corresponds to MLE, the best possible fit to the data, and is a discrete analog of overfitting. While all these situations are possible, we claim that generically the evidence is well–behaved. Indeed, while small β increases the fit to the data, it also increases the phase space volume of all allowed distributions and thus decreases probability of each particular one [remember that ⟨qi⟩β has an extra β counts in each bin, thus distributions with qi < β/(N +κ) are strongly suppressed]. The fight between the “goodness of fit” and the phase space volume should then result in some non–trivial βcl, set by factors ∝N in the exponent of the integrand. Figure 4 shows how the prior, Eq. (11), performs on some of the many distributions we tested. The left panel describes learning of distributions that are typical in the prior Pβ({qi}) and, therefore, are also likely in P({qi}; β). Thus we may expect a reasonable performance, but the real results exceed all expectations: for all three cases, the actual relative error drops to the 10% level at N as low as 30 (recall that K = 1000, so we only have ∼0.03 data points per bin on average)! To put this in perspective, simple estimates like fixed β ones, MLE, and MLE corrected as in Eq. (10) with K∗equal to the number of nonzero ni’s produce an error so big that it puts them off the axes until N > 100. 5 Our results have two more nice features: the estimator seems to know its error pretty well, and it is almost completely unbiased. One might be puzzled at how it is possible to estimate anything in a 1000–bin distribution with just a few samples: the distribution is completely unspecified for low N! The point is that we are not trying to learn the distribution — in the absence of additional prior information this would, indeed, take N ≫K — but to estimate just one of its characteristics. It is less surprising that one number can be learned well with only a handful of measurements. In practice the algorithm builds its estimate based on the number of coinciding samples (multiple coincidences are likely only for small β), as in the Ma’s approach to entropy estimation from simulations of physical systems [18]. What will happen if the algorithm is fed with data from a distribution {˜qi} that is strongly atypical in P({qi}; β)? Since there is no {˜qi} in our prior, its estimate may suffer. Nonetheless, for any {˜qi}, there is some β which produces distributions with the same mean entropy as S[˜qi]. Such β should be determined in the usual fight between the “goodness of fit” and 5More work is needed to compare our estimator to more complex techniques, like in Ref. [15, 16]. (a) (b) −0.2 0 0.6 β = 0.0007 S = 1.05 bits ( S − S ) / S ^ −0.2 0 0.6 β = 0.02 S = 5.16 bits ( S − S ) / S ^ 10 30 100 300 1000 3000 10000 −0.3 0 0.1 β = 1.0 S = 9.35 bits ( S − S ) / S ^ N −0.4 0 0.3 β = 0.02 K = 2000 (half empty) S = 5.16 bits ( S − S ) / S ^ −0.2 0 0.4 Zipf’s law: qi ~ 1/i K = 1000 S = 7.49 bits ( S − S ) / S ^ 10 30 100 300 1000 3000 10000 −0.2 0 0.4 qi ~ 50 − 4 (ln i)2 K = 1000 S = 4.68 bits ( S − S ) / S ^ N Figure 4: Learning entropies with the prior Eq. (11) and P(β) = 1. The actual relative errors of the estimator are plotted; the error bars are the relative widths of the posteriors. (a) Distributions from Fig. 1. (b) Distributions atypical in the prior. Note that while bS may be safely calculated as just ⟨S⟩βcl, one has to do an honest integration over β to get c S2 and the error bars. Indeed, since Pβ(S) is almost a δ-function, the uncertainty at any fixed β is very small (see Fig. 3). the Occam factors, and the correct value of entropy will follow. However, there will be an important distinction from the “correct prior” cases. The value of β indexes available phase space volumes, and thus the smoothness (complexity) of the model class [19]. In the case of discrete distributions, smoothness is the absence of high peaks. Thus data with faster decaying Zipf plots (plots of bins’ occupancy vs. occupancy rank i) are rougher. The priors Pβ({qi}) cannot account for all possible roughnesses. Indeed, they only generate distributions for which the expected number of bins ν with the probability mass less than some q is given by ν(q) = KB(q, β, κ −β), where B is the familiar incomplete Beta function, as in Eq. (5). This means that the expected rank ordering for small and large ranks is qi ≈ 1 − βB(β, κ −β)(K −1) i K 1/(κ−β) , i ≪K , (14) qi ≈ βB(β, κ −β)(K −i + 1) K 1/β , K −i + 1 ≪K . (15) In an undersampled regime we can observe only the first of the behaviors. Therefore, any distribution with qi decaying faster (rougher) or slower (smoother) than Eq. (14) for some β cannot be explained well with fixed βcl for different N. So, unlike in the cases of learning data that are typical in Pβ({qi}), we should expect to see βcl growing (falling) for qualitatively smoother (rougher) cases as N grows. N 1/2 full Zipf rough units ·10−2 ·10−1 ·10−3 10 1.7 1907 16.8 30 2.2 0.99 11.5 100 2.4 0.86 12.9 300 2.2 1.36 8.3 1000 2.1 2.24 6.4 3000 1.9 3.36 5.4 10000 2.0 4.89 4.5 Table 1: βcl for solutions shown on Fig. 4(b). Figure 4(b) and Tbl. 1 illustrate these points. First, we study the β = 0.02 distribution from Fig. 1. However, we added a 1000 extra bins, each with qi = 0. Our estimator performs remarkably well, and βcl does not drift because the ranking law remains the same. Then we turn to the famous Zipf’s distribution, so common in Nature. It has ni ∝1/i, which is qualitatively smoother than our prior allows. Correspondingly, we get an upwards drift in βcl. Finally, we analyze a “rough” distribution, which has qi ∝50 −4(ln i)2, and βcl drifts downwards. Clearly, one would want to predict the dependence βcl(N) analytically, but this requires calculation of the predictive information (complexity) for the involved distributions [19] and is a work for the future. Notice that, the entropy estimator for atypical cases is almost as good as for typical ones. A possible exception is the 100– 1000 points for the Zipf distribution—they are about two standard deviations off. We saw similar effects in some other “smooth” cases also. This may be another manifestation of an observation made in Ref. [4]: smooth priors can easily adapt to rough distribution, but there is a limit to the smoothness beyond which rough priors become inaccurate. To summarize, an analysis of a priori entropy statistics in common power–law Bayesian estimators revealed some very undesirable features. We are fortunate, however, that these minuses can be easily turned into pluses, and the resulting estimator of entropy is precise, knows its own error, and gives amazing results for a very large class of distributions. Acknowledgements We thank Vijay Balasubramanian, Curtis Callan, Adrienne Fairhall, Tim Holy, Jonathan Miller, Vipul Periwal, Steve Strong, and Naftali Tishby for useful discussions. I. N. was supported in part by NSF Grant No. PHY99-07949 to the Institute for Theoretical Physics. References [1] D. MacKay, Neural Comp. 4, 415–448 (1992). [2] V. Balasubramanian, Neural Comp. 9, 349–368 (1997). [3] W. Bialek, C. Callan, and S. Strong, Phys. Rev. Lett. 77, 4693–4697 (1996). [4] I. Nemenman and W. Bialek, Advances in Neural Inf. Processing Systems 13, 287–293 (2001). [5] J. Skilling, in Maximum entropy and Bayesian methods, J. Skilling ed. (Kluwer Academic Publ., Amsterdam, 1989), pp. 45–52. [6] D. Wolpert and D. Wolf, Phys. Rev. E 52, 6841–6854 (1995). [7] I. Nemenman, Ph.D. Thesis, Princeton, (2000), ch. 3, http://arXiv.org/abs/physics/0009032. [8] P. de Laplace, marquis de, Essai philosophique sur les probabilit´es (Courcier, Paris, 1814), trans. by F. Truscott and F. Emory, A philosophical essay on probabilities (Dover, New York, 1951). [9] G. Hardy, Insurance Record (1889), reprinted in Trans. Fac. Actuaries 8 (1920). [10] G. Lidstone, Trans. Fac. Actuaries 8, 182–192 (1920). [11] H. Jeffreys, Proc. Roy. Soc. (London) A 186, 453–461 (1946). [12] R. Krichevskii and V. Trofimov, IEEE Trans. Inf. Thy. 27, 199–207 (1981). [13] F. Willems, Y. Shtarkov, and T. Tjalkens, IEEE Trans. Inf. Thy. 41, 653–664 (1995). [14] T. Schurmann and P. Grassberger, Chaos 6, 414–427 (1996). [15] S. Strong, R. Koberle, R. de Ruyter van Steveninck, and W. Bialek, Phys. Rev. Lett. 80, 197– 200 (1998). [16] S. Panzeri and A. Treves, Network: Comput. in Neural Syst. 7, 87–107 (1996). [17] K. Sjlander, K. Karplus, M. Brown, R. Hughey, A. Krogh, I. S. Mian, and D. Haussler, Computer Applications in the Biosciences (CABIOS) 12, 327–345 (1996). [18] S. Ma, J. Stat. Phys. 26, 221 (1981). [19] W. Bialek, I. Nemenman, N. Tishby, Neural Comp. 13, 2409-2463 (2001).
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Audio-Visual Sound Separation Via Hidden Markov Models John Hershey Department of Cognitive Science University of California San Diego jhershey@cogsci.ucsd.edu Michael Casey Mitsubishi Electric Research Labs Cambridge, Massachussets casey@merl.com Abstract It is well known that under noisy conditions we can hear speech much more clearly when we read the speaker's lips. This suggests the utility of audio-visual information for the task of speech enhancement. We propose a method to exploit audio-visual cues to enable speech separation under non-stationary noise and with a single microphone. We revise and extend HMM-based speech enhancement techniques, in which signal and noise models are factori ally combined, to incorporate visual lip information and employ novel signal HMMs in which the dynamics of narrow-band and wide band components are factorial. We avoid the combinatorial explosion in the factorial model by using a simple approximate inference technique to quickly estimate the clean signals in a mixture. We present a preliminary evaluation of this approach using a small-vocabulary audio-visual database, showing promising improvements in machine intelligibility for speech enhanced using audio and visual information. 1 Introduction We often take for granted the ease with which we can carryon a conversation in the proverbial cocktail party scenario: guests chatter, glasses clink, music plays in the background: the room is filled with ambient sound. The vibrations from different sources and their reverberations coalesce translucently yielding a single time series at each ear, in which sounds largely overlap even in the frequency domain. Remarkably the human auditory system delivers high-quality impressions of sounds in conditions that perplex our best computational systems. A variety of strategies appear to be at work in this, including binaural spatial analysis, and inference using prior knowledge of likely signals and their contexts. In speech perception, vision often plays a crucial role, because we can follow in the lips and face the very mechanisms that modulate the sound, even when the sound is obscured by acoustic noise. It has been demonstrated that the addition of visual cues can enhance speech recognition as much as removing 15 dB of noise [1]. Vision provides speech cues that are complementary to audio cues such as components of consonants and vowels that are likely to be obscured by acoustic noise [2]. Visual information is demonstrably beneficial to HMM-based automatic speech recognition (ASR) systems, which typically suffer tremendously under moderate acoustical noise [3]. We introduce a method of audio-visual speech enhancement using factorial hidden Markov models (fHMMs). We focus on speech enhancement rather than speech recognition for two reasons: first, speech conveys useful paralinguistic information, such as prosody, emotion, and speaker identity, and second, speech contains useful cues for separation from noise, such as pitch. In automatic speech recognition (ASR) systems, these cues are typically discarded in an effort to reduce irrelevant variance among speakers and utterances within a phonetic class. Whereas the benefit of vision to speech recognition is well known, we may well wonder if visual input offers similar benefits to speech enhancement. In [4] a nonparametric density estimator was used to adapt audio and video transforms to maximize the mutual information between the face of a target speaker and an audio mixture containing both the target voice and a distracter voice. These transforms were then used to construct a stationary filter for separating the target voice from the mixture without any prior knowledge or training. In [5] a multi-layer perceptron is trained to map noisy estimates of formants to clean ones, employing lip parameters (width, height and area of the lip opening) extracted from video as additional input. The re-estimated formant contours were used to filter the speech to enhance the signal. In both cases video information improved signal separation. Neither system, however, made use of the dynamics of speech. In speech recognition, HMMs are commonly used because of the advantages of modeling signal dynamics. This suggests the following strategy: train an audiovisual HMM on clean speech, infer the likelihoods of its state sequences, and use the inferred state probabilities of the signal and noise to estimate a sequence of filters to clean the data. In cases where background noise also has regularity, such as the combination of two voices, another HMM can be used to model the background noise. Ephraim [6] first proposed an approach to factorially combining two HMMs in such an enhancement system. In [7] an efficient variational learning rule for the factorial HMM is formulated, and in [8, 9] fHMM speech enhancement was recently revived using some clever tricks to allow more complex models. The fHMM approach is amenable to audio-visual speech enhancement in many different forms. In the simplest formulation, which we pursue here, the signal observation model includes visual features. These visual inputs constrain the signal HMM and produce more accurate filters. Below we present a prototype architecture for such a system along with preliminary results. 1 1.1 Factorial Speech Models One of the challenges of using speech HMMs for enhancement is to model speech in sufficient detail. Typically, speech models, following the practice in ASR, ignore narrow-band, spectral details (corresponding to upper cepstral components) which carry pitch information, because they tend to vary across speakers and utterances for the same word or phoneme. Instead such systems focus on the smooth, or wideband, spectral characteristics (corresponding to lower cepstral components) such as are produced by the articulation of the mouth. Such wide-band spectral patterns loosely represent formant patterns, a well-known cue for vowel discrimination. In cases where the pitch or other narrow-band properties, of the background signals differ from the foreground speech, and have predictable dynamics, such as with lWe defer a detailed mathematical development to subsequent publications. Contact jhershey@cogsci.ucsd.edu for further information two simultaneous speech signals, these components may be helpful in separating the two signals. Figure 1 illustrates the analysis of two words into wide-band and narrow-band components. "one" "two" Full band: Narrow band: Wide band: Figure 1: full-band, narrow-band, and wide-band log spectrograms of two words. The wide-band log spectrograms (bottom) are derived by low-pass filtering the log spectra (across the frequency domain), and the narrow-band log spectrograms (middle) derived by high pass filtering the log spectra The full log spectrogram (top) is the sum of the two. However, the wide-band and narrow-band variations in speech are only loosely coupled. For instance, a given formant is likely to be uttered with many different pitches and a given pitch may be used to utter any formant. Thus a model of the full spectrum of speech would have to have enough states to represent every combination of pitches and formants. Such a model requires a large amount of training data and imposes serious computational burdens. For instance in [8] a model with 8000 states is employed. When combined with a similarly complex noise model, the composite model has 64 million states. This is expensive in terms of computation as well as the number of data points required for inference. To parsimoniously model the complexity of speech, we employ a factorial HMM for a single speech signal, in which wide and narrow-band components are represented in sub-models with independent dynamics. We therefore train the two submodels independently using Gaussian observation probability density functions (p.d.f.) on the wide-band or narrow-band log spectra, with diagonal covariances for the sake of simplicity. Figure 2(a) depicts the graphical model for a single wide or narrow-band component. Discrete States Continuous Observations (a) simple HMM Narrow-Band Slate Wide-Band Stale Combined Observmions (b) factorial speech HMM Figure 2: single HMMs are trained separately on wide-band and narrow-band speech signals (a) and then combined factorially in (b) by adding the means and variances of their observation distributions To combine the sub-models, we have to specify the observation p.d.f. for a combination of a wide and a narrow-band state, over the log-spectrum of speech prior to liftering. Because the observation densities of each component are Gaussian, and the log-spectra of the wide and narrow-band components add in the log spectrum, the composite state has a Gaussian observation p.d.f., whose mean and variance is the sum of the component observation means and variances. Although the states of the two sub-models are marginally independent they are typically conditionally dependent given the observation sequence. In other words we assume that the state dependencies between the sub-models for a given speech signal can be explained entirely via the observations. Figure 2(b) depicts the combination of the wide and narrow-band models, where the observation p.d.f. 's are a function of two state variables. When combining the signal and noise models (or two different speech models) in contrast, the signals add in the frequency domain, and hence in the log spectral domain they longer simply add. In the spectral domain the amplitudes of the two signals have log-normal distributions, and the relative phases are unknown. There is no closed form distribution for the sum of two random variables with log-normal amplitudes and a uniformly distributed phase difference. Disregarding phase differences we apply a well-known approximation to the sum of two lognormal random variables, in which we match the mean and variance of a lognormal random variable to the sum of the means and variances of the two component lognormal random variables [10]. Phase uncertainty can also be incorporated into an approximation; however in practice the costs appear to outweigh the benefits.2 Figure 3(a) depicts the combination of two factorial speech models, where the observation p.d.f.s are a function of two state variables. (a) dual factorial HMM Video Observations -'-- .. . Audio Observations ~ (1f 6 0 (b) speech fHMM with video Figure 3: combining two speech fHMMs (a) and adding video observations to a speech fHMM (b). Using the log-normal observation distribution of the composite model we can estimate the likelihood of the speech and noise states for each frame using the well known forward-backward recursion. For each frame of the test data we can compute the expected value of the amplitude of each model in each frequency bin. Taking 2The uncertainty of the phase differences can be incorporated by modeling the sum as a mixture of lognormals that uniformly samples phase differences. Each mixture element is approximated by taking as its mean the length of the sum of the mean amplitudes when added in the complex plane according a particular phase difference, and as its variance the sum of the two variances. This estimation is facilitated by the assumption of diagonal covariances in the log spectral domain. the expected value of the signal in the numerator and the expected value of the signal plus noise in the denominator yields a Wiener filter which is applied to the original noisy signal enhancing the desired component. When we have two speech signals one person's noise is another's signal and we can separate both by the same method. 2 Incorporating vision We incorporate vision after training the audio models in order to test the improvement yielded by visual input while holding the audio model constant. A video observation distribution is added to each state in the model by obtaining the probability of each state in each frame of the audio training data using the forward-backward procedure, then estimating the parameters of the video observation distributions for each state, in the manner of the Baum-Welch observation re-estimation formula. This procedure is iterated until it converges. In this way we construct a system in which the visual observations are modular. Figure 3(b) depicts the structure ofthe resulting speech model. Such a method in which audio and visual features are integrated early in processing is only one of several approaches. We envision other late integration approaches in which audio and visual dynamics are more loosely coupled. What method of audio-visual integration may be best for this task is an open question. 3 Efficient inference In the models described above, in which we factorially combine two speech models, each of which is itself factorial, the complexity of inference in the composite model, using the forward-backward recursion, can easily become unmanageable. If K is the number of states in each subcomponent, then K4 is the number of states in the composite HMM. In our experiments K is on the order of 40 states, so there are 2,560,000 states in the composite model. Naively each composite state must be searched when computing the probabilities of state sequences necessary for inference. Interesting approximation schemes for similar models are developed in [8, 9]. We develop an approximation as follows. Rather than computing the forward-backward procedure on the composite HMM, we compute it sequentially on each sub-HMM to derive the probability of each state in each frame. Of course, in order to evaluate the observation probabilities of the current sub-HMMs for a given frame, we need to consider the state probabilities of the other three sub-HMMs, because their means and variances are combined in the observation model. These state probabilities and their associated observation probabilities comprise a mixture model for a given frame. The composite mixture model still has K4 states, so to defray this complexity during forward-backward analysis of the current sub-HMM, for each frame we approximate the observation mixtures of each of the other three sub-HMMs with a single Gaussian, whose mean and variance matches that of the mixture. Thus we only have to consider the K states of the current model, and use the summarized means and variances of the other three HMMs as auxiliary inputs to the observation model. We initialize the state probabilities in each frame with the equilibrium distribution for each sub-HMM. In our experiments, after a handful of iterations, the composite state probabilities tend to converge. This method is closely related to a structured variational approximation for factorial HMMs [7] and can be also be seen as an approximate belief propagation or sum-product algorithm [11]. 4 Data We used a small-vocabulary audio-visual speech database developed by Fu Jie Huang at Carnegie Mellon University3 [12]. These data consist of audio and video recordings of 10 subjects (7 males and 3 females) saying 78 isolated words commonly used for numbers and time, such aS,"one" "Monday", "February", "night", etc. The sequence of 78 words is repeated in 10 different takes. Half of these takes were used for training, and one of the remaining takes was used as the test set. The data set included outer lip parameters extracted from video using an automatic lip tracker, including height of the upper and lower lips relative to the corners the width from corner to corner. We interpolated these lip parameters to match the audio frame rate, and calculate time derivatives. Audio consisted of 16-bit, 44.1 kHz recordings which we resample to 8000 kHz. The audio was framed at 60 frames per second, with an overlap of 50%, yielding 264 samples per frame. 4 The frames were analyzed into cepstra: the wide-band log spectrum is derived from the lower 20 cepstral components and the wide-band log spectrum from the upper cepstra. 5 Results Speaker dependent wide and narrow-band HMMs having 40 states each were trained on data from two subjects (" Anne" and" Chris") selected from the training set. A PCA basis was used to reduce the log spectrograms to a more manageable size of 30 dimensions during training. This resulted in some non-zero covariances near the diagonal in the learned observation covariance matrices, which we discarded. An entropic prior and parameter extinction were used to sparsify the transition matrices during training [13]. The narrow-band model learned states that represented different pitches and had transition probabilities that were non-zero mainly between neighboring pitches. The narrow-band model's video observation probability distributions were largely overlapping, reflecting the fact that video tells us little about pitch. The wide-band model learned states that represented different formant structures. The video observation distributions for several states in the wide-band model were clearly separated, reflecting the information that video provides about the formant structure. Subjectively the enhanced signals sound well separated from each other for the most part. Figure 4(a) (bottom) shows the estimated spectrograms for a mixture of two different words spoken by the same speaker - an extremely difficult task. To quantify these results we evaluate the system using speech recognizer, on the slightly easier task of separating the speech of the two different speakers, whose voices were in different but overlapping pitch ranges. A test set was generated by mixing together 39 randomly chosen pairs of words, one from each subject, such that no word was used twice. Each word pair was mixed at five different signal to noise ratios (SNRs), with the SNR provided to the system at test time.5 The total number of test mixtures for each subject was thus 195. 3see http://amp.ece.cmu.edu/projects/ Audio VisualSpeechProcessing/ 4Sine windows were used in analysis and synthesis such that their product forms windows that sum to unity when overlapped 50%. The windowed frames were analyzed using a 264-point fast Fourier transform (FFT). The phases of the resulting spectra were discarded. 5Estimation of the SNR is necessary in practice; however this subject has been treated The separated test sounds were estimated by the system under two conditions: with and without the use of video information. We evaluated the estimates on the test set using a speech recognition system developed by Bhiksha Raj, using the eMU Sphinx ASR engine.6 Existing speech HMMs trained on 60 hours of broadcast news data were used for recognition. 7 The models were adapted in an unsupervised manner to clean speech from each speaker, by learning a single affine transformation of all the state means, using a maximum likelihood linear regression procedure [14]. The recognizer adapted to each speaker was tested with the enhanced speech produced by the speech model for that speaker, as well as with no enhancement. Results are shown in figure 4(b). Recognition was greatly facilitated by the enhancement, with additional gains resulting from the use of video. It is somewhat surprising that the gains for video occur mostly in areas of higher SNR, whereas in human speech perception they occur under lower SNR. Little subjective difference was noted with the use of video in the case of two speakers. However in other experiments, when both voices came from the same speaker, the video was crucial in disambiguating which signal came from which voice. "one" "two" Originals Mixture Separated SNR dB (a) signal separation spectrograms (b) automatic speech recognition Figure 4: spectrograms of separated speech signals for a mixture two words spoken by the same speaker (a), and speech recognition performance for 39 mixtures of two words spoken by different speakers (b) 6 Discussion We have presented promising techniques for audio-visual speech enhancement. We introduced a factorial HMM to track both formant and pitch information, as well as video, in a unified probabilistic model, and demonstrated its effectiveness in speech enhancement. We are not aware of any other HMM-based audio-visual elsewhere [6] and is beyond the scope of this paper. 6see http://www.speech.cs.cmu.edu/sphinxj. 7These models represented every combination of three phones (triphones) using 6000 states tied across trip hone models, with a 16-element Gaussian mixture observation model for each state. The data were processed at 8 kHz in 25ms windows overlapped by 15ms, with a frame rate of 100 frames per second, and analyzed into 31 Mel frequency components from which 13 cepstral coefficients were derived. These coefficients with the mean vector removed, and supplemented with their time differences, comprised the observed features speech enhancement systems in the literature. The results are tentative given the small sample of voices used; however they suggest that further study with a larger sample of voices is warranted. It would be useful to compare the performance of a factorial speech model to that of each factor in isolation, as well as to a fullspectrum model. Measures of quality and intelligibility by human listeners in terms of speech and emotion recognition, as well as speaker identity, will also be helpful in further demonstrating the utility of these techniques. We look forward to further development of these techniques in future research. Acknowledgments We wish to thank Mitsubishi Electric Research Labs for hosting this research. Special thanks to Bhiksha Raj for devising and producing the evaluation using speech recognition, and to Matt Brand for his entropic HMM toolkit. References [1] W. H. Sumby and I. Pollack. Visual contribution to speech intelligibility in noise. Journal of the Acoustical Society of America, 26:212- 215, 1954. [2] Jordi Robert-Ribes, Jean-Luc Schwartz, Tahar Lallouache, and Pierre Escudier. Complementarity and synergy in bimodal speech. Journel of the Acoustical Society of America, 103(6):3677- 3689, 1998. [3] Stepmane Dupont and Juergen Luettin. Audio-visual speech modeling for continuous speech recognition. IEEE transactions on Multimedia, 2(3):141- 151, 2000. [4] John W. Fisher, Trevor Darrell, William T. Freeman, and Paul Viola. Learning joint statistical models for audio-visual fusion and segregation. In Advances in Neural Information Processing Systems 13. 200l. [5] Laurent Girin, Jean-Luc Schwartz, and Gang Feng. Audio-visual enhancement of speech in noise. Journel of the Acoustical Society of America, 109(6):3007- 3019, 200l. [6] Yariv Ephraim. Statistical-model based speech enhancement systems. Proceedings of the IEEE, 80(10):1526- 1554, 1992. [7] Z. Ghahramani and M. Jordan. Factorial hidden markov models. In David S. Touretzky, Michael C. Mozer, and M.E. Hasselmo, editors, Advances in Neural Information Processing Systems 8, 1996. [8] Sam T. Roweis. One microphone source separation. In Advances in Neural Information Processing Systems 13. 200l. [9] Hagai Attias, John C. Platt, Alex Acero, and Li Deng. Speech denoising and dereverberation using probabilistic models. In Advances in Neural Information Processing Systems 13. 200l. [10] M. J. F . Gales. Model-Based Techniques for Noise Robust Speech Recognition. PhD thesis, Cambridge University, 1996. [11] F. R. Kschischang, B. Frey, and H.-A. Loeliger. Factor graphs and the sum-product algorithm. IEEE Trans. Inform. Theory, 47(2):498- 519, 200l. [12] F. J. Huang and T. Chen. Real-time lip-synch face animation driven by human voice. In IEEE Workshop on Multimedia Signal Processing, Los Angeles, California, Dec 1998. [13] Matt Brand. Structure learning in conditional probability models via an entropic prior and parameter extinction. Neural Computation, 11(5):1155- 1182, 1999. [14] C. J. Leggetter and P. C. Woodland. Maximum likelihood linear regression for speaker adaptation of the parameters of continuous density hidden markov models. Computer Speech and Language, 9: 171- 185, 1995.
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Spectral Relaxation for K-means Clustering Hongyuan Zha & Xiaofeng He Dept. of Compo Sci. & Eng. The Pennsylvania State University University Park, PA 16802 {zha,xhe}@cse.psu.edu Chris Ding & Horst Simon NERSC Division Lawrence Berkeley National Lab. UC Berkeley, Berkeley, CA 94720 {chqding,hdsimon}@lbl.gov Ming Gu Dept. of Mathematics UC Berkeley, Berkeley, CA 95472 mgu@math.berkeley.edu Abstract The popular K-means clustering partitions a data set by minimizing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function. 1 Introduction K-means is a very popular method for general clustering [6]. In K-means clusters are represented by centers of mass of their members, and it can be shown that the K-means algorithm of alternating between assigning cluster membership for each data vector to the nearest cluster center and computing the center of each cluster as the centroid of its member data vectors is equivalent to finding the minimum of a sum-of-squares cost function using coordinate descend. Despite the popularity of Kmeans clustering, one of its major drawbacks is that the coordinate descend search method is prone to local minima. Much research has been done on computing refined initial points and adding explicit constraints to the sum-of-squares cost function for K-means clustering so that the search can converge to better local minimum [1,2]. In this paper we tackle the problem from a different angle: we find an equivalent formulation of the sum-of-squares minimization as a trace maximization problem with special constraints; relaxing the constraints leads to a maximization problem that possesses optimal global solutions. As a by-product we also have an easily computable lower bound for the minimum of the sum-of-squares cost function. Our work is inspired by [9, 3] where connection to Gram matrix and extension of Kmeans method to general Mercer kernels were investigated. The rest of the paper is organized as follows: in section 2, we derive the equivalent trace maximization formulation and discuss its spectral relaxation. In section 3, we discuss how to assign cluster membership using pivoted QR decomposition, taking into account the special structure of the partial eigenvector matrix. Finally, in section 4, we illustrate the performance of the clustering algorithms using document clustering as an example. Notation. Throughout, II . II denotes the Euclidean norm of a vector. The trace of a matrix A, i.e., the sum of its diagonal elements, is denoted as trace(A). The Frobenius norm of a matrix IIAIIF = Jtrace(AT A). In denotes identity matrix of order n. 2 Spectral Relaxation Given a set of m-dimensional data vectors ai, i = 1, ... ,n, we form the m-by-n data matrix A = [a1,"" an]. A partition II of the date vectors can be written in the following form (1) where E is a permutation matrix, and Ai is m-by-si, i.e., the ith cluster contains the data vectors in A. For a given partition II in (1), the associated sum-of-squares cost function is defined as k Si Si ss(II) = L L Ila~i) - mi11 2 , m· = "a(i)ls· 'l ~ S 2, i=l s=l s=l i.e., mi is the mean vector of the data vectors in cluster i. Let e be a vector of appropriate dimension with all elements equal to one, it is easy to see that mi = Aiel Si and Si SSi == L Ila~i) - mil1 2 = IIAi - mieTII} = IIAi(Isi - eeT ISi)II}· s=l Notice that lSi - eeT I Si is a projection matrix and (Isi - eeT I Si)2 = lSi - eeT lSi, it follows that SSi = trace(Ai(Isi - eeT I si)Af) = trace((Isi - eeT I si)AT Ai). Therefore, ss(II) = t, SSi = t, (trace(AT Ai) - (~) AT Ai (~) ) . Let the n-by-k orthonormal matrix X be X = :~ (elVsl elVSi. Sk (2) The sum-of-squares cost function can now be written as ss(II) = trace(AT A) - trace(XT AT AX), and its minimization is equivalent to max{ trace(XT AT AX) I X of the form in (2)}. REMARK. Without loss of generality, let E = I in (1). If we let Xi be the cluster indicator vector, i.e., xT = [0, ... ,0,1, ... ,1,0, .. . ,0]. '---v-----" Si Then it is easy to see that trace(XT AT AX) = t xT AT AXi = t IIAxil1 2 i=l XTXi i=l IIxil1 2 Using the partition in (1), the right-hand side of the above can be written as a weighted sum of the squared Euclidean norms of the mean vector of each clusters. REMARK. If we consider the elements of the Gram matrix AT A as measuring similarity between data vectors, then we have shown that Euclidean distance leads to Euclidean inner-product similarity. This inner-product can be replaced by a general Mercer kernel as is done in [9, 3]. Ignoring the special structure of X and let it be an arbitrary orthonormal matrix, we obtain a relaxed maximization problem max trace(XT AT AX) XTX=h It turns out the above trace maximization problem has a closed-form solution. Theorem. (Ky Fan) Let H be a symmetric matrix with eigenvalues Al ::::: A2 ::::: ... ::::: An, and the corresponding eigenvectors U = [Ul, .. . , Un]. Then Al + ... Ak = max trace(XT H X) . XTX=Ik (3) Moreover, the optimal X* is given by X* = [Ul' ... ' Uk]Q with Q an arbitrary orthogonal matrix. It follows from the above theorem that we need to compute the largest k eigenvectors of the Gram matrix AT A. As a by-product, we have min{m,n} minss(II) ::::: trace(AT A) max trace(XT AT AX) = L 0-; (A), (4) n XT X=h i=k+l where oi(A) is the i largest singular value of A. This gives a lower bound for the minimum of the sum-of-squares cost function. REMARK. It is easy to see from the above derivation that we can replace A with A - aeT , where a is an arbitrary vector. Then we have the following lower bound min{m,n} mJnss(II) ::::: m~ L u;(A - aeT ). i=k+l REMARK. One might also try the following approach: notice that T2 1", '" 2 IIAi - mie IIF = 2Si ~ ~ Ilaj - aj'11 . aj EAi aj' EAi Let W = ( Ilai - ajl12 )i,j=l' and and Xi = [Xij]j=l with 1 if aj E Ai Xij = { o otherwise Then k T n ss(II) = ~ '" Xi WXi > ~ min ZTWZ = ~ '" Ai(W). 2 ~ XT Xi 2 ZT Z=h 2 ~ i=l " i=n-k+l Unfortunately, some of the smallest eigenvalues of W can be negative. Let X k be the n-by-k matrix consisting of the k largest eigenvectors of AT A. Each row of X k corresponds to a data vector, and the above process can be considered as transforming the original data vectors which live in a m-dimensional space to new data vectors which now live in a k-dimensional space. One might be attempted to compute the cluster assignment by applying the ordinary K-means method to those data vectors in the reduced dimension space. In the next section, we discuss an alternative that takes into account the structure of the eigenvector matrix X k [5]. REMARK. The similarity of the projection process to principal component analysis is deceiving: the goal here is not to reconstruct the data matrix using a low-rank approximation but rather to capture its cluster structure. 3 Cluster Assignment Using Pivoted QR Decomposition Without loss of generality, let us assume that the best partition of the data vectors in A that minimizes ss(II) is given by A = [AI"'" Ak], each submatrix Ai corresponding to a cluster. Now write the Gram matrix of A as ATA=[A~A' ArA, ~ 1+E=:B+E. o 0 ArAk If the overlaps among the clusters represented by the submatrices Ai are small, then the norm of E will be small as compare with the block diagonal matrix B in the above equation. Let the largest eigenvector of AT Ai be Yi , and AT AiYi = fJiYi , IIYil1 = 1, i = 1, ... , k, then the columns of the matrix span an invariant subspace of B. Let the eigenvalues and eigenvectors of AT A be A1:::: A2:::: ... :::: An, AT AXi = AiXi, i = 1, ... ,n. Assume that there is a gap between the two eigenvalue sets {fl1,'" flk} and {Ak+1 ,'" An}, i.e., o < J = min{lfli - Aj II i = 1, ... ,k, j = k + 1, ... ,n}. Then Davis-Kahan sin(0) theorem states that IlynXk+1,'" ,xn]11 < IIEII/J [11, Theorem 3.4]. After some manipulation, it can be shown that X k == [Xl, ... ,Xk] = Yk V + O(IIEII), where V is an k-by-k orthogonal matrix. Ignoring the O(IIEII) term, we see that v v cluster 1 cluster k where we have used y'[ = [Yil , ... ,Yis.], and VT = [V1' ... ,Vk]. A key observation is that all the Vi are orthogonal to each other: once we have selected a Vi, we can jump to other clusters by looking at the orthogonal complement of Vi' Also notice that IIYil1 = 1, so the elements of Yi can not be all small. A robust implementation of the above idea can be obtained as follows: we pick a column of X k T which has the lar;est norm, say, it belongs to cluster i, we orthogonalize the rest of the columns of X k against this column. For the columns belonging to cluster i the residual vector will have small norm, and for the other columns the residual vectors will tend to be not small. We then pick another vector with the largest residual norm, and orthogonalize the other residual vectors against this residual vector. The process can be carried out k steps, and it turns out to be exactly QR decomposition with column pivoting applied to X k T [4], i.e., we find a permutation matrix P such that X'[P = QR = Q[Rl1,Rd, where Q is a k-by-k orthogonal matrix, and Rl1 is a k-by-k upper triangular matrix. We then compute the matrix R = Rj} [Rl1 ' Rd pT = [Ik' Rj} R12]PT. Then the cluster membership of each data vector is determined by the row index of the largest element in absolute value of the corresponding column of k REMARK. Sometimes it may be advantageous to include more than k eigenvectors to form Xs T with s > k. We can still use QR decomposition with column pivoting to select k columns of Xs T to form an s-by-k matrix, say X. Then for each column z of Xs T we compute the least squares solution of t* = argmintERk liz - Xtll. Then the cluster membership of z is determined by the row index of the largest element in absolute value of t* . 4 Experimental Results In this section we present our experimental results on clustering a dataset of newsgroup articles submitted to 20 newsgroups.1 This dataset contains about 20,000 articles (email messages) evenly divided among the 20 newsgroups. We list the names of the news groups together with the associated group labels. lThe newsgroup dataset together with the bow toolkit for processing it can be downloadedfrorn http : //www . cs.cmu.edu/afs/cs/project/theo-ll/www/naive-bayes.html. 0·~.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0·1L, -~--,c-----O~ ' _--,c-'-_~-----' p-{)R p-Kmeans Figure 1: Clustering accuracy for five newsgroups NG2/NG9/NG10/NG15/NG18: p-QR vs. p-Kmeans (left) and p-Kmeans vs. Kmeans (right) NG1: alt.atheism NG2: comp.graphics NG3: comp.os.ms-vindovs.misc NG4: comp.sys.ibm.pc.hardvare NG5:comp.sys.mac.hardvare NG6: comp.vindovs.x NG7:misc.forsale NG8: rec.autos NG9:rec.motorcycles NG10: rec.sport.baseball NGll:rec.sport.hockey NG12: sci. crypt NG13:sci.electronics NG14: sci.med NG15:sci.space NG16: soc.religion.christian NG17:talk.politics.guns NG18: talk.politics.mideast NG19:talk.politics.misc NG20: talk.religion.misc We used the bow toolkit to construct the term-document matrix for this dataset, specifically we use the tokenization option so that the UseNet headers are stripped, and we also applied stemming [8]. The following three preprocessing steps are done: 1) we apply the usual tf.idf weighting scheme; 2) we delete words that appear too few times; 3) we normalized each document vector to have unit Euclidean length. We tested three clustering algorithms: 1) p-QR, this refers to the algorithm using the eigenvector matrix followed by pivoted QR decomposition for cluster membership assignment; 2) p-Kmeans, we compute the eigenvector matrix, and then apply K-means on the rows of the eigenvector matrix; 3) K-means, this is K-means directly applied to the original data vectors. For both K-means methods, we start with a set of cluster centers chosen randomly from the (projected) data vectors, and we aslo make sure that the same random set is used for both for comparison. To assess the quality of a clustering algorithm, we take advantage of the fact that the news group data are already labeled and we measure the performance by the accuracy of the clustering algorithm against the document category labels [10]. In particular, for a k cluster case, we compute a k-by-k confusion matrix C = [Cij] with Cij the number of documents in cluster i that belongs to newsgroup category j. It is actually quite subtle to compute the accuracy using the confusion matrix because we do not know which cluster matches which newsgroup category. An optimal way is to solve the following maximization problem max{ trace(CP) I P is a permutation matrix}, and divide the maximum by the total number of documents to get the accuracy. This is equivalent to finding perfect matching a complete weighted bipartite graph, one can use Kuhn-Munkres algorithm [7]. In all our experiments, we used a greedy algorithm to compute a sub-optimal solution. Table 1: Comparison of p-QR, p-Kmeans, and K-means for two-way clustering Newsgroups p-QR p-Kmeans K-means NG1/NG2 89.29 ± 7.51 % 89.62 ± 6.90% 76.25 ± 13.06% NG2/NG3 62.37 ± 8.39% 63.84 ± 8.74% 61.62 ± 8.03% NG8/NG9 75.88 ± 8.88% 77.64 ± 9.00% 65.65 ± 9.26% NG10/NG11 73.32 ± 9.08% 74.86 ± 8.89% 62.04 ± 8.61% NG1/NG15 73.32 ± 9.08% 74.86 ± 8.89% 62.04 ± 8.61% NG18/NG19 63.86 ± 6.09% 64.04 ± 7.23% 63.66 ± 8.48% Table 2: Comparison of p-QR, p-Kmeans, and K-means for multi-way clustering Newsgroups p-QR p-Kmeans K-means NG2/NG3/NG4/NG5/NG6 (50) 40.36 ± 5.17% 41.15 ± 5.73% 35.77 ± 5.19% NG2/NG3/NG4/NG5/NG6 UOO) 41.67 ± 5.06% 42.53 ± 5.02% 37.20 ± 4.39% NG2/NG9/NG10/NG15/NG18 l50j 77.83 ± 9.26% 70.13 ± 11.67% 58.10 ± 9.60% NG2/NG9/NG10/NG15/NG18 (100) 79.91 ± 9.90% 75.56 ± 10.63% 66.37 ± 10.89% NG1/NG5/NG7/NG8/NG11/ (50) 60.21 ± 4.88% 58.18 ± 4.41% 40.18 ± 4.64% NG12/NG13/NG14/NG15/NG17 NG1/NG5/NG7 /NG8/NG 11/ (100) 65.08 ± 5.14% 58.99 ± 5.22% 48.33 ± 5.64% NG12/NG13/NG14/NG15/NG17 EXAMPLE 1. In this example, we look at binary clustering. We choose 50 random document vectors each from two newsgroups. We tested 100 runs for each pair of newsgroups, and list the means and standard deviations in Table 1. The two clustering algorithms p-QR and p-Kmeans are comparable to each other, and both are better and sometimes substantially better than K-means. EXAMPLE 2. In this example, we consider k-way clustering with k = 5 and k = 10. Three news group sets are chosen with 50 and 100 random samples from each newsgroup as indicated in the parenthesis. Again 100 runs are used for each tests and the means and standard deviations are listed in Table 2. Moreover, in Figure 1, we also plot the accuracy for the 100 runs for the test NG2/NG9/NG10/NG15/NG18 (50). Both p-QR and p-Kmeans perform better than Kmeans. For news group sets with small overlaps, p-QR performs better than p-Kmeans. This might be explained by the fact that p-QR explores the special structure of the eigenvector matrix and is therefore more efficient. As a less thorough comparison with the information bottleneck method used in [10], there for 15 runs of NG2/NG9/NGlO/NG15/NG18 (100) mean accuracy 56.67% with maximum accuracy 67.00% is obtained. For 15 runs of the 10 newsgroup set with 50 samples, mean accuracy 35.00% with maximum accuracy about 40.00% is obtained. EXAMPLE 3. We compare the lower bound given in (4). We only list a typical sample from NG2/NG9/NGlO/NG15/NG18 (50). The column with "NG labels" indicates clustering using the newsgroup labels and by definition has 100% accuracy. It is quite clear that the news group categories are not completely captured by the sum-of-squares cost function because p-QR and "NG labels" both have higher accuracy but also larger sum-of-squares values. Interestingly, it seems that p-QR captures some of this information of the newsgroup categories. p-QR p-Kmeans K-means NG labels lower bound accuracy 86.80% 83.60% 57.60% 100% N/A ssm) 224.1110 223.8966 228.8416 224.4040 219.0266 Acknowledgments This work was supported in part by NSF grant CCR-9901986 and by Department of Energy through an LBL LDRD fund. References [1] P. S. Bradley and Usama M. Fayyad. (1998). Refining Initial Points for K-Means Clustering. Proc. 15th International Conf. on Machine Learning, 91- 99. [2] P. S. Bradley, K. Bennett and A. Demiritz. Constrained K-means Clustering. Microsoft Research, MSR-TR-2000-65, 2000. [3] M. Girolani. (2001). Mercer Kernel Based Clustering in Feature Space. To appear in IEEE Transactions on Neural Networks. [4] G. Golub and C. Van Loan. (1996). Matrix Computations. Johns Hopkins University Press, 3rd Edition. [5] Ming Gu, Hongyuan Zha, Chris Ding, Xiaofeng He and Horst Simon. (2001) . Spectral Embedding for K- Way Graph Clustering. Technical Report, Department of Computer Science and Engineering, CSE-OI-007, Pennsylvania State University. [6] J.A. Hartigan and M.A. Wong. (1979). A K-means Clustering Algorithm. Applied Statistics, 28:100- 108. [7] L. Lovasz and M.D. Plummer. (1986) Matching Theory. Amsterdam: North Holland. [8] A. McCallum. Bow: A toolkit for statistical language modeling, text retrieval, classification and clustering. http : //www . CS. cmu. edu/ mccallum/bow. [9] B. Schi:ilkopf, A. Smola and K.R. Miiller. (1998). Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation, 10: 1299- 1219. [10] N. Slonim and N. Tishby. (2000). Document clustering using word clusters via the information bottleneck method. Proceedings of SIGIR-2000. [11] G.W. Stewart and J.G. Sun. (1990). Matrix Perturbation Theory. Academic Press, San Diego, CA.
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Gaussian Process Regression with Mismatched Models Peter Sollich Department of Mathematics, King's College London Strand, London WC2R 2LS, U.K. Email peter.sollich@kcl.ac . uk Abstract Learning curves for Gaussian process regression are well understood when the 'student' model happens to match the 'teacher' (true data generation process). I derive approximations to the learning curves for the more generic case of mismatched models, and find very rich behaviour: For large input space dimensionality, where the results become exact, there are universal (student-independent) plateaux in the learning curve, with transitions in between that can exhibit arbitrarily many over-fitting maxima; over-fitting can occur even if the student estimates the teacher noise level correctly. In lower dimensions, plateaux also appear, and the learning curve remains dependent on the mismatch between student and teacher even in the asymptotic limit of a large number of training examples. Learning with excessively strong smoothness assumptions can be particularly dangerous: For example, a student with a standard radial basis function covariance function will learn a rougher teacher function only logarithmically slowly. All predictions are confirmed by simulations. 1 Introduction There has in the last few years been a good deal of excitement about the use of Gaussian processes (GPs) as an alternative to feedforward networks [1]. GPs make prior assumptions about the problem to be learned very transparent, and even though they are non-parametric models, inference- at least in the case of regression considered below- is relatively straightforward. One crucial question for applications is then how 'fast' GPs learn, i.e. how many training examples are needed to achieve a certain level of generalization performance. The typical (as opposed to worst case) behaviour is captured in the learning curve, which gives the average generalization error t as a function of the number of training examples n. Good bounds and approximations for t(n) are now available [1, 2, 3, 4, 5], but these are mostly restricted to the case where the 'student' model exactly matches the true 'teacher' generating the datal. In practice, such a match is unlikely, and so it is lThe exception is the elegant work of Malzahn and Opper [2], which uses a statistical physics framework to derive approximate learning curves that also apply for any fixed target function. However, this framework has not yet to my knowledge been exploited to important to understand how GPs learn if there is some model mismatch. This is the aim of this paper. In its simplest form, the regression problem is this: We are trying to learn a function B* which maps inputs x (real-valued vectors) to (real-valued scalar) outputs B*(x). We are given a set of training data D , consisting of n input-output pairs (xl, yl) ; the training outputs yl may differ from the 'clean' teacher outputs B*(xl ) due to corruption by noise. Given a test input x, we are then asked to come up with a prediction B(x), plus error bar, for the corresponding output B(x). In a Bayesian setting, we do this by specifying a prior P(B) over hypothesis functions, and a likelihood P(DIB) with which each B could have generated the training data; from this we deduce the posterior distribution P(BID) ex P(DIB)P(B). For a GP, the prior is defined directly over input-output functions B; this is simpler than for a Bayesian feedforward net since no weights are involved which would have to be integrated out. Any B is uniquely determined by its output values B(x) for all x from the input domain, and for a GP, these are assumed to have a joint Gaussian distribution (hence the name). If we set the means to zero as is commonly done, this distribution is fully specified by the covariance function (B(x)B(xl))o = C(X,XI). The latter transparently encodes prior assumptions about the function to be learned. Smoothness, for example, is controlled by the behaviour of C(x, Xl) for Xl -+ x: The Ornstein-Uhlenbeck (OU) covariance function C(x, Xl) = exp( -Ix - xliiI) produces very rough (non-differentiable) functions, while functions sampled from the radial basis function (RBF) prior with C(x, Xl) = exp[-Ix - x/12 1(212)] are infinitely differentiable. Here I is a lengthscale parameter, corresponding directly to the distance in input space over which we expect significant variation in the function values. There are good reviews on how inference with GPs works [1, 6], so I only give a brief summary here. The student assumes that outputs y are generated from the 'clean' values of a hypothesis function B(x) by adding Gaussian noise of xindependent variance (J2. The joint distribution of a set of training outputs {yl} and the function values B(x) is then also Gaussian, with covariances given (under the student model) by (ylym) = C(xl,xm) + (J2Jlm = (K)lm, (yIB(x)) = C(xl,x) = (k(X))1 Here I have defined an n x n matrix K and an x-dependent n-component vector k(x) . The posterior distribution P(BID) is then obtained by conditioning on the {yl}; it is again Gaussian and has mean and variance (B(x))oID == B(xID) = k(X)TK-1y (1) ((B(x) - B(X))2)oID C(x,x) - k(X)TK-1k(x) (2) From the student's point of view, this solves the inference problem: The best prediction for B(x) on the basis of the data D is B(xID) , with a (squared) error bar given by (2). The squared deviation between the prediction and the teacher is [B(xID) - B*(x)]2; the average generalization error (which, as a function of n, defines the learning curve) is obtained by averaging this over the posterior distribution of teachers, all datasets, and the test input x: E = ((([B(xID) - B*(xWk ID)D)x (3) Now of course the student does not know the true posterior of the teacher; to estimate E, she must assume that it is identical to the student posterior, giving from (2) E = ((([B(xID) - B(X)]2)oID)D)x = ((C(x,x) - k(xfK-1k(X)){xl})x (4) consider systematically the effects of having a mismatch between the teacher prior over target functions and the prior assumed by the student. where in the last expression I have replaced the average over D by one over the training inputs since the outputs no longer appear. If the student model matches the true teacher model, E and € coincide and give the Bayes error, i.e. the best achievable (average) generalization performance for the given teacher. I assume in what follows that the teacher is also a GP, but with a possibly different covariance function C* (x, x') and noise level (}";. This allows eq. (3) for E to be simplified, since by exact analogy with the argument for the student posterior (()* (x) k iD = k* (x) TK :;-1y , ((); (x) )O. ID = (()* (x ))~. I D +C* (x, x) - k* (x) TK :;-1 k* (x) and thus, abbreviating a(x) = K-1k(x) - K ;;-1k*(x), E = ((a(x)TyyTa(x) + C*(x,x) - k*(X)TK:;-1k*(x))D)x Conditional on the training inputs, the training outputs have a Gaussian distribution given by the true (teacher) model; hence (yyT){yl}l{xl} = K *, giving E = ((C*(x,x) - 2k*(x)TK-1k(x) + k(X)T K -1K *K -1k(x)){xl})x (5) 2 Calculating the learning curves An exact calculation of the learning curve E(n) is difficult because of the joint average in (5) over the training inputs X and the test input x . A more convenient starting point is obtained if (using Mercer's theorem) we decompose the covariance function into its eigenfunctions ¢i(X) and eigenvalues Ai, defined w.r.t. the input distribution so that (C(x, X')¢i(X') )x' = Ai¢i(X) with the corresponding normalization (¢i(X)¢j(x))x = bij. Then 00 00 i=1 i=1 For simplicity I assume here that the student and teacher covariance functions have the same eigenfunctions (but different eigenvalues). This is not as restrictive as it may seem; several examples are given below. The averages over the test input x in (5) are now easily carried out: E.g. for the last term we need ((k(x)k(x)T)lm)x = L AiAj¢i(Xl)(¢i(X)¢j (x))x¢j (xm) = L A7¢i(Xl )¢i(Xm) ij i Introducing the diagonal eigenvalue matrix (A)ij = Aibij and the 'design matrix' (<I»li = ¢i(Xl ), this reads (k(x)k(x)T)x = <I>A2<I>T. Similarly, for the second term in (5) , (k(x)k;(x))x = <I>AA*<I>T, and (C*(x,x))x = trA*. This gives, dropping the training inputs subscript from the remaining average, E = (tr A* - 2tr<I>AA*<I>TK-1 + tr <I>A2<I>TK - 1K *K - 1) In this new representation we also have K = (}"21 + <I>A<I>T and similarly for K* ; for the inverse of K we can use the Woodbury formula to write K -1 = (}"-2 [1 (}"- 2<I>g<I> T], where 9 = (A - 1 + (}"- 2<I> T <I> )- 1. Inserting these results, one finds after some algebra that E = (}";(}"-2 [(tr g) - (tr gA -1 g)] + (tr gA*A -29) which for the matched case reduces to the known result for the Bayes error [4] € = (tr g) (7) (8) Eqs. (7,8) are still exact. We now need to tackle the remaining averages over training inputs. Two of these are of the form (tr QM9) ; if we generalize the definition of Q to Q = (A -1 + vI + wM + (/-2IJ>TIJ»-1 and define 9 = (tr Q) , then they reduce to (trQMQ) = -agjaw. (The derivative is taken at v = w = 0; the idea behind introducing v will become clear shortly.) So it is sufficient to calculate g. To do this, consider how Q changes when a new example is added to the training set. One has Q(n + 1) - Q(n) = [Q-1(n) + (/-21jJ1jJTJ -1 _ Q(n) = _ Q(n)1jJ1jJTQ(n) (9) (/2 + 1jJTQ(n)1jJ in terms of the vector 1jJ with elements (1jJ)i = <Pi(xn+d, using again the Woodbury formula. To obtain the change in 9 we need the average of (9) over both the new training input Xn+1 and all previous ones. This cannot be done exactly, but we can approximate by averaging numerator and denominator separately; taking the trace then gives g(n + 1) - g(n) = -(trQ2(n))j[(/2 + g(n)]. Now, using our auxiliary parameter v, -(trQ2) = agjav; if we also approximate n as continuous, we get the simple partial differential equation agjan = (agjaV)j((/2 + g) with the initial condition gln=o = tr (A -1 + vI + WM)-1. Solving this using the method of characteristics [7] gives a self-consistent equation for g, 9 = tr [A -1 + (v + (/2: g) 1+ wM r1 The Bayes error (8) is E = glv=w=o and therefore obeys E = trG, G -1 = A -1 + _n_ I (/2 + E (10) (11) within our approximation (called 'LC' in [4]). To obtain E, we differentiate both sides of (10) w.r.t. w, set v = w = 0 and rearrange to give (tr QM9) = -agjaw = (tr MG2)j[1 - (tr G 2)nj((/2 + E)2] Using this result in (7), with M = A -1 and M = A -1 A*A -1, we find after some further simplifications the final (approximate) result for the learning curve: , (/; tr G 2 + n-1 ((/2 + E)2 tr A*A -2G2 E = E ----'---::------::c-::---':-:---;:---:-;:---:----::--::c-::-(/2trG2 +n-1((/2 +E)2trA-1G2 (12) which transparently shows how in the matched case E and E become identical. 3 Examples I now apply the result for the learning curve (11,12) to some exemplary learning scenarios. First, consider inputs x which are binary vectors2 with d components Xa E {-I, I}, and assume that the input distribution is uniform. We consider covariance functions for student and teacher which depend on the product x . Xl only; this includes the standard choices (e.g. OU and RBF) which depend on the Euclidean distance Ix - xII, since Ix - x/12 = 2d - 2x . Xl. All these have the same eigenfunctions [9], so our above assumption is satisfied. The eigenfunctions are indexed by subsets p of {I, 2 ... d} and given explicitly by <pp(x) = ITaEP Xa' The 2This scenario may seem strange, but simplifies the determination of the eigenfunctions and eigenvalues. For large d, one expects other distributions with continuously varying x and the same first- and second-order statistics ((Xa) = 0, (XaXb) = 8ab ) to give similar results [8]. corresponding eigenvalues depend only on the size s = Ipl of the subsets and are therefore (~)-fold degenerate; letting e = (1,1 ... 1) be the 'all ones' input vector, they have the values As = (C(x, e)¢>p(x))x (which can easily be evaluated as an average over two binomially distributed variables, counting the number of + 1 's in x overall and among the Xa with a E p). With the As and A; determined, it is then a simple matter to evaluate the predicted learning curve (11,12) numerically. First, though, focus on the limit of large d, where much more can be said. If we write C(X,XI) = f(x· xl/d), the eigenvalues become, for d -+ 00, As = d-sf(s)(O) and the contribution to C(x, x) = f(l) from the s-th eigenvalue block is As == (~)As -+ f(s)(O)/s!, consistent with f(l) = 2::o f(s)(0)/s! The As, because of their scaling with d, become infinitely separated for d -+ 00. For training sets of size n = O(dL), we then see from (11) that eigenvalues with s > L contribute as if n = 0, since As » n / (u 2 + €); they have effectively not yet been learned. On the other hand, eigenvalues with s < L are completely suppressed and have been learnt perfectly. We thus have a hierarchical learning scenario, where different scalings of n with d-as defined by L-correspond to different 'learning stages'. Formally, we can analyse the stages separately by letting d -+ 00 at a constant ratio a = n/(f) of the number of examples to the number of parameters to be learned at stage L (note (f) = O(dL) for large d). An independent (replica) calculation along the lines of Ref. [8] shows that our approximation for the learning curve actually becomes exact in this limit. The resulting a-dependence of to can be determined explicitly: Set h = 2:s::=:L As (so that fa = f(I)) and similarly for fi. Then for large a , to = fL+1 + (fL+1 + u;)a- l + O(a- 2 ) (13) This implies that, during successive learning stages, (teacher) eigenvalues are learnt one by one and their contribution eliminated from the generalization error, giving plateaux in the learning curve at to = fi, f2, .... These plateaux, as well as the asymptotic decay (13) towards them, are universal [8], i.e. student-independent. The (non-universal) behaviour for smaller a can also be fully characterized: Consider first the simple case of linear percept ron learning (see e.g. [7]), which corresponds to both student and teacher having simple dot-product covariance functions C (x, Xl) = C * (x, Xl) = X· xl/d. In this case there is only a single learning stage (only Al = A~ = 1 are nonzero), and to = r(a) decays from r(O) = 1 to r(oo) = 0, with an over-fitting maximum around a = 1 if u2 is sufficiently small compared to u;. In terms of this function r(a), the learning curve at stage L for general covariance functions is then exactly given by to = fL+1 + ALr(a) if in the evaluation of r(a) the effective noise levels &2 = (f L+1 + ( 2 ) / AL and &; = (fL+1 + u;) / A L are used. Note how in &;, the contribution fL+1 from the not-yet-Iearned eigenvalues acts as effective noise, and is normalized by the amount of 'signal' AL = fL - fL+l available at learning stage L. The analogous definition of &2 implies that, for small u 2 and depending on the choice of student covariance function, there can be arbitrarily many learning stages L where &2 « &;, and therefore arbitrarily many over-fitting maxima in the resulting learning curves. From the definitions of &2 and &; it is clear that this situation can occur even if the student knows the exact teacher noise level, i.e. even if u 2 = u;. Fig. 1(left) demonstrates that the above conclusions hold not just for d -+ 00; even for the cases shown, with d = 10, up to three over-fitting maxima are apparent. Our theory provides a very good description of the numerically simulated learning curves even though, at such small d, the predictions are still significantly different from those for d -+ 00 (see Fig. 1 (right) ) and therefore not guaranteed to be exact. In the second example scenario, I consider continuous-valued input vectors, uni10 100 n 234 a 234 a 234 a Figure 1: Left: Learning curves for RBF student and teacher, with uniformly distributed, binary input vectors with d = 10 components. Noise levels: Teacher u; = 1, student u2 = 10-4, 10-3 , ... , 1 (top to bottom). Length scales: Teacher l* = d1/2, student l = 2d1/2. Dashed: numerical simulations, solid: theoretical prediction. Right: Learning curves for u 2 = 10- 4 and increasing d (top to bottom: 10, 20, 30, 40, 60, 80, [bold] 00). The x-axis shows a = n/(f), for learning stages L = 1,2,3; the dashed lines are the universal asymptotes (13) for d -+ 00. formly distributed over the unit interval [0,1]; generalization to d dimensions (x E [O , I]d) is straightforward. For covariance functions which are stationary, i.e. dependent on x and x' only through x - x', and assuming periodic boundary conditions (see [4] for details), one then again has covariance function-independent eigenfunctions. They are indexed by integers3 q, with cPq(x) = e21riqx; the corresponding eigenvalues are Aq = J dx C(O, x)e-27riqx . For the ('periodified') RBF covariance function C(x,x' ) = exp[-(x X ' )2/(2l2)], for example, one has Aq ex exp(-ip/2), where ij = 27rlq. The OU case C(x, x') = exp( -Ix - x/l/l), on the other hand, gives Aq ex (1 + ij2) - 1, thus Aq ex q- 2 for large q. I also consider below covariance functions which interpolate in smoothness between the OU and RBF limits: E.g. the MB2 (modified Bessel) covariance C(x, x') = e- a (1 + a), with a = Ix - x/l /l, yields functions which are once differentiable [5]; its eigenvalues Aq ex (1 + ij2)-2 show a faster asymptotic power law decay, Aq ex q-4, than those of the OU covariance function. To subsume all these cases I assume in the following analysis of the general shape of the learning curves that Aq ex q-r (and similarly A~ ex q-r.). Here r = 2 for OU, r = 4 for MB2, and (due to the faster-than-power law decay of its eigenvalues) effectively r = 00 for RBF. From (11,12), it is clear that the n-dependence of the Bayes error E has a strong effect on the true generalization error E. From previous work [4], we know that E(n) has two regimes: For small n, where E » u2 , E is dominated by regions in input space which are too far from the training examples to have significant correlation with them, and one finds E ex n-(r-1). For much larger n, learning is essentially against noise, and one has a slower decay E ex (n/u2)-(r- 1)/r . These power laws can be derived from (11) by approximating factors such as [A;;-l + n/ (u 2 + E)]- l as equal to either Aq or to 0, depending on whether n / (u 2 + E) < or > A;;-l. With the same technique, one can estimate the behaviour of E from (12). In the small n-regime, one finds E ~ C1 u; + C2n-(r. -1), with prefactors C1, C2 depending on the student. Note 3Since Aq = A_q, one can assume q ~ 0 if all Aq for q > 0 are taken as doubly degenerate. £ 0.1 £ 0.1 £ 0.1 10 100 n 10 100 1000 n Figure 2: Learning curves for inputs x uniformly distributed over [0,1]. Teacher: MB2 covariance function, lengthscale I. = 0.1, noise level (7; = 0.1; student lengthscale I = 0.1 throughout. Dashed: simulations, solid: theory. Left: OU student with (72 as shown. The predicted plateau appears as (72 decreases. Right: Students with (72 = 0.1 and covariance function as shown; for clarity, the RBF and OU results have been multiplied by v'IO and 10, respectively. Dash-dotted lines show the predicted asymptotic power laws for MB2 and OU; the RBF data have a persistent upward curvature consistent with the predicted logarithmic decay. Inset: RBF student with (72 = 10-3 , showing the occurrence of over-fitting maxima. that the contribution proportional to (7; is automatically negligible in the matched case (since then E = € » (72 = (7; for small n); if there is a model mismatch, however, and if the small-n regime extends far enough, it will become significant. This is the case for small (72; indeed, for (72 -+ 0, the 'small n' criterion € » (72 is satisfied for any n. Our theory thus predicts the appearance of plateaux in the learning curves, becoming more pronounced as (72 decreases; Fig. 2 (left ) confirms this4. Numerical evaluation also shows that for small (72, over-fitting maxima may occur before the plateau is reached, consistent with simulations; see inset in Fig. 2(right). In the large n-regime (€ « (72), our theory predicts that the generalization error decays as a power law. If the student assumes a rougher function than the teacher provides (r < r.) , the asymptotic power law exponent E ex: n-(r-l)/r is determined by the student alone. In the converse case, the asymptotic decay is E ex: n-(r.-l) / r and can be very slow, actually becoming logarithmic for an RBF student (r -+ CXl). For r = r., the fastest decay for given r. is obtained, as expected from the properties of the Bayes error. The simulation data in Fig. 2 are compatible with these predictions (though the simulations cover too small a range of n to allow exponents to be determined precisely). It should be stressed that the above results imply that there is no asymptotic regime of large training sets in which the learning curve assumes a universal form, in contrast to the case of parametric models where the generalization error decays as E ex: lin for sufficiently large n independently of model mismatch (as long as the problem is learnable at all). This conclusion may seem counterintuitive, but becomes clear if one remembers that a GP covariance function with an infinite number of nonzero eigenvalues Ai always has arbitrarily many eigenvalues 4If (J2 = 0 exactly, the plateau will extend to n -+ 00. With hindsight, this is clear: a GP with an infinite number of nonzero eigenvalues has no limit on the number of its 'degrees of freedom' and can fit perfectly any amount of noisy training data, without ever learning the true teacher function. that are arbitrarily close to zero (since the Ai are positive and 2:iAi = (C(x,x)) is finite). Whatever n, there are therefore many eigenvalues for which Ail» n/u2 , corresponding to degrees of freedom which are still mainly determined by the prior rather than the data (compare (11)). In other words, a regime where the data completely overwhelms the mismatched prior- and where the learning curve could therefore become independent of model mismatch- can never be reached. In summary, the above approximate theory makes a number of non-trivial predictions for GP learning with mismatched models, all borne out by simulations: for large input space dimensions, the occurrence of multiple over-fitting maxima; in lower dimensions, the generic presence of plateaux in the learning curve if the student assumes too small a noise level u 2 , and strong effects of model mismatch on the asymptotic learning curve decay. The behaviour is much richer than for the matched case, and could guide the choice of (student) priors in real-world applications of GP regression; RBF students, for example, run the risk of very slow logarithmic decay of the learning curve if the target (teacher) is less smooth than assumed. An important issue for future work- some of which is in progress- is to analyse to which extent hyperparameter tuning (e.g. via evidence maximization) can make GP learning robust against some forms of model mismatch, e.g. a misspecified functional form of the covariance function. One would like to know, for example, whether a data-dependent adjustment of the lengthscale of an RBF covariance function would be sufficient to avoid the logarithmically slow learning of rough target functions. References [1] See e.g. D J C MacKay, Gaussian Processes, Tutorial at NIPS 10; recent papers by Csat6 et al. (NIPS 12), Goldberg/Williams/Bishop (NIPS 10), Williams and Barber/Williams (NIPS 9), Williams/Rasmussen (NIPS 8); and references below. [2] D Malzahn and M Opper. In NIPS 13, pages 273- 279; also in NIPS 14. [3] C A Michelli and G Wahba. In Z Ziegler, editor, Approximation theory and applications, pages 329- 348. Academic Press, 1981; M Opper. In I K KwokYee et al., editors, Theoretical Aspects of Neural Computation, pages 17-23. Springer, 1997. [4] P Sollich. In NIPS 11, pages 344-350. [5] C K I Williams and F Vivarelli. Mach. Learn., 40:77-102, 2000. [6] C K I Williams. In M I Jordan, editor, Learning and Inference in Graphical Models, pages 599-621. Kluwer Academic, 1998. [7] P Sollich. J. Phys. A, 27:7771- 7784,1994. [8] M Opper and R Urbanczik. Phys. Rev. Lett., 86:4410- 4413, 2001. [9] R Dietrich, M Opper, and H Sompolinsky. Phys. Rev. Lett., 82:2975-2978, 1999.
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Kernel Machines and Boolean Functions Adam Kowalczyk Telstra Research Laboratories Telstra, Clayton, VIC 3168 a.kowalczyk@trl.oz.au Alex J. Smola, Robert C. Williamson RSISE, MLG and TelEng ANU, Canberra, ACT, 0200 Alex.Smola, Bob.Williamson @anu.edu.au Abstract We give results about the learnability and required complexity of logical formulae to solve classification problems. These results are obtained by linking propositional logic with kernel machines. In particular we show that decision trees and disjunctive normal forms (DNF) can be represented by the help of a special kernel, linking regularized risk to separation margin. Subsequently we derive a number of lower bounds on the required complexity of logic formulae using properties of algorithms for generation of linear estimators, such as perceptron and maximal perceptron learning. 1 Introduction The question of how many Boolean primitives are needed to learn a logical formula is typically an NP-hard problem, especially when learning from noisy data. Likewise, when dealing with decision trees, the question what depth and complexity of a tree is required to learn a certain mapping has proven to be a difficult task. We address this issue in the present paper and give lower bounds on the number of Boolean functions required to learn a mapping. This is achieved by a constructive algorithm which can be carried out in polynomial time. Our tools for this purpose are a Support Vector learning algorithm and a special polynomial kernel. In Section 2 we define the classes of functions to be studied. We show that we can treat propositional logic and decision trees within the same framework. Furthermore we will argue that in the limit boosted decision trees correspond to polynomial classifiers built directly on the data. Section 3 contains our main result linking the margin of separation to a simple complexity measure on the class of logical formulae (number of terms and depth). Subsequently we apply this connection to devise test procedures concerning the complexity of logical formulae capable of learning a certain dataset. More specifically, this will involve the training of a perceptron to minimize the regularized risk functional. Experimental results and a discussion conclude the paper. Some proofs have been omitted due to space constraints. They can be found in an extended version of this paper (available at http://www.kernel-machines.org). 2 Polynomial Representation of Boolean Formulae We use the standard assumptions of supervised learning: we have a training set
. Based on these observations we attempt to find a function ! " which incorporates the information given by the training set. Here goodness of fit is measured relative to some predefined loss function or a probabilistic model. What makes the situation in this paper special is that we assume that $#&%(' where %*) +, and moreover .) 0/1 . In other words, we attempt to learn a binary function on Boolean variables. A few definitions will be useful. The set of all polynomials of degree 2436587 on %9' will be denoted1 by :<;=?> . They are given by the expansion @ )BA C?DE<F C C where GH I J I 5K% ' and J I J 243 (1) for every ) ML
N NL ' 5PO , where F C 5RQ and we use a compact notation C S) LUTWV YXX
X L T[Z ' for every M\
N N [ \ ' 5Y%' for monomials on %]' , with the usual convention of L_^ S) - for every La`+ . In order to avoid further notation we always assume in such expansions that F Ccb ) + for all I 58G . The subset dfefg > 4:<;h=N> of all polynomials of the form M@ )ji -lk4m A C?DE C where G I J I 5n% ' and J I J 2o3 (2) will be called disjunctive normal forms. It is linked by the following lemma to the set of disjunctive normal forms depg > q rts u v commonly used in the propositional logic. The latter consist of all clauses wx % ' gzyf{|z} ~ } which can be expressed by disjunctions of terms, each being a conjunction of up to 3 logical primitives L T ) -
and L T ) +
. Lemma 1 Assume for each \ 5 h-h
there exists an \ 5 h-h
such that L T ) i L T[ for all ) ML
N [ NL ' 5KO (3) Then for every n5ndepg > there exists wx5ndfefg > q rts u v such that for every 5KO @ `Bif and only if w @ l~ } And vice versa, for every such w there exists satisfying the above relation. Standing Assumption: Unless stated otherwise, we will in the rest of the paper assume that (3) of the above lemma holds. This is not a major restriction, since we can always satisfy (3) by artificially enlarging Oo%6' into ML T i L T ML T 58O 4% ' . Now we consider another special subclass of polynomials, d ~ > O a:<;= > , called decision trees . These are polynomials which have expansions of type (1) where M\ all coefficients F C 5 i -h
and \\ for every 5O exactly one monomial C , I 5 G , ‘fires’, i.e. exactly one of the numbers C T D equals 1 and all the others are 0. Eq. (1) shows that each decision tree can be expressed as half of a difference between two disjunctive normal forms such that for any given input, one and only one of the conjunctions comprising them will be true. There exists also an obvious link to popular decision trees (on Boolean variables) used for classification in machine learning, cf. [4, 12]. Here the depth of a leaf equals the degree of the corresponding monomial, and the coefficient F C 5
/1 corresponds to the class associated with the leaf. 1Such binary polynomials are widely used under the name of score tables, e.g. typically loan applications are assessed by financial institutions by an evaluation of such score tables. 3 Reproducing Kernel Hilbert Space and Risk Kernel The next step is to map the complexity measure applied to decision trees, such as depth or number of leaves, to a Reproducing Kernel Hilbert Space (RKHS), as used in Support Vector machines. This is defined as )a:<;h=M> with the scalar product corresponding to the norm defined via the quadratic form on ) A CWDE C F C (4) Here T + with \ 5 +_
3 are complexity weights for each degree of the polynomials and the coefficients F C are the coefficients of expansion (1). Lemma 2 (Existence of Kernel) The RKHS kernel realizing the dot product corresponding to the quadratic form (4) with X 58:<;= > has the following efficient functional form: M ) > A ^
(5) Proof The norm is well defined by (4) for all 5 :<;=?> and the space :<;=?> is complete. Furthermore it is easy to check that (4) defines a homogeneous quadratic form on :<;=?> . Via the polarization identity we can reconstruct a bilinear form (dot product) from (4). This gives us the desired Hilbert space. From [1] we obtain that there exists a unique kernel @? corresponding to X . The key observation for derivation of its form (5) is that given @ 5O and + there are exactly non-vanishing monomials of the form C C ) L"! V L ! V X
XX L"!$# L ! # , where 2&% (' % ' XX
X ' % 2 are positions of 1’s in the sequence I . Note that for the special case where )*)
with ) + and 3 ` , (5) simply leads to a binomial expansion and we obtain M ) > A ^ ) ) -lk ) (6) The larger ) , the less severely we will penalize higher order polynomials, which provides us with an effective means of controlling the complexity of the estimates. Note that this is applicable to the case when 3 `aJ cJ , and always holds for 31) . Due to the choice of the F C in depg > and d ~ > O we obtain JNJ JNJ ) -lk,+ A C?DE- C for n5ndepg > and JNJ JNJ ) A C?DE C for Y5Kd ~ > O Next we introduce regularized risk functionals. They follow the standard assumptions made in soft-margin SVM and regularization networks. For our training set M T T of size . and a regularization constant / + we define 0 / S) JNJ JNJ k /
A T i T T 021 / ) JNJ JNJ k /
A T i T T 1 for every Y5K:<;h= > , where 1 S) +, for every 15nQ . The first risk is typically used by regularization networks [8], the other by support vector machines [5]. Note that for all 5 :<;= > we have 0 / H` 021 / . Furthermore, if Y5Kdepg > d ~ > O , then J M T J` - and hence 0 / @` 021 / ` A C?DE C k+ /
(7) where ) \J0 T b ) M T (8) denotes the number of classification errors (on the training set). Note that in (7) equalities hold throughout for 5od ~ > O and in such a case the risks are fully determined by the depths of the leaves of the decision tree and the number of classification errors. Furthermore, in the particular case of decision trees and all coefficients - T ) - , i.e. when JNJ JNJ equals to the number of leaves of the decision tree Y5Kd ~ > O , the regularized risks 0 / ) 0 1 / are exactly equal to the “cost complexity” employed to prune decision trees by CART algorithm [4]. In other words, the basis of the pruning algorithm in CART is the minimisation of the regularised risk in the class of subtrees of the maximal tree, with the regularisation constant / selected by a heuristic applied to a validation set. Our reasoning in the following relies on the idea that if we can find some function x5n:<;h=> which minimizes 0 / or 0 1 / , then the minimizer of the risk functionals, when chosen from the more restrictive set 5d ~ > O or 5dfefg > , must have a risk functional at least as large as the one found by optimizing over :<;h= > . This can then be translated into a lower bound on the complexity of since d ~ > O dfefg > 4:<;h=N> . 4 Complexity Bounds The last part missing to establish a polynomial-time device to lower-bound the required complexity of a logical formula is to present actual algorithms for minimizing 0 / or 021 / . In this section we will study two such methods: the kernel perceptron and the maximum margin perceptron and establish bounds on execution time and regularized risk. Kernel Perceptron Test The / -perceptron learning algorithm is a direct modification of ordinary linear perceptron learning rule. In the particular case of / ) + it becomes the ordinary perceptron learning rule in the feature space Q . For / + it implements perceptron learning rule in the extended feature space QKQ ; cf. [7, 6] for details. Algorithm 1 Regularized kernel perceptron ( / -perceptron) Given: a Mercer kernel and a constant / `o+ . Initialize: ) + and % T ) + for \ ) - [ N [ . . while an update is possible do find such that T T % T T @k / % 2 + , then update: % % k - and k- end while We introduce the special notation: ) T M T T , ) !#" T$ M T T and &% ! ' M@ S) T T % T T @ for every 5oO and ( % 5oQ . Note that &% ! ' 5o:<;h= > and JNJ % ! ' JNJ ) T T % T % M T . A modification of the standard proof of convergence of linear perceptron [11] combined with the extended feature space trick [13] gives the following result. Theorem 3 Assume that the coefficients ( %) % T 5KQ were generated after -th update of the / -perceptron and ) % ! D
^ ! " T T % ! ' T ,JNJ % ! ' JNJ 1 Then 2 k / , k / . and 0 / ` 0 1 / ` J[J % ! ' J[J k / JNJ ( % JNJ ` k / for every n5n:<;=?> . (9) Note that defined above is the maximal margin of separation of the training data by polynomials from :<;h= > (treated as elements of the RKHS). Maximum Margin Perceptron Test Below we state formally the soft margin version of maximal margin perceptron algorithm. This is a simplified (homogeneous) version of the algorithm introduced in [9]. Algorithm 2 Greedy Maximal Margin Perceptron ( / -MMP) Given: ) + , / ` + a and a Mercer kernel . Initialize: ) M k / for ) -h
N [ . ; \ )
! " , ) T , T ) + ; ) T @k T / and % ) T for ) -
N N [ . ; while TN TN X TN 2 i ) do for for every ) -h
N [ . do i T k T T T @k / T T ; i ; 3 ) k i m ; % T T k i T % ; # > # if + , else
! #
!$# # > # if ' + & + ' % ' - , else + . end for find \ )
, then set i T k T T k m T i T T ; end while The proof of the following theorem uses the extended feature space [13]. Theorem 4 Given + ' ) and / `+ . Assume that the vector ( % ) % T 5 Q was generated after -th iteration of the “while loop” of the / -MMP learning rule. Then 2 H k 0 k4m / ) k / . i k / (10) 0 / .` 021 / @` JNJ % ! ' JNJ k / J[J ( % J[J ` k / k ) k k4m / (11) for every Y5K:<;h= > . If the algorithm halts after -th update, then J[J &% ! ' J[J k / JNJ ( % JNJ ` ! " D rtq 021 / i ) (12) Note that condition (10) ensures the convergence of the algorithm in a finite time. The above theorem for /x) + ensures that solution generated by Algorithm 2 converges to the (hard) maximum margin classifier. Further, it can be shown that the bound (11) holds for every ( % ) % T such that each % T ` + and % T ) - . Bounds on classification error The task of finding a linear perceptron minimizing the number of classification errors on the training set is known to be NP-hard. On this basis it is reasonable to expect that finding a decision tree or disjunctive normal form of upper bounded complexity and minimizing the number of errors is also hard. In this section we provide a lower bound on the number of errors for such classifiers. The following estimates on
, i.e. the number of classification errors (8), can be derived from Theorems 3 and 4: Theorem 5 Let / ) + and 5xd ~ > . If the vector ( % ) % T 5xQ has been generated after -th iteration of the “while loop” of the / -perceptron learning rule, then
` / + JNJ % ! ' J[J k / JNJ ( % JNJ i JNJ JNJ ` / + ) k / i JNJ JNJ (13) On the other hand, if ( %) % T 5nQ has been generated after -th iteration of the “while loop” of the / -MMP learning rule, then
$` / + JNJ % ! ' JNJ k / J[J ( % J[J i J[J JNJ (14) ` / + k / k ) k k m / i JNJ J[J (15) Additionally, the estimate (14) holds for every ( %B) % T 5 Q such that % T ) - and each % T ` + . Note that % T equals in (13), while it is 1 in (14). The following result is derived form some recent results of Ben David and Simon [2] on efficient learning of perceptrons. Theorem 6 Given + and integer 3 + . There exists an algorithm which runs in time polynomial in both the input dimension and the number of training samples . , that given the labelled training sample M T T , \ ) - [ N [ . , it outputs a polynomial x5Y:<;h= > such that 2 for every in x5nd ~ > dfefg > . Following [2] we give an explicit formulation of the algorithm : for each subset of 2 + elements of the training set T T T
find the maximal margin hyperplane, if one exists. Using the standard quadratic programming approach this can be done in time polynomial in both and . [3]. Next, define 5Q as the vector of the hyperplane with the lowest error rate on the whole training set. Finally, set X S) X X 58:<;=?> . 5 Experimental Results and Discussion We have used a standard machine learning benchmark of noisy 7 bit LED display for 10 digits, 0 though 9, originally introduced in [4]. We generated 500 examples for training and 5000 for independent test, under assumption of 10% probability of a bit being reversed. The task set was to discriminate between two classes, digits 0-4 and digits 5-9. Each “noisy digit” data vector L0 [ N [ NL was complemented by an additional 7 bits vector i L
N N [i L to ensure that our Standing Assumption of Section 2 holds true. For a sake of simplicity we used fixed complexity weights, T ) - , \ ) +,
N N [ 3 , and / ) + , which for a decision tree n5nd ~ > gives a simple formula for the risk 0 1 / ) 0 / ) [number of leaves] k [number of errors] Four different algorithms have been applied to this data: M\ Decision Trees, version C4.5 [12] (available from www.cse.unsw.edu.au/ quinlan/), \\ regularized kernel perceptron (Algorithm 1) with the generated coefficients scaled ( %R ( % z JNJ % ! ' J[J k / J[J ( % J[J , where is the number of updates to the convergence, \\ \ greedy maximal margin classifier (Algorithm 2) and M\z mask perceptron [10] which for this data generates a polynomial Y5K:<;h=N> using some greedy search heuristics. Table 1 gives the experimental results. Table 1: Results for recognition of two groups of digits on faulty LED-display. Algorithm Risk (no. of leaves /SV/terms ) Error rate %: train/test 31) 31) 3) 3) Decision tree 110 (4 leaves) 80 (17 leaves) 21.3 / 22.9 12.0 / 15.8 Kernel SVM 44.4(413 SV) 40.8 (382 SV) 12.2 / 15.1 11.2 / 14.8 Kernel percep. 53.1 (294 SV) 54.9 (286 SV) 11.8 / 16.3 13.8 / 17.1 Mask percep. 53.2(10 terms) 49.1 (26 terms) 12.8 / 15.7 11.8 / 15.6 The lower bound on risk from maximal margin criterion (Eq. 11) are 44.3 and 40.7 for 3 ) and 3() , respectively. Similarly, the lower bound on risk from kernel perceptron criterion (Eq. 9) were 39.7 and 36.2, respectively. Risks for SVM solutions approach this bound and for kernel perceptron they are reasonably close. Comparison with the risks obtained for decision trees show that our lower bounds are meaningful (for the “un-pruned” decision trees risks were only slightly worse). The mask perceptron results show that simple (low number of terms) polynomial solutions with risks approaching our lower bounds can be practically found. The Bayes-optimal classifier can be evaluated on this data set, since we know explicitly the distribution from which data is drawn. Its error rates are 11.2% and 13.8% on the training and test sets, respectively. SVM solutions have error rates closest to the Bayesian classifier (the test error rate for 31) exceeds the one of the Bayes-optimal classifier by only 7%). Boosted Decision Trees An obvious question to ask is what happens if we take a large enough linear combination of decision trees. This is the case, for instance, in boosting. We can show that :<;= > is spanned by d ~ > O . In a nutshell, the proof relies on the partition of the identity into ) A 1 C
where
) i L
i L
i L ' and solving this expansion for C , where the remainder turns out to be a decision tree. This means that in the limit, boosting decision trees finds a maximum margin solution in :<;= > , a goal more directly achievable via a maximum margin perceptron on :<;= > . Conclusion We have shown that kernel methods with their analytical tools are applicable well outside of their traditional domain, namely in the area of propositional logic, which traditionally has been an area of discrete, combinatorial rather then continuous analytical methods. The constructive lower bounds we proved offer a fresh approach to some seemingly intractable problems. For instance, such bounds can be used as points of reference for practical applications of inductive techniques like as decision trees. The use of Boolean kernels introduced here allows a more insightful comparison of performance of logic based and analytical, linear machine learning algorithms. This contributes to the research in the theory of learning systems as illustrated by the result on existence of polynomial time algorithm for estimation of minimal number of training errors for decision trees and disjunctive normal forms. A potentially more practical link, to boosted decision trees, and their convergence to the maximum margin solutions has to be investigated further. The current paper sets foundations for such research. Boolean kernels can potentially stimulate more accurate (kernel) support vector machines by providing more intuitive construction of kernels. This is the subject of ongoing research. Acknowledgments A.K. acknowledges permission of the Chief Technology Officer, Telstra to publish this paper. A.S. was supported by a grant of the DFG Sm 62/1-1. Parts of this work were supported by the ARC and an R& D grant from Telstra. Thanks to P. Sember and H. Ferra for help in preparation of this paper. References [1] N. Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society, 68:337 – 404, 1950. [2] S. Ben-David and H. U. Simon. Efficient learning of linear perceptron. In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 189–195, Cambridge, MA, 2001. MIT Press. [3] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, MA, 1995. [4] L. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone. Classification and Regression Trees. Wadsworth Int., Belmont, Ca., 1984. [5] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273 – 297, 1995. [6] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines and other kernel-based learning methods. Cambridge University Press, Cambridge, 2000. [7] Y. Freund and R. E. Schapire. Large margin classification using the perceptron algorithm. In J. Shavlik, editor, Machine Learning: Proceedings of the Fifteenth International Conference, San Francisco, CA, 1998. Morgan Kaufmann. [8] F. Girosi, M. Jones, and T. Poggio. Regularization theory and neural networks architectures. Neural Computation, 7(2):219–269, 1995. [9] A. Kowalczyk. Maximal margin perceptron. In A. Smola, P.Bartlett, B. Sch¨olkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, pages 61–100, Cambridge, MA, 2000. MIT Press. [10] A. Kowalczyk and H. Ferr`a. Developing higher-order networks with empirically selected units. IEEE Transactions on Neural Networks, 5:698–711, 1994. [11] A. B. Novikoff. On convergence proofs on perceptrons. Symposium on the Mathematical Theory of Automata, 12:615–622, 1962. [12] J.R. Quinlan. Simplifying decision trees. Int. J. Man-Machine Studies, 27:221–234, (1987). [13] J. Shawe-Taylor and N. Christianini. Margin distribution and soft margin. In A. J. Smola, P. L. Bartlett, B. Sch¨olkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, pages 349–358, Cambridge, MA, 2000. MIT Press.
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Efficient Resources Allocation for Markov Decision Processes Remi Munos CMAP, Ecole Polytechnique, 91128 Palaiseau, France http://www.cmap.polytechnique.fr/....munos remi.munos@polytechnique.fr Abstract It is desirable that a complex decision-making problem in an uncertain world be adequately modeled by a Markov Decision Process (MDP) whose structural representation is adaptively designed by a parsimonious resources allocation process. Resources include time and cost of exploration, amount of memory and computational time allowed for the policy or value function representation. Concerned about making the best use of the available resources, we address the problem of efficiently estimating where adding extra resources is highly needed in order to improve the expected performance of the resulting policy. Possible application in reinforcement learning (RL) , when real-world exploration is highly costly, concerns the detection of those areas of the state-space that need primarily to be explored in order to improve the policy. Another application concerns approximation of continuous state-space stochastic control problems using adaptive discretization techniques for which highly efficient grid points allocation is mandatory to survive high dimensionality. Maybe surprisingly these two problems can be formulated under a common framework: for a given resource allocation, which defines a belief state over possible MDPs, find where adding new resources (thus decreasing the uncertainty of some parameters -transition probabilities or rewards) will most likely increase the expected performance of the new policy. To do so, we use sampling techniques for estimating the contribution of each parameter's probability distribution function (Pdf) to the expected loss of using an approximate policy (such as the optimal policy of the most probable MDP) instead of the true (but unknown) policy. Introduction Assume that we model a complex decision-making problem under uncertainty by a finite MDP. Because of the limited resources used, the parameters of the MDP (transition probabilities and rewards) are uncertain: we assume that we only know a belief state over their possible values. IT we select the most probable values of the parameters, we can build a MDP and solve it to deduce the corresponding optimal policy. However, because of the uncertainty over the true parameters, this policy may not be the one that maximizes the expected cumulative rewards of the true (but partially unknown) decision-making problem. We can nevertheless use sampling techniques to estimate the expected loss of using this policy. Furthermore, if we assume independence of the parameters (considered as random variables), we are able to derive the contribution of the uncertainty over each parameter to this expected loss. As a consequence, we can predict where adding new resoUrces (thus decreasing the uncertainty over some parameters) will decrease mostly this loss, thus improving the MDP model of the decision-making problem so as to optimize the expected future rewards. As possible application, in model-free RL we may wish to minimize the amount of real-world exploration (because each experiment is highly costly). Following [1] we can maintain a Dirichlet pdf over the transition probabilities of the corresponding MDP. Then, our algorithm is able to predict in which parts of the state space we should make new experiments, thus decreasing the uncertainty over some parameters (the posterior distribution being less uncertain than the prior) in order to optimize the expected payoff. Another application concerns the approximation of continuous (or large discrete) state-space control problems using variable resolution grids, that requires an efficient resource allocation process in order to survive the "curse of dimensionality" in high dimensions. For a given grid, because of the interpolation process, the approximate back-up operator introduces a local interpolation error (see [4]) that may be considered as a random variable (for example in the random grids of [6]). The algorithm introduced in this paper allows to estimate where we should add new grid-points, thus decreasing the uncertainty over the local interpolation error, in order to increase the expected performance of the new grid representation. The main tool developed here is the calculation of the partial derivative of useful global measures (the value function or the loss of using a sub-optimal policy) with respect to each parameter (probabilities and rewards) of a MDP. 1 Description of the problem We consider a MDP with a finite state-space X and action-space A. A transition from a state x, action a to a next state y occurs with probability p(Ylx, a) and the corresponding (deterministic) reward is r(x, a). We introduce the back-up operator T a defined, for any function W : X --t JR, as T aW(x) == (' LP(Ylx, a)W(y) + r(x, a) y (1) (with some discount factor 0 < (' < 1). It is a contraction mapping, thus the dynamic programming (DP) equation V(x) == maxaEA T aV(x) has a unique fixed point V called the value function. Let. us define the Q-values Q(x, a) == T aV (x). The optimal policy 1[* is the mapping from any state x to· the action 1[*(x) that maximizes the Q-values: 1[*(x) == maxaEA Q(x, a). The parameters of the MDP - the probability and the reward functions - are not perfectly known: all we know is a pdf over their possible values. This uncertainty comes from the limited amount of allocated resources for estimating those parameters. Let us choose a specific policy 1r (for example the optimal policy of the MDP with the most probable parameters). We can estimate the expected loss of using 1r instead of the true (but unknown) optimal policy 1[*. Let us write J-t == {Pj} the set of all parameters (p and r functions) of a MDP. We assume that we know a probability distribution function pdf(J-Lj) over their possible values. For a MDP MJ.t defined by its parameters P, we write pJL(yIx, a), r JL (x, a), V JL, QJL, and 7f1-!' respectively its transition probabilities, rewards, value function, Q-values, and optimal policy. 1.1 Direct gain optimization We define the gain ]JL(x; 7f) in the MDP MJL as the expected sum of discounted rewards obtained starting from state x and using policy 7f: ]JL(x; 1f) == E[2: rykrJL(Xk' 7f(xk))lxo == x; 7f] (2) k where the expectation is taken for sequences of states Xk --t Xk+l occurring with probability pP(Xk+llxk, 7fJL(Xk)). By definition, the optimal gain in MJL is VJL(x) == ]JL (x; 7fJL) which is obtained for the optimal policy 7fIL. Let ~ (x) == ]JL (x; if) be the approximate gain obtained for some approximate policy .7r in the same MDP MIL. We define the loss to occur LJL(x) from X when one uses the approximate policy 7r instead of the optimal one 7fJL in MJL: LIL(X) == VIL(x) - ~(x) (3) An example of approximate policy 1? would be the optimal policy of the most probable MDP, defined by the most probable parameters fi(ylx, a) and r(x, a). We also consider the problem of maximizing the global gain from a set of initial states chosen according to some probability distribution P(x). Accordingly, we define the global gain of a policy 11"": ]JL(7f) == Ex ]JL(x; 7f)P(x) and the global loss LIL of using some approximate policy 7r instead of the optimal one nIL (4) Thus, knowing the pdf over all parameters J-l we can define the expected global loss L == EJL[LIL]. Next, we would like to define what is the contribution of each parameter uncertainty to this loss, so we know where we should add new resources (thus reducing some parameters uncertainty) in order to decrease the expected global loss. We would like to estimate, for each parameter J-lj, E[8L I Add 8u units of resource for Pj] (5) 1.2 Partial derivative of the loss ill order to quantify (5) we need to be more explicit about the pdf over JL. First, we assume the independence of the parameters JLj (considered as random variables). Suppose that pdf(JLj) == N (0, Uj) (normal distribution of mean 0 and standard deviation Uj). We would like to estimate the variation 8L of the expected loss L when we make a small change of the uncertainty over Pj (consequence of adding new resources), for example when changing the standard deviation of 8aj in pdf(J.tj). At the limit of an infinitesimal variation we obtain the partial derivative Z;., which 3 when computed for all parameters J-lj, provides the respective contributions of each parameter's uncertainty to the global loss. Another example is when the pdf(pj) is a uniform distribution of support [-bj , bj]. Then the partial contribution of JLj'S uncertainty to the global loss can be expressed as gf. More generally, we can define a finite number of characteristic scalar mea3 surements of the pdf uncertainty (for example the entropy or the moments) and (6) compute the partial derivative of the expected global loss with respect to these coefficients. Finally, knowing the actual resources needed to estimate a parameter J..tj with some uncertainty defined by pdf(J..tj ), we are able to estimate (5). 1.3 Unbiased estimator We sample N sets of parameters {J..ti }i=1..N from the pd!(J..t), which define N-MDPs Mi. For convenience, we use the superscript i to refer to the i-th MDP sample and the subscript j for the j-th parameter of a variable. We solve each MDP using standard DP techniques (see [5]). This expensive computation that can be speed-up in two ways: first, by using the value function and policy computed for the first MDP as initial values for the other MDPs; second, since all MDPs have the same structure, by computing once for all an efficient ordering (using a topological sort, possibly with loops) of the states that will be used for value iteration. For each MDP, we compute the global loss L i of using the policy 'if and estimate the expected global loss: L ~ -1 2:::1 L i . In order to estimate the contribution of a p-arameter's uncertainty to L, we derive the partial derivative of L with respect to the characteristic coefficients of pdf(J-tj ). In the case of a reward parameter J..tj that follows a normal distribution N(O, Uj), we can write J..tj == Uj€j where €j follows N(O, 1). The partial derivative of the expected loss L with respect to Uj is aaL == a8 E/L~N(o.u)[L/L] = a8 Ee~N(o.l)[LUe] = Ee~N(o.1)[8aLue ~j] ~ ~ ~ ~ from which we deduce the unbiased estimator 8L '" ~t 8L i Jt; (7) aUj N i=l aJ..tj Uj where ~;; is the partial derivative of the global loss Li of MDP M i with respect to the parameter J..tj (considered as a variable). For other distributions, we can define similar results to (6) and deduce analogous estimators (for uniform distributions, we have the same estimator with bj instead of Uj). The remainder of the paper is organized as follow. Section 2 introduces useful tools to derive the partial contribution of each parameter -transition probability and reward- to the value function in a Markov Chain, Section 3 establishes the partial contribution of each parameter to the global loss, allowing to calculate the estimator (7), and Section 4 provides an efficient algorithm. All proofs are given in the full length paper [2]. 2 Non-local dependencies 2.1 Influence of a markov chain In [3] we introduced the notion of influence of a Markov Chain as a way to measure value function/rewards correlations between states. Let us consider a set of values V satisfying a Bellman equation Vex) == , LP(ylx)V(y) + rex) y We define the discounted cumulative k-chained transition probabilities Pk(ylx): (8) po(ylx) Pl(ylx) Ix=y (= 1 (if x = y) or 0 (if x =1= y)) IP(ylx) LP1(ylw)Pl(wlx) w LP1(ylw)Pk-l(wlx) w The influence I(ylx) of a state y on another state x is defined as I(ylx) = 2::%:oPk(ylx). Intuitively, I(ylx) measures the expected discounted number of visits of state y starting from x; it is also the partial derivative of the value function Vex) with respect to the reward r(y). Indeed Vex) can be expressed, as a linear combination of the rewards at y weighted by the influence I(ylx) Vex) = LI(Ylx)r(y) (9) y We can also define the influence of a state y on a function f: I(ylf(·)) = 2::x l(ylx)f(x) and the influence of a function f on another function 9 : l(f(·)\g(·)) = Y":y I(ylg(·))f(y)· In [3], we showed that the influence satisfies I(ylx) = , LP(ylw)I(wlx) + lx=y (10) w which is a fixed-point equation of a contractant operator (in I-norm) thus has a unique solution -the influence- that can be computed by successive iterations. Similarly, the influence I(ylf(·)) can be obtained as limit of the iterations I(ylf(·)) +-, LP(Ylw)I(wlf(·)) + fey) w Thus the computation of the influence I(ylf(·)) is cheap (equivalent to solving a Markov chain). 2.2 Total derivative of V We wish to express the contribution of all parameters - transition probabilities p and rewards r - (considered as variables) to the value function V by defining the total derivative of V as a function of those P¥ameters. We recall that the total derivative of a function f of several variables Xl, ..,' X n is df = 88f dXI + ... + a8t dxn . Xl Xn We already know that the partial derivative of Vex) with respect to the reward r(z) is the influence I(zjx) = ~~~1. Now, the dependency with respect to the transition probabilities has to be expressed more carefully because the probabilities p(wlz) for a given z are dependent (they sum to one). A way to express that is provided in the theorem that follows whose proof is in [2]. Theorelll 1 For a given state z, let us alter the probabilities p(wlz), for all w, with some c5'p(wlz) value, such that 2::w c5'p(wlz) = o. Then Vex) is altered by c5'V(x) = I(zlx)[,2::wV(w)c5'p(wlz)]. We deduce the total derivative of v: dV(x) = L1(zlx)[, L V(w)dp(wlz) + dr(z)] z w under the constraint 2::w dp(wiz) = 0 for all z. 3 Total derivative of the loss , For a given MDP M with parameters J..L (for notation simplification we do not write the JL superscript in what follows), we want to estimate the loss of using an approximate policy 7? instead of the optimal one 1f. First, we define the one-step loss l(x) at a state x as the difference between the gain obtained by choosing the best action 7f(x) then using the optimal policy 1f and the gain obtained by choosing action n(x) then the same optimal policy 7f l(x) == Q(x,1f(x)) - Q(x,ir(x)) (11) Now we consider the loss L(x), defined by (3), for an initial state x when we use the approximate policy n. We can prove that L(x) is the expected discounted cumulative one-step losses l(Xk) for reachable states Xk: L(x) == E[L I'k l(Xk)lxo == x;n] k with the expectation taken in the same sense as in (2). 3.1 Decomposition of the one-step loss We use (9) to decompose the Q-values Q(x, a) == I' LP(wlx, a) L I(ylw)r(y, 1f(Y)) + r(x, a) w y == r(x,a) + Lq(Ylx,a)r(y,7f(y)) y using the partial contributions q(ylx,a) == I'Ewp(wlx,a)I(ylw) where I(ylw) is the influence of y on w in the Markov chain derived from the MDP M by choosing policy 7f. Similarly, we decompose the one-step loss l(x) == Q(x,7f(x)) - Q(x, n(x)) == r(x,1f(x)) - r(x,7f(x)) + L [q(ylx,1f(x)) - q(ylx,n(x))] r(y,7f(Y)) y == r(x, 7f(x)) -r(x, 7?(x)) + Ll(Ylx)r(y, 7f(Y)) y as a function of the partial contributions l(ylx) == q(ylx,1f(x)) - q(ylx, n(x)) (see figure 1). o q (ylx ,IT ) q (ylx ,11- ) Figure 1: The reward r(y,1r(Y)) at state y contributes to the one-step loss l(x) = Q(x, 1r(x)) - Q(x, 1?(x)) with the proportion l(ylx) q(ylx, 1I"(x)) - q(ylx, 1?(x)). 3.2 Total derivative of the one-step loss and global loss Similarly to section (2.2), we wish to express the contribution of all parameters transition probabilities p and rewards r - (considered as variables) to the one-step loss function by defining the total derivative of I as a function of those parameters. Theorem 2 Let us introduce the (formal) differential back-up operator dTa defined, for any function W : X ~ JR, as dTaW(x) == ry L W(y)dp(ylx, a) + dr(x, a) y (similar to the back-up operator (1) but using dp and dr instead of p and r). The total derivative of the one-step loss is dl(x).==L1(zlx)dT7f(z)V(z) + dT7f(x)V(x) - dT;Cx)V(x) z under the constraint E y dp(ylx, a) == 0 for all x and a. Theorem 3 Let us introduce the one-step-loss back-up operator S and its formal differential version dS defined, for any function W : X ~ JR, as SW(x) ry LP(Ylx, 7T"(x))W(y) + l(x) y dSW(x) ry L dp(ylx, 7T"(x))W(y) + dl(x) y Then, the loss L(x) at x satisfies Bellman's equation L == SL. The total derivative of the loss L (x) and global loss L are, respectively dL(x) L I(zlx)dSL(z) Z dL L I(zIP(·))dSL(z) z from which (after regrouping the contribution to each parameter) we deduce the partial derivatives of the global loss with respect to the rewards and transition probabilities 4 A fast algorithm We use the sampling technique introduced in section 1.3. In order to compute the estimator (7) we calculate the partial derivatives ~~; based on the result of the previous section, with the following algorithm~ Given the pdf over the parameters j.L, select a policy 7? (for example the optimal policy of the most probable MDP). For i == 1..N, solve each MDP M i and deduce its value function Vi, Q-values Qi, and optimal policy 7ri . Deduce the one-step loss li(x) from (11). Compute the influence I(xIP(·)) (which depends on the transition probabilities pi of M i ) and the influence I(li(xl·)IP(·)) from which we deduce a ~(Li ). Then calculate Li(x) by solving Bellman's equation Li = SLi and deduce r x,a 8P,r~:,a). These partial derivatives enable to compute the unbiased estimator (7). The complexity of solving a discounted MDP with K states, each one connected to M next states, is O(KM), as is the complexity of computing the influences. Thus, the overall complexity of this algorithm is O(NKM). Conclusion· Being able to compute the contribution of each parameter -transition probabilities and rewards- to the value function (theorem 1) and to the loss of the expected rewards to occur if we use an approximate policy (theorem 3) enables us to use sampling techniques to estimate what are the parameters whose uncertainty are the most harmful to the expected gain.. A relev-ant resource allocation process would consider adding new computational resources to reduce uncertainty over the true value of those parameters. In the examples given in the introduction, this would be doing new experiments in model-free RL for defining more precisely the transition probabilities of some relevant states. In discretization techniques for continuous control problems, this would be adding new grid points in order to improve the quality of the interpolation at relevant areas of the state space in order to maximize the expected gain of the new policy. Initial experiments for variable resolution discretization using random grids show improved performance compared to [3]. Acknowledgments I am grateful to Andrew Moore, Drew Bagnell and Auton's Lab members for motivating discussions. References [1] Richard Dearden, Nir Friedman, and David Andre. Model based bayesian exploration. Proceeding of Uncertainty in Artificial Intelligence, 1999. [2] Remi Munos. Decision-making under uncertainty:. Efficiently estimating where extra ressources are needed. Technical report, Ecole Polytechnique, 2002. [3] Remi Munos and Andrew Moore. Influence and variance of a markov chain : Application to adaptive discretizations in optimal control. Proceedings of the 38th IEEE Conference on Decision and Control, 1999. [4] Remi Munos and Andrew W. Moore. Rates of convergence for variable resolution schemes in optimal control. International Conference on Machine Learning, 2000. [5] Martin L. Puterman. Markov Decision Processes, Discrete Stochastic Dynamic Programming. A Wiley-Interscience Publication, 1994. [6] John Rust. Using Randomization to Break the Curse of Dimensionality. Computational Economics. 1997.
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Information Geometrical Framework for Analyzing Belief Propagation Decoder Shiro Ikeda Kyushu Inst. of Tech., & PRESTO, JST Wakamatsu, Kitakyushu, Fukuoka, 808-0196 Japan shiro@brain.kyutech.ac.jp Toshiyuki Tanaka Tokyo Metropolitan Univ. Hachioji, Tokyo, 192-0397 Japan tanaka@eei.metro-u.ac.jp Shun-ichi Amari RIKEN BSI Wako, Saitama, 351-0198 Japan amari@brain.riken.go.jp Abstract The mystery of belief propagation (BP) decoder, especially of the turbo decoding, is studied from information geometrical viewpoint. The loopy belief network (BN) of turbo codes makes it difficult to obtain the true “belief” by BP, and the characteristics of the algorithm and its equilibrium are not clearly understood. Our study gives an intuitive understanding of the mechanism, and a new framework for the analysis. Based on the framework, we reveal basic properties of the turbo decoding. 1 Introduction Since the proposal of turbo codes[2], they have been attracting a lot of interests because of their high performance of error correction. Although the thorough experimental results strongly support the potential of this iterative decoding method, the mathematical background is not sufficiently understood. McEliece et al.[5] have shown its relation to the Pearl’s BP, but the BN for the turbo decoding is loopy, and the BP solution gives only an approximation. The problem of the turbo decoding is a specific example of a general problem of marginalizing an exponential family distribution. The distribution includes higher order correlations, and its direct marginalization is intractable. But the partial model with a part of the correlations, can be marginalized with BP algorithm exactly, since it does not have any loop. By collecting and exchanging the BP results of the partial models, the true “belief” is approximated. This structure is common among various iterative methods, such as Gallager codes, Beth´e approximation in statistical physics[4], and BP for loopy BN. We investigate the problem from information geometrical viewpoint[1]. It gives a new framework for analyzing these iterative methods, and shows an intuitive understanding of them. Also it reveals a lot of basic properties, such as characteristics of the equilibrium, the condition of stability, the cost function related to the decoder, and the decoding error. In this paper, we focus on the turbo decoding, because its structure is simple, but the framework is general, and the main results can be generalized. 2 Information Geometrical Framework 2.1 Marginalization, MPM Decoding, and Belief Let us consider a distribution of
which is defined as follows !"$#%& !'(" )'*
+'," -. /$ (1) where, " # is the linear function of 0 1/2 , and each " 3 is the higher order correlations of 0 1 2 . The problem of turbo codes and similar iterative methods are to marginalize this distribution. Let 4 denote the operator of marginalization as, 4 5 & 6 798 ;: 1=<> 1 . The marginalization is equivalent to take the expectation of as ?@6 798 ;A BC & D?E&F
/F HG In the case of MPM (maximization of the posterior marginals) decoding, 1JI 0LKM $' M 2 and the sign of each FN1 is the decoding result. In the belief network, 1 I 0PO M 2 and FN1 is the belief. In these iterative methods, the marginalization of eq.(1) is not tractable, but the marginalization of the following distribution is tractable. Q & SR/T* % U" # )'," Q & >'VT.
KW Q XT/ ZY M
\[] ^T I`_ G (2) Each Q & SRT includes only one of the 0 " 3 & $2 in eq.(1), and additional parameter T is used to adjust linear part of . The iterative methods are exchanging information through T for each aQ , and finally approximate 4 5 & . 2.2 The Case of Turbo Decoding bcNd9e%fghNi\f jNk\d l m
noXprqsXtvu w x m
noXprqsXtNy w{z | | | | | | | | l+} w x } w\z ~LX
| l+} w x } w{z | bcNd9e%fk\i\f+jNk\d sXoXprqsXtLu z sXoXprqsXt y x l+} w$x l+} w z | | | Figure 1: Turbo codes In the case of turbo codes, is the information bits, from which the turbo encoder generates two sets of parity bits, ,v\
/v/ , and & +
) , %UN I 0LKM {' M 2 (Fig.1). Each parity bit is expressed as the form : 1 1 , where the product is taken over a subset of 0LM
V2 . The codeword & S is then transmitted over a noisy channel, which we assume BSC (binary symmetric channel) with flipping probability M+ . The receiver observes S , 1 v9N I 0vKM $' M 2 . The ultimate goal of the turbo decoding is the MPM decoding of based on S + . Since the channel is memoryless, the following relation holds S * %
' @ ' @ SK ¡' N¢ r£¤> ¥ O M MK§¦{¨ ©«ª > ¬£&> 6
798 ¯® © ° ±²'(°N³± G By assuming the uniform prior on , the posterior distribution is given as follows & S + B + S + % &
' ' § U" # & !'("
+ !'(" / G (3) Here is the normalizing factor, and "$#v& ( S , " Q ( Q Q Y M . Equation(3) is equivalent to eq.(1), where [^ . When is large, marginalization of r S is intractable since it needs summation over terms. Turbo codes utilize two decoders which solve the MPM decoding of aQ SRTY M in eq.(2). The distribution is derived from Q and the prior of which has the form of SRT !XT K £¤XT G & SR/T is a factorizable distribution. The marginalization of Q is feasible since its BN is loop free. The parameter T serves as the window of exchanging the information between the two decoders. The MPM decoding is approximated by updating T iteratively in “turbo” like way. 2.3 Information Geometrical View of MPM Decoding Let us consider the family of all the probability distributions over . We denote it by , which is defined as & & S¥ O I 0LKM $' M 2 A B M G We consider an ° –flat submanifold # in . This is the submanifold of # SR defined as # # & SR
" # !' ` KW # ) E
I _ G (4) Since " # & * , every distribution of # can be rewritten as follows # & SR
U" # !'
KW # ]'
KW # / G It shows that every distribution of # is decomposable, or factorizable. From the information geometry[1], we have the following theorem of –projection. Theorem 1. Let be an ° –flat submanifold in , and let & I . The point in that minimizes the KL-divergence from & to , is denoted by, 45 & ¨ © !#" B%$'& )(+* $R & -,U and is called the –projection of & to . The –projection is unique. It is easy to show that the marginalization corresponds to the –projection to # [7]. Since MPM decoding and marginalization is equivalent, MPM decoding is also equivalent to the –projection to # . 2.4 Information Geometry of Turbo Decoding Let ./ 5 & denote the parameters in of the –projected distribution, .05# ¨ 1 2 3 © 4 ɀ (8* $R & SR
', G The turbo decoding process is written as follows, 1. Let T O for O , and M . 2. Project & SR/T onto # as ./5 & SR/T , and calculate T by T .0P5 & SRT K T G 3. Project & SR/T onto # as . 5 & SRT , and calculate T by T ./5 & SR/T K T G 4. If .0P5 & SRT .0P5 & SR/T , go to step 2. The turbo decoding approximates the estimated parameter , the projection of S onto # , as T 'VT , where the estimated distribution is # & SR
&" # & !' T P ' T
KVW # XT 'VT / G (5) An intuitive understanding of the turbo decoding is as follows. In step 2, T in eq.(5) is replaced with " & . The distribution becomes SRT , and T is estimated by projecting it onto # . In step 3, XT
in eq.(5) is replaced with "
& , and T is estimated by – projection of + SRT . We now define the submanifold corresponding to each decoder, Q Q SRT "$#v& !'(" Q !' T P KVW Q XT T
I`_ Y M G T is the coordinate system of Q . Q is also an ° –flat submanifold. and Q # hold because " Q & includes cross terms of and "
N& J" in general. The information geometrical view of the turbo decoding is schematically shown in Fig.2. 3 The Properties of Belief Propagation Decoder 3.1 Equilibrium When the the turbo decoding converges, equilibrium solution defines three important distributions, SRT , SRT , and #L SR . They satisfy the following two conditions: M G 4 5 +& SR/T H 4 5 SRT # & SR
G (6) G T 'VT G (7) Let us define a manifold as & I A B & A B # & SR
G From its definition, for any & I , the expectation of is the same, and its – projection to # coincides with # SR . This is an –flat submanifold[1], and we call an equimarginal submanifold. Since eq.(6) holds, #L SR
$ SRT $ SRT I is satisfied. Let us define an ° –flat version of the submanifold as , which connects # SR , & SR/T , and & SR/T in log-linear manner # & SR +& SR T SR T A Q < # Q M G Since eq.(7) holds, & is included in the . It can be proved by taking # KM , M .
"! #%$'&)( +* "! #, -/. -/. 0 ! & ! # 1! & 1! # 2'3'457698 : ;=<>7? -/. @ "! & $ # ( +* ! & , Figure 2: Turbo decoding A BDCFEHGJI KLCFEHGJI BNM BPO BNQ RTSVUXWZY\[^]`_ a+b=cVdfehg ejik'l cJm=n k'o RpqUWrY9s ] t _ R t UXWZY=s ] p _ R UWvuw Wrx+w y pqxzw y t _ Figure 3: and
Theorem 2. When the turbo decoding procedure converges, the convergent probability distributions # & SR
, + SRT , and & SR/T belong to equimarginal submanifold , while its ° –flat version
includes these three distributions and also the posterior distribution & (Fig.3). If includes + , & SR
is the true marginalization of S + . However, does not necessarily include . This fact means that r S and # & SR
are not necessarily equimarginal, which is the origin of the decoding error. 3.2 Condition of Stability The expectation parameters are defined as follows with W # in eq.(4) and W Q in eq.(2) ?#L %6 798 EA B #v& SR L{ 4 W #L $ Z? Q XT%6
798 A B Q & SRTL{}| W Q XT^Y M G Equation (6) is rewritten as follows with these parameters, ? # ?>+T ? T G We give a sufficiently small perturbation ~ to T and apply one turbo decoding step. The –projection from & SRT ' ~ to # gives, ? # ' *? XT ' ~ #v ³ XT ~ G Here, #L is the Fisher information matrix of #v& SR , and Q T is that of Q & SRT , Y M . Note that # is a diagonal matrix. The Fisher information matrix is defined as follows # L{ 44" W # L{ 4 ? # { Q XT{ |\| W Q XTL{ | ? Q XT ZY M G T in step 2 will be, T T '
# $³ XT K ~ G Here, is an identity matrix of size . Following the same line for step 3, we derive the theorem which coincides with the result of Richardson[6]. Theorem 3. Let 1 be the eigenvalues of the matrix defined as
#% {³ T K Z
#% $³ T K G When 1{ M holds for all , the equilibrium point is stable. 3.3 Cost Function and Characteristics of Equilibrium We give the cost function which plays an important role in turbo decoding. XT /T W #L K W T !' W XT / G Here, *T ' T . This function is identical to the “free energy” defined in [4]. Theorem 4. The equilibrium state T G
G G /T is the critical point of . Proof. Direct calculation gives { | *? # K ? XT« , { | *? # K ?>XT . For the equilibrium, ?#L ? T ? T holds, and the proof is completed. When XT Q K T Q is small, T T K T T K #% ³ #% ³ {}| { | G This shows how the algorithm works, but it does not give the characteristics of the equilibrium point. The Hessian of is { | | { | | { | | { | | # K ² # # # K G And by transforming the variables as, T ' T and T K T , we have { 44 { 4
{ 4 { M # K ²H' X² K X² K K ²' G Most probably, { 44 is positive definite but { is always negative, and is generally saddle at equilibrium. 3.4 Perturbation Analysis For the following discussion, we define a distribution & SR
! as SR ! " # !' ` ]' KW !/ ] W !H ® © A B &"$#v& !'
' / a& %6
798 (&" & $ \" & / G This distribution includes #v& SR ( ), & S ( , ), and Q & SR/T ( ,;T , Q ), where § M M / , @ M O / , and O M / . The expectation parameter ? ! is defined as, ? ! L{ 4 W ! A B SR ! G Let us consider , where every distribution SR ! I has the same expectation parameter, that is, ?S !H ?S holds. Here, we define, ?S
H ?S + !a . From the Taylor expansion, we have, FL1 ! FL1 !' A { PFN1/ 9 ' A Q { Q FN1\ " Q ' M A Q# $ { Q { $ FL1 " Q $ 'A # Q { Q { F 1 " Q ' M A 3 # % { 3 { % F 1 9 3 % ' !&'(&)P!' &1 *&+) G (8) The indexes 0
L r2 are for , 0 Y+ N2 are for , and a6
798 + K . After adding some definitions, that is, F 1 !EF 1 , and { F 1 J 1 , where 0 1 2 is the Fisher information matrix of & SR
!a which is a diagonal matrix, we substitute 1 with function of Q up to its 2nd order, and neglect the higher orders of Q . And we have, 1 K 1 1 A Q 1 Q Q K 1 1 A Q# $ { Q K A 3 3 3 3 Q { 3 { $ K A r $ {
F 1 " Q $ (9) where, 1 1 M 1=1 , and 1 Q L{
F 1 . Let a , and since & SR
$ + SR I holds, T and T K K T . Also when we put V , § K T holds. From eq.(9), we have the following result, K 1 # Q K 1=1 1 Q K 1 1 { Q K A 3 3 3 3 Q {%3 { Q K A r Q {N FN1\ G (10) Next, let , and we consider & SR !
% I , where is the parameter which satisfies this equation. Since SR!! & S is not necessarily included in , is generally not equal to . From eq.(9), 1 K 1 # K 1 1 A Q 1 Q K 1 1 A Q { Q K A 3 3 3 3 Q { 3 { Q K A r Q {N FN1 G From the condition T 1 'VT 1 and eq.(10), we have the following approximation, 1 K 1 1 A Q < $ { Q K A 3 3 3 3 Q { 3 { $ K A r $ { FL1 G This result gives the approximation accuracy of the BP decoding. Let the true belief be ? , and we evaluate the difference between ? and ?S
on # . The result is summarized in the following theorem. Theorem 5. The true expectation of , which is ? ? )
% , is approximated as, ? ? !' M A Q < $ { Q K A 3 3 3 3 Q { 3 { $ K A r $ { ?S G (11) Where ? is the solution of the turbo decoding. Equation (11) is related to the –embedded–curvature of (Fig.3). The result can be extended to general case where [ [3, 8]. 4 Discussion We have shown a new framework for understanding and analyzing the belief propagation decoder. Since the BN of turbo codes is loopy, we don’t have enough theoretical results for BP algorithm, while a lot of experiments show that it works surprisingly well in such cases. The mystery of the BP decoders is summarized in 2 points, the approximation accuracy and the convergence property. Our results elucidate the mathematical background of the BP decoding algorithm. The information geometrical structure of the equilibrium is summarized in Theorem 2. It shows the ° –flat submanifold
plays an important role. Furthermore, Theorem 5 shows that the relation between
and the –flat submanifold causes the decoding error, and the principal component of the error is the curvature of . Since the curvature strongly depends on the codeword, we can control it by the encoder design. This shows a room for improvement of the “near optimum error correcting code”[2]. For the convergent property, we have shown the energy function, which is known as Beth´e free energy[4, 9]. Unfortunately, the fixed point of the turbo decoding algorithm is generally a saddle of the function, which makes further analysis difficult. We have only shown a local stability condition, and the global property is one of our future works. This paper gives a first step to the information geometrical understanding of the belief propagation decoder. The main results are for the turbo decoding, but the mechanism is common with wider class, and the framework is valid for them. We believe further study in this direction will lead us to better understanding and improvements of these methods. Acknowledgments We thank Chiranjib Bhattacharyya who gave us the opportunity to face this problem. We are also grateful to Yoshiyuki Kabashima and Motohiko Isaka for useful discussions. References [1] S. Amari and H. Nagaoka. (2000) Methods of Information Geometry, volume 191 of Translations of Mathematical Monographs. American Mathematical Society. [2] C. Berrou and A. Glavieux. (1996) Near optimum error correcting coding and decoding: Turbo-codes. IEEE Transactions on Communications, 44(10):1261–1271. [3] S. Ikeda, T. Tanaka, and S. Amari. (2001) Information geometry of turbo codes and low-density parity-check codes. submitted to IEEE transaction on Information Theory. [4] Y. Kabashima and D. Saad. (2001) The TAP approach to intensive and extensive connectivity systems. In M. Opper and D. Saad, editors, Advanced Mean Field Methods – Theory and Practice, chapter 6, pages 65–84. The MIT Press. [5] R. J. McEliece, D. J. C. MacKay, and J.-F. Cheng. (1998) Turbo decoding as an instance of Pearl’s “belief propagation” algorithm. IEEE Journal on Selected Areas in Communications, 16(2):140–152. [6] T. J. Richardson. (2000) The geometry of turbo-decoding dynamics. IEEE Transactions on Information Theory, 46(1):9–23. [7] T. Tanaka. (2001) Information geometry of mean-field approximation. In M. Opper and D. Saad, editors, Advanced Mean Field Methods – Theory and Practice, chapter 17, pages 259–273. The MIT Press. [8] T. Tanaka, S. Ikeda, and S. Amari. (2002) Information-geometrical significance of sparsity in Gallager codes. in T. G. Dietterich et al. (eds.), Advances in Neural Information Processing Systems, vol. 14 (this volumn), The MIT Press. [9] J. S. Yedidia, W. T. Freeman, and Y. Weiss. (2001) Bethe free energy, Kikuchi approximations, and belief propagation algorithms. Technical Report TR2001–16, Mitsubishi Electric Research Laboratories.
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Speech Recognition using SVMs Nathan Smith Cambridge University Engineering Dept Cambridge, CB2 1PZ, U.K. ndsl 002@eng.cam.ac.uk Mark Gales Cambridge University Engineering Dept Cambridge, CB2 1PZ, U.K. mjfg@eng.cam.ac.uk Abstract An important issue in applying SVMs to speech recognition is the ability to classify variable length sequences. This paper presents extensions to a standard scheme for handling this variable length data, the Fisher score. A more useful mapping is introduced based on the likelihood-ratio. The score-space defined by this mapping avoids some limitations of the Fisher score. Class-conditional generative models are directly incorporated into the definition of the score-space. The mapping, and appropriate normalisation schemes, are evaluated on a speaker-independent isolated letter task where the new mapping outperforms both the Fisher score and HMMs trained to maximise likelihood. 1 Introduction Speech recognition is a complex, dynamic classification task. State-of-the-art systems use Hidden Markov Models (HMMs), either trained to maximise likelihood or discriminatively, to achieve good levels of performance. One of the reasons for the popularity of HMMs is that they readily handle the variable length data sequences which result from variations in word sequence, speaker rate and accent. Support Vector Machines (SVMs) [1] are a powerful, discriminatively-trained technique that have been shown to work well on a variety of tasks. However they are typically only applied to static binary classification tasks. This paper examines the application of SVMs to speech recognition. There are two major problems to address. First, how to handle the variable length sequences. Second, how to handle multi-class decisions. This paper only concentrates on dealing with variable length sequences. It develops our earlier research in [2] and is detailed more fully in [7]. A similar approach for protein classification is adopted in [3]. There have been a variety of methods proposed to map variable length sequences to vectors of fixed dimension. These include vector averaging and selecting a 'representative' number of observations from each utterance. However, these methods may discard useful information. This paper adopts an approach similar to that of [4] which makes use of all the available data. Their scheme uses generative probability models of the data to define a mapping into a fixed dimension space, the Fisher score-space. When incorporated within an SVM kernel, the kernel is known as the Fisher kernel. Relevant regularisation issues are discussed in [5]. This paper examines the suitability of the Fisher kernel for classification in speech recognition and proposes an alternative, more useful, kernel. In addition some normalisation issues associated with using this kernel for speech recognition are addressed. Initially a general framework for defining alternative score-spaces is required. First, define an observation sequence as 0 = (01 , . . . Ot, ... OT) where Ot E ~D , and a set of generative probability models of the observation sequences as P = {Pk(OI(h)}, where 9 k is the vector of parameters for the kth member of the set. The observation sequence 0 can be mapped into a vector of fixed dimension [4], i{J~ (0) (1) f(·) is the score-argument and is a function of the members of the set of generative models P. i{Jft is the score-mapping and is defined using a score-operator F. i{J~(0) is the score and occupies the fixed-dimension score-space. Our investigation of score-spaces falls into three divisions. What are the best generative models, scorearguments and score-operators to use? 2 Score-spaces As HMMs have proved successful in speech recognition, they are a natural choice as the generative models for this task. In particular HMMs with state output distributions formed by Gaussian mixture models. There is also the choice of the score-argument. For a two-class problem, let Pi(019i ) represent a generative model, where i = {g, 1, 2} (g denotes the global2-class generative model, and 1 and 2 denote the class-conditional generative models for the two competing classes). Previous schemes have used the log of a single generative model, Inpi(019i) representing either both classes as in the original Fisher score (i = g) [4], or one of the classes (i = 1 or 2) [6]. This score-space is termed the likelihood score-space, i{J~k(O). The score-space proposed in this paper uses the log of the ratio of the two classconditional generative models, In(P1(019d /P2(0192)) where 9 = [9{,9J]T. The corresponding score-space is called the likelihood-ratio score-space, i{J~(0) . Thus, i{J~k(O) i{J~(0) (2) (3) The likelihood-ratio score-space can be shown to avoid some of the limitations of the likelihood score-space, and may be viewed as a generalisation of the standard generative model classifier. These issues will be discussed later. Having proposed forms for the generative models and score-arguments, the scoreoperators must be selected. The original score-operator in [4] was the 1st-order derivative operator applied to HMMs with discrete output distributions. Consider a continuous density HMM with N emitting states, j E {I . . . N}. Each state, j, has an output distribution formed by a mixture of K Gaussian components, N(J-tjk' ~jd where k E {I ... K}. Each component has parameters of weight Wjk, mean J-tjk and covariance ~jk. The 1st-order derivatives of the log probability of the sequence 0 with respect to the model parameters are given below1, where the derivative operator has been defined to give column vectors, T L ')'jk(t)S~,jkl (4) t = l lFor fuller details of the derivations see [2). V Wjk Inp(OIO) where S[t,jk] Ijdt) is the posterior probability of component k of state j at time t. Assuming the HMM is left-to-right with no skips and assuming that a state only appears once in the HMM (i.e. there is no state-tying), then the 1st-order derivative for the self-transition probability for state j, ajj, is, t[/j(t) 1] t=l ajj Tajj(l- ajj) (8) The 1st-order derivatives for each Gaussian parameter and self-transition probability in the HMM can be spliced together into a 'super-vector' which is the score2 . From the definitions above, the score for an utterance is a weighted sum of scores for individual observations. If the scores for the same utterance spoken at different speaking rates were calculated, they would lie in different regions of score-space simply because of differing numbers of observations. To ease the task of the classifier in score-space, the score-space may be normalised by the number of observations, called sequence length normalisation. Duration information can be retained in the derivatives of the transition probabilities. One method of normalisation redefines score-spaces using generative models trained to maximise a modified log likelihood function, In( 010). Consider that state j has entry time Tj and duration dj (both in numbers of observations) and output probability bj(Ot) for observation Ot [7]. So, N 1 T;+dj- 1 In(OIO) L d- ((dj -1) lnajj + Inaj(j+1) + L (Inbj(Ot))) (9) j=l J t=Tj It is not possible to maximise In(OIO) using the EM algorithm. Hill-climbing techniques could be used. However, in this paper, a simpler normalisation method is employed. The generative models are trained to maximise the standard likelihood function. Rather than define the score-space using standard state posteriors Ij(t) (the posterior probability of state j at time t), it is defined on state posteriors normalised by the total state occupancy over the utterance. The standard component posteriors 1 j k (t) are replaced in Equations 4 to 6 and 8 by their normalised form 'Yjk(t), A . (t) _ Ij(t) (WjkN(Ot; ILjk, ~jk) ) ~k T K 2::T=l/j(T) 2::i = l wjiN(ot; ILji' ~ji) (10) In effect, each derivative is divided by the sum of state posteriors. This is preferred to division by the total number of observations T which assumes that when the utterance length varies, the occupation of every state in the state sequence is scaled by the same ratio. This is not necessarily the case for speech. The nature of the score-space affects the discriminative power of classifiers built in the score-space. For example, the likelihood score-space defined on a two-class 2Due to the sum to unity constraints, one of the weight parameters in each Gaussian mixture is discarded from the definition of the super-vector, as are the forward transitions in the left-to-right HMM with no skips. generative model is susceptible to wrap-around [7]. This occurs when two different locations in acoustic-space map to a single point in score-subspace. As an example, consider two classes modelled by two widely-spaced Gaussians. If an observation is generated at the peak of the first Gaussian, then the derivative relative to the mean of that Gaussian is zero because S [t ,jk] is zero (see Equation 4). However, the derivative relative to the mean of the distant second Gaussian is also zero because of a zero component posterior f jdt). A similar problem occurs with an observation generated at the peak of the second Gaussian. This ambiguity in mapping two possible locations in acoustic-space to the zero of the score-subspace of the means represents a wrapping of the acoustic space onto this subspace. This also occurs in the subspace of the variances. Thus wrap-around can increase class confusion. A likelihood-ratio score-space defined on these two Gaussians does not suffer wraparound since the component posteriors for each Gaussian are forced to unity. So far, only 1st-order derivative score-operators have been considered. It is possible to include the zeroth, 2nd and higher-order derivatives. Of course there is an interaction between the score-operator and the score-argument. For example, the zeroth-order derivative for the likelihood score-space is expected to be less useful than its counter-part in the likelihood-ratio score-space because of its greater sensitivity to acoustic conditions. A principled approach to using derivatives in score-spaces would be useful. Consider the simple case of true class-conditional generative models P1(OIOd and P2(OI02) with respective estimates of the same functional form P1 (0 10d and P2 (0102 ) . Expressing the true models as Taylor series expansions about the parameter estimates 01 and O2 (see [7] for more details, and [3]), Inpi (OIOi ) + (Oi - Oi ) TV' 9i Inpi (OIOi ) 1 A T T A A ( 3) +"2(Oi - Oi ) [V' 9i V' 9i Inpi(OIOi)](Oi - Oi ) + 0 Oi (·) will, V'~i' vec(V' 9i V'~) T . . . ]T Inpi (OIOi) (11) The output from the operator in square brackets is an infinite number of derivatives arranged as a column vector. Wi is also a column vector. The expressions for the two true models can be incorporated into an optimal minimum Bayes error decision A AT AT rule as follows, where 0 [01 , 02]T, W = [wi, WJjT, and b encodes the class priors, Inp1(OIOd -lnp2(OI02) + b a wi[l, V'~1' vec(V' 91 V'~1) T ... ]T Inp1 (OIOdwJ [l , V'~,' vec(V' 92 V'~) T ... ]T Inp2(OI02) + b a T[ T ( T ) T ]T I P1(OIOd b w 1, V' 9' vec V' 9 V' 9 . . . n A + P2(OI02) a w T iplr(o) + b a (12) iplr(o) is a score in the likelihood-ratio score-space formed by an infinite number of derivatives with respect to the parameter estimates O. Therefore, the optimal decision rule can be recovered by constructing a well-trained linear classifier in iplr(o). In this case, the standard SVM margin can be interpreted as the log posterior margin. This justifies the use of the likelihood-ratio score-space and encourages the use of higher-order derivatives. However, most HMMs used in speech recognition are 1st-order Markov processes but speech is a high-order or infinite-order Markov process. Therefore, a linear decision boundary in the likelihood-ratio score-space defined on 1st-order Markov model estimates is unlikely to be sufficient for recovering the optimal decision rule due to model incorrectness. However, powerful non-linear classifiers may be trained in such a likelihood-ratio score-space to try to compensate for model incorrectness and approximate the optimal decision rule. SVMs with nonlinear kernels such as polynomials or Gaussian Radial Basis Functions (GRBFs) may be used. Although gains are expected from incorporating higher-order derivatives into the score-space, the size of the score-space dramatically increases. Therefore, practical systems may truncate the likelihood-ratio score-space after the 1st-order derivatives, and hence use linear approximations to the Taylor series expansions3. However, an example of a 2nd-order derivative is V' J-Ljk (V'~;k Inp(OIO)) , T V' J-L;k (V'~;k Inp(OIO)) ~ - L 'Yjk(t)"2';;k1 (13) t= l For simplicity the component posterior 'Y j k (t) is assumed independent of J-L j k. Once the score-space has been defined, an SVM classifier can be built in the score-space. If standard linear, polynomial or GRBF kernels are used in the score-space, then the space is assumed to have a Euclidean metric tensor. Therefore, the score-space should first be whitened (i.e. decorrelated and scaled) before the standard kernels are applied. Failure to perform such score-space normalisation for a linear kernel in score-space results in a kernel similar to the Plain kernel [5]. This is expected to perform poorly when different dimensions of score-space have different dynamic ranges [2]. Simple scaling has been found to be a reasonable approximation to full whitening and avoids inverting large matrices in [2] (though for classification of single observations rather than sequences, on a different database). The Fisher kernel in [4] uses the Fisher Information matrix to normalise the score-space. This is only an acceptable normalisation for a likelihood score-space under conditions that give a zero expectation in score-space. The appropriate SVM kernel to use between two utterances Oi and OJ in the normalised score-space is therefore the Normalised kernel, kN(Oi, OJ) (where ~s c is the covariance matrix in score-space), (14) 3 Experimental Results The ISOLET speaker-independent isolated letter database [8] was used for evaluation. The data was coded at a 10 msec frame rate with a 25.6 msec windowsize. The data was parameterised into 39-dimensional feature vectors including 12 MFCCs and a log energy term with corresponding delta and acceleration parameters. 240 utterances per letter from isolet{ 1,2,3,4} were used for training and 60 utterances per letter from isolet5 for testing. There was no overlap between the training and test speakers. Two sets of letters were tested, the highly confusible E-set, {B C D E G P T V Z}, and the full 26 letters. The baseline HMM system was well-trained to maximise likelihood. Each letter was modelled by a 10-emitting state left-to-right continuous density HMM with no skips, and silence by a single emitting-state HMM with no skips. Each state output distribution had the same number of Gaussian components with diagonal covariance matrices. The models were tested using a Viterbi recogniser constrained to a silence-letter-silence network. 31t is useful to note that a linear decision boundary, with zero bias, constructed in a single-dimensional likelihood-ratio score-space formed by the zeroth-order derivative operator would, under equal class priors, give the standard minimum Bayes error classifier. The baseline HMMs were used as generative models for SVM kernels. A modified version of SV Mlight Version 3.02 [9] was used to train 1 vI SVM classifiers on each possible class pairing. The sequence length normalisation in Equation 10, and simple scaling for score-space normalisation, were used during training and testing. Linear kernels were used in the normalised score-space, since they gave better performance than GRBFs of variable width and polynomial kernels of degree 2 (including homogeneous, inhomogeneous, and inhomogeneous with zero-mean score-space). The linear kernel did not require parameter-tuning and, in initial experiments, was found to be fairly insensitive to variations in the SVM trade-off parameter C. C was fixed at 100, and biased hyperplanes were permitted. A variety of score-subspaces were examined. The abbreviations rn, v, wand t refer to the score-subspaces \7 J-Ljk Inpi(OIOi), \7veC (I;jk) Inpi(OIOi), \7Wjk Inpi(OIOi) and \7 ajj Inpi(OIOi) respectively. 1 refers to the log likelihood Inpi(OIOi) and r to the log likelihood-ratio In[p2(OI02) /Pl(OIOd]. The binary SVM classification results (and, as a baseline, the binary HMM results) were combined to obtain a single classification for each utterance. This was done using a simple majority voting scheme among the full set of 1v1 binary classifiers (for tied letters, the relevant 1v1 classifiers were inspected and then, if necessary, random selection performed [2]). Table 1: Error-rates for HMM baselines and SVM score-spaces (E-set) Num compo HMM SVM score-space per class min. Bayes majority lik-ratio lik lik per state error voting (stat. sign.) (I-class) (2-class) 1 11.3 11.3 6.9 ~99.8~! 7.6 6.1 2 8.7 8.7 5.0 (98.9%) 6.3 9.3 4 6.7 6.7 5.7 (13.6%) 8.0 23.2 6 7.2 7.2 6.1 (59.5%) 7.8 30.6 Table 1 compares the baseline HMM and SVM classifiers as the complexity of the generative models was varied. Statistical significance confidence levels are given in brackets comparing the baseline HMM and SVM classifiers with the same generative models, where 95% was taken as a significant result (confidence levels were defined by (100 - P), where P was given by McNemar's Test and was the percentage probability that the two classifiers had the same error rates and differences were simply due to random error; for this, a decision by random selection for tied letters was assigned to an 'undecided' class [7]). The baseline HMMs were comparable to reported results on the E-set for different databases [10]. The majority voting scheme gave the same performance as the minimum Bayes error scheme, indicating that majority voting was an acceptable multi-class scheme for the E-set experiments. For the SVMs, each likelihood-ratio score-space was defined using its competing class-conditional generative models and projected into a rnr score-space. Each likelihood (I-class) score-space was defined using only the generative model for the first of its two classes, and projected into a rnl score-space. Each likelihood (2-class) score-space was defined using a generative model for both of its classes, and projected into a rnl score-space (the original Fisher score, which is a projection into its rn score-subspace, was also tested but was found to yield slightly higher error rates). SVMs built using the likelihood-ratio score-space achieved lower error rates than HMM systems, as low as 5.0%. The likelihood (I-class) score-space performed slightly worse than the likelihood-ratio score-space because it contained about half the information and did not contain the log likelihood-ratio. In both cases, the optimum number of components in the generative models was 2 per state, possibly reflecting the gender division within each class. The likelihood (2-class) score-space performed poorly possibly because of wrap-around. However, there was an exception for generative models with 1 component per class per state (in total the models had 2 components per state since they modelled both classes). The 2 components per state did not generally reflect the gender division in the 2-class data, as first supposed, but the class division. A possible explanation is that each Gaussian component modelled a class with bi-modal distribution caused by gender differences. Most of the data modelled did not sit at the peaks of the two Gaussians and was not mapped to the ambiguous zero in score-subspace. Hence there was still sufficient class discrimination in score-space [7]. This task was too small to fully assess possible decorrelation in error structure between HMM and SVM classifiers [6]. Without scaling for score-space normalisation, the error-rate for the likelihood-ratio score-space defined on models with 2 components per state increased from 5.0% to 11.1%. Some likelihood-ratio mr score-spaces were then augmented with 2nd-order derivatives ~ J-tjk (~~jk lnp( 018)). The resulting classifiers showed increases in error rate. The disappointing performance was probably due to the simplicity of the task, the independence assumption between component posteriors and component means, and the effect of noise with so few training scores in such large score-spaces. It is known that some dimensions of feature-space are noisy and degrade classification performance. For this reason, experiments were performed which selected subsets of the likelihood-ratio score-space and then built SVM classifiers in those score-subspaces. First, the score-subspaces were selected by parameter type. Error rates for the resulting classifiers, otherwise identical to the baseline SVMs, are detailed in Table 2. Again, the generative models were class-conditional HMMs with 2 components per state. The log likelihood-ratio was shown to be a powerful discriminating feature4 • Increasing the number of dimensions in score-space allowed more discriminative classifiers. There was more discrimination, or less noise, in the derivatives of the component means than the component variances. As expected in a dynamic task, the derivatives of the transitions were also useful since they contained some duration information. Table 2: Error rates for subspaces of the likelihood-ratio score-space (E-set) score-space error rate, % score-space dimensionality r 8.5 1 v 7.2 1560 m 5.2 1560 mv 5.0 3120 mvt 4.4 3140 wmvtr 4.1 3161 Next, subsets of the mr and wmvtr score-spaces were selected according to dimensions with highest Fisher-ratios [7]. The lowest error rates for the mr and wmvtr score-spaces were respectively 3.7% at 200 dimensions and 3.2% at 500 dimensions (respectively significant at 99.1% and 99.7% confidence levels relative to the best HMM system with 4 components per state). Generally, adding the most discriminative dimensions lowered error-rate until less discriminative dimensions were added. For most binary classifiers, the most discriminative dimension was the log likelihoodratio. As expected for the E-set, the most discriminative dimensions were dependent on initial HMM states. The low-order MFCCs and log energy term were the most important coefficients. Static, delta and acceleration streams were all useful. 4The error rate at 8.5% differed from that for the HMM baseline at 8.7% because of the non-zero bias for the SVMs. The HMM and SVM classifiers were run on the full alphabet. The best HMM classifier, with 4 components per state, gave 3.4% error rate. Computational expense precluded a full optimisation of the SVM classifier. However, generative models with 2 components per state and a wmvtr score-space pruned to 500 dimensions by Fisher-ratios, gave a lower error rate of 2.1% (significant at a 99.0% confidence level relative to the HMM system). Preliminary experiments evaluating sequence length normalisation on the full alphabet and E-set are detailed in [7]. 4 Conclusions In this work, SVMs have been successfully applied to the classification of speech data. The paper has concentrated on the nature of the score-space when handling variable length speech sequences. The standard likelihood score-space of the Fisher kernel has been extended to the likelihood-ratio score-space, and normalisation schemes introduced. The new score-space avoids some of the limitations of the Fisher score-space, and incorporates the class-conditional generative models directly into the SVM classifier. The different score-spaces have been compared on a speakerindependent isolated letter task. The likelihood-ratio score-space out-performed the likelihood score-spaces and HMMs trained to maximise likelihood. Acknowledgements N. Smith would like to thank EPSRC; his CASE sponsor, the Speech Group at IBM U.K. Laboratories; and Thorsten Joachims, University of Dortmund, for BV Mlight. References [1] V. Vapnik. The Nature of Statistical Learning Theory. Springer-Verlag, 1995. [2] N. Smith, M. Gales, and M. Niranjan. Data-dependent kernels in SVM classification of speech patterns. Tech. Report CUED/F-INFENG/TR.387, Cambridge University Eng.Dept., April 2001. [3] K. Tsuda et al. A New Discriminative Kernel from Probabilistic Models. In T.G. Dietterich, S. Becker and Z. Ghahramani, editors Advances in Neural Information Processing Systems 14, MIT Press, 2002. [4] T. Jaakkola and D. Haussler. Exploiting Generative Models in Discriminative Classifiers. In M.S. Kearns, S.A. Solia, and D.A. Cohn, editors, Advances in Neural Information Processing Systems 11. MIT Press, 1999. [5] N. Oliver, B. Scholkopf, and A. Smola. Advances in Large-Margin Classifiers, chapter Natural Regularization from Generative Models. MIT Press, 2000. [6] S. Fine, J. Navratil, and R. Gopinath. A hybrid GMM/ SVM approach to speaker identification. In Proceedings, volume 1, International Conference on Acoustics, Speech, and Signal Processing, May 2001. Utah, USA. [7] N. Smith and M. Gales. Using SVMs to classify variable length speech patterns. Tech. Report CUED/ F-INFENG/ TR.412, Cambridge University Eng.Dept., June 2001. [8] M. Fanty and R. Cole. Spoken Letter Recognition. In R.P. Lippmann, J .E. Moody, and D.S. Touretzky, editors, Neural Information Processing Systems 3, pages 220-226. Morgan Kaufmann Publishers, 1991. [9] T. Joachims. Making Large-Scale SVM Learning Practical. In B. Scholkopf, C. Burges, and A. Smola, editors, Advances in Kernel Methods - Support Vector Learning. MIT-Press, 1999. [10] P.C. Loizou and A.S. Spanias. High-Performance Alphabet Recognition. IEEE Transactions on Speech and Audio Processing, 4(6):430-445, Nov. 1996.
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Learning Discriminative Feature Transforms to Low Dimensions in Low Dimensions Kari Torkkola Motorola Labs, 7700 South River Parkway, MD ML28, Tempe AZ 85284, USA Kari.Torkkola@motorola.com http://members.home.net/torkkola Abstract The marriage of Renyi entropy with Parzen density estimation has been shown to be a viable tool in learning discriminative feature transforms. However, it suffers from computational complexity proportional to the square of the number of samples in the training data. This sets a practical limit to using large databases. We suggest immediate divorce of the two methods and remarriage of Renyi entropy with a semi-parametric density estimation method, such as a Gaussian Mixture Models (GMM). This allows all of the computation to take place in the low dimensional target space, and it reduces computational complexity proportional to square of the number of components in the mixtures. Furthermore, a convenient extension to Hidden Markov Models as commonly used in speech recognition becomes possible. 1 Introduction Feature selection or feature transforms are important aspects of any pattern recognition system. Optimal feature selection coupled with a particular classifier can be done by actually training and evaluating the classifier using all combinations of available features. Obviously this wrapper strategy does not allow learning feature transforms, because all possible transforms cannot be enumerated. Both feature selection and feature transforms can be learned by evaluating some criterion that reflects the “importance” of a feature or a number of features jointly. This is called the filter configuration in feature selection. An optimal criterion for this purpose would naturally reflect the Bayes error rate. Approximations can be used, for example, based on Bhattacharyya bound or on an interclass divergence criterion. These are usually accompanied by a parametric estimation, such as Gaussian, of the densities at hand [6, 12]. The classical Linear Discriminant Analysis (LDA) assumes all classes to be Gaussian with a shared single covariance matrix [5]. Heteroscedastic Discriminant Analysis (HDA) extends this by allowing each of the classes have their own covariances [9]. Maximizing a particular criterion, the joint mutual information (MI) between the features and the class labels [1, 17, 16, 13], can be shown to minimize the lower bound of the classification error [3, 10, 15]. However, MI according to the popular definition of Shannon can be computationally expensive. Evaluation of the joint MI of a number of variables is plausible through histograms, but only for a few variables [17]. As a remedy, Principe et al showed in [4, 11, 10] that using Renyi’s entropy instead of Shannon’s, combined with Parzen density estimation, leads to expressions of mutual information with computational complexity of , where is the number of samples in the training set. This method can be formulated to express the mutual information between continuous variables and discrete class labels in order to learn dimension-reducing feature transforms, both linear [15] and non-linear [14], for pattern recognition. One must note that regarding finding the extrema, both definitions of entropy are equivalent (see [7] pages 118,406, and [8] page 325). This formulation of MI evaluates the effect of each sample to every other sample in the transformed space through the Parzen density estimation kernel. This effect can also called as the “information force”. Thus large/huge databases are hard to use due to the complexity. To remedy this problem, and also to alleviate the difficulties in Parzen density estimation in high-dimensional spaces ( ), we present a formulation combining the mutual information criterion based on Renyi entropy with a semi-parametric density estimation method using Gaussian Mixture Models (GMM). In essence, Parzen density estimation is replaced by GMMs. In order to evaluate the MI, evaluating mutual interactions between mixture components of the GMMs suffices, instead of having to evaluate interactions between all pairs of samples. An approach that maps an output space GMM back to input space and again to output space through the adaptive feature transform is taken. This allows all of the computation to take place in the target low dimensional space. Computational complexity is reduced proportional to the square of the number of components in the mixtures. This paper is structured as follows. An introduction is given to the maximum mutual information (MMI) formulation for discriminative feature transforms using Renyi entropy and Parzen density estimation. We discuss different strategies to reduce its computational complexity, and we present a formulation based on GMMs. Empirical results are presented using a few well known databases, and we conclude by discussing a connection to Hidden Markov Models. 2 MMI for Discriminative Feature Transforms Given a set of training data ,
as samples of a continuous-valued random variable , , and class labels as samples of a discrete-valued random variable , ! "
$# &% ' )( , the objective is to find a transformation (or its parameters * ) to + ,. 0/1 such that + 3254 *,$6 that maximizes 7 8 9 , the mutual information (MI) between transformed data 9 and class labels . The procedure is depicted in Fig. 1. To this end, we need to express 7 as a function of the data set, 7 +
, in a differentiable form. Once that is done, we can perform gradient ascent on 7 as follows *;:=<?>@2A*;:CBED F 7 F * 25*:GBED H I J > F 7 F + F + F * (1) To derive an expression for MI using a non-parametric density estimation method we apply Renyi’s quadratic entropy instead of Shannon’s entropy as described in [10, 15] because of its computational advantages. Estimating the density K + of 9 as a sum of spherical Gaussians each centered at a sample + , the expression of Renyi’s quadratic entropy of 9 is L;M 9 2 N)O P'QSRUT6K + + 2 N)O P'Q RVTEWX H I Y J > H I Z J >\[ + N + Y ] 7 [ + N + Z ] 7 _^` + 2 N)O P'Q H I Y J > H I Z J > [ + Y N + Z '] 7 (2) Above, use is made of the fact that the convolution of two Gaussians is a Gaussian. Thus Renyi’s quadratic entropy can be computed as a sum of local interactions as defined by the kernel, over all pairs of samples. In order to use this convenient property, a measure of mutual information making use of quadratic functions of the densities would be desirable. Between a discrete variable and a continuous variable 9 such a measure has been derived in [10, 15] as follows: 7 8 9 2 I " RVT K + + B I " RVT6K K + + N I " RVT K + K K + + (3) We use for the number of samples in class K , Y for th sample regardless of its class, and Z for the same sample, but emphasizing that it belongs to class K , with index within the class. Expressing densities as their Parzen estimates with kernel width ] results in 7 + $
2 H I J >
I Y J >
I J > [ + Y N + ] 7 B H I J > H I Y J > H I J > [ + Y N + ] 7 N H I J > I Z J > H I Y J > [ + Z N + Y ] 7 (4) Mutual information 7 +
can now be interpreted as an information potential induced by samples of data in different classes. It is now straightforward to derive partial F 7 F + which can accordingly be interpreted as an information force that other samples exert to sample + . The three components of the sum give rise to following three components of the information force: > Samples within the same class attract each other, All samples regardless of class attract each other, and Samples of different classes repel each other. This force, coupled with the latter factor F + F * inside the sum in (1), tends to change the transform in such a way that the samples in transformed space move into the direction of the information force, and thus increase the MI criterion 7 +
. See [15] for details. g(w,x) Class labels: c High-dimensional data: x Mutual Information I(c,y) (=Information potential) Gradient Low dimensional features: y w I ∂ ∂ Figure 1: Learning feature transforms by maximizing the mutual information between class labels and transformed features. Each term in (4) consists of a double sum of Gaussians evaluated using the pairwise distance between the samples. The first component consists of a sum of these interactions within each class, the second of all interactions regardless of class, and the third of a sum of the interactions of each class against all other samples. The bulk of computation consists of evaluating these ' Gaussians, and forming the sums of those. Information force, the gradient of 7 , makes use of the same Gaussians, in addition to pairwise differences of the samples [15]. For large , complexity of is a problem. Thus, the rest of the paper explores possibilities of reducing the computation to make the method applicable to large databases. 3 How to Reduce Computation? In essence, we are trying to learn a transform that minimizes the class density overlap in the output space while trying to drive each class into a singularity. Since kernel density estimate results in a sum of kernels over samples, a divergence measure between the densities necessarily requires operations. The only alternatives to reduce this complexity are either to reduce , or to form simpler density estimates. Two straightforward ways to achieve the former are clustering or random sampling. In this case clustering needs to be performed in the high-dimensional input space, which may be difficult and computationally expensive itself. A transform is then learned to find a representation that discriminates the cluster centers or the random samples belonging to different classes. Details of the densities may be lost, more so with random sampling, but at least this might bring the problem down to a computable level. The latter alternative can be accomplished by a GMM, for example. A GMM is learned in the low-dimensional output space for each class, and now, instead of comparing samples against each other, comparing samples against the components of the GMMs suffices. However, as the parameters of the transform are being learned iteratively, the +
will change at each iteration, and the GMMs need to be estimated again. There is no guarantee that the change to the transform and to the + _
is so small that simple re-estimation based on previous GMMs would suffice. However, this depends on the optimization method used. A further step in reducing computation is to compare GMMs of different classes in the output space against each other, instead of comparing the actual samples. In addition to the inconvenience of re-estimation, we lack now the notion of “mapping”. Nothing is being transformed by 4 from the input space to the output space, such that we could change the transform in order to increase the MI criterion. Although it would be possible now to evaluate the effect of each sample to each mixture component, and the effect of each component to the MI, that is, 2 Y T T , due to the double summing, we will pursue the mapping strategy outlined in the following section. 4 Two GMM Mapping Strategies IO-mapping. If the GMM is available in the high-dimensional input space, those models can be directly mapped into the output space by the transform. Let us call this case the IO-mapping. Writing the density of class K as a GMM with mixture components and Z as their mixture weights we get K
2 I Z J > Z [ N Z Z (5) We consider now only linear transforms. The transformed density in the low-dimensional output space is then simply K + 2 I Z J > Z [ + N Z Z (6) Now, the mutual information in the output space between class labels and the densities as transformed GMMs can be expressed as a function of , and it will be possible to evaluate F 7 F to insert into (1). A great advantage of this strategy is that once the input space GMMs have been created (by the EM-algorithm, for example), the actual training data needs not be touched at all during optimization! This is thus a very viable approach if the GMMs are already available in the high-dimensional input space (see Section 7), or if it is not too expensive or impossible to estimate them using the EM-algorithm. However, this might not be the case. OIO-mapping. An alternative is to construct a GMM model for the training data in the low-dimensional output space. Since getting there requires a transform, the GMM is constructed after having transformed the data using, for example, a random or an informed guess as the transform. Density estimated from the samples in the output space for class K is K + 2 I Z J > Z [ + N Z Z (7) Once the output space GMM is constructed, the same samples are used to construct a GMM in the input space using the same exact assignments of samples to mixture components as the output space GMMs have. Running the EM-algorithm in the input space is now unnecessary since we know which samples belong to which mixture components. Similar strategy has been used to learn GMMs in high dimensional spaces [2]. Let us now use the notation of Eq.(5) to denote this density also in the input space. As a result, we have GMMs in both spaces and a transform mapping between the two. The transform can be learned as in the IO-mapping, by using the equalities Z 2 Z and Z 2 Z . This case will be called OIO-mapping. The biggest advantage is now avoiding to operate in the high-dimensional input space at all, not even the one time in the beginning of the procedure. 5 Learning the Transform through Mapped GMMs We present now the derivation of adaptation equations for a linear transform that apply to either mapping. The first step is to express the MI as a function of the GMM that is constructed in the output space. This GMM is a function of the transform matrix , through the mapping of the input space GMM to the output space GMM. The second step is to compute its gradient F 7 F and to make use of it in the first half of Equation (1). 5.1 Information Potential as a Function of GMMs GMM in the output space for each class is already expressed in (7). We need the following equalities: K + 2 K + , where denotes the class prior, and K + 2 H J > K + . Let us denote the three terms in (3) as H , , and N
. Then we have H 2 I " RVT K + + 2 H I J > RUT WX I !J > [ + N $ _^` 2 H I J > I J > I Z J > Z [ N Z B Z (8) To compact the notation, we change the indexing, and make the substitutions Y
2 Y N , Y 2 Y B , [ C 2 [ Y
Y
, C 2 Y Y [ , where C % ! ( , and is the total number of mixture components, and 2 Y " " C . Now we can write H , , and
in a convenient form. H 2 H I J > )2 H I J > H I J > H I J >
2 H I J > H I J > (9) 5.2 Gradient of the Information Potential As each Gaussian mixture component is now a function of the corresponding input space component and the transform matrix , it is straightforward (albeit tedious) to write the gradient F 7 F . Since each of the three terms in 7 is composed of different sums of [ , we need its gradient as F F [ C 2 F F [ Y Y 2 F F [ Y Y
(10) where the input space GMM parameters are Y 2 Y N and Y
2 Y B with the equalities Y
2 Y and Y
2 Y
. [ expresses the convolution of two mixture components in the output space. As we also have those components in the high-dimensional input space, the gradient expresses how this convolution in the output space changes, as that maps the mixture components to the output space, is being changed. The mutual information measure is defined in terms of these convolutions, and maximizing it tends to find a that (crudely stated) minimizes these convolutions between classes and maximizes them within classes. The desired gradient of the Gaussian with respect to the transform matrix is as follows: F F [ 2 N [ > Y
7.N Y
Y
> Y
Y B Y
Y (11) The total gradient F 7 F can now be obtained simply by replacing [ C in (8) and (9) by the above gradient. In evaluating 7 , the bulk of computation is in evaluating the , the componentwise convolutions. Computational complexity is now . In addition, the F 7 F requires pairwise sums and differences of the mixture parameters in the input space, but these need only be computed once. 6 Empirical Results The first step in evaluating this approach is to compare its performance to the computationally more expensive MMI feature transforms that use Parzen density estimation. To this end, we repeated the pattern recognition experiments of [15] using exactly the same LVQ-classifier. These experiments were done using five publicly available databases that are very different in terms of the amount of data, dimension of data, and the number of training instances. For details of the data sets, please see [15]. OIO-mapping was used with 3-5 diagonal Gaussians per class to learn a dimension-reducing linear transform. Gradient ascent was used for optimization1. Results are presented in Tables 1 - 5. The last column denotes the original dimensionality of the data set. As a figure of the overall performance, the average over all five databases and all reduced dimensions, which ranged from one up to the original dimension minus one, was 69.6% for PCA, 77.8% for the MMI-Parzen combination, and 77.0% for the MMI-GMM combination (30 tests altogether). For LDA this figure cannot be calculated since some databases had a small " and LDA can only produce " NA features. The results are very satisfactory since the best we could hope for is performance equal to the MMI-Parzen combination. Thus a very significant reduction in computation caused only a minor drop in performance with this classifier. 7 Discussion We have presented a method to learn discriminative feature transforms using Maximum Mutual Information as the criterion. Formulating MI using Renyi entropy, and Gaussian 1Example video clips can be viewed at http://members.home.net/torkkola/mmi. Table 1: Accuracy on the Phoneme test data set using LVQ classifier. Output dimenson 1 2 3 4 6 9 20 PCA 7.6 70.0 76.8 81.1 84.2 87.3 90.0 LDA 5.1 66.0 74.7 80.2 82.8 86.0 MMI-Parzen 15.5 68.5 75.2 80.2 82.6 85.3 MMI-GMM 21.4 70.4 76.8 80.2 82.6 87.7 Table 2: Accuracy on the Landsat test data set using LVQ classifier. Output dimension 1 2 3 4 9 15 36 PCA 41.2 81.5 85.8 87.8 89.4 90.3 90.4 LDA 42.5 75.7 86.2 87.2 88.8 90.0 MMI-Parzen 65.1 82.0 86.4 86.2 87.6 89.5 MMI-GMM 65.0 80.4 86.1 88.3 87.4 89.1 Table 3: Accuracy on the Letter test data set using LVQ classifier. Output dimension 1 2 3 4 6 8 16 PCA 4.5 16.0 36.0 53.2 75.2 82.5 92.4 LDA 13.4 38.0 53.1 68.1 80.3 86.3 MMI-Parzen 16.4 50.3 62.8 70.9 82.4 88.6 MMI-GMM 15.7 42.4 48.3 68.5 80.9 86.6 Table 4: Accuracy on the Pipeline data set using LVQ classifier. Output dimension 1 2 3 4 5 7 12 PCA 41.5 88.0 87.8 89.7 96.4 97.2 99.0 LDA 98.4 98.8 MMI-Parzen 99.4 99.1 98.9 99.2 98.9 99.0 MMI-GMM 91.3 98.8 99.1 98.9 99.1 98.7 Table 5: Accuracy on the Pima data set using LVQ classifier. Output dimension 1 2 3 4 5 6 8 PCA 64.4 73.0 75.2 74.1 75.6 74.7 74.7 LDA 65.8 MMI-Parzen 72.0 77.5 78.7 78.5 78.3 78.3 MMI-GMM 73.9 79.7 79.4 77.9 76.7 77.5 Mixture Models as a semi-parametric density estimation method, allows all of the computation to take place in the low-dimensional transform space. Compared to previous formulation using Parzen density estimation, large databases become now a possibility. A convenient extension to Hidden Markov Models (HMM) as commonly used in speech recognition becomes also possible. Given an HMM-based speech recognition system, the state discrimination can be enhanced by learning a linear transform from some highdimensional collection of features to a convenient dimension. Existing HMMs can be converted to these high-dimensional features using so called single-pass retraining (compute all probabilities using current features, but do re-estimation using a the high-dimensional set of features). Now a state-discriminative transform to a lower dimension can be learned using the method presented in this paper. Another round of single-pass retraining then converts existing HMMs to new discriminative features. A further advantage of the method in speech recognition is that the state separation in the transformed output space is measured in terms of the separability of the data represented as Gaussian mixtures, not in terms of the data itself (actual samples). This should be advantageous regarding recognition accuracies since HMMs have the same exact structure. References [1] R. Battiti. Using mutual information for selecting features in supervised neural net learning. Neural Networks, 5(4):537–550, July 1994. [2] Sanjoy Dasgupta. Experiments with random projection. In Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, pages 143–151, Stanford, CA, June30 - July 3 2000. [3] R.M. Fano. Transmission of Information: A Statistical theory of Communications. Wiley, New York, 1961. [4] J.W. Fisher III and J.C. Principe. A methodology for information theoretic feature extraction. In Proc. of IEEE World Congress On Computational Intelligence, pages 1712–1716, Anchorage, Alaska, May 4-9 1998. [5] K. Fukunaga. Introduction to statistical pattern recognition (2nd edition). Academic Press, New York, 1990. [6] Xuan Guorong, Chai Peiqi, and Wu Minhui. Bhattacharyya distance feature selection. In Proceedings of the 13th International Conference on Pattern Recognition, volume 2, pages 195 – 199. IEEE, 25-29 Aug. 1996. [7] J.N. Kapur. Measures of information and their applications. Wiley, New Delhi, India, 1994. [8] J.N. Kapur and H.K. Kesavan. Entropy optimization principles with applications. Academic Press, San Diego, London, 1992. [9] Nagendra Kumar and Andreas G. Andreou. Heteroscedastic discriminant analysis and reduced rank HMMs for improved speech recognition. Speech Communication, 26(4):283–297, 1998. [10] J.C. Principe, J.W. Fisher III, and D. Xu. Information theoretic learning. In Simon Haykin, editor, Unsupervised Adaptive Filtering. Wiley, New York, NY, 2000. [11] J.C. Principe, D. Xu, and J.W. Fisher III. Pose estimation in SAR using an information-theoretic criterion. In Proc. SPIE98, 1998. [12] George Saon and Mukund Padmanabhan. Minimum bayes error feature selection for continuous speech recognition. In Todd K. Leen, Thomas G. Dietterich, and Volker Tresp, editors, Advances in Neural Information Processing Systems 13, pages 800– 806. MIT Press, 2001. [13] Janne Sinkkonen and Samuel Kaski. Clustering based on conditional distributions in an auxiliary space. Neural Computation, 14:217–239, 2002. [14] Kari Torkkola. Nonlinear feature transforms using maximum mutual information. In Proceedings of the IJCNN, pages 2756–2761, Washington DC, USA, July 15-19 2001. [15] Kari Torkkola and William Campbell. Mutual information in learning feature transformations. In Proceedings of the 17th International Conference on Machine Learning, pages 1015–1022, Stanford, CA, USA, June 29 - July 2 2000. [16] N. Vlassis, Y. Motomura, and B. Krose. Supervised dimension reduction of intrinsically low-dimensional data. Neural Computation, 14(1), January 2002. [17] H. Yang and J. Moody. Feature selection based on joint mutual information. In Proceedings of International ICSC Symposium on Advances in Intelligent Data Analysis, Rochester, New York, June 22-25 1999.
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Probabilistic Abstraction Hierarchies Eran Segal Computer Science Dept. Stanford University eran@cs.stanford.edu Daphne Koller Computer Science Dept. Stanford University koller@cs.stanford.edu Dirk Ormoneit Computer Science Dept. Stanford University ormoneit@cs.stanford.edu Abstract Many domains are naturally organized in an abstraction hierarchy or taxonomy, where the instances in “nearby” classes in the taxonomy are similar. In this paper, we provide a general probabilistic framework for clustering data into a set of classes organized as a taxonomy, where each class is associated with a probabilistic model from which the data was generated. The clustering algorithm simultaneously optimizes three things: the assignment of data instances to clusters, the models associated with the clusters, and the structure of the abstraction hierarchy. A unique feature of our approach is that it utilizes global optimization algorithms for both of the last two steps, reducing the sensitivity to noise and the propensity to local maxima that are characteristic of algorithms such as hierarchical agglomerative clustering that only take local steps. We provide a theoretical analysis for our algorithm, showing that it converges to a local maximum of the joint likelihood of model and data. We present experimental results on synthetic data, and on real data in the domains of gene expression and text. 1 Introduction Many domains are naturally associated with a hierarchical taxonomy, in the form of a tree, where instances that are close to each other in the tree are assumed to be more “similar” than instances that are further away. In biological systems, for example, creating a taxonomy of the instances is often one of the first steps in understanding the system. In particular, much of the work on analyzing gene expression data [3] has focused on creating gene hierarchies. Similarly, in text domains, creating a hierarchy of documents is a common task [12, 7]. In many of these applications, the hierarchy is unknown; indeed, discovering the hierarchy is often a key part of the analysis. The standard algorithms applied to the problem typically use an agglomerative bottom-up approach [3] or a divide-and-conquer top-down approach [8]. Although these methods have been shown to be useful in practice, they suffer from several limitations: First, they proceed via a series of local improvements, making them particularly prone to local maxima. Second, at least the bottom-up approaches are typically applied to the raw data; models (if any), are constructed as a post-processing step. Thus, domain knowledge about the type of distribution from which data instances are sampled is rarely used in the formation of the hierarchy. In this paper, we present probabilistic abstraction hierarchies (PAH), a probabilistically principled general framework for learning abstraction hierarchies from data which overcomes these difficulties. We use a Bayesian approach, where the different models correspond to different abstraction hierarchies. The prior is designed to enforce our intuitions about taxonomies: nearby classes have similar data distributions. More specifically, a model in a PAH is a tree, where each node in the tree is associated with a class-specific probabilistic model (CPM). Data is generated only at the leaves of the tree, so that a model basically defines a mixture distribution whose components are the CPMs at the leaves of the tree. The CPMs at the internal nodes are used to define the prior over models: We prefer models where the CPM at a child node is close to the CPM at its parent, relative to some distance function between CPMs. Our framework allows a wide range of notions of distance between models; we essentially require only that the distance function be convex in the parameters of the two CPMs. For example, if a CPM is a Gaussian distribution, we might use a simple squared Euclidean distance between the parameters of the two CPMs. We present a novel algorithm for learning the model parameters and the tree structure in this framework. Our algorithm is based on the structural EM (SEM) approach of [4], but utilizes “global” optimization steps for learning the best possible hierarchy and CPM parameters (see also [5, 13] for similar global optimization steps within SEM). Each step in our procedure is guaranteed to increase the joint probability of model and data, and hence (like SEM) our procedure is guaranteed to converge to a local optimum. Our approach has several advantages. (1) It provides principled probabilistic semantics for hierarchical models. (2) It is model based, which allows us to exploit domain structural knowledge more easily. (3) It utilizes global optimization steps, which tend to avoid local maxima and help make the model less sensitive to noise. (4) The abstraction hierarchy tends to pull the parameters of one model closer to those of nearby ones, which naturally leads to a form of parameter smoothing or shrinkage [12]. We present experiments for PAH on synthetic data and on two real data sets: gene expression and text. Our results show that the PAH approach produces hierarchies that are more robust to noise in the data, and that the learned hierarchies generalize better to test data than those produced by hierarchical agglomerative clustering. 2 Probabilistic Abstraction Hierarchy Let be the domain of some random observation, e.g., the set of possible assignments to a set of features. Our goal is to take a set of instances in , and to cluster them into some set of classes. Standard “flat” clustering approaches — for example, Autoclass [1] or the -means algorithm — are special cases of a generative mixture model. In such models, each data instance belongs to one of the classes, each of which is associated with a different class-specific probabilistic model (CPM). Each data instance is sampled independently by first selecting one of the classes according to a multinomial distribution, and then randomly selecting the data instance itself from the CPM of the chosen class. In standard clustering models, there is no relation between the individual CPMs, which can be arbitrarily different. In this paper, we propose a model where the different classes are related to each other via an abstraction hierarchy, such that classes that are “nearby” in the hierarchy have similar probabilistic models. More precisely, we define: Definition 2.1 A probabilistic abstraction hierarchy (PAH) is a tree with nodes
and undirected edges , such that has exactly leaves . Each node , ! " , is associated with a CPM # , which defines a distribution over ; we use $ to denote #
# . We also have a multinomial distribution over the leaves ; we use % to denote the parameters of this distribution. Our framework does not, in principle, place restrictions on the form of the CPMs; we can use any probabilistic model that defines a probability distribution over . For example, # may be a Bayesian network, in which case its specification would include the parameters, and perhaps also the network structure; in a different setting, # may be a hidden Markov model. In practice, however, the choice of CPMs has ramifications both for the overall hierarchical model and the algorithm. As discussed above, we assume that data is generated only from the leaves of the tree. Thus, we augment with an additional hidden class variable & for each data item, which takes the values denoting the leaf that was chosen to generate this item. Given a PAH , an element ')( , and a value * for & , we define +-,.' *0/2123+-,.&45*6/ %718+-,9':/;#=<>1 , where +-,9&?@*A/;%1 is the multinomial distribution over the leaves and +-,.'/# < 1 is the conditional density of the data item given the CPM at leaf * . The induced g1 g2 M5 g3 g4 M4 g1 g2 g3 g4 M3 g1 g2 g3 g4 M2 g1 g2 g3 g4 M1 g1 g2 g3 g4 f(g2) f(g2) f(g2) f(g2) f(g2) M2 M3 M4 M1 M4 M6 (=M3) M5 (=M4) M1 M2 M5 (=M4) M2 M3 M6 (=M4) M4 M6 (=M3) M1 M3 (a) (b) Figure 1: (a) A PAH with 3 leaves over a 4-dimensional continuous state space, along with a visualization of the Gaussian distribution for the 3rd dimension. (b) Two different weight-preserving transformations for a tree with 4 leaves # # . distribution of ' given , from which the data are generated, is simply +-,.'-/ 21 , where * is summed out from +-,9' * /
21 . As we mentioned, the role of the internal nodes in the tree is to enforce an intuitive interpretation of the model as an abstraction hierarchy, by enforcing similarity between CPMs at nearby leaves. We achieve this goal by defining a prior distribution over abstraction hierarchies that penalizes the distance between neighboring CPMs # and # using a distance function 7,.# # 1 . Note that we do not require that be a distance in the mathematical sense; instead, we only require that it be symmetric (as we chose to use undirected trees), non-negative, and that ,9# # 1 iff # # .1 One obvious choice is to define 7,.# # 1- IDKL ,.# # 1 IDKL ,.# ># 1 , where IDKL ,9#># 1 is the KLdistance between the distributions that # and # define over . This distance measure has the advantage of being applicable to any pair of CPMs over the same space, even if their parameterization is different. Given a definition of , we define the prior over PAHs as +-,.21
, ! ,9# # 11 , where represents the extent to which differences in distances are penalized (larger represents a larger penalty).2 Given a set of data instances " with domain , our goal is to find a PAH that maximizes +-,. /#" 1$ !+-, 21+-," / 21 or equivalently, %'&( +-, 21 )%*&+( +-," / 21 . By maximizing this expression, we are trading off the fit of the mixture model over the leaves to the data " , and the desire to generate a hierarchy in which nearby models are similar. Fig. 1(a) illustrates a typical PAH with Gaussian CPM distributions, where a CPM close to the leaves of the tree is more specialized and thus has fairly peaked distributions. Conversely, CPMs closer to the root of the tree, acting to bridge between their neighbors, are expected to have less peaked distributions and peak only around parts of the distribution which are common to an entire subtree. 3 Learning the Models Our goal in this section is to learn a PAH from a data set " -,/. 0 1,2. 3 0 . This learning task is fairly complex, as many aspects are unknown: the structure of the tree , the CPMs #
# at the nodes of , the parameters % , and the assignment of the instances in " to leaves of . Hence, the likelihood function has multiple local maxima, and no general method exists for finding the global maximum. In this section, we provide an efficient algorithm for finding a locally optimal . 1Two models are considered identical if 4!5 687:9<; =5?>A@CBEDF; =5 >A@HGIB . 2Care must be taken to ensure that ; =KJLB is a proper probability distribution, but this will always be the case for the choice of M we use in this paper. We also note that, if desired, we can modify this prior to incorporate a prior over the parameters of the @:N ’s. To simplify the algorithm, we assume that the structure of the CPMs # # is fixed. This reduces the choice of each # to a pure numerical optimization problem. The general framework of our algorithm extends to cases where we also have to solve the model selection problem for each # , but the computational issues are somewhat different. We first discuss the case of complete data, where for each data instance ,/. 0 ( " , we are given the leaf from which it was generated. For this case, we show how to learn the structure of the tree and the setting of the parameters % and $ . This problem, of constructing a tree over a set of points that is not fixed, is very closely related to the Steiner tree problem [10], virtually all of whose variants are NP-hard. We propose a heuristic approach that decouples the joint optimization problem into two subproblems: optimizing the CPM parameters given the tree structure, and learning a tree structure given a set of CPMs. Somewhat surprisingly, we show that our careful choice of additive prior allows each of these subproblems to be tackled very effectively using global optimization techniques. We begin with the task of learning the CPMs. Thus, assume that we are given both the structure of the tree and the assignment of each data instance ,/. " 0 ( " to one of the leaves, denoted & . " 0 . It remains to find % $ that minimize : %'&( +-, " / 21 %'&( +-,.21 . Substituting the definitions into , we get that D
; = /> !B N !" $# N ; =&%' />-@ N B)( * N,+ -/.01 2 M =@ N43 @ B/5 (1) The first term, involving the multinomial parameters % , separates from the rest, so that the optimization of relative to % reduces to straightforward maximum likelihood estimation. To optimize the CPM parameters, the key property turns out to be the convexity of the function, which holds in a wide variety of choices of CPMs and ; in particular, it holds for the models used in our experiments. The convexity property allows us to find the global minimum of using a simple iterative procedure. In each iteration, we optimize the parameters of one of the # ’s, fixing the parameters of the remaining CPMs # ( 76 ). This procedure is repeated for each of the # ’s in a round robin fashion, until convergence. By the joint convexity of , this iterative procedure is guaranteed to converge to the global minimum of . An examination of (1) shows that the optimization of each CPM # involves only the data cases assigned to # (if is a leaf) and the parameters of the CPMs # that are neighbors of # in the tree, thereby simplifying the computation substantially. We now turn our attention to the second subproblem, of learning the structure of the tree given the learned CPMs. We first consider an empty tree containing only the (unconnected) leaf nodes , and find the optimal parameter settings for each leaf CPM as described above. Note that these CPMs are unrelated, and the parameters of each one are computed independently of other CPMs. Given this initial set of CPMs for the leaf nodes , the algorithm tries to learn a good tree structure relative to these CPMs. The goal is to find the lowest weight tree, subject to the restriction that the tree structure must keep the same set of leaves . Due to the decomposability of %'&( +-, 21 , the penalty of the tree can be measured via the sum of the edge weights ,9# # 1 . This problem is also a variant of the Steiner tree problem. As a heuristic substitute, we follow the lines of [5] and use a minimum spanning tree (MST) algorithm for constructing low-weight trees. At each iteration, the algorithm starts out with a tree over some set of nodes . It takes the leaves of this tree, and constructs an MST over them. Of course, in the resulting tree, some of the # are no longer leaves. This problem is corrected by a transformation that “pushes” a leaf down the tree, duplicating its model; this transformation preserves the weight (score) of the tree. By using only , the algorithm simply “throws away” the entire structure of the previous tree. However, we can also construct new MSTs built from all nodes
of the previous tree. For all nodes for 98 :8 which end up as internal nodes, we perform the same transformation described above. In both cases, this transformation is not unique, as it depends on the order in which the steps are executed; see Fig. 1(b). The algorithm therefore generates an entire pool of candidate trees (from both and ), generated using different random resolutions of ambiguities in the weight-preserving transformation. For each such tree, the CPM learning algorithm is used to find an optimal setting of the parameters. The trees are evaluated relative to our score ( %'&( +-,. / " 1 ), and the highest scoring tree is kept. The tree just constructed has a new set of CPMs, so we can repeat this process. To detect termination, the algorithm also keeps the tree from the previous iteration, and terminates when the score of all trees in the newly constructed pool is lower than the score of the best tree from previous iteration. Finally, we address the fact that the data we have is incomplete, in that the assignments & . " 0 of data instances to classes is not determined. We address the problem of incomplete data using the standard Expectation Maximization (EM) algorithm [2] and the structural EM algorithm [4] which extends EM to the problem of model selection. Starting from an initial model, the algorithm iterates the following two steps: The E-step computes the distribution over the unobserved variables given the observed data and the current model. In our case, the distribution over the unobserved variables is computed by evaluating +-,.& . " 0 2/ ,/. " 0 21 for all 8 " 83/ ")/ . The M-step learns new models that increase the expected log likelihood of the data, relative to the distribution computed in the E-step. In our case, the M-step is precisely the algorithm for complete data described above, but using a soft assignment of data instances to nodes in the tree. The full algorithm is shown in Fig. 2. A simple analysis along the lines of [4] can be used to show that the log-probability %'&( +-,.5/<" 1 increases at every M-step. We therefore obtain the following theorem: Theorem 3.1 The algorithm in Fig. 2 converges to a local maximum of %*&+( +-, /<" 1 . 1. Initialize J D = @
3 5 5 5 3 @ 3 B and the models at the leaves. Randomly initialize . 2. Repeat until convergence: (a) -step: i. Choose an MST over some subset of
3 5 5 5 3 , using M =@ N 3 @ --B as edge weights. ii. Transform the MST so that
3 5 5 5 3 become leaves. (b) -step: For
D 3 5 5 5 3 , compute the posterior probabilities for the indicator variable
. For : =
!D B 9 DF; =
D > 3 JLB F; =&%'
>-@ NB N45 (c) @ -step: Update the CPMs and . Let = 3 B D %'
3
! "
. Then: N 9 D "
=
D B # 9 D $ % & $ ' IE ( ; = = 3 B 3 # >) B ,5 Figure 2: Abstraction Hierarchy Learning Algorithm 4 Experimental Results We focus our experimental results on genomic expression data, although we also provide some results on a text dataset. In gene expression data, the level of mRNA transcript of every gene in the cell is measured simultaneously, using DNA microarray technology. This genomic expression data provides researchers with much insight towards understanding the overall cellular behavior. The most commonly used method for analyzing this data is clustering, a process which identifies clusters of genes that share similar expression patterns (e.g., [3]), and which are therefore also often involved in similar cellular processes. We apply PAH to this data, using CPMs of the form # +* ,, /.102 1 , in which case KL-distance is simply: IDKL ,.# ># 1 35476 89 , 8 8 1 0 , which is simply the sum of squared distances between the means of the corresponding Gaussian components, normalized by their variance. We therefore define 7,.# # 1 IDKL ,9# # 1 . The most popular clustering method for genomic expression data to date is hierarchical agglomerative clustering (HAC) [3], which builds a hierarchy among the genes by iteratively merging the closest genes relative to some distance metric. We use the same distance metric for HAC. (Note that in HAC the metric is used as the distance between data cases whereas in our algorithm it is used as the distance between models.) To perform a direct comparison between PAH and HAC, we often need to obtain a probabilistic model from HAC. To do so, we create CPMs from the genes that HAC assigned to each internal node. In both PAH and HAC, we then assign each gene (in the training set or the test set) to the hierarchy by choosing the best (highest likelihood) CPM among all the nodes in the tree (including internal nodes) and recording the probability +-, /# 1 that this CPM assigns to the gene. Structure Recovery. A good algorithm for learning abstraction hierarchies should recover the true hierarchy as well as possible. To test this, we generated a synthetic data set, and measured the ability of each method to recover the distances between pairs of instances (genes) in the generating model, where distance here is the length of the path between two genes in the hierarchy. We generated the data set by sampling from the leaves of a PAH; to make the data realistic, we sampled from a PAH that we learned from a real gene expression data set. To allow a comparison with HAC, we generated one data instance from each leaf. We generated data for 80 (imaginary) genes and 100 experiments, for a total of 8000 measurements. For robustness, we generated 5 different such data sets and ran PAH and HAC for each data set. We used the correlation and the error between the pairwise distances in the original and the learned tree as measures of similiarity. The correlation was
for PAH, compared to a much worse
for HAC. The average error was
for PAH and
for HAC. These results show that PAH recovers an abstraction hierarchy much better than HAC. Generalization. We next tested the ability of the different methods to generalize to unobserved (test) data, measuring the extent to which each method captures the underlying structure in the data. We ran these tests on the yeast data set of [6]. We selected 953 genes with significant changes in expression, using their full set of 93 experiments. Again, we ran PAH and HAC and evaluated performance using 5 fold cross validation. For PAH we also used different settings for (the coefficient of the penalty term in +-, 21 ), which explores the performance in the range of only fitting the data ( : ) and greatly favoring hierarchies in which nearby models are similar (large ). In both cases, we learned a model using training data, and evaluated the log-likelihood of test instances as described above. The results, summarized in Fig. 3(a), clearly show that PAH generalizes much better to previously unobserved data than HAC and that PAH works best at some tradeoff between fitting the data and generating a hierarchy in which nearby models are similar. Robustness. Our goal in constructing a hierarchy is to extract meaningful biological conclusions from the data. However, data is invariably partial and noisy. If our analysis produces very different results for slightly different training data, the biological conclusions are unlikely to be meaningful. Thus, we want genes that are assigned to nearby nodes in the tree, to be close together also in hierarchies learned from perturbed data sets. We tested robustness to noise by learning a model from the original data set and from perturbed data sets in which we permuted a varying percentage of the expression measuments. We then compared the distances (the path length in the tree) between the nodes assigned to every pair of genes in trees learned from the original data and trees learned from perturbed data sets. The results are shown in Fig. 3(b), demonstrating that PAH preserves the pairwise distances extremely well even when
of the data is perturbed (and performs reasonably well for + permutation), while HAC completely deteriorates when
of the data is permuted. -104 -102 -100 -98 -96 -94 -92 -90 0 2 4 6 8 10 12 14 16 18 20 Lambda Average log probability PAH HAC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 Data permuted (%) Correlation coefficient PAH HAC (a) (b) Figure 3: (a) Generalization to test data (b) Robustness to noise Training set Test set Model p avg. L1 difference avg. L1 difference PAH 90% 5 5 5 5 HAC 5
5 5 5 PAH 80% 5 5 5 5 HAC 5 5 5 5 PAH 70% 5 5 5 5 HAC 5 5 5 5 global variant artificial intellig methodolog vector gradient sigmoid feedforward bayesian network mont carlo mcmc spline nonLinear sample method markov pearl causal causat influenc counterfactu influenc causal causat statist hmm model hidden learn train forward graphic acyclic acyclic graph graph maximum likelihood maximum likelihood algorithm parameter parameter Em maxim featur boltzmann machin (a) (b) Figure 4: (a) Robustness of PAH and HAC to different subsets of training instances. (b) Word hierarchy learned on Cora data. A second important test is robustness to our particular choice of training data: a particular training set reflects only a subset of the experiments that we could have performed. In this experiment, we used the Yeast Compendium data of [9], which measures the expression profiles triggered by specific gene mutations. We selected 450 genes and all 298 arrays, focusing on genes that changed significantly. For each of three values of ranging from to + , we generated ten different training sets by sampling (without replacement) percent of the 450 genes, the rest of which form a test set. We then placed both training and test genes within the hierarchy. For each data set, every pair of genes either appear together in the training set, the test set, or do not appear together (i.e., one appears in the training set and the other in the test set). We compared, for each pair of genes, their distances in training sets in which they appear together and their distances in test sets in which they appear together. The results are summarized in Fig. 4(a). Our results on the training data show that PAH consistently constructs very similar hierarchies, even from very different subsets of the data. By contrast, the hierarchies constructed by HAC are much less consistent. The results on the test data are even more striking. PAH is very consistent about its classification into the hierachy even of test instances — ones not used to construct the hierarchy. In fact, there is no significant difference between its performance on the training data and the test data. By contrast, HAC places test instances in very different configurations in different trees, reducing our confidence in the biological validity of the learned structure. Intuitiveness. To get qualitative insight into the hierarchies produced, we ran PAH on 350 documents from the Probabilistic Methods category in the Cora dataset (cora.whizbang.com) and learned hierarchies among the (stemmed) words. We constructed a vector for each word with an entry for each document whose value is the TFIDFweighted frequency of the word within the document. Fig. 4(b) shows parts of the learned hierarchy, consisting of 441 nodes, where we list high confidence words for each node. PAH organized related words into the same region of the tree. Within each region, many words were arranged in a way which is consistent with our intuitive notion of abstraction. 5 Discussion We presented probabilistic abstraction hierarchies, a general framework for learning abstraction hierarchies from data, which relates different classes in the hierarchy by a tree whose nodes correspond to class-specific probability models (CPMs). We utilize a Bayesian approach, where the prior favors hierarchies in which nearby classes have similar data distributions, by penalizing the distance between neighboring CPMs. A unique feature of PAH is the use of global optimization steps for constructing the hierarchy and for finding the optimal setting of the entire set of parameters. This feature differentiates us from many other approaches that build hierarchies by local improvements of the objective function or approaches that optimize a fixed hierarchy [7]. The global optimization steps help in avoiding local maxima and in reducing sensitivity to noise. Our approach leads naturally to a form of parameter smoothing, and provides much better generalization for test data and robustness to noise than other clustering approaches. In principle, we can use any probabilistic model for the CPM as long as it defines a probability distribution over the state space. We have recently [14] applied this approach to the substantially more complex problem of clustering proteins based on their amino acid sequence using profile HMMs [11]. Acknowledgements. We thank Nir Friedman for useful comments. This work was supported by NSF Grant ACI-0082554 under the NSF ITR program, and by the Sloan Foundation. Eran Segal was also supported by a Stanford Graduate Fellowship (SGF). References [1] P. Cheeseman and J. Stutz. Bayesian Classification (AutoClass): Theory and Results. AAAI Press, 1995. [2] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B 39:1–39, 1977. [3] M. Eisen, P. Spellman, P. Brown, and D. Botstein. Cluster analysis and display of genome-wide expression patterns. PNAS, 95:14863–68, 1998. [4] N. Friedman. The Bayesian structural EM algorithm. In Proc. UAI, 1998. [5] N. Friedman, M. Ninio, I. Pe’er, and T. Pupko. A structural EM algorithm for phylogentic inference. In Proc. RECOMB, 2001. [6] A.P. Gasch et al. Genomic expression program in the response of yeast cells to environmental changes. Mol. Bio. Cell, 11:4241–4257, 2000. [7] T. Hofmann. The cluster-abstraction model: Unsupervised learning of topic hierarchies from text data. In Proc. IJCAI, 1999. [8] T. Hofmann. The cluster-abstraction model: Unsupervised learning of topic hierarchies from text data. In Proc. International Joint Conference on Artificial Intelligence, 1999. [9] T. R. Hughes et al. Functional discovery via a compendium of expression profiles. Cell, 102(1):109–26, 2000. [10] F.K. Hwang, D.S.Richards, and P. Winter. The Steiner Tree Problem. Annals of Discrete Mathematics, Vol. 53, North-Holland, 1992. [11] A. Krogh, M. Brown, S. Mian, K. Sjolander, and D. Haussler. Hidden markov models in computational biology: Applications to protein modeling. Mol. Biology, 235:1501–1531, 1994. [12] A. McCallum, R. Rosenfeld, T. Mitchell, and A. Ng. Improving text classification by shrinkage in a hierarchy of classes. In Proc. ICML, 1998. [13] M. Meila and M.I. Jordan. Learning with mixtures of trees. Machine Learning, 1:1–48, 2000. [14] E. Segal and D. Koller. Probabilistic hierarchical clustering for biological data. In RECOMB, 2002.
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Information-Geometrical Significance of Sparsity in Gallager Codes Toshiyuki Tanaka Department of Electronics and Information Engineering Tokyo Metropolitan University Tokyo 192-0397, Japan tanaka@eei.metro-u.ac.jp Shiro Ikeda Kyushu Institute of Technology & JST Fukuoka 808-0196, Japan shiro@brain.kyutech.ac.jp Shun-ichi Amari RIKEN, Brain Science Institute Saitama 351-0198, Japan amari@brain.riken.go.jp Abstract We report a result of perturbation analysis on decoding error of the belief propagation decoder for Gallager codes. The analysis is based on information geometry, and it shows that the principal term of decoding error at equilibrium comes from the m-embedding curvature of the log-linear submanifold spanned by the estimated pseudoposteriors, one for the full marginal, and K for partial posteriors, each of which takes a single check into account, where K is the number of checks in the Gallager code. It is then shown that the principal error term vanishes when the parity-check matrix of the code is so sparse that there are no two columns with overlap greater than 1. 1 Introduction Recent progress on error-correcting codes has attracted much attention because their decoders, exhibiting performance very close to Shannon's limit, can be implemented as neural networks. Important examples are turbo codes and Gallager codes [1]. It is now well understood that application of belief propagation to the respective graphical representations of the decoding problems for both codes yields practically efficient decoding algorithms which are the same as the existing ones (the turbo decoding [2] and the sum-product decoding [3], respectively). They are, however, not exact but approximate, since the associated graphical representations have loops in both cases. An important problem posed is to quantify the effect that comes from the existence of loops in the underlying graph. The so-called TAP approach [4] in statistical physics is an alternative way to formulate the same decoding algorithm [5]. Since this approach also assumes that the underlying graph is locally loop-free, one is faced with the same problem as above. In this paper, we analyze the properties of the belief propagation decoder to Gallager codes, expecting that better theoretical understanding of the properties of the belief propagation s information vector GT generator matrix t codeword r received vector A parity-check matrix z syndrome vector BSC(σ) BSC(σ ′) Figure 1: Gallager code decoder will help understand the properties and efficiency of belief propagation in general, applied to loopy graphs, as well as those of the TAP approach. We specifically make use of the information geometry [6] and report a result of perturbation analysis on decoding error of the belief propagation decoder. 2 Gallager codes Gallager code is defined by its parity-check matrix A, which has the form A = [C1 | C2], (1) where C1 and C2 are K × M and K × K matrices, both of which are taken to be very sparse. C2 is assumed invertible. We define the generator matrix of the Gallager code to be GT = I C−1 2 C1 (2) where I is the M × M identity matrix. AGT = O mod 2 holds by definition. The whole model of communication with the Gallager code is shown in Fig. 1. An information vector s of length M is encoded into a codeword t = GT s mod 2 of length N ≡K +M. The codeword t is then transmitted over a channel. We assume that the transmission channel is a binary symmetric channel (BSC) with bit-error probability σ. The received vector is then r = t+n mod 2, where n is the noise vector. Decoder tries to find the most probable x satisfying the parity-check equation Ax = z mod 2, (3) where z ≡Ar mod 2 is the syndrome vector. Since At = AGT s = 0 mod 2, we have z = An mod 2. Therefore, the solution x serves as an estimate of the noise vector n. If we are successful in finding the true noise vector n, we can reconstruct, from r, the original codeword t by t = r + n mod 2, and then the information vector s. Since Eq. (3) is underdetermined, one has to take into account the prior knowledge of the noise in order to solve it properly. The decoding problem can be cast into the Bayes framework. In the sequel, we transform expression of a bit from binary (1, 0) to bipolar (−1, 1). The prior for x is p(x) = exp β1 · x −Nψ(β) , ψ(β) ≡log(eβ + e−β), (4) where 1 is an N-dimensional vector whose elements are all 1, i.e., 1 ≡[1, . . . , 1]. β is a parameter which is related with the bit-error probability σ of the transmission channel by σ = 1 2(1 −tanh β). (5) For the sake of analytical tractability, we assume that the syndrome vector z is observed via another BSC channel with bit-error probability σ′ (see Fig. 1). This leads p(z|x) ∝exp h ρ K X r=1 zr Y i∈ (r) xi i , (6) where (r) is the set of all indices of nonzero elements in row r of the parity-check matrix A, i.e., (r) ≡{i | Ari = 1}, and ρ is defined by σ′ = (1/2)(1 −tanh ρ). If we take the limit ρ →+∞, then we recover the conventional situation of observing the syndrome in a deterministic way. In what follows, we consider the case in which ρ is finite, or equivalently, the case with soft parity-check constraints. Since experimental findings suggest that it is usually the case for decoding results of Gallager codes to violate no parity-check constraints [3], we believe that making the parity-check constraints soft does not alter essential properties of the problem. 3 Decoding The posterior distribution of x for given observed syndrome z is derived from the prior p(x) and the conditional p(z | x) by applying the Bayes formula, and the result is p(x|z) ∝exp h c0(x) + ρ K X r=1 cr(x) i , (7) where we let c0(x) ≡β1 · x, cr(x) ≡zr Y i∈ (r) xi (r = 1, . . . , K). (8) The objective of decoder of Gallager codes is to obtain the marginal-posterior-mode (MPM) estimate based on the posterior p(x|z): ˆxi = arg max xi X x\xi p(x|z). (9) The MPM estimation provides the Bayes-optimum decoder minimizing expected bit-error probability of the decoding results. However, the marginalization is in general computationally hard, which renders the decoding problem intractable. An approximate decoding algorithm, originally proposed by Gallager himself [1], is known to be efficient in practice. It has been recently rediscovered by MacKay [3] by application of the belief propagation to the underlying graphical model. Murayama et al. [5] also formulated the same algorithm based on the so-called TAP approach [4]. The decoder implementing the algorithm is called the belief propagation decoder, and is analyzed in the following. We define a family of distributions with a set of parameters ζ = (ζ1, . . . , ζN)T and v = (v1, . . . , vK ): S = n p(x; ζ, v) p(x; ζ, v) = exp ζ · x + v · c(x) −ϕ(ζ, v) o , (10) where c(x) ≡ c1(x), . . . , cK (x) T . The family S includes the factorizable test distribution p0(x; θ0) (= p(x; θ0, 0)), the true posterior p(x|z) (= p(x; β1, ρ1)), and K partial posteriors pr(x; θr) (= p(x; θr, ρer); er ≡(0, . . . , 0, 1 ˆr , 0, . . . , 0)T ). We then define the expectation parameter η(ζ, v) by η(ζ, v) ≡ X x x p(x; ζ, v). (11) The marginalization in Eq. (9) corresponds to evaluating the expectation parameter of the true posterior. We now introduce the equimarginal family M(θ0) ≡ p(x; ζ, v) η(ζ, v) = η(θ0, 0) , (12) and define the marginalization operator 5 as follows: For p ∈S, 5◦p ≡θ0 if p ∈M(θ0). Knowing θ0 = 5◦p is regarded as being equivalent to knowing the expectation parameter of p, since η(θ0, 0) is easily evaluated from θ0; in other words, the marginalization is tractable for distributions belonging to the factorizable model: M0 ≡ p0(x; θ0) ≡p(x; θ0, 0) = exp(θ0 · x −ϕ0(θ0)) (13) The basic assumption of iterative decoding is that the marginalization is also tractable for the models corresponding to constituent decoders with single checks, with factorizable priors: Mr ≡pr(x; θ) ≡p(x; θ, ρer) = exp(θ · x + ρcr(x) −ϕr(θ)) (14) The algorithm of the belief propagation decoder is described in the notation employed here as follows: Initialization: Let t = 0 and θ0 r = β1, r = 1, . . . , K. Horizontal step: Evaluate the marginalization of pr(x; θt r) ∈Mr to produce a guess ζ t r based on the current prior information θt r and the check zr: ζ t r = 5 ◦pr(x; θt r), r = 1, . . . , K, (15) and calculate a net contribution (the ' cavity field' [7]) from the check zr by subtracting the prior information: ξt r = ζ t r −θt r. (16) (It should be noted that (ξt r)i = 0 holds for i ̸∈ (r) as it should be, since the constituent decoder with check zr cannot provide any contribution regarding information of xi, i ̸∈ (r).) Vertical step: Compose the 'leave-one-out' estimates [7] θt+1 r = β1 + X r′̸=r ξt r′, r = 1, . . . , K, (17) and the pseudoposterior θt+1 = β1 + K X r=1 ξt r. (18) Iterate the above steps until convergence is achieved. The desired decoding result η(β1, ρ1) is then approximated by η(θ∗, 0), where θ∗is the convergent value of {θt}. 4 Information-geometrical characterization of equilibrium Assume that the convergence is achieved and let (θ∗, ξ∗ 1, . . . , ξ ∗ K) be the equilibrium values of (θt, ξt 1, . . . , ξt K ). Then, from Eqs. (15) and (16), we have 5 ◦pr(x; θ∗−ξ∗ r ) = θ∗, r = 1, . . . , K. (19) This means that p0(x; θ∗) and pr(x; θ∗−ξ∗ r ), r = 1, . . . , K, are equimarginal, that is, pr(x; θ∗−ξ∗ r ) ∈M(θ∗), r = 1, . . . , K (20) S M0 MK M1 M2 p0(x; θ∗) p1(x; θ∗−ξ∗ 1) pK (x; θ∗−ξ∗ K ) p2(x; θ∗−ξ∗ 2) p(x|z) M(θ ∗) E(θ∗) Figure 2: Geometric structure of belief propagation decoder holds. Another property of the equilibrium is the log-linear relation log p(x|z) −log p0(x; θ∗) = K X r=1 log pr(x; θ∗−ξ∗ r ) −log p0(x; θ∗) + const. (21) or, in the (ζ, v) coordinate, (β1, ρ1) −(θ∗, 0) = K X r=1 (θ∗−ξ∗ r , ρer) −(θ∗, 0) . (22) This means that the true posterior p(x|z) belongs to the 'log-linear submanifold' E(θ∗), the affine subspace in the (ζ, v)-coordinate rooted at (θ∗, 0) and spanned by (−ξ∗ r , ρer), r = 1, . . . , K. These two properties do not imply p(x|z) ∈M(θ∗). In fact, if we were to assume, instead of the log-linear relation (21), the linear relation p(x|z) −p0(x; θ∗) = K X r=1 pr(x; ξ∗ r ) −p0(x; θ∗) , (23) then we would have p(x|z) ∈M(θ∗) and thus η(β1, ρ1) = η(θ∗, 0). This is not the case because of the difference between the linear and log-linear relations. To what degree the log-linear relation deviates from the linear relation determines the decoding error of belief propagation decoder. The structure is best described on the basis of information geometry [6]. Figure 2 illustrates the geometric structure of the belief propagation decoder. It should be noted that the geometrical structure shown here is essentially the same as that for the turbo decoding [8, 9]. 5 Main result Based on the information geometry, we have evaluated decoding error, the difference between the true expectation η(β1, ρ1) and its estimate by the belief propagation decoder η(θ∗, 0), via perturbation analysis. Taking into account the terms up to second order, we have η(β1, ρ1) −η(θ∗, 0) = ρ2 2 X r,s;r̸=s Brsη(θ∗, 0) + O(ρ3), (24) where Brs ≡ ∂ ∂vr − N X k=1 gkk Ak r ∂ ∂θk ∂ ∂vs − N X j=1 g j j A j s ∂ ∂θ j , (25) and Ai r ≡∂ηi(θ∗, 0) ∂vr = Covθ∗,0 xi, cr(x) . (26) {Brs} are the elements of the m-embedding curvature tensor of the manifold E(θ∗) in S. gii ≡1/(1−ηii(θ∗, 0)2) are the diagonal elements of the inverse of the Fisher information of p0(x; θ∗). This is the generalization of the result obtained for the turbo decoding [8]. Explicit calculation gives the following theorem. Theorem 1. The decoding error of belief propagation decoder is given, within the secondorder with respect to ρ, by ηi(β1, ρ1) −ηi(θ∗, 0) = ρ2(1 −η2 i ) " −ηi X r,s r̸=s,i∈ (r)∩ (s) zrzs X j∈( (r)∩ (s))\i (1 −η2 j) × Y k∈( (r)∩ (s))\i, j η2 k Y l∈( (r)− (s))∪( (s)− (r)) ηl + X r,s r̸=s,i∈ (r)− (s) zrzs 1 − Y j∈ (r)∩ (s) η2 j − X j∈ (r)∩ (s) (1 −η2 j) Y k∈( (r)∩ (s))\ j η2 k × Y l∈[( (r)− (s))\i]∪[ (s)− (r)] ηl # + O(ρ3) (27) where ηi ≡ηi(θ∗, 0). From this theorem, it immediately follows that: Corollary 2. If the parity-check matrix A has no two columns with overlap greater than 1, then the principal error term, given in Eq. (27), vanishes. These are the main result of this paper. 6 Discussion The general result given in Eq. (24) shows that the principal error term is not coordinate invariant, since the summation with respect to r and s in the right-hand side of Eq. (24) excludes terms with r = s. This corresponds to the empirical fact that the performance does depend on the design of the code, that is, the choice of the parity-check matrix A. Explicit evaluation of the principal error term, as in Theorem 1, makes it possible to improve the performance of a code, just in the same way as the perturbational approach to improving the naive mean-field approximation [10, 11, 12, 13, 14, 15, 16, 17]. It is believed [3] that Gallager codes have smaller average probability of decoding error if we avoid any two columns of the parity-check matrix A to have overlap greater than 1. An intuitive explanation to this belief is that such avoidance prevents loops with length 4 from appearing in the graphical representation. Since short loops are expected to do harm in proper functioning of belief propagation, their existence may raise the possibility of decoding errors. Our result supports this belief by showing analytically that the principal term of decoding error vanishes when the parity-check matrix of the code is so sparse and prepared with care so that there are no two columns with overlap greater than 1. Loops with length longer than 4 do not contribute to the decoding error at least via the principal term, but they may have effects via higher-order terms. Our analysis presented here can be extended in a straightforward manner to higher-order perturbation analysis in order to quantify these effects. It should be noted that our approach taken in this paper is different from the common approach to analyzing the properties of the belief propagation decoder in the literature, in that we do not consider ensembles of codes. A typical reasoning found in the literature (e.g., [18]) is first to consider an ensemble of random parity-check matrices, to state that the probability (over the ensemble) of containing short loops in the associated graph decreases down to zero as the size of the parity-check matrix tends to infinity, and to assume that the behavior of the belief propagation decoder for codes with longer loops is the same as that of belief propagation for loop-free case. The statistical-mechanical approach to performance analysis of Gallager-type codes [5] also assumes random ensembles. Our analysis, on the other hand, does not assume ensembles but allows, although asymptotically, performance evaluation of the belief propagation decoder to Gallager codes with any single instance of the parity-check matrix with finite size. Acknowledgments The authors would like to thank Dr. Yoshiyuki Kabashima for his helpful suggestions and comments. References [1] R. G. Gallager, Low Density Parity Check Codes, Ph. D. Thesis, Mass. Inst. Tech., 1960. [2] R. J. McEliece, D. J. C. MacKay, and J. Cheng, “Turbo decoding as an instance of Pearl' s `belief propagation' algorithm,” IEEE J. Select. A. Commun., vol. 16, no. 2, pp. 140–152, 1998. [3] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp. 399–431, 1999. [4] D. J. Thouless, P. W. Anderson, and R. G. Palmer, “Solution of `Solvable model of a spin glass' ,” Phil. Mag., vol. 35, no. 3, pp. 593–601, 1977. [5] T. Murayama, Y. Kabashima, D. Saad, and R. Vicente, “Statistical physics of regular low-density parity-check error-correcting codes,” Phys. Rev. E, vol. 62, no. 2, pp. 1577–1591, 2000. [6] S. Amari and H. Nagaoka (Transl. by D. Harada), Methods of Information Geometry, Translations of Mathematical Monographs, vol. 191, American Math. Soc., 2000. [7] Y. Kabashima and D. Saad, “The TAP approach to intensive and extensive connectivity systems,” in M. Opper and D. Saad (eds.), Advanced Mean Field Methods — Theory and Practice, The MIT Press, 2001, pp. 65–84. [8] S. Ikeda, T. Tanaka, and S. Amari, “Information geometrical framework for analyzing belief propagation decoder,” in T. G. Dietterich et al. (eds.), Advances in Neural Information Processing Systems, vol. 14 (this volume), The MIT Press, 2002. [9] S. Ikeda, T. Tanaka, and S. Amari, “Information geometry of turbo codes and low-density paritycheck codes,” submitted to IEEE Trans. Inform. Theory, 2001. [10] H. J. Kappen and F. B. Rodriguez, “Efficient learning in Boltzmann machines using linear response theory,” Neural Computation, vol. 10, no. 5, pp. 1137–1156, 1998. [11] H. J. Kappen and F. B. Rodriguez, “Boltzmann machine learning using mean field theory and linear response correction,” in M. I. Jordan et al. (eds.), Advances in Neural Information Processing Systems, vol. 10, The MIT Press, 1998, pp. 280–286. [12] T. Tanaka, “A theory of mean field approximation,” in M. S. Kearns et al. (eds.), Advances in Neural Information Processing Systems, vol. 11, The MIT Press, 1999, pp. 351–357. [13] T. Tanaka, “Information geometry of mean-field approximation,” Neural Computation, vol. 12, no. 8, pp. 1951–1968, 2000. [14] J. S. Yedidia, “An idiosyncratic journey beyond mean field theory,” in M. Opper and D. Saad (eds.), Advanced Mean Field Methods — Theory and Practice, The MIT Press, 2001, pp. 21–35. [15] H. J. Kappen and W. J. Wiegerinck, “Mean field theory for graphical models,” in M. Opper and D. Saad (eds.), Advanced Mean Field Methods — Theory and Practice, The MIT Press, 2001, pp. 37–49. [16] S. Amari, S. Ikeda, and H. Shimokawa, “Information geometry of α-projection in mean field approximation,” in M. Opper and D. Saad (eds.), Advanced Mean Field Methods — Theory and Practice, The MIT Press, 2001, pp. 241–257. [17] T. Tanaka, “Information geometry of mean-field approximation,” in M. Opper and D. Saad (eds.), Advanced Mean Field Methods — Theory and Practice, The MIT Press, 2001, pp. 259– 273. [18] T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decodeing,” IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 599–618, 2001.
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Adaptive Sparseness Using Jeffreys Prior M´ario A. T. Figueiredo Institute of Telecommunications, and Department of Electrical and Computer Engineering. Instituto Superior T´ecnico 1049-001 Lisboa, Portugal mtf @lx.it.pt Abstract In this paper we introduce a new sparseness inducing prior which does not involve any (hyper)parameters that need to be adjusted or estimated. Although other applications are possible, we focus here on supervised learning problems: regression and classification. Experiments with several publicly available benchmark data sets show that the proposed approach yields state-of-the-art performance. In particular, our method outperforms support vector machines and performs competitively with the best alternative techniques, both in terms of error rates and sparseness, although it involves no tuning or adjusting of sparsenesscontrolling hyper-parameters. 1 Introduction The goal of supervised learning is to infer a functional relation , based on a set of (maybe noisy) training examples
! . Usually, the inputs are vectors, "#%$ &'")(*+&'")(-,.0/2143 5 , . When is continuous (typically 6143 5 ), we are in the context of regression, whereas in classification is of categorical nature (e.g., 718 :9#;:; ). Usually, the structure of < is assumed fixed and the objective is to estimate a vector of parameters = defining it; accordingly we write >?@ = . To achieve good generalization (i.e. to perform well on yet unseen data) it is necessary to control the complexity of the learned function (see [1] - [4], and the many references therein). In Bayesian approaches, complexity is controlled by placing a prior on the function to be learned, i.e., on = . This should not be confused with a generative (informative) Bayesian approach, since it involves no explicit modelling of the joint probability A @ B . A common choice is a zero-mean Gaussian prior, which appears under different names, like ridge regression [5], or weight decay, in the neural learning literature [6]. Gaussian priors are also used in non-parametric contexts, like the Gaussian processes (GP) approach [2], [7], [8], [9], which has roots in earlier spline models [10] and regularized radial basis functions [11]. Very good performance has been reported for methods based on Gaussian priors [8], [9]. Their main disadvantage is that they do not control the structural complexity of the resulting functions. That is, if one of the components of = (say, a weight in a neural network) happens to be irrelevant, a Gaussian prior will not set it exactly to zero, thus This work was partially supported by the Portuguese Foundation for Science and Technology (FCT), Ministry of Science and Technology, under project POSI/33143/SRI/2000. pruning that parameter, but to some small value. Sparse estimates (i.e., in which irrelevant parameters are set exactly to zero) are desirable because (in addition to other learning-theoretic reasons [4]) they correspond to a structural simplification of the estimated function. Using Laplacian priors (equivalently, -penalized regularization)is known to promote sparseness [12] - [15]. Support vector machines (SVM) take a non-Bayesian approach to the goal of sparseness [2], [4]. Interestingly, however, it can be shown that the SVM and -penalized regression are closely related [13]. Both in approaches based on Laplacian priors and in SVMs, there are hyper-parameters which control the degree of sparseness of the obtained estimates. These are commonly adjusted using cross-validation methods which do not optimally utilize the available data, and are time consuming. We propose an alternative approach which involves no hyperparameters. The key steps of our proposal are: (i) a hierarchical Bayes interpretation of the Laplacian prior as a normal/independent distribution (as used in robust regression [16]); (ii) a Jeffreys’ non-informative second-level hyper-prior (in the same spirit as [17]) which expresses scale-invariance and, more importantly, is parameter-free [18]; (iii) a simple expectation-maximization (EM) algorithm which yields a maximum a posteriori (MAP) estimate of = (and of the observation noise variance, in the case of regression). Our method is related to the automatic relevance determination (ARD) concept [7], [19], which underlies the recently proposed relevance vector machine (RVM) [20], [21]. The RVM exhibits state-of-the-art performance, beating SVMs both in terms of accuracy and sparseness [20], [21]. However, we do not resort to a type-II maximum likelihood approximation [18] (as in ARD and RVM); rather, our modelling assumptions lead to a marginal a posteriori probability function on = whose mode is located by a very simple EM algorithm. Like the RVM, but unlike the SVM, our classifier produces probabilistic outputs. Experimental evaluation of the proposed method, both with synthetic and real data, shows that it performs competitively with (often better than) GP-based methods, RVM, and SVM. 2 Regression We consider functions of the type @ = = / , i.e., that are linear with respect to = (whose dimensionality we will denote by ). This includes: (i) classical linear regression, $; & & , .0/ ; (ii) nonlinear regression via a set of basis functions, ) $ -.0/ ; (iii) kernel regression, $; )@ @ -.0/ , where )@ is some (symmetric) kernel function [2] (as in SVM and RVM regression), not necessarily verifying Mercer’s condition. We follow the standard assumption that " ) " =
" , for ; , where $ . is a set of independent zero-mean Gaussian variables with variance . With $ .0/ , the likelihood function is then A = = , where is the "! design matrix which depends on the " s and on the adopted function representation, and #$ % '& denotes a Gaussian density of mean % and covariance & , evaluated at # . With a zero-mean Gaussian prior with covariance ( , A = ( ) = * ( , the posterior A = is still Gaussian with mean and mode at + = (-, . / , / When ( is proportional to identity, say ( 0/ 1 , this is called ridge regression [5]. With a Laplacian prior for = , A = 2 43 " A 5 " 2 , with A 5 " 2 76 98;:=< :9
2> 5 " , the posterior A = is not Gaussian. The maximum a posteriori (MAP) estimate is given by + = @?BACEDGFIH KJ; = 9L J .M 2>J = J : (1) where J;#J is the Euclidean ( ) norm, and J;#J " " is the norm. In linear regression this is called the LASSO (least absolute shrinkage and selection operator) [14]. The main effect of the penalty is that some of the components of + = may be exactly zero. If is an orthogonal matrix, (1) can be solved separately for each 5 " , leading to the soft threshold estimation rule, widely used in wavelet-based signal/image denoising [22]. Let us consider an alternative model: let each 5 " have a zero-mean Gaussian prior A 5'" " 5'" * "- , with its own variance " (like in ARD and RVM). Now, rather than adopting a type-II maximum likelihood criterion (as in ARD and RVM), let us consider hyper-priors for the " s and integrate them out. Assuming exponential hyper-priors A " 2 M 81:=< 9 " BMB (for "
* , because these are variances) we obtain A 5 " A 5 " "- A " " M 8;:=< :9 G 5'" This shows that the Laplacian prior is equivalent to a 2-level hierachical-Bayes model: zero-mean Gaussian priors with independent exponentially distributed variances. This decomposition has been exploited in robust least absolute deviation (LAD) regression [16]. The hierarchical decomposition of the Laplacian prior allows using the EM algorithm to implement the LASSO criterion in (1) by simply regarding $ . as hidden/missing data. In fact, the complete log-posterior (with a flat prior for , and where diag , , ), C A = 9 C 9 J' 9 = J 9 = / = (2) is easy to maximize with respect to = and . The E-step reduces to the computation of the conditional expectation of , given current (at iteration ! ) estimates " $# %& and + = #'%& . This leads to ( #'%& *) $ 1 " #'%& + = #'%& . + diag + 5 ( #'%& , + 5 ( # %& , . The M-step is then defined by the two following update equations: " ,#'% & ; J' 9 + = #'%& J (3) and + = #'% & 2 " ,#'% & ( #'%& . / , / (4) This EM algorithm is not the most efficient way to solve (1); see, e.g., the methods proposed in [23], [14]. Our main goal is to open the way to the adoption of different hyper-priors. One question remains: how to adjust , which controls the degree of sparseness of the estimates? Our proposal is to remove from the model, by replacing the exponential hyperprior by a non-informative Jeffreys hyper-prior: A +" . / , " . This prior expresses ignorance with respect to scale (see [17], [18]) and, most importantly, it is parameter-free. Of course this is no longer equivalent to a Laplacian prior on = , but to some other prior. As will be shown experimentally, this prior strongly induces sparseness and yields state-of-the-art performance. Computationally, this choice leads to a minor modification of the EM algorithm described above: matrix ( #'%& is now given by ( #'%& diag + 5 !( #'%& , + 5 +( #'%& , . Since several of the + 5 " s may go to zero, it is not convenient to deal with ( #'%& . However, we can re-write the M-step as + = # % & 10 #'%& " ,#'% & 20 #'%& / 30 # %& , 0 #'%& / where 0 #'%& diag + 5 ( #'%& + 5 ( # %& , thus avoiding the inversion of the elements of + = #'%& . Moreover, it is not necessary to invert the matrix, but simply to solve the corresponding linear system, whose dimension is only the number of non-zero elements in 0 #'%& . 3 Regression experiments Our first example illustrates the use of the proposed method for variable selection in standard linear regression. Consider a sequence of 20 true = s, having from 1 to 20 non-zero components (out of 20): from $ * * *. to $ . . For each = , we obtain 100 random ( *G!-M * ) design matrices, following the procedure in [14], and for each of these, we obtain data points with unit noise variance. Fig. 1 (a) shows the mean number of estimated non-zero components, as a function of the true number. Our method exhibits a very good ability to find the correct number of nonzero components in = , in an adaptive manner. B 5 10 15 20 25 5 10 15 20 25 True # of nonzero parameters Estim. # of nonzero parameters - 8 - 6 - 4 - 2 0 2 4 6 8 - 0.2 0 0.2 0.4 0.6 0.8 1 1.2
Figure 1: (a) Mean number of nonzero components in + = versus the number of nonzero components in = (the dotted line is the identity). (b) Kernel regression. Dotted line: true function F H )& & . Dots: 50 noisy observations ( 4* ; ). Solid line: the estimated function. Circles: data points corresponding to the non-zero parameters. We now consider two of the experimental setups of [14]: = $ ;: * * MB* * *. , with , and =
$ * * * * * * * . , with M . In both cases, M * , and the design matrices are generated as in [14]. In table 3, we compare the relative modelling error ( )1) $ J; + = 9- = J . ) improvement (with respect to the least squares solution) of our method and of several methods studied in [14]. Our method performs comparably with the best method for each case, although it involves no tuning or adjustment of parameters, and is computationally faster. Table 1: Relative ( ) improvement in modeling error of several mehods. Method =
= Proposed method M LASSO (CV) ; LASSO (GCV) * Subset selection ; We now study the performance of our method in kernel regression, using Gaussian kernels, i.e., @ " 81:=< :9"J 9 " J B M ! . We begin by considering the synthetic example studied in [20] and [21], where the true function is F H & +& (see Fig. 1 (b)). To compare our results to the RVM and the variational RVM (VRVM), we ran the algorithm on 25 generations of the noisy data. The results are summarized in Table 2 (which also includes the SVM results from [20]). Finally, we have also applied our method to the wellknown Boston housing data-set (20 random partitions of the full data-set into 481 training samples and 25 test samples); Table 2 shows the results, again versus SVM, RVM, and VRVM regression (as reported in [20]). In these tests, our method performs better than RVM, VRVM, and SVM regression, although it doesn’t require any tuning. Table 2: Mean root squared errors and mean number of kernels for the “ F H & & ” function and the Boston housing examples. “ F H& & ” function Boston housing Method MSE No. kernels New method * * * SVM * * ; M * RVM * * VRVM * * Method MSE No. kernels New method M SVM ; * M M M RVM ; * ; ; ; VRVM ; * * 4 Classification In classification the formulation is somewhat more complicated, with the standard approach being generalized linear models [24]. For a two-class problem ( 1 9#; ; ), the probability that an observation belongs to, say class 1, is given by a nonlinear function 3 5 $ * ;. (called the link), ) ; = / ) , where can have one of the forms referred in the first paragraph of Section 2 (linear, nonlinear, kernel). Although the most common choice for is the logistic function, ; 8;:=< 9 , , in this paper, we adopt the probit model , where
, & * ; :& (5) the standard Gaussian cumulative distribution function (cdf). The probit model has a simple interpretation in terms of hidden variables [25], which we will exploit. Consider a hidden variable = / , where A * ; . Then, if the classification rule is > ; if * , and 9#; if * , we obtain the probit model: ) >;K = / *: = / Given training data ) + , consider the corresponding vector of hidden/missing variables $ +:.0/ . If we had , we would have a simple linear regression likelihood A = = . This fact suggests using the EM algorithm to estimate = , by treating as missing data. To promote sparseness, we will adopt the same hierarchical prior on = that we have used for regression: A 5 " " . 5 " * "- and A "- ; " (the Jeffreys prior). The complete log posterior (with the hidden vectors and ) is C A = 29 = / / = 9M = / / 9 = / = (6) which is similar to (2), except for the noise variance which is not needed here, and for the fact that now is missing. The expected value of is similar to the regression case; accordingly we define the same diagonal matrix 0 #'%& diag + 5 !( # %& + 5 ( #'%& . In addition, we also need ) $ + = #'%& . (notice that the complete log-posterior is linear with respect to ), which can be expressed in closed form, for each " , as ")( #'%& +) $ +" + = # %& . + = / #'%& " + = / #'%& "- 1 * ;+ ; 9 9 + = / #'%& "- if " 2; + = / #'%& " 9 + = / # %& )" * ; 9 + = / #'%& "- if " 29#; . (7) These expressions are easily derived after noticing that " is (conditionally) Gaussian with mean + = / #'%& "* , but left-truncated at zero if " ; , and right-truncated at zero if " 9#; . With #'%& $ ( #'%& ( # %& . / , the M-step is similar to the regression case, + = #'% & 10 #'%& 20 #'%& / 30 # %& , 0 #'%& / #'%& with #'%& playing the role of observed data. 5 Classification experiments In all the experiments we use kernel classifiers, with Gaussian kernels, i.e., )@" > 8;:=< 9"J 98 " J B M : where is a parameter that controls the kernel width. Our first experiment is mainly illustrative and uses Ripley’s synthetic data1; the optimal error rate for this problems is [3]. Table 3 shows the average test set error (on 1000 test samples) and the final number of kernels, for 20 classifiers learned from 20 random subsets of size 100 from the original 250 training samples. For comparison, we also include results (from [20]) for RVM, variational RVM (VRVM), and SVM classifiers. On this data set, our method performs competitively with RVM and VRVM and much better than SVM (specially in terms of sparseness). To allow the comparisons, we chose @* , as in [20]. Table 3 also reports the numbers of errors achieved by the proposed method and by several state-of-the-art techniques on three well-known benchmark problems: the Pima Indians diabetes2, the Leptograpsus crabs2, and the Wisconsin breast cancer 3 (WBC). For the WBC, we report average results over 30 random partitions (300/269 training/testing, as in [26]). All the inputs are normalized to zero mean and unit variance, and the kernel width was set to , for the Pima and crabs problems, and to ; M for the WBC. On the Pima and crabs data sets, our algorithm outperforms all the other techniques. On the WBC data set, our method performs nearly as well as the best available alternative. The running time of our learning algorithm (in MATLAB, on a PIII-800MHz) is less than 1 second for crabs, and about 2 seconds for the Pima and WBC problems. Finally, notice that the classifiers obtained with our algorithm are much sparser than the SVM classifiers. Table 3: Numbers of test set errors for the four data sets studied (see text for details). The numbers in square brackets in the “method” column indicate the bibliographic reference from which the results are quoted. The numbers in parentheses indicate the (mean) number of kernels used by the classifiers (when available). Method Ripley’s Pima Crabs WBC Proposed method 94 (4.8) 61 (6) 0 (5) 8.5 (5) SVM [20] 106 (38) 64 (110) N/A N/A RVM [20] 93 (4) 65 (4) N/A N/A VRVM [20] 92 (4) 65 (4) N/A N/A SVM [26] N/A 64 4 9 Neural network [9] N/A 75 3 N/A Logistic regression [9] N/A 66 4 N/A Linear discriminant [26] N/A 67 3 19 Gaussian process [9], [26] N/A 68, 67 3 8 1Available (divided into training/test sets) at: http://www.stats.ox.ac.uk/pub/PRNN/ 2Available at http://www.stats.ox.ac.uk/pub/PRNN/ 3Available at: http://www.ics.uci.edu/ mlearn/MLSummary.html 6 Concluding remarks We have introduced a new sparseness inducing prior related to the Laplacian prior. Its main feature is the absence of any hyper-parameters to be adjusted or estimated. Experiments with several publicly available benchmark data sets, both for regression and classification, have shown state-of-the-art performance. In particular, our approach outperforms support vector machines and Gaussian process classifiers both in terms of error rate and sparseness, although it involves no tuning or adjusting of sparseness-controlling hyper-parameters. Future research includes testing on large-scale problems, like handwritten digit classification. One of the weak points of our approach, when used with kernel-based methods, is the need to solve a linear system in the M-step (of dimension equal to the number of training points) whose computational requirements make it impractical to use with very large training data sets. This issue is of current interest to researchers in kernel-based methods (e.g., [27]), and we also intend to focus on it. References [1] V. Cherkassky and F. Mulier, Learning from Data: Concepts, Theory, and Methods. New York: Wiley, 1998. [2] N. Cristianini and J. Shawe-Taylor, Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, 2000. [3] B. Ripley, Pattern Recognition and Neural Networks. Cambridge University Press, 1996. [4] V. Vapnik, Statistical Learning Theory. New York: John Wiley, 1998. [5] A. Hoerl and R. Kennard, “Ridge regression: Biased estimation for nonorthogonal problems,” Technometrics, vol. 12, pp. 55–67, 1970. [6] C. Bishop, Neural Networks for Pattern Recognition. Oxford University Press, 1995. [7] R. Neal, Bayesian Learning for Neural Networks. New York: Springer Verlag, 1996. [8] C. Williams, “Prediction with Gaussian processes: from linear regression to linear prediction and beyond,” in Learning and Inference in Graphical Models, Kluwer, 1998. [9] C. Williams and D. Barber, “Bayesian classification with Gaussian priors,” IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 20, no. 12, pp. 1342–1351, 1998. [10] G. Kimeldorf and G. Wahba, “A correspondence between Bayesian estimation of stochastic processes and smoothing by splines,” Annals of Mathematical Statistics, vol. 41, pp. 495–502, 1990. [11] T. Poggio and F. Girosi, “Networks for approximation and learning,” Proceedings of the IEEE, vol. 78, pp. 1481–1497, 1990. [12] S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Journal of Scientific Computation, vol. 20, no. 1, pp. 33–61, 1998. [13] F. Girosi, “An equivalence between sparse approximation and support vector machines,” Neural Computation, vol. 10, pp. 1445–1480, 1998. [14] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society (B), vol. 58, 1996. [15] P. Williams, “Bayesian regularization and pruning using a Laplace prior,” Neural Computation, vol. 7, pp. 117–143, 1995. [16] K. Lange and J. Sinsheimer, “Normal/independent distributions and their applications in robust regression,” Journal of Computational and Graphical Statistics, vol. 2, pp. 175–198, 1993. [17] M. Figueiredo and R. Nowak, “Wavelet-based image estimation: an empirical Bayes approach using Jeffreys’ noninformative prior,” IEEE Transactions on Image Processing, vol. 10, pp. 1322-1331, 2001. [18] J. Berger, Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, 1980. [19] D. MacKay, “Bayesian non-linear modelling for the 1993 energy prediction competition,” in Maximum Entropy and Bayesian Methods, G. Heidbreder, ed., pp. 221–234, Kluwer, 1996. [20] C. Bishop and M. Tipping, “Variational relevance vector machines,” in Proceedings of the 16th Conference in Uncertainty in Artificial Intelligence, pp. 46–53, Morgan Kaufmann, 2000. [21] M. Tipping, “The relevance vector machine,” in Advances in Neural Information Processing Systems – NIPS 12 (S. Solla, T. Leen, and K.-R. M¨uller, eds.), pp. 652–658, MIT Press, 2000. [22] D. L. Donoho and I. M. Johnstone, “Ideal adaptation via wavelet shrinkage,” Biometrika, vol. 81, pp. 425–455, 1994. [23] M. Osborne, B. Presnell, and B. Turlach, “A new approach to variable selection in least squares problems,” IMA Journal of Numerical Analysis, vol. 20, pp. 389–404, 2000. [24] P. McCullagh and J. Nelder, Generalized Linear Models. London: Chapman and Hall, 1989. [25] J. Albert and S. Chib, “Bayesian analysis of binary and polychotomous response data,” Journal of the American Statistical Association, vol. 88, pp. 669–679, 1993. [26] M. Seeger, “Bayesian model selection for support vector machines, Gaussian processes and other kernel classifiers,” in Advances in Neural Information Processing – NIPS 12 (S. Solla, T. Leen, and K.-R. M¨uller, eds.), pp. 603–609, MIT Press, 2000. [27] C. Williams and M. Seeger, “Using the Nystrom method to speedup kernel machines,” in NIPS 13, MIT Press, 2001.
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An Efficient Clustering Algorithm Using Stochastic Association Model and Its Implementation Using Nanostructures Takashi Morie, Tomohiro Matsuura, Makoto Nagata, and Atsushi Iwata Graduate School of Advanced Sciences of Matter, Hiroshima University Higashi-hiroshima, 739-8526 Japan. http://www.dsl.hiroshima-u.ac.jp morie@dsl.hiroshima-u.ac.jp Abstract This paper describes a clustering algorithm for vector quantizers using a “stochastic association model”. It offers a new simple and powerful softmax adaptation rule. The adaptation process is the same as the on-line K-means clustering method except for adding random fluctuation in the distortion error evaluation process. Simulation results demonstrate that the new algorithm can achieve efficient adaptation as high as the “neural gas” algorithm, which is reported as one of the most efficient clustering methods. It is a key to add uncorrelated random fluctuation in the similarity evaluation process for each reference vector. For hardware implementation of this process, we propose a nanostructure, whose operation is described by a single-electron circuit. It positively uses fluctuation in quantum mechanical tunneling processes. 1 Introduction Vector quantization (VQ) techniques are used in a wide range of applications, including speech and image processing, data compression. VQ techniques encode a data manifold using only a finite set of reference vectors
. A data vector is represented by the best-matching or “winning” reference vector , which minimizes the average distortion error: ! #"$ &%(' )+* -, (1) where ' .) is the probability distribution of data vectors over manifold . Various clustering algorithms to obtain the best reference vectors have been reported. Here, we treat on-line training, in which the data point distribution is not given a priori, but instead a stochastic sequence of incoming sample data points drives the adaptation procedure. The straightforward approach is the well-known on-line K-means clustering algorithm, in which only the nearest reference vector to the sample vector is adjusted; / 0132450 6 7 .8+9"$0:& (2) where, 2 is the step size and 50 is the Kronecker delta. However, this simple clustering algorithm is often stuck in a local minimum. To avoid this difficulty, a common approach is to introduce a “soft-max” adaptation rule that not only adjusts the “winning” reference vector but affects other reference vectors depending on their proximity to . The maximum-entropy (ME) algorithm [1] adjusts all reference vectors 0 depending on the Euclidean distance to ; / 0 324
1
.8+9"$ 0 & (3) where parameter defines the proximity. The Kohonen’s self-organization map (SOM) algorithm [2] is another well-known model; / 01324 . 8+9"$0 (4) In this model, every reference vector is assigned to a site of a lattice. Each time a sample vector is presented, not only the “winning” reference vector is adjusted but also the reference vectors assigned to the lattice sites adjacent to the winner are updated according to function , which is typically chosen to be a Gaussian: !
" %
(5) where # is a parameter that defines the proximity. The neural-gas (NG) clustering algorithm [3] is a powerful soft-max adaptation rule, in which all reference vectors are adjusted depending on the “neighborhood ranking”; / 0)2 $% '& 0 . ++96 .7 8+ " 0 (6) where & 0 . +# is the ranking, which depends on and the whole set . The function % (& 0 is typically as follows: )% (& 7 +* " % (7) where parameter , defines the proximity. This algorithm exhibits faster convergence to smaller distortion errors, however consumes higher computational power especially for sorting. An efficient version of the NG clustering that adjusts only several reference vectors having upper ranking was also proposed [4]. In the next section, we propose a new efficient soft-max adaptation algorithm. It employs the stochastic association model that we have proposed related to single-electron circuits [5], [6]. In Sec. 3, it is demonstrated from simulation results that this new clustering algorithm is as powerful as the other algorithms. In Sec. 4, we propose a nanostructure based on a single-electron circuit for implementing the stochastic association model. 2 Stochastic association algorithm A usual associative memory is defined as a system that deterministically extracts the vector most similar to the input vector from the stored reference vectors. This just corresponds to the process choosing the winning reference vector for a certain data vector in all conventional clustering algorithms. In our stochastic association (SA) model, the association probability depends on the similarity between the input and the reference vectors. The SA algorithm extracts not only the reference vector most similar to the input but also other similar reference vectors with the probability depending on the similarity. In the SA algorithm, stochastic fluctuation is added in the evaluation process of distortion error 0 between data vector and reference vector 0 . We propose this algorithm inspired Di wi Dn wn Ri Rn rn Distance Pr A (r ) n i ϕ(r −D ) n n Figure 1: Probability distribution in evaluation of the distortion error between the data vector and each reference vector. by the quantum mechanical property of single-electron circuits as described in Sec. 4, and we expect that such fluctuation helps to avoid getting stuck in local minima of . The distortion error 0 can be the squared Euclidean distance "# 0 % or the Manhattan distance #" 0 . The evaluation result is represented by 0 0 (8) where is a random variable with probability distribution function 6 . Therefore, the evaluation result 0 is also considered as a random variable. The probability that 0 has value 0 is represented by
0) 04 0 " 0: (9) The winning reference vector # is determined by - 0
0 (10) The probability that reference vector becomes the winner when has value for a certain data vector is given by the product of " - and the probability that 0! #")%$ '& as shown in Fig. 1. Therefore, the probability that becomes the winner is given by integrating it with ;
&( *) ) *+ , - . , " - 0/ 0#1 0 2 0 . , (11) 2 0 . , 43 ) 576 . " 0:(*8 (12) If the winning reference vector is updated as expressed by eq. (2), the SA model can provide a new soft-max adaptation rule. Figure 2 shows an architecture for clustering processing using the SA model. The distortion error between the input vector and each stored reference vector is evaluated in parallel with stochastic fluctuation. The winner-take-all circuit deterministically extracts the winner, and the winning reference vector is only updated with a constant value. As in the K-means algorithm, only one reference vector is adjusted for each adaptation step and the update value for the selected reference vector is independent of similarity or proximity. However, unlike the K-means algorithm, the adjusted vector is not always the most similar reference vector, and sometimes other similar vectors are adjusted. The total adjusting tendency in the SA algorithm seems similar to the NG or ME algorithm because the probability of reference vector selection is determined by the neighborhood ranking and the distances between each reference vector and a given data vector. Reference vectors distortion error evaluation with stochastic fluctuation Input vector v Winner-Take-All update only one vector wi wc ξ Figure 2: Architecture for clustering processing using the SA model. t=0 tmax = 5000 tmax = 50000 (a) SA (b) ME Figure 3: Test problem and clustering results by SA and ME algorithms. Data samples uniformly distribute in square regions, and points represent reference vectors. Both algorithms use the same initial state. 3 Simulation results In order to test the performance of the SA algorithm in minimizing the distortion error and to compare it with the other soft-max approaches, we performed the same simulation of model clustering described by Ref. [3]. The data clusters are of square shape within a twodimensional input space as shown in Fig. 3. In the simulation, the number of clusters was 15, and that of reference vectors was 60. We averaged the results of 50 simulation runs for each of which not only the initialization of the reference vectors were chosen randomly but also the 15 clusters were placed randomly. The SA algorithm in this simulation used the squared Euclidean distance as a distortion error 0 and the normal distribution as the probability distribution of the stochastic fluctuation; .67 % 3
+" % % (13) 50000 Total number of adaptation steps tmax 2 1 0 Neural-gas (NG) Stochastic association (SA) Maximum-entropy(ME) SOM Performance α algorithm parameter initial final
ME 1 10000 SOM 2 0.01 NG 10 0.01 SA 0.2 0.0001 All 0.5 0.005 Figure 4: Clustering performance of SA algorithm comparing with other clustering methods. The optimized parameters used in the simulation are also shown. Figure 3 shows an example of clustering by the SA algorithm compared with that by the ME algorithm. The result of the SA algorithm demonstrates nearly perfect clustering for 8 . In contrast, the clustering result by the ME algorithm is not so good although the parameters used were optimized. Here, all the clustering algorithms including the SA algorithm use an annealing procedure to escape local minima. The parameters were gradually reduced during adaptation: .8+7 0 "! 0 # " #%$'&)( *%+ # ,) 9(2 (14) where 8 is the total number of adaptation steps. The values optimized by numerous preliminary simulations are shown in Fig. 4, which were used in the simulation described here. In order to compare the performance of the algorithms, we used a performance measure , . 8 7" /10 ! / , where 2/ is the minimal distortion error in this problem. The relationships between 8 and , for the four algorithms are shown in Fig. 4. The clustering performance of the SA algorithm is nearly equal to that of the NG algorithm, which is the most efficient clustering method in this test problem. The number of adaptation steps to reach the steady state and the distortion error at the steady state in the SA algorithm are nearly the same as those in the NG algorithm. We also performed other simulations, one of which was vector quantization of a real image (‘Lena’, 256 3 256 pixels, 8-bit grayscale). In this case, the SOM demonstrated the best performance, and the SA algorithm also had the nearly equal performance. Consequently, comparing with the other soft-max algorithms, the SA algorithm has nearly the best clustering performance. Moreover, it does not require a sorting process unlike the NG algorithm nor a searching process of adjacent lattice sites unlike the SOM; only one reference vector is adjusted per adaptation step. Thus, the computational power required by the SA algorithm is much less than that required by the other soft-max algorithm. If the number of reference vectors is , the total updating steps of reference vectors in the SA algorithm are
! times as many as those in the other algorithms. Thus, the SA algorithm is the most efficient clustering method. 200 300 10-4 10-8 10-6 Temperature (K) eM moving time tM (sec) H-H state L-H state (d) t 0 C1 C1 C2 C2 C3 Cj Cj : 0.1aF C1 : 0.06aF C2 : 0.02aF C3 (parasitic) : 0.002aF Co : 100aF : Electron e D5 Dc Co M Av D1 D1 D5 Dv3 Dv1 Ah Vr1Vr2 Vd1 Vd2 (b) Ne ~ Σ |Vdi − Vri| i Vbg Vr1 Vr2 Vr3 Vd1 Vd2 Vd3 Av Ah Co MOSFET (a) Energy (meV) Position of eM D1 400 -400 0 Dc -400 400 0 Position of eM D1 Dc Data matched (H-H state) Data unmatched (L-H state) Position of eM Energy D1 D1 Dc Position of eM (c) Energy Energy (meV) Figure 5: Nanostructure evaluating Hamming distance. (a) Schematic of nanostructure, where dot arrays are extremely enlarged compared with a MOSFET to emphasize the dot structures. (b) Single-electron circuit. (c) Potential profile in dot array 2 . (d) moving time for bit comparator operation. 4 Nanostructure implementing SA model The key for implementing the SA model is adding random fluctuation as expressed by eq. (8). We have already proposed single-electron circuits and nanostructures evaluating Hamming distance for the SA model [5]-[9]. Figure 5(a) and (b) show a nanostructure and the corresponding single-electron circuit, respectively, which are the most sophisticated version of our circuits and structures [9]. The nanostructure consists of plural (
) dot structures arranged on a MOS transistor gate electrode. Each dot structure consists of 1-D dot arrays 2 ( -
& - -
) and 2 (
% ), where & means the number of dots at a side of 2 . (From Monte Carlo single-electron circuit simulation, & should be more than 3). The dot diameter assumed is around 1 nm. The capacitance corresponds to the gate capacitance of an ultrasmall MOS transistor. An electron is introduced in array 2 , which is for example performed by using Fowler-Nordheim tunneling from the grounded plate over . Electron , which is initially located at , can move along array 2 through tunneling junctions , but it cannot move to 2 through the normal capacitor % . Digital (High/Low) voltages 0 and 5 0 (
) are applied at both edges of 2 , which correspond to elements of data and reference vectors, respectively. Each dot structure simply works as an exclusive-NOR logic gate (bit comparator) with random fluctuation as explained below. If the two digital data bits ( or ) are matched, electron stabilizes at center dot , otherwise moves to an off-center position. After stabilizing , by changing voltages 0 , 5 0 and back-gate voltage , vertical dot array 2 detects whether stays at or not; only if stays at , 2 is polarized and an electron is induced at the gate electrode of . The total number of induced electrons ( ) is proportional to the number of dot structures with matched bits; thus the Hamming distance can be measured by counting the induced electrons using the ultrasmall MOS transistor. (If one of the input digital data is applied through an inverter, the number of unmatched bits can be calculated). The detail of operation stabilizing is as follows: Because of the charging energy of itself, the total energy as a function of the position of in array 2 has two peaks at the midpoints of each side of the array, and has minimal values at and both of
as shown in Fig. 5(c). The energy barrier height for at is assumed larger than the thermal energy at room temperature. In L-L state, the energy at
rises up, thus is most strongly stabilized at . On the other hand, in H-L(L-H) or H-H state, the energy barrier is lower than that of L-L state, thus can more easily overcome the barrier by using thermal noise. Figure 5(d) shows the relation between operation temperature and time ( 8 ) required until moves to
, which was obtained by Monte Carlo single-electron circuit simulation. The moving process assisted by thermal noise is purely stochastic, thus 8 scatters in a wide range. However, because the energy barrier height in H-L(L-H) states is lower than that in H-H state as shown in Fig. 5(c), there exists a certain time span 8 / within which in H-L(L-H) states moves to
while in H-H state stays at . At room temperature (300K), 8 / is several microseconds in this case although 8 / depends on the tunneling resistance. If the detection process starts after 8 / , nearly perfect exclusive-NOR (bit comparison) operation is achieved. On the other hand, if the start timing is shifted from 8 / , arbitrary amount of fluctuation can be included in the bit comparison result. Thus, we utilize quantum mechanical tunneling processes assisted by thermal noise in this structure, which is similar to a phenomenon known as stochastic resonance. Although digital data are treated in the above explanation, analog data can be treated in the same circuit by using pulse-width modulation (PWM) signals, which have a digital amplitude and an analog pulse width [10]. Therefore, instead of the Hamming distance, the Manhattan distance can be evaluated by using this nanostructure. Because random fluctuation is naturally added in our nanostructure, it can implement the calculation expressed by eq. (8). The annealing procedure described by eqs. (13) and (14) can be performed by changing the time scale in the stabilization operation; that means the scaling of pulse-width modulation. The proposed nanostructure has not yet been fabricated using the present VLSI technology, but the basic technology related to nanocrystalline floating-dot MOSFET devices, which are closely related to our structure, is now being developed [11]-[13]. Furthermore, wellcontrolled self-assembly processes using molecular manipulation technology, especially using DNA [14], would be utilized to fabricate our nanostructure. Thus, it could be constructed in the near future. 5 Conclusions The stochastic association algorithm offers a simple and powerful soft-max adaptation rule for vector quantizers. Although it is the same as the simple on-line K-means clustering method except for adding random fluctuation in the distortion error evaluation process, our new method has an efficient adaptation performance as high as the neural-gas (NG) or the SOM algorithms. Moreover, our method needs no additional process such as sorting and only one reference vector is adjusted at each adaptation step; thus the computational effort is much smaller compared with the conventional soft-max clustering algorithms. By employing the nanostructure proposed in this paper, very high performance clustering hardware could be constructed. Acknowledgments The authors wish to thank Prof. Masataka Hirose for his support and encouragement. This work has been supported in part by Grants-in-aid for the Core Research for Evolutional Science and Technology (CREST) from Japan Science and Technology Corporation(JST). References [1] K. Rose, E. Gurewitz, and G. C. Fox, “Statistical Mechanics and Phase Transitions in Clustering,” Physical Review Letters, vol. 65, no. 8, pp. 945–948, 1990. [2] T. Kohonen, Self-Organization and Associative Memory, Springer-Verlag, Berlin, 1984. [3] T. M. Martinetz, S. G. Berkovich, and K. J. Schulten, ““Neural-Gas” Network for Vector Quantization and its Apllication to Time-Series Prediction,” IEEE Trans. Neural Networks, vol. 4, pp. 558–569, 1993. [4] S. Rovetta and R. Zunino, “Efficient Training of Neural Gas Vector Quantizers with Analog Circuit Implementation,” IEEE Trans. Circuits & Syst., vol. 46, pp. 688–698, 1999. [5] M. Saen, T. Morie, M. Nagata, and A. Iwata, “A Stochastic Associative Memory Using SingleElectron Tunneling Devices,” IEICE Trans. Electron., vol. E81-C, no. 1, pp. 30–35, 1998. [6] T. Yamanaka, T. Morie, M. Nagata, and A. Iwata, “A Single-Electron Stochastic Associative Processing Circuit Robust to Random Background-Charge Effects and Its Structure Using Nanocrystal Floating-Gate Transistors,” Nanotechnology, vol. 11, no. 3, pp. 154–160, 2000. [7] T. Morie, T. Matsuura, S. Miyata, T. Yamanaka, M. Nagata, and A. Iwata, “Quantum Dot Structures Measuring Hamming Distance for Associative Memories,” Superlattices & Microstructures, vol. 27, no. 5/6, pp. 613–616, 2000. [8] T. Matsuura, T. Morie, M. Nagata, and A. Iwata, “A Multi-Quantum-Dot Associative Circuit Using Thermal-Noise Assisted Tunneling,” in Ext. Abs. of Int. Conf. on Solid State Devices and Materials, pp. 306–307, Sendai, Japan, Aug. 2000. [9] T. Morie, T. Matsuura, M. Nagata, and A. Iwata, “Quantum Dot Structures Measuring Hamming Distance for Associative Memories,” in Extended Abstracts, 4th International Workshop on Quantum Functional Devices (QFD2000), pp. 210–213, Kanazawa, Japan, Nov. 2000. [10] A. Iwata and M. Nagata, “A Concept of Analog-Digital Merged Circuit Architecture for Future VLSI’s,” IEICE Trans. Fundamentals., vol. E79-A, no. 2, pp. 145–157, 1996. [11] S. Tiwari, F. Rana, H. Hanafi,A. Hartstein, E. F. Crabb´e, and K. Chan, “A Silicon Nanocrystals Based Memory,” Appl. Phys. Lett., vol. 68, no. 10, pp. 1377–1379, 1996. [12] A. Kohno, H. Murakami, M. Ikeda, H. Nishiyama, S. Miyazaki, and M. Hirose, “Transient Characteristics of Electron Charging in Si-Quantum-Dot Floating Gate MOS Memories,” in Ext. Abs. of Int. Conf. on Solid State Devices and Materials, pp. 124–125, Sendai, Japan, Aug. 2000. [13] R. Ohba, N. Sugiyama, J. Koga, K. Uchida, and A. Toriumu, “Novel Si Quantum Memory Structure with Self-Alighed Stacked Nanocrystalline Dots,” in Ext. Abs. of Int. Conf. on Solid State Devices and Materials, pp. 122–123, Sendai, Japan, Aug. 2000. [14] R. A. Kiehl, “Nanoelectronic Array Architecture,” in Extended Abstracts, 4th International Workshop on Quantum Functional Devices (QFD2000), pp. 49–51, Kanazawa, Japan, Nov. 2000.
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Grammar Transfer in a Second Order Recurrent Neural Network Michiro N egishi Department of Psychology Rutgers University 101 Warren St. Smith Hall #301 Newark, NJ 07102 negishi@psychology.rutgers.edu Stephen Jose Hanson Psychology Department Rutgers University 101 Warren St. Smith Hall #301 Newark, NJ 07102 jose@psychology.rutgers.edu Abstract It has been known that people, after being exposed to sentences generated by an artificial grammar, acquire implicit grammatical knowledge and are able to transfer the knowledge to inputs that are generated by a modified grammar. We show that a second order recurrent neural network is able to transfer grammatical knowledge from one language (generated by a Finite State Machine) to another language which differ both in vocabularies and syntax. Representation of the grammatical knowledge in the network is analyzed using linear discriminant analysis. 1 Introduction In the field of artificial grammar learning, people are known to be able to transfer grammatical knowledge to a new language which consists of a new vocabulary [6]. Furthermore, this effect persists even when the new strings violate the syntactic rule slightly as long as they are similar to the old strings [1]. It has been shown in the past studies that recurrent neural networks also have the ability to generalize previously acquired knowledge to novel inputs. For instance, Dienes et al. ([2]) showed that a neural network can generalize abstract knowledge acquired in one domain to a new domain. They trained the network to predict the next input symbol in grammatical sequences in the first domain, and showed that the network was able to learn to predict grammatical sequences in the second domain more effectively than it would have learned them without the prior learning. During the training in the second domain, they had to freeze the weights of a part of the network to prevent catastrophic forgetting. They used this simulation paradigm to emulate and analyze domain transfer, effect of similarity between training and test sequences, and the effect of n-gram information in human data. Hanson et al. ([5]) also showed that a prior learning of a grammar facilitates the learning of a new grammar in the cases where either the syntax or the vocabulary was kept constant. In this study we investigate grammar transfer by a neural network, where both syntax and vocabularies are different from the source grammar to the target grammar. Unlike Dienes et al.'s network, all weights in the network are allowed to change during the learning of the target grammar, which allows us to investigate interference as well as transfer from the source grammar to the target grammar. 2 Simulation Design 2.1 The Grammar Transfer Task In the following simulations, a neural network is trained with sentences that are generated by a Finite State Machine (FSM) and is tested whether the learning of sentences generated by another FSM is facilitated. Four pairs of FSMs used for the grammar transfer task are shown in Fig. 2. In each FSM diagram, symbols (e.g. A, B, C, ... ) denote words, numbers represent states, a state number with an incoming arrow with no state numbers at the arrow foot (e.g. state 1 in the left FSM in Fig. 2A) signifies the initial state, and numbers in circles (e.g. state 3 in the left FSM in Fig. 2A) signify the accepting states. In each pair of diagrams, transfer was tested in both directions: from the left FSM to the right FSM, and to the opposite direction. Words in a sentence are generated by an FSM and presented to the network one word at a time. At each time, the next word is selected randomly from next possible words (or end of sentence where possible) at the current FSM state with the equal probability, and the FSM state is updated to the next state. The sentence length is limited to 20 words, excluding START. The task for the network is to predict the correct termination of sentences. If the network is to predict that the sentence ends with the current input, the activity of the output node of the network has to be above a threshold value, otherwise the output has to be below another threshold value. Note that if a FSM is at an accepting state but can further transit to another state, the sentence mayor may not end. Therefore, the prediction may succeed or fail. However, the network will eventually learn to yield higher values when the FSM is at an accepting state than when it is not. After the network learns each training sentence, it is tested with randomly generated 1000 sentences and the training session is completed only when the network makes correct end point judgments for all sentences. Then the network is trained with sentences generated by another FSM. The extent of transfer is measured by the reduction of the number of sentences required to train the network on an FSM after a prior learning of another FSM, compared to the number of sentences required to train the network on the current FSM from scratch. 2.2 The Network Architecture and the Learning Algorithm The network is a second order recurrent neural network, with an added hidden layer that receives first order connections from the input layer (Fig. 1). The network has an input layer with seven nodes (A, B, C, ... F, and START), an output layer with one node, an input hidden layer with four nodes, a state hidden layer with four nodes, and a feedback layer with four nodes. Recurrent neural networks are often used for modeling syntactic processing [3]. Second order networks are suited for processing languages generated by FSMs [4]. Learning is carried out by the weight update rule for recurrent networks developed by Williams and Zipser ([7]), extended to second order connections ([4]) where necessary. The learning rate and the momentum are 0.2 and 0.8, respectively. High and low thresholds are initialized to 0.20 and 0.17 respectively and are adapted after the network have processed the test sentences as follows. The high threshold is modified to the minimum value yielded for all end points in the test sentences minus a margin (0.01). The low threshold is modified to the high threshold minus another margin (0.02). These thresholds are used in the next training and test. Output Layer State Hidden Layer ~--'1c:----=, Feedback '-+-+_-+---" Layer Input Layer Figure 1: A second order recurrent network used in simulations. The network consists of an input layer that receives words, an output layer that predicts sentence ends, two hidden layers (an input hidden layer and a state hidden layer), and a feedback layer that receives a copy of the state hidden layer activities. 3 The Simulation Results 3.1 The Transfer Effects Numbers of required trainings and changes in number of trainings averaged over 20 networks with different initial weights are shown in Fig. 2. Numbers in parentheses are standard errors of number of trainings. Changes are shown with either a "+" sign (increase) or a "-" sign (reduction). For instance, Fig. 2A shows that it required 14559 sentence presentations for the network to learn the left FSM after the network was trained on the right FSM. On the other hand, it required 20995 sentence presentation for the network to learn the left FSM from the scratch. Therefore there was 30.7% reduction in the transfer direction from right to left. Note that the network was trained only once on sentences from the source grammar to the criteria and then only once on the sentences from the target grammar. Thus after the completion of the target grammar learning, the knowledge about the source grammar is disrupted to some extent. To show that the network eventually learns both grammars, number of required training was examined for more than one cycle. After ten cycles, number of required trainings was reduced to 0.13% (not shown). 3.2 Representation of Grammatical Knowledge To analyze the representation of grammatical knowledge in the network, Linear Discriminant Analysis (LDA) was applied to hidden layer activities. LDA is a technique which finds sets of coefficients that defines a linear combination of input variables that can be used to discriminate among sets of input data that belong to different categories. Linear combinations of hidden layer node activities using these coefficients provide low-dimensional views of hidden layer activities that best separate specified categories (e.g. grammatical functions). In this respect, LDA is similar to Principal Component Analysis (PCA) except that PCA finds dimensions along which the data have large variances, whereas LDA finds dimensions which differentiate the specified categories. A B c D 20995 (1119) • -30.7% / \ 14559 (1332) 40943 (2905) , -30.5% < 28448 (7924) 20995 (1l20) +6l2% 9731 (961) +3.81 ¥o 9374 (674) 25314 (3959) " -40.9% • 42826 (4575) 67521 D i. " 1 F F -~.:~:) /:121~ . ~ ~ ~<------~E 35097 68858 ~ (10255) (3423) 20995 44114 (1119) (3604) D E 2 -------,+ 1~ 2.ma GJ--/ ] , • -----) 1 D F -25.8% ~ ( F GGE 15575 39372 CV (1149) (2271) Figure 2: Initial savings observed in various grammar transfer tasks. Numbers are required number of training averaged over 20 networks with different initial weights. Numbers in parentheses are standard errors. Numbers shown with "%" are change in number of training due to transfer. A negative change means reduction (positive transfer) and a positive change means increase (negative transfer, or interference). e 8 e 0 ~ m 0 e 0 ID ee N ~ I "E "§ 15 2 0 0 5 0 00 -I rJ' 0 0 000 00 00 0 00 0 0 0 0 0 0°00 0 0 0 <i> <i> <i> o~oo 8 I!J 0 fJO 0 8 0 0 0 0 0 0 <i> 0 <i> <i> <i> 0 0 0 0 -2 go 0 -3 -3 -2 -I 0 Linear OiSO"iminanl 1 Figure 3: State space organization for a grammar transfer task (a case of Fig_ 2B)_ State space activities corresponding to FSM states 1, 2, and 3 are plotted with squares, diamonds, and circles, respectively_ State space activities that belong to the target FSM have dots in the plots, whereas those that belong to the source FSM do not have fill in patterns_ -I o o esponding State 3 region corresponding 'GO inputs ~ .ACAC ··o 0 o 0 o 0 °00 Stat 3 region corresponding ~;::;;;=~~~~~p..,.:l:.~JI~:.O~ in~Sbts .. B8B ... o o ~ 0 ~ ~~a~~~~J ,~~~~Tc~~,in g (state 3) -3 ~----~------~----~------~----~------~ -3 -2 -I 0 Linear Oisaiminanl 1 Figure 4: Trajectories corresponding to loops in Fig_ 2B in the state hidden layer state space_ The broken line corresponds to a hypothetical shared discrimination cue a hypothetical boundary described in 4_ It is the between white diamonds and white circles (i. e. states 2 and 3 in the source grammar), as well as it can be one of the discrimination boundaries between diamonds with dots and squares with dots (i. e. states 2 and 1 in the target grammar). The triangular shape shows the three FSM state trajectory corresponding to inputs BCCBCC .... Ellipses show to state space activities involved in one state loops (at state 1 and at state 3) and two state loops (at state 2 and 3). 4 Discussion In the first grammar transfer task (Fig. 2A), only the initial and the accepting states in the FSMs were different, so the frequency distribution of subsequences of words were very similar except for short sentences. In this case, 31 % saving was observed in one transfer direction although there was little change in required training in the other direction. In the second grammar transfer task, directions of all arcs in the FSMs were reversed. Therefore the mirror images of sentences accepted in one grammar were accepted in the other grammar. Although the grammars were very different, there were significant amount of overlaps in the permissible short subsequences. In this case, there were 31% and 41 % savings in training. In the third and fourth grammar transfer tasks, the source and the target grammars shared less subsequences. In the third case (Fig. 2C) for instance, the subsequences were very different because the source grammar had two one-state loops (at states 1 and 3) with the same word A, whereas two one-state loops in the target grammar consisted of different words (D and E). In this case, there was little change in the number of learnings required in one transfer direction but there was 67% increase in the other direction. In the fourth case (Fig 2. D), there was 26% reduction in one direction but there was 12% increase in the other direction in the number of learnings required. From these observations we hypothesize that, as in the case of syntax transfer ([5]), if the acquired grammar allows frequent subsequence of words that appears in the target grammar (after the equivalent symbol sets are substituted) the transfer is easier and thus there are more savings. What is the source of savings in grammar transfer? It is tempting to say that, as in the vocabulary transfer task ([5]), the source of savings is the organization of the state hidden layer activity which directly reflects the FSM states. Fig. 3 shows the state space organization after the grammar transfer shown in Fig. 2B. Fig. 4 shows the change in the state hidden layer activities drawn over the state space organization. The triangular lines are the trajectories as the network receives BCCBCC, which creates the 3-state loops (231)(231) in the FSM. Regions of trajectories corresponding to the 2-state loop (23) and two I-state loops (1) and (3) are also shown in Fig. 4, although the trajectory lines are not shown to avoid a cluttered figure. It can be seen that state space activities that belong to different FSM state loops tend to be distinct even when they belong to the same FSM state, although there seem to be some tendencies that they are allocated in vicinities. Unlike in the vocabulary transfer, regions belonging to different FSM loops tend to be interspersed by regions that belong to the other grammar, causing state space structure to be more fragmented. Furthermore, we found that there was no significant correlation between the correct rate of the linear discrimination with respect to FSM states (which reflects the extent to which the state space organization reflects the FSM states) and savings (not shown). One could reasonably argue that the saving is not due to transfer of grammatical knowledge but is due to some more low-level processing specific to neural networks. For instance, the network may have to move weight values to an appropriate range at the first stage of the source grammar learning, which might become unnecessary for the leaning of the target grammar. We conducted a simulation to examine the effect of altering the initial random weights using the source and target grammars. The space limitation does not permit us to present the details, but we did not observe the effect of initializing the bias and the weights to appropriate ranges. If neither the state space organization nor the lower-level statistics was not the source of savings, what was transferred? As already mentioned, state space organization observed in grammar transfer task is more fragmented than that observed in vocabulary transfer task (Fig. 3). These fragmented regions have to be discriminated as far as each region (which represents a combination of the current network state and the current vocabulary) has to yield a different network state. State hidden nodes provide clues for the discrimination by placing boundaries in the network state space. Boundary lines collectively define regions in the state space which correspond to sets of state-vocabulary combinations that should be treated equivalently in terms of the given task. These boundaries can be shared: for instance, a hypothetical boundary shown by a broken line in the Fig. 4 can be the discrimination boundary between white diamonds and white circles (i. e. states 2 and 3 in the source grammar), as well as it can be one of the discrimination boundaries between diamonds with dots and squares with dots (i. e. states 2 and 1 in the target grammar). We speculate that shared boundaries may be the source of savings. That is, boundaries created for the source grammar learning can be used, possibly with some modifications, as one of the boundaries for the target grammar. In other words, the source of savings may not be as high level as FSM state space but some lower level features at the syntactic processing level. 5 Conclusion We investigated the ability of a recurrent neural network to transfer grammatical knowledge of a previously acquired language to another. We found that the network was able to transfer the grammatical knowledge to a new grammar with a slightly different syntax defined over a new vocabulary (grammar transfer). The extent of transfer seemed to depend on the subsequences of symbols generated by the two grammars, after the equivalence sets are translated, although the results presented in this paper are admittedly very restricted in the type of syntax covered and the size of syntactic rules and vocabularies. We hypothesize that the ability of the network to transfer grammatical knowledge comes from sharing discrimination boundaries of input and vocabulary combinations. In sum, we hope to have demonstrated that neural networks do not simply learn associations among input symbols but they acquire structural knowledge from inputs. References [1] Brooks, L. R., and Vokey, J . R. (1991) Abstract analogies and abstracted grammars: Comments on Reber (1989) and Mathews et al. (1090). Journal of Experimental Psychology: General, 120, 316-323. [2] Dienes, Z., Altmann, and G., Gao, S-J. (1999) Mapping across domains without feedback: A neural network model of transfer of implicit knowledge, Cognitive Science 23, 53-82. [3] Elman, J. L. (1991) Distributed representation, simple recurrent neural networks, and grammatical structure. Machine Learning, 7, 195-225. [4] Giles, C. L., Miller, C. B., Chen, D., Chen, H. H. , Sun, G. Z., and Lee, Y. C. (1992) Learning and Extracting Finite State Automata with Second-Order Recurrent Neural Networks, it Neural Computation, 4, 393-495. [5] Hanson, S. J., Negishi, M., (2001) The emergence of explicit knowledge (symbols & rules) in (associationist) neural networks, Submitted. [6] Reber, A. (1969) Transfer of syntactic structure in synthetic languages. Journal of Experimental Psychology, 81 , 115-119. [7] Williams, R. J. and Zipser, D. (1989) A learning algorithm for continually running fully recurrent neural networks, Neural Computation, 1 (2) , 270.
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Prod uct Analysis: Learning to model observations as products of hidden variables Brendan J. Freyl, Anitha Kannanl , Nebojsa Jojic2 1 Machine Learning Group, University of Toronto, www.psi.toronto.edu 2 Vision Technology Group, Microsoft Research Abstract Factor analysis and principal components analysis can be used to model linear relationships between observed variables and linearly map high-dimensional data to a lower-dimensional hidden space. In factor analysis, the observations are modeled as a linear combination of normally distributed hidden variables. We describe a nonlinear generalization of factor analysis, called "product analysis", that models the observed variables as a linear combination of products of normally distributed hidden variables. Just as factor analysis can be viewed as unsupervised linear regression on unobserved, normally distributed hidden variables, product analysis can be viewed as unsupervised linear regression on products of unobserved, normally distributed hidden variables. The mapping between the data and the hidden space is nonlinear, so we use an approximate variational technique for inference and learning. Since product analysis is a generalization of factor analysis, product analysis always finds a higher data likelihood than factor analysis. We give results on pattern recognition and illuminationinvariant image clustering. 1 Introduction Continuous-valued latent representations of observed feature vectors can be useful for pattern classification via Bayes rule, summarizing data sets, and producing lowdimensional representations of data for later processing. Linear techniques, including principal components analysis (Jolliffe 1986), factor analysis (Rubin and Thayer 1982) and probabilistic principal components analysis (Tipping and Bishop 1999), model the input as a linear combination of hidden variables, plus sensor noise. The noise models are quite different in all 3 cases (see Tipping and Bishop (1999) for a discussion). For example, whereas factor analysis can account for different noise variances in the coordinates of the input, principal components analysis assumes that the noise variances are the same in different input coordinates. Also, whereas factor analysis accounts for the sensor noise when estimating the combination weights, principal components analysis does not. Often, the input coordinates are not linearly related, but instead the input vector is the result of a nonlinear generative process. In particular, data often can be accurately described as the product of unknown random variables. Examples include the combination of "style" and "content" (Tenenbaum and Freeman 1997), and the combination of a scalar light intensity and a reflectance image. We introduce a generalization of factor analysis, called "product analysis" , that performs maximum likelihood estimation to model the input as a linear combination of products of hidden variables. Although exact EM is not tractable because the hidden variables are nonlinearly related to the input, the form of the product analysis model makes it well-suited to a variational inference technique and a variational EM algorithm. Other approaches to learning nonlinear representations include principal surface analysis (1984) and nonlinear autoencoders (Baldi and Hornik 1989; Diamantaras and Kung 1996), which minimize a reconstruction error when the data is mapped to the latent space and back; mixtures of linear models (Kambhatla and Leen 1994; Ghahramani and Hinton 1997; Tipping and Bishop 1999), which approximate nonlinear relationships using piece-wise linear patches; density networks (MacKay 1995), which use Markov chain Monte Carlo methods to learn potentially very complex density functions; generative topographic maps (Bishop, SvensE'm and Williams 1998), which use a finite set of fixed samples in the latent space for efficient inference and learning; and kernel principal components analysis (Sch6Ikopf, Smola and Muller 1998), which finds principal directions in nonlinear functions of the input. Our goals in developing product analysis is to introduce a technique that • produces a density estimator of the data • separates sensor noise from the latent structure • learns a smooth, nonlinear map from the input to the latent space • works for high-dimensional data and high-dimensional latent spaces • is particularly well-suited to products of latent variables • is computationally efficient While none of the other approaches described above directly addresses all of these goals, product analysis does. 2 Factor analysis model Of the three linear techniques described above, factor analysis has the simplest description as a generative model of the data. The input vector x is modeled using a vector of hidden variables z. The hidden variables are independent and normally distributed with zero mean and unit variance: p(z) = N(z; 0, I). (1) The input is modeled as a linear combination of the hidden variables, plus independent Gaussian noise: p(xlz) = N(x; Az, \]f). (2) The model parameters are the factor loading matrix A and the diagonal matrix of sensor noise variances, \]f. Factor analysis (d. (Rubin and Thayer 1982)) is the procedure for estimating A and lJI using a training set. The marginal distribution over the input is p(x) = N(x; 0, AA T + lJI), so factor analysis can be viewed as estimating a lowrank parameterization of the covariance matrix of the data. 3 Product analysis model In the "product analyzer", the input vector x is modeled using a vector of hidden variables z, which are independent and normally distributed with zero mean and unit variance: p(z) = N(z; 0, I). (3) In factor analysis, the input is modeled as a linear combination of the hidden variables. In product analysis, the input is modeled as a linear combination of monomials in the hidden variables. The power of variable Zk in monomial i is Sik. So, the ith monomial is Ji(z) = II ZZik. (4) k Denoting the vector of Ji(z) 's by f(z) , the density of the input given z is p(xlz) = N(x; Af(z) , lJI). (5) The model parameters are A and the diagonal covariance matrix lJI. Here, we learn A, maintaining the distribution over z constant. Alternatively, if A is known apriori, we can learn the distribution over z, maintaining A to be fixed. The matrix S = {Sik} can be specified beforehand, estimated from the data using cross-validation, or averaged over in a Bayesian fashion. When S = I, J(z) = z and the product analyzer simplifies to the factor analyzer. If, for some i, Sik = 0, for all k, Ji(z) = 1 and this monomial will account for a constant offset in the input. 4 Product analysis Exact EM in the product analyzer is intractable, since the sufficient statistics require averaging over the posterior p(zlx), for which we do not have a tractable expression. Instead, we use a variational approximation (Jordan et al. 1998), where for each training case, the posterior p(zlx) is approximated by a factorized Gaussian distribution q(z) and the parameters of q(z) are adjusted to make the approximation accurate. Then, the approximation q(z) is used to compute the sufficient statistics for each training case in a generalized EM algorithm (Neal and Hinton 1993). The q-distribution is specified by the variational parameters 'f/ and ~: q(z) = N(z; 'f/, ~), (6) where ~ is a diagonal covariance matrix. q is optimized by minimizing the relative entropy (Kullback-Leibler divergence), 1 q(z) J( = z q(z) In p(zlx) . (7) In fact , minimizing this entropy is equivalent to maximizing the following lower bound on the log-probability of the observation: B = 1 q(z) In p~7~~) ~ lnp(x) (8) Pulling lnp(x) out of the integral, the bound can be expressed as B = lnp(x) - 1 q(z) In p~~~~) = lnp(x) - K. (9) Since lnp(x) does not directly depend on the variational parameters, maximizing B is equivalent to minimizing K. Note that since K :::=: 0, B :S lnp(x). Using Lagrange multipliers, it is easy to show that the bound is maximized when q(z) = p(zlx), in which case K = 0 and B = lnp(x). Substituting the expressions for p(z), p(xlz) and q(z) into (8), and using the fact that f(z)T ATw-1 Af(z) = tr(f(z)T ATw-1 Af(z)) = tr(ATw-1 Af(z)f(z)T), we have B = ~ (In I 27fe<I> I -In 127fWI -In 127f11 -l1Tl1- XTW-1x + 2xTw-1 AE[f(z)] + tr(A TW-l AE[f(z)f(z)T])), (10) where E[] denotes an expectation with respect to q(z). The expectations are simplified as follows: E[!i(Z)] = E[II ZZik] = II E[ZZik] = II m Sik (T)k, ¢k), k k k E[Ji(Z)!j(z)] = E[II z:idSik ] = IIE[z:ik+sik ] = II mSik+Sik(T)k,¢k), k k k (11) where mn(T), ¢) is the nth moment under a Gaussian with mean T) and variance ¢. Closed forms for the mn(T), ¢) are found by setting derivatives of the Gaussian moment generating function to zero: (12) After substituting the closed forms for the moments, B is a polynomial in the T)k 'S and the (Pk's. For each training case, B is maximized with respect to the T)k'S and the ¢k'S using, e.g., conjugate gradients. The model parameters A and W that maximize the sum of the bounds for the training cases can be computed directly, since W does not affect the solution for A, B is quadratic in A, and the optimal W can be written in terms of A and the variational parameters. If the power of each latent variable is restricted to be 0 or 1 in each monomial, o :S Sik :S 1, the above expressions simplify to k k In this case, we can directly maximize B with respect to each T)k in turn, since B is quadratic in each T)k. 5 Experimental results: 5.1 Classification results on the Wisconsin breast cancer database: We obtained results on using product analysis for classification of malignant and benign cancer using the breast cancer database provided by Dr. Wolberg from the Univ. of Wisconsin. Each observation in the database is characterized by nine cytological a) b) c) --.---,-~~ ..... Figure 1: a) Data from training set. Mean images learned using b) product analysis c) mixture of gaussians features, namely, lump thickness, uniformity of cell and shape, marginal adhesion, single epithelial cell size, bare nuclei, bland chromotin, normal nucleoli and mitoses. Each feature is assigned an integer between 1 and 10. In their earlier work (Wolberg and Mangasarian 1990), the authors used linear programming for classification. The objective was to find a hyperplane that separates the classes of malignant and benign cancer. In the absence of a separating plane, average sum of misclassifications of each class is minimized. Our approach is learn one density model for the benign feature vectors and a second density model for the malignant feature vectors and then use Bayes rule to classify an input vector. With separate models, classification involves assigning the observation to the model that provides the largest probability for occurrence of that observation as given by, P( l ) P(xlclass)P(class) c as s Ix = --=--:----:-:-----:-=-:-:----'---:-'---=--:'---c---'-----,-----'-----:-=---:-----::-------,P(xlbenign)P(benign) + P(xlmalignant)P(malignant) To compare with the result reported in (Wolberg and Mangasarian 1990), 4.1 % error rate on 369 instances, we used the same set for our learning scheme and found that the product analysis produced 4% misclassfication. In addition, to compare the recognition rate of product analysis with the recognition rate of factor analysis, we divided the data set into 3 sets for training, validation and testing. The parameters of the model are learned using the training set, and tested on the validation set. This is repeated for 20 times, remembering the parameters that provided the best classification rate on the validation set. Finally, the parameters that provided the best performance on the validation set is used to classify the test set, only once. Since the data is limited, we perform this experimentation on 4 different random breakups of data into training, validation and test set. For product analysis model, we chose 3 hidden variables without optimization but for factor analysis, we chose the optimum number of factors. The average error rate on the 4 breakups was 5% using product analysis and 5.59% using factor analysis. Figure 2: Images generated from the learned mixture of product analyzers Figure 3: First row: Observation. Second row: corresponding image normalized for translation and lighting after lighting & transformation invariant model is learned 5.2 Mixture of lighting invariant appearance models: Often, objects are imaged under different illuminants. To learn an appearance model, we want to automatically remove the lighting effects and infer lighting-normalized images. Since ambient light intensity and reflectances of patches on the object multiply to produce a lighting-affected image, we can model lighting-invariance using a product analyzer. P(x,z) = P(xlz)P(z), where x is the vector of pixel intensities of the observation, Zl is the random variable describing the light intensity, and the remaining Zi are the pixel intensities in the lighting normalized image. We learn the distribution over z, where f(z) = [ZlZ2, ZlZ3, ... ZlZN+l]T and A is identity. By infering Zl, we can remove its effect on observation. The mixture model of product analyzer has joint distribution 7rcP(xlz)P(z), where 7rc is the probability of each class. It can be used to infer various kinds of images (e.g. faces of different people) under different lighting conditions. We trained this model on images with 2 different poses of the same person(Fig. la). The variation in the images is governed by change in pose, light, and background clutter. Fig. Ib and Fig. lc compares the components learned using a mixture of product analyzers and a mixture of Gaussians. Due to limited variation in the pose and large variation in lighting, the mixture of gaussians is unable to extract the mean images. However, mixture of product analyzers is able to capture the distributions well. (Fig. 3). 5.3 Transformation and lighting invariant appearance models: Geometric transformations like shift and shearing can occur when scenes are imaged. Transformation invariant mixtures of Guassians and factor analyzers (Frey and Jojic 2002; Jojic et al. 2001) enable infering transformation-neutral image. Here, we add lighting-invariance to this framework enabling clustering based on interesting features such as pose, without concern for transformation and lighting effects. 6 Conclusions We introduced a density model that explains observations as products of hidden variables and we presented a variational technique for inference and learning in this model. On the Wisonsin breast cancer data, we found that product analysis outperforms factor analysis, when used with Bayes rule for pattern classification. We also found that product analysis was able to separate the two hidden causes, lighting and image noise in noisy images with varying illumination and varying pose. References Baldi, P. and Hornik, K 1989. Neural networks and principal components analysis: Learning from examples without local minima. N eural Networks, 2:53- 58. Bishop, C. M., Svensen, M., and Williams, C. K 1. 1998. Gtm: the generative topographic mapping. Neural Computation, 10(1):215- 235. Diamantaras, K 1. and Kung, S. Y. 1996. Principal Component Neural Networks. Wiley, New York NY. Frey, B. J. and Jojic, N. 2002. Transformation invariant clustering and linear component analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence. To appear. Available at http://www.cs.utoronto.ca/~frey. Ghahramani, Z. and Hinton, G. E. 1997. The EM algorithm for mixtures of factor analyzers. University of Toronto Technical Report CRG-TR-96-1. Hastie, T . 1984. Principal Curves and Surfaces. Stanford University, Stanford CA. Doctoral dissertation. Jojic, N., Simard, P., Frey, B. J. , and Heckerman, D. 2001. Separating appearance from deformation. To appear in Proceedings of the IEEE International Conference on Computer Vision. Jolliffe, 1. T. 1986. Principal Component Analysis. Springer-Verlag, New York NY. Jordan, M. 1., Ghahramani, Z. , Jaakkola, T. S. , and Saul, L. K 1998. An introduction to variational methods for graphical models. In Jordan, M. 1. , editor, Learning in Graphical Models. Kluwer Academic Publishers, Norwell MA. Kambhatla, N. and Leen, T. K. 1994. Fast non-linear dimension reduction. In Cowan, J . D., Tesauro, G., and Alspector, J., editors, Advances in N eural Information Processing Systems 6, pages 152- 159. Morgan Kaufmann, San Francisco CA. MacKay, D. J. C. 1995. Bayesian neural networks and density networks. Nuclear Instruments and Methods in Physics Research, 354:73- 80. Neal, R. M. and Hinton, G. E. 1993. A new view of the EM algorithm that justifies incremental and other variants. Unpublished manuscript available over the internet by ftp at ftp:/ /ftp. cs. utoronto. ca/pub/radford/em. ps. Z. Rubin, D. and Thayer, D. 1982. EM algorithms for ML factor analysis. Psychometrika, 47(1):69- 76. Scholkopf, B., Smola, A., and Miiller, K-R. 1998. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299- 1319. Tenenbaum, J. B. and Freeman, W. T. 1997. Separating style and content. In Mozer, M. C. , Jordan, M. 1. , and Petsche, T., editors, Advances in Neural Information Processing Systems 9. MIT Press, Cambridge MA. Tipping, M. E. and Bishop, C. M. 1999. Mixtures of probabilistic principal component analyzers. N eural Computation, 11(2):443- 482. Wolberg, W. H. and Mangasarian, O. L. 1990. Multisurface method of pattern separation for medical diagnosis applied to breast cytology. In Proceedings of the National Academy of Sciences.
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Matching Free Trees with Replicator Equations Marcello Pelillo Dipartimento di Informatica Universit`a Ca’ Foscari di Venezia Via Torino 155, 30172 Venezia Mestre, Italy E-mail: pelillo@dsi.unive.it Abstract Motivated by our recent work on rooted tree matching, in this paper we provide a solution to the problem of matching two free (i.e., unrooted) trees by constructing an association graph whose maximal cliques are in one-to-one correspondence with maximal common subtrees. We then solve the problem using simple replicator dynamics from evolutionary game theory. Experiments on hundreds of uniformly random trees are presented. The results are impressive: despite the inherent inability of these simple dynamics to escape from local optima, they always returned a globally optimal solution. 1 Introduction Graph matching is a classic problem in computer vision and pattern recognition, instances of which arise in areas as diverse as object recognition, motion and stereo analysis [1]. In many problems (e.g., [2, 11, 19]) the graphs at hand have a peculiar structure: they are connected and acyclic, i.e. they are free trees. Note that, unlike “rooted” trees, in free trees there is no distinguished node playing the role of the root, and hence no hierarchy is imposed on them. Standard graph matching techniques, such as [8], yield solutions that are not constrained to preserve connectedness and hence cannot be applied to free trees. A classic approach to solving the graph matching problem consists of transforming it into the equivalent problem of finding a maximum clique in an auxiliary graph structure, known as the association graph [1]. This framework is attractive because it casts graph matching as a pure graph-theoretic problem, for which a solid theory and powerful algorithms have been developed. Note that, although the maximum clique problem is known to be hard, powerful heuristics exist which efficiently find good approximate solutions [4]. Motivated by our recent work on rooted tree matching [15], in this paper we propose a solution to the free tree matching problem by providing a straightforward way of deriving an association graph from two free trees. We prove that in the new formulation there is a one-to-one correspondence between maximal (maximum) cliques in the derived association graph and maximal (maximum) subtree isomorphisms. As an obvious corollary, the computational complexity of finding a maximum clique in such graphs is therefore the same as the subtree isomorphism problem, which is known to be polynomial in the number of nodes [7]. Following [13, 15], we use a recent generalization of the Motzkin-Straus theorem [12] to formulate the maximum clique problem as a quadratic programming problem. To (approximately) solve it we employ replicator equations, a class of simple continuous- and discretetime dynamical systems developed and studied in evolutionary game theory [10, 17]. We illustrate the power of the approach via experiments on hundreds of (uniformly) random trees. The results are impressive: despite the counter-intuitive maximum clique formulation of the tree matching problem, and the inherent inability of these simple dynamics to escape from local optima, they always found a globally optimal solution. 2 Subtree isomorphisms and maximal cliques Let be a graph, where is the set of nodes and is the set of (undirected) edges. The order of is the number of nodes in , while its size is the number of edges. Two nodes
are said to be adjacent (denoted
) if they are connected by an edge. The adjacency matrix of is the symmetric matrix "!# defined as "! $% if & ! ' otherwise ( The degree of a node
, denoted )+*-,./
0 , is the number of nodes adjacent to it. A path is any sequence of distinct nodes
012
435(2(-(6
87 such that for all 9: % (2(2(; ,
=< 3>?
; in this case, the length of the path is . If
01@A
B7 the path is called a cycle. A graph is said to be connected if any two nodes are joined by a path. The distance between two nodes
and , denoted by C8/
5;+ , is the length of the shortest path joining them (by convention C8/
5;+ DFE , if there is no such path). Given a subset of nodes GIHJ , the induced subgraph LK GNM is the graph having G as its node set, and two nodes are adjacent in LK GNM if and only if they are adjacent in . A connected graph with no cycles is called a free tree, or simply a tree. Trees have a number of interesting properties. One which turns out to be very useful for our characterization is that in a tree any two nodes are connected by a unique path. Let O 3 J 3 P 3 and O0QRJ.QS TQ# be two trees. Any bijection UWVYX 3[Z X\Q , with X[3 H]83 and X Q H? Q , is called a subtree isomorphism if it preserves both the adjacency relationships between the nodes and the connectedness of the matched subgraphs. Formally, this means that, given
5;[^X_3 , we have
W if and only if U5/
0 ]U= and, in addition, the induced subgraphs O 3 K X 3 M and O0Q`K XaQ2M are connected. A subtree isomorphism is maximal if there is no other subtree isomorphism U4bcV0X^b 3 Z X^b Q with XR3 a strict subset of Xb 3 , and maximum if XR3 has largest cardinality. The maximal (maximum) subtree isomorphism problem is to find a maximal (maximum) subtree isomorphism between two trees. A word of caution about terminology is in order here. Despite name similarity, we are not addressing the so-called subtree isomorphism problem, which consists of determining whether a given tree is isomorphic to a subtree of a larger one. In fact, we are dealing with a generalization thereof, the maximum common subtree problem, which consists of determining the largest isomorphic subtrees of two given trees. We shall continue to use our own terminology, however, as it emphasizes the role of the isomorphism U . The free tree association graph (FTAG) of two trees Od3 eB3fP N3- and O Q gh Q Q is the graph i j where W 3 @.Q (1) and, for any two nodes /
5;kN and /8 l in , we have =
kN /8 l nmoC.=
pWC8/kqPlr ( (2) Note that this definition of the association graph is stronger than the standard one used for matching arbitrary relational structures [1]. A subset of vertices of is said to be a clique if all its nodes are mutually adjacent. A maximal clique is one which is not contained in any larger clique, while a maximum clique is a clique having largest cardinality. The maximum clique problem is to find a maximum clique of . The following theorem, which is the basis of the work reported here, establishes a one-toone correspondence between the maximum subtree isomorphism problem and the maximum clique problem. Theorem 1 Any maximal (maximum) subtree isomorphism between two trees induces a maximal (maximum) clique in the corresponding FTAG, and vice versa. Proof (outline). Let U&VYX_3 Z X Q be a maximal subtree isomorphism between trees On3 and O Q , and let h:P denote the corresponding FTAG. Let G H be defined as G r=
U5/
0 ; >Vr
X 3 . From the definition of a subtree isomorphism it follows that U maps the path between any two nodes
X 3 onto the path joining U5=
B and U= . This clearly implies that C.=
\C.hU=
B PU5/+ ; for all
? X_3 , and therefore G is a clique. Trivially, G is a maximal clique because U is maximal, and this proves the first part of the theorem. Suppose now that Gr=
3 ;k 3 4#/
7 ;k 7 is a maximal clique of , and let X 3 -
43#Y;
B7 H&B3 and X Q -k>3f0;k 7 HW Q . Define UVrXR3 Z X Q as U5/
pWk , for all 9d % (-(2(; . From the definition of the FTAG and the hypothesis that G is a clique, it is simple to see that U is a one-to-one and onto correspondence between X 3 and XaQ , which trivially preserves the adjacency relationships between nodes. The fact that U is a maximal isomorphism is a straightforward consequence of the maximality of G . To conclude the proof we have to show that the subgraphs that we obtain when we restrict ourselves to X 3 and XaQ , i.e. O 3 K X 3 M and O0Q K XaQ2M , are trees, and this is equivalent to showing that they are connected. Suppose by contradiction that this is not the case, and let
! X[3 be two nodes which are not joined by a path in On3 K XL3 M . Since both
and
! are nodes of OY3 , however, there must exist a path
1 B3n(-(2(
g
! joining them in O3 . Let , for some % (2(-(
, be a node on this path which is not in X 3 . Moreover, let a be the -th node on the path k 1 3n(2(2( A k ! which joins k and k ! in O Q (remember that C8/
! NC8/k ;k ! , and hence C.=k ;k ! > ). It is easy to show that the set r 2 G H is a clique, thereby contradicting the hypothesis that G is a maximal clique. This can be proved by exploiting the obvious fact that if is a node on the path joining any two nodes
and , then C.=
: C8/
5 8 C8 ;+ . The “maximum” part of the statement is proved similarly. The FTAG is readily derived by using a classical representation for graphs, i.e., the socalled distance matrix which, for an arbitrary graph h:P of order , is the ^ matrix =C "! where C "! ?C8/
! , the distance between nodes
and
! . Efficient, classical algorithms are available for obtaining such a matrix [6]. Note also that the distance matrix of a graph can easily be constructed from its adjacency matrix . In fact, denoting by 7 ! the /9 ! -th entry of the matrix 7 , the -th power of , we have that C ! equals the least for which 7 !#" ' (there must be such an since a tree is connected). 3 Matching free trees with replicator dynamics Let j be an arbitrary graph of order , and let $Y7 denote the standard simplex of IR 7 : $07L%& IR 7 V(' b &^ % and *) ' Y9d % (2(2(; where ' is the vector whose components equal 1, and a prime denotes transposition. Given a subset of vertices G of , we will denote by & its characteristic vector which is the point in $ 7 defined as $ % G if 9p@G ' otherwise where G denotes the cardinality of G . Now, consider the following quadratic function &Y & b & % & b & (3) where g= ! is the adjacency matrix of . The following theorem, recently proved by Bomze [3], expands on the Motzkin-Straus theorem [12], a remarkable result which establishes a connection between the maximum clique problem and quadratic programming. Theorem 2 Let G be a subset of vertices of a graph , and let & be its characteristic vector. Then, G is a maximal (maximum) clique of if and only if & is a local (global) maximizer in $ 7 . Moreover, all local (and hence global) maximizers of in $ 7 are strict and are characteristic vectors of maximal cliques of . Unlike the original Motzkin-Straus formulation, which is plagued by the presence of “spurious” solutions [14], the previous result guarantees us that all maximizers of on $07 are strict, and are characteristic vectors of maximal/maximum cliques in . In a formal sense, therefore, a one-to-one correspondence exists between maximal cliques and local maximizers of in $ 7 on the one hand, and maximum cliques and global maximizers on the other hand. We now turn our attention to a class of simple dynamical systems that we use for solving our quadratic optimization problem. Let A/kN "! be a non-negative real-valued @ matrix, and consider the following continuous-time dynamical system: . P
; . P
; ; 7 ! 3 +!
; !r
; (4) where a dot signifies derivative with respect to time, and its discrete-time counterpart: 8 % : 8 ; P ; 7 ! 3 ! ; !
; (5) where
; : 7 ! 3 k ! !
; ( (6) Both (4) and (5) are called replicator equations in evolutionary game theory, since they are used to model evolution over time of relative frequencies of interacting, self-replicating entities [10, 17]. It is readily seen that the simplex $ 7 is invariant under these dynamics, which means that every trajectory starting in $7 will remain in $47 for all future times, and their stationary points coincide. We are now interested in the dynamical properties of replicator dynamics; it is these properties that will allow us to solve our original tree matching problem. The following result is known in mathematical biology as the fundamental theorem of natural selection [10, 17] and, in its original form, traces back to R. A. Fisher. Theorem 3 If ib then the function &b & is strictly increasing along any nonconstant trajectory under both continuous-time (4) and discrete-time (5) replicator dynamics. Furthermore, any such trajectory converges to a stationary point. Finally, a vector & $ 7 is asymptotically stable under (4) and (5) if and only if & is a strict local maximizer of &4b & on $07 . In light of their dynamical properties, replicator equations naturally suggest themselves as a simple heuristic for solving the maximal subtree isomorphism problem. Indeed, let O 3 Ih 3 3 and O0Q I.Q`P TQ- be two free trees, and let denote the adjacency matrix of their FTAG . By letting W % (7) where is the identity matrix, we know that the replicator dynamical systems (4) and (5), starting from an arbitrary initial state, will iteratively maximize the function defined in (3) over the simplex and will eventually converge with probability 1 to a strict local maximizer which, by virtue of Theorem 2, will then correspond to the characteristic vector of a maximal clique in the association graph. As stated in Theorem 1, this will in turn induce a maximal subtree isomorphism between O 3 and O0Q . Clearly, in theory there is no guarantee that the converged solution will be a global maximizer of , and therefore that it will induce a maximum isomorphism between the two original trees, but see below. Recently, there has been much interest around the following exponential version of replicator equations, which arises as a model of evolution guided by imitation [9, 10, 17]: ;
;
7 ! 3 !
;
% (8) where is a positive constant. As tends to 0, the orbits of this dynamics approach those of the standard, “first-order” replicator model (4), slowed down by the factor ; moreover, for large values of the model approximates the so-called “best-reply” dynamics [9, 10]. A customary way of discretizing equation (8) is given by the following difference equations:
% :
;
7 ! 3 !
;
( (9) From a computational perspective, exponential replicator dynamics are particularly attractive as they may be considerably faster and even more accurate than the standard, first-order model (see [13] and the experiments reported in the next section). 4 Results and conclusions We tested our algorithms over large random trees. Random structures represent a useful benchmark not only because they are not constrained to any particular application, but also because it is simple to replicate experiments and hence to make comparisons with other algorithms. In this series of experiments, the following protocol was used. A hundred 100-node free trees were generated uniformly at random using a procedure described by Wilf in [18]. Then, each such tree was subject to a corruption process which consisted of randomly deleting a fraction of its nodes (in fact, the to-be-deleted nodes were constrained to be the terminal ones, otherwise the resulting graph would have been disconnected), thereby obtaining a tree isomorphic to a proper subtree of the original one. Various levels of corruption (i.e., percentage of node deletion) were used, namely 2%, 10%, 20%, 30% and 40%. This means that the order of the pruned trees ranged from 98 to 60. Overall, therefore, 500 pairs of trees were obtained, for each of which the corresponding FTAG was constructed as described in Section 2. To keep the order of the association graph as low as possible, its vertex set was constructed as follows: %r=
kN @ b @ b b V)+*2,8/
0 ) *2,B/kN assuming b b b , the edge set being defined as in (2). It is straightforward to see that when the first tree is isomorphic to a subtree of the second, Theorem 1 continues to hold. This simple heuristic may significantly reduce the dimensionality of the search space. We also performed some experiments with unpruned FTAG’s but no significant difference in performance was noticed apart, of course, heavier memory requirements. Both the discrete-time first-order dynamics (5) and its exponential counterpart (9) (with % ' ) were used. The algorithms were started from the simplex barycenter and stopped when either a maximal clique (i.e., a local maximizer of ) was found or the distance between two successive points was smaller than a fixed threshold. In the latter case the converged vector was randomly perturbed, and the algorithms restarted from the perturbed point. Note that this situation corresponds to convergence to a saddle point. After convergence, we calculated the proportion of matched nodes, i.e., the ratio between the cardinality of the clique found and the order of the smaller subtree, and then we averaged. Figure 1(a) shows the results obtained using the linear dynamics (5) as a function of the corruption level. As can be seen, the algorithm was always able to find a correct maximum isomorphism, i.e. a maximum clique in the FTAG. Figure 1(b) plots the corresponding (average) CPU time taken by the processes, with corresponding error bars (simulations were performed on a machine equipped with a 350MHz AMDK6-2 processor). In Figure 2, the results pertaining to the exponential dynamics (8) are shown. In terms of solution’s quality the algorithm performed exactly as its linear counterpart, but this time it was dramatically faster. This confirms earlier results reported in [13]. Before concluding, we note that our approach can easily be extended to tackle the problem of matching attributed (free) trees. In this case, the attributes result in weights being placed on the nodes of the association graph, and a conversion of the maximum clique problem to a maximum weight clique problem [15, 5]. Moreover, it is straightforward to formulate errortolerant versions of our framework along the lines suggested in [16] for rooted attributed trees, where many-to-many node correspondences are allowed. All this will be the subject of future investigations. Finally, we think that the results presented in this paper (together with those reported in [13, 15]) raise intriguing questions concerning the connections between (standard) notions of computational complexity and the “elusiveness” of global optima in continuous settings. Acknowledgments. The author would like to thank M. Zuin for his support in performing the experiments. References [1] D. H. Ballard and C. M. Brown. Computer Vision. Prentice-Hall, Englewood Cliffs, NJ, 1982. [2] H. Blum and R. N. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition, 10:167–180, 1978. [3] I. M. Bomze. Evolution towards the maximum clique. J. Glob. Optim., 10:143–164, 1997. [4] I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization (Suppl. Vol. A), pages 1–74. Kluwer, Boston, MA, 1999. [5] I. M. Bomze, M. Pelillo, and V. Stix. Approximating the maximum weight clique using replicator dynamics. IEEE Trans. Neural Networks, 11(6):1228–1241, 2000. 9 5 100 Percentage of correct matches 500 1000 1500 2000 2500 3000 Average CPU time (in secs) Figure 1: Results obtained over 100-node random trees with various levels of corruption, using the first-order dynamics (5). Top: Percentage of correct matches. Bottom: Average computational time taken by the replicator equations. [6] T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press, Cambridge, MA, 1990. [7] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NPCompleteness. W. H. Freeman, San Francisco, CA, 1979. [8] S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. IEEE Trans. Pattern Anal. Machine Intell. 18:377-388, 1996. [9] J. Hofbauer. Imitation dynamics for games. Collegium Budapest, preprint, 1995. [10] J. Hofbauer and K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, UK, 1998. [11] T.-L. Liu, D. Geiger, and R. V. Kohn. Representation and self-similarity of shapes. In Proc. ICCV’98—6th Int. Conf. Computer Vision, pages 1129–1135, Bombay, India, 1998. [12] T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of Tur´an. Canad. J. Math., 17:533–540, 1965. 9 5 100 Percentage of correct matches 100 200 300 400 Average CPU time (in secs) Figure 2: Results obtained over 100-node random trees with various levels of corruption, using the exponential dynamics (9) with . Top: Percentage of correct matches. Bottom: Average computational time taken by the replicator equations. [13] M. Pelillo. Replicator equations, maximal cliques, and graph isomorphism. Neural Computation, 11(8):2023–2045, 1999. [14] M. Pelillo and A. Jagota. Feasible and infeasible maxima in a quadratic program for maximum clique. J. Artif. Neural Networks, 2:411–420, 1995. [15] M. Pelillo, K. Siddiqi, and S. W. Zucker. Matching hierarchical structures using association graphs. IEEE Trans. Pattern Anal. Machince Intell., 21(11):1105–1120, 1999. [16] M. Pelillo, K. Siddiqi, and S. W. Zucker. Many-to-many matching of attributed trees using association graphs and game dynamics. In C. Arcelli, L. P. Cordella, and G. Sanniti di Baja, editors, Visual Form 2001, pages 583–593. Springer, Berlin, 2001. [17] J. W. Weibull. Evolutionary Game Theory. MIT Press, Cambridge, MA, 1995. [18] H. Wilf. The uniform selection of free trees. J. Algorithms, 2:204–207, 1981. [19] S. C. Zhu and A. L. Yuille. FORMS: A flexible object recognition and modeling system. Int. J. Computer Vision, 20(3):187–212, 1996.
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Multiplicative Updates for Classification by Mixture Models Lawrence K. Saul and Daniel D. Lee Department of Computer and Information Science Department of Electrical Engineering University of Pennsylvania, Philadelphia, PA 19104 Abstract We investigate a learning algorithm for the classification of nonnegative data by mixture models. Multiplicative update rules are derived that directly optimize the performance of these models as classifiers. The update rules have a simple closed form and an intuitive appeal. Our algorithm retains the main virtues of the Expectation-Maximization (EM) algorithm—its guarantee of monotonic improvement, and its absence of tuning parameters—with the added advantage of optimizing a discriminative objective function. The algorithm reduces as a special case to the method of generalized iterative scaling for log-linear models. The learning rate of the algorithm is controlled by the sparseness of the training data. We use the method of nonnegative matrix factorization (NMF) to discover sparse distributed representations of the data. This form of feature selection greatly accelerates learning and makes the algorithm practical on large problems. Experiments show that discriminatively trained mixture models lead to much better classification than comparably sized models trained by EM. 1 Introduction Mixture models[11] have been widely applied to problems in classification. In these problems, one must learn a decision rule mapping feature vectors ( ) to class labels ( ) given labeled examples. Mixture models are typically used to parameterize class-conditional distributions,
, and then to compute posterior probabilities,
, from Bayes rule. Parameter estimation in these models is handled by an Expectation-Maximization (EM) algorithm[3], a learning procedure that monotonically increases the joint log likelihood, , summed over training examples (indexed by ). A virtue of this algorithm is that it does not require the setting of learning rates or other tuning parameters. A weakness of the above approach is that the model parameters are optimized by maximum likelihood estimation, as opposed to a discriminative criterion more closely related to classification error[14]. In this paper, we derive multiplicative update rules for the parameters of mixture models that directly maximize the discriminative objective function, . This objective function measures the conditional log likelihood that the training examples are correctly classified. Our update rules retain the main virtues of the EM algorithm—its guarantee of monotonic improvement, and its absence of tuning parameters—with the added advantage of optimizing a discriminative cost function. They also have a simple closed form and appealing intuition. The proof of convergence combines ideas from the EM algorithm[3] and methods for generalized and improved iterative scaling[2, 4]. The approach in this paper is limited to the classification of nonnegative data, since from the constraint of nonnegativity emerges an especially simple learning algorithm. This limitation, though, is not too severe. An abundance of interesting data occurs naturally in this form: for example, the pixel intensities of images, the power spectra of speech, and the word-document counts of text. Real-valued data can also be coerced into this form by addition or exponentiation. Thus we believe the algorithm has broad applicability. 2 Mixture models as generative models Mixture models are typically used as generative models to parameterize probability distributions over feature vectors . Different mixture models are used to model different classes of data. The parameterized distributions take the form:
(1) where the rows of the nonnegative weight matrix are constrained to sum to unity, , and the basis functions are properly normalized distributions, such that for all . The model can be interpreted as the latent variable model,
(2) where the discrete latent variable indicates which mixture component is used to generate the observed variable . In this setting, one identifies and . The basis functions, usually chosen from the exponential family, define “bumps” of high probabilityin the feature space. A popular choice is the multivariate Gaussian distribution: ! " $#&%('*),+.-0/21 1 3 40"65 # 1 3 7 (3) with means 3 and covariance matrices " . Gaussian distributions are extremely versatile, but not always the most appropriate. For sparse nonnegative data, a more natural choice is the exponential distribution: 98;: < :>= 5?@BA
CA (4) with parameter vectors < . Here, the value of 3 indexes the elements of < and . The parameters of these basis functions must be estimated from data. Generative models can be viewed as a prototype method for classification, with the parameters of each mixture component defining a particular basin of attraction in the feature space. Intuitively, patterns are labeled by the most similar prototype, chosen from among all possible classes. Formally, unlabeled examples are classified by computing posterior probabilities from Bayes’ rule,
EDGF
H IH (5) where
denote the prior probabilities of each class. Examples are classified by the label with the highest posterior probability. An Expectation-Maximization (EM) algorithm can be used to estimate the parameters of mixture models. The EM algorithm optimizes the joint log likelihood,
(6) summed over training examples. If basis functions are not shared across different classes, then the parameter estimation for
can be done independently for each class label . This has the tremendous advantage of decomposing the original learning problem into several smaller problems. Moreover, for many types of basis functions, the EM updates have a simple closed form and are guaranteed to improve the joint log likelihood at each iteration. These properties account for the widespread use of mixture models as generative models. 3 Mixture models as discriminative models Mixture models can also be viewed as purely discriminative models. In this view, their purpose is simply to provide a particular way of parameterizing the posterior distribution, . In this paper, we study posterior distributions of the form: (7) The right hand side of this equation defines a valid posterior distribution provided that the mixture weights and basis functions . are nonnegative. Note that for this interpretation, the mixture weights and basis functions do not need to satisfy the more stringent normalization constraints of generative models. We will deliberately exploit this freedom, an idea that distinguishes our approach from previous work on discriminatively trained mixture models[6] and hidden Markov models[5, 12]. In particular, the unnormalized basis functions we use are able to parameterize “saddles” and “valleys” in the feature space, as well as the “bumps” of normalized basis functions. This makes them more expressive than their generative counterparts: examples can not only be attracted to prototypes, but also repelled by opposites. The posterior distributions in eq. (7) must be further specified by parameterizing the basis functions as a function of . We study basis functions of the form = @
(8) where denotes a real-valued vector and denotes a nonnegative and possibly “expanded” representation[14] of the original feature vector. The exponential form in eq. (8) allows us to recover certain generative models as a special case. For example, consider the multivariate Gaussian distribution in eq. (3). By defining the “quadratically expanded” feature vector: # ' # # # ' 5 # (9) we can equate the basis functions in eqs. (3) and (8) by choosing the parameter vectors to act on in the same way that the means 3 and covariance matrices " act on . The exponential distributions in eq. (4) can be recovered in a similar way. Such generative models provide a cheap way to initialize discriminative models for further training. 4 Learning algorithm Our learning algorithm directly optimizes the performance of the models in eq. (7) as classifiers. The objective function we use for discriminative training is the conditional log likelihood, 9 (10) summed over training examples. Let denote the binary matrix whose th element denotes whether the th training example belongs to the th class. Then we can write the objective function as the difference of two terms, 1 5 , where: = @
(11) 5 E = @
(12) The competition between these terms give rise to a scenario of contrastive learning. It is the subtracted term, 5 , which distinguishes the conditional log likelihood optimized by discriminative training from the joint log likelihood optimized by EM. Our learning algorithm works by alternately updating the mixture weights and the basis function parameters. Here we simply present the update rules for these parameters; a derivation and proof of convergence are given in the appendix. It is easiest to write the basis function updates in terms of the nonnegative parameters = @BA . The updates then take the simple multiplicative form: / 5
7 (13) = @ A = @ A / : 5 : 7 where + : : (14) It is straightforward to compute the gradients in these ratios and show that they are always nonnegative. (This is a consequence of the nonnegativity constraint on the feature vectors: : for all examples and feature components 3 .) Thus, the nonnegativityconstraints on the mixture weights and basis functions are enforced by these multiplicative udpates. The updates have a simple intuition[9] based on balancing opposing terms in the gradient of the conditional log likelihood. In particular, note that the fixed points of this update rule occur at stationary points of the conditional log likelihood—that is, where and : , or equivalently, where 5 and : 5 : . The learning rate is controlled by the ratios of these gradients and—additionally, for the basis function updates—by the exponent , which measures the sparseness of the training data. The value of is the maximum sum of features that occurs in the training data. Thus, sparse feature vectors leads to faster learning, a crucial point to which we will return shortly. It is worth comparing these multiplicative updates to others in the literature. Jebara and Pentland[6] derived similar updates for mixture weights, but without emphasizing the special form of eq. (13). Others have investigated multiplicative updates by the method of exponentiated gradients (EG)[7]. Our updates do not have the same form as EG updates: in particular, note that the gradients in eqs. (13–14) are not exponentiated. If we use one basis function per class and an identity matrix for the mixture weights, then the updates reduce to the method of generalized iterative scaling[2] for logistic or multinomial regression (also known as maximum entropy modeling). More generally, though, our multiplicative updates can be used to train much more powerful classifiers based on mixture models. 5 Feature selection As previously mentioned, the learning rate for the basis function parameters is controlled by the sparseness of the training data. If this data is not intrinsically sparse, then the multiplicative upates in eqs. (13–14) can be impractically slow (just as the method of iterative 01-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 NMF basis vectors pixel image 20 40 60 80 0 5 10 NMF feature vector Figure 1: Left: nonnegative basis vectors for handwritten digits discovered by NMF. Right: sparse feature vector for a handwritten “2”. The basis vectors are ordered by their contribution to this image. scaling). In this case, it is important to discover sparse distributed representations of the data that encode the same information. On large problems, such representations can accelerate learning by several orders of magnitude. The search for sparse distributed representations can be viewed as a form of feature selection. We have observed that suitably sparse representations can be discovered by the method of nonnegative matrix factorization (NMF)[8]. Let the raw nonnegative (and possibly nonsparse) data be represented by the matrix , where is its raw dimensionality and is the number of training examples. Algorithms for NMF yield a factorization , where is a nonnegative marix and is a nonnegative matrix. In this factorization, the columns of are interpreted as basis vectors, and the columns of as coefficients (or new feature vectors). These coefficients are typically very sparse, because the nonnegative basis vectors can only be added in a constructive way to approximate the original data. The effectiveness of NMF is best illustrated by example. We used the method to discover sparse distributed representations of the MNIST data set of handwritten digits[10]. The data set has 60000 training and 10000 test examples that were deslanted and cropped to form grayscale pixel images. The raw training data was therefore represented by a matrix, with
and . The left plot of Fig. 1 shows the basis vectors discovered by NMF, each plotted as a image. Most of these basis vectors resemble strokes, only a fraction of which are needed to reconstruct any particular image in the trainingset. For example, only about twenty basis vectors make an appreciable contribution to the handwritten “2” shown in the right plot of Fig. 1. The method of NMF thus succeeds in discovering a highly sparse representation of the original images. 6 Results Models were evaluated on the problem of recognizing handwritten digits from the MNIST data set. From the grayscale pixel images, we generated two sets of feature vectors: one by NMF, with nonnegative features and dimensionality ; the other, by principal components analysis (PCA), with real-valued features and dimensionality
. These reduced dimensionality feature vectors were used for both training and testing. Baseline mixture models for classification were trained by EM algorithms. Gaussian mixture models with diagonal covariance marices were trained on the PCA features, while exponential mixture models (as in eq. (4)) were trained on the NMF features. The mixture models were trained for up to 64 iterations of EM, which was sufficient to ensure a high degree of convergence. Seven baseline classifiers were trained on each feature set, with different numbers of mixture components per digit (
). The error rates of these models, indicated by EM-PCA40 and EM-NMF80, are shown in Table 1. Half as many PCA features were used as NMF features so as to equalize the number of fitted parameters in different basis functions. Mixture models on the NMF features were also trained discriminatively by the multiplicative updates in eqs. (13–14). Models with varying numbers of mixture components per digit (
) were trained by 1000 iterations of these updates. Again, this was sufficient to ensure a high degree of convergence; there was no effort at early stopping. The models were initialized by setting and for randomly selected feature vectors. The results of these experiments, indicated by DT-NMF80, are also shown in Table 1. The results show that the discriminatively trained models classify much better than comparably sized models trained by EM. The ability to learn more compact classifiers appears to be the major advantage of discriminative training. A slight disadvantage is that the resulting classifiers are more susceptible to overtraining. model EM-PCA40 EM-NMF80 DT-NMF80 1 10.2 10.1 15.7 14.7 5.5 5.8 2 8.5 8.3 12.3 10.7 4.0 4.4 4 6.8 6.4 9.3 8.2 2.8 3.5 8 5.3 5.1 7.8 7.0 1.7 3.2 16 4.0 4.4 6.2 5.7 1.0 3.4 32 3.1 3.6 5.0 5.1 64 1.9 3.1 3.9 4.2 Table 1: Classification error rates (%) on the training set (
and the test set ( ) for mixture models with different numbers of mixture components per digit ( ). Models in the same row have roughly the same number of fitted parameters. It is instructive to compare our results to other benchmarks on this data set[10]. Without making use of prior knowledge, better error rates on the test set have been obtained by support vector machines ( ), k-nearest neighbor (
), and fully connected multilayer neural networks ( ). These results, however, either required storing large numbers of trainingexamples or trainingsignificantly larger models. For example, the nearest neighbor and support vector classifiers required storing tens of thousands of training examples (or support vectors), while the neural network had over 120,000 weights. By contrast, the discriminatively trained mixture model (with ) has less than 6500 iteratively adjusted parameters, and most of its memory footprint is devoted to preprocessing by NMF. We conclude by describing the problems best suited to the mixture models in this paper. These are problems with many classes, large amounts of data, and little prior knowledge of symmetries or invariances. Support vector machines and nearest neighbor algorithms do not scale well to this regime, and it remains tedious to train large neural networks with unspecified learning rates. By contrast, the compactness of our models and the simplicity of their learning algorithm make them especially attractive. A Proof of convergence In this appendix, we show that the multiplicative updates from section 4 lead to monotonic improvement in the conditional log likelihood. This guarantee of convergence (to a stationary point) is proved by computing a lower bound on the conditional log likelihood for updated estimates of the mixture weights and basis function parameters. We indicate these updated estimates by H and H , and we indicate the resulting values of the conditional log likelihoodand its component terms by H , H , and H 5 . The proof of convergence rests on three simple inequalities applied to H . The first term in the conditional log likelihoodcan be lower bounded by Jensen’s inequality. The same bound is used here as in the derivation of the EM algorithm[3, 13] for maximum likelihood estimation: H 9 H = F @
H = F @
(15) The right hand side of this inequality introduces an auxiliary probability distribution for each example in the training set. The bound holds for arbitrary distributions, provided they are properly normalized: for all . The second term in the conditional log likelihood occurs with a minus sign, so for this term we require an upper bound. The same bounds can be used here as in derivations of iterative scaling[1, 2, 4, 13]. Note that the logarithm function is upper bounded by: 1 for all . We can therefore write: H 5 1 5 H = F @
=
H = F @
=
1 (16) To further bound the right hand side of eq. (16), we make the followingobservation: though the exponentials = F @
are convex functions of the parameter vector H with elements H : , they are concave functions of the “warped” parameter vector = F @ with elements = F @BA , where is defined by eq. (14). (The validity of this observation hinges on the nonnegativity of the feature vectors .) It follows that for any example in the training set, the exponential = F @
is upper bounded by its linearized expansion around = F @BA = @BA , given by: = F @
= @
: = F @BA 1 = @ A "! :.= @
= @BA # (17) The last term in parentheses in eq. (17) is the derivative of = @
with respect to the independent variable = @BA , computed by the chain rule. Tighter bounds are possible than eq. (17), but at the expense of more complicated update rules. Combining the above inequalities with a judicious choice for the auxiliary parameters , we obtain a proof of convergence for the multiplicative updates in eqs. (13–14). Let: =
5 # = @
(18) 5 =
5 # = @
(19) Eq. (18) sets the auxiliary parameters , while eq. (19) defines an analogous distribution 5 for the opposing term in the conditional log likelihood. (This will prove to be a useful notation.) Combining these definitions with eqs. (15–17) and rearranging terms, we obtain the following inequality: H 1 / H H 1 7 1 5 H 1 H : : ! = F @BA 5 @BA 1 # (20) Both sides of the inequality vanish (yielding an equality) if H and H . We derive the update rules by maximizing the right hand side of this inequality. Maximizing the right hand side with respect to H while holding the basis function parameters fixed yields the update, eq. (13). Likewise, maximizing the right hand side with respect to H while holding the mixture weights fixed yields the update, eq. (14). Since these choices for H and H lead to positive values on the right hand side of the inequality (except at fixed points), it follows that the multiplicative updates in eqs. (13–14) lead to monotonic improvement in the conditional log likelihood. References [1] M. Collins, R. Schapire, and Y. Singer (2000). Logistic regression, adaBoost, and Bregman distances. In Proceedingsof the Thirteenth Annual Conferenceon Computational Learning Theory. [2] J. N. Darroch and D. Ratcliff (1972). Generalized iterative scaling for log-linear models. Annals of Mathematical Statistics 43:1470–1480. [3] A. P. Dempster, N. M. Laird, and D. B. Rubin (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. B 39: 1–37. [4] S. Della Pietra, V. Della Pietra, and J. Lafferty (1997). Inducing features of random fields. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(4): 380–393. [5] P.S. Gopalakrishnan, D. Kanevsky, A. Ndas and D. Nahamoo (1991). An inequality for rational functions with applications to some statistical estimation problems. IEEE Transactions on Information Theory 37: 107–113. [6] T. Jebara and A. Pentland (1998). Maximum conditional likelihood via bound maximization and the CEM algorithm. In M. Kearns, S. Solla, and D. Cohn (eds.). Advancesin Neural Information Processing Systems 11, 494–500. MIT Press: Cambridge, MA. [7] J. Kivinen and M. Warmuth (1997). Additive versus exponentiated gradient updates for linear prediction. Journal of Information and Computation 132: 1–64. [8] D. D. Lee and H. S. Seung (1999). Learning the parts of objects with nonnegative matrix factorization. Nature 401: 788–791. [9] D. D. Lee and H. S. Seung (2000). Algorithms for nonnegative matrix factorization. In T. Leen, T. Dietterich, and V. Tresp (eds.). Advances in Neural Information Processing Systems 13. MIT Press: Cambridge, MA. [10] Y.LeCun, L. Jackel, L.Bottou, A.Brunot, C.Cortes, J. Denker, H.Drucker, I.Guyon, U. Muller, E.Sackinger, P.Simard, and V.Vapnik (1995). A comparison of learning algorithms for handwritten digit recognition. In F.Fogelman and P.Gallinari (eds.). International Conferenceon Artificial Neural Networks, 1995, Paris: 53–60. [11] G. McLachlan and K. Basford (1988). Mixture Models: Inference and Applications to Clustering. Marcel Dekker. [12] Y. Normandin (1991). Hidden Markov Models, Maximum Mutual Information Estimation and the Speech Recognition Problem. Ph.D. Thesis, McGill University, Montreal. [13] J. A. O’Sullivan (1998). Alternating minimization algorithms: from Blahut-Arimoto to Expectation-Maximization. In A. Vardy (ed.). Codes, Curves, and Signals: Common Threads in Communications. Kluwer: Norwell, MA. [14] V. Vapnik (1999). The Nature of Statistical Learning Theory. Springer Verlag.
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The Infinite Hidden Markov Model Matthew J. Beal Zoubin Ghahramani Carl Edward Rasmussen Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR, England http://www.gatsby.ucl.ac.uk m.beal,zoubin,edward @gatsby.ucl.ac.uk Abstract We show that it is possible to extend hidden Markov models to have a countably infinite number of hidden states. By using the theory of Dirichlet processes we can implicitly integrate out the infinitely many transition parameters, leaving only three hyperparameters which can be learned from data. These three hyperparameters define a hierarchical Dirichlet process capable of capturing a rich set of transition dynamics. The three hyperparameters control the time scale of the dynamics, the sparsity of the underlying state-transition matrix, and the expected number of distinct hidden states in a finite sequence. In this framework it is also natural to allow the alphabet of emitted symbols to be infinite— consider, for example, symbols being possible words appearing in English text. 1 Introduction Hidden Markov models (HMMs) are one of the most popular methods in machine learning and statistics for modelling sequences such as speech and proteins. An HMM defines a probability distribution over sequences of observations (symbols)
by invoking another sequence of unobserved, or hidden, discrete state variables . The basic idea in an HMM is that the sequence of hidden states has Markov dynamics—i.e. given , is independent of for all ! #"$ &% —and that the observations ' are independent of all other variables given . The model is defined in terms of two sets of parameters, the transition matrix whose (*)+-, element is .0/ *12 3)54 6(87 and the emission matrix whose (:9+-, element is .0/ ;9<4 (=7 . The usual procedure for estimating the parameters of an HMM is the Baum-Welch algorithm, a special case of EM, which estimates expected values of two matrices > and ? corresponding to counts of transitions and emissions respectively, where the expectation is taken over the posterior probability of hidden state sequences [6]. Both the standard estimation procedure and the model definition for HMMs suffer from important limitations. First, maximum likelihood estimation procedures do not consider the complexity of the model, making it hard to avoid over or underfitting. Second, the model structure has to be specified in advance. Motivated in part by these problems there have been attempts to approximate a full Bayesian analysis of HMMs which integrates over, rather than optimises, the parameters. It has been proposed to approximate such Bayesian integration both using variational methods [3] and by conditioning on a single most likely hidden state sequence [8]. In this paper we start from the point of view that the basic modelling assumption of HMMs—that the data was generated by some discrete state variable which can take on one of several values—is unreasonable for most real-world problems. Instead we formulate the idea of HMMs with a countably infinite number of hidden states. In principle, such models have infinitely many parameters in the state transition matrix. Obviously it would not be sensible to optimise these parameters; instead we use the theory of Dirichlet processes (DPs) [2, 1] to implicitly integrate them out, leaving just three hyperparameters defining the prior over transition dynamics. The idea of using DPs to define mixture models with infinite number of components has been previously explored in [5] and [7]. This simple form of the DP turns out to be inadequate for HMMs.1 Because of this we have extended the notion of a DP to a two-stage hierarchical process which couples transitions between different states. It should be stressed that Dirichlet distributions have been used extensively both as priors for mixing proportions and to smooth n-gram models over finite alphabets [4], which differs considerably from the model presented here. To our knowledge no one has studied inference in discrete infinite-state HMMs. We begin with a review of Dirichlet processes in section 2 which we will use as the basis for the notion of a hierarchical Dirichlet process (HDP) described in section 3. We explore properties of the HDP prior, showing that it can generate interesting hidden state sequences and that it can also be used as an emission model for an infinite alphabet of symbols. This infinite emission model is controlled by two additional hyperparameters. In section 4 we describe the procedures for inference (Gibbs sampling the hidden states), learning (optimising the hyperparameters), and likelihood evaluation (infinite-state particle filtering). We present experimental results in section 5 and conclude in section 6. 2 Properties of the Dirichlet Process Let us examine in detail the statistics of hidden state transitions from a particular state ( to *12 , with the number of hidden states finite and equal to . The transition probabilities given in the ( +-, row of the transition matrix can be interpreted as mixing proportions for *12 that we call . Imagine drawing > samples from a discrete indicator variable which can take on values
with proportions given by . The joint distribution of these indicators is multinomial .0/ 4 7 with > / 8 )7 (1) where we have used the Kronecker-delta function ( / 7 iff , and otherwise) to count the number of times > that *12 0) has been drawn. Let us see what happens to the distribution of these indicators when we integrate out the mixing proportions under a conjugate prior. We give the mixing proportions a symmetric Dirichlet prior with positive concentration hyperparameter .0/ 4 27 "!$#&%'#)(*+&,./ 0/ 0/ 7$ 1 /2 7 1 / 0/ 7 43
5 76 (2) where is restricted to be on the simplex of mixing proportions that sum to 1. We can analytically integrate out under this prior to yield: 1That is, if we only applied the mechanism described in section 2, then state trajectories under the prior would never visit the same state twice; since each new state will have no previous transitions from it, the DP would choose randomly between all infinitely many states, therefore transitioning to another new state with probability 1. .0/ ' 4 7 .0/ ' 4 78.0/2 4 27 1 / 27 1 /-> 27 1 /*> 0/ 7 1 / 0/ 7 (3) Thus the probability of a particular sequence of indicators is only a function of the counts > > . The conditional probability of an indicator given the setting of all other indicators (denoted 6 ) is given by .0/ )54 6 27 > 6 0/ >
(4) where > 6 is the counts as in (1) with the '+-, indicator removed. Note the self-reinforcing property of (4): is more likely to choose an already popular state. A key property of DPs, which is at the very heart of the model in this paper, is the expression for (4) when we take the limit as the number of hidden states tends to infinity: .0/ )54 6 27 61 3 ) ' i.e. represented 3 61 3 for all unrepresented ) , combined (5) where is the number of represented states (i.e. for which > 6 ), which cannot be infinite since > is finite. can be interpreted as the number of pseudo-observations of / . / , i.e. the strength of belief in the symmetric prior.2 In the infinite limit acts as an “innovation” parameter, controlling the tendency for the model to populate a previously unrepresented state. 3 Hierarchical Dirichlet Process (HDP) We now consider modelling each row of the transition and emission matrices of an HMM as a DP. Two key results from the previous section form the basis of the HDP model for infinite HMMs. The first is that we can integrate out the infinite number of transition parameters, and represent the process with a finite number of indicator variables. The second is that under a DP there is a natural tendency to use existing transitions in proportion to their previous usage, which gives rise to typical trajectories. In sections 3.1 and 3.2 we describe in detail the HDP model for transitions and emissions for an infinite-state HMM. 3.1 Hidden state transition mechanism Imagine we have generated a hidden state sequence up to and including time " , building a table of counts > for transitions that have occured so far from state ( to ) , i.e. >! " 6 / 8 (87 / 1 )7 . Given that we are in state ( , we impose on state *12 a DP (5) with parameter whose counts are those entries in the ( +-, row of > , i.e. we prefer to reuse transitions we have used before and follow typical trajectories (see Figure 1): .0/ *12 )54 ( > 27 > "$# >% ) (6) Note that the above probabilities do not sum to 1—under the DP there is a finite probability 0/</ " > 7 of not selecting one of these transitions. In this case, the model defaults to a second different DP (5) on *12 with parameter & whose counts are given by a vector > ' . We refer to the default DP and its associated counts as the oracle. Given that we have defaulted to the oracle DP, the probabilities of transitioning now become .0/ *12 )54 ( > ' &
7 ( *) +-, /.0 ) 1 1 ) ' i.e. ) represented 1 +-, /.0 ) 1 1 )32 ' i.e. ) is a new state (7) 2Under the infinite model, at any time, there are an infinite number of (indistinguishable) unrepresented states available, each of which have infinitesimal mass proportional to 4 . nii + α nij + β + α β nij + β + α jΣ jΣ self
transition oracle nij nij + β + α jΣ existing
transition j=i nj o nj o + γ γ nj o + γ jΣ jΣ existing
state new
state a) b) c) d) Figure 1: (left) State transition generative mechanism. (right a-d) Sampled state trajectories of length (time along horizontal axis) from the HDP: we give examples of four modes of behaviour. (a)
4
, explores many states with a sparse transition matrix. (b) 4
, retraces multiple interacting trajectory segments. (c) ! 4"# , switches between a few different states. (d) $
4%&
#'&
, has strict left-to-right transition dynamics with long linger time. Under the oracle, with probability proportional to & an entirely new state is transitioned to. This is the only mechanism for visiting new states from the infinitely many available to us. After each transition we set > )( >% and, if we transitioned to the state ) via the oracle DP just described then in addition we set >!' ( > ' . If we transitioned to a new state then the size of > and > ' will increase. Self-transitions are special because their probability defines a time scale over which the dynamics of the hidden state evolves. We assign a finite prior mass * to self transitions for each state; this is the third hyperparameter in our model. Therefore, when first visited (via & in the HDP), its self-transition count is initialised to * . The full hidden state transition mechanism is a two-level DP hierarchy shown in decision tree form in Figure 1. Alongside are shown typical state trajectories under the prior with different hyperparameters. We can see that, with just three hyperparameters, there are a wealth of types of possible trajectories. Note that & controls the expected number of represented hidden states, and influences the tendency to explore new transitions, corresponding to the size and density respectively of the resulting transition count matrix. Finally * controls the prior tendency to linger in a state. The role of the oracle is two-fold. First it serves to couple the transition DPs from different hidden states. Since a newly visited state has no previous transitions to existing states, without an oracle (which necessarily has knowledge of all represented states as it created them) it would transition to itself or yet another new state with probability 1. By consulting the oracle, new states can have finite probability of transitioning to represented states. The second role of the oracle is to allow some states to be more influential (more commonly transitioned to) than others. 3.2 Emission mechanism The emission process ,+ is identical to the transition process -+ *12 in every respect except that there is no concept analogous to a self-transition. Therefore we need only introduce two further hyperparameters /. and &). for the emission HDP. Like for state transitions we keep a table of counts ? 10 " 6 / 28 (=7 / 28 97 which is the number of times before " that state ( has emitted symbol 9 , and ? ' 0 is the number of times symbol miq miq + βe βe miq + βe mq mq o + γe γe mq o + γe qΣ qΣ qΣ qΣ existing
emission existing
symbol new
symbol oracle 0 0.5 1 1.5 2 2.5 x 10 4 0 500 1000 1500 2000 2500 0 20 40 60 80 100 10 0 10 1 10 2 Figure 2: (left) State emission generative mechanism. (middle) Word occurence for entire Alice novel: each word is assigned a unique integer identity as it appears. Word identity (vertical) is plotted against the word position (horizontal) in the text. (right) (Exp 1) Evolution of number of represented states (vertical), plotted against iterations of Gibbs sweeps (horizontal) during learning of the ascending-descending sequence which requires exactly 10 states to model the data perfectly. Each line represents initialising the hidden state to a random sequence containing "
!
distinct represented states. (Hyperparameters are not optimised.) 9 has been emitted using the emission oracle. For some applications the training sequence is not expected to contain all possible observation symbols. Consider the occurence of words in natural text e.g. as shown in Figure 2 (middle) for the Alice novel. The upper envelope demonstrates that new words continue to appear in the novel. A property of the DP is that the expected number of distinct symbols (i.e. words here) increases as the logarithm of the sequence length. The combination of an HDP for both hidden states and emissions may well be able to capture the somewhat super-logarithmic word generation found in Alice. 4 Inference, learning and likelihoods Given a sequence of observations, there are two sets of unknowns in the infinite HMM: the hidden state sequence , and the five hyperparameters * & . & . defining the transition and emission HDPs. Note that by using HDPs for both states and observations, we have implicitly integrated out the infinitely many transition and emission parameters. Making an analogy with non-parametric models such as Gaussian Processes, we define a learned model as a set of counts > >!' ? ?3' and optimised hyperparameters * & . & . . We first describe an approximate Gibbs sampling procedure for inferring the posterior over the hidden state sequence. We then describe hyperparameter optimisation. Lastly, for calculating the likelihood we introduce an infinite-state particle filter. The following algorithm summarises the learning procedure: 1. Instantiate a random hidden state sequence
. 2. For
# - Gibbs sample given hyperparameter settings, count matrices, and observations. - Update count matrices to reflect new
; this may change , the number of represented hidden states. 3. End 4. Update hyperparameters 4 # 4 #
given hidden state statistics. 5. Goto step 2. 4.1 Gibbs sampling the hidden state sequence Define > and ? as the results of removing from > and ? the transition and emission counts contributed by . Define similar items > ' and ? ' related to the transition and emission oracle vectors. An exact Gibbs sweep of the hidden state from " takes / 7 operations, since under the HDP generative process changing affects the probability of all subsequent hidden state transitions and emissions.3 However this computation can be reasonably approximated in / 7 , by basing the Gibbs update for only on the state of its neigbours 6 *12 and the total counts > ? > ' ?3' .4 In order to facilitate hyperparameter learning and improve the mixing time of the Gibbs sampler, we also sample a set of auxiliary indicator variables *12 . alongside ; each of these is a binary variable denoting whether the oracle was used to generate *12 respectively. 4.2 Hyperparameter optimisation We place vague Gamma priors5 on the hyperparameters * & . &). . We derive an approximate form for the hyperparameter posteriors from (3) by treating each level of the HDPs separately. The following expressions for the posterior for * , , and '. are accurate for large & , while the expressions for & and & . are exact: .0/ * 4 7 /
' 7 / 3 3 7 # # 6 1 / * 27 1 / * 7 1 /*> * 7 1 / " >% * 27 .0/2 . 4 27 / 3 3 7 # . # 1 /2 . 7 1 / " 0 ? 0 . 7 .0/ & 4 7 / 1 ' 1 7 & # 1 / &
7 1 / ' &7 .0/ & . 4 27 / 1 ' 1 7 & # 1 / &). 7 1 / ' . & . 7 where is the number of represented states that are transitioned to from state ( (including itself); similarly . is the number of possible emissions from state ( . ' and ' . are the number of times the oracle has been used for the transition and emission processes, calculated from the indicator variables . . We solve for the maximum a posteriori (MAP) setting for each hyperparameter; for example /.MAP is obtained as the solution to following equation using gradient following techniques such as Newton-Raphson: " # . / .MAP / .MAP 7!
! / " 0 ?3 0 .MAP 7#"
3 / 3
7'/ .MAP 4.3 Infinite-state particle filter The likelihood for a particular observable sequence of symbols involves intractable sums over the possible hidden state trajectories. Integrating out the parameters in any HMM induces long range dependencies between states. In particular, in the DP, making the transition ( + ) makes that transition more likely later on in the sequence, so we cannot use standard tricks like dynamic programming. Furthermore, the number of distinct states can grow with the sequence length as new states are generated. If the chain starts with distinct states, at time " there could be " possible distinct states making the total number of trajectories over the entire length of the sequence / 7%$ / $ . 3Although the hidden states in an HMM satisfy the Markov condition, integrating out the parameters induces these long-range dependencies. 4This approximation can be motivated in the following way. Consider sampling parameters & from the posterior distribution '()&+* , .-%/ of parameter matrices, which will depend on the count matrices. By the Markov property, for a given & , the probability of only depends on 10 , 2 and 43 , and can therefore be computed without considering its effect on future states. 5 56 7 (48 :9;/)<9=?>A@B(48C/+D 5 = 0 .E 0GF1H , with 8 and 9 the shape and inverse-scale parameters. We propose estimating the likelihood of a test sequence given a learned model using particle filtering. The idea is to start with some number of particles distributed on the represented hidden states according to the final state marginal from the training sequence (some of the may fall onto new states).6 Starting from the set of particles , the tables from the training sequences > > ' ? ?3' , and " the recursive procedure is as specified below, where .0/ 4 6
7 " / 7 : 1. Compute <'(42 * / for each particle . 2. Calculate (#
>
/ '(42* 2 :2 10 / . 3. Resample particles 6 (#
%> / ( / . 4. Update transition and emission tables , for each particle. 5. For each sample forward dynamics: 3 6 '( 43 * / ; this may cause particles to land on novel states. Update and . 6. If ! , Goto 1 with "
. The log likelihood of the test sequence is computed as " +#%$& . Since it is a discrete state space, with much of the probability mass concentrated on the represented states, it is feasible to use ' / 7 particles. 5 Synthetic experiments Exp 1: Discovering the number of hidden states We applied the infinite HMM inference algorithm to the ascending-descending observation sequence consisting of 30 concatenated copies of (*),+.-0/213/4-5+.) . The most parsimonious HMM which models this data perfectly has exactly 10 hidden states. The infinite HMM was initialised with a random hidden state sequence, containing distinct represented states. In Figure 2 (right) we show how the number of represented states evolves with successive Gibbs sweeps, starting from a variety of initial . In all cases converges to 10, while occasionally exploring 9 and 11. Exp 2: Expansive A sequence of length 76 was generated from a 4-state 8-symbol HMM with the transition and emission probabilities as shown in Figure 3 (top left). Exp 3: Compressive A sequence of length 86 was generated from a 4-state 3-symbol HMM with the transition and emission probabilities as shown in Figure 3 (bottom left). In both Exp 2 and Exp 3 the infinite HMM was initialised with a hidden state sequence with :9 distinct states. Figure 3 shows that, over successive Gibbs sweeps and hyperparameter learning, the count matrices for the infinite HMM converge to resemble the true probability matrices as shown on the far left. 6 Discussion We have shown how a two-level Hierarchical Dirichlet Process can be used to define a nonparametric Bayesian HMM. The HDP implicity integrates out the transition and emission parameters of the HMM. An advantage of this is that it is no longer necessary to constrain the HMM to have finitely many states and observation symbols. The prior over hidden state transitions defined by the HDP is capable of producing a wealth of interesting trajectories by varying the three hyperparameters that control it. We have presented the necessary tools for using the infinite HMM, namely a linear-time approximate Gibbs sampler for inference, equations for hyperparameter learning, and a particle filter for likelihood evaluation. 6Different particle initialisations apply if we do not assume that the test sequence immediately follows the training sequence. True transition and emission probability matrices used for Exp 2 0 0 0 0 0 0 0 0 True transition and emission probability matrices used for Exp 3 0 0 0 0 0 0 Figure 3: The far left pair of Hinton diagrams represent the true transition and emission probabilities used to generate the data for each experiment 2 and 3 (up to a permutation of the hidden states; lighter boxes correspond to higher values). (top row) Exp 2: Expansive HMM. Count matrix pairs ) are displayed after
!
!
sweeps of Gibbs sampling. (bottom row) Exp 3: Compressive HMM. Similar to top row displaying count matrices after "
!
!
sweeps of Gibbs sampling. In both rows the display after a single Gibbs sweep has been reduced in size for clarity. On synthetic data we have shown that the infinite HMM discovers both the appropriate number of states required to model the data and the structure of the emission and transition matrices. It is important to emphasise that although the count matrices found by the infinite HMM resemble point estimates of HMM parameters (e.g. Figure 3), they are better thought of as the sufficient statistics for the HDP posterior distribution over parameters. We believe that for many problems the infinite HMM’s flexibile nature and its ability to automatically determine the required number of hidden states make it superior to the conventional treatment of HMMs with its associated difficult model selection problem. While the results in this paper are promising, they are limited to synthetic data; in future we hope to explore the potential of this model on real-world problems. Acknowledgements The authors would like to thank David Mackay for suggesting the use of an oracle, and Quaid Morris for his Perl expertise. References [1] C. E. Antoniak. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Annals of Statistics, 2(6):1152–1174, 1974. [2] T. S. Ferguson. A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1(2):209–230, March 1973. [3] D. J. C. MacKay. Ensemble learning for hidden Markov models. Technical report, Cavendish Laboratory, University of Cambridge, 1997. [4] D. J. C. MacKay and L. C. Peto. A hierarchical Dirichlet language model. Natural Language Engineering, 1(3):1–19, 1995. [5] R. M. Neal. Markov chain sampling methods for Dirichlet process mixture models. Technical Report 9815, Dept. of Statistics, University of Toronto, 1998. [6] L. R. Rabiner and B. H. Juang. An introduction to hidden Markov models. IEEE Acoustics, Speech & Signal Processing Magazine, 3:4–16, 1986. [7] C. E. Rasmussen. The infinite Gaussian mixture model. In Advances in Neural Information Processing Systems 12, Cambridge, MA, 2000. MIT Press. [8] A. Stolcke and S. Omohundro. Hidden Markov model induction by Bayesian model merging. In S. J. Hanson, J. D. Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 11–18, San Francisco, CA, 1993. Morgan Kaufmann.
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EM-DD: An Improved Multiple-Instance Learning Technique Qi Zhang Department of Computer Science Washington University St. Louis, MO 63130-4899 qz@cs. wustl. edu Sally A. Goldman Department of Computer Science Washington University St. Louis, MO 63130-4899 sg@cs. wustl. edu Abstract We present a new multiple-instance (MI) learning technique (EMDD) that combines EM with the diverse density (DD) algorithm. EM-DD is a general-purpose MI algorithm that can be applied with boolean or real-value labels and makes real-value predictions. On the boolean Musk benchmarks, the EM-DD algorithm without any tuning significantly outperforms all previous algorithms. EM-DD is relatively insensitive to the number of relevant attributes in the data set and scales up well to large bag sizes. Furthermore, EMDD provides a new framework for MI learning, in which the MI problem is converted to a single-instance setting by using EM to estimate the instance responsible for the label of the bag. 1 Introduction The multiple-instance (MI) learning model has received much attention. In this model, each training example is a set (or bag) of instances along with a single label equal to the maximum label among all instances in the bag. The individual instances within the bag are not given labels. The goal is to learn to accurately predict the label of previously unseen bags. Standard supervised learning can be viewed as a special case of MI learning where each bag holds a single instance. The MI learning model was originally motivated by the drug activity prediction problem where each instance is a possible conformation (or shape) of a molecule and each bag contains all likely low-energy conformations for the molecule. A molecule is active if it binds strongly to the target protein in at least one of its conformations and is inactive if no conformation binds to the protein. The problem is to predict the label (active or inactive) of molecules based on their conformations. The MI learning model was first formalized by Dietterich et al. in their seminal paper [4] in which they developed MI algorithms for learning axis-parallel rectangles (APRs) and they also provided two benchmark "Musk" data sets. Following this work, there has been a significant amount of research directed towards the development of MI algorithms using different learning models [2,5,6,9,12]. Maron and Raton [7] applied the multiple-instance model to the task of recognizing a person from a series of images that are labeled positive if they contain the person and negative otherwise. The same technique was used to learn descriptions of natural scene images (such as a waterfall) and to retrieve similar images from a large image database using the learned concept [7]. More recently, Ruffo [11] has used this model for data mining applications. While the musk data sets have boolean labels, algorithms that can handle realvalue labels are often desirable in real-world applications. For example, the binding affinity between a molecule and receptor is quantitative, and hence a real-value classification of binding strength is preferable to a binary one. Most prior research on MI learning is restricted to concept learning (i.e. boolean labels). Recently, MI learning with real-value labels has been performed using extensions of the diverse density (DD) and k-NN algorithms [1] and using MI regression [10]. In this paper, we present a general-purpose MI learning technique (EM-DD) that combines EM [3] with the extended DD [1] algorithm. The algorithm is applied to both boolean and real-value labeled data and the results are compared with corresponding MI learning algorithms from previous work. In addition, the effects of the number of instances per bag and the number of relevant features on the performance of EM-DD algorithm are also evaluated using artificial data sets. A second contribution of this work is a new general framework for MI learning of converting the MI problem to a single-instance setting using EM. A very similar approach was also used by Ray and Page [10]. 2 Background Dietterich et al. [4], presented three algorithms for learning APRs in the MI model. Their best performing algorithm (iterated-discrim) , starts with a point in the feature space and "grows" a box with the goal of finding the smallest box that covers at least one instance from each positive bag and no instances from any negative bag. The resulting box was then expanded (via a statistical technique) to get better results. However, the test data from Muskl was used to tune the parameters of the algorithm. These parameters are then used for Muskl and Musk2. Auer [2] presented an algorithm, MULTINST, that learns using simple statistics to find the halfspaces defining the boundaries of the target APR and hence avoids some potentially hard computational problems that were required by the heuristics used in the iterated-discrim algorithm. More recently, Wang and Zucker [11] proposed a lazy learning approach by applying two variant of the k nearest neighbor algorithm (k-NN) which they refer to as citation-kNN and Bayesian k-NN. Ramon and De Raedt [9] developed a MI neural network algorithm. Our work builds heavily upon the Diverse Density (DD) algorithm of Maron and Lozano-Perez [5,6]. When describing the shape of a molecule by n features , one can view each conformation of the molecule as a point in a n-dimensional feature space. The diverse density at a point p in the feature space is a probabilistic measure of both how many different positive bags have an instance near p, and how far the negative instances are from p. Intuitively, the diversity density of a hypothesis h is just the likelihood (with respect to the data) that h is the target. A high diverse density indicates a good candidate for a "true" concept. We now formally define the general MI problem (with boolean or real-value labels) and DD likelihood measurement originally defined in [6] and extended to real-value labels in [1]. Let D be the labeled data which consists of a set of m bags B = {B1, ... , B m } and labels L = {l\, ... ,£m }, i.e., D = {< B 1,£l >, ... ,< Bm, £m >}. Let bag Bi = {Bil " '" Bij , ... Bin} where Bij denote the lh instance in bag i. Assume the labels of the instances in Bi are £i 1, ... , £ij, ... , £in . For boolean labels, £i = £i1 V £i2 V ... V £in, and for real-value labels, £i = max{ £il, £i2, ... , £in}. The diverse density of hypothesized target point h is defi d (h) (h I ) Pr(D I h) Pr(h) Pr(B , L I h) Pr(h) A . ne as D D = Pr D = () = (). ssummg a Pr D Pr B , L uniform prior on the hypothesis space and independence of < B i , £i > pairs given h, using Bayes' rule, the maximum likelihood hypothesis, hDD , is defined as: n n arg maxPr(D I h) = arg max IT Pr(Bi , £i I h) = arg min I) -log Pr(£i I h , B i )) hEH hEH i=l hEH i=l where Label (Bi I h) is the label that would be given to B i if h were the correct hypothesis. As in the extended DD algorithm [1], Pr(£i I h, Bi) is estimated as l-I£i - Label (Bi I h) I in [1]. When the labels are boolean (0 or 1) , this formulation is exactly the most-likely-cause estimator used in the original DD algorithm [5]. For most applications the influence each feature has on the label varies greatly. This variation is modeled in the DD algorithm by associating with each attribute an (unknown) scale factor. Hence the target concept really consists of two values per dimension, the ideal attribute value and the scale value. Using the assumption that binding strength drops exponentially as the similarity between the conformation to the ideal shape increases, the following generative model was introduced by Maron and Lozano-Perez [6] for estimating the label of bag B i for hypothesis h = {h 1 , ... , hn , Sl , ... , sn} : Label(Bi I h) =max{ exP[- t (Sd(Bijd - hd)) 2]} J d=l (1) where Sd is a scale factor indicating the importance of feature d, hd is the feature value for dimension d, and B ijd is the feature value of instance Bij on dimension d. Let NLDD(h, D) = 2::7=1 (-log Pr(£i I h, B i)) , where NLDD denote the negative logarithm of DD. The DD algorithm [6] uses a two-step gradient descent search to find a value of h that minimizes NLDD (and hence maximizes DD). Ray and Page [10] developed multiple-instance regression algorithm which can also handle real-value labeled data. They assumed an underlying linear model for the hypothesis and applied the algorithm to some artificial data. Similar to the current work, they also used EM to select one instance from each bag so multiple regression can be applied to MI learning. 3 Our algorithm: EM-DD We now describe EM-DD and compare it with the original DD algorithm. One reason why MI learning is so difficult is the ambiguity caused by not knowing which instance is the important one. The basic idea behind EM-DD is to view the knowledge of which instance corresponds to the label of the bag as a missing attribute which can be estimated using EM approach in a way similar to how EM is used in the MI regression [10]. EM-DD starts with some initial guess of a target point h obtained in the standard way by trying points from positive bags, then repeatedly performs the following two steps that combines EM with DD to search for the maximum likelihood hypothesis. In the first step (E-step) , the current hypothesis h is used to pick one instance from each bag which is most likely (given our generative model) to be the one responsible for the label given to the bag. In the second step (M -step), we use the two-step gradient ascent search (quasi-newton search dfpmin in [8]) of the standard DD algorithm to find a new hi that maximizes DD(h). Once this maximization step is completed, we reset the proposed target h to hi and return to the first step until the algorithm converges. Pseudo-code for EM-DD is given in Figure 1. We now briefly provide intuition as to why EM-DD improves both the accuracy and computation time of the DD algorithm. Again, the basic approach of DD is to use a gradient search to find a value of h that maximizes DD(h). In every search step, the DD algorithm uses all points in each bag and hence the maximum that occurs in Equation (1) must be computed. The prior diverse density algorithms [1,5,6,7] used a softmax approximation for the maximum (so that it will be differentiable), which dramatically increases the computation complexity and introduces additional error based on the parameter selected in softmax. In comparison, EM-DD converts the multiple-instance data to single-instance data by removing all but one point per bag in the E -step, which greatly simplifies the search step since the maximum that occurs in Equation (1) is removed in the E -step. The removal of softmax in EMDD greatly decreases the computation time. In addition, we believe that EM-DD helps avoid getting caught in local minimum since it makes major changes in the hypothesis when it switches which point is selected from a bag. We now provide a sketch of the proof of convergence of EM-DD. Note that at each iteration t , given a set of instances selected in the E-step, the M-step will find a unique hypothesis (ht ) and corresponding DD (ddt). At iteration t + 1, if ddt+1 ::; ddt , the algorithm will terminate. Otherwise, ddt+1 > ddt , which means that a different set of instances are selected. For the iteration to continue, the DD will decrease monotonically and the set of instances selected can not repeat. Since there are only finite number of sets to instances that can be selected at the E-step, the algorithm will terminate after a finite number of iterations. However, there is no guarantee on the convergence rate of EM algorithms. We found that the NLDD(h, D) usually decreases dramatically after the first several iterations and then begins to flatten out. From empirical tests we found that it is often beneficial to allow NLDD to increase slightly to escape a local minima and thus we used the less restrictive termination condition: Idd1 ddo I < 0.01 . ddo or the number of iterations is greater than 10. This modification reduces the training time while gaining comparable results. However, for this modification no convergence proof can be given without restricting the number of iterations. 4 Experimental results In this section we summarize our experimental results. We begin by reporting our results for the two musk benchmark data sets provided by Dietterich et al. [4]. These data sets contain 166 feature vectors describing the surface for low-energy conformations of 92 molecules for Muskl and 102 molecules for Musk2 where roughly half of the molecules are known to smell musky and the remainder are not. The Musk1 data set is smaller both in having fewer bags (i.e molecules) and many fewer instances per bag (an average of 6.0 for Musk1 versus 64.7 for Musk2). Prior to this work, the highly-tuned iterated-discrim algorithm of Dietterich et al. still gave the best performance on both Musk1 and Musk2. Maron and Lozano-Perez [6] Main(k, D) partition D = {D1 ' D2, ... , D10 }; 111 O-fold cross validation for (i = l ;i:::; 10;i++) Dt = D - Di; IIDt training data, Di validation data pick k random positive bags B 1 , ... , B k from Dt ; let Ho be the union of all instances from selected bags; for every instance Ij E H 0 hj = EM-DD (Ij, Dt); ei = mino:<;:j:<;:IIHoll{error(hj,Di)}; return avg(e1,e2, ... , e1o) ; EM-DD(I, Dt ) Let h = {h1' ... , hn , Sl, ... , sn}; Ilinitial hypothesis For each dimension d = 1, ... , n hd = Id; Sd = 0.1 ; nlddo = +00; nldd1 = NLDD(h, Dt); while (nldd1 < nlddo) for each bag Bi E Dt liE-step pi = argmaxBijEBi Pr(Bij E h); hi = argmaXhEH flP r(fi I h,pi); 11M-step nlddo = nldd1; nldd1 = NLDD(hl,Dt); h = hi; return h; Figure 1: Pseudo-code for EM-DD where k indicates the number of different starting bags used, Pr(Bij E h) = exp[- I:~=1 (sd(Bijd - hd))2]. Pr(fi I h,p,!) is calculate as either 1-lfi - Pr(pi E h) I (linear model) or exp[-( fi - Pr(pi E h) )2] (Gaussian-like model) , where Pr(pi E h) = maxBijEBi Pr(Bij E h). summarize the generally held belief that "The performance reported for iterateddiscrim APR involves choosing parameters to maximize the test set performance and so probably represents an upper bound for accuracy on this (Musk1) data set." EM-DD without tuning outperforms all previous algorithms. To be consistent with the way in which past results have been reported for the musk benchmarks we report the average accuracy of la-fold cross-validation (which is the value returned by Main in Figure l. EM-DD obtains an average accuracy of 96.8% on Musk1 and 96.0% on Musk2. A summary of the performance of different algorithms on the Musk1 and Musk2 data sets is given in Table l. In addition, for both data sets, there are no false negative errors using EM-DD , which is important for the drug discovery application since the final hypothesis would be used to filter potential drugs and a false negative error means that a potential good drug molecule would not be tested and thus it is good to minimize such errors. As compared to the standard DD algorithm , EM-DD only used three random bags for Muskl and two random bags for Musk2 (versus all positive bags used in DD) as the starting point of the algorithm. Also, unlike the results reported in [6] in which the threshold is tuned based on leave-one-out cross validation, for our reported results the threshold value (of 0.5) is not tuned. More importantly, EM-DD runs over 10 times faster than DD on Musk1 and over 100 times faster when applied to Musk2. Table 1: Comparison of performance on Musk1 and Musk2 data sets as measured by giving the average accuracy across 10 runs using 10-fold cross validation. Algorithm EM-DD Iterated-discrim [4] Citation-kNN [11] Bayesian-kNN [11] Diverse density [6] Multi-instance neural network [9] Multinst [2] Musk1 accuracy 96.8% 92.4% 92.4% 90.2% 88.9% 88.0% 76.7% Musk 2 accuracy 96.0% 89.2% 86.3% 82.4% 82.5% 82.0% 84.0% In addition to its superior performance on the musk data sets, EM-DD can handle real-value labeled data and produces real-value predictions. We present results using one real data set (Affinity) 1 that has real-value labels and several artificial data sets generated using the technique of our earlier work [1]. For these data sets, we used as our starting points the points from the bag with the highest DD value. The result are shown in Table 2. The Affinity data set has 283 features and 139 bags with an average of 32.5 points per bag. Only 29 bags have labels that were high enough to be considered as "positive." Using the Gaussian-like version of our generative model we obtained a squared loss of 0.0185 and with the linear model we performed slightly better with a loss of 0.0164. In contrast using the standard diverse density algorithm the loss was 0.042l. EM-DD also gained much better performance than DD on two artificial data (160.166.1a-S and 80.166.1a-S) where both algorithms were used2 . The best result on Affinity data was obtained using a version of citation-kNN [1] that works with real-value data with the loss as 0.0124. We think that the affinity data set is well-suited for a nearest neighbor approach in that all of the negative bags have labels between 0.34 and 0.42 and so the actual predictions for the negative bags are better with citation-kNN. To study the sensitivity of EM-DD to the number ofrelevant attributes and the size of the bags, tests were performed on artificial data sets with different number of relevant features and bag sizes. As shown in Table 2, similar to the DD algorithm [1], the performance of EM-DD degrades as the number of relevant features decreases. This behavior is expected since all scale factors are initialized to the same value and when most of the features are relevant less adjustment is needed and hence the algorithm is more likely to succeed. In comparison to DD , EM-DD is more robust against the change of the number of relevant features. For example, as shown in Figure 2, when the number of relevant features is 160 out of 166, both EM-DD and DD algorithms perform well with good correlation between the actual labels and predicted labels. However, when the number of relevant features decreases to 80, almost no correlation between the actual and predicted labels is found using DD , while EM-DD can still provide good predictions on the labels. Intuitively, as the size of bags increases, more ambiguity is introduced to the data and the performance of algorithms is expected to go down. However, somewhat ] Jonathan Greene from CombiChem provided us with the Affinity data set. However, due to the proprietary nature of it we cannot make it publicly available. 2See Amar et al. [1] for a description of these two data sets. Table 2: Performance on data with real-value labels measured as squared loss. Data set # reI. features #pts per bag EM-DD DD [1] Affinity 32.5 .0164 .0421 160.166.1a-S 160 4 .0014 .0052 160.166.1b-S 160 15 .0013 160.166.1c-S 160 25 .0012 80.166.1a-S 80 4 .0029 .1116 80.166.1b-S 80 15 .0023 80.166.1c-S 80 25 .0022 40.166.1a-S 40 4 .0038 40.166.1b-S 40 15 .0026 40.166.1c-S 40 25 .0037 surprisingly, the performance of EM-DD actually improves as the number of examples per bag increases. We believe that this is partly due to the fact that with few points per bag the chance that a bad starting point has the highest diverse density is much higher than when the bags are large. In addition, in contrast to the standard diverse density algorithm, the overall time complexity of EM-DD does not go up as the size of the bags increased, since after the instance selection (E-step) , the time complexities of the dominant M-step are essentially the same for data sets with different bag sizes. The fact that EM-DD scales up well to large bag sizes in both performance and running time is very important for real drug-discovery applications in which the bags can be quite large. 5 Future directions There are many avenues for future work. We believe that EM-DD can be refined to obtain better performance by finding alternate ways to select the initial hypothesis and scale factors. One option would be to use the result from a different learning algorithm as the starting point then use EM-DD to refine the hypothesis. We are currently studying the application of the EM-DD algorithm to other domains such as content-based image retrieval. Since our algorithm is based on the diverse density likelihood measurement we believe that it will perform well on all applications in which the standard diverse density algorithm has worked well. In addition, EM-DD and MI regression [10] presented a framework to convert the multiple-instance data to single-instance data, where supervised learning algorithms can be applied. We are currently working on using this general methodology to develop new MI learning techniques based on supervised learning algorithms and EM. Acknowledgments The authors gratefully acknowledge the support NSF grant CCR-9988314. We thank Dan Dooly for many useful discussions. We also thank Jonathan Greene who provided us with the Affinity data set. References [1] Amar, R.A., Dooly, D.R., Goldman, S.A. & Zhang, Q. (2001). Multiple-Instance Learning of Real-Valued Data. Pr'oceedings 18th International Confer'ence on Machine Learning, pp. 3- 10. San Francisco, CA: Morgan Kaufmann. [2] Auer, P. (1997) On learning from mult-instance examples: Empirical evaluation of a theoretical approach. Proceedings 14th International Conference on Machine Learning, 160.166.1a-S (DD) 80.166.1a-S (DD) 0 . 8 0 . 8 0 . 6 0 . 6 , . 0 . 4 - ~-: :- T.;-~ --- - '- ' ':~.,::. - ..' 0 . 4 : .... 0.2 . ~. " 0.2 ' . ..... 0.2 0 . 4 0.6 0 . 8 0.2 0 . 4 0.6 0 . 8 Actual Actual 160.166.1a-S (EM-DD) 80.166.1a-S (EM-DD) 0 . 8 " ,': 0 . 8 .::::.; ":" .. ~ ~ 0 . 6 :'," ~ ~ 0 . 6 ~ 0 . 4 ~ 0 . 4 0.2 ", .. , .. :' . 0.2 ':': .... 0.2 0 . 4 0.6 0 . 8 0.2 0 . 4 0.6 0 . 8 Actual Actual Figure 2: Comparison of EM-DD and DD on real-value labeled artificial data with different number of relevant features. The x-axis corresponds to the actual label and y-axis gives t he predicted label. pp. 21-29. San Francisco, CA: Morgan Kaufmann. [3] Dempster, A.P., Laird, N.M. , & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistics Society, Series B, 39(1): 1-38. [4] Dietterich, T. G., Lathrop, R. H., & Lozano-Perez, T. (1997). Solving the multipleinstance problem with axis-parallel rectangles. Artificial Intelligence, 89(1-2): 31-7l. [5] Maron, O. (1998). Learning from Ambiguity. Doctoral dissertation, MIT, AI Technical Report 1639. [6] Maron, O. & Lozano-Perez, T. (1998). A framework for multiple-instance learning. Neural Information Processing Systems 10. Cambridge, MA: MIT Press. [7] Maron, O. & Ratan, A. (1998). Multiple-instance learning for natural scene classification. Proceedings 15th International Conference on Machine Learning, pp. 341-349. San Francisco, CA: Morgan Kaufmann. [8] Press, W.H., Teukolsky, S.A., Vetterling, W .T., and Flannery, B.P. (1992). Numerical Recipes in C: the art of scientific computing. Cambridge University Press, New York, second edition. [9] Ramon, J. & L. De Raedt. (2000). Multi instance neural networks. Proceedings of ICML-2000 workshop on "Attribute- Value and Relational Learning. [10] Ray, S. & Page, D. (2001) . Multiple-Instance Regression. Proceedings 18th International Conference on Machine Learning, pp. 425-432. San Francisco, CA: Morgan Kaufmann. [11] RufIo, G. (2000) . Learning single and multiple instance decision tr'ees for' computer' security applications. Doctoral dissertation. Department of Computer Science, University of Turin, Torino, Italy. [12] Wang, J. & Zucker, J.-D. (2000). Solving the Multiple-Instance Learning Problem: A Lazy Learning Approach. Proceedings 17th International Conference on Ma chine Learning, pp. 1119-1125. San Francisco, CA: Morgan Kaufmann.
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Fast Parameter Estimation Using Green's Functions K. Y. Michael Wong Department of Physics Hong Kong University of Science and Technology Clear Water Bay, Hong Kong phkywong@ust.hk FuIi Li Department of Applied Physics Xian Jiaotong University Xian, China 710049 flli@xjtu. edu. en Abstract We propose a method for the fast estimation of hyperparameters in large networks, based on the linear response relation in the cavity method, and an empirical measurement of the Green's function. Simulation results show that it is efficient and precise, when compared with cross-validation and other techniques which require matrix inversion. 1 Introduction It is well known that correct choices of hyperparameters in classification and regression tasks can optimize the complexity of the data model, and hence achieve the best generalization [1]. In recent years various methods have been proposed to estimate the optimal hyperparameters in different contexts, such as neural networks [2], support vector machines [3, 4, 5] and Gaussian processes [5]. Most of these methods are inspired by the technique of cross-validation or its variant, leave-one-out validation. While the leave-one-out procedure gives an almost unbiased estimate of the generalization error, it is nevertheless very tedious. Many of the mentioned attempts aimed at approximating this tedious procedure without really having to sweat through it. They often rely on theoretical bounds, inverses to large matrices, or iterative optimizations. In this paper, we propose a new approach to hyperparameter estimation in large systems. It is known that large networks are mean-field systems, so that when one example is removed by the leave-one-out procedure, the background adjustment can be analyzed by a self-consistent perturbation approach. Similar techniques have been applied to the neural network [6], Bayesian learning [7] and the support vector machine [5]. They usually involve a macroscopic number of unknown variables, whose solution is obtained through the inversion of a matrix of macroscopic size, or iteration. Here we take a further step to replace it by a direct measurement of the Green's function via a small number of learning processes. The proposed procedure is fast since it does not require repetitive cross-validations, matrix inversions, nor iterative optimizations for each set of hyperparaemters. We will also present simulation results which show that it is an excellent approximation. The proposed technique is based on the cavity method, which was adapted from disordered systems in many-body physics. The basis of the cavity method is a self-consistent argument addressing the situation of removing an example from the system. The change on removing an example is described by the Green's function, which is an extremely general technique used in a wide range of quantum and classical problems in many-body physics [8]. This provides an excellent framework for the leave-one-out procedure. In this paper, we consider two applications of the cavity method to hyperparameter estimation, namely the optimal weight decay and the optimal learning time in feedforward networks. In the latter application, the cavity method provides, as far as we are aware of, the only estimate of the hyperparameter beyond empirical stopping criteria and brute force cross-validation. 2 Steady-State Hyperparameter Estimation Consider the network with adjustable parameters w. An energy function E is defined with respect to a set of p examples with inputs and outputs respectively given by {IL and y'", JL = 1, ... ,p, where (IL is an N-dimensional input vector with components e;, j = 1,· ·· ,N, and N » 1 is macroscopic. We will first focus on the dynamics of a single-layer feedforward network and generalize the results to multilayer networks later. In single-layer networks, E has the form E = L f(X'",y'") + R(w). (1) '" Here f( x'" , y'") represents the error function with respect to example JL. It is expressed in terms of the activation x'" == w· (IL. R( w) represents a regularization term which is introduced to limit the complexity of the network and hence enhance the generalization ability. Learning is achieved by the gradient descent dynamics dWj(t) _ _ ~_oE_ dt (2) The time-dependent Green's function Gjk(t, s) is defined as the response of the weight Wj at time t due to a unit stimulus added at time s to the gradient term with respect to weight Wk, in the limit of a vanishing magnitude of the stimulus. Hence if we compare the evolution of Wj(t) with another system Wj(t) with a continuous perturbative stimulus Jhj(t), we would have dWj(t) = _~ oE Jh() dt Now. + J t , J (3) and the linear response relation Wj(t) = Wj(t) + L J dsGjk(t,s)Jhk(s). k (4) Now we consider the evolution ofthe network w;'"(t) in which example JL is omitted from the training set. For a system learning macroscopic number of examples, the changes induced by the omission of an example are perturbative, and we can assume that the system has a linear response. Compared with the original network Wj(t), the gradient of the error of example JL now plays the role of the stimulus in (3). Hence we have (5) Multiplying both sides by ~f and summing over j, we obtain 1-'( ) I-'() J [1 '" I-'G ( ) I-']OE(XI-'(S)'YI-') h t - x t + ds N "7:~j jk t ,s ~k oxl-'(s)' Here hl-'(t) == V;\I-'(t) . ~ is called the cavity activation of example ft. dynamics has reached the steady state, we arrive at hI-' I-' OE(XI-' , yl-') = x +, oxl-' ' where, = limt--+oo J dS[Ljk ~fGjk (t, s)~r]j N is the susceptibility. (6) When the (7) At time t, the generalization error is defined as the error function averaged over the distribution of input (, and their corresponding output y, i.e., (8) where x == V; . (is the network activation. The leave-one-out generalization error is an estimate of 109 given in terms ofthe cavity activations hI-' by fg = LI-' 10 (hI-' ,yl-')jp. Hence if we can estimate the Green's function, the cavity activation in (7) provides a convenient way to estimate the leave-one-out generalization error without really having to undergo the validation process. While self-consistent equations for the Green's function have been derived using diagrammatic methods [9], their solutions cannot be computed except for the specific case of time-translational invariant Green's functions, such as those in Adaline learning or linear regression. However, the linear response relation (4) provides a convenient way to measure the Green's function in the general case. The basic idea is to perform two learning processes in parallel, one following the original process (2) and the other having a constant stimulus as in (3) with 6hj (t) = TJ6jk, where 8j k is the Kronecka delta. When the dynamics has reached the steady state, the measurement Wj - Wj yields the quantity TJ Lk J dsGjk(t, s). A simple averaging procedure, replacing all the pairwise measurements between the stimulation node k and observation node j, can be applied in the limit of large N. We first consider the case in which the inputs are independent and normalized, i.e., (~j) = 0, (~j~k) = 8j k. In this case, it has been shown that the off-diagonal Green's functions can be neglected, and the diagonal Green's functions become selfaveraging, i.e., Gjk(t, s) = G(t, s)8jk , independent of the node labels [9], rendering , = limt--+oo J dsG(t, s). In the case that the inputs are correlated and not normalized, we can apply standard procedures of whitening transformation to make them independent and normalized [1]. In large networks, one can use the diagrammatic analysis in [9] to show that the (unknown) distribution of inputs does not change the self-averaging property of the Green's functions after the whitening transformation. Thereafter, the measurement of Green's functions proceeds as described in the simpler case of independent and normalized inputs. Since hyperparameter estimation usually involves a series of computing fg at various hyperparameters, the one-time preprocessing does not increase the computational load significantly. Thus the susceptibility, can be measured by comparing the evolution of two processes: one following the original process (2), and the other having a constant stimulus as in (3) with 8hj (t) = TJ for all j. When the dynamics has reached the steady state, the measurement (Wj - Wj) yields the quantity TJ,. We illustrate the extension to two-layer networks by considering the committee machine, in which the errorfunction takes the form E(2::a !(xa), y) , where a = 1,· ··, nh is the label of a hidden node, Xa == wa . [is the activation at the hidden node a, and! represents the activation function. The generalization error is thus a function of the cavity activations of the hidden nodes, namely, E9 = 2::JL E(2::a !(h~), yJL) /p, where h~ = w~JL . (IL . When the inputs are independent and normalized, they are related to the generic activations by hJLJL+'" aE(2:: c !(X~) , yJL) a - Xa ~ "lab a JL ' b Xb (9) where "lab = limt~ oo J dsGab(t, s) is the susceptibility tensor. The Green's function Gab(t, s) represents the response of a weight feeding hidden node a due to a stimulus applied at the gradient with respect to a weight feeding node b. It is obtained by monitoring nh + 1 learning processes, one being original and each of the other nh processes having constant stimuli at the gradients with respect to one of the hidden nodes, viz., dw~~) (t) _ 1 aE dt - - N ------=:(b) + 'f)rSab , b = 1, ... ,nh· aWaj (10) When the dynamics has reached the steady state, the measurement (w~7 - Waj) yields the quantity 'f)'Yab. We will also compare the results with those obtained by extending the analysis of linear unlearning leave-one-out (LULOO) validation [6]. Consider the case that the regularization R(w) takes the form of a weight decay term, R(w) = N 2::ab AabWa . Wb/2. The cavity activations will be given by hJL = JL + '" ( iJ 2::jk ~'j(A + Q)~}bk~r ) aE(2::c !(xn, yJL)) a Xa ~ ," 1 a JL ' b 1 - 11 2::cjdk ~'j !'(xn(A + Q)~ , dd'(x~)~r Xb (11) where E~ represents the second derivative of E with respect to the student output for example /1, and the matrix Aaj,bk = AabrSjk and Q is given by Qaj,bk = ~ 2: ~'j f'(x~)f'(x~)~r· (12) JL The LULOO result of (11) differs from the cavity result of (9) in that the susceptibility "lab now depends on the example label /1, and needs to be computed by inverting the matrix A + Q. Note also that second derivatives of the error term have been neglected. To verify the proposed method by simulations, we generate examples from a noisy teacher network which is a committee machine nh (1 ) yJL = ~ erf yf2Ba · f + (Jzw (13) Here Ba is the teacher vector at the hidden node a. (J is the noise level. ~'j and zJL are Gaussian variables with zero means and unit variances. Learning is done by the gradient descent of the energy function (14) and the weight decay parameter ,X is the hyperparameter to be optimized. The generalization error fg is given by where the averaging is performed over the distribution of input { and noise z. It can be computed analytically in terms of the inner products Q ab = wa . Wb, Tab = Ba . Bb and Rab = Ba . Wb [10]. However, this target result is only known by the teacher, since Tab and Rab are not accessible by the student. Figure 1 shows the simulation results of 4 randomly generated samples. Four results are compared: the target generalization error observed by the teacher, and those estimated by the cavity method, cross-validation and extended LULOO. It can be seen that the cavity method yields estimates of the optimal weight decay with comparable precision as the other methods. For a more systematic comparison, we search for the optimal weight decay in 10 samples using golden section search [11] for the same parameters as in Fig. 1. Compared with the target results, the standard deviations of the estimated optimal weight decays are 0.3, 0.25 and 0.24 for the cavity method, sevenfold cross-validation and extended LULOO respectively. In another simulation of 80 samples of the singlelayer perceptron, the estimated optimal weight decays have standard deviations of 1.2, 1.5 and 1.6 for the cavity method, tenfold cross-validation and extended LULOO respectively (the parameters in the simulations are N = 500, p = 400 and a ranging from 0.98 to 2.56). To put these results in perspective, we mention that the computational resources needed by the cavity method is much less than the other estimations. For example, in the single-layer perceptrons, the CPU time needed to estimate the optimal weight decay using the golden section search by the teacher, the cavity method, tenfold cross-validation and extended LULOO are in the ratio of 1 : 1.5 : 3.0 : 4.6. Before concluding this section, we mention that it is possible to derive an expression of the gradient dEg I d,X of the estimated generalization error with respect to the weight decay. This provides us an even more powerful tool for hyperparameter estimation. In the case of the search for one hyperparameter, the gradient enables us to use the binary search for the zero of the gradient, which converges faster than the golden section search. In the single-layer experiment we mentioned, its precision is comparable to fivefold cross-validations, and its CPU time is only 4% more than the teacher's search. Details will be presented elsewhere. In the case of more than one hyperparameters, the gradient information will save us the need for an exhaustive search over a multidimensional hyperparameter space. 3 Dynamical Hyperparameter Estimation The second example concerns the estimation of a dynamical hyperparameter, namely the optimal early stopping time, in cases where overtraining may plague the generalization ability at the steady state. In perceptrons, when the examples are noisy and the weight decay is weak, the generalization error decreases in the early stage of learning, reaches a minimum and then increases towards its asymptotic value [12, 9]. Since the early stopping point sets in before the system reaches the steady state, most analyses based on the equilibrium state are not applicable. Cross-validation stopping has been proposed as an empirical method to control overtraining [13]. Here we propose the cavity method as a convenient alternative. 0.52 e Q) <= 0 ~ 0.46 .!::! m Q) <= Q) 0> 0.40 e Q) <= 0 ~ .!::! m Q) <= Q) 0> 0.40 0 G----8 target G----EJ cavity 0-0 LULOO weight decay A (c) o (b) (d) 2 weight decay A Figure 1: (a-d) The dependence ofthe generalization error of the multilayer perceptron on the weight decay for N = 200, p = 700, nh = 3, (J = 0.8 in 4 samples. The solid symbols locate the optimal weight decays estimated by the teacher (circle), the cavity method (square), extended LULOO (diamond) and sevenfold cross-validation (triangle) . In single-layer perceptrons, the cavity activations of the examples evolve according to (6), enabling us to estimate the dynamical evolution of the estimated generalization error when learning proceeds. The remaining issue is the measurement of the time-dependent Green's function. We propose to introduce an initial homogeneous stimulus, that is, Jhj (t) = 1]J(t) for all j. Again, assuming normalized and independent inputs with (~j) = 0 and (~j~k) = Jjk , we can see from (4) that the measurement (Wj(t) - Wj(t)) yields the quantity 1]G(t, 0). We will first consider systems that are time-translational invariant, i.e., G(t, s) = G(t - s, 0). Such are the cases for Adaline learning and linear regression [9], where the cavity activation can be written as h'"(t) = x'"(t) + J dsG(t - s, 0) OE(X'"(S), y'"). ox,"(s) This allows us to estimate the generalization error Eg(t) via Eg(t) (16) 2:.," E(h'"(t), y'")/p, whose minimum in time determines the early stopping point. To verify the proposed method in linear regression, we randomly generate examples from a noisy teacher with y'" = iJ . f'" + (Jzw Here iJ is the teacher vector with B2 = 1. e; and z'" are independently generated with zero means and unit variances. Learning is done by the gradient descent of the energy function E(t) = 2:.,"(y'" - w(t) . f'")2/2. The generalization error Eg(t) is the error averaged over the distribution of input [ and their corresponding output y, i.e., Eg(t) = ((iJ . [ + (JZ - w· [)2/2). As far as the teacher is concerned, Eg(t) can be computed as Eg(t) = (1 - 2R(t) + Q(t) + (J2)/2. where R(t) = w(t) . iJ and Q(t) = W(t)2. Figure 2 shows the simulation results of 6 randomly generated samples. Three results are compared: the teacher's estimate, the cavity method and cross-validation. Since LULOO is based on the equilibrium state, it cannot be used in the present context. Again, we see that the cavity method yields estimates of the early stopping time with comparable precision as cross-validation. The ratio of the CPU time between the cavity method and fivefold cross-validation is 1 : 1.4. For nonlinear regression and multilayer networks, the Green's functions are not time-translational invariant. To estimate the Green's functions in this case, we have devised another scheme of stimuli. Preliminary results for the determination of the early stopping point are satisfactory and final results will be presented elsewhere. 1 .1 e Q.i c:: a ~ 0.9 .!::! ~ <l> c:: <l> 0> 0.7 e Q.i c:: a ~ 0.9 .!::! c;; Q.i c:: <l> 0> 0.7 0 2 0 2 0 2 4 time t time t time t Figure 2: (a-f) The evolution of the generalization error of linear regression for N = 500, p = 600 and (J = 1. The solid symbols locate the early stopping points estimated by the teacher (circle), the cavity method (square) and fivefold crossvalidation (diamond). 4 Conclusion We have proposed a method for the fast estimation of hyperparameters in large networks, based on the linear response relation in the cavity method, combined with an empirical method of measuring the Green's function. Its efficiency depends on the independent and identical distribution of the inputs, greatly reducing the number of networks to be monitored. It does not require the validation process or the inversion of matrices of macroscopic size, and hence its speed compares favorably with cross-validation and other perturbative approaches such as extended LULOO. For multilayer networks, we will explore further speedup of the Green's function measurement by multiplexing the stimuli to the different hidden units into a single network, to be compared with a reference network. We will also extend the technique to other benchmark data to study its applicability. Our initial success indicates that it is possible to generalize the method to more complicated systems in the future. The concept of Green's functions is very general, and its measurement by comparing the states of a stimulated system with a reference one can be adopted to general cases with suitable adaptation. Recently, much attention is paid to the issue of model selection in support vector machines [3, 4, 5]. It would be interesting to consider how the proposed method can contribute to these cases. Acknowledgements We thank C. Campbell for interesting discussions and H. Nishimori for encouragement. This work was supported by the grant HKUST6157/99P from the Research Grant Council of Hong Kong. References [1] C. M. Bishop, Neural Networks for Pattern Recognition, Clarendon Press, Oxford (1995). [2] G. B. Orr and K-R. Muller, eds., Neural Networks: Tricks of the Trade, Springer, Berlin (1998). [3] O. Chapelle and V. N. Vapnik, Advances in Neural Information Processing Systems 12, S. A. Solla, T. KLeen and K-R. Muller, eds., MIT Press, Cambridge, 230 (2000). [4] S. S. Keerthi, Technical Report CD-OI-02, http://guppy.mpe.nus.edu.sg/ mpessk/nparm.html (2001). [5] M. Opper and O. Winther, Advances in Large Margin Classifiers, A. J. Smola, P. Bartlett, B. Sch6lkopf and D. Schuurmans, eds., MIT Press, Cambridge, 43 (1999). [6] J. Larsen and L. K Hansen, Advances in Computational Math ematics 5, 269 (1996). [7] M. Opper and O. Winther, Phys. Rev. Lett. 76, 1964 (1996). [8] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGrawHill, New York (1971). [9] K Y. M. Wong, S. Li and Y. W. Tong, Phys. Rev. E 62, 4036 (2000). [10] D. Saad and S. A. Solla, Phys. Rev. Lett. 74, 4337 (1995). [11] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, Cambridge (1990). [12] A. Krogh and J. A. Hertz, J. Phys. A 25, 1135 (1992). [13] S. Amari, N. Murata, K-R. Muller, M. Finke and H. H. Yang, IEEE Trans. on Neural Networks 8, 985 (1997).
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Fast and Robust Classification using Asymmetric AdaBoost and a Detector Cascade Paul Viola and Michael Jones Mistubishi Electric Research Lab Cambridge, MA viola@merl.com and mjones@merl.com Abstract This paper develops a new approach for extremely fast detection in domains where the distribution of positive and negative examples is highly skewed (e.g. face detection or database retrieval). In such domains a cascade of simple classifiers each trained to achieve high detection rates and modest false positive rates can yield a final detector with many desirable features: including high detection rates, very low false positive rates, and fast performance. Achieving extremely high detection rates, rather than low error, is not a task typically addressed by machine learning algorithms. We propose a new variant of AdaBoost as a mechanism for training the simple classifiers used in the cascade. Experimental results in the domain of face detection show the training algorithm yields significant improvements in performance over conventional AdaBoost. The final face detection system can process 15 frames per second, achieves over 90% detection, and a false positive rate of 1 in a 1,000,000. 1 Introduction In many applications fast classification is almost as important as accurate classification. Common examples include robotics, user interfaces, and classification in large databases. In this paper we demonstrate our approach in the domain of low latency, sometimes called “real-time”, face detection. An extremely fast face detector is a critical component in many applications. User-interfaces can be constructed which detect the presence and number of users. Teleconference systems can automatically devote additional bandwidth to participant’s faces. Video security systems can record facial images of individuals after unauthorized entry. Recently we presented a real-time face detection system which scans video images at 15 frames per second [8] yet achieves detection rates comparable with the best published results (e.g. [7]) 1 Face detection is a scanning process, in which a face classifier is evaluated at every scale and location within each image. Since there are about 50,000 unique scales 1In order to achieve real-time speeds other systems often resort to skin color filtering in color images or motion filtering in video images. These simple queues are useful but unreliable. In large image databases color and motion are often unavailable. Our system detects faces using only static monochrome information. and locations in a typical image, this amounts to evaluating the face classifier 750,000 times per second. One key contribution of our previous work was the introduction of a classifier cascade. Each stage in this cascade was trained using AdaBoost until the required detection performance was achieved [2]. In this paper we present a new training algorithm designed specifically for a classifier cascade called asymmetric AdaBoost. The algorithm is a generalization of that given in Singer and Shapire [6]. Many of the formal guarantees presented by Singer and Shapire also hold for this new algorithm. The paper concludes with a set of experiments in the domain of face detection demonstrating that asymmetric AdaBoost yields a significant improvement in detection performance over conventional boosting. 2 Classifier Cascade In the machine learning community it is well known that more complex classification functions yield lower training errors yet run the risk of poor generalization. If the main consideration is test set error, structural risk minimization provides a formal mechanism for selecting a classifier with the right balance of complexity and training error [1]. Another significant consideration in classifier design is computational complexity. Since time and error are fundamentally different quantities, no theory can simply select the optimal trade-off. Nevertheless, for many classification functions computation time is directly related to the structural complexity. In this way temporal risk minimization is clearly related to structural risk minimization. This direct analogy breaks down in domains where the distribution over the class labels is highly skewed. For example, in the domain of face detection, there are at most a few dozen faces among the 50,000 sub-windows in an image. Surprisingly in these domains it is often possible to have the best of both worlds: high detection rates and extremely fast classification. The key insight is that while it may be impossible to construct a simple classifier which can achieve a low training/test error, in some cases it is possible to construct a simple classifier with a very low false negative rate. For example, in the domain of face detection, we have constructed an extremely fast classifier with a very low false negative rate (i.e. it almost never misses a face) and a 50% false positive rate. Such a detector might be more accurately called a classification pre-filter: when an image region is labeled ’nonface’ then it can be immediately discarded, but when a region is labeled ’face’ then further classification effort is required. Such a pre-filter can be used as the first stage in a cascade of classifiers (see Figure 1). In our face detection application (described in more detail in Section 5) the cascade has 38 stages. Even though there are many stages, most are not evaluated for a typical nonface input window since the early stages weed out many non-faces. In fact, over a large test set, the average number of stages evaluated is less than 2. In a cascade, computation time and detection rate of the first few stages is critically important to overall performance. The remainder of the paper describes techniques for training cascade classifiers which are efficient yet effective. 3 Using Boosting to Train the Cascade In general almost any form of classifier can be used to construct a cascade; the key properties are that computation time and the detection rate can be adjusted. Examples include support vector machines, perceptrons, and nearest neighbor classifiers. In the case of an SVM computation time is directly related to the number of support vectors and detection rate is related to the margin threshold [1]. T F T F T F 1 2 3 Reject Sub−window All Sub−windows Further Processing Figure 1: Schematic depiction of a detection cascade. A sequence of classifiers are applied to every example. The initial classifier eliminates a large number of negative examples with very little processing. Subsequent stages eliminate additional negatives but require additional computation. Extremely few negative examples remain after several stages. In our system each classifier in the cascade is a single layer perceptron whose input is a set of computationally efficient binary features. The computational cost of each classifier is then simply the number of input features. The detection rate is adjusted by changing the threshold (or bias). Much of the power of our face detection system comes from the very large and varied set of features available. In our experiments over 6,000,000 different binary features were available for inclusion in the final classifiers (see Figure 4 for some example features). The efficiency of each classifier, and hence the efficiency of the cascade, is ensured because a very small number of features are included in the early stages; the first stage has 1 (!) feature, the second stage 5 features, then 20, and then 50. See Section 5 for a brief description of the feature set. The main contribution of this paper is the adaptation of AdaBoost for the task of feature selection and classifier learning. Though it is not widely appreciated, AdaBoost provides a principled and highly efficient mechanism for feature selection[2, 6]. If the set of weak classifiers is simply the set of binary features (this is often called boosting stumps) each round of boosting adds a single feature to the set of current features. AdaBoost is an iterative process in which each round selects a weak classifier, , which minimizes:
! (1) Following the notation of Shapire and Singer,
is the weight on example at round " , $#&% (')*'+ is the target label of the example, is the example, and !, is a confidence rated binary classifier[6]. After every round the weights are updated as follows:
-/. 01
(2) The classifier 2 takes on two possible values 3-465879;: <;=>= <?=A@ and CBDE5F7>9;: <G@H@ <G@>= , where IKJ*L is the weight of the examples given the label which have true label M . These predictions insure that the weights on the next round are balanced: that the relative weights of positive and negative examples one each side of the classification boundary are equal. Minimizing minimizes the weighted exponential loss at round " . Minimizing in each round is also a greedy technique for minimizing N which is an upper bound on the training error of the strong classifier. It has also been observed that the example weights are directly related to example margin, which leads to a principled argument for AdaBoost’s generalization capabilities [5]. The key advantage of AdaBoost as a feature selection mechanism, over competitors such as the wrapper method [3], is the speed of learning. Given the constraint that the search over features is greedy, AdaBoost efficiently selects the feature which minimizes N , a surrogate for overall classification error. The entire dependence on previously selected features is efficiently and compactly encoded using the example weights. As a result, the addition of the 100th feature requires no more effort than the selection of the first feature. 2 4 Asymmetric AdaBoost One limitation of AdaBoost arises in the context of skewed example distributions and cascaded classifiers: AdaBoost minimizes a quantity related to classification error; it does not minimize the number of false negatives. Given that the final form of the classifier is a weighted majority of features, the detection and false positive rates are adjustable after training. Unfortunately feature selection proceeds as if classification error were the only goal, and the features selected are not optimal for the task of rejecting negative examples. One naive scheme for “fixing” AdaBoost is to modify the initial distribution over the training examples. If we hope to minimize false negatives then the weight on positive examples could be increased so that the minimum error criteria will also have very few false negatives. We can formalize this intuitive approach as follows. Recall that AdaBoost is a scheme which minimizes: $ / A (3) Each term in the summation is bounded above by a simple loss function: >
0$ ' if ) otherwise (4) where is the class assigned by the boosted classifier. As a result, minimizing N minimizes an upper bound on simple loss. We can introduce a related notion of asymmetric loss:
01 if ' and 6(' ) . if (' and 1E' ) otherwise (5) where false negatives cost times more than false positives. Note that
0 >! * 5879 "
0 . If we take the bound in Equation 4 and multiply both sides by we obtain a bound on the asymmetric loss: $# H " % . Minimization of this bound can be achieved using AdaBoost by pre-weighting each example by > 5879 " . The derivation is identical to that of Equation 3. Expanding 2Given that there are millions of features and thousands of examples, the boosting process requires days of computation. Many other techniques while feasible for smaller problems are likely to be infeasible for this sort of problem. Equation 2 repeatedly for
0 in terms of
B3. 0 we arrive at,
-/. 1 # H 5879 " N ) (6) where the second term in the numerator arises because of the initial asymmetric weighting. Noticing that the left hand side must sum to 1 yields the following equality, $ !/ A D 5F7>9 " (7) Therefore AdaBoost minimizes the required bound on asymmetric loss. Unfortunately this naive technique is only somewhat effective. The main reason is AdaBoost’s balanced reweighting scheme. As a result the initially asymmetric example weights are immediately lost. Essentially the AdaBoost process is too greedy. The first classifier selected absorbs the entire effect of the initial asymmetric weights. The remaining rounds are entirely symmetric. We propose a closely related approach that results in the minimization of the same bound, which nevertheless preserves the asymmetric loss throughout all rounds. Instead of applying the necessary asymmetric multiplier > 5879 " at the first round of an round process, the nth root . 5F7>9 " is applied before each round. Referring to Equation 6 we can see the final effect is the same; this preserves the bound on asymmetric loss. But the effect on the training process is quite different. In order to demonstrate this approach we generated an artificial data set and learned strong classifiers containing 4 weak classifiers. The results are shown inFigure 2. In this figure we can see that all but the first weak classifier learned by the naive rule are poor, since they each balance positive and negative errors. The final combination of these classifiers cannot yield high detection rates without introducing many false positives. All the weak classifiers generated by the proposed Asymmetric Adaboost rule are consistent with asymmetric loss and the final strong classifier yields very high detection rates and modest false positive rates. One simple reinterpretation of this distributed scheme for asymmetric reweighting is as a reduction in the positive confidence of each weak classifier ,$ . 5F7>9 . This forces each subsequent weak classifier to focus asymmetrically on postive examples. 5 Experiments We performed two experiments in the domain of frontal face detection to demonstrate the advantages of asymmetric AdaBoost. Experiments follow the general form, though differ in details, from those presented in Viola and Jones [8]. In each round of boosting one of a very large set of binary features are selected. These features, which we call rectangle features, are briefly described in Figure 4. In the first experiment a training and test set containing faces and non-faces of a fixed size were acquired (faces were scaled to a size pixels). The training set consisted of 1500 face examples and 5000 non-face examples. Test data included 900 faces and 5000 nonfaces. The face examples were manually cropped from a large collection of Web images while the non-face examples were randomly chosen patches from Web images that were known not to contain any faces. Naive asymetric AdaBoost and three parameterizations of Asymmetric AdaBoost were used to train classifiers with 4 features on this data. Figure 3 shows the ROC curves on Figure 2: Two simple examples: positive examples are ’x’, negative ’o’ and weak classifiers are linear separators. On the left is the naive asymetric result. The first feature selected is labelled ’1’. Subsequent features attempt to balance positive and negative errors. Notice that no linear combination of the 4 weak classifiers can achieve a low false positive and low false negative rate. On the right is the asymetric boosting result. After learning 4 weak classifier the positives are well modelled and most of the negative are rejected. 0.97 0.975 0.98 0.985 0.99 0.995 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 NAIVE T11-F10 T15-F10 T20-F10 Figure 3: ROC curves for four boosted classifier with 4 features. The first is naive asymmetric boosting. The other three results are for the new asymmetric approach, each using slightly different parameters. The ROC curve has been cropped to show only the region of interest in training a cascaded detector, the high detection rate regime. Notice that that at 99% detection asymmetric Adaboost cuts the false positive by about 20%. This will significantly reduce the work done by later stages in the cascade. A B C D Figure 4: Left: Example rectangle features shown relative to the enclosing detection window. The sum of the pixels which lie within the white rectangles are subtracted from the sum of pixels in the gray rectangles. A threshold operation is then applied to yield a binary output. Two-rectangle features are shown in (A) and (B). Figure (C) shows a threerectangle feature, and (D) a four-rectangle feature. Right: The first two example feature selected by the boosting process. Notice that the first feature relies on the fact that the horizontal region of the eyes is darker than the horizontal region of the cheeks. The second feature, whose selection is conditioned on the first, acts to distinguish horizontal edges from faces by looking for a strong vertical edge near the nose. test data for the three classifiers. The key result here is that at high detection rates the false positive rate can be reduced significantly. In the second experiment, naive and asymmetric AdaBoost were used to train two different complete cascaded face detectors. Performance of each cascade was determined on a realworld face detection task, which requires scanning of the cascade across a set of large images which contain embedded faces. The cascade training process is complex, and as a result comparing detection results is useful but potentially risky. While the data used to train the two cascades were identical, the performance of earlier stages effects the selection of non-faces used to train later stages. As a result different non-face examples are used to train the corresponding stages for the Naive and Asymmetric results. Layers were added to each of the cascades until the number of false positives was reduced below 100 on a validation set. For normal boosting this occurred with 34 layers. For asymmetric AdaBoost this occurred with 38 layers. Figure 5 shows the ROC curves for the resulting face detectors on the MIT+CMU [4] test set. 3 Careful examination of the ROC curves show that the asymmetric cascade reduces the number of false positives significantly. At a detection rate of 91% the reduction is by a factor of 2. 6 Conclusions We have demonstrated that a cascade classification framework can be used to achieve fast classification, high detection rates, and very low false positive rates. The goal for each classifier in the cascade is not low error, but instead extremely high detection rates and modest false positive rates. If this is achieved, each classifier stage can be used to filter out and discard many negatives. 3Note: the detection and false positive rates for the simple 40 feature experiment and the more complex cascaded experiment are not directly comparable, since the test sets are quite different. 0 50 100 150 200 250 300 0.8 0.85 0.9 0.95 correct detection rate false positives ROC curves for face detector with different boosting algorithms Asymmetric Boosting Normal Boosting Figure 5: ROC curves comparing the accuracy of two full face detectors, one trained using normal boosting and the other with asymmetric AdaBoost. Again, the detector trained using asymmetric AdaBoost is more accurate over a wide range of false positive values. Many modern approaches for classification focus entirely on the minimization of errors. Questions of relative loss only arise in the final tuning of the classifier. We propose a new training algorithm called asymmetric AdaBoost which performs learning and efficient feature selection with the fundamental goal of achieving high detection rates. Asymmetric AdaBoost is a simple modification of the “confidence-rated” boosting approach of Singer and Shapire. Many of their derivations apply to this new approach as well. Experiments have demonstrated that asymmetric AdaBoost can lead to significant improvements both in classification speed and in detection rates. References [1] Corinna Cortes and Vladimir Vapnik. Support-vector networks. Machine Learning, 20, 1995. [2] Yoav Freund and Robert E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In Computational Learning Theory: Eurocolt ’95, pages 23–37. Springer-Verlag, 1995. [3] G. John, R. Kohavi, and K. Pfleger. Irrelevant features and the subset selection problem. In Machine Learning Conference, pages 121–129. Morgan Kaufmann, 1994. [4] H. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. In IEEE Patt. Anal. Mach. Intell., volume 20, pages 22–38, 1998. [5] R. E. Schapire, Y. Freund, P. Bartlett, and W. S. Lee. Boosting the margin: a new explanation for the effectiveness of voting methods. Ann. Stat., 26(5):1651–1686, 1998. [6] Robert E. Schapire and Yoram Singer. Improved boosting algorithms using confidence-rated predictions. Machine Learning, 37:297–336, 1999. [7] H. Schneiderman and T. Kanade. A statistical method for 3D object detection applied to faces and cars. In Computer Vision and Pattern Recognition, 2000. [8] Paul Viola and Michael J. Jones. Robust real-time object detection. In Proc. of IEEE Workshop on Statistical and Computational Theories of Vision, 2001.
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Information-geometric decomposition spike analysis Hiroyuki Nakahara; Shun-ichi Amari Lab. for Mathematical Neuroscience, RIKEN Brain Science Institute 2-1 Hirosawa, Wako, Saitama, 351-0198 Japan {him, amari} @brain.riken.go.jp Abstract We present an information-geometric measure to systematically investigate neuronal firing patterns, taking account not only of the second-order but also of higher-order interactions. We begin with the case of two neurons for illustration and show how to test whether or not any pairwise correlation in one period is significantly different from that in the other period. In order to test such a hypothesis of different firing rates, the correlation term needs to be singled out 'orthogonally' to the firing rates, where the null hypothesis might not be of independent firing. This method is also shown to directly associate neural firing with behavior via their mutual information, which is decomposed into two types of information, conveyed by mean firing rate and coincident firing, respectively. Then, we show that these results, using the 'orthogonal' decomposition, are naturally extended to the case of three neurons and n neurons in general. 1 Introduction . In Based on the theory of hierarchical structure and related invariant decomposition of interactions by information geometry [3], the present paper briefly summarizes methods useful for systematically analyzing a population of neural firing [9]. Many researches have shown that the mean firing rate of a single neuron may carry significant information on sensory and motion signals. Information conveyed by populational firing, however, may not be only an accumulation of mean firing rates. Other statistical structure, e.g., coincident firing [13, 14], may also carry behavioral information. One obvious step to investigate this issue is to single out a contribution by coincident firing between two neurons, i.e., the pairwise correlation [2, 6]. In general, however, it is not sufficient to test a pairwise correlation of neural firing, because there can be triplewise and higher correlations. For example, three variables (neurons) are not independent in general even when they are pairwise independent. We need to establish a systematic method of analysis, including these higher-order • also affiliated with Dept. of Knowledge Sci., Japan Advanced Inst. of Sci. & Tech. correlations [1, 5,7, 13]. We propose one approach, the information-geometric measure that uses the dual orthogonality of the natural and expectation parameters in exponential family distributions [4]. We represent a neural firing pattern by a binary random vector x. The probability distribution of firing patterns can be expanded by a log linear model, where the set {p( x)} of all the probability distributions forms a (2n - I)-dimensional manifold 8 n. Each p(x) is given by 2n probabilities pi1···in=Prob{X1=i1,···,Xn=in}, ik=O,I, subjectto L Pi1···in=1 il ,"',in and expansion in log p( x) is given by logp(x) = L BiXi + L BijXiXj + L BijkXiXjXk··· + B1 ... nX1 ... Xn - 'Ij;, i<j i<j<k where indices of Bijk, etc. satisfy i < j < k, etc. We can have a general theory of this n neuron case [3, 9], however, to be concrete given the limited space, we mainly discuss two and three neuron cases in the present paper. Our method shares some features with previous studies (e.g. [7]) in use of the log linear model. Yet, we make explicit use of the dual orthogonality so that the method becomes more transparent and more systematic. In the present paper, we are interested in addressing two issues: (1) to analyze correlated firing of neurons and (2) to connect such a technique with behavioral events. In (1), previous studies often assumed independent firing as the null hypothesis. However, for example, when we compare firing patterns in two periods, as control and 'test' periods, there may exist a weak correlation in the control period. Hence, benefiting from the 'orthogonal' coordinates, we develop a method applicable to the null hypothesis of non-independent firing, irrespective of firing rates. It is equally important to relate such a method with investigation of behavioral significance as (2). We show that we can do so, using orthogonal decomposition of the mutual information (MI) between firing and behavior [11, 12]. In the following, we discuss first the case of two neurons and then the case of three neurons, demonstrating our method with artificial simulated data. The validity of our method has been shown also with experimental data[9, 10] but not shown here due to the limited space. 2 Information-geometric measure: case of two neurons We denote two neurons by Xl and X 2 (Xi = 1, ° indicates if neuron i has a spike or not in a short time bin). Its joint probability p(x), x = (X1,X2), is given by Pij = Prob{x1 = i;X2 = j} > 0, i,j = 0, 1. Among four probabilities, {POO ,P01,P10,Pl1}, only three are free. The set of all such distributions of x forms a three-dimensional manifold 8 2. Any three of Pij can be used as a coordinate system of 8 2. There are many different coordinate systems of 8 2 . The coordinates of the expectation parameters, called 17-coordinates, 'TI = (171,172,1712), is given by 17i = Prob {Xi = I} = E [Xi], i = 1,2, 173 = 1712 = E [X1 X2] = P12, where E denotes the expectation and 17i and 1712 correspond to the mean firing rates and the mean coincident firing, respectively. As other coordinate systems, we can also use the triplet, (171,172, Cov [Xl, X 2]) , where Cov [Xl , X 2] is the covariance,and/or the triplet (171,172, p), where p is the correlation coefficient (COR), p = J '112 -,/11 '12 , often called N-JPSTH [2]. '/11 (l - '7d'72 (1 - '72) Which quantity would be convenient to represent the pairwise correlational component? It is desirable to define the degree of the correlation independently from the marginals (171,172), To this end, we use the 'orthogonal' coordinates (171 , 172, B), originating from information geometry of 8 2 , so that the coordinate curve of B is always orthogonal to those of 171 and 172. The orthogonality of two directions in 8 2 (8n in general) is defined by the Riemannian metric due to the Fisher information matrix [8, 4]. Denoting any coordinates in 8 n by ~ = (6, ... , ~n)' the Fisher information matrix G is given by (1) where l (x;~) = logp (x; ~). The orthogonality between ~i and ~j is defined by 9ij(~) = O. In case of 8 2 , we desire to have E [tel (X;171 , 172, B) 8~il(x;171'172,B)] = o (i = 1, 2). When B is orthogonal to (171, 172), we say that B represents pure correlations independently of marginals. Such B is given by the following theorem. Theorem 1. The coordinate B = log PuPoo (2) P01PlO is orthogonal to the marginals 171 and 172. We have another interpretation of B. Let's expand p(x) by logp(x) = L;=l BiXi + B12X1X2 'IjJ. Simple calculation lets us get the coefficients, B1 = log Pia, B2 = paa log EQl, 'IjJ = -logpoo, and B = B12 (as Eq 2). The triplet () = (B1' B2, B12 ) forms paa another coordinate system, called the natural parameters, or B-coordinates. We remark that B12 is 0 when and only when Xl and X 2 are independent. The triplet C == (171,172,B12 ) forms an 'orthogonal' coordinate system of 8 2 , called the mixed coordinates [4]. We use the Kullback-Leibler divergence (KL) to measure the discrepancy between two probabilities p(x) and q(x) , defined by D[p:q] = LxP(x)log~t~}. In the following, we denote any coordinates of p by e etc (the same for q). Using the orthogonality between 17- and B-coordinates, we have the decomposition in the KL. Theorem 2. D [p : q] = D [p : r*] + D [r* : q], D [q : p] = D [q : r**] + D [r** : p] , (3) where r* and r** are given by Cr > = (17f, 17~, Bj) and Cr » = (17f, 17g, B~), respectively. The squared distance ds 2 between two nearby distributions p(x , ~) and p(x,~, +d~) is given by the quadratic form of d~, ds2 = L 9ij(~)d~id~j, i,jE(1,2,3) which is approximately twice the KL, i.e., ds 2 ~ 2D [P(x , ~) : p(x,~ + ~)]. Now suppose ~ is the mixed coordinates C. Then, the Fisher information matrix [ gll gl2 0 1 is of the form gfj = gf2 g~2 0 and we have ds2 = dsi + ds~, where dsi = o 0 g~3 g~3(dB3) 2, ds~ = Li,j E(1,2) 9fjd17id17j, corresponding to Eq. 3. This decomposition comes from the choice of the orthogonal coordinates and gives us the merits of simple procedure in statistical inference. First, let us estimate the parameter TI = (1}1,1}2) and B from N observed data Xl, ... , XN. The maximum likelihood estimator (mle) ( , which is asymptotically unbiased and efficient, is easily obtained by 1)' . = l..#{x· = I} and 8 = log fh?(1-=-fh-.ib+~12) using • N • (1]1-1]12)(1]2-1]12) , fj12 = tt#{XIX2 = I}. The covariance of estimation error, f::J.TI and f::J.B, is given asymptotically by Cov [ ~~ ] = ttGZ1. Since the cross terms of G or G-1 vanish for the orthogonal coordinates, we have Cov [f::J.TI, f::J.B] = 0, implying that the estimation error f::J.TI of marginals and that of interaction are mutually independent. Such a property does not hold for other non-orthogonal parameterization such as the COR p, the covariance etc. Second, in practice, we often like to compare many spike distributions, q(x(t)) (i.e, (q(t)) for (t = 1", T), with a distribution in the control period p( x) , or (P. Because the orthogonality between TI and B allows us to treat them independently, these comparisons become very simple. These properties bring a simple procedure of testing hypothesis concerning the null hypothesis Ho : B = Bo against (4) where Bo is not necessarily zero, whereas Bo = 0 corresponds to the null hypothesis of independent firing, which is often used in literature in different setting. Let the log likelihood of the models Ho and HI be, respectively, lo = maxlogp(Xl ' ... , XN ; TI, Bo) and h = maxlogp(Xl' ... , XN; TI, B). TI TI,e The likelihood ratio test uses the test statistics A = 2log ~. By the mle with respect to TI and e, which can be performed independently, we have lo = logp(x,r"Bo), (5) where r, are the same in both models. A similar situation holds in the case of testing TI = Tlo against TI =I Tlo for unknown B. Under the hypothesis Ho, A is approximated for a large N as A = 2 t log P(Xi;~' B~) ';::;j N gi3 (8 - BO)2 '" X2(1). i=l p(Xi; TI, B) (6) Thus, we can easily submit our data to a hypothetical testing of significant coincident firing against null hypothesis of any correlated firing, independently from the mean firing rate modulation1 . We now turn to relate the above approach with another important issue, which is to relate such a coincident firing with behavior. Let us denote by Y a variable of discrete behavioral choices. The MI between X = (X1,X2 ) and Y is written by [ p(x,y)] J(X, Y) = Ep(x,y) log p(x)p(y) = Ep(Y) [D [P(Xly) : p(X)]]. Using the mixed coordinates for p(Xly) and p(X) , we have D [P(Xly) : p(X)] D [«(Xly) : «(X)] = D [«(Xly) : ('J + D [(I : «(X)J, where (' = ('(X,y) ((1 (Xly), (2 (X Iy), (3 (X)) = (1}1 (Xly), 1}2(Xly), B3(X)). 1 A more proper formulation in this hypothetical testing can be derived, resulting in using p value from X2(2) distribution, but we omit it here due to the limited space [9] Theorem 3. J(X, Y) = It (X, Y) + h(X, Y) , (7) where It (X, Y), h(X, Y) are given by It (X, Y) = Ep(Y) [D [«(Xly) : ('(X,y)]] ,h(X, Y) = Ep(Y) [D [('(X,y) : «(X)]] . Obviously, the similar result holds with respect to p(YIX). By this theorem, J is the sum of the two terms: It is by modulation of the correlation components of X, while h is by modulation of the marginals of X. This observation helps us investigate the behavioral significance by modulating either coincident firing or mean firing rates. 0 . 1 ,-----~-~--~-~-____, u; ~ A (al / 'J, ~ 0 .05~ .•.••.• __ I~) • • _ ~, .•.•.•.•.•.•.• ~C) • • .• .• .• .• .• o -........................ _12 . .. a 100 300 500 0.1 ,-------~--~-~-____, \ll ~0.05 B . , .. , .. . , . .. , .. ". , .. ' .. , .. " . . .. ' .. , . ... . . , .. . . , .. ,_ .#' ...... .. , ... . ' .. -" .. . , •• , !112 W : ~ ~ 'J, _________ .. __ '_12 '- - - .-- - --- -"" -_ .. °0:------=:-: 10:::0~~~~3=00=---~---:-: 500 00 100 300 500 00 100 "300 500 lime Ims) lime Ims) Figure 1: Demonstration of information-geometric measure in two neuron case, using simulated neural data, where two behavioral choices (sl, s2) are assumed. A,B. (1]1 , 1]2 , 1]12) with respect to sl, s2. C,D. COR,B, computed by using ", L-iP(Si)",(Si) with P(Si) = 1/2 (i = 1, 2). E. p-values. F. MI. Fig 1 succinctly demonstrates results in this section. Figs 1 A, B are supposed to show mean firing rates of two neurons and mean coincident firing for two different stimuli (sl, s2). The period (a) is assumed as the control period, i.e., where no stimuli is shown yet, whereas the stimulus is shown in the periods (b,c). Fig 1 C, D gives COR, B. They look to change similarly over periods, which is reasonable because both COR and B represent the same correlational component, but indeed change slightly differently over periods (e.g., the relative magnitudes between the periods (a) and (c) are different for COR and B) , which is also reasonable because both represent the correlational component as in different coordinate systems. Using B in Fig 1 D, Fig 1 E shows p-values derived from X2 (1) (i.e., P > 0.95 in Fig 1 E is 'a significance with P < 0.05') for two different null hypotheses, one of the averaged firing in the control period (by solid line) and the other of independent firing (by dashed line), which is of popular use in literature. In general, it becomes complicated to test the former hypothesis, using COR. This is because the COR, as the coordinate component, is not orthogonal to the mean firing rates so that estimation errors among the COR and mean firing rates are entangled and that the proper metric among them is rather difficult to compute. Once using B, this testing becomes simple due to orthogonality between B and mean firing rates. Notably, we would draw completely different conclusions on significant coincident firing given each null hypothesis in Fig 1 E. This difference may be striking when we are to understand the brain function with these kinds of data. Fig 1 F shows the MI between firing and behavior, where behavioral event is with respect to stimuli, and its decomposition. There is no behavioral information conveyed by the modulation of coincident firing in the period (b) (i.e., h = 0 in the period (b)). The increase in the total MI (i.e., I) in the period (c), compared with the period (b), is due not to the MI in mean firing (h) but to the MI correlation (h). Thus, with a great ease, we can directly inspect a function of neural correlation component in relation to behavior. 3 Three neuron case With more than two neurons, we need to look not only into a pairwise interaction but also into higher-order interactions. Our results in the two neuron case are naturally extended to n neuron case and here, we focus on three neuron case for illustration. For three neurons X = (X1,X2,X3), we let p(x), x = (X1,X2,X3), be their joint probability distribution and put Pijk = Prob {Xl = i, X2 = j, X3 = k}, i, j, k = 0,1. The set of all such distributions forms a 7-dimensional manifold 8 3 due to "L.Pijk = 1. The 1]-coordinates 'fI = ('fI1; 'fI2; 'fI3) = (1]1,1]2,1]3; 1]12,1]23,1]13; 1]123) is defined by 1]i = E [Xi] (i = 1,2,3), 1Jij = E [XiXj] (i, j = 1,2,3; i i- j), 1]123 = E [X1X2X3]. To single out the purely triplewise correlation, we utilize the dual orthogonality of 8- and 1]-coordinates. By using expansion of log p( x) = "L. 8iXi + "L.8ijXiXj + 8123X1X2X3 'ljJ, we obtain 8-coordinates, () = (()1;()2;()3) = (81,82,83; 812,823 ,813; 8123). It's easy to get the expression of these coefficients (e.g. 8 = log P111 PIOO POIOP001). Information geometry gives the following theorem. ,123 P110PIOIP0l1POOO Theorem 4. 8123 represents the pure triplewise interaction in the sense that it is orthogonal to any changes in the single and pairwise marginals, i.e., 'fIl and 'fI2. We use the following two mixed coordinates to utilize the dual orthogonality, (I = ('fIl; ()2; ()3), (2 = ('fIl; 'fI2; ()3). Here (2 is useful to single out the triple wise interaction (()3 = 8123), while (I is to single out the pairwise and triplewise interactions together (()2; ()3). Note that 8123 is not orthogonal to {8ij }. In other words, except the case of no triple wise interaction (8123 = 0), 8ij do not directly represent the pairwise correlation of two random variables Xi, X j. The case of independent firing is given by 1]ij = 1]i1]j, 1]123 = 1]11]21]3 or equivalently by ()2 = 0, ()3 = o. The decomposition in the KL is now given as follows. Theorem 5. D [p : q] = D [p : p] + D [p : q] = D [p : fi] + D [p : q] = D [p : p] + D [p : fi] + D [p : q] . (8) where, using the mixed coordinates, we have (g = ('fIi; 'fI~; ()§), (f = ('fIi; ()~; ()§). A hypothetical testing is formulated similarly to the two neuron case. We can examine a significance of the triplewise interaction by A2 = 2ND [p : p] ~ N g~7 (~) (8f238i23)2 ~ X2(1). For a significance of triplewise and pairwise interactions together, we have Al = 2ND [p : fi] ~ N "L.J,j=4 gfj(f)((f - (f)((f - (f) ~ X2(4). For the decomposition of the MI between firing X and behavior Y, we have Theorem 6. J(X, Y) = h (X, Y) + h(X, Y) = h(X, Y) + J4(X, Y) (9) where h(X, Y) = Ep(Y) [D [(I(X ly) : ( I(X,y)]] , h(X, Y) = Ep(Y) [D [(I (X,y) : ( I(X)]] , h(X, Y) = Ep(Y) [D [(2(X ly) : ( 2(X,y)]] , 14 (X, Y) = Ep(Y) [D [(2(X,y) : ( 2(X)]], By the first equality, I is decomposed into two parts: II is conveyed by the pairwise and triplewise interactions of firing, and h by the mean firing rate modulation. By the second equality, I is decomposed differently: h, conveyed by the triplewise interaction, and 14 , by the other terms. 8 ~~I----' [: r:- ~ : --0. 10 100 300 500 700 0.04 '" (a) 0.02 00 100 (e) (d) 300 500 700 ~,-~ ~ ~~j ~.~~ it • .. -2 0 100 300 500 700 0 100 300 500 700 N205 - - -i: -. -. --, --) ---95 N~0501 1 -F- - - - - - - - - - - .':,f·- - - - • - - - - - - - - .- - - -I·95 x t. ." ,t: t' ..... ~I/"'\:""''! ~I ...... I ~....,:..;..".u,,'rV! '/ J.~'" ~ oo'-'--"1"" oo:"-'-'-'-'-"'-'----::3~ 00c--~---'c5-:-:00-~---=1l700 0 1 00 300 500 700 1- _____ _ E time (ms) time (ms) Figure 2: Demonstration in three neuron case. A '11 = ('111> '112, '113) ~ ('T/i ,'T/ij,'T/ijk) from top to bottom, since we treated a homogeneous case in this simulation for simplicity. B. COR. C. (}12,(}13 , (}23' D (}123 . E p-value,...., X2 (1). F p-value,...., X2(4). We emphasize that all the above decompositions come from the choice of the 'orthogonal' coordinates. Fig 2 highlights some of the results in this section. Fig 2 A shows the mean firing rates (see legend). The period (a) is assumed as the control period. Fig 2 B indicates that COR changes only in the periods (c,d), while Fig 2 C indicates that (}123 changes only in the period (d). Taken together, we observe that the triplewise correlation (}123 can be modulated independently from COR. Fig 2 E indicates the p-value from X2(1) against the null hypothesis of the activity in the control period. The triple wise coincident firing becomes significant only in the period (d). Fig 2 F indicates the p-value from X2(4) . The coincident firing, taking the triplewise and pairwise interaction together, becomes significant in both periods (c,d). We cannot observe these differences in modulation of pairwise and triplewise interactions over periods (c, d), when we inspect only COR. Remark: For a general n neuron case, we can use the k-cut mixed coordinates, (k = ('111 ' ... , '11k; 0k+l, ... , On) = ('I1k- ; 0k+)' Using the orthogonality between 'I1kand 0k+, the similar results hold. To meet the computational complexity involved in this general case, some practical difficulties should be resolved in practice [9]. 4 Discussions We presented the information-geometric measures to analyze spike firing patterns, using two and three neuron cases for illustration. The choice of 'orthogonal' coordinates provides us with a simple, transparent and systematic procedure to test significant firing patterns and to directly relate such a pattern with behavior. We hope that this method simplifies and strengthens experimental data analysis. Acknowledgments HN thanks M. Tatsuno, K. Siu and K. Kobayashi for their assistance. HN is supported by Grants-in-Aid 13210154 from the Ministry of Edu. Japan. References [1] M. Abeles, H. Bergman, E. Margalit, and E. Vaadia. Spatiotemporal firing patterns in the frontal cortex of behaving monkeys. J Neurophysiol, 70(4):162938.,1993. [2] A. M. H. J. Aertsen, G. 1. Gerstein, M. K. Habib, and G. Palm. Dynamics of neuronal firing correlation: Modulation of "effective connectivity". Journal of Neurophysiology, 61(5):900- 917, May 1989. [3] S. Amari. Information geometry on hierarchical decomposition of stochastic interactions. IEEE Transaction on Information Theory, pages 1701- 1711,2001. [4] S. Amari and H. Nagaoka. Methods of Information Geometry. AMS and Oxford University Press, 2000. [5] S. Griin. Unitary joint-events in multiple-neuron spiking activity: detection, significance, and interpretation. Verlag Harri Deutsch, Reihe Physik, Band 60. Thun, Frankfurt/Main, 1996. [6] H. Ito and S. Tsuji. Model dependence in quantification of spike interdependence by joint peri-stimulus time histogram. Neural Computation, 12:195- 217, 2000. [7] L. Martignon, G. Deco, K. Laskey, M. Diamond, W. A. Freiwald, and E. Vaadia. Neural coding: Higher-order temporal patterns in the neurostatistics of cell assemblies. Neural Computation, 12(11):2621- 2653, 2000. [8] H. Nagaoka and S. Amari. Differential geometry of smooth families of probability distributions. Technical report, University of Tokyo, 1982. [9] H. Nakahara and S. Amari. Information geometric measure for neural spikes. in prepration. [10] H. Nakahara, S. Amari, M. Tatsuno, S. Kang, K. Kobayashi, K. Anderson, E. Miller, and T. Poggio. Information geometric measures for spike firing. Society for Neuroscience Abstracts, 27:821.46 (page.2178), 2001. [11] M. W. Oram, N. G. Hatsopoulos, B. J. Richmond, and J. P. Donoghue. Excess synchrony in motor cortical neurons provides redundant direction information with that from coarse temporal measures. J Neurophysiol., 86(4):1700- 1716, 2001. [12] S. Panzeri and S. R. Schultz. A unified approach to the study of temporal, correlational, and rate coding. Neural Computation, 13(6):1311-49., 2001a. [13] A. Riehle, S. Griin, M. Diesmann, and A. Aertsen. Spike synchronization and rate modulation differentially involved in motor cortical function. Science, 278:1950- 1953, 12 Dec 1997. [14] E. Vaadia, I. Haalman, M. Abeles, H. Bergman, Y. Prut, H. Slovin, and A. Aertsen. Dynamics of neuronal interactions in monkey cortex in relation to behavioural events. Nature, 373:515- 518, 9 Feb 1995.
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Learning Body Pose via Specialized Maps Romer Rosales Department of Computer Science Boston University, Boston, MA 02215 rrosales@cs.bu.edu Stan Sclaroff Department of Computer Science Boston University, Boston, MA 02215 sclaroff@cs.bu.edu Abstract A nonlinear supervised learning model, the Specialized Mappings Architecture (SMA), is described and applied to the estimation of human body pose from monocular images. The SMA consists of several specialized forward mapping functions and an inverse mapping function. Each specialized function maps certain domains of the input space (image features) onto the output space (body pose parameters). The key algorithmic problems faced are those of learning the specialized domains and mapping functions in an optimal way, as well as performing inference given inputs and knowledge of the inverse function. Solutions to these problems employ the EM algorithm and alternating choices of conditional independence assumptions. Performance of the approach is evaluated with synthetic and real video sequences of human motion. 1 Introduction In everyday life, humans can easily estimate body part locations (body pose) from relatively low-resolution images of the projected 3D world (e.g., when viewing a photograph or a video). However, body pose estimation is a very difficult computer vision problem. It is believed that humans employ extensive prior knowledge about human body structure and motion in this task [10]. Assuming this, we consider how a computer might learn the underlying structure and thereby infer body pose. In computer vision, this task is usually posed as a tracking problem. Typically, models comprised of 2D or 3D geometric primitives are designed for tracking a specific articulated body [13, 5, 2, 15]. At each frame, these models are fitted to the image to optimize some cost function. Careful manual placement of the model on the first frame is required, and tracking in subsequent frames tends to be sensitive to errors in initialization and numerical drift. Generally, these systems cannot recover from tracking errors in the middle of a sequence. To address these weaknesses, more complex dynamic models have been proposed [14, 13,9]; these methods learn a prior over some specific motion (such as walking). This strong prior however, substantially limits the generality of the motions that can be tracked. Departing from the aforementioned tracking paradigm, in [8] a Gaussian probability model was learned for short human motion sequences. In [17] dynamic programming was used to calculate the best global labeling according to the learned joint probability density function of the position and velocity of body features. Still, in these approaches, the joint locations, correspondences, or model initialization must be provided by hand. In [1], the manifold of human body dynamics was modeled via a hidden Markov model and learned via entropic minimization. In all of these approaches models were learned. Although the approach presented here can be used to model dynamics, we argue that when general human motion dynamics are intended to be learned, the amount of training data, model complexity, and computational resources required are impractical. As a consequence, models with large priors towards specific motions (e.g., walking) are generated. In this paper we describe a non-linear supervised learning algorithm, the Specialized Maps Architecture (SMA), for recovering articulated body pose from single monocular images. This approach avoids the need for initialization and tracking per se, and reduces the above mentioned disadvantages. 2 Specialized Maps There at least two key characteristics of the problem we are trying to solve which make it different from other supervised learning problems. First, we have access to the inverse map. We are trying to learn unknown probabilistic maps from inputs to outputs space, but we have access to the map (in general probabilistic) from outputs to inputs. In our pose estimation problem, it is easy to see how we can artificially, using computer graphics (CG), produce some visual features (e.g., body silhouettes) given joint positions1 . Second, it is one-to-many: one input can be associated with more than one output. Features obtained from silhouettes (and many other visual features) are ambiguous. Consider an occluded arm, or the reflective ambiguity generated by symmetric poses. This last observation precludes the use of standard algorithms for supervised learning that fit a single mapping function to the data. Given input and output spaces ~c and ~t, and the inverse function ( : ~t -+ ~c, we describe a solution for these supervised learning problems. Our approach consists in generating a series of m functions ¢k : ~c -+ ~t. Each of these functions is specialized to map only certain inputs (for a specialized sub-domain) better than others. For example, each sub-domain can be a region of the input space. However, the specialized sub-domain of ¢k can be more general than just a connected region in the input space. Several other learning models use a similar concept of fitting surfaces to the observed data by splitting the input space into several regions and approximating simpler functions in these regions (e.g., [11,7, 6]). However, in these approaches, the inverse map is not incorporated in the estimation algorithm because it is not considered in the problem definition and the forward model is usually more complex, making inference and learning more difficult. The key algorithmic problems are that of estimating the specialized domains and functions in an optimal way (taking into account the form of the specialized functions), and using the knowledge of the inverse function to formulate efficient inferIThus, ( is a computer graphics rendering, in general called forward kinematics ence and learning algorithms. We propose to determine the specialized domains and functions using an approximate EM algorithm and to perform inference using, in an alternating fashion, the conditional independence assumptions specified by the forward and inverse models. Fig. l(a) illustrates a learned forward model. Figure 1: SMA diagram illustrating (a) an already learned SMA model with m specialized functions mapping subsets of the training data, each subset is drawn with a different color (at initializations, coloring is random) and (b) the mean-output inference process in which a given observation is mapped by all the specialized functions, and then a feedback matching step, using (, is performed to choose the best of the m estimates. 3 Probabilistic Model Let the training sets of output-input observations be \)! = {1jI1, ... , 1jIN }, and Y = {Vl , ... ,VN} respectively. We will use Zi = (1jIi,Vi) to define the given output-input training pair, and Z = {ZI ' ... , ZN} as our observed training set. We introduce the unobserved random variable y = (Yl , ... , Yn). In our model any Yi has domain the discrete set C = {l, ... , M} oflabels for the specialized functions, and can be thought as the function number used to map data point i; thus M is the number of specialized mapping functions. Our model uses parameters 8 = (81 , ... , 8M , A), 8k represents the parameters of the mapping function k; A = (AI"", AM), where Ak represents P(Yi = kI8): the prior probability that mapping function with label i will be used to map an unknown point. As an example, P(Yi lzi, 8) represents the probability that function number Yi generated data point number i. Using Bayes' rule and assuming independence of observations given 8, we have the log-probability of our data given the modellogp(ZI8), which we want to maximize: argm;x 2:)og LP(1jIi lvi, Yi = k,8)P(Yi = kI8)p(Vi), i k (1) where we used the independence assumption p(vI8) = p(v). This is also equivalent to maximizing the conditional likelihood of the model. Because of the log-sum encountered, this problem is intractable in general. However, there exist practical approximate optimization procedures, one of them is Expectation Maximization (EM) [3,4, 12]. 3.1 Learning The EM algorithm is well known, therefore here we only provide the derivations specific to SMA's. The E-step consists of finding P(y = klz, 8) = P(y). Note that the variables Yi are assumed independent (given Zi)' Thus, factorizing P(y): p(y) = II P(t)(Yi) = II[(AYiP(1/Jilvi,Yi,B))/(2:AkP(1/Jilvi,Yi = k,B))] (2) kEC However, p( 1/Ji lVi, Yi = k, B) is still undefined. For the implementation described in this paper we use N(1/Ji; ¢k(Vi,Bk), ~k)' where Bk are the parameters of the k-th specialized function, and ~k the error covariance of the specialized function k. One way to interpret this choice is to think that the error cost in estimating 1/J once we know the specialized function to use, is a Gaussian distribution with mean the output of the specialized function and some covariance which is map dependent. This also led to tractable further derivations. Other choices were given in [16]. The M-step consists of finding B(t) = argmaxoEj>(t) [logp(Z,yIB)]. In our case we can show that this is equivalent to finding: argmJn 2: 2: P(t)(Yi = k)(1/Ji - ¢k(Vi, Bk))T~kl(Zi - ¢k(Zi,Bk))· (3) i k This gives the following update rules for Ak and ~k (where Lagrange multipliers were used to incorporate the constraint that the sum of the Ak'S is 1. 1 - 2: P(Yi = klzi' B) n . (4) In keeping the formulation general, we have not defined the form of the specialized functions ¢k. Whether or not we can find a closed form solution for the update of Bk depends on the form of ¢k. For example if ¢k is a non-linear function, we may have to use iterative optimization to find Bit). In case ¢k yield a quadratic form, then a closed form update exists. However, in general we have: (6) In our experiments, ¢k is a I-hidden layer perceptron. Thus, the M-step is an approximate, iterative optimization procedure. 4 Inference Once learning is accomplished, each specialized function maps (with different levels of accuracy) the input space. We can formally state the inference process as that of maximum-a-posteriori (MAP) estimation where we are interested in finding the most likely output h given an input configuration x: h* = argmaxp(hlx) = argmax '" p(hly, x)P(y), h h ~ (7) Y Any further treatment depends on the properties of the probability distributions involved. If p(hlx, y) = N(h ; ¢y(x), ~y), the MAP estimate involves finding the maximum in a mixture of Gaussians. However, no closed form solution exists and moreover, we have not incorporated the potentially useful knowledge of the inverse function C. 4.1 MAP by Using the Inverse Function ( The access to a forward kinematics function ( (called here the inverse function) allows to formulate a different inference algorithm. We are again interested in finding an optimal h* given an input x (e.g. , an optimal body pose given features taken from an image). This can be formulated as: h* = arg maxp(hlx) = argmaxp(xlh) "p(hly, x)P(y) , h h ~ (8) y simply by Bayes' rule, and marginalizing over all variables except h. Note that we have made the distribution p(xlh) appear in the solution. This is important because we can know use our knowledge of ( to define this distribution. This solution is completely general within our architecture, we did not make any assumptions on the form of the distributions or algorithms used. 5 Approximate Inference using ( Let us assume that we can approximate Lyp(hly, x)P(y) by a set of samples generated according to p(hly,x)P(y) and a kernel function K(h,hs). Denote the set of samples HSpl = {hs}s=l...s. An approximate to Lyp(hly,x)P(y) is formally built by ~ L;=l K(h, hs), with the normalizing condition J K(h, hs)dh = 1 for any given hs. We will consider two simple forms of K. If K(h, hs) = J(h - h s), we have: h = argmaxhP(xlh) L;=l J(h - hs). After some simple manipulations, this can be reduced to the following equivalent discrete optimization problem whose goal is to find the most likely sample s*: (9) where the last equivalence used the assumption p(xlh) = N(x; ((h), ~d. A S If K(h, h s) = N(h; hs , ~Spl)' we have: h = argmaxhP(xlh) L S=l N(h; hs , ~Spl). This case is hard to use in practice, because contrary to the case above (Eq. 9), in general, there is no guarantee that the optimal h is among the samples. 5.1 A Deterministic Approximation based on the Functions Mean Output The structure of the inference in SMA, and the choice of probabilities p(hlx, y) allows us to construct a newer approximation that is considerably less expensive to compute, and it is deterministic. Intuitively they idea consists of asking each of the specialized functions ¢k what their most likely estimate for h is, given the observed input x. The opinions of each of these specialized functions are then evaluated using our distribution p(xlh) similar to the above sampling method. This can be justified by the observation that the probability of the mean is maximal in a Gaussian distribution. Thus by considering the means ¢k(X), we would be considering the most likely output of each specialized function. Of course, in many cases this approximation could be very far from the best solution, for example when the uncertainty in the function estimate is relatively high relative to the difference between means. We use Fig. l(b) to illustrate the mean-output (MO) approximate inference process. When generating an estimate of body pose, denoted h, given an input x (the gray point with a dark contour in the lower plane), the SMA generates a series of output hypotheses tl q, = {h!h obtained using hk = (/Jk(x), with k E C (illustrated by each of the points pointed to by the arrows). Given the set tlq" the most accurate hypothesis under the mean-output criteria is the one that minimizes the function: k* (10) where in the last equation we have assumed p(xlh) is Gaussian. 5.2 Bayesian Inference Note that in many cases, there may not be any need to simply provide a point estimate, in terms of a most likely output h. In fact we could instead use the whole distribution found in the inference process. We can show that using the above choices for K we can respectively obtain. 1 s p(hlx) = S 2: N (x; ((hs ), ~d, 8= 1 (11) s p(hlx) = N(h; h8' ~Spz) 2:N(x; ((h) , ~d· (12) 8=1 6 Experiments The described architecture was tested using a computer graphics rendering as our ( inverse function. The training data set consisted of approx. 7,000 frames of human body poses obtained through motion capture. The output consisted of 20 2D marker positions (i. e., 3D markers projected to the image plane using a perspective model) but linearly encoded by 8 real values using Principal Component Analysis (PCA). The input (visual features) consisted of 7 real-valued Hu moments computed on synthetically generated silhouettes of the articulated figure. For training/testing we generated 120,000 data points: our 3D poses from motion capture were projected to 16 views along the view-sphere equator. We took 8,000 for training and the rest for testing. The only free parameter in this test, related to the given SMA, was the number of specialized functions used; this was set to 15. For this, several model selection approaches could be used instead. Due to space limitations, in this paper we show results using the mean-output inference algorithm only, readers are referred to http://cs-people.bu.edu/rrosales/SMABodyInference where inference using multiple samples is shown. Fig. 2(left) shows the reconstruction obtained in several single images coming from three different artificial sequences. The agreement between reconstruction and observation is easy to perceive for all sequences. Note that for self-occluding configurations, reconstruction is harder, but still the estimate is close to ground-truth. No human intervention nor pose initialization was required. For quantitative results, Fig. 2(right) shows the average marker error and variance per body orientation in percentage of body height. Note that the error is bigger for orientations closer to a and 7r radians. This intuitively agrees with the notion that at those angles (side-views), there is less visibility of the body parts. We consider this performance promising, given the complexity of the task and the simplicity of the approach. By choosing poses at random from training set, the RMSE was 17% of body height. In related work, quantitative performance have been usually ignored, in part due to the lack of ground-truth and standard evaluation data sets. Penormance regarding cameraviewpoinl (16 101al) 2.9 ,-----~~--,-:.----'.----.--:----.;...-_---,----, 2.75 14 16 Figure 2: Left: Example reconstruction of several test sequences with CG-generated silhouettes. Each set consists of input images and reconstruction (every 5th frame). Right: Marker root-mean-square-error and variance per camera viewpoint (every 27r/32 rads.). Units are percentage of body height. Approx. 110,000 test poses were used. 6.1 Experiments using Real Visual Cues Fig. 3 shows examples of system performance with real segmented visual data, obtained from observing a human subject. Reconstruction for several relatively complex sequences are shown. Note that even though the characteristics of the segmented body differ from the ones used for training, good performance is still achieved. Most reconstructions are visually close to what can be thought as the right pose reconstruction. Body orientation is also generally accurate. 7 Conclusion In this paper, we have proposed the Specialized Mappings Architecture (SMA). A learning algorithm was developed for this architecture using ideas from ML estimation and latent variable models. Inference was based on the possibility of alternatively use different sets of conditional independence assumptions specified by the forward and inverse models. The incorporation of the inverse function in the model allows for simpler forward models. For example the inverse function is an architectural alternative to the gating networks of Mixture of Experts [11]. SMA advantages for body pose estimation include: no iterative methods for inference are used, the Figure 3: Reconstruction obtained from observing a human subject (every 10th frame). algorithm for inference runs in constant time and scales only linearly O(M) with respect to the number of specialized functions M; manual initialization is not required; compared to approaches that learn dynamical models, the requirements for data are much smaller, and also large priors to specific motions are prevented thus improving generalization capabilities. References [1] M. Brand. Shadow puppetry. In ICCV, 1999. [2] C. Bregler. Tracking people with twists and exponential maps. In CVPR, 1998. [3] 1. Csiszar and G. Thsnady. Information geometry and alternating minimization procedures. Statistics and Decisions, 1:205- 237, 1984. [4] A. Dempster, N. Laird, and D. Rubin. Maximum likelihood estimation from incomplete data. Journal of the Royal Statistical Society (B), 39(1), 1977. [5] J. Deutscher, A. Blake, and 1. Reid. Articulated body motion capture by annealed particle filtering. In CVPR, 2000. [6] J.H. Friedman. Multivatiate adaptive regression splines. The Annals of Statistics, 19,1-141, 1991. [7] G. Hinton, B. Sallans, and Z. Ghahramani. A hierarchical community of experts. Learning in Graphical Models, M. Jordan (editor), 1998. [8] N. Howe, M. Leventon, and B. Freeman. Bayesian reconstruction of 3d human motion from single-camera video. In NIPS-1 2, 2000. [9] M. Isard and A. Blake. Contour tracking by stochastic propagation of conditional density. In ECCV, 1996. [10] G. Johansson. Visual perception of biological motion and a model for its analysis. Perception and Psychophysics, 14(2): 210-211, 1973. [11] M. 1. Jordan and R. A. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6, 181-214, 1994. [12] R. Neal and G. Hinton. A view of the em algorithm that justifies incremental, sparse, and other variants. Learning in Graphical Models, M. Jordan (editor), 1998. [13] Dirk Ormoneit, Hedvig Sidenbladh, Michael J . Black, and Trevor Hastie. Learning and tracking cyclic human motion. In NIPS-1 3, 200l. [14] Vladimir Pavlovic, James M. Rehg, and John MacCormick. Learning switching linear models of human motion. In NIPS-13, 200l. [15] J. M. Regh and T. Kanade. Model-based tracking of self-occluding articulated objects. In ICC V, 1995. [16] R. Rosales and S. Sclaroff. Specialized mappings and the estimation of body pose from a single image. In IEEE Human Motion Workshop, 2000. [17] Y. Song, Xiaoling Feng, and P. Perona. Towards detection of human motion. In CVPR, 2000.
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Circuits for VLSI Implementation of Temporally-Asymmetric Hebbian Learning Adria Bofill Alan F. Murray DanlOn P. Thompson Dept. of Electrical Engineering The University of Edinburgh Edinburgh, EH93JL, UK adria. bofill@ee.ed.ac. uk alan. murray@ee.ed.ac.uk damon. thompson @ee.ed.ac. uk Abstract Experimental data has shown that synaptic strength modification in some types of biological neurons depends upon precise spike timing differences between presynaptic and postsynaptic spikes. Several temporally-asymmetric Hebbian learning rules motivated by this data have been proposed. We argue that such learning rules are suitable to analog VLSI implementation. We describe an easily tunable circuit to modify the weight of a silicon spiking neuron according to those learning rules. Test results from the fabrication of the circuit using a O.6J.lm CMOS process are given. 1 Introduction Hebbian learning rules modify weights of synapses according to correlations between activity at the input and the output of neurons. Most artificial neural networks using Hebbian learning are based on pulse-rate correlations between continuousvalued signals; they reduce the neural spike trains to mean firing rates and thus precise timing does not carry information. With this approach the spiking nature of biological neurons is just an efficient solution that evolution has produced to transmit analog information over an unreliable medium. In recent years, recorded data have indicated that synaptic strength modifications are also induced by timing differences between pairs of presynaptic and postsynaptic spikes [1][2]. A class of learning rules derived from these experimental data is illustrated in Figure 1 [2]-[4]. The "causal/non-causal" basis of these Hebbian learning algorithms is present in all variants of this spike-timing dependent weight modification rule. When the presynaptic spike arrives at the synapse a few milliseconds presynaptic spike presynaptic spike tpre' ,tpre postsynaptic spike postsynaptic spike !post tpost ' !'.w tpre - tpost tpre - tpost (a) (b) Figure 1: Two temporally-asymmetric Hebbian learning rules drawing on experimental data. The curves show the shape of the weight change (~W) for differences between the firing times of the presynaptic (tpre) and the postsynaptic (tpost) neurons. When the presynaptic spike arrives at the synapse a few ms before the postsynaptic neuron fires , the weight of the synapse is increased. If the postsynaptic neuron fires first, the weight is decreased. before an output spike is generated, the synaptic efficiency increases. In contrast, when the postsynaptic neuron fires first , the efficiency of the synapse is weakened. Hence, only those synapses that receive spikes that appear to contribute to the generation of the postsynaptic spike are reinforced. In [5] a similar spike-timing difference based learning rule has been used to learn input sequence prediction in a recurrent network. Studies reported in [4] indicate that the positive (potentiation) element of the learning curve must be smaller than the negative (depression) to obtain stable competitive weight modification. Pulse signal representation has been used extensively in hardware implementations of artificial neural networks [6] [7]. Such systems use pulses as a mere technological solution to benefit from the robustness of binary signal transmission while making use of analog circuitry for the elementary computation units. However, they do not exploit the relative timing differences between individual pulses to compute. Also, analog hardware is not well-suited to the complexity of most artificial neural network algorithms. The learning rules presented in Figure 1 are suitable for analog VLSI because: (a) the signals involved in the weight modification are local to the neuron, (b) no temporal averaging of the presynaptic or postsynaptic activity is needed and (c) they are remarkably simple compared to complex neural algorithms that impose mathematical constraints in terms of accuracy and precision. An analog VLSI implementation of a similar, but more complex, spike-timing dependent learning rule can be found in [8]. We describe a circuit that implements the spike-timing dependent weight change described above along with the test results from a fabricated chip. We have focused on the implementation of the weight modification circuits, as VLSI spiking neurons with tunable membrane time constant and refractory period have already been proposed in [9] and [10]. 2 Learning circuit description Figure 2 shows the weight change circuit and Figure 3 the form of signals required to drive learning. These driving signals are generated by the circuits described in Figure 4. The voltage across the weight capacitor, Cw in Figure 2, is modified according to the spike-timing dependent weight change rule discussed above. The weight change, ~ W, is defined as -~ Vw so that the leakage of the capacitor leads Vw in the direction of weight decay. The circuits presented allow the control of: (a) the abruptness of the transition between potentiation and depression at the origin, (b) the difference between the areas under the curve in the potentiation and depression regions, (c) the absolute value of the area under each side of the curve and (d) the time constant of the curve decay. PI Figure 2: Weight change circuit postsynaptic spike up down n '-----__ (a) (b) Figure 3: Stimulus for the weight change circuit The weight change circuit of Figure 2 works as follows. When a falling edge of either a postsynaptic or a presynaptic spike occurs, a short activation pulse is generated which causes Cdec to be charged to V pea k through transistor Nl. The charge accumulated in Cdec will leak to ground with a rate set by Vdecay ' The resulting voltage at the gate of N3 produces a current flowing through P2-P3-N4. If a presynaptic spike is active after the falling edge of a postsynaptic spike an activelow up pulse is applied to the gate of transistor P5. Thus, the current flowing through N3 is mirrored to transistor P4 causing an increase in the voltage across Cw that corresponds to a decrease in the weight. In contrast, when a presynaptic spike precedes a postsynaptic spike an active-high down pulse is generated and the current in N3 is mirrored to N5-N6 resulting in a discharge of Cw . As the current in N2 is constant, the current integrated by Cw displays an exponential decay, if Vpeak is such that N3 is in sub-threshold mode. Hence, the rate of decay of the learning curve is fixed by the ratio hlCdec. The abruptness of the transition zone between potentiation and depression is set by the duration of both the presynaptic and postsynaptic spike. Finally, an imbalance between the areas under the positive and negative side of the curve can be introduced via Vdep and Vpot . The effect of all these circuit parameters is exemplified by the test results shown in the following section. act down post_spike (a) (b) Figure 4: Learning drivers. (a) Delayed act pulse generator. (b) Asynchronous controller for up and down signals The circuit of Figure 4(a) , present in both the presynaptic and postsynaptic neurons, generates a short act pulse with the falling edge of the output spike. The act pulses are ORed at each synapse to produce the activation pulse applied to the weight change circuit of Figure 2. The other two driving signals, up and down, are produced by a small asynchronous controller using standard and asymmetric C-elements [11] shown in Figure 4(b). The internal signal q indicates if the last falling edge to occur corresponds to a pre (q = 1) or a postsynaptic spike (q = 0). This ensures that an up signal that decreases the weight is only generated when a presynaptic spike is active after the falling edge of a postsynaptic spike. Similarly down is activated only when the postsynaptic spike is active following a presynaptic spike falling edge. Using the current flowing through N3 (Figure 2) to both increase and decrease the weight allows us to match the curve at the potentiation and depression regions at the expense of having to introduce the driving circuits of Figure 4. 3 Results from the temporally-asymmetric Hebbian chip The circuit in Figure 2 has been fabricated in a O.6J.lm standard CMOS process. The driving signals (down, up and activation) are currently generated off-chip. The circuit can be operated in the p,s timescale, however, here we only present test results with time constants similar to those suggested by experimental data and studied using software models in [3]-[5]. 3.5"==;;;r----;;.==---~--~==~ 2.5 t -I =2ms pre post \ t - t = 5 pre post t -I =7.5ms pre post 0.5 1.5 Time(s) (a) Vdecay = 515mV V peak = 694mV T = 1ms ' p T act = SOj.tS V =OV pot Vdd - Vdep = OV 2.5 >' 1.5 0.5 00 t Vdecay = 515mV V peak = 694m V T = 1ms 'p Tact = 50j.ts V =OV pot Vdd - Vdep = OV t -I =5m post pre / - t = post pre ms / 0.5 Time ( s ) (b) t - t = 7.5ms po" i 1.5 2.5 Figure 5: Test results. Linearity. (a) The voltage across Cw is initially set to OV and increased by a sequence of consecutive pairs of pre and postsynaptic spikes. The delays between presynaptic and postsynaptic firing times were set to 2ms, 5ms and 7.5ms (b) The order of pre and postsynaptic spikes is reversed to decrease Vw . In both plots the duration of the spikes, T sp , and the activation pulse, Ta ct , is set to 1ms and and 50p,s respectively. The learning window plots shown in Figures 6-8 were constructed with test data from a sequence of consecutive presynaptic and postsynaptic spikes with different delays. Before every new pair of presynaptic and postsynaptic spikes, the voltage in Cw was reset to Vw =2V. The weight change curves are similar for other initial "reset" weight voltages owing to the linearity of the learning circuit for different Vw values as shown in Figure 5. A power supply voltage of Vdd=5V is used in all test results shown. 100 > 50 E >' 0 <l r ~ -50 - 100 Vdecay =516mV Tsp =1ms Tact = 501-15 V =OV pot Vdd - Vdep = OV Vpeak =716mV --- V =711mV ,,'k -V =701mV "" -25 -20 -15 - 10 -5 0 5 10 15 20 25 tp,. - tpo,t (ms) (a) 80 ,--.--,--,--,--,--,--,---, 60 40 > E 20 >~ l' ; -20 <l -40 -60 Tsp= 1ms Tact = 5O~IS Vdd - Vdep = OV V = OV pot - - ......... ~--.=.::.---.:." ,I " " " , , Vdecay =482mV Vpeak = 699mV - - v = 499mV decay Vpeak = 701mV - Vdecay =517mV Vpeak = 702mV - ';",-;"--' ::"= '--"'- , -80~-~-~-~-~-~-~-~-----' -40 -30 -20 -10 0 10 20 30 40 tpre - \ OSI (m s) (b) Figure 6: Test results. (a) M aximum weight change.(b) Learning window decay. The decay of both tails of the learning window is set by Vdecay. A wide range of time constants can be set. Note, however, that Vpeak needs to be increased slightly for faster decay rates to maintain exactly the same peak value. The maximum weight change is easily tuned with Vp eak as shown in Figure 6( a). Changing the value of Vp eak modifies by the same amount the absolute value of the peaks at both sides of the curve. The decay of the learning window is controlled by Vdecay' An increase in Vdecay causes both tails of the learning window to decay faster as seen in Figure 6 (b). As mentioned above, matching between both sides of the learning window is possible because the same source of current is used to both increase and decrease the weight. 100 ~-~-~~~~-~-~-~-~ 80 60 40 > .:. 20 >'0 I' -20 ;: <l -40 -60 -80 V decay = SOOmV V peak = 705mV T = 1ms ' p Tact = 50l1s V =OV pot Vdd - Vdep = OV - - - V - V = 3.8mV _ , _ dd dep Vdd - Vdep = 9.3mV -1~go'---_--c1~ 5 ---C10~~-5~-0~-~5-~-~ 15:--~ 20 'pre - 'post (ms) (a) 100 80 60 40 > E 20 >' 0 <l , -20 ;: <l -40 -60 -80 -100 -20 V decay = SOOmV V peak = 705mV Tsp= 1ms ,,_ Tact = 50~lS ,, ;' Vdd -Vdep=OV // .-."';::: ,-;// V =OV pot - - - V =47mV - -- VPOt = 9~V pot - 15 - 10 -5 0 5 10 15 'pre - 'post (ms) (b) 20 Figure 7: Test results. Imbalance between potentiation and depression. The imbalance between the areas under the potentiation and depression regions of the learning window is a critical parameter of this class of learning rules [3] [4]. The circuit proposed can adjust the peak of the curve for potentiation and depression independently (Figure 7). Vp ot can be used to reduce the area under the potentiation region while keeping unchanged the depression part of the curve, thus setting the overall area under the curve to a negative value (Figure 7(b)). Similarly, with Vdd Vdep the area of the depression region can also be reduced (Figure 7 ( a) ) . 100 ,---,--~--,,---,--~--, 100 50 > E 50 i 100 ;: 100 <l 50 50 Tact = 50flS Vdd - Vdep =OV V =OV pot __ T sp = 100llS Vpeak = 790mV Vdecay = 499mV ---- T =1ms " Vpeak = 699mV Vdecay = 482mV 100 '---~--~--L--~--~-~ -15 -10 -5 0 5 10 15 'pre - 'post (ms) Figure 8: Test results. Abruptness at the origin The abruptness of the learning window at the origin (short delays between pre and postsynaptic spikes) is set by the duration of the spikes. Data in Figure 8 show that the two peaks of the learning window are separated by 2 times the durations of the spikes (Tsp ). 4 Discussion and future work Drawn from experimental data, several temporally-asymmetric Hebbian learning rules have been proposed recently. These learning rules only strengthen the weights when there is a causal relation between presynaptic and postsynaptic activity. Purely random time coincidences between spikes will tend to decrease the weights. Synaptic weight normalization is thus achieved via competition to drive postsynaptic spikes [4]. Predictive sequence learning has been achieved using a similar time-difference learning rule based on the same data [5]. Other pulse-based learning rules have also been used to study how delay tuning could be achieved in the sound source localization system of the barn owl [12]. A simple circuit to implement a general weight change block based on such learning rules has been designed and partially fabricated. The main characteristics of the learning rule, namely the abruptness at the origin, the rate of the decay of the learning window, the imbalance between the potentiation and depression regions and the rate of learning, can be tuned easily. The design also ensures that the circuit can operate at different timescales. As shown, the fabricated circuits have good linearity over a wide range of weight voltage values. We are currently developing a second chip with a small network of temporally asymmetric Hebbian spiking neurons using the circuit described in this paper. The structure of the network will be reconfigurable. The small network will be used to carry out movement planning experiments by learning of temporal sequences. We envisage the application of networks of temporally-asymmetric Hebbian learning silicon neurons as higher level processing stages for the integration of sensor and motor activities in neuromorphic system. We will concentrate on auditory applications and adaptive, spike-based motion estimation. In both types of application, naturally-occurring correlations in data can be exploited to drive the pulse timingbased learning process. Acknowledgelllents We thank Robin Woodburn, Patrice Fleury and Martin Reekie for fruitful discussions during the design and tape out of the chip. We also acknowledge that the circuits presented incorporate some of the insights into neuromorphic engineering that one of the authors gained at the Telluride Workshop on Neuromorphic Engineering 2000 (http://www.ini.unizh.ch/telluride2000 /). References [1] Markram, H., Lubke, J., Frotscher, M. & Sakmann, B.(1997) Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs. Science 275, 213-215. [2] Zhang, L.L , Tao, H.W., Holt, C.E. , Harris, W.A. & Poo, M-m.(1998) A critical window for cooperation and competition among developing retinotectal synapses. Nature 395, 3744. [3] Abbott, L.F. & Song, S.(1999) Temporally Asymmetric Hebbian Learning, Spike Timing and Neuronal Response Variability. In Kearns, M.S., Solla, S.A., & Cohn, D.A. (eds.), Advances in Neural Information Processing S ystems 11, 69-75. Cambridge, MA: MIT Press. [4] Song, S., Miller, K.D. & Abbott, L.F.(2000) Competitive Hebbian Learning Through Spike-Timing Dependent Synaptic Plasticity. Nature Neuroscience 3, 919-926. [5] Rao, R.P.N., & Sejnowski, T .J.(2000) Predictive Sequence Learning in Recurrent Neocortical Circuits. In Solla, S.A., Leen, T .K. & Muller, K-R. (eds.), Advances in Neural Information Processing Systems 12, 164-170. Cambridge, MA: MIT Press. [6] Murray, A.F. & Smith A.V.W.(1987) Asynchronous Arithmetic for VLSI Neural Systems. Electronic Letters 23 , 642-643. [7] Murray, A.F. & Tarrasenko, L.(1994) Neural Computing: An Analogue VLSI Approach. Chapman-Hall. [8] Hafliger, P., Mahowald, M. & Watts, L. (1996) A Spike Based Learning Neuron in Analog VLSL In Mozer, M.C., Jordan, M.L , & Petsche, T. (eds.) , Advances in Neural Information Processing Systems 9, 692-698. Cambridge, MA: MIT Press. [9] Indiveri, G.(2000) Modeling Selective Attention Using a Neuromorphic Analog VLSI Device. Neural computation 12, 2857-2880. [10] van Schaik, A., Fragniere, E. & Vittoz, E.(1996) An Analogue Electronic Model of Ventral Cochlear Nucleus Neurons. In Proceedings of the 5th International Conference on Microelectronics for Neural, Fuzzy and Bio-inspired S ystems; Microneuro '96, 52-59. Los Alamitos, CA: IEEE Computer Society Press. [11] Shams, M., Ebergen, J.C. & Elmasry, M.L (1998) Modeling and Comparing CMOS Implementations of the C-elment. IEEE Transactions on Very Large Scale Intergration (VLSI) Systems, Vol. 6, No.4, 563-567. [12] Gerstner, W. , Kempter, R. , van Hemmen, J.1. & Wagner, H.(1999) Hebbian Learning of Pulse Timing in the Barn Owl Auditory System. In Mass, W. & Bishop, C.M. (eds.), Pulsed Neural Networks. Cambridge, MA: MIT Press.
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Model Based Population Tracking and Automatic Detection of Distribution Changes Igor V. Cadez ∗ Dept. of Information and Computer Science, University of California, Irvine, CA 92612 icadez@ics.uci.edu P. S. Bradley digiMine, Inc. 10500 NE 8th Street, Bellevue, WA 98004-4332 paulb@digimine.com Abstract Probabilistic mixture models are used for a broad range of data analysis tasks such as clustering, classification, predictive modeling, etc. Due to their inherent probabilistic nature, mixture models can easily be combined with other probabilistic or non-probabilistic techniques thus forming more complex data analysis systems. In the case of online data (where there is a stream of data available) models can be constantly updated to reflect the most current distribution of the incoming data. However, in many business applications the models themselves represent a parsimonious summary of the data and therefore it is not desirable to change models frequently, much less with every new data point. In such a framework it becomes crucial to track the applicability of the mixture model and detect the point in time when the model fails to adequately represent the data. In this paper we formulate the problem of change detection and propose a principled solution. Empirical results over both synthetic and real-life data sets are presented. 1 Introduction and Notation Consider a data set D = {x1, x2, . . . , xn} consisting of n independent, identically distributed (iid) data points. In context of this paper the data points could be vectors, sequences, etc. Further, consider a probabilistic mixture model that maps each data set to a real number, the probability of observing the data set: P(D|Θ) = n Y i=1 P(xi|Θ) = n Y i=1 K X k=1 πkP(xi|θk), (1) where the model is parameterized by Θ = {π1, . . . , πK, θ1, . . . , θK}. Each P(.|θk) represents a mixture component, while πi represents mixture weights. It is often more convenient ∗Work was done while author was at digiMine, Inc., Bellevue, WA. to operate with the log of the probability and define the log-likelihood function as: l(Θ|D) = log P(D|Θ) = n X i=1 log P(xi|Θ) = n X i=1 LogPi which is additive over data points rather than multiplicative. The LogPi terms we introduce in the notation represent each data point’s contribution to the overall log-likelihood and therefore describe how well a data point fits under the model. For example, Figure 3 shows a distribution of LogP scores using a mixture of conditionally independent (CI) models. Maximizing probability1 of the data with respect to the parameters Θ can be accomplished by the Expectation-Maximization (EM) algorithm [6] in linear time in both data complexity (e.g., number of dimensions) and data set size (e.g., number of data points). Although EM guarantees only local optimality, it is a preferred method for finding good solutions in linear time. We consider an arbitrary but fixed parametric form of the model, therefore we sometimes refer to a specific set of parameters Θ as the model. Note that since the logarithm is a monotonic function, the optimal set of parameters is the same whether we use likelihood or log-likelihood. Consider an online data source where there are data sets Dt available at certain time intervals t (not necessarily equal time periods or number of data points). For example, there could be a data set generated on a daily basis, or it could represent a constant stream of data from a monitoring device. In addition, we assume that we have an initial model Θ0 that was built (optimized, fitted) on some in-sample data D0 = {D1, D2, . . . , Dt0}. We would like to be able to detect a change in the underlying distribution of data points within data sets Dt that would be sufficient to require building of a new model Θ1. The criterion for building a new model is loosely defined as “the model does not adequately fit the data anymore”. 2 Model Based Population Similarity In this section we formulate the problem of model-based population similarity and tracking. In case of mixture models we start with the following observations: • The mixture model defines the probability density function (PDF) that is used to score each data point (LogP scores), leading to the score for the overall population (log-likelihood or sum of LogP scores). • The optimal mixture model puts more PDF mass over dense regions in the data space. Different components allow the mixture model to distribute its PDF over disconnected dense regions in the data space. More PDF mass in a portion of the data space implies higher LogP scores for the data points lying in that region of the space. • If model is to generalize well (e.g., there is no significant overfitting) it cannot put significant PDF mass over regions of data space that are populated by data points solely due to the details of a specific data sample used to build the model. • Dense regions in the data space discovered by a non-overfitting model are the intrinsic property of the true data-generating distribution even if the functional form of the model is not well matched with the true data generating distribution. In the latter case, the model might not be able to discover all dense regions or might not model the correct shape of the regions, but the regions that are discovered (if any) are intrinsic to the data. 1This approach is called maximum-likelihood estimation. If we included parameter priors we could equally well apply results in this paper to the maximum a posteriori estimation. • If there is confidence that the model is not overfitting and that it generalizes well (e.g., cross-validation was used to determine the optimal number of mixture components), the new data from the same distribution as the in-sample data should be dense in the same regions that are predicted by the model. Given these observations, we seek to define a measure of data-distribution similarity based on how well the dense regions of the data space are preserved when new data is introduced. In model based clustering, dense regions are equivalent to higher LogP scores, hence we cast the problem of determining data distribution similarity into one of determining LogP distribution similarity (relative to the model). For example, Figure 3 (left) shows a histogram of one such distribution. It is important to note several properties of Figure 3: 1) there are several distinct peaks from which distribution tails off toward smaller LogP values, therefore simple summary scores fail to efficiently summarize the LogP distribution. For example, log-likelihood is proportional to the mean of LogP distribution in Figure 3, and the mean is not a very useful statistic when describing such a multimodal distribution (also confirmed experimentally); 2) the histogram itself is not a truly non-parametric representation of the underlying distribution, given that the results are dependent on bin width. In passing we also note that the shape of the histogram in Figure 3 is a consequence of the CI model we use: different peaks come from different discrete attributes, while the tails come from continuous Gaussians. It is a simple exercise to show that LogP scores for a 1-dimensional data set generated by a single Gaussian have an exponential distribution with a sharp cutoff on the right and tail toward the left. To define the similarity of the data distributions based on LogP scores in a purely nonparametric way we have at our disposal the powerful formalism of Kolmogorov-Smirnov (KS) statistics [7]. KS statistics make use of empirical cumulative distribution functions (CDF) to estimate distance between two empirical 1-dimensional distributions, in our case distributions of LogP scores. In principle, we could compare the LogP distribution of the new data set Dt to that of the training set D0 and obtain the probability that the two came from the same distribution. In practice, however, this approach is not feasible since we do not assume that the estimated model and the true data generating process share the same functional form (see Section 3). Consequently, we need to consider the specific KS score in relation to the natural variability of the true data generating distribution. In the situation with streaming data, the model is estimated over the in-sample data D0. Then the individual in-sample data sets D1, D2, . . . , Dt0 are used to estimate the natural variability of the KS statistics. This variability needs to be quantified due to the fact that the model may not truly match the data distribution. When the natural variance of the KS statistics over the in-sample data has been determined, the LogP scores for a new dataset Dt, t > t0 are computed. Using principled heuristics, one can then determine whether or not the LogP signature for Dt is significantly different than the LogP signatures for the in-sample data. To clarify various steps, we provide an algorithmic description of the change detection process. Algorithm 1 (Quantifying Natural Variance of KS Statistics): Given an “in-sample” dataset D0 = {D1, D2, . . . , Dt0}, proceed as follows: 1. Estimate the parameters Θ0 of the mixture model P(D|Θ) over D0 (see equation (1)). 2. Compute LogP(Di) = ni X ˆi=1 log P(xˆi|Θ0), xˆi ∈Di, ni = |Di|, i = 1, . . . , t0. (2) 3. For 1 ≤i, j ≤t0, compute LKS(i, j) = log [PKS(Di, Dj)]. See [7] for details on PKS computation. 4. For 1 ≤i ≤t0, compute the KS measure MKS(i) as MKS(i) = Pt0 j=1 LKS(i, j) t0 . (3) 5. Compute µM = Mean[MKS(i)] and σM = STD[MKS(i)] to quantify the natural variability of MKS over the “in-sample” data. Algorithm 2 (Evaluating New Data): Given a new dataset Dt, t > t0, µM and σM proceed as follows: 1. Compute LogP(Dt) as in (2). 2. For 1 ≤i ≤t0, compute LKS(i, t). 3. Compute MKS(t) as in (3). 4. Apply decision criteria using MKS(t), µM, σM to determine whether or not Θ0 is a good fit for the new data. For example, if |MKS(t) −µM| σM > 3, (4) then Θ0 is not a good fit any more. Note, however, that the 3-σ interval be interpreted as a confidence interval only in the limit when number of data sets goes to infinity. In applications presented in this paper we certainly do not have that condition satisfied and we consider this approach as an “educated heuristic” (gaining firm statistical grounds in the limit). 2.1 Space and Time Complexity of the Methodology The proposed methodology was motivated by a business application with large data sets, hence it must have time complexity that is close to linear in order to scale well. In order to assess the time complexity, we use the following notation: nt = |Dt| is the number of data points in the data set Dt; t0 is the index of the last in-sample data set, but is also the number of in-sample data sets; n0 = |D0| = Pt0 t=1 nt is the total number of in-sample data points (in all the in-sample data sets); n = n0/t0 is the average number of data points in the in-sample data sets. For simplicity of argument, we assume that all the data sets are approximately of the same size, that is nt ≈n. The analysis presented here does not take into account the time and space complexity needed to estimated the parameters Θ of the mixture model (1). In the first phase of the methodology, we must score each of the in-sample data points under the model (to obtain the LogP distributions) which has time complexity of O(n0). Calculation of KS statistics for two data sets is done in one pass over the LogP distributions, but it requires that the LogP scores be sorted, hence it has time complexity of 2n + 2O(n log n) = O(n log n). Since we must calculate all the pairwise KS measures, this step has time complexity of t0(t0 −1)/2 O(n log n) = O(t2 0n log n). In-sample mean and variance of the KS measure are obtained in time which is linear in t0 hence the asymptotic time complexity does not change. In order to evaluate out-of-sample data sets we must keep LogP distributions for each of the in-sample data sets as well as several scalars (e.g., mean and variance of the in-sample KS measure) which requires O(n0) memory. To score an out-of-sample data set Dt, t > t0, we must first obtain the LogP distribution of Dt which has time complexity of O(n) and then calculate the KS measure relative to each of the in-sample data sets which has time complexity O(n log n) per in-sample data set, or t0O(n log n) = O(t0n log n) for the full in-sample period. The LogP distribution for Dt can be discarded once the pairwise KS measures are obtained. −5.5 −5 −4.5 −4 −3.5 −3 −2.5 0 500 1000 1500 2000 2500 3000 3500 LogP Count −5.5 −5 −4.5 −4 −3.5 −3 −2.5 0 500 1000 1500 2000 2500 3000 3500 LogP Count −5.5 −5 −4.5 −4 −3.5 −3 −2.5 0 500 1000 1500 2000 2500 3000 3500 LogP Count −5.5 −5 −4.5 −4 −3.5 −3 −2.5 0 500 1000 1500 2000 2500 3000 3500 LogP Count Figure 1: Histograms of LogP scores for two data sets generated from the first model (top row) and two data sets generated from the second model (bottom row). Each data set contains 50,000 data points. All histograms are obtained from the model fitted on the in-sample period. Overall, the proposed methodology requires O(n0) memory, O(t2 0n log n) time for preprocessing and O(t0n log n) time for out-of-sample evaluation. Further, since t0 is typically a small constant (e.g., t0 = 7 or t0 = 30), the out-of-sample evaluation practically has time complexity of O(n log n). 3 Experimental Setup Experiments presented consist of two parts: experiments on synthetic data and experiments on the aggregations over real web-log data. 3.1 Experiments on Synthetic Data Synthetic data is a valuable tool when determining both applicability and limitations of the proposed approach. Synthetic data was generated by sampling from a a two component CI model (the true model is not used in evaluations). The data consist of a two-state discrete dimension and a continuous dimension. First 100 data sets where generated by sampling from a mixture model with parameters: [π1, π2] = [0.6, 0.4] as weights, θ1 = [0.8, 0.2] and θ2 = [0.4, 0.6] as discrete state probabilities, [µ1, σ2 1] = [10, 5] and [µ2, σ2 2] = [0, 7] as mean and variance (Gaussian) for the continuous variable. Then the discrete dimension probability of the second cluster was changed from θ2 = [0.4, 0.6] to θ′ 2 = [0.5, 0.5] keeping the remaining parameters fixed and an additional 100 data sets were generated by sampling from this altered model. This is a fairly small change in the distribution and the underlying LogP scores appear to be very similar as can be seen in Figure 1. The figure shows LogP distributions for the first two data sets generated from the first model (top row) and the first two data sets generated from the second model (bottom row). Plots within each 0 20 40 60 80 100 120 140 160 180 200 −6 −5 −4 −3 −2 −1 0 Data set Dt <log(KS probability)> (a) 0 20 40 60 80 100 120 140 160 180 200 −15 −10 −5 0 Data set Dt <log(KS probability)> (b) 0 20 40 60 80 100 120 140 160 180 200 −14 −12 −10 −8 −6 −4 −2 0 Data set Dt <log(KS probability)> (c) 0 20 40 60 80 100 120 140 160 180 200 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 Data set Dt <log(KS probability)> (d) Figure 2: Average log(KS probability) over the in-sample period for four experiments on synthetic data, varying the number of data points per data set: a) 1,000; b) 5,000; c) 10,000; d) 50,000. The dotted vertical line separates in-sample and out-of-sample periods. Note that y-axes have different scales in order to show full variability of the data. row should be more similar than plots from different rows, but this is difficult to discern by visual inspection. Algorithms 1 and 2 were evaluated by using the first 10 data sets to estimate a two component model. Then pairwise KS measures were calculated between all possible data set pairs relative to the estimated model. Figure 2 shows average KS measures over in-sample data sets (first 10) for four experiments varying the number of data points in each experiment. Note that the vertical axes are different in each of the plots to better show the range of values. As the number of data points in the data set increases, the change that occurs at t = 101 becomes more apparent. At 50,000 data points (bottom right plot of Figure 2) the change in the distribution becomes easily detectable. Since this number of data points is typically considered to be small compared to the number of data points in our real life applications we expect to be able to detect such slight distribution changes. 3.2 Experiments on Real Life Data Figure 3 shows a distribution for a typical day from a content web-site. There are almost 50,000 data points in the data set with over 100 dimensions each. The LogP score distribution is similar to that of synthetic data in Figure 1 which is a consequence of the CI model used. Note, however, that in this data set the true generating distribution is not known and is unlikely to be purely a CI model. Therefore, the average log KS measure over insample data has much lower values (see Figure 3 right, and plots in Figure 2). Another way to phrase this observation is to note that since the true generating data distribution is most likely not CI, the observed similarity of LogP distributions (the KS measure) is much lower since there are two factors of dissimilarity: 1) different data sets; 2) inability of the CI model to capture all the aspects of the true data distribution. Nonetheless, the first 31 −100 −80 −60 −40 −20 0 20 40 60 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 LogP Count 0 10 20 30 40 50 60 70 80 90 100 −800 −700 −600 −500 −400 −300 −200 −100 Data set Dt <log(KS probability)> Figure 3: Left: distribution of 42655 LogP scores from mixture of conditional independence models. The data is a single-day of click-stream data from a commercial web site. Right: Average log(KS probability) over the 31 day in-sample period for a content website showing a glitch on day 27 and a permanent change on day 43, both detected by the proposed methodology. data sets (one month of data) that were used to build the initial model Θ0 can be used to define the natural variability of the KS measures against which additional data sets can be compared. The result is that in Figure 3 we clearly see a problem with the distribution on day 27 (a glitch in the data) and a permanent change in the distribution on day 43. Both of the detected changes correspond to real changes in the data, as verified by the commercial website operators. Automatic description of changes in the distribution and criteria for automatic rebuilding of the model are beyond scope of this paper. 4 Related Work Automatic detection of various types of data changes appear in the literature in several different flavors. For example, novelty detection ([4], [8]) is the task of determining unusual or novel data points relative to some model. This is closely related to the outlier detection problem ([1], [5]) where the goal is not only to find unusual data points, but the ones that appear not to have been generated by the data generating distribution. A related problem has been addressed by [2] in the context of time series modeling where outliers and trends can contaminate the model estimation. More recently mixture models have been applied more directly to outlier detection [3]. The method proposed in this paper addesses a different problem. We are not interested in new and unusual data points; on the contrary, the method is quite robust with respect to outliers. An outlier or two do not necessarily mean that the underlying data distribution has changed. Also, some of the distribution changes we are interested in detecting might be considered uninteresting and/or not-novel; for example, a slight shift of the population as a whole is something that we certainly detect as a change but it is rarely considered novel unless the shift is drastic. There is also a set of online learning algorithms that update model parameters as the new data becomes available (for variants and additional references, e.g. [6]). In that framework there is no such concept as a data distribution change since the models are constantly updated to reflect the most current distribution. For example, instead of detecting a slight shift of the population as a whole, online learning algorithms update the model to reflect the shift. 5 Conclusions In this paper we introduced a model-based method for automatic distribution change detection in an online data environment. Given the LogP distribution data signature we further showed how to compare different data sets relative to the model using KS statistics and how to obtain a single measure of similarity between the new data and the model. Finally, we discussed heuristics for change detection that become principled in the limit as the number of possible data sets increases. Experimental results over synthetic and real online data indicate that the proposed methodology is able to alert the analyst to slight distributional changes. This methodology may be used as the basis of a system to automatically re-estimate parameters of a mixture model on an “as-needed” basis – when the model fails to adequately represent the data after a certain point in time. References [1] V. Barnett and T. Lewis. Outliers in statistical data. Wiley, 1984. [2] A. G. Bruce, J. T. Conor, and R. D. Martin. Prediction with robustness towards outliers, trends, and level shifts. In Proceedings of the Third International Conference on Neural Networks in Financial Engineering, pages 564–577, 1996. [3] I. V. Cadez, P. Smyth, and H. Mannila. Probabilistic modeling of transaction data with applications to profiling, visualization, and prediction. In F. Provost and R. Srikant, editors, Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 37–46. ACM, 2001. [4] C. Campbell and K. P. Bennett. A linear programming approach to novelty detection. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 395–401. MIT Press, 2001. [5] T. Fawcett and F. J. Provost. Activity monitoring: Noticing interesting changes in behavior. In Proceedings of the Fifth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 53–62, 1999. [6] R. Neal and G. Hinton. A view of the em algorithm that justifies incremental, sparse and other variants. In M. I. Jordan, editor, Learning in Graphical Models, pages 355–368. Kluwer Academic Publishers, 1998. [7] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C: The Art of Scientific Computing, Second Edition. Cambridge University Press, Cambridge, UK, 1992. [8] B. Sch¨olkopf, R. C. Williamson, A. J. Smola, J. Shawe-Taylor, and J. C. Platt. Support vector method for novelty detection. In S. A. Solla, T. K. Leen, and K.-R. Mller, editors, Advances in Neural Information Processing Systems 12, pages 582–588. MIT Press, 2000.
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Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering Mikhail Belkin and Partha Niyogi Depts. of Mathematics and Computer Science The University of Chicago Hyde Park, Chicago, IL 60637. (misha@math.uchicago.edu,niyogi@cs.uchicago.edu) Abstract Drawing on the correspondence between the graph Laplacian, the Laplace-Beltrami operator on a manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in a higher dimensional space. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Several applications are considered. In many areas of artificial intelligence, information retrieval and data mining, one is often confronted with intrinsically low dimensional data lying in a very high dimensional space. For example, gray scale n x n images of a fixed object taken with a moving camera yield data points in rn: n2 . However, the intrinsic dimensionality of the space of all images of the same object is the number of degrees of freedom of the camera - in fact the space has the natural structure of a manifold embedded in rn: n2 . While there is a large body of work on dimensionality reduction in general, most existing approaches do not explicitly take into account the structure of the manifold on which the data may possibly reside. Recently, there has been some interest (Tenenbaum et aI, 2000; Roweis and Saul, 2000) in the problem of developing low dimensional representations of data in this particular context. In this paper, we present a new algorithm and an accompanying framework of analysis for geometrically motivated dimensionality reduction. The core algorithm is very simple, has a few local computations and one sparse eigenvalue problem. The solution reflects the intrinsic geometric structure of the manifold. The justification comes from the role of the Laplacian operator in providing an optimal embedding. The Laplacian of the graph obtained from the data points may be viewed as an approximation to the Laplace-Beltrami operator defined on the manifold. The embedding maps for the data come from approximations to a natural map that is defined on the entire manifold. The framework of analysis presented here makes this connection explicit. While this connection is known to geometers and specialists in spectral graph theory (for example, see [1, 2]) to the best of our knowledge we do not know of any application to data representation yet. The connection of the Laplacian to the heat kernel enables us to choose the weights of the graph in a principled manner. The locality preserving character of the Laplacian Eigenmap algorithm makes it relatively insensitive to outliers and noise. A byproduct of this is that the algorithm implicitly emphasizes the natural clusters in the data. Connections to spectral clustering algorithms developed in learning and computer vision (see Shi and Malik, 1997) become very clear. Following the discussion of Roweis and Saul (2000) , and Tenenbaum et al (2000), we note that the biological perceptual apparatus is confronted with high dimensional stimuli from which it must recover low dimensional structure. One might argue that if the approach to recovering such low-dimensional structure is inherently local, then a natural clustering will emerge and thus might serve as the basis for the development of categories in biological perception. 1 The Algorithm Given k points Xl , ... , Xk in ]]{ I, we construct a weighted graph with k nodes, one for each point, and the set of edges connecting neighboring points to each other. 1. Step 1. [Constructing the Graph] We put an edge between nodes i and j if Xi and Xj are "close" . There are two variations: (a) [-neighborhoods. [parameter [ E ]]{] Nodes i and j are connected by an edge if Ilxi Xj 112 < f. Advantages: geometrically motivated, the relationship is naturally symmetric. Disadvantages: often leads to graphs with several connected components, difficult to choose f. (b) n nearest neighbors. [parameter n E 1'::1] Nodes i and j are connected by an edge if i is among n nearest neighbors of j or j is among n nearest neighbors of i. Advantages: simpler to choose, tends to lead to connected graphs. Disadvantages: less geometrically intuitive. 2. Step 2. [Choosing the weights] Here as well we have two variations for weighting the edges: (a) Heat kernel. [parameter t E ]]{]. If nodes i and j are connected, put Ilxi-X i 112 Wij = et The justification for this choice of weights will be provided later. (b) Simple-minded. [No parameters]. Wij = 1 if and only if vertices i and j are connected by an edge. A simplification which avoids the necessity of choosing t. 3. Step 3. [Eigenmaps] Assume the graph G, constructed above, is connected, otherwise proceed with Step 3 for each connected component. X 10- 3 8,-----------, 0,---------------..., 6 ~~ + 10 20 30 I 40 o 20 40 4i" • 2 : o ~. ,. -2 $ -4 \ Co .: -6 '~ _8 L-~~~~----~ -5 o -4L-~ __ ~ ____ ~ -2 o 2 Figure 1: The left panel shows a horizontal and a vertical bar. The middle panel is a two dimensional representation of the set of all images using the Laplacian eigenmaps. The right panel shows the result of a principal components analysis using the first two principal directions to represent the data. Dots correspond to vertical bars and '+' signs correspond to horizontal bars. Compute eigenvalues and eigenvectors for the generalized eigenvector problem: Ly = )'Dy (1) where D is diagonal weight matrix, its entries are column (or row, since W is symmetric) sums of W , Dii = Lj Wji. L = D - W is the Laplacian matrix. Laplacian is a symmetric, positive semidefinite matrix which can be thought of as an operator on functions defined on vertices of G. Let Yo , ... , Y k -1 be the solutions of equation 1, ordered according to their eigenvalues with Yo having the smallest eigenvalue (in fact 0). The image of X i under the embedding into the lower dimensional space :Il{m is given by (y 1 ( i) , . . . ,y m (i)). 2 Justification Recall that given a data set we construct a weighted graph G = (V, E) with edges connecting nearby points to each other. Consider the problem of mapping the weighted connected graph G to a line so that connected points stay as close together as possible. We wish to choose Yi E :Il{ to minimize 2)Yi - Yj )2Wij i ,j under appropriate constraints. Let y = (Y1, Y2 , ... ,Yn)T be the map from the graph to the real line. First, note that for any y , we have (2) where as before, L = D W. To see this, notice that Wij 1S symmetric and Dii = Lj Wij . Thus Li,j(Yi - Yj)2Wij can be written as 2)Y; + yJ - 2YiYj )Wij = LY; Dii + LyJ Djj - 2 LYiYj Wij = 2yT Ly i ,j j i ,j -bO -.,!,!.,'i.~ "" - sa y - "" """.,"",oulo .-n a y s h o uld .",,;11 Figure 3: Fragments labeled by arrows in figure 2, from left to right. The first contains infinitives of verbs, the second contains prepositions and the third mostly modal and auxiliary verbs. We see that syntactic structure is well-preserved. Therefore, the minimization problem reduces to finding argminyTDY=lyT Ly. The constraint yT Dy = 1 removes an arbitrary scaling factor in the embedding. Matrix D provides a natural measure on the vertices of the graph. From eq. 2, we see that L is a positive semidef:·st· -. inite matrix and the vector y that minimizes the objective function is given by the minimum eigenvalue solution to the generalized eigenvalue problem Ly = )'Dy. Figure 2: 300 most frequent words of the Brown corpus represented in the spectral domain. Let 1 be the constant function taking value 1 at each vertex. It is easy to see that 1 is an eigenvector with eigenvalue O. If the graph is connected, 1 is the only eigenvector for ). = O. To eliminate this trivial solution which collapses all vertices of G onto the real number 1, we put an additional constraint of orthogonality to obtain Yopt = argmm yT Dy=l yT Ly yTDl=O Thus, the solution Y opt is now given by the eigenvector with the smallest non-zero eigenvalue. More generally, the embedding of the graph into lR!.m (m > 1) is given by the n x m matrix Y = [Y1Y2 ... Yml where the ith row, denoted by Yl, provides the embedding coordinates of the ith vertex. Thus we need to minimize This reduces now to L IIYi - 1j 112Wij = tr(yT LY) i ,j Yopt = argminYT DY=I tr(yT LY) For the one-dimensional embedding problem, the constraint prevents collapse onto a point. For the m-dimensional embedding problem, the constraint presented above prevents collapse onto a subspace of dimension less than m. 2.1 The Laplace-Beltrami Operator The Laplacian of a graph is analogous to the Laplace-Beltrami operator on manifolds. Consider a smooth m-dimensional manifold M embedded in lR k. The Riemannian structure (metric tensor) on the manifold is induced by the standard Riemannian structure on lR k. Suppose we have a map f : M ----+ lR . The gradient V f( x) (which in local coordinates can be written as V f( x) = 2::7=1 ltax.) is a vector field on the manifold, such that for small ox (in a local coordinate chart) \ ,/ \ Figure 4: 685 speech datapoints plotted in the two dimensional Laplacian spectral representation. If(x + ox) - f(x)1 ~ I(Vf(x) ,ox)1 ~ IIVf1111ox11 Thus we see that if IIV fll is small, points near x will be mapped to points near f( x). We therefore look for a map that best preserves locality on average by trying to find Minimizing f IIVf(x)112 corresponds directly to minimizing Lf = ~ 2::ij (li M ' f j )2W ij on a graph. Minimizing the squared gradient reduces to finding eigenfunctions of the Laplace-Beltrami operator.c. Recall that .c d;j div V(I) , where div is the divergence. It follows from the Stokes theorem that -div and V are formally adjoint operators, i.e. if f is a function and X is a vector field fM (X, V f) = fM div(X)f. Thus 1M IIV fl12 = 1M .c(l)f We see that .c is positive semidefinite and the f that minimizes fM IIV fl12 has to be an eigenfunction of .c. 2.2 Heat Kernels and the Choice of Weight Matrix The Laplace-Beltrami operator on differentiable functions on a manifold M is intimately related to the heat flow. Let f : M ----+ lR be the initial heat distribution, u(x, t) be the heat distribution at time t (u(x ,O) = f(x) ). The heat equation is the partial differential equation ~~ = £u. The solution is given by u(x , t) = fM Ht(x, y)f(y) where Ht is the heat kernel - the Green's function for this PDE. Therefore, Locally, the heat kernel is approximately equal to the Gaussian, Ht(x, y) ~ n Ilx-yl12 . . (47rt)-"2e--4-t where Ilx yll (x and yare m local coordmates) and tare both sufficiently small and n = dim M. Notice that as t tends to 0, the heat kernel Ht(x , y) becomes increasingly localized and tends to Dirac's b-function, i.e., lim fM Ht(x, y)f(y) = f(x). Therefore, for small t from the definition of the derivat---+D tive we have 1 [ n (llx-Yl1 2 ] £f(x;) ~ -I, f(x) - (47rt)-"2 JM e- 4t-f(y)dy If Xl , ... , Xk are data points on M, the last expression can be approximated by Xj O< IIXj -X ill<t: The coefficient t is global and will not affect the eigenvectors of the discrete Laplacian. Since the inherent dimensionality of M may be unknown, we put a = t(47rt)¥-. Noticing that the Laplacian of the constant function is zero, we immediately have .1 = 0: Ilx ·-x . 11 2 e ' 4t ' • Notice, however, that we do not Xj O< IIXj -X ill« have to worry about a , since the graph Laplacian L will choose the correct multiplier for us. Finally we see how to choose the edge weights for the adjacency matrix W: 3 Examples if Ilxi Xj II < f otherwise Exalllple 1 - A Toy Vision Exalllple: Consider binary images of vertical and horizontal bars located at arbitrary points in the 40 x 40 visual field. We choose 1000 images, each containing either a vertical or a horizontal bar (500 containing vertical bars and 500 horizontal bars) at random. Fig. 1 shows the result of applying the Laplacian Eigenmaps compared to PCA. Exalllple 2 Words in the Brown Corpus: Fig. 2 shows the results of an experiment conducted with the 300 most frequent words in the Brown corpus - a collection of texts containing about a million words available in electronic format. Each word is represented as a vector in a 600 dimensional space using information about the frequency of its left and right neighbors (computed from the bigram statistics of the corpus). Exalllple 3 - Speech: In Fig. 4 we consider the low dimensional representations arising from applying the Laplacian Eigenmap algorithm to a sentence of speech pP}jf= J3'~ _o. o .<=1 00 pO.<=! -ge l Figure 5: A blowup of the three selected regions in figure 4, from left to right. Notice the phonetic homogeneity of the chosen regions. Note that points marked with the same symbol may arise from occurrences of the same phoneme at different points in the utterance. The symbol "sh" stands for the fricative in the word she; "aa" ," ao" stand for vowels in the words dark and all respectively; "kcl" ," dcl" ," gcl" stand for closures preceding the stop consonants "k" ," d" ," g" respectively. "h#" stands for silence. sampled at 1kHz. Short-time Fourier spectra were computed at 5 ms intervals yielding 685 vectors of 256 Fourier coefficients for every 30 ms chunk of the speech signal. Each vector is labeled according to the identity of the phonetic segment it belonged to. Fig. 4 shows the speech data points plotted in the two dimensional Laplacian representation. The two "spokes" correspond predominantly to fricatives and closures respectively. The central portion corresponds mostly to periodic sounds like vowels, nasals, and semivowels. Fig. 5 shows three different regions of the representation space. References [1] Fan R. K. Chung, Spectral Graph Theory, Regional Conference Series in Mathematics, number 92, 1997 [2] Fan R. K. Chung, A. Grigor'yan, S.-T. Yau, Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs, Communications on Analysis and Geometry, to appear, [3] S. Rosenberg, The Laplacian on a Riemmannian Manifold, Cambridge University Press, 1997, [4] Sam T. Roweis, Lawrence K. Saul, Nonlin ear Dimensionality Reduction by Locally Linear Embedding, Science, vol 290, 22 Dec. 2000, [5] Jianbo Shi, Jitendra Malik, Normalized Cuts and Image Segmentation, IEEE Transactions on PAMI, vol 22, no 8, August 2000 [6] J. B. Tenenbaum, V. de Silva, J. C. Langford, A Global Geometric Framework for Nonlinear Dimensionality Reduction, Science, Vol 290, 22 Dec. 2000
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A Bayesian Model Predicts Human Parse Preference and Reading Times in Sentence Processing Srini Narayanan Daniel Jurafsky SRI International and ICSI Berkeley University of Colorado, Boulder snarayan@cs.berkeley.edu jurafsky@colorado.edu Abstract Narayanan and Jurafsky (1998) proposed that human language comprehension can be modeled by treating human comprehenders as Bayesian reasoners, and modeling the comprehension process with Bayesian decision trees. In this paper we extend the Narayanan and Jurafsky model to make further predictions about reading time given the probability of difference parses or interpretations, and test the model against reading time data from a psycholinguistic experiment. 1 Introduction Narayanan and Jurafsky (1998) proposed that human language comprehension can be modeled by treating human comprehenders as Bayesian reasoners, and modeling the comprehension process with Bayesian decision trees. In this paper, we show that the model accounts for parse-preference and reading time data from a psycholinguistic experiment on reading time in ambiguous sentences. Parsing, (generally called ‘sentence processing’ when we are referring to human parsing), is the process of building up syntactic interpretations for a sentence from an input sequence of written or spoken words. Ambiguity is extremely common in parsing problems, and previous research on human parsing has focused on showing that many factors play a role in choosing among the possible interpretations of an ambiguous sentence. We will focus in this paper on a syntactic ambiguity phenomenon which has been repeatedly investigated: the main-verb (MV), reduced relative (RR) local ambiguity (Frazier & Rayner, 1987; MacDonald, Pearlmutter, & Seidenberg, 1994; McRae, Spivey-Knowlton, & Tanenhaus, 1998, inter alia) In this ambiguity, a prefix beginning with a noun-phrase and an ambiguous verb-form could either be continued as a main clause (as in 1a), or turn out to be a relative clause modifier of the first noun phrase (as in 1b). 1. a. The cop arrested the forger. b. The cop arrested by the detective was guilty of taking bribes. Many factors are known to influence human parse preferences. One such factor is the different lexical/morphological frequencies of the simple past and participial forms of the ambiguous verbform (arrested, in this case). Trueswell (1996) found that verbs like searched, with a frequency-based preference for the simple past form, caused readers to prefer the main clause interpretation, while verbs like selected, had a preference for a participle reading, and supported the reduced relative interpretation. The transitivity preference of the verb also plays a role in human syntactic disambiguation. Some verbs are preferably transitive, where others are preferably intransitive. The reduced relative interpretation, since it involves a passive structure, requires that the verb be transitive. MacDonald, Pearlmutter, and Seidenberg (1994), Trueswell, Tanenhaus, and Kello (1994) and other have shown that verbs which are biased toward an intransitive interpretation also bias readers toward a main clause interpretation. Previous work has shown that a competition-integration model developed by SpiveyKnowlton (1996) could model human parse preference in reading ambiguous sentences (McRae et al., 1998). While this model does a nice job of accounting for the reading-time data, it and similar ‘constraint-based’ models rely on a complex set of feature values and factor weights which must be set by hand. Narayanan and Jurafsky (1998) proposed an alternative Bayesian approach for this constraint-combination problem. A Bayesian approach offers a well-understood formalism for defining probabilistic weights, as well as for combining those weights. Their Bayesian model is based on the probabilistic beam-search of Jurafsky (1996), in which each interpretation receives a probability, and interpretations were pruned if they were much worse than the best interpretation. The model predicted large increases in reading time when unexpected words appeared which were only compatible with a previously-pruned interpretation. The model was thus only able to characterize very gross timing effects caused by pruning of interpretations. In this paper we extend this model’s predictions about reading time to other cases where the best interpretation turns out to be incompatible with incoming words. In particular, we suggest that any evidence which causes the probability of the best interpretation to drop below its next competitor will also cause increases in reading time. 2 The Experimental Data We test our model on the reading time data from McRae et al. (1998), an experiment focusing on the effect of thematic fit on syntactic ambiguity resolution. The thematic role of noun phrase “the cop” in the prefix “The cop arrested” is ambiguous. In the continuation ”The cop arrested the crook”, the cop is the agent. In the continuation “The cop arrested by the FBI agent was convicted for smuggling drugs”, the cop is the theme. The probabilistic relationship between the noun and the head verb (“arrested”) biases the thematic disambiguation decision. For example, “cop” is a more likely agent for “arrest”, while “crook” is a more likely theme. McRae et al. (1998) showed that this ‘thematic fit’ between the noun and verb affected phrase-by-phrase reading times in sentences like the following: 2. a. The cop / arrested by / the detective / was guilty / of taking / bribes. b. The crook / arrested by / the detective / was guilty / of taking / bribes. In a series of experiment on 40 verbs, they found that sentences with good agents (like cop in 2a) caused longer reading times for the phrase the detective than sentences with good themes (like crook in 2b). Figure 1 shows that at the initial noun phrase, reading time is lower for good-agent sentences than good-patient sentences. But at the NP after the word “by”, reading time is lower for good-patient sentences than good-agent sentences.1 1In order to control for other influences on timing, McRae et al. (1998) actually report reading time deltas between a reduced relative and non-reduced relative for. It is these deltas, rather than raw reading times, that our model attempts to predict. 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 The cop/crook arrested by the detective Increased Reading Times (compared to control) Good Agent Good Patient Figure 1: Self-paced reading times (from Figure 6 of McRae et al. (1998)) After introducing our model in the next section, we show that it predicts this cross-over in reading time; longer reading time for the initial NP in good-patient sentences, but shorter reading time for the post-“by” NP in good-patient sentences. 3 The Model and the Input Probabilities In the Narayanan and Jurafsky (1998) model of sentence processing, each interpretation of an ambiguous sentence is maintained in parallel, and associated with a probability which can be computed via a Bayesian belief net. The model pruned low-probability parses, and hence predicted increases in reading time when reading a word which did not fit into any available parse. The current paper extends the Narayanan and Jurafsky (1998) model’s predictions about reading time. The model now also predicts extended reading time whenever an input word causes the best interpretation to drop in probability enough to switch in rank with another interpretation. The model consists of a set of probabilities expressing constraints on sentence processing, and a network that represents their independence relations: Data Source P(Agent verb, initial NP) McRae et al. (1998) P(Patient verb, initial NP) McRae et al. (1998) P(Participle verb) British National Corpus counts P(SimplePast verb) British National Corpus counts P(transitive verb) TASA corpus counts P(intransitive verb) TASA corpus counts P(RR initial NP, verb-ed, by) McRae et al. (1998) (.8, .2) P(RR initial NP, verb-ed, by,the) McRae et al. (1998) (.875. .125) P(Agent initial NP, verb-ed, by, the, NP) McRae et al. (1998) (4.6 average) P(MC SCFG prefix) SCFG counts from Penn Treebank P(RR SCFG prefix) SCFG counts from Penn Treebank The first constraint expresses the probability that the word “cop”, for example, is an agent, given that the verb is “arrested”. The second constraint expresses the probability that it is a patient. The third and fourth constraints express the probability that the ”-ed” form of the verb is a participle versus a simple past form (for example P(Participle “arrest”)=.81). These were computed from the POS-tagged British National Corpus. Verb transitivity probabilities were computed by hand-labeling subcategorization of 100 examples of each verb in the TASA corpus. (for example P(transitive “entertain”)=.86). Main clause prior probabilities were computed by using an SCFG with rule probabilities trained on the Penn Treebank version of the Brown corpus. See Narayanan and Jurafsky (1998) and Jurafsky (1996) for more details on probability computation. 4 Construction Processing via Bayes nets Using Belief nets to model human sentence processing allows us to a) quantitatively evaluate the impact of different independence assumptions in a uniform framework, b) directly model the impact of highly structured linguistic knowledge sources with local conditional probabilitytables, while well known algorithms for updating the Belief net (Jensen (1995)) can compute the global impact of new evidence, and c) develop an on-line interpretation algorithm, where partial input corresponds to partial evidence on the network, and the update algorithm appropriately marginalizes over unobserved nodes. So as evidence comes in incrementally, different nodes are instantiated and the posterior probability of different interpretations changes appropriately. The crucial insight of our Belief net model is to view specific interpretations as values of latent variables that render top-down ( ) and bottom-up evidence ( ) conditionally independent (d-separate them (Pearl, 1988)). Thus syntactic, lexical, argument structure, and other contextual information acts as prior or causal support for an interpretation, while bottom-up phonological or graphological and other perceptual information acts as likelihood, evidential, or diagnostic support. To apply our model to on-line disambiguation, we assume that there are a set of interpretations (
) that are consistent with the input data. At different stages of the input, we compute the posterior probabilities of the different interpretations given the top down and bottom-up evidence seen so far. 2 Semantic_fit Arg Tense P(Tense|v) P(Arg|v) P(A | v, ty(Subj)) P(T | v, ty(Subj)) AND AND MV thm RRthm Tense = past Sem_fit = Agent Tense = pp Sem_fit = Theme V = examine-ed type_of(Subj) = witness Arg = trans Figure 2: The Belief net that represents lexical and thematic support for the two interpretations. Figure 2 reintroduces the basic structure of our belief net model from Narayanan and Jurafsky (1998). Our model requires conditional probabilitydistributions specifying the preference of every verb for different argument structures, as well its preference for different tenses. We also compute the semantic fit between possible fillers in the input and different conceptual roles of a given predicate. As shown in Figure 2, the and interpretations require the conjunction of specific values corresponding to tense, semantic fit and argument structure features. Note that only the interpretation requires the transitive argument structure. 2In this paper, we will focus on the support from thematic, and syntactic features for the Reduced Relative (RR) and Main Verb (MV) interpretations at different stages of the input for the examples we saw earlier. So we will have two interpretations "!$# where %&')( * +, -*).,/10 2$3 4%& )( * + -* . /,06575 . NP S VP #1[] the witness examined V [.48] S−> NP [V ... VP NP S VP NP [.14] NP−> NP XP [.92] S−> NP ... examined The witness MAIN VERB REDUCED RELATIVE Figure 3: The partial syntactic parse trees for the and the interpretations assuming an generating grammar. S S NP NP Det N NP VP V N Det The witness examined The witness examined XP MAIN CLAUSE REDUCED RELATIVE VP XP Figure 4: The Bayes nets for the partial syntactic parse trees The conditional probability of a construction given top-down syntactic evidence is relatively simple to compute in an augmented-stochastic-context-free formalism (partial parse trees shown in Figure 3 and the corresponding bayes net in Figure 4). Recall that the prior probability gives the conditional probability of the right hand side of a rule given the left hand side. The Inside/Outside algorithm applied to a fixed parse tree structure is obtained exactly by casting parsing as a special instance of belief propagation. The correspondences are straightforward a) the parse tree is interpreted as a belief network. b)the non-terminal nodes correspond to random variables, the range of the variables being the non-terminal alphabet, c) the grammar rules define the conditional probabilities linking parent and child nodes, d) the nonterminal at the root, as well as the terminals at the leaves represent conditioningevidence to the network, and e) Conditioningon this evidence produces exactly the conditional probabilities for each nonterminal node in the parse tree and the joint probability distribution of the parse.3 The overall posterior ratio requires propagating the conjunctive impact of syntactic and lexical/thematic sources on our model. Furthermore, in computing the conjunctive impact of the lexical/thematic and syntactic support to compute and , we use the NOISY-AND (assumes exception independence) model (Pearl, 1988) for combining conjunctive sources. In the case of the and interpretations. At various points, we compute the posterior support for the different interpretations using the following equation.
. The first term is 3One complication is that the the conditional distribution in a parse tree %&! , #"( $ / is not the product distribution %&! ( $ /-%&!"( $ / (it is the conjunctive distribution). However, it is possible to generalize the belief propagation equations to admit conjunctive distributions %&! , #"( $ / and %&$" 3 ( % / . The diagnostic (inside) support becomes & &' / 0)(+*-, ./& &0 /1& &2 /-%&0 12 ( ' / and the causal support becomes 3&' / 0546(879, :;3&!< /1& &!=)/-%&' #= ( </ (details can be found at http://www.icsi.berkeley.edu/ snarayan/scfg.ps). the syntactic support while the second is the lexical and thematic support for a particular interpretation ( ). 5 Model results 0 1 2 3 4 5 6 7 8 9 MV/RR NP verbed by the NP Model Good Agent Human Good Agent Model Good Patient Human Good Patient Figure 5: Completion data We tested our model on sentences with the different verbs in McRae et al. (1998). For each verb, we ran our model on sentences with Good Agents (GA) and Good Patients (GP) for the initial NP. Our model results are consistent with the on-line disambiguation studies with human subjects (human performance data from McRae et al. (1998)) and show that a Bayesian implementation of probabilistic evidence combination accounts for garden-path disambiguation effects. Figure 5 shows the first result that pertains to the model predictions of how thematic fit might influence sentence completion times. Our model shows close correspondence to the human judgements about whether a specific ambiguous verb was used in the Main Clause (MV) or reduced relative (RR) constructions. The human and model predictions were conducted at the verb (The crook arrested), by (the crook arrested by), the (the crook arrested by the) and Agent NP (the crook arrested by the detective). As in McRae et al. (1998) the data shows that thematic fit clearly influenced the gated sentence completion task. The probabilistic account further captured the fact that at the by phrase, the posterior probability of producing an RR interpretation increased sharply, thematic fit and other factors (a) 2.1 0.541 0.1 0.7 0.13 0.04 0 0.5 1 1.5 2 2.5 The crook/detective arrested by the detective P(MC)/P(RR) Good Agent initial NP Good Patient Initial NP 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 The cop arrested by the detective P(X) Good Agent Main Clause Good Agent RR (b) Figure 6: a) MV/RR for the ambiguous region showing a flip for the Good Agent (ga) case. b) P(MV) and P(RR) for the Good Patient and Good Agent cases. influenced both the sharpness and the magnitude of the increase. The second result pertains to on-line reading times. Figure 6 shows how the human reading time reduction effects (reduced compared to unreduced interpretations) increase for Good Agents (GA) but decrease for Good Patients in the ambiguous region. This explains the reading time data in Figure 1. Our model predicts this larger effect from the fact that the most probable interpretation for the Good Agent case flips from the MV to the RR interpretation in this region. No such flip occurs for the Good Patient (GP) case. In Figure 6(a), we see that the GP results already have the MV/RR ratio less than one (the RR interpretation is superior) while a flip occurs for the GA sentences (from the initial state where MV/RR to the final state where MV/RR ). Figure 6 (b) shows a more detailed view of the GA sentences showing the crossing point where the flip occurs. This finding is fairly robust ( of GA examples) and directly predicts reading time difficulties. 6 Conclusion We have shown that a Bayesian model of human sentence processing is capable of modeling reading time data from a syntactic disambiguation task. A Bayesian model extends current constraint-satisfaction models of sentence processing with a principled way to weight and combine evidence. Bayesian models have not been widely applied in psycholinguistics. To our knowledge, this is the first study showing a direct correspondence between the time course of maintaining the best a posteriori interpretation and reading time difficulty. We are currently exploring how our results on flipping of preferred interpretation could be combined with Hale (2001)’s proposal that reading time correlates with surprise (a surprising (low probability) word leads to large amounts of probability mass to be pruned) to arrive at a structured probabilistic account of a wide variety of psycholinguistic data. References Frazier, L., & Rayner, K. (1987). Resolution of syntactic category ambiguities: Eye movements in parsing lexically ambiguous sentences. Journal of Memory and Language, 26, 505–526. Hale, J. (2001). A probabilistic earley parser as a psycholinguistic model. Proceedings of NAACL2001. Jensen, F. (1995). Bayesian Networks. Springer-Verlag. Jurafsky, D. (1996). A probabilistic model of lexical and syntactic access and disambiguation. Cognitive Science, 20, 137–194. MacDonald, M.C., Pearlmutter, N.J., & Seidenberg, M.(1994). The lexical nature of syntactic ambiguity resolution. Psychological Review, 101, 676-703. McRae, K., Spivey-Knowlton, M., & Tanenhaus, M. K.(1998). Modeling the effect of thematic fit (and other constraints) in on-line sentence comprehension. Journal of Memory and Language,38, 283–312. Narayanan, S., & Jurafsky, D. (1998). Bayesian models of human sentence processing. In COGSCI98, pp. 752–757 Madison, WI. Lawrence Erlbaum. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufman, San Mateo, Ca. Spivey-Knowlton, M.(1996). Integration of visual and linguistic information: Human data and model simulations. Ph.D. Thesis, University of Rochester, 1996. Trueswell, J. C.(1996). The role of lexical frequency in syntactic ambiguity resolution. Journal of Memory and Language, 35, 566-585. Trueswell, J. C., Tanenhaus, M. K., & Kello, C. (1994). Verb-specific constraints in sentence processing: Separating effects of lexical preference from garden-paths. Journal of Experimental Pyschology: Learning, Memory and Cognition, 19(3), 528–553.
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